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HACETTEPE JOURNAL OF MATHEMATICS AND STATISTICS
Volume 49 Issue 2 (2020)
E-ISSN: 2651-477X
Honorary EditorLawrence Michael BrownEditor in ChiefMathematicsAyşe Çiğdem Özcan (Hacettepe University, Mathematics - [email protected])StatisticsÇağdaş Hakan Aladağ (Hacettepe University, Statistics - [email protected])Associate EditorsBülent Saraç (Hacettepe University, Mathematics - [email protected])Aslı Yıldız (Hacettepe University, Mathematics - [email protected])Ceren Eda Can (Hacettepe University, Statistics - [email protected])Derya Ersel (Hacettepe University, Statistics - [email protected])Production EditorsGülbanu Tekbulut (Hacettepe University - [email protected])O. Oğulcan Tuncer (Hacettepe University, Mathematics - [email protected])Talha Arıkan (Hacettepe University, Mathematics - [email protected])Nurbanu Bursa (Hacettepe University, Statistics - [email protected])Erhan Pişirir (Hacettepe University, Statistics - [email protected])Derya Turfan (Hacettepe University, Statistics - [email protected])Area EditorsMathematicsEvrim Akalan (Associative Rings and Algebras - [email protected])Okay Çelebi (Partial Differential Equations - [email protected])Olgür Çelikbaş (Commutative Rings and Algebras - [email protected])Angel Del Rio (Algebra, Group Theory - [email protected])Gülin Ercan (Algebra, Group Theory - [email protected])Sergio Estrada (Homological Algebra - [email protected])Rodrigo Hernandez Gutierrez (General Topology - [email protected])Varga Kalantarov (Partial Differential Equations - [email protected])Emre Mengi (Numerical Analysis - [email protected])Cihan Orhan (Analysis, Summability - [email protected])Murad Özaydın (Differential Geometry, Global Analysis - [email protected])Abdullah Özbekler (Ordinary Differential Equations - [email protected])Ekin Özman (Number Theory, Algebraic Geometry - [email protected])Serap Öztop Kaptanoğlu (Abstract Harmonic Analysis - [email protected] )Mehmetçik Pamuk (Topology, Manifolds and Cell Complexes - [email protected])Bülent Ünal (Differential Geometry - [email protected])Yunus E. Zeytuncu (Functions of Several Complex Variables - [email protected])
StatisticsAli Allahverdi (Operational Research & Statistics - [email protected])Olcay Arslan (Robust Statistics - [email protected])Narayanaswany Balakrishnan (Applied Statistics, Theory of Statistics - [email protected])Adil Baykasoğlu (Operational Research - [email protected])Sat Gupta (Sampling, Time Series - [email protected])Yasemin Kayhan Atılgan (Computational Statistics & Mathematical Statistics - [email protected])Birdal Şenoğlu (Experimental Design, Statistical Distributions - [email protected])Tahir Hanalioğlu (Stochastic Processes Theory, Probability Theory - [email protected])Zeynep Işıl Kalaylıoğlu (Bayesian Inference, Model Selection - [email protected])
Contents
Mathematics
Research Articles
1 Statistical cluster point and statistical limit point sets of subsequences of a given sequenceby Harry I. Miller and Leila Miller-Van Wieren . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494
2 A note on the paper “Best constants for the Hardy-Littlewood maximal operator on finitegraphs”by Zaryab Hussain and Sadia Talib . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498
3 Prime geodesic theorem for the modular surfaceby Muharem Avdispahić . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505
4 Approximation by Cheney-Sharma Chlodovsky operatorsby Dilek Söylemez and Fatma Taşdelen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510
5 Well-posedness and exponential stability of a thermoelastic-Bresse system with second soundand delayby Gang Li, Yue Luan and Wenjun Liu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523
6 The Laguerre pseudospectral method for the two-dimensional Schrödinger equation with sym-metric nonseparable potentialsby Haydar Alıcı . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539
7 Lucas polynomial solution of nonlinear differential equations with variable delaysby Sevin Gümgüm, Nurcan Baykuş Savaşaneril, Ömür Kıvanç Kürkçü and Mehmet Sezer . . . . . . . . . 553
8 Convolutions of the bi-periodic Fibonacci numbersby Takao Komatsu and José L. Ramírez . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565
9 On centrally-extended multiplicative (generalized)-(α, β)-derivations in semiprime ringsby Najat Muthana and Zakeiah Alkhamisi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578
10 Euler sums and non-integerness of harmonic type sumsby Haydar Göral and Doǧa Can Sertbaş . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586
11 Almost L-Dunford-Pettis sets in Banach lattices and its applicationsby Abderrahman Retbi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599
12 Ideal based trace graph of matricesby T. Tamizh Chelvam and M. Sivagami . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608
13 A note for some parabolic multilinear commutators generated by a class of parabolic maximaland linear opertors with rough kernel on the parabolic generalized local Morrey spacesby Ferit Gürbüz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617
14 Construction of arithmetic secret sharing schemes by using torsion limitsby Seher Tutdere and Osmanbey Uzunkol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 638
15 Representations and T ∗-extensions of δ-Bihom-Jordan-Lie algebrasby Abdelkader Ben Hassine, Liangyun Chen and Juan Li . . . . . . . . . . . . . . . . . . . . . . . . . 648
16 Complex fuzzy soft matrices with applicationsby Madad Khan, Saima Anis, Seok-Zun Song and Young Bae Jun . . . . . . . . . . . . . . . . . . . . . 676
17 New analogues of the Filbert and Lilbert matrices via products of two k-tuples asymmetricentriesby Emrah Kılıç, Neşe Ömür and Sibel Koparal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684
18 Graphical calculus of Hopf crossed modulesby Kadir Emir . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695
19 On ∗-differential identities in prime rings with involutionby Shakir Ali, Ali N.A. Koam and Moin A. Ansari . . . . . . . . . . . . . . . . . . . . . . . . . . . . 708
20 On some subclasses k-uniformly Janowski starlike and convex functions associated with t-symmetric pointsby Khalida Inayat Noor, Nasir Khan, Muhammad Arif and Janusz Sokól . . . . . . . . . . . . . . . . . 716
21 On total mean curvatures of foliated half-lightlike submanifolds in semi-Riemannian manifoldsby Fortuné Massamba and Samuel Ssekajja . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 727
22 Pair of generalized derivations acting on multilinear polynomials in prime ringsby Basudeb Dhara, Sukhendu Kar and Priyadwip Das . . . . . . . . . . . . . . . . . . . . . . . . . . . 740
23 Computation of Zagreb indices and Zagreb polynomials of Sierpiński graphsby Hafiz Muhammad Afzal Siddiqui . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754
24 Oscillatory behavior of n-th order nonlinear delay differential equations with a nonpositiveneutral termby S.R. Grace, I. Jadlovská and A. Zafer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 766
25 Hopf algebra structure on superspace SP2|1q
by Salih Celik . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 777
26 On LPI ringsby Rachida El Khalfaoui, Najib Mahdou and Abdeslam Mimouni . . . . . . . . . . . . . . . . . . . . . . 784
27 A minimal family of sub-basesby Yiliang Li, Jinjin Li, Yidong Lin, Jun-e Feng and Hongkun Wang . . . . . . . . . . . . . . . . . . . 793
28 On C -coherent rings, strongly C -coherent rings and C -semihereditary ringsby Zhu Zhanmin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 808
29 Slant submersions in paracontact geometryby Yılmaz Gündüzalp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 822
30 Lorentz-Schatten classes of direct sum of operatorsby Pembe Ipek Al . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 835
31 On submanifolds of Kenmotsu manifold with Torqued vector fieldby Halil İbrahim Yoldaş, Şemsi Eken Meriç and Erol Yaşar . . . . . . . . . . . . . . . . . . . . . . . . 843
32 Rings of frame maps from P(R) to frames which vanish at infinityby Ali Akbar Estaji and Ahmad Mahmoudi Darghadam . . . . . . . . . . . . . . . . . . . . . . . . . . 854
Statistics
Research Articles
33 Visual research on the trustability of classical variable selection methods in Cox regressionby Nihal Ata Tutkun and Yasemin Kayhan Atilgan . . . . . . . . . . . . . . . . . . . . . . . . . . . . 869
34 Gaussian copula of stable random vectors and applicationby Phuc Ho Dang and Truc Giang Vo Thi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 887
35 An adaptation of pseudo-score confidence interval method for linear mixed modelsby Hatice Tul Kubra Akdur, Deniz Ozonur and Hulya Bayrak . . . . . . . . . . . . . . . . . . . . . . . 902
Hacettepe Journal ofMathematics & Statistics
Hacet. J. Math. Stat.Volume 49 (2) (2020), 494 – 497
DOI : 10.15672/hujms.712019
Research Article
Statistical cluster point and statistical limit pointsets of subsequences of a given sequence
Harry I. Miller, Leila Miller-Van Wieren∗
Faculty of Engineering and Natural Sciences, International University of Sarajevo, Sarajevo, 71000,Bosnia-Herzegovina
AbstractJ.A. Fridy [Statistical limit points, Proc. Amer. Math. Soc., 1993] considered statisticalcluster points and statistical limit points of a given sequence x. Here we show that almostall subsequences of x have the same statistical cluster point set as x. Also, we show ananalogous result for the statistical limit points of x.
Mathematics Subject Classification (2010). 40D25, 40G99, 28A12
Keywords. sequences, subsequences, statistical cluster points, statistical limit points
1. IntroductionFridy [1] has proven that Γx, the set of statistical cluster points of x = (xn), is always
a closed set and Γx is non-empty if x is bounded. However Λx, the set of statistical limitpoints of x, need not be closed. In [2] H.I. Miller studied statistical convergence andrelations between statistical convergence of a sequence x and statistical convergence ofthe subsequences of x. In particular, in [2], it is shown that if L is the statistical limit ofx, then almost all subsequences of x have L as their statistical limit. Here we combinetwo notions, statistical cluster points and subsequences, showing that Γx is equal to thestatistical cluster point set of almost all subsequences of x. This is a continuation of theresults in [3] that also combine statistical cluster points and subsequences. Namely, in [3]it is shown that if Γx = ∅ and F is a non-empty closed subset of Γx, then there existsa subsequence y of x such that Γy = F . Additionally we show that Λx is equal to thestatistical limit point set of almost all subsequences of x. This is a continuation of theresults in [4] that also combine statistical limit points and subsequences.
2. Preliminaries
If t ∈ (0, 1] , then t has a unique binary expansion t =∞∑
n=1
en
2n, en ∈ 0, 1 , with infinitely
many ones. Next if x = (xn) is a sequence of reals, for each t ∈ (0, 1] , let x(t) denote thesubsequence of x obtained by the following rule: xn is in the subsequence if and only ifen = 1. Clearly the mapping t → x(t) is a one-to-one onto mapping between (0, 1] and thecollection of all subsequences of x.
∗Corresponding Author.Email addresses: [email protected] (H.I. Miller), [email protected] (L.M. Wieren)Received: 10.07.2016; Accepted: 06.10.2016
Statistical cluster point and statistical limit point sets... 495
If K is a subset of the positive integers N , then following Fridy [1], Kn denotes the setk ∈ K : k ≤ n and |Kn| denotes the number of elements in Kn. The natural densityof K (see [5]) is given by δ(K) = limn→∞ n−1|Kn|, provided this limit exists. In the casethat δ(K) = 0 we say that K is thin, and otherwise we say that K is non-thin.
Statistical convergence of a sequence is defined as follows.We say that L is the statistical limit of the sequence x, if for every ϵ > 0,
limn→∞
1n
|k ≤ n : |xk − L| ≥ ϵ| = 0.
Statistical convergence and its connection to subsequences is studied in [2].Statistical limit points and statistical cluster points of a sequence x are defined as
follows.We say that a number λ is a statistical limit point of a sequence of reals x = (xn) if
limk→∞ xnk= λ for some non-thin subsequence of (xn).
We say that a number γ is a statistical cluster point of a sequence of reals (xn) if forevery ϵ > 0 the set k ∈ N : |xk − γ| < ϵ is non-thin.
In [1], given a sequence x, three sets are considered. Lx, the set of limit points of x;Λx, the set of statistical limit points of x, and Γx, the set of statistical cluster points of x.Also, if x is bounded, then Γx is closed and non-empty.
In this paper we want to examine, Γx and its relation to Γx(t). Additionally we alsoconsider Λx and its relation to Λx(t).
3. ResultsOur main result is the following.
Theorem 3.1. If x = (xn) is a bounded sequence, then Γx = Γx(t) for almost all t ∈ (0, 1](in the sense of Lebesgue measure).
Proof. Since Γx is closed, it is either finite or separable, i.e. there is a countable subset ofΓx, ln : n ∈ N such that its closure is Γx. We consider only the second case, the proofin the first case is much simpler.
First we show that Γx ⊆ Γx(t) for almost all t. It is sufficient to show that m(Bn) = 1 forn = 1, 2, . . . where Bn = t ∈ (0, 1] : ln ∈ Γx(t). This is true since in that case m(B) = 1for B =
∩∞n=1 Bn and then ln : n ∈ N ⊆ Γx(t) for all t ∈ B and consequently Γx ⊆ Γx(t)
for all t ∈ B.Since ln ∈ Γx, then for every ϵ > 0, k ∈ N : |xk − ln| < ϵ is non-thin . If ϵ = 1
j wecan denote the above set by kj
1, kj2, kj
3, . . .. Then, since it is non-thin there exists δj > 0such that
1p
|i : kji ≤ p| > δj
for infinitely many p. We can assume that p = kjM for infinitely many sufficiently large
M . Now for each j, by the Law of Large Numbers, the limiting frequency of xkj
ii =
1, 2, . . . among the sequence x(t) is 12 for almost all t ∈ (0, 1], i.e. if t =
∑∞m=1
em2m , then
limm→∞1m
∑mi=1 tj
ki= 1
2 for almost all t ∈ (0, 1]. That is, m(Dj) = 1, where
Dj = t ∈ (0, 1] : limm→∞
1m
m∑i=1
tjki
= 12
(3.1)
for all j. Hence if D =∩∞
j=1 Dj , m(D) = 1. Now we will check that ln is a statisticalcluster point for each t in D.
To see this we will show that i ∈ N : |x(t)i − ln| < 1j is non-thin for every j ∈ N and
every t ∈ Dj .
496 Harry I. Miller, Leila Miller-Van Wieren
Consider the earlier mentioned p = kjM for M large enough. Then the number of
such i ≤ p, with |xi − ln| < 1j is greater than pδj . Now take t ∈ Dj . By (3.1),
limm→∞1m
∑mi=1 tj
ki= 1
2 . So for large M , p = kjM , we have
1p
|i ≤ p : |x(t)i − ln| <1j
| >δj
4,
i.e. this holds for infinitely many p, i.e. i ∈ N : |x(t)i − ln| < 1j is non-thin for every
j ∈ N and every t ∈ Dj . Hence ln is a statistical cluster point for every t ∈ D. Thiscompletes the proof that Γx ⊆ Γx(t) for almost all t.
Next we show that Γx(t) ⊆ Γx for almost all t. We will show that this inclusion holdsfor all normal t ∈ (0, 1], i.e. for all t =
∑∞n=1
en2n for which limn→∞
1n
∑ni=1 ei = 1
2 . It iswell known that almost all t ∈ (0, 1] are normal (see [5] ).
Suppose that l is a statistical cluster point of x(t) for some normal t. Then for anyϵ > 0, i : |(x(t))i − l| < ϵ is non-thin, i.e. there exists δϵ > 0 such that
1n
|i ≤ n : |(x(t))i − l| < ϵ| > 2δϵ
for infinitely many n. This implies that1n
|i ≤ n : |xi − l| < ϵ| >12
δϵ
for infinitely many n, and hence l is a statistical cluster point of x. Therefore Γx(t) ⊆ Γx
for all normal t, and consequently for almost all t ∈ (0, 1]. Therefore we conclude thatΓx(t) = Γx for almost all t ∈ (0, 1].
Next, we will prove an analogous result for the set of statistical limit points of x and itssubsequences. The set Λx is not necessarily closed (see [4]). However the following usefultheorem was proved by Kostyrko, Mačaj, Šalat and Strauch [4].
Theorem 3.2. For every bounded sequence x, the set Λx is an Fσ-set in R.
In the proof of the above theorem, the authors show that
Λx =∞∪
j=1Λ(x,
1j
)
where Λ(x, 1j ) = l, ∃ki, i = 1, 2 . . . , limi→∞ xki
= l, δ(ki) ≥ 1j where δ denotes the
upper statistical density (i.e. δ(ki) = lim supi→∞i
ki) and Λ(x, 1
j ) is closed for all j.Here is our second result.
Theorem 3.3. If x = (xn) is a bounded sequence, then Λx = Λx(t) for almost all t ∈ (0, 1](in the sense of Lebesgue measure).
Proof. We proceed in a similar manner as in the proof of Theorem 3.1.First we show that Λx ⊆ Λx(t) for almost all t.As mentioned earlier, Λx =
∪∞j=1 Tj , where
Tj = Λ(x,1j
) = l, ∃ki, i = 1, 2 . . . , limi→∞
xki= l, δ(ki) ≥ 1
j.
Suppose j ∈ N is fixed. Using the above notation (from [4]), Tj is closed and separableso there exists a set lij : i ∈ N such that its closure is Tj . Let i ∈ N . If l = lij , thenby the Law of Large Numbers, l ∈ Λ(x(t), 1
4j ), for all t ∈ Bij , where m(Bij) = 1. LetBj =
∩∞i=1 Bij . Then m(Bj) = 1. Hence lij : i ∈ N ⊆ Λ(x(t), 1
4j ) for every t ∈ Bj . Nowsince Tj and Λ(x(t), 1
4j ) are both closed we get that Tj ⊆ Λ(x(t), 14j ) for every t ∈ Bj .
Statistical cluster point and statistical limit point sets... 497
Therefore Λx =∪∞
j=1 Tj ⊆∪∞
j=1 Λ(x(t), 14j ) = Λx(t) for all t ∈
∩∞j=1 Bj . Since
m(∩∞
j=1 Bj) = 1, we have shown that Λx ⊆ Λx(t) for almost all t.Next we show that Λx(t) ⊆ Λx for almost all t. Again we show that this inclusion holds
for all normal t ∈ (0, 1]. Suppose that l is a statistical limit point of x(t) for some normalt. Then x(t) has a non-thin subsequence that converges to l (in the normal sense). It iseasy to see that this subsequence x(t)i = xki
is then also a non-thin subsequence of x andtherefore l is also a statistical limit point of x. This completes the proof.
4. Concluding remarksWe mentioned that m(ν) = 1, where ν is the set of normal numbers in (0, 1]. However
ν is a set of first Baire category. In light of this we suspect that a category analogue ofour Theorem 3.1 is not true.
Also, one could examine possible analogues of our results using permutations ratherthan subsequences.
References[1] J.A. Fridy, Statistical limit points, Proc. Amer. Math. Soc. 118 1187–1192, 1993.[2] H.I. Miller, Measure theoretical subsequence characterization of statistical conver-
gence, Trans. Amer. Math. Soc. 347 (5) 1811–1819, 1995.[3] H.I. Miller and L. Miller-Van Wieren, Some statistical cluster point theorems, Hacet.
J. Math. Stat. 44 (6) 1405–1409, 2015.[4] P. Kostyrko, M. Mačaj, T. Šalat, and O. Strauch, On statistical limit points, Proc.
Amer. Math. Soc. 129 (9), 2647–2654, 2000.[5] J.C. Oxtoby, Measure and Category: A survey of the analogies between topological
and measure spaces. Second edition, Springer-Verlag, New York-Berlin,1980.
Hacettepe Journal ofMathematics & Statistics
Hacet. J. Math. Stat.Volume 49 (2) (2020), 498 – 504
DOI : 10.15672/hujms.471101
Research Article
A note on the paper “Best constants for theHardy−Littlewood maximal operator on finite
graphs”
Zaryab Hussain∗1, Sadia Talib2
1Department of Mathematics, University of Central Punjab, Faisalabad-38000, Pakistan2Department of Mathematics, Government College University, Faisalabad-38000, Pakistan
Abstract
Let Gmn be a simple, connected and finite graph. Suppose ϕ : N → R+ is a positive and
increasing function. We consider the action of generalized maximal operator MϕGm
non ℓp
spaces and find optimal bound for the quasi norm ∥MϕGm
n∥p for the case 0 < p ≤ 1. In
addition we find bounds for the norm ∥MϕGm
n∥p for the case 1 < p < ∞. We also prove
some general results for 0 < p ≤ 1.
Mathematics Subject Classification (2010). 05C12, 42B25
Keywords. generalized maximal operator, ℓp-estimates, Gmn graph
1. IntroductionLet G (V, E) be a connected, finite and simple graph where V (G) is the set of vertices
and E(G) is the set of edges between the vertices of graph G. Let dG : V (G) × V (G) → Rbe the geodesic metric space defined for u, v ∈ V (G) as the number of edges in shortestpath between u and v written as dG(u, v). The set NG(u) = x ∈ V (G) | dG(u, x) = 1 isthe neighborhood of u in graph G, cardinality of neighborhood set is called degree of uand is denoted as dG(u). For any function f : V (G) → R we can consider the generalizedmaximal operator [1] Mϕ
G : ℓp → ℓp, such as
MϕGf(j) = sup
r≥0
1ϕ (|B(j, r)|)
∑s∈B(j,r)
|f(s)| (1.1)
where ϕ : N → R+ is a positive, increasing function and B(j, r) = x ∈ V (G) | dG(j, x) ≤ ris the ball of radius r with center at j. Note that M t
G is the classical Hardy-Littlewoodmaximal operator and M t1− s
r
G , where 0 < s < r, is the fractional maximal operator. Asdistance takes only natural numbers as values, the radius r ≥ 0 considered in the definition
∗Corresponding Author.Email addresses: [email protected] (Z. Hussain), [email protected] (S. Talib)Received: 16.10.2018; Accepted: 27.12.2018
Note on "Best constants for the Hardy-Littlewood maximal operator on finite graphs" 499
of generalized maximal operator can be taken to be a natural number also the diameterof the graph of n vertices is at most n − 1, so we can write the equation (1.1) such as
MϕGf(j) = max
r=0,1,...,n−1
1ϕ (|B(j, r)|)
∑s∈B(j,r)
|f(s)| . (1.2)
For 0 < p < ∞, the norm of MϕG is define as
∥MϕG∥p := sup
f =0
∥MϕGf∥p
∥f∥p
where ∥f∥p =( ∑
s∈V (G)|f(s)|p
) 1p
.
In the paper [2] authors proved that if 0 < p ≤ 1, then
∥MKn∥p =(
1 + n − 1np
) 1p
,
if 1 < p < ∞, then (1 + n − 1
np
) 1p
≤ ∥MKn∥p ≤(
1 + n − 1n
) 1p
, (1.3)
where Kn is a complete graph. In this paper we generalize the results given in [2].
2. PreliminariesDefinition 2.1. A family of graphs Gm
n as those simple graphs having n vertices withone vertex say k (central vertex) of degree n − 1 and all other vertices of degree m, where1 ≤ m ≤ n − 1.
Gmn is a very large family of graphs as it contains both star graph Sn ∼ G1
n as well ascomplete graph Kn = Gn−1
n as end-points, it has also many more important graphs in it.
Figure 1. G49 graph
For example if we take m = 3, then G3n ∼ Wn (wheel graph). For k ∈ V (central vertex)
the B (k, r) for Gmn is
B(k, r) =
k , for r = 0,
V, for r ≥ 1.
For j ∈ V other than k, the B (j, r) for Gmn is
B(j, r) =
j , for r = 0,
j∪
NGmn
(j), for r = 1,
V, for r ≥ 2.
500 Z. Hussain, S. Talib
Suppose j ∈ V , the generalized maximal operator for Gmn is
MϕGm
nf(j) =
max
1
ϕ(1) |f(j)| , 1ϕ(n)
∑x∈V
|f(x)|
, if j = k,
max
1ϕ(1) |f(j)| , 1
ϕ(m+1)∑
v∈B(j,1)|f(v)| , 1
ϕ(n)∑
x∈V|f(x)|
, if j = k.
(2.1)Let see a particular example for the norm of generalized maximal operator on G2
7.
Example 2.2. Let G27 ∼ F7 (friendship graph of 7 vertices) with V = 1, 2, 3, 4, 5, 6, 7
be the vertex set, 1 is the central vertex. There are 9 edges in this graph 1-2, 1-3, 1-4, 1-5,1-6, 1-7, 2-3, 4-5 and 6-7, now it is easy to draw this graph. Take Dirac delta as function,ϕ(t) = t2 and p = 1
2 , then we have
M t2
G27δ1(j) =
1, for j = 1,
19 , for j = 2, 3, 4, 5, 6, 7,
and
M t2
G27δ2(j) =
1, for j = 2,
19 , for j = 3,
149 , for j = 1, 4, 5, 6, 7.
Hence ∥M t2
G27δ1∥ 1
2= 9 and ∥M t2
G27δ2∥ 1
2= 4.1927. By symmetry, we also have the esti-
mates for the remaining vertices: ∥M t2
G27δ3∥ 1
2= ∥M t2
G27δ4∥ 1
2= ∥M t2
G27δ5∥ 1
2= ∥M t2
G27δ6∥ 1
2=
∥M t2
G27δ7∥ 1
2= 4.1927, so ∥M t2
G27∥ 1
2= 9. This calculation can be obtained directly from
Proposition 3.1.The operator Mϕ
Gn−1n
(Gn−1
n = Kn)
is the smallest, in the pointwise ordering, among allMϕ
G, with G a graph of n vertices. That is for each f : V → R and every j ∈ V , we havethat
Mϕ
Gn−1n
f(j) ≤ MϕGf(j). (2.2)
Consequently for every 0 < p < ∞,
∥Mϕ
Gn−1n
∥pp ≤ ∥Mϕ
G∥pp. (2.3)
Lemma 2.3 ([2]). Let G be the graph, and Ω : ℓp(G) → ℓp(G) be a sublinear operatorwith 0 < p ≤ 1. Then,
∥Ω∥p = maxj∈V
∥Ωδj∥p.
3. Main resultsProposition 3.1. If 0 < p ≤ 1, then
∥MϕGm
n∥p =
( 1ϕp(1)
+ n − 1ϕp(m + 1)
) 1p
and if 1 < p < ∞, then( 1ϕp(1)
+ n − 1ϕp(m + 1)
) 1p
≤ ∥MϕGm
n∥p ≤
(1
ϕp(1)+ (n − 1) max
(m + 1)p−1
ϕp(m + 1),
np−1
ϕp(n)
) 1p
.
Note on "Best constants for the Hardy-Littlewood maximal operator on finite graphs" 501
Proof. Let f : V → R be a function such that ∥f∥p = 1. Suppose that k ∈ V (Gmn ) is the
central vertex of the graph define δk, then for 0 < p < ∞ we have
∥MϕGm
nδk∥p =
(MϕGm
nδk(k)
)p+
∑i∈V \k
(Mϕ
Gmn
δk(i))p
1p
=( 1
ϕp(1)+ n − 1
ϕp(m + 1)
) 1p
.
Now suppose r ∈ V (Gmn ) such that r = k, we define δr, then we have
∥MϕGm
nδr∥p =
((Mϕ
Gmn
δr(r))p
+∑
i∈NGmn
(r)\k
(Mϕ
Gmn
δr(i))p
+∑
b∈k∪
x: x ∈NGmn
(r)
(Mϕ
Gmn
δr(b))p) 1
p
=( 1
ϕp(1)+ m − 1
ϕp(m + 1)+ n − m
ϕp(n)
) 1p
.
As ∥δk∥p = 1 so we have for 0 < p < ∞
∥MϕGm
n∥p ≥ max
( 1ϕp(1)
+ n − 1ϕp(m + 1)
) 1p
,
( 1ϕp(1)
+ m − 1ϕp(m + 1)
+ n − m
ϕp(n)
) 1p
.
Due to the monotonicity of ϕ, the maximum is always attained at the first term, so
∥MϕGm
n∥p ≥
( 1ϕp(1)
+ n − 1ϕp(m + 1)
) 1p
.
For 0 < p ≤ 1 using Lemma 2.3 we get
∥MϕGm
n∥p =
( 1ϕp(1)
+ n − 1ϕp(m + 1)
) 1p
.
Now we will prove the upper bound for 1 < p < ∞
∥MϕGm
nf∥p =
(MϕGm
nf(k)
)p+
∑i∈V \k
(Mϕ
Gmn
f(i))p
1p
=
max
1ϕp(1)
|f(k)|p,1
ϕp(n)
(∑w∈V
|f(w)|)p
+∑
i∈V \kmax
1ϕp(1)
|f(i)|p,
1ϕp(m + 1)
∑x∈B(j,1)
|f(x)|
p
,1
ϕp(n)
(∑w∈V
|f(w)|)p
1p
after applying Hölder’s inequality we get
∥MϕGm
n∥p ≤ sup
max
1ϕp(1)
|f(k)|p,1
ϕp(n)np−1
+
∑i∈V \k
max
1ϕp(1)
|f(i)|p,1
ϕp(j)jp−1
1p
502 Z. Hussain, S. Talib
where 1ϕp(j)jp−1 = max
1
ϕp(m+1) (m + 1)p−1 , 1ϕp(n)np−1
. If 1
ϕp(1) |f(k)|p ≤ 1ϕp(n)np−1 and
1ϕp(1) |f(i)|p ≤ 1
ϕp(j)jp−1 for all vertices then we have
∥MϕGm
n∥p ≤
1ϕp(n)
np−1 +∑
i∈V \k
1ϕp(j)
jp−1
1p
=(
np−1
ϕp(n)+ (n − 1) jp−1
ϕp(j)
) 1p
.
If 1ϕp(1) |f(k)|p ≤ 1
ϕp(n)np−1 and 1ϕp(1) |f(i)|p > 1
ϕp(j)jp−1 for some i then
∥MϕGm
n∥p ≤ sup
( 1ϕp(n)
np−1 +∑
i∈ 1ϕp(1) |f(i)|p> 1
ϕp(j) jp−1
1ϕp(1)
|f(i)|p
+∑
i∈ 1ϕp(1) |f(i)|p≤ 1
ϕp(j) jp−1
1ϕp(j)
jp−1) 1
p
≤(
np−1
ϕp(n)+ 1
ϕp(1)+ (n − 2) jp−1
ϕp(j)
) 1p
.
If 1ϕp(1) |f(k)|p ≥ 1
ϕp(n)np−1 and 1ϕp(1) |f(i)|p > 1
ϕp(j)jp−1 for some i then
∥MϕGm
n∥p ≤ sup
( 1ϕp(1)
|f(k)|p +∑
i∈ 1ϕp(1) |f(i)|p> 1
ϕp(j) jp−1
1ϕp(1)
|f(i)|p
+∑
i∈ 1ϕp(1) |f(i)|p≤ 1
ϕp(j) jp−1
1ϕp(j)
jp−1) 1
p
= sup
∑y∈k
∪i
1ϕp(1)
|f(y)|p +∑
i∈ 1ϕp(1) |f(i)|p≤ 1
ϕp(j) jp−1
1ϕp(j)
jp−1
1p
≤(
1ϕp(1)
+ (n − 2) jp−1
ϕp(j)
) 1p
.
If 1ϕp(1) |f(k)|p ≥ 1
ϕp(n)np−1 and 1ϕp(1) |f(i)|p ≤ 1
ϕp(j)jp−1 then we have
∥MϕGm
n∥p ≤ sup
1ϕp(1)
|f(k)|p +∑
i∈ 1ϕp(1) |f(i)|p≤ 1
ϕp(j) jp−1
1ϕp(j)
jp−1
1p
≤(
1ϕp(1)
+ (n − 1) jp−1
ϕp(j)
) 1p
.
Now we will prove some general results. For rest of the paper we assume 0 < p ≤ 1.
Theorem 3.2. For the general graph G with n vertices we have
∥Mϕ
Gn−1n
∥pp ≤ ∥Mϕ
G∥pp ≤ ∥Mϕ
G1n∥p
p.
Note on "Best constants for the Hardy-Littlewood maximal operator on finite graphs" 503
Proof. Lower bound of this theorem is trivial. We have to prove only the upper bound.Let j ∈ V and define δj , then we have
∥MϕGδj∥p
p =(Mϕ
Gδj(j))p
+∑
x∈V \j
(Mϕ
Gδj(x))p
= 1ϕp(1)
+∑
x∈V \j
1ϕ (|B(j, r)|)
∑w∈B(j,r)
δj(w)
p
clearly 2 ≤ |B(j, r)| for the radius r ≥ 1, so we get
∥MϕGδj∥p
p ≤ 1ϕp(1)
+ n − 1ϕp(2)
by using Lemma 2.3, we get∥Mϕ
G∥pp ≤ ∥Mϕ
G1n∥p
p.
Theorem 3.3. G = Gn−1n if and only if ∥Mϕ
G∥pp = ∥Mϕ
Gn−1n
∥pp.
Proof. If G = Gn−1n then ∥Mϕ
G∥p = ∥Mϕ
Gn−1n
∥p is a trivial case. We have only to provethe converse part, for that let G = Gn−1
n then there exist two different vertices x and y inV such that dG (x, y) > 1. Let consider two sets X = B (x, 1) = j ∈ V : dG (x, j) ≤ 1and Y = B (y, 1) = j ∈ V : dG (y, j) ≤ 1. It is clear that |X| , |Y | ≥ 2. Thus, weconsider two cases.
Case 1. min |X| , |Y | ≤ n2 .
We assume that |X| ≤ n2 . Let k ∈ X such that it is different from x and we define δk,
then
∥MϕGδk∥p
p =∑v∈V
(Mϕ
Gδk(v))p
=(Mϕ
Gδk(k))p
+(Mϕ
Gδk(x))p
+∑
v∈V \x,k
(Mϕ
Gδk(v))p
since MϕGδk(v) ≥ 1
ϕ(n) for each v ∈ V , so we get
∥MϕG∥p
p ≥ ∥MϕGδk∥p
p ≥ 1ϕp(1)
+ 1ϕp (|X|)
+ n − 2ϕp(n)
≥ 1ϕp(1)
+ 1ϕp(
n2) + n − 2
ϕp(n)
> ∥Mϕ
Gn−1n
∥pp,
which completes the proof of case 1.
Case 2. min |X| , |Y | > n2 .
It is easy to see that X ∩ Y = ∅. Let k ∈ X ∩ Y and define δk, then
∥MϕGδk∥p
p =(Mϕ
Gδk(k))p
+(Mϕ
Gδk(x))p
+(Mϕ
Gδk(y))p
+∑
v∈V \x,y,k
(Mϕ
Gδk(v))p
≥ 1ϕp(1)
+ 1ϕp (|X|)
+ 1ϕp (|Y |)
+ n − 3ϕp(n)
.
504 Z. Hussain, S. Talib
Clearly |X| , |Y | ≤ n − 1, so
∥MϕG∥p
p ≥ ∥MϕGδk∥p
p ≥ 1ϕp(1)
+ 2ϕp (n − 1)
+ n − 3ϕp(n)
> ∥Mϕ
Gn−1n
∥pp.
Theorem 3.4. G ∼ G1
n if and only if ∥MϕG∥p
p = ∥MϕG1
n∥p
p.
Proof. G ∼ G1n ⇒ ∥Mϕ
G∥p = ∥MϕG1
n∥p is trivial. Now suppose that G G1
n and hencen ≥ 3 then ∃ x = y in V such that dG(x), dG(y) > 1. Suppose that ∥f∥1 ≤ 1, for everyfunction f : V → R then, either Mϕ
Gf(j) = |f(j)|ϕ(1) or Mϕ
Gf(j) ≤ 1ϕ(dG(j)+1) . Take the set
X =
j ∈ V : MϕGf(j) = f(j)
ϕ(1)
, then we have
∥MϕGf∥p
p =∑j∈X
(Mϕ
Gf(j))p
+∑j ∈X
(Mϕ
Gf(j))p
≤∑j∈X
(Mϕ
Gf(j))p
+∑j ∈X
1ϕp (dG(j) + 1)
.
If x, y ∈ X, then
∥MϕGf∥p
p ≤ 1ϕp(1)
+ n − 2ϕp(2)
.
If x ∈ X, then since X = ∅, we have
∥MϕGf∥p
p ≤ 1ϕp(1)
+ 1ϕp(dG(j) + 1)
+ n − 2ϕp(2)
≤ 1ϕp(1)
+ 1ϕp(3)
+ n − 2ϕp(2)
.
Similarly case when y ∈ X. So
∥MϕG∥p
p ≤ sup 1
ϕp(1)+ n − 2
ϕp(2),
1ϕp(1)
+ 1ϕp(3)
+ n − 2ϕp(2)
< ∥Mϕ
G1n∥p
p,
which completes our arguments. If we put ϕ(t) = t and m = n−1 in the result of Proposition 3.1, then we get the expres-
sion (1.3). Moreover, if we put ϕ(t) = t in the results of Theorems 3.2, 3.3 and 3.4 thenwe get the same results as proved in [2], which shows that this work is a generalization of [2].
Acknowledgment. The authors would like to thank the referee for the useful correc-tions and invaluable comments which improved the first version of this paper. Moreoverthey said special thanks to their respected teacher Dr. Irshad Ahmad for his support andguidance.
References[1] I. Ahmad and W. Nazeer, Optimal Couples of Rearrangement Invariant Spaces for
Generalized Maximal Operators, J. Funct. Spaces 2014, Article ID 647123, 5 pages,2014.
[2] J. Soria and P. Tradacete, Best constants for the Hardy-Littlewood maximal operatoron finite graphs, J. Math. Anal. Appl. 436 (2), 661–682, 2016.
Hacettepe Journal ofMathematics & Statistics
Hacet. J. Math. Stat.Volume 49 (2) (2020), 505 – 509
DOI : 10.15672/hujms.568323
Research Article
Prime geodesic theorem for the modular surfaceMuharem Avdispahić
University of Sarajevo, Department of Mathematics, Zmaja od Bosne 33-35, 71000 Sarajevo, Bosnia andHerzegovina
AbstractUnder the generalized Lindelöf hypothesis, the exponent in the error term of the prime ge-odesic theorem for the modular surface is reduced to 5
8 +ε outside a set of finite logarithmicmeasure.Mathematics Subject Classification (2010). 11M36, 11F72, 58J50
Keywords. prime geodesic theorem, Selberg zeta function, modular group
1. IntroductionLet Γ = PSL (2,Z) be the modular group and H the upper half-plane equipped with the
hyperbolic metric. The norms N(P0) of primitive conjugacy classes P0 in Γ are sometimescalled pseudo-primes. The length of the primitive closed geodesic on the modular surfaceΓ \ H joining two fixed points, which are the same for all representatives of P0, equalslog(N(P0)). The statement about the number πΓ(x) of classes P0 such that N(P0) ≤ x,for x > 0, is known as the prime geodesic theorem, PGT.
The main tool in the proof of PGT is the Selberg zeta function, defined by
ZΓ(s) =∏
P0
∞∏k=0
(1 −N(P0)−s−k), Re(s) > 1,
and meromorphicaly continued to the whole complex plane.The relationship between the prime geodesic theorem and the distribution of zeros of
the Selberg zeta function resembles to a large extent the relationship between the primenumber theorem and the zeros of the Riemann zeta.
The function ZΓ satisfies the analogue of the Riemann hypothesis. The zeros 12 + iγ =
12 ± i
√λ− 1
4 of the Selberg ZΓ lying on Re(s) = 12 correspond to the eigenvalues λ ≥ 1
4 ofthe essentially self-adjoint Laplace-Beltrami operator ∆ = −y2
(∂2
∂x2 + ∂2
∂y2
)on Γ\H. See,
e.g., [6] for some important applications of the modular group and the modular surfacePSL(2,Z) \ H in physics.
It is an outstanding problem whether the error term in the prime geodesic theorem isO(x
12 +ε) as it would be the case in the prime number theorem once the Riemann hypothesis
be proved. The obstacles in establishing an analogue of von Koch’s theorem [13, p. 84] inthis setting comes from the fact that ZΓ is a meromorphic function of order 2, while theRiemann zeta is of order 1 ([12, relation (6.14) on p. 113]).
Email address: [email protected]: 31.05.2018; Accepted: 05.01.2019
506 M. Avdispahić
In the case of Fuchsian groups Γ ⊂ PSL (2,R), the best estimate of the remainder termin PGT is still O
(x
34
log x
)obtained by Randol [18] (see also [2, 7] for different proofs). We
note that its analogue O(x
4d20+d0
2d0+1 (log x)−1)
is valid also for strictly hyperbolic manifolds
of higher dimensions, where d0 = d−12 and d ≥ 3 is the dimension of a manifold [4, Theorem
2.1].The attempts to reduce the exponent 3
4 in PGT were successful only in special cases.The chronological list of improvements for the modular group Γ = PSL(2,Z) includes3548 + ε (Iwaniec [15]), 7
10 + ε (Luo and Sarnak [17]), 71102 + ε (Cai [8]) and the present 25
36 + ε(Soundararajan and Young [19]).
Iwaniec [14] remarked that the generalized Lindelöf hypothesis for Dirichlet L-functionswould imply 2
3 + ε.We proved in [3] that 2
3 + ε is valid outside a set of finite logarithmic measure. In thepresent note, we relate the error term in the Gallagherian PGT (i.e., PGT with an errorterm valid outside a set of finite logarithmic measure) on PSL(2,Z) to the subconvexitybound for Dirichlet L- functions. This enables us to replace 2
3 + ε by 58 + ε under the gen-
eralized Lindelöf hypothesis. More precisely, the main result of this paper is the followingtheorem.
Theorem 1.1. Let Γ = PSL(2,Z) be the modular group, ε > 0 arbitrarily small and θ besuch that
L
(12
+ it, χD
)≪ (1 + |t|)A |D|θ+ε
for some fixed A > 0, where D is a fundamental discriminant. There exists a set B offinite logarithmic measure such that
πΓ (x) =∫ x
0
dt
log t+O
(x
58 + θ
4 +ε)
(x → ∞, x /∈ B) .
Inserting the Conrey-Iwaniec [9] value θ = 16 into Theorem 1.1, we obtain
Corollary 1.2.πΓ (x) = li (x) +O
(x
23 +ε
)(x → ∞, x /∈ B) .
Any improvement of θ immediately results in the obvious improvement of the error termin PGT. Taking into account that the Lindelöf hypothesis allows θ = 0, we get
Corollary 1.3. Under the generalized Lindelöf hypothesis for Dirichlet L-functions in theconductor aspect, we have
πΓ (x) = li (x) +O(x
58 +ε
)(x → ∞, x /∈ B) .
Remark 1.4. The obtained error term in PGT for strictly hyperbolic Fuchsian groupsis O
(x
710 (log x)− 4
5 (log log x)15 +ε
)outside a set of finite logarithmic measure [1]. This is
in accordance with the above mentioned Luo-Sarnak unconditional exponent 710 + ε in
PGT for Γ = PSL(2,Z). In the case of a cocompact Kleinian group or a noncompactcongruence group for some imaginary quadratic number field, the respective Gallagherianbound is O
(x
2113 (log x)− 11
13 (log log x)2
13 +ε)
[4, Theorem 1.2].
2. PreliminariesThe motivation for Theorem 1.1 comes from several sources, including Gallagher [11],
Iwaniec [15] and Balkanova and Frolenkov [5].
PGT for the modular surface 507
Recall that πΓ (x) = li (x) + O(x
58 + θ
4 +ε)
is equivalent to ψΓ (x) = x + O(x
58 + θ
4 +ε),
where ψΓ (x) =∑
N(P0)k≤x
logN (P0) is the Γ analogue of the classical Chebyshev function
ψ.Under the Riemann hypothesis, Gallagher improved von Koch’s remainder term in the
prime number theorem from ψ(x) = x+O(x
12 (log x)2
)to ψ(x) = x+O
(x
12 (log log x)2
)outside a set of finite logarithmic measure.
Following Koyama [16], we shall apply the next lemma due to Gallagher [10] to oursetting.
Lemma 2.A. Let A be a discrete subset of R and η ∈ (0, 1). For any sequence c(ν) ∈ C,ν ∈ A, let the series
S (u) =∑ν∈A
c (ν) e2πiνu
be absolutely convergent. Then
∫ U
−U|S (u)|2 du ≤
(πη
sin πη
)2 ∫ +∞
−∞
∣∣∣∣∣∣∣U
η
∑t≤ν≤t+ η
U
c (ν)
∣∣∣∣∣∣∣2
dt.
Iwaniec [15] established the following explicit formula with an error term for ψΓ onΓ = PSL(2,Z).
Lemma 2.B. For 1 ≤ T ≤ x12
(log x)2 , one has
ψΓ (x) = x+∑
|γ|≤T
xρ
ρ+O
(x
T(log x)2
),
where ρ = 12 + iγ denote zeros of ZΓ lying on Re(s) = 1
2 and counted with their multiplic-ities.
Recently, O. Balkanova and D. Frolenkov [5] have proved the following estimate.
Lemma 2.C. ∑|γ|≤Y
xiγ ≪ max(x
14 + θ
2Y12 , x
θ2Y)
log3 Y ,
∑|γ|≤Y
xiγ ≪ Y log2 Y if Y >x
12 + 7
6 θ
κ (x),
where ρ = 12 + iγ are the zeros of ZΓ, θ is the subconvexity exponent for Dirichlet
L−functions, and κ (x) is the distance from√x+ 1√
xto the nearest integer.
3. Proof of Theorem 1.1
Proof. Inserting T = x12
(log x)2 into Lemma 2.B, we obtain
ψΓ (x) = x+∑
|γ|≤T
xρ
ρ+O
(x
12 (log x)4
). (3.1)
We would like to bound the expression∑
|γ|≤Y
xρ
ρ , where Y ∈ (0, T ) is a parameter to be
determined later on.
508 M. Avdispahić
Let n = ⌊log x⌋ and Bn =x ∈
[en, en+1) :
∣∣∣∣∣ ∑|γ|≤Y
xiγ
ρ
∣∣∣∣∣ > xεY12
. Looking at the loga-
rithmic measure of Bn, we get
µ∗Bn =∫Bn
dx
x=∫Bn
x2εYdx
x1+2εY≤
en+1∫en
∣∣∣∣∣∣∑
|γ|≤Y
xiγ
ρ
∣∣∣∣∣∣2
dx
x1+2εY(3.2)
≤ 1e2nεY
en+1∫en
∣∣∣∣∣∣∑
|γ|≤Y
xiγ
ρ
∣∣∣∣∣∣2dx
x.
After substitution x = en · e2π(u+ 14π ), the last integral becomes
2π
14π∫
− 14π
∣∣∣∣∣∣∑
|γ|≤Y
e(n+ 12 )iγ
ρe2πiγu
∣∣∣∣∣∣2
du.
Applying Lemma 2.A, with η = U = 14π and cγ = e(n+ 1
2 )iγ
ρ for |γ| ≤ Y , cγ = 0 otherwise,we get
14π∫
− 14π
∣∣∣∣∣∣∑
|γ|≤Y
e(n+ 12 )iγ
ρe2πiγu
∣∣∣∣∣∣2
du ≤( 1
4sin 1
4
)2 +∞∫−∞
∑t<γ≤t+1
|γ|≤Y
1|ρ|
2
dt. (3.3)
Note that∑
t<γ≤t+11
|ρ| = O (1) since # γ : t < |γ| ≤ t+ 1 = O (t) by the Weyl law.Thus,
+∞∫−∞
∑t<γ≤t+1
|γ|≤Y
1|ρ|
2
dt = O
Y∫
12
dt
= O (Y ) . (3.4)
The relations (3.2), (3.3) and (3.4) imply µ∗Bn ≪ Ye2nεY
= 1e2nε . Hence, the set B = ∪Bn
has a finite logarithmic measure.
For x /∈ B, we have∣∣∣∣∣ ∑|γ|≤Y
xiγ
ρ
∣∣∣∣∣ ≤ xεY12 , i.e.
∣∣∣∣∣∣∑
|γ|≤Y
xρ
ρ
∣∣∣∣∣∣ ≤ x12 +εY
12 . (3.5)
Now, we rely on Lemma 2.C to estimate∣∣∣∣∣ ∑Y <|γ|≤T
xρ
ρ
∣∣∣∣∣. Let us put S (x, T ) =∑
|γ|≤Txiγ .
By Abel’s partial summation, we have
∑Y <|γ|≤T
xiγ
ρ= S (x, T )
12 + iT
− S (x, Y )12 + iY
+ i
T∫Y
S (x, u)(12 + iu
)2du.
Multiplying the last relation by x12 and recalling that Lemma 2.C yields
∑|γ|≤Y
xiγ ≪
x14 + θ
2 +εY12 for Y < T = x
12
(log x)2 , we get
PGT for the modular surface 509
∣∣∣∣∣∣∑
Y <|γ|≤T
xρ
ρ
∣∣∣∣∣∣ ≪ x34 + θ
2 +ε
T12
+ x34 + θ
2 +ε
Y12
+T∫
Y
x34 + θ
2 +εu12
u2 du ≪ x34 + θ
2 +ε
Y12
. (3.6)
Combining (3.5) and (3.6), we see that the optimal choice for the parameter Y isY ≈ x
14 + θ
2 . Then,∑
|γ|≤T
xρ
ρ = O(x
12 +εY
12)
= O(x
58 + θ
4 +ε)
for x /∈ B.
The relation (3.1) becomes
ψΓ (x) = x+O(x
58 + θ
4 +ε)
(x → ∞, x /∈ B) ,
as asserted.
References[1] M. Avdispahić, Prime geodesic theorem of Gallagher type, arXiv:1701.02115, 2017.[2] M. Avdispahić, On Koyama’s refinement of the prime geodesic theorem, Proc. Japan
Acad. Ser. A Math. Sci. 94 (3), 21–24, 2018.[3] M. Avdispahić, Gallagherian PGT on PSL(2,Z), Funct. Approx. Comment. Math.
58 (2), 207–213, 2018.[4] M. Avdispahić, Errata and addendum to "On the prime geodesic theorem for hyperbolic
3-manifolds" Math. Nachr. 291 (2018), no. 14-15, 2160–2167, Math. Nachr. 292 (4),691–693, 2019.
[5] O. Balkanova and D. Frolenkov, Bounds for a spectral exponential sum, J. LondonMath. Soc. 99 (2), 249–272, 2019.
[6] R. Blumenhagen and E. Plauschinn, Introduction to conformal field theory: Withapplications to string theory, in: Lect. Notes Phys. 779, Springer, Berlin Heidelberg,2009.
[7] P. Buser, Geometry and spectra of compact Riemann surfaces, Progress in Mathe-matics 106, Birkhäuser, Boston-Basel-Berlin, 1992.
[8] Y. Cai, Prime geodesic theorem, J. Théor. Nombres Bordeaux, 14 (1), 59–72, 2002.[9] J.B. Conrey and H. Iwaniec, The cubic moment of central values of automorphic
L-functions, Ann. of Math. (2) 151 (3), 1175–1216, 2000.[10] P.X. Gallagher, A large sieve density estimate near σ = 1, Invent. Math. 11, 329–339,
1970.[11] P.X. Gallagher, Some consequences of the Riemann hypothesis, Acta Arith. 37, 339–
343, 1980.[12] D.A. Hejhal, The Selberg trace formula for PSL(2, R). Vol I, Lecture Notes in Math-
ematics 548, Springer, Berlin, 1976.[13] A.E. Ingham, The distribution of prime numbers, Cambridge University Press, 1932.[14] H. Iwaniec, Non-holomorphic modular forms and their applications, in: Modular
forms (Durham, 1983), Ellis Horwood Ser. Math. Appl.: Statist. Oper. Res., 157–196, Horwood, Chichester, 1984.
[15] H. Iwaniec, Prime geodesic theorem, J. Reine Angew. Math. 349, 136–159, 1984.[16] S. Koyama, Refinement of prime geodesic theorem, Proc. Japan Acad. Ser A Math.
Sci. 92 (7), 77–81, 2016.[17] W. Luo and P. Sarnak, Quantum ergodicity of eigenfunctions on PSL2(Z)\H2, Inst.
Hautes Études Sci. Publ. Math. 81, 207–237, 1995.[18] B. Randol, On the asymptotic distribution of closed geodesics on compact Riemann
surfaces, Trans. Amer. Math. Soc. 233, 241–247, 1977.[19] K. Soundararajan and M.P. Young, The prime geodesic theorem, J. Reine Angew.
Math. 676, 105–120, 2013.
Hacettepe Journal ofMathematics & Statistics
Hacet. J. Math. Stat.Volume 49 (2) (2020), 510 – 522
DOI : 10.15672/hujms.458188
Research Article
Approximation by Cheney-Sharma Chlodovskyoperators
Dilek Söylemez∗1, Fatma Taşdelen2
1Ankara University, Elmadag Vocational School, Department of Computer Pogramming, 06780, Ankara,Turkey
2Ankara University, Faculty of Science, Department of Mathematics, 06100 Tandogan, Ankara, Turkey
AbstractThe main purpose of this paper is to construct Cheney-Sharma Chlodovsky operators. Westudy approximation properties of the new operators with the help of weighted Korovkin-type theorem and universal Korovkin-type theorem. We also give the rate of convergenceby means of the modulus of continuity. Furthermore, we give A-statistical convergenceproperty of these operators.
Mathematics Subject Classification (2010). 41A36, 40A35
Keywords. Cheney-Sharma Chlodovsky operators, weighted approximation, rate ofapproximation, A−statistical approximation
1. IntroductionApproximation theory is concerned with how functions can best be approximated with
simpler functions. Since Korovkin theorem was obtained, the studies of the linear methodsof approximation given by sequences of positive and linear operators have become an im-portant area in approximation theory. The study of the statistical convergence in approx-imation theory for sequences of linear positive operators was attempted by Gadjiev andOrhan [23]. They proved a Korovkin type theorem by considering statistical convergenceinstead of ordinary convergence. Later, many authors gave several approximation resultsvia summability methods, for example, we refer the readers to [1,2,4,16,28,29,31,32,34].
In 1932, Chlodovsky [14], introduced the classical Bernstein-Chlodovsky polynomialsas a generalization of Bernstein polynomials on unbounded set.For every n ∈ N , f ∈ C [0, ∞) and x ∈ [0, ∞) , these polynomials Cn : C [0, ∞) → C [0, ∞)defined by
Cn(f, x) :=
n∑
k=0f(
knbn
) (nk
) (xbn
)k (1 − x
bn
)n−k, if 0 ≤ x ≤ bn
f (x) , if x > bn
,
where 0 ≤ x ≤ bn and bn is a positive sequence with the properties;limn→∞ bn = ∞, limn→∞ bn/n = 0.
∗Corresponding Author.Email addresses: [email protected] (D. Söylemez), [email protected] (F. Taşdelen)Received: 10.10.2018; Accepted: 06.01.2019
Approximation by Cheney-Sharma Chlodovsky operators 511
Recently some authors have studied some Chlodovsky type polynomials which may befound in [6, 12,24–27,30].
It is known that the Abel-Jensen equalities are given by the following formulas (see [5],p.326)
(u + v + nβ)n−1 =n∑
k=0
(n
k
)u (u + kβ)k−1 v [v + (n − k) β]n−k−1 (1.1)
and
(u + v) (u + v + mβ)m−1 =m∑
k=0
(m
k
)u (u + kβ)k−1 v [v + (m − k) β]m−k−1 , (1.2)
where u, v and β ∈ R. By means of these equalities Cheney-Sharma [13] introduced thefollowing Bernstein type operators for f ∈ C[0, 1], x ∈ [0, 1] and n ∈ N
Qn (f ; x) = (1 + nβ)1−nn∑
k=0f
(k
n
)(n
k
)x (x + kβ)k−1 (1.3)
× [1 − x + (n − k) β]n−k
and
Gn (f ; x) = (1 + nβ)1−nn∑
k=0f
(k
n
)(n
k
)x (x + kβ)k−1 (1.4)
× (1 − x) [1 − x + (n − k) β]n−k−1 ,
where β is a nonnegative real parameter. It is obvious that for β = 0 these operatorsreduce to the classical Bernstein operators. Cheney-Sharma proved that if nβn → ∞ asn → ∞, then for f ∈ C[0, 1], these operators uniformly convergence to f on [0, 1] . In[7], the authors showed that Cheney-Sharma operators preserve the Lipschitz constantand order of a Lipschitz continuous function as well as the properties of the function ofmodulus of continuity. They also gave a result of Gn (f ; x) under the convexity of f .Kantorovich type generalization of the Cheney-Sharma operators was studied in [33]. Forthese operators, we refer the readers to [3, 8, 15,35,36].
In the present paper, we construct the Chlodovsky-type generalization of the Cheney-Sharma operators is given by (1.4) and we prove that the weighted uniform convergenceof G∗
nf to f . We also give approximation results using universal Korovkin-type theoremand obtain rate of approximation in terms of the usual modulus of continuity. Finally, westudy A-statistical convergence behaviours of new positive linear operators expressed asbelow.
We introduce the Cheney-Sharma Chlodovsky operators as
G∗n (f ; x) = (1 + nβ)1−n
n∑k=0
f
(k
nbn
)(n
k
)x
bn
(x
bn+ kβ
)k−1(1.5)
×(
1 − x
bn
)[1 − x
bn+ (n − k) β
]n−k−1
for 0 ≤ x ≤ bn and f(x) for x > bn. Here bn is a positive sequence with the properties;limn→∞ bn = ∞, limn→∞ bn/n = 0. If we take bn = 1, then we obtain the Cheney-Sharmaoperators (1.4) .
Now, let us recall the concept of A-statistical convergence. Let A = (ank) be a summa-bility matrix and let x = (xk) be a sequence.
If the series(Ax)n :=
∑k
ankxk
512 D. Söylemez, F. Taşdelen
is convergent for each n then, we say that Ax := (Ax)n is the A-transformation of x.If the sequence Ax converges to a number L then, we say that x is A-summable to L. Asummability matrix A is said to be regular if limn(Ax)n = L whenever limk xk = L [9].
Let A = (ank) be a nonnegative regular summability matrix and let K be a subset ofpositive integer. Then K is said to have A-density δA(K) if the limit
δA(K) := limn
∑k∈K
ank
exists [10, 11, 19]. The sequence x = (xk) is said to be A-statistically convergent to realnumber α if for any ε > 0
limk
∑k:|xk−α|≥ε
ank = 0.
In this case, we write stA − lim x = α [18,20]. Note that x is A-statistically convergent toα if and only if for any ε > 0, δA(Kε) = 0, where Kε := k ∈ N : |xk − α| ≥ ε. If A is theidentity matrix I, then I-statistical convergence reduces to ordinary convergence, and, ifA = C1, the Cesáro matrix of order one, then it coincides with statistical convergence.
Throughout the paper, we will consider the following class of functions. Let ρ (x) =1 + x2,
Bρ
(R+)
=
f : R+ → R, |f (x)| ≤ Mf ρ (x) , x ≥ 0
,
where Mf is a constant depending on f .
Cρ
(R+)
=
f ∈ Bρ
(R+)
; f is continuous on R+
,
Ckρ
(R+)
=
f ∈ Cρ
(R+)
; limx→∞f (x)ρ (x)
= kf
,
where kf is a constant depending on f .It is obvious that Ck
ρ
(R+) ⊂ Cρ
(R+) ⊂ Bρ
(R+) . The space Bρ
(R+) is a normed linear
space with the norm ∥f∥ρ = supx∈R+|f(x)|ρ(x) .
2. Approximation by (G∗n)
Korovkin theorem was extended to unbounded intervals and a weighted Korovkin typetheorem in a subspace of continuous functions on the real axis R was proved in [21, 22].Let us recall the weighted form of the Korovkin Theorem ([21,22]).
Lemma 2.1. The positive linear operators Ln, n ≥ 1, act from Cρ(R+) to Bρ
(R+) if and
only if the inequality|Ln (ρ; x)| ≤ Knρ (x) , x ≥ 0
holds, where Kn is a positive constant.
Theorem 2.2. Let the sequence of linear positive operators (Ln)n≥1 acting from Cρ(R+)
to Bρ(R+) satisfy the three conditions
limn→∞
∥Ln (tv; x) − xv∥ρ = 0, v = 0, 1, 2. (2.1)
Then for any function f ∈ Ckρ
(R+)
limn→∞
∥Lnf − f∥ρ = 0.
Lemma 2.3. Let ei (t) = ti, i = 0, 1, 2. For the operators (1.5) , we haveG∗
n (e0; x) = 1, (2.2)
G∗n (e1; x) = x (2.3)
Approximation by Cheney-Sharma Chlodovsky operators 513
and
G∗n (e2; x) ≤ x (x + 2bnβ) (1 + nβ)
+ xbn
n(nβ)2 (1 + nβ) + x (x + 2bnβ) nβ (2.4)
+ xbn
n(nβ)3 + x
bn
n.
Proof. Firstly, we show that G∗n (ei; x) for i = 0,
G∗n (e0; x) = (1 + nβ)1−n
n∑k=0
(n
k
)x
bn
(x
bn+ kβ
)k−1 (1 − x
bn
)[1 − x
bn+ (n − k) β
]n−k−1.
If we take u = xbn
, v = 1 − xbn
in (1.2) , we get
G∗n (e0; x) = 1. (2.5)
Morever, taking G∗n (ei; x) for i = 1, one can have
G∗n (e1; x) = (1 + nβ)1−n
n∑k=1
(n − 1k − 1
)bn
x
bn
(x
bn+ kβ
)k−1
×(
1 − x
bn
)[1 − x
bn+ (n − k) β
]n−k−1. (2.6)
If we take k → k + 1 and considering the equalityxbn
+ β + kβ = 1 + nβ −[1 − x
bn+ (n − k − 1) β
], we reach to
G∗n (e1; x) = (1 + nβ)1−n bn
n−1∑k=0
(n − 1
k
)x
bn
(x
bn+ β + kβ
)k−1(1 + nβ)
×(
1 − x
bn
)[1 − x
bn+ (n − k − 1) β
]n−k−2
− (1 + nβ)1−n bn
n−1∑k=0
(n − 1
k
)x
bn
(x
bn+ β + kβ
)k−1
×(
1 − x
bn
)[1 − x
bn+ (n − k − 1) β
]n−k−1. (2.7)
In order to find the first sum, we replace in the identity (1.2), u = xbn
+ β, v = 1 − xbn
, nby n − 1, we get
(1 + β) (1 + nβ)n−2 =(
x
bn+ β
) n−1∑k=0
(n − 1
k
)(x
bn+ β + kβ
)k−1
×(
1 − x
bn
)[1 − x
bn+ (n − k − 1) β
]n−k−2. (2.8)
If we use (2.8) in (2.7) , we have
G∗n (e1; x) = (1 + β) xbn
x + bnβ− (1 + nβ)1−n bn
n−1∑k=0
(n − 1
k
)x
bn
(x
bn+ β + kβ
)k−1
×(
1 − x
bn
)[1 − x
bn+ (n − k − 1) β
]n−k−1. (2.9)
514 D. Söylemez, F. Taşdelen
On the other hand, if we take u, v and n by xbn
+ β, 1 − xbn
and n − 1, respectively in (1.1),we get
(1 + nβ)n−1 =n−1∑k=0
(n − 1
k
)(x
bn+ β
)(x
bn+ β + kβ
)k−1
×[1 − x
bn+ (n − k − 1) β
]n−k−1. (2.10)
By considering (2.10) in (2.9) , so we have
G∗n (e1; x) = x. (2.11)
Finally, we estimate G∗n (ei; x) for i = 2,
G∗n (e2; x) = (1 + nβ)1−n n − 1
n(bn)2
n−2∑k=0
(n − 2
k
)x
bn
(x
bn+ kβ + 2β
)k+1
×(
1 − x
bn
)[1 − x
bn+ (n − k − 2) β
]n−k−3
+ (1 + nβ)1−n bn
n
n∑k=1
k
nbn
(n
k
)x
bn
(x
bn+ kβ
)k−1
×(
1 − x
bn
)[1 − x
bn+ (n − k) β
]n−k−1
= (1 + nβ)1−n n − 1n
(bn)2 x
bn
n−2∑k=0
(n − 2
k
)(x
bn+ kβ + 2β
)k+1
×[1 − x
bn+ (n − k − 2) β
]n−k−2
− (1 + nβ)1−n n − 1n
(bn)2 x
bn
n−2∑k=0
(n − 2
k
)x
bn
(x
bn+ kβ + 2β
)k+1
× (n − k − 2) β
[1 − x
bn+ (n − k − 2) β
]n−k−3
+ bn
nG∗
n (e1; x) .
In order to find an upper estimate for the G∗n (e2; x) , we need to give the following equality,
as in [13]
L
(j, n,
x
bn, 1 − x
bn
):=
n∑k=0
(n
k
)(x
bn+ kβ
)k+j−1 (1 − x
bn+ (n − k) β
)n−k
, (2.12)
where n ∈ N, 0 ≤ x ≤ bn and bn is a sequence of positive numbers such that limn→∞ bn =∞, limn→∞
bnn = 0.
It is clear that the function L(j, n, x
bn, 1 − x
bn
)satisfies the following reduction formula
L
(j, n,
x
bn, 1 − x
bn
)= x
bnL
(j − 1, n,
x
bn, 1 − x
bn
)+ nβL
(j, n − 1,
x
bn+ β, 1 − x
bn
).
Approximation by Cheney-Sharma Chlodovsky operators 515
Therefore from (2.12), we get
G∗n (e2; x) = x
bn(1 + nβ)1−n n − 1
n(bn)2 L
(2, n − 2,
x
bn+ 2β, 1 − x
bn
)− x
bn(1 + nβ)1−n n − 1
n(bn)2 (n − 2) βL
(2, n − 3,
x
bn+ 2β, 1 − x
bn+ β
)+ bn
nG∗
n (e1; x)
= K1 (n, x) + K2 (n, x) + bn
nx.
From Lemma 1 in [13], we know that
L (2, n, x, y) =∞∫
0
e−sds
∞∫0
e−u [x (x + y + nβ + sβ + uβ)n (2.13)
+nβ2u (x + y + nβ + sβ + uβ)n−1]
du
and so we have,
L
(2, n − 2,
x
bn+ 2β, 1 − x
bn
)
= (1 + nβ)n−2∞∫
0
e−sds
∞∫0
e−u
[(x
bn+ 2β
)(1 + sβ + uβ
1 + nβ
)n−2]
du
+ (1 + nβ)n−3∞∫
0
e−sds
∞∫0
e−u (n − 2) β2u
(1 + sβ + uβ
1 + nβ
)n−3du
= α (n, x) + β (n, x) . (2.14)
Taking into account this inequality (1 + v)n−2 ≤ exp (nv) , we have
α (n, x) ≤ (1 + nβ)n−2∞∫
0
e−sds
∞∫0
e−u[(
x
bn+ 2β
)exp
(n
(sβ + uβ
1 + nβ
))]du
=(
x
bn+ 2β
)(1 + nβ)n−2
∞∫0
e−s(
11+nβ
)ds
∞∫0
e−u(
11+nβ
)du
=(
x
bn+ 2β
)(1 + nβ)n−2 (1 + nβ)2 (2.15)
and
β (n, x) ≤ (1 + nβ)n−3∞∫
0
e−sds
∞∫0
e−uu (n − 2) β2 exp(
n
(sβ + uβ
1 + nβ
))du
= (n − 2) β2 (1 + nβ)n−3∞∫
0
e−s(
11+nβ
)ds
∞∫0
ue−u(
11+nβ
)du
= (1 + nβ)n (n − 2) β2. (2.16)
516 D. Söylemez, F. Taşdelen
Taking (2.15) and (2.16) in (2.14), we get
K1 (n, x) ≤(
x
bn+ 2β
)x
bn(1 + nβ)1−n n − 1
n(bn)2 (1 + nβ)n
+ x
bn(1 + nβ)1−n n − 1
n(bn)2 (1 + nβ)n (n − 2) β2
≤ x (x + 2bnβ) (1 + nβ) + xbn
n(nβ)2 (1 + nβ) . (2.17)
On the other hand
K2 (n, x) ≤ x
bn(1 + nβ)1−n n − 1
n(bn)2 (n − 2) β
× L
(2, n − 3,
x
bn+ 2β, 1 − x
bn+ β
)(2.18)
and
L
(2, n − 3,
x
bn+ 2β, 1 − x
bn+ β
)
=∞∫
0
e−sds
∞∫0
e−u(
x
bn+ 2β
)(1 + nβ + sβ + uβ)n−3 du
+∞∫
0
e−sds
∞∫0
e−u (n − 2) β2 (1 + nβ + sβ + uβ)n−4 du
= γ (n, x) + ξ (n, x) . (2.19)Here
γ (n, x) ≤ (1 + nβ)n−3(
x
bn+ 2β
) ∞∫0
e−sds
∞∫0
e−u exp(
n
(sβ + uβ
1 + nβ
))du
= (1 + nβ)n−3(
x
bn+ 2β
) ∞∫0
e−s(
11+nβ
)ds
∞∫0
e−u(
11+nβ
)du
= (1 + nβ)n−3(
x
bn+ 2β
)(1 + nβ)2 (2.20)
and
ξ (n, x) =∞∫
0
e−sds
∞∫0
ue−u (n − 2) β2 (1 + nβ + sβ + uβ)n−4 du
≤ (1 + nβ)n−4 (n − 2) β2 (1 + nβ) (1 + nβ)2 . (2.21)
So, taking (2.19) , (2.20) and (2.21) in (2.18) , we have
K2 (n, x) ≤ x (x + 2bnβ) nβ + xbn
n(nβ)3 .
Hence, we can writeG∗
n (e2; x) ≤ x (x + 2bnβ) (1 + nβ)
+ xbn
n(nβ)2 (1 + nβ) + x (x + 2bnβ) nβ
+ xbn
n(nβ)3 + x
bn
n. (2.22)
So, we obtain the desired results of Lemma 2.1.
Approximation by Cheney-Sharma Chlodovsky operators 517
Now, in order to get an approximation result, we consider β as a sequence of positivereal numbers such that β = βn and limn→∞nβn = 0.
We show here that the operators G∗n (f) satisfy the required conditions for the weighted
Korovkin type theorem.
Theorem 2.4. Suppose that n ∈ N, (bn) and (βn) are sequence of positive numbers suchthat limn→∞ bn = ∞, limn→∞ bn/n = 0 and limn→∞nβn = 0. Then for each f ∈ Ck
ρ
(R+) ,
we havelim
n→∞∥G∗
nf − f∥ρ = 0.
Proof. By Lemma 2.2, (G∗n) are linear operators acting from Cρ
(R+) to Bρ
(R+). Indeed,
from (2.5) and (2.22) , we easily obtain that
|G∗n (ρ; x)| ≤
(1 + x2
)Kn.
On the other hand, using (2.5) , (2.11) and (2.22) , one can write
∥G∗n (e0) − 1∥ρ = 0,
∥G∗n (e1) − x∥ρ = 0,
and ∥∥∥G∗n (e2) − x2
∥∥∥ρ
≤ sup∣∣∣∣∣ x2
1 + x2 (2nβn)∣∣∣∣∣
+∣∣∣∣ 2x
1 + x2 bnβn (1 + nβn)∣∣∣∣
+ x
1 + x2bn
n(nβn)2 (1 + 2nβn) + x
1 + x2bn
n
,
by the hypothesis, we havelim
n→∞
∥∥∥G∗n (e2) − x2
∥∥∥ρ
= 0.
So the proof is completed.
In the following theorem, we give approximation results of the operators (G∗n) with the
help of universal Korovkin-type theorem.
Theorem 2.5. Let (G∗n) be the operators given by (1.5) . Then, for any
f ∈ C [0, ∞) ∩ E, the following relation holds
limn→∞
G∗n (f ; x) = f (x)
uniformly on each compact subset of [0, ∞) , where
E :=
f : x ∈ [0, ∞) ,f (x)
1 + x2 is convergent as x → ∞
.
Proof. Using universal Korovkin-type theorem [5], it is sufficient to prove that the oper-ators (G∗
n) verify the conditions
limn→∞
G∗n (ei; x) = xi, i = 0, 1, 2,
uniformly on each compact subset of [0, ∞). From (2.2) − (2.4) , we obtain that thementioned conditions are supplied. So the proof is completed.
518 D. Söylemez, F. Taşdelen
3. Rate of convergenceIn this section, we compute rate of convergence of the operators (G∗
n) by means of themodulus of continuity.
Let f ∈ C [0, ∞) such that f is uniformly continuous for any δ > 0. The usual modulusof continuity for f is defined as
ω (f, δ) = sup|x−y|≤δ
x,y∈[0,∞)
|f (x) − f (y)| .
For f ∈ C [0, ∞) and any x, y ∈ [0, ∞) , we have
|f (x) − f (y)| ≤ ω (f ; |y − x|) ,
and for any δ > 0
|f (x) − f (y)| ≤ ω (f, δ)( |x − y|
δ+ 1
). (3.1)
Now, we give the following result.
Theorem 3.1. Suppose that n ∈ N, (bn) and (βn) are sequences of positive numberssuch that limn→∞ bn = ∞, limn→∞ bn/n = 0 and limn→∞nβn = 0. Then for each f ∈C [0, ∞) ∩ E, we have
|G∗n (f ; x) − f (x)| ≤ 2ω (f ; δn (x)) ,
where δn (x) = x2 (2nβn) + 2xbnβn (1 + nβn) + x bnn (nβn)2 (1 + 2nβn) + x bn
n .
Proof. Using the linearity and positivity of G∗n (f ; x) and then applying (3.1) , we have
|G∗n (f ; x) − f (x)| ≤ G∗
n (|f (t) − f (x)| ; x)
≤ (1 + nβ)1−nn∑
k=0ω (f, δ)
(1 + |t − x|
δ
)(n
k
)x
bn
(x
bn+ kβ
)k−1
×(
1 − x
bn
)[1 − x
bn+ (n − k) β
]n−k−1
= ω (f, δ) (1 + nβ)1−nn∑
k=0
(1 + |t − x|
δ
)(n
k
)x
bn
(x
bn+ kβ
)k−1
×(
1 − x
bn
)[1 − x
bn+ (n − k) β
]n−k−1,
from (2.2) , one has
|G∗n (f ; x) − f (x)| ≤ ω (f, δ)
1 + 1
δ(1 + nβn)1−n
n∑k=0
|t − x|(
n
k
)x
bn
(x
bn+ kβ
)k−1
×(
1 − x
bn
)[1 − x
bn+ (n − k) β
]n−k−1
.
By the Cauchy-Schwarz inequality, we reach to
|G∗n (f ; x) − f (x)| ≤ ω (f, δ)
1 + 1
δ
G∗
n
((t − x)2 ; x
) 12
.
On the other hand, using (2.2) , (2.3) and (2.4), we get
G∗n (φ; x) = G∗
n (e2; x) − 2xG∗n (e1; x) + x2G∗
n (e0; x)
≤ x2 (2nβn) + 2xbnβn (1 + nβn) + xbn
n(nβn)2 (1 + 2nβn) + x
bn
n.
Approximation by Cheney-Sharma Chlodovsky operators 519
Let choose δ = δn (x) := x2 (2nβn)+2xbnβn (1 + nβn)+x bnn (nβn)2 (1 + 2nβn)+x bn
n , thenwe have
|G∗n (f ; x) − f (x)| ≤ 2ω
(f,√
δn (x))
.
This concludes the proof.
4. A-Statistical approximation by (G∗n)
In this section, we obtain A-statistical convergence of the Cheney-Sharma-Chlodovskyoperators to identity operator on the weighted spaces. Let us recall the weighted Korovkintype approximation theorem for the A-statistical convergence was given by Duman andOrhan in [17].
Theorem 4.1. Let A = (ank) be a nonnegative regular summability matrix and let ρ1,ρ2weight functions such that ρ1
lim|x|→∞
ρ1 (x)ρ2 (x)
= 0. (4.1)
Assume that (Tn)n≥1 is a sequence of positive linear operators from Cρ1 (R) into Bρ2 (R) .One has
stA − limn
∥Tnf − f∥ρ2= 0,
for all f ∈ Cρ1 (R) if and only if
stA − limn
∥TnFv − Fv∥ρ1= 0, k = 0, 1, 2,
whereFv (x) = xvρ1 (x)
1 + x2 , v = 0, 1, 2.
Corollary 4.2. [17]. Let A = (ank) be a nonnegative regular summability matrix and let(Ln) be a sequence of positive linear operators acting from Cρ0
(R+) into Bρλ
(R+) , λ > 0
one hasstA − lim
n∥Lnf − f∥ρλ
= 0, f ∈ Cρ0
(R+)
,
if and only ifstA − lim
n∥Lnei − ei∥ρ0
= 0, i = 0, 1, 2, (4.2)
where ρ0 (x) = 1 + x2 and ρλ (x) = 1 + x2+λ, λ > 0.
Using this theorem the following Korovkin type statistical theorem can be proved for(G∗
n) :
Theorem 4.3. Let A = (ank) be a nonnegative regular summability matrix and let (bn) and(βn) be sequences of positive numbers such that stA − lim
n→∞bn/n = 0 , stA − lim
n→∞nβn = 0.
Then for each f ∈ Ckρ
(R+) , we have
stA − limn→∞
∥G∗nf − f∥ρλ
= 0,
where ρ0 (x) = 1 + x2 and ρλ (x) = 1 + x2+λ, λ > 0.
Proof. Using Corollary 4.2, it is sufficient to prove that the operators (G∗n) verify the
conditions given in (4.1) . Indeed, from (2.2) and (2.3) , it is clear that
stA − limn
∥G∗n (e0; .) − e0∥ρ0
= 0
andstA − lim
n∥G∗
n (e1; .) − e1∥ρ0= 0.
520 D. Söylemez, F. Taşdelen
Now using (2.4) , one can have∥∥∥G∗n (e2) − x2
∥∥∥ρ0
≤ sup∣∣∣∣∣ x2
1 + x2 (2nβn)∣∣∣∣∣
+∣∣∣∣ 2x
1 + x2 bnβn (1 + nβn)∣∣∣∣
+ x
1 + x2bn
n(nβn)2 (1 + 2nβn) + x
1 + x2bn
n
,
which implies∥∥∥G∗n (e2) − x2
∥∥∥ρ0
≤ 2nβn + 2bnβn (1 + nβn) + bn
n(nβn)2 (1 + 2nβn) + bn
n= Kn.
Now, for a given ϵ > 0, let us define the following sets:
M :=
k : ∥G∗n (e2; .) − e2∥ρ0
≥ ϵ
,
M1 :=
k : 2nβn ≥ ϵ
5
,
M2 :=
k : 2bnβn (1 + nβn) ≥ ϵ
5
,
M3 := 1
n(nβ)2 (1 + 2nβ) ≥ ϵ
5
,
M4 :=
bn
n(nβn)2 (1 + 2nβn) ≥ ϵ
5
,
M5 :=
bn
n≥ ϵ
5
.
Then, we see that M ⊆ M1 ∪ M2 ∪ M3 ∪ M4 ∪ M5. Therefore, we get∑n:∥G∗
n(e2;.)−e2∥ρ0≥ϵ
ak,n ≤∑
n∈M1
ak,n +∑
n∈M2
ak,n +∑
n∈M3
ak,n +∑
n∈M4
ak,n +∑
n∈M5
ak,n (4.3)
and, taking the limit k → ∞ in (4.3) , we havestA − lim
n∥G∗
n (e2; .) − e2∥ρ0= 0.
This proves the theorem.
Remark 4.4. In Theorem 4.3, we have stronger results than approximation theoremsgiven in section 2. Indeed, Theorem 4.3 may be useful when nβn and bn/n does notconvergent to zero as n → ∞.
The following example shows that there exist some sequences (βn) and (bn) such thatA-satistical convergence holds but ordinary convergence does not hold for nβn and bn/n.
Example 4.5. Let (bn) and (βn) be the sequences defined by
bn =
en2, if n is a perfect square√
n, otherwise.
andβn =
n, if n is a perfect square1
n2 , otherwise.It is easy to see that bn/n and nβn are not convergent but statistically convergent, i.e.,C1-statistically convergent.
Approximation by Cheney-Sharma Chlodovsky operators 521
References[1] A.M. Acu, C.V.Muraru, D.F. Sofonea and A.V. Radu, Some approximation properties
of a Durrmeyer variant of q-Bernstein–Schurer operators, Math. Methods Appl. Sci.39 (18), 5636–5650, 2016.
[2] O. Agratini and O. Doğru, Weighted Approximation by q−szász King type operators,Taiwanese J. Math. 14 (4), 1283–1296, 2010.
[3] O. Agratini and I.A. Rus, Iterates of a class of dicsrete linear operators via contractionprinciple, Comment. Math. Univ. Carolin. 44 (3), 555–563, 2003.
[4] P.N. Agrawal, V. Gupta, A.S. Kumar and A. Kajla. Generalized Baskakov-Szász typeoperators, Appl. Math. Comput. 236, 311–324, 2014.
[5] F. Altomare and M. Campiti, Korovkin-type approximaton theory and its applications,Walter de Gruyter, Berlin-New York, 1994.
[6] G. Başcanbaz-Tunca and F. Taşdelen, On Chlodovsky form of the Meyer-König andZeller operators, An. Univ. Vest Timiş. Ser. Mat.-Inform. 49 (2), 137–144, 2011.
[7] G. Başcanbaz-Tunca, A. Erençin and F. Taşdelen, Some properties of Bernstein typeCheney and Sharma operators, Gen. Math. 24 (1-2), 17–25, 2016.
[8] G. Başcanbaz-Tunca, A. Erençin and H. G. İnce-İlarslan, Bivariate Cheney andSharma Operators on Simplex, Hacet. J. Math. Stat. 47 (4), 793–804, 2018.
[9] J. Boos, Classical and Modern Methods in Summability, Oxford University Press,Oxford, 2000.
[10] R.C. Buck, The measure theoretic approach to density, Amer. J. Math. 68, 560-580,1946.
[11] R.C. Buck, Generalized asymptotic density, Amer. J. Math. 75, 335–346, 1953.[12] I. Büyükyazıcı, Approximation by Stancu-Chlodowsky polynomials, Comput. Math.
Appl. 59, 274–282, 2010.[13] E.W. Cheney and A. Sharma, On a generalization of Bernstein polynomials, Riv.
Mat. Univ. Parma 2 (5), 77–84, 1964.[14] I. Chlodovsky, Sur le development des fonctions dUfines dans un interval infinien
series de polyn omes de S.N. Bernstein, Compositio Math. 4, 380–392, 1937.[15] M. Craciun, Approximation operators constructed by means of Sheffer sequences, Rev.
Anal. Numér. Théor. Approx. 30 (2), 135–150, 2001.[16] O. Duman, M.K. Khan, and C. Orhan, A-statistical convergence of approximating
operators, Math. Inequal. Appl. 6 (4), 689–699, 2003.[17] O. Duman and C. Orhan, Statistical approximation by positive linear operators, Studia
Math. 161 (2), 187–197, 2004.[18] H. Fast, Sur la convergence statistique, Colloq. Math. 2, 241–244, 1951.[19] J.A. Fridy, On statistical convergence, Analysis, 5 (4), 301–313, 1985.[20] J.A. Fridy and H.I. Miller, A matrix characterization of statistical convergence, Anal-
ysis 11, 59–66, 1991.[21] A.D. Gadjiev, The convergence problem for a sequence of positive linear operators on
unbounded sets and theorems analogues to that of P.P. Korovkin, Dokl. Akad. NaukSSSR 218, 1974.
[22] A.D. Gadjiev, Theorems of the type of P. P. Korovkin’s theorems (in Russian), Math.Z. 205, 781–786, 1976. Translated in Maths Notes, 20 (5-6), 995–998, 1977.
[23] A.D. Gadjiev and C. Orhan, Some approximation theorems via statistical convergence,Rocky Mountain J. Math. 32, 129–138, 2002.
[24] A.D. Gadjiev, R.O. Efendiev and E. Ibikli, Generalized Bernstein Chlodowsky poly-nomials, Rocky Mountain J. Math. 28 (4), 1267–1277, 1998.
[25] A. Izgi, Rate of approximation new-type generalization of Chlodovsky polynomial, Gen.Math. 21 (2), 125–140, 2013.
522 D. Söylemez, F. Taşdelen
[26] H. Karsli and E. Ibikli, Convergence rate of a new Bezier variant of Chlodowskyoperators to bounded variation functions, J. Comput. Appl. Math. 212 (2), 431–443,2008.
[27] H. Karsli and V. Gupta, Some approximation properties of q−Chlodowsky operators,Appl. Math. Comput. 195 (1), 220–229, 2008.
[28] A.S. Kumar and T. Acar, Approximation by generalized Baskakov–Durrmeyer–Stancutype operators, Rend. Circ. Mat. Palermo, 65 (3), 411–424, 2016.
[29] M. Mursaleen, K.J. Ansari and A. Khan, Some approximation results by(p, q) −analogue of Bernstein–Stancu operators, Appl. Math. Comput. 264, 392–402,2015.
[30] M. Mursaleen, A.H. Al-Abieda and A.M. Acu, Approximation by Chlodowsky type ofSzász operators based on Boas-Buck type polynomials, Turk. J. Math 42, 2243–2259,2018.
[31] M.A. Özarslan, O. Duman, and H.M. Srivastava, Statistical approximation resultsfor Kantorovich-type operators involving some special polynomials, Math. Comput.Model. 48 (3-4), 388–401, 2008.
[32] D. Söylemez and M. Unver, Korovkin Type Theorems for Cheney-Sharma Operatorsvia Summability Methods, Results Math. 72, 1601–1612, 2017.
[33] M.W. Müller, Approximation by Cheney-Sharma-Kantorovic Polynomials in the Lp-Metric, Rocky Mountain J. Math. 19 (1), 281–291, 1989.
[34] M. Unver, Abel transform of positive linear operators, in: ICNAAM 2013, AIP Con-ference Proceedings 1558, 1148–1151, 2013.
[35] D.D. Stancu and C. Cismaşiu, On an approximating linear positive operator ofCheney-Sharma, Rev. Anal. Numèr. Thèor. Approx. 26 (1-2), 221–227, 1997.
[36] D.D. Stancu and E.I. Stoica, On the use Abel-Jensen type combinatorial formulas forconstruction and investigation of some algebraic polynomial operators of approxima-tion, Stud. Univ. Babeş-Bolyai Math. 54 (4), 167–182, 2009.
Hacettepe Journal ofMathematics & Statistics
Hacet. J. Math. Stat.Volume 49 (2) (2020), 523 – 538
DOI : 10.15672/hujms.568332
Research Article
Well-posedness and exponential stability of athermoelastic-Bresse system with second sound
and delay
Gang Li, Yue Luan, Wenjun Liu∗
College of Mathematics and Statistics, Nanjing University of Information Science and Technology,Nanjing 210044, China
AbstractIn this paper, we consider a one-dimensional thermoelastic-Bresse system with a delayterm, where the heat conduction is given by Cattaneo’s law effective in the shear angledisplacement. We prove that the system is well-posed by using the semigroup method,and show, using the multiplier method, that the dissipation induced by the heat is strongenough to exponentially stabilize the system in the presence of a “small" delay when thestable number is zero.
Mathematics Subject Classification (2010). 35L53, 35L05, 93C20, 93D20
Keywords. thermoelastic-Bresse system, second sound, exponential decay, time delay
1. IntroductionIn this paper, we consider the following thermoelastic-Bresse system with a constant
internal delay:
ρ1φtt − k(ψ + φx + lω)x − lk0(ωx − lφ) + µφt(x, t− τ0) = 0, (x, t) ∈ (0, 1) × (0,+∞),ρ2ψtt − bψxx + k(φx + ψ + lω) + γθx = 0, (x, t) ∈ (0, 1) × (0,+∞),ρ1ωtt − k0(ωx − lφ)x + lk(φx + ψ + lω) = 0, (x, t) ∈ (0, 1) × (0,+∞),ρ3θt + qx + γψtx = 0, (x, t) ∈ (0, 1) × (0,+∞),τqt + βq + θx = 0, (x, t) ∈ (0, 1) × (0,+∞),φ(x, 0) = φ0(x), φt(x, 0) = φ1(x), θ(x, 0) = θ0(x), x ∈ [0, 1],ψ(x, 0) = ψ0(x), ψt(x, 0) = ψ1(x), q(x, 0) = q0(x), x ∈ [0, 1],ω(x, 0) = ω0(x), ωt(x, 0) = ω1(x), φt(x,−t) = f0(x, t), x ∈ [0, 1], t ∈ (0, τ),φ(0, t) = φx(1, t) = ψx(0, t) = ψ(1, t) = 0, t ∈ [0,+∞),ωx(0, t) = ω(1, t) = θ(0, t) = q(1, t) = 0, t ∈ [0,+∞).
(1.1)
∗Corresponding Author.Email addresses: [email protected] (G. Li), [email protected] (Y. Luan), [email protected]
(W. Liu)Received: 23.09.2017; Accepted: 11.01.2019
524 G. Li, Y. Luan, W. Liu
This is a thermoelastic system of Bresse type ([3, 12, 13]) which governs the mechanicaldeformations in elastic structures of circular arch type, where the heat flux is given byCattaneo’s law. It is composed of five functions, three of which representing the mechanicaldeformations: the longitudinal displacement ω, the vertical displacement φ and the shearangle displacement ψ; θ is the difference temperature, q is the heat flux ([15,20,21]). Thecoefficients ρi(i = 1, 2, 3), k, l, k0, b, k, γ, τ, β are positive constants, µ is a real number, andτ0 > 0 represents the time delay.
With respect to asymptotic behavior of solutions for thermoelastic Bresse systems, someresults can be obtained. Fatori and Rivera [8] considered Bresse system with thermaldissipation effective only in one equation wrote as
ρ1φtt − k(ψ + φx + lω)x − lk0(ωx − lφ) = 0, (x, t) ∈ (0, 1) × (0,+∞),ρ2ψtt − bψxx + k(φx + ψ + lω) + γθx = 0, (x, t) ∈ (0, 1) × (0,+∞),ρ1ωtt − k0(ωx − lφ)x + lk(φx + ψ + lω) = 0, (x, t) ∈ (0, 1) × (0,+∞),θt − k1θxx +mψxt = 0, (x, t) ∈ (0, 1) × (0,+∞),
and showed that there exist exponential stability if and only if the wave propagation isequal. They also showed that, in general, the system is not exponentially stable but thatthere exists polynomial stability with rates that depend on the wave propagations andthe regularity of the initial data. In [10], Keddi et al. studied the well-posedness andthe asymptotic stability of a one-dimensional thermoelastic Bresse system, where the heatconduction is given by Cattaneo’s law effective in the shear angle displacement, wrote as
ρ1φtt − k(ψ + φx + lω)x − lk0(ωx − lφ) = 0, (x, t) ∈ (0, 1) × (0,+∞),ρ2ψtt − bψxx + k(φx + ψ + lω) + γθx = 0, (x, t) ∈ (0, 1) × (0,+∞),ρ1ωtt − k0(ωx − lφ)x + lk(φx + ψ + lω) = 0, (x, t) ∈ (0, 1) × (0,+∞),ρ3θt + qx + γψtx = 0, (x, t) ∈ (0, 1) × (0,+∞),τqt + βq + θx = 0, (x, t) ∈ (0, 1) × (0,+∞).
They established the well-posedness of the system and proved that the system was expo-nentially stable depending on the stable number of the system, and showed that in general,the system was polynomially stable. If l ≡ 0, Bresse system reduces to the well-knownTimoshenko system (see [1, 5–7,14,22]).
Time delays so often arise in many physical, chemical, biological, thermal and econom-ical phenomena (see [4,9,16–19,23–25,27,29–34]). The presence of delay may be a sourceof instability. In recent years, the control of partial differential equations with time delayeffects has become an active area of research. For example, Kafini et al [9] studied theTimoshenko system of thermoelasticity of type III with delay of the form
ρ1ϕtt − k(ϕx + ψ)x + µ1ϕt(x, t) + µ2ϕt(x, t− τ) = 0, (x, t) ∈ (0, 1) × (0,+∞),ρ2ψtt − bψxx + k(ϕx + ψ) + βθtx = 0, (x, t) ∈ (0, 1) × (0,+∞),ρ3θtt − δθxx + γψtx −Kθtxx = 0, (x, t) ∈ (0, 1),×(0,+∞),θ(·, 0) = θ0, θt(·, 0) = θ1, ψ(·, 0) = ψ0, x ∈ [0, 1],ψt(·, 0) = ψ1, ϕ(·, 0) = ϕ0, ϕt(·, 0) = ϕ1, x ∈ [0, 1],ϕt(x, t− τ) = f0(x, t− τ), t ∈ [0, τ ],ϕ(0, t) = ϕ(1, t) = ψ(0, t) = ψ(1, t) = θx(0, t) = θx(1, t) = 0. t ∈ [0,+∞),
and proved that under suitable conditions on the initial data the energy decays expo-nentially in the case of equal wave speeds in spite of the existence of the delay. Andthey also got the result that the energy decays polynomially under different wave speeds
Well-posedness and exponential stability 525
assumption. In [2], Apalara and Messaoudi considered the following one-dimensional lin-ear thermoelastic system of Timoshenko type with delay, where the heat flux is given byCattaneo’s law:
ρ1φtt − k(ψ + φx)x + µφt(x, t− τ0) = 0, (x, t) ∈ (0, 1) × (0,+∞),ρ2ψtt − bψxx + k(φx + ψ) + γθx = 0, (x, t) ∈ (0, 1) × (0,+∞),ρ3θt + qx + γψtx = 0, (x, t) ∈ (0, 1) × (0,+∞),τqt + βq + θx = 0, (x, t) ∈ (0, 1) × (0,+∞),φ(x, 0) = φ0(x), φt(x, 0) = φ1(x), ψ(x, 0) = ψ0(x), ψt(x, 0) = ψ1(x), x ∈ [0, 1],θ(x, 0) = θ0(x), q(x, 0) = q0(x), φt(x,−t) = f0(x, t), x ∈ [0, 1],φ(0, t) = φ(1, t) = ψx(0, t) = ψx(1, t) = θ(0, t) = θ(1, t) = 0, t ∈ (0,+∞).
They proved an exponential decay result under a smallness condition on the delay anda stability number, and reproduced the polynomial decay of Santos et al. [28] using themultiplier method in the case of absence of delay.
Based on the above results, in this paper, we study the thermoclastic-Bresse system(1.1) with second sound and delay. Introducing a delay term in the internal feedback ofthermoclastic-Bresse system with second sound makes our problem different from thoseconsidered so far in the literature (such as [10]). For our purpose, we use the idea ofApalara and Messaoudi in [2] to take into account the effect of the delay. We first usethe semigroup method to prove the well-posedness result of the system. Then, we show,using the multiplier method, that the dissipation induced by the heat is strong enough tostabilize the system in the presence of a “small" delay when the stable number is zero.
The remaining part of this paper is organized as follows. In Section 2, we establish thewell-posedness result of the system. In Section 3, we give the exponential decay result bymodifying some classical multipliers.
2. Well-posednessIn this section, we use the semigroup techniques to prove the well-posedness of problem
(1.1). In order to exhibit the dissipative nature of system (1.1), as in [23], we introducethe new variable
z(x, ρ, t) = φt(x, t− ρτ0) x ∈ (0, 1), ρ ∈ (0, 1), t > 0.
A simple differentiation shows that the variable satisfies
τ0zt(x, ρ, t) + zρ(x, ρ, t) = 0 x ∈ (0, 1), ρ ∈ (0, 1), t > 0.
526 G. Li, Y. Luan, W. Liu
Hence, problem (1.1) is equivalent to the following:
ρ1φtt − k(ψ + φx + lω)x − lk0(ωx − lφ) + µz(x, 1, t) = 0, (x, t) ∈ (0, 1) × (0,+∞),ρ2ψtt − bψxx + k(φx + ψ + lω) + γθx = 0, (x, t) ∈ (0, 1) × (0,+∞),ρ1ωtt − k0(ωx − lφ)x + lk(φx + ψ + lω) = 0, (x, t) ∈ (0, 1) × (0,+∞),ρ3θt + qx + γψtx = 0, (x, t) ∈ (0, 1) × (0,+∞),τqt + βq + θx = 0, (x, t) ∈ (0, 1) × (0,+∞),τ0zt(x, ρ, t) + zρ(x, ρ, t) = 0, (x, t) ∈ (0, 1) × (0,+∞),φ(x, 0) = φ0(x), φt(x,−t) = f0(x, t), θ(x, 0) = θ0(x), x ∈ [0, 1],ψ(x, 0) = ψ0(x), ψt(x, 0) = ψ1(x), q(x, 0) = q0(x), x ∈ [0, 1], t ∈ (0, τ),ω(x, 0) = ω0(x), ωt(x, 0) = ω1(x), z(x, 0, t) = φt(x, t), x ∈ [0, 1], t ∈ (0,+∞),φ(0, t) = φx(1, t) = ψx(0, t) = ψ(1, t) = 0, t ∈ [0,+∞),ωx(0, t) = ω(1, t) = θ(0, t) = q(1, t) = 0, t ∈ [0,+∞).
(2.1)Now, we let
Φ = (φ, u, ψ, v, ω, w, θ, q, z),
then system (2.1) can be written as an evolutionary equation:
Φ′(t) + (A + B) Φ(t) = 0, t > 0,Φ(0) = Φ0 = (φ0, φ1, ψ0, ψ1, ω0, ω1, θ0, q0, z0)T ,
(2.2)
where A is a linear operator defined by
AΦ =
−u
− k
ρ1(φx + ψ + lω)x − k0l
ρ1(ωx − lφ) + |µ|
ρ1u+ µ
ρ1z(·, 1)
−v
− b
ρ2ψxx + b
ρ2(φx + ψ + lω) + γ
ρ2θx
−wk0ρ1
(ωx − lφ)x + kl
ρ1(φx + ψ + lω)
1ρ3qx + γ
ρ3vx
β
τq + 1
τθx
1τ0zρ
,
Well-posedness and exponential stability 527
and the operator B : D(B) = H → H is defined by
BΦ = |µ|ρ1
0−u00000
.
We give the following spaces:H1
∗ (0, 1) = f ∈ H1(0, 1) : f(0) = 0,H1
∗ (0, 1) = f ∈ H1(0, 1) : f(1) = 0,H2
∗ (0, 1) = H2(0, 1) ∩H1∗ (0, 1),
H2∗ (0, 1) = H2(0, 1) ∩ H1
∗ (0, 1),and the energy space:
H = H1∗ (0, 1) × L2(0, 1) × H1
∗ (0, 1) × L2(0, 1) × H1∗ (0, 1)
×L2(0, 1) × L2(0, 1) × L2(0, 1) × L2((0, 1) × (0, 1)),equipped with the inner product
(Φ, Φ)H =k∫ 1
0(φx + ψ + lω)
(φx + ψ + lω
)dx+ k0
∫ 1
0(ωx − lφ)(ωx − lφ) dx
+ ρ1
∫ 1
0uudx+ b
∫ 1
0ψxψxdx+ ρ2
∫ 1
0vvdx
+ ρ1
∫ 1
0ωωdx+ ρ3
∫ 1
0θθdx+ τ
∫ 1
0qqdx+ τ0|µ|
∫ 1
0
∫ 1
0zzdρdx.
H is a Hilbert space for l small enough. In this case, the above inner product is equivalentto the natural inner product defined on H. To this end, the operator A with its domain is
D(A) =
Ψ ∈ H | φ ∈ H2
∗ (0, 1);ψ, ω ∈ H2∗ (0, 1);u, θ ∈ H1
∗ (0, 1),v, w, q ∈ H1
∗ (0, 1); φx(1) = 0, ψx(0) = ωx(0) = 0,z, zρ ∈ L2((0, 1), L2(0, 1)), z(x, 0) = φ(x)
.In what follows, we have the well-posedness result of problem (2.2).
Theorem 2.1. Assume that Φ0 ∈ H, then problem (2.2) exists a unique solution U ∈C(R+,H). Moreover, if Φ0 ∈ D(A) then Φ ∈ C
(R+, D(A)
)∩ C1(R+,H).
Proof. It is easy to see thatD(A) is dense in H. For Φ = (φ, u, ψ, v, ω, w, θ, q, z)T ∈ D(A),a direct computation gives that
(AΦ,Φ)H = |µ|∫ 1
0u2dx+ β
∫ 1
0q2dx+ µ
∫ 1
0uz(·, 1)dx+ |µ|
∫ 1
0
∫ 1
0zzρdρdx. (2.3)
By using Young’s inequality, the third term in the right hand side of (2.3) gives
−µ∫uz(·, 1)dx ≤ |µ|
2
∫ l
0z2(·, 1)dx+ |µ|
2
∫ 1
0u2dx,
which implies that
µ
∫ 1
0uz(0, 1)dx ≥ −|µ|
2
∫ l
0z2(·, 1)dx− |µ|
2
∫ 1
0u2dx.
528 G. Li, Y. Luan, W. Liu
Also, using integration by parts and the fact that z(x, 0) = u(x), the last term in theright-hand side of (2.3) gives∫ 1
0
∫ 1
0zzρdρdx = 1
2
∫ 1
0z2(·, 1)dx− 1
2
∫ 1
0u2dx.
Consequently, (2.3) yields
(AΦ,Φ)H ≥ β
∫ 1
0q2dx.
Hence A is monotone. Next, we will prove that the operator I + A is surjective.For all G = (g1, g2, g3, g4, g5, g6, g7, g8, g9)T ∈ H, we solve the equation
(I + A)Φ = G. (2.4)That is
−u+ φ = g1 ∈ H1∗ (0, 1),
−k(φx + ψ + lω)x − k0l(ωx − lφ) + (|µ| + ρ1)u+ µz(·, 1) = ρ1g2 ∈ L2(0, 1),−v + ψ = g3 ∈ H1
∗ (0, 1),−bψxx + k(φx + ψ + lω) + γθx + ρ2v = ρ2g4 ∈ L2(0, 1),−w + ω = g5 ∈ H1
∗ (0, 1),−k0(ωx − lφ)x + kl(φx + ψ + lω) + ρ1w = ρ1g6 ∈ L2(0, 1),qx + γvx + ρ3θ = ρ3g7 ∈ L2(0, 1),(β + τ)q + θx = τg8 ∈ L2(0, 1),zρ + τ0z = τ0g9 ∈ L2((0, 1) × (0, 1)).
(2.5)
From (2.5)8, we know that
θ = τ
∫ x
0g8(f)dy − (β + τ)
∫ x
0q(y)dy, (2.6)
which conclude θ(0, t) = 0. Inserting u = φ− g1, v = ψ− g3, w = ω− g5, the last equationin (2.5) together with the fact that z(x, 0) = u(x), one has
z(x, ρ) = φ(x)e−τ0ρ − e−τ0ρg1(x) + τ0e−τ0ρ
∫ ρ
0sτ0sg9(x, s)ds.
It can be easily shown that φ, ψ, ω and q satisfy
−k(φx + ψ + lω)x − k0l(ωx − lφ) + (|µ| + ρ1 + µe−τ0)φ = h1 ∈ L2(0, 1),−bψxx + k(φx + ψ + lω) + ρ2ψ − γ(β + τ)q = h2 ∈ L2(0, 1),−k0(ωx − lφ)x + kl(φx + ψ + lω) + ρ1w = h3 ∈ L2(0, 1),
−qx + ρ3(β + τ)∫ s
0q(y)dy − γψx = h4 ∈ L2(0, 1),
(2.7)
where
h1 = (ρ1 + |µ| + µ)g1 + ρ1g2 − µτ0e−τ0
∫ 1
0eτ0sg9(x, s)ds,
h2 = ρ2(g3 + g4) − τγg8,
h3 = ρ1(g5 + g6),
h4 = −γg3x − ρ3
(g7 − τ
∫ x
0g8(y)dy
).
The variational formulation corresponding to (2.7) takes the form
B((φ,ψ, ω, q), (φ, ψ, ω, q)
)= F (φ, ψ, ω, q), (2.8)
Well-posedness and exponential stability 529
where B : [H1∗ (0, 1) × H1
∗ (0, 1) × H1∗ (0, 1) × L2(0, 1)]2 → R is the bilinear form defined by
B((φ,ψ, ω, q),
(φ, ψ, ω, q
))=k
∫ 1
0(φx + ψ + lω)
(φx + ψ + lω
)dx+ (β + τ)
∫ 1
0qqdx+ b
∫ 1
0ψxψxdx
+ ρ2
∫ 1
0ψψdx− γ(β + τ)
∫ 1
0qψdx+ γ(β + τ)
∫ 1
0ψqdx+ ρ1
∫ 1
0φφdx
+ ρ1
∫ 1
0ωωdx+ k0
∫ 1
0(ωx − lφ) (ωx − lφ) dx
+ ρ3(β + τ)2∫ 1
0
(∫ x
0q(y)dy
∫ x
0q(y)dy
)dx,
and F : [H1∗ (0, 1) × H1
∗ (0, 1) × H1∗ (0, 1) × L2(0, 1)] → R is the linear functional given by
F (φ, ψ, ω, q) =∫ 1
0h1φdx+
∫ 1
0h2ψdx+
∫ 1
0h3ωdx+ (β + τ)
∫ 1
0h4
∫ x
0q(y)dydx.
Now, for V = H1∗ (0, 1) × H1
∗ (0, 1) × H1∗ (0, 1) × L2(0, 1) equipped with the norm
∥(φ,ψ, ω, q)∥V = ∥(φx + ψ + lω)∥22 + ∥ωx − lφ∥2
2 + ∥ψx∥22 + ∥q∥2
2,
and combining with∫ 1
0
(φ2
x + ψ2x + ω2
x
)dx ≤ c
∫ 1
0
((φx + ψ + lω)2 + (ωx − lφ)2 + ψ2
x
)dx,
for l small enough, it follows that B and F are bounded. Furthermore, using the definitionof B, we get
B((φ,ψ, ω, q), (φ,ψ, ω, q)) =k∫ 1
0(φx + ψ + lω)2dx+ (β + τ)
∫ 1
0q2dx+ b
∫ 1
0ψ2
xdx
+ ρ2
∫ 1
0ψ2dx+ ρ1
∫ 1
0φ2dx+ ρ1
∫ 1
0ω2dx
+ k0
∫ 1
0(ωx − lφ)2dx+ ρ3(β + τ)2
∫ 1
0
(∫ x
0q(y)dy
)2dx
≥c∥(φ,ψ, ω, q)∥2V .
Thus, B is coercive. Consequently, Lax-Milgram Lemma provides that system (2.7) has aunique solution φ ∈ H1
∗ (0, 1), ψ ∈ H1∗ (0, 1), ω ∈ H1
∗ (0, 1), q ∈ L2(0, 1). Substituting φ, ψ,ω, q into (2.5)1, (2.5)3, (2.5)5 and (2.5)8 respectively, we get u ∈ H1
∗ (0, 1), v ∈ H1∗ (0, 1),
w ∈ H1∗ (0, 1), θ ∈ H1
∗ (0, 1).If(ψ, ω, q
)≡ (0, 0, 0) ∈ H1
∗ (0, 1) × H1∗ (0, 1) × L2(0, 1), then (2.8) reduces to
k
∫ 1
0(φx + ψ + lω)φxdx− k0l
∫ 1
0(ωx − lφ)φdx+ ρ1
∫ 1
0φφdx =
∫ 1
0h1φdx, (2.9)
for all φ in H1∗ (0, 1), which implies
− kφxx = kψx + l(k + k0)ωx − (k0l2 + ρ1)φ+ h1 ∈ L2(0, 1). (2.10)
Consequently, by the regularity theory for the linear elliptic equations, we obtain
φ ∈ H2∗ (0, 1).
Moreover, (2.9) is also true for any ϕ ∈ C1([0, 1]), ϕ(0) = 0 which is in H1∗ (0, 1). Hence,
taking any ϕ ∈ C1([0, 1]), ϕ(0) = 0, one has
k
∫ 1
0φxϕxdx−
∫ 1
0
(kψx + l(k + k0)ωx −
(k0l
2 + ρ1)φ+ h1
)ϕdx = 0.
530 G. Li, Y. Luan, W. Liu
Thus, using integration by parts and taking into account (2.10), we get
φx(1)ϕ(1) = 0, ∀ϕ ∈ C1([0, 1]), ϕ(0) = 0.Therefore,
φx(1) = 0.Similarly, we get
−bψxx = −kφx − (k + ρ− 2)ψ − lkω − γ(β + τ)q + h2 ∈ L2(0, 1),
−kωxx = −l(k + k0)φx − lkψ +(ρ1 + l2k0
)ω + h3 ∈ L2(0, 1),
−qx = γψx − (β + τ)ρ3
∫ x
0q(y)dy + h4 ∈ L2(0, 1).
Thus, we haveψ, ω ∈ H2
∗ (0, 1), q ∈ H1∗ (0, 1), ωx(0) = ψx(0) = 0.
Hence, there exists a unique Φ ∈ D(A) such that (2.4) is satisfied, which conclude thatthe operator A is maximal. With this, it is easy to obtain that A is a maximal monotoneoperator. On the other hand, it is obvious that operator B is Lipschitz continuous. Con-sequently, A + B is the infinitesimal generator of a linear contraction C0-semigroup on H.This completes the proof (see [26] and [11]).
3. Exponential stabilityIn this section, we state and prove our stability result for the solution of system (2.1)
by using the multiplier technique. We first introduce the following energy functional:
E(t) =12
∫ 1
0
[ρ1φ
2t + ρ2ψ
2t + ρ1ω
2t + bψ2
x + ρ3θ2 + τq2 + k(φx + ψ + lω)2
]dx
+ 12
∫ 1
0
[k0(ωx − lφ)2 + |µ|τ0
∫ 1
0z2(x, ρ, t)dρ
]dx. (3.1)
Our main result of this section reads as follows.
Theorem 3.1. Let (φ,ψ, ω, θ, q, z) be the solution of (2.1), assume that k = k0 and
ξ :=(τ − ρ1
kρ3
)(ρ2b
− ρ1k
)− τρ1γ
2
bkρ3= 0. (3.2)
Then for |µ| small enough, the energy functional (3.1) satisfies
E(t) ≤ k1e−k2t, ∀t ≥ 0, (3.3)
where k1, k2 are two positive constants.
We need the following lemmas to show that the associated energy non-increase in time.
Lemma 3.2. Let (φ,ψ, ω, θ, q, z) be the solution of (2.1), the energy functional defined by(3.1) satisfies
E′(t) = −β∫ 1
0q2dx+ |µ|
∫ 1
0φ2
tdx.
Proof. (2.1)1, (2.1)2, (2.1)3, (2.1)4 and (2.1)5, by multiplying φt, ψt, ωt, θ and q respec-tively, then integrating over (0, 1) and summing up, using the boundary conditions, weget
12d
dt
ρ1
∫ 1
0φ2
tdx+ ρ2
∫ 1
0ψ2
t dx+ ρ1
∫ 1
0ω2
t dx+ b
∫ 1
0ψ2
xdx+ ρ3
∫ 1
0θ2dx+ τ
∫ 1
0q2dx
+ 1
2d
dt
∫ 1
0k0(ωx − lφ)2dx+ k
∫ 1
0(φx + ψ + lω)2dx+ |µ|τ0
∫ 1
0
∫ 1
0z2(x, ρ, t)dρdx
Well-posedness and exponential stability 531
= − β
∫ 1
0q2dx+ |µ|
∫ 1
0φ2
tdx. (3.4)
Now, multiplying (2.1)6 by |µ|z and integrating over (0, 1) × (0, 1), bearing in mindz(x, 0, t) = φt(x, t), we obtain
|µ|τ02
d
dt
∫ 1
0
∫ 1
0z2(x, ρ, t)dρdx = |µ|
2
∫ 1
0φ2
tdx− |µ|2
∫ 1
0z2(x, 1, t)dx. (3.5)
The result follows by the combination of (3.4)-(3.5) and Young’s inequality. Lemma 3.3. Let (φ,ψ, ω, θ, q, z) be the solution of (2.1). The functional
F1(t) = −ρ1
∫ 1
0(φφt + ωωt)dx,
satisfies
F ′1(t) ≤ − ρ1
∫ 1
0φ2
tdx− ρ1
∫ 1
0ω2
t dx+ C
∫ 1
0ψ2
xdx+ k0
∫ 1
0(ωx − lφ)2dx
+ C(ε1)∫ 1
0(φx + ψ + lω)2dx+ ε1
∫ 1
0z2(x, 1, t)dx, (3.6)
for all ε1 > 0.
Proof. By differentiating F1 and using (2.1)1 and (2.1)3, we conclude thatF ′
1(t)
= − ρ1
∫ 1
0φ2
tdx− ρ1
∫ 1
0ω2
t dx−∫ 1
0φ(k(φx + ψ + lω)x + lk0(ωx − lφ) − µz(x, 1, t))dx
− ρ
∫ 1
0ω(k0(ωx − lφ)x − lk(φx + ψ + lω))dx
= − ρ1
∫ 1
0φ2
tdx− ρ1
∫ 1
0ω2
t dx+ k
∫ 1
0(φx + ψ + lω)2dx− k
∫ 1
0ψ(φx + ψ + lω)dx
+ k0
∫ 1
0(ωx − lφ)2dx− µ
∫ 1
0φz(x, 1, t)dx.
Using Young’s and Poincaré inequalities, (3.6) is established. Lemma 3.4. Let (φ,ψ, ω, θ, q, z) be the solution of (2.1). The functional
F2(t) = ρ2
∫ 1
0ψψtdx
satisfies the estimate
F ′2(t) ≤ − b
2
∫ 1
0ψ2
xdx+ ρ2
∫ 1
0ψ2
t dx+ k2
b
∫ 1
0(φx + ψ − lω)2dx+ C
∫ 1
0θ2dx. (3.7)
Proof. Taking the derivative of F2 with respect to t and using (2.1)2, it follows that
F ′2(t) = ρ2
∫ 1
0ψ2
xdx−∫ 1
0ψ(bψxx − k(φx − ψ + lω) − γθx)dx.
Using Young’s and Poincaré inequalities, we obtain (3.7). Lemma 3.5. Let (φ,ψ, ω, θ, q, z) be the solution of (2.1). The functional
F3(t) = −ρ2ρ3γ
∫ 1
0θ
∫ x
0ψt(y)dydx,
satisfies
F ′3(t) ≤ − ρ2
2
∫ 1
0ψ2
t dx+ ε3
∫ 1
0(φx + ψ + lω)2dx+ ε3
∫ 1
0ψ2
xdx+ C(ε3)∫ 1
0θ2dx
532 G. Li, Y. Luan, W. Liu
+ C
∫ 1
0q2dx. (3.8)
for all ε3 > 0.
Proof. By differentiating F3 and using (2.1)2 and (2.1)4, we get
F ′3(t) =ρ2
γ
∫ 1
0(qx + γψtx)
∫ x
0ψt(y)dydx− ρ3
γ
∫ 1
0θ
∫ x
0(bψxx − k(φx + ψ + lω))dydx
= − ρ2
∫ 1
0ψ2
t dx− ρ2γ
∫ 1
0qψtdx+ ρ3
∫ 1
0θ2dx− bρ3
γ
∫ 1
0θψxdx+ bρ3
γ
∫ 1
0θψxdx
+ kρ3γ
∫ 1
0(φx + ψ + lω)
∫ x
0θ(y)dydx.
The result thanks to Young’s inequality. Lemma 3.6. Let (φ,ψ, ω, θ, q, z) be the solution of (2.1). The functional
F4(t) = τρ3
∫ 1
0θ
∫ x
0q(y)dydx,
satisfies
F ′4(t) ≤ −ρ3
2
∫ 1
0θ2dx+ ε4
∫ 1
0ψ2
t dx+ C(ε4)∫ 1
0q2dx, (3.9)
for all ε4 > 0.
Proof. Differentiating F4 with respect to t, using (2.1)4 and (2.1)5, one has
F ′4(t) =τ
∫ 1
0(−qx − γψtx)
∫ x
0q(y)dydx+ ρ3
∫ 1
0θ
∫ x
0(−βq − θx)dydx
= − ρ3
∫ 1
0θ2dx+ τ
∫ 1
0q2dx+ τγ
∫ 1
0qψtdx− βρ3
∫ 1
0θ
∫ x
0qydydx.
Then, we use Cauchy-Schwarz and Young’s inequalities with ε4 > 0 to obtain (3.9). Lemma 3.7. Let (φ,ψ, ω, θ, q, z) be the solution of (2.1). The functional
F5(t) = −ρ1
∫ 1
0φt(ωx − lφ)dx− ρ1
∫ 1
0ωt(φx + ψ + lω)dx,
satisfies
F ′5(t) ≤ − lk0
2
∫ 1
0(ωx − lφ)2dx− lρ1
2
∫ 1
0ω2
t dx+ lρ1
∫ 1
0φ2
tdx+ lk
∫ 1
0(φx + ψ + lω)2dx
+ ρ12l
∫ 1
0ψ2
t dx+ 2lk0
∫ 1
0µ2z2(x, 1, t)dx, (3.10)
for all ε5 > 0.
Proof. Differentiating F5 with respect to t, using (2.1)1 and (2.1)3, it follows that
F ′5(t) = −
∫ 1
0(k(φx + ψ + lω)x + lk0(ω − lφ) − µz(x, 1, t))(ωx − lφ)dx
−∫ 1
0(k0(ωx − lφ)x − lk(φx + ψ + lω))(φx + ψ + lω)dx
− ρ1
∫ 1
0φt(ωx − lφ)tdx− ρ1
∫ 1
0ωt(φx + ψ + lω)tdx
= − lk0
∫ 1
0(ωx − lφ)2dx+ lk
∫ 1
0(φx + ψ + lω)2dx− ρ1
∫ 1
0ω2
t dx+ lρ1
∫ 1
0φ2
tdx
− ρ1
∫ 1
0ψtωtdx+
∫ 1
0µz(x, 1, t)(ωx − lφ)dx.
Well-posedness and exponential stability 533
(3.10) follows Young’s inequality with the fact that k = k0. Lemma 3.8. Let (φ,ψ, ω, θ, q, z) be the solution of (2.1). The functional
F6(t) = τ0
∫ 1
0
∫ 1
0e−τ0ρz(x, ρ, t)dρdx
satisfies
F ′6(t) ≤ −m
(∫ 1
0z2(x, 1, t)dx+ τ0
∫ 1
0
∫ 1
0z2(x, ρ, t)dρdx
)+∫ 1
0φ2
tdx, (3.11)
where m = mine−τ0 , e−τ0ρ.Proof. Similarly computation, using (2.1)6, we have
F ′6(t) = d
dρ
∫ 1
0
∫ 1
0e−τ0ρz2(x, ρ, t)dρdx− τ0
∫ 1
0
∫ 1
0e−τ0ρz2(x, ρ, t)dρdx
−∫ 1
0
[e−τ0z2(x, 1, t) − z2(x, 0, t)
]dx− τ0
∫ 1
0
∫ 1
0e−τ0ρz2(x, ρ, t)dρdx.
It is obvious that result (3.11). Lemma 3.9. Let (φ,ψ, ω, θ, q, z) be the solution of (2.1). The functional
F7(t) = −ρ1
∫ 1
0(ωx − lφ)
∫ x
0ωt(y)dydx− ρ1
∫ 1
0φt
∫ x
0(φx + ψ + lω)(y)dydx
satisfies
F ′7(t) ≤ − ρ1
2
∫ 1
0φ2
tdx− k0
∫ 1
0(ωx − lφ)2dx+
(k + 1
2
)∫ 1
0(φx + ψ + lω)2dx
+ ρ1
∫ 1
0ω2
t dx+ ρ12
∫ 1
0ψ2
t dx+ 12µ2∫ 1
0z2(x, 1, t)dx. (3.12)
Proof. Differentiating F7 with respect to t, using (2.1)1 and (2.1)3, we get
F ′7(t) = − ρ1
∫ 1
0(ωx − lφ)t
∫ x
0ωt(y)dydx− ρ1
∫ 1
0φt
∫ 1
0(φx + ψ + lω)t(y)dydx
−∫ 1
0(ωx − lφ)
∫ x
0(k0(ωx − lφ)x − lk(φx + ψ + lω))dydx
−∫ 1
0(k(φx + ψ + lω)x + lk0(ωx − lφ) − µz(x, 1, t))
∫ x
0(φx + ψ + lω)(y)dydx
=ρ1
∫ 1
0ω2
t dx+∫ 1
0µz(x, 1, t)
∫ x
0(φx + ψ + lω)(y)dydx
− k0
∫ 1
0(ωx − lφ)2dx+ l(k − k0)
∫ 1
0(ωx − lφ)
∫ x
0(φx + ψ + lω)(y)dydx
+ k
∫ 1
0(φx + ψ + lω)2dx− ρ1
∫ 1
0φ2
tdx− ρ1
∫ 1
0φt
∫ x
0ψt(y)dydx.
The result follows Young’s and Cauchy-Schwarz inequalities with the fact that k = k0. Lemma 3.10. Let (φ,ψ, ω, θ, q, z) be the solution of (2.1). The functional
F8(t) =ρ2
∫ 1
0ψt(φx + ψ + lω)dx+ bρ1
k
∫ 1
0φtψxdx+ bρ3
γ
(ρ1k
− ρ2b
)∫ 1
0θφtdx
− b
γ
(ρ1k
− ρ2b
)∫ 1
0q(φx + ψ + lω)dx− bl2ρ2
k0
∫ 1
0ψtψdx+ blρ1
k0
∫ 1
0ωtψdx
satisfies
F ′8(t) ≤ − k
2
∫ 1
0(φx + ψ + lω)2dx+ 2b2l2
k
∫ 1
0ψ2
xdx+ C(ε)∫ 1
0ψ2
t dx+ ε8
∫ 1
0ω2
t dx
534 G. Li, Y. Luan, W. Liu
+ C(ε8)∫ 1
0q2dx+ C(ε8)
∫ 1
0θ2dx+ ε8
∫ 1
0(ωx − lφ)2dx+ C
∫ 1
0µz2(x, 1, t)dx,
(3.13)
for all ε8 > 0.
Proof. A differentiation of above functional gives
F ′8(t)
=∫ 1
0(bψxx − k(φx + ψ + lω) − γθx)(φx + ψ + lω)dx
+ ρ2
∫ 1
0ψt(φx + ψ + lω)dx+ b
k
∫ 1
0(k(φx + ψ + lω)x + lk0(ωx − lφ) − µz(x, 1, t))ψxdx
+ bρ1kγ
∫ 1
0φx(−ρ3θt − qx)dx+ b
γ
(ρ1k
− ρ2b
)∫ 1
0(−qx − γψtx)φtdx
+ bρ3γρ1
(ρ1k
− ρ2b
)∫ 1
0θ(k(φx + ψ + lω)x + lk0(ωx − lφ) − µz(x, 1, t))dx
− b
γτ
(ρ1k
− ρ2b
)∫ 1
0(−βq − θx)(φx + ψ + lω)dx− b
γ
(ρ1k
− ρ2b
)∫ 1
0q(φx + ψ + lω)tdx
− bl2
k0
∫ 1
0(bψxx − k(φx + ψ + lω) − γθx)ψdx− bl2ρ2
k0
∫ 1
0ψ2
t dx
+ bl
k0
∫ 1
0(k0(ωx − lφ)x − lk(φx + ψ + lω))ψdx+ blρ1
k0
∫ 1
0ωtψtdx
= − k
∫ 1
0(φx + ψ + lω)2dx+
(ρ2 − bl2ρ2
k0
)∫ 1
0ψ2
t dx+(lρ2 + blρ1
k0
)∫ 1
0ψtωtdx
+(
−γ − ρ3kb
γρ1
(ρ1k
− ρ2b
)+ b
γτ
(ρ1k
− ρ2b
))∫ 1
0θx(φx + ψ + lω)dx
− b
γ
(ρ1k
− ρ2b
)∫ 1
0qψtdx− bl
γ
(ρ1k
− ρ2b
)∫ 1
0qωtdx+ blk0ρ3
rρ1
(ρ1k
− ρ2b
)∫ 1
0θ(ωx − lφ)dx
− γbl2
k0
∫ 1
0θψxdx+ bβ
γτ
(ρ1k
− ρ2b
)∫ 1
0q(φx + ψ + lω)dx+ b2l2
k0
∫ 1
0ψ2
xdx
+ bl
(k0k
− 1)∫ 1
0ψx(ωx − lφ)dx− b
k
∫ 1
0µz(x, 1, t)ψxdx− bρ3
γρ1
(ρ1k
− ρ2b
)∫ 1
0θµz(x, 1, t)dx.
Noting that k = k0 and ξ = 0, the above equation turns into
F ′8(t)
= −k∫ 1
0(φx + ψ + lω)2dx+
(ρ2 − blρ2
k0
)∫ 1
0ψ2
t dx+(lρ2 + blρ1
k0
)∫ 1
0ψtωtdx
+ b
γ
(ρ1k
− ρ2b
)∫ 1
0qψtdx− bl
γ
(ρ1k
− ρ2b
)∫ 1
0qωtdx+ blk0ρ3
γρ1
(ρ1k
− ρ2b
)∫ 1
0θ(ωx − lφ)dx
− γbl2
k0
∫ 1
0θψxdx+ bβ
γτ
(ρ1k
− ρ2b
)∫ 1
0q(φx + ψ + lω)dx+ b2l2
k0
∫ 1
0ψ2
xdx
− b
k
∫ 1
0µz(x, 1, t)ψxdx− bρ3
γρ1
(ρ1k
− ρ2b
)∫ 1
0θµz(x, 1, t)dx.
Using Young’s inequality, we get (3.13).
Now, we are ready to prove an exponential decay result under a smallness condition onthe delay.
Well-posedness and exponential stability 535
Proof of Theorem 3.1We define a Lyapunov functional
L(t) := NE(t) +8∑
i=1NiFi(t), (3.14)
which it is equivalent to the energy functional E. Now, gathering the estimates in Lemmas3.3-3.10, we obtain
L′(t) ≤ −(N1ρ1 + ρ1
2N7 − ρ1lN5 −N6 − µN
)∫φ2
tdx
−(N1ρ1 + lρ1
2N5 − ρ1N7 − ε8N8
)∫ 1
0ω2
t dx
−(b
2N2 − CN1 − ε3N3 − 2b2l2
kN7
)∫ 1
0ψ2
xdx
−(ρ32N4 − CN2 − C(ε3)N3 − C(ε8)N8
)∫ 1
0θ2dx
−(ρ22N3 − ρ2N2 − ε4N4 − ρ1
2lN5 − ρ1
2N7 − C(ε8)N8
)∫ 1
0ψ2
t dx
−(lk02N5 − k0N1 + k0N7 − ε8N8
)∫ 1
0(ωx − lφ)2dx
− (Nβ − CN3 − C(ε4)N4 − C(ε8)N8)∫ 1
0q2dx
−(k
2N8 − C(ε1)N1 − k2
bN2 − ε3N3 − lkN5 −
(k + 1
2
)N7
)∫ 1
0(φx + ψ + lω)2dx
−(mN6 −N1ε1 − 2lk0µ
2N5 − µ2
2N7 − CµN8
)∫ 1
0z2(x, 1, t)dx
−N6mτ0
∫ 1
0
∫ 1
0z2(x, ρ, t)dρdx.
Then, we letN6 = 1, N1 = N7 = lN5 = 3
ρ1,
the choices yield
L′(t) ≤ −(1
2−Nµ
)∫ 1
0φ2
tdx−(3
2−N8ε8
)∫ 1
0ω2
t dx
−(b
2N2 −
(C + 2b2l2
k
)3ρ1
− ε3N3
)∫ 1
0ψ2
xdx
−(ρ22N3 − ρ2N2 − ε4N4 − 3 − C(ε8)N8
)∫ 1
0ψ2
t dx−(3k0
2ρ1− ε8N8
)∫ 1
0(ωx − lφ)2dx
−(ρ32N4 − CN2 − C(ε3)N3 − C(ε8)N8
)∫ 1
0θ2dx−mτ0
∫ 1
0z2(x, ρ, t)dρdx
−(k
2N8 −
(C(ε1) +
(k + 1
2
)+ k
) 3ρ1
− k2
bN2 − ε3N3
)∫ 1
0(φx + ψ + lω)2dx
− (Nβ − CN3 − C(ε4)N4 − C(ε8)N8)∫ 1
0q2dx
−(m−
(ε1 + 2k0µ
2 + 12µ2) 3ρ1
− CµN8
)∫ 1
0z2(x, 1, t)dx.
536 G. Li, Y. Luan, W. Liu
As follows, we need to choose our constants carefully. we let ε1 = ρ16m and choose N2 large,
such thatb
2N2 −
(C + 2b2l2
k
)3ρ1
≥ α1 > 0.
At this point, we take N8 large enough so thatk
2N8 −
(C(ε1) +
(k + 1
2
)+ k
) 3ρ1
− k2
bN2 ≥ α2 > 0,
and then select ε8 such that ε8 ≤ min
32N8
, 3k02ρ1N8
. We choose N3 large enough so that
ρ22N3 − ρ2N2 − ε4N4 − 3 − C(ε8)N8 ≥ α3 > 0,
and choose ε3 such that α1 − ε3N3 > 0 and α2 − ε3N3 > 0. We then choose N4 largeenough so that
ρ32N4 − CN2 − C(ε3)N3 − C(ε8)N8 > 0,
and choose ε4 such that α3 − ε4N4 > 0. We set N so large to satisfiesNβ − CN3 − C(ε4)N4 − C(ε8)N8 > 0.
Finally, by taking |µ| so small thatm
2−(
2k0µ2 + 1
2µ2) 3ρ1
− CµN8 > 0.
Utilizing the definition of E(t), we haveL′(t) ≤ −c1E(t).
On the other hand, exploiting (3.14), we get(N − c2)E(t) ≤ L(t) ≤ (N + c2)E(t),
which deduces thatL′(t) ≤ −k2L(t), ∀t > 0.
A simple integration over (0, 1) leads to
L(t) ≤ L(0)e−k2t.
It gives the desired result in Theorem 3.1 when combined with the equivalence of L(t) andE(t).
Acknowledgment. This work was supported by the National Natural Science Foun-dation of China [grant number 11771216], the Natural Science Foundation of JiangsuProvince [grant number BK20151523], the Six Talent Peaks Project in Jiangsu Province[grant number 2015-XCL-020] and the Qing Lan Project of Jiangsu Province.
References[1] T.A. Apalara, Well-posedness and exponential stability for a linear damped Timo-
shenko system with second sound and internal distributed delay, Electron. J. Differ-ential Equations 2014 (254), 1–15, 2014.
[2] T.A. Apalara and S.A. Messaoudi, An exponential stability result of a Timoshenkosystem with thermoelasticity with second sound and in the presence of delay, Appl.Math. Optim. 71 (3), 449–472, 2015.
[3] M. Bresse, Cours de Mecanique Appliquee par M. Bresse rsistance des matriaux etstabilit des constructions, Mallet-Bachelier, Paris, 1859.
[4] A.C. Casal and J.I. Díaz, On the complex Ginzburg-Landau equation with a delayedfeedback, Math. Models Methods Appl. Sci. 16 (1), 1–17, 2006.
Well-posedness and exponential stability 537
[5] M.M. Cavalcanti et al., Uniform decay rates for the energy of Timoshenko systemwith the arbitrary speeds of propagation and localized nonlinear damping, Z. Angew.Math. Phys. 65 (6), 1189–1206, 2014.
[6] M.M. Chen, W.J. Liu and W.C. Zhou, Existence and general stabilization of theTimoshenko system of thermo-viscoelasticity of type III with frictional damping anddelay terms, Adv. Nonlinear Anal. 7 (4), 547–569, 2018.
[7] Z.J.Chen, W.J. Liu and D.Q. Chen, General decay rates for a laminated beam withmemory, Taiwanese J. Math. 23 (5), 1227–1252, 2019.
[8] L.H. Fatori and J.E.M. Rivera, Rates of decay to weak thermoelastic Bresse system,IMA J. Appl. Math. 75 (6), 881–904, 2010.
[9] M. Kafini et al., Well-posedness and stability results in a Timoshenko-type system ofthermoelasticity of type III with delay, Z. Angew. Math. Phys. 66 (4), 1499–1517,2015.
[10] A.A. Keddi, T.A. Apalara and S.A. Messaoudi, Exponential and polynomial decay in athermoelastic-Bresse system with second sound, Appl. Math. Optim. 77 (2), 315–341,2018.
[11] V. Komornik, Exact controllability and stabilization, The multiplier method. Masson-John Wiley, Paris, 1994.
[12] J.E. Lagnese, G. Leugering and E.J.P.G. Schmidt, Modelling of dynamic networks ofthin thermoelastic beams, Math. Methods Appl. Sci. 16 (5),327–358, 1993.
[13] J.E. Lagnese, G. Leugering and E.J.P.G. Schmidt, Modeling, analysis and control ofdynamic elastic Multi-Link structures, Systems & Control: Foundations & Applica-tions, Boston, MA, 1994.
[14] G. Li, X.Y. Kong and W.J. Liu, General decay for a laminated beam with structuraldamping and memory: the case of non-equal wave speeds, J. Integral Equations Appl.30 (1), 95–116, 2018.
[15] G. Li, Y. Luan, J.Y. Yu and F.D. Jiang, Well-posedness and exponential stabil-ity of a flexible structure with second sound and time delay, Appl. Anal. DOI:10.1080/00036811.2018.1478081.
[16] G. Li, D.H. Wang and B.Q. Zhu, Well-posedness and decay of solutions for a trans-mission problem with history and delay, Electron. J. Differential Equations, 2016(23), 1–21, 2016.
[17] G.W. Liu, Well-posedness and exponential decay of solutions for a transmission prob-lem with distributed delay, Electron. J. Differential Equations, 2017 (174), 1-13, 2017.
[18] W.J. Liu and M.M. Chen, Well-posedness and exponential decay for a porous ther-moelastic system with second sound and a time-varying delay term in the internalfeedback, Contin. Mech. Thermodyn. 29 (3), 731–746, 2017.
[19] W.J. Liu, K.W. Chen and J. Yu, Asymptotic stability for a non-autonomous full vonKármán beam with thermo-viscoelastic damping, Appl. Anal. 97 (3), 400–414, 2018.
[20] W.J. Liu, D.H. Wang and D.Q. Chen, General decay of solution for a transmis-sion problem in infinite memory-type thermoelasticity with second sound, J. Therm.Stresses 41 (6), 758–775, 2018.
[21] W.J. Liu, J.Y. Yu and G. Li, Exponential stability of a flexible structure with secondsound, Ann. Polon. Math. DOI: 10.4064/ap171116-31-8.
[22] W.J. Liu and W.F. Zhao, Stabilization of a thermoelastic laminated beam with pasthistory, Appl. Math. Optim. 80, 103-133, 2019.
[23] S. Nicaise and C. Pignotti, Stability and instability results of the wave equation witha delay term in the boundary or internal feedbacks, SIAM J. Control Optim. 45 (5),1561–1585, 2006.
[24] S. Nicaise, C. Pignotti and J. Valein, Exponential stability of the wave equation withboundary time-varying delay, Discrete Contin. Dyn. Syst. Ser. S, 4 (3), 693–722, 2011.
538 G. Li, Y. Luan, W. Liu
[25] S. Nicaise, J. Valein and E. Fridman, Stability of the heat and of the wave equationswith boundary time-varying delays, Discrete Contin. Dyn. Syst. Ser. S, 2 (3), 559–581,2009.
[26] A. Pazy, Semigroups of linear operators and applications to partial differential equa-tions. Springer, New York, 1983.
[27] Y. Qin, J. Ren and T. Wei, Global existence, asymptotic stability, and uniform at-tractors for non-autonomous thermoelastic systems with constant time delay, J. Math.Phys. 53 (6), 063701, 1–20, 2012.
[28] M.L. Santos, D.S. Almeida Júnior and J.E. Muñoz Rivera, The stability number of theTimoshenko system with second sound, J. Differential Equations, 253 (9), 2715–2733,2012.
[29] D.H. Wang, G. Li and B.Q. Zhu, Exponential energy decay of solution for a trans-mission problem with viscoelastic term and delay, Mathematics, 4 (42), 1–13, 2016.
[30] D.H. Wang, G. Li and B.Q. Zhu, Well-posedness and general decay of solution for atransmission problem with viscoelastic term and delay, J. Nonlinear Sci. Appl. 9 (3),1202–1215, 2016.
[31] B. Wu, S.Y. Wu, J. Yu and Z.W. Wang, Determining the memory kernel from a fixedpoint measurement data for a parabolic equation with memory effect, Comput. Appl.Math. 37 (2), 1877–1893, 2018.
[32] S.T. Wu, Asymptotic behavior for a viscoelastic wave equation with a delay term,Taiwanese J. Math. 17 (3), 765–784, 2013.
[33] X.B. Zhang and H.L. Zhu, Hopf bifurcation and chaos of a delayed finance system,Complexity, Article ID: 6715036, 1–18, 2019.
[34] X.B. Zhang, H.Y. Zhao and Z.S. Feng, Spatio-temporal complexity of a delayed diffu-sive model for plant invasion, Comput. Math. Appl. 76 (11-12), 2575–2612, 2018.
Hacettepe Journal ofMathematics & Statistics
Hacet. J. Math. Stat.Volume 49 (2) (2020), 539 – 552
DOI : 10.15672/hujms.459593
Research Article
The Laguerre pseudospectral method for thetwo-dimensional Schrödinger equation with
symmetric nonseparable potentialsHaydar Alıcı
Department of Mathematics, Harran University, 63290, Şanlıurfa, Turkey
AbstractThe Hermite pseudospectral method is one of the natural techniques for the numericaltreatment of the problems defined over unbounded domains such as two-dimensional time-independent Schrödinger equation on the whole real plane. However, it is shown here thatfor the symmetric potentials, transformation of the problem over the first quadrant andthe application of the Laguerre pseudospectral method reduce the cost by a factor of fourwhen compared to the Hermite pseudospectral method.
Mathematics Subject Classification (2010). 65L60, 81Q05, 65L15, 34L40, 42C10
Keywords. the Laguerre pseudospectral method, two dimensional Schrödingerequation, symmetric potentials
1. IntroductionIn [1], the eigenvalues and wavefunctions of the two-dimensional time-independent
Schrödinger equation[− ∂2
∂x2 − ∂2
∂y2 + V (x, y)]
Ψ(x, y) = EΨ(x, y), (x, y) ∈ R2 (1.1)
with the boundary conditionsΨ(x, ±∞) = 0, −∞ < x < ∞Ψ(±∞, y) = 0, −∞ < y < ∞
(1.2)
are approximated by means of the Hermite pseudospectral method (HPM) for severalnonseparable quantum mechanical potentials V (x, y). In this article, we consider thereflection symmetric potentials
V (x, y) = V (−x, y) = V (x, −y) = V (−x, −y) (1.3)for which the spectrum of the system (1.1)-(1.2) can be decomposed into four subsetscontaining the states E2n,2m, E2n,2m+1, E2n+1,2m and E2n+1,2m+1. Although the Hermitepseudospectral method can still be used with further modifications in the Lagrange inter-polant so that each eigenvalue subset can be treated by a separate basis set, it does notseem practical. Therefore, the main aim of this paper is to find more natural basis setswhich can treat the states E2n,2m, E2n,2m+1, E2n+1,2m and E2n+1,2m+1 separately. In this
Email address: [email protected]: 13.09.2018; Accepted: 14.01.2019
540 H. Alıcı
case, one deals with four matrices of size N2 ×N2 instead of a matrix of size (2N)2 ×(2N)2
reducing the cost by a factor of two in each direction which is a considerable amount intwo dimensions.
First of all the reflection symmetric character of the potential suggests the use of trans-formations
ξ = (αx)2, η = (αy)2, α > 0 (1.4)where the parameter α may be regarded as an optimization parameter and introduced fornumerical purposes. With (1.4), the Schrödinger equation takes the form[
ξ∂2
∂ξ2 + η∂2
∂η2 + 12
∂
∂ξ+ 1
2∂
∂η− 1
4α2 V(√
ξ/α,√
η/α)]
Ψ = − E
4α2 Ψ (1.5)
where (ξ, η) ∈ (0, ∞) × (0, ∞). Accordingly the boundary conditions in (1.2) read asΨ(ξ, ∞) = 0, 0 < ξ < ∞Ψ(∞, η) = 0, 0 < η < ∞.
(1.6)
Now, introducing the transformation
Ψ(ξ, η) = ξs1ηs2e− 12 (ξ+η)Φ(ξ, η), s1, s2 ≥ 0 (1.7)
satisfying the boundary conditions in (6) where the polynomial terms are introduced tocope with the artificial singularity at ξ = 0 and η = 0, we rewrite the last equation as[
ξ∂2
∂ξ2 + η∂2
∂η2 +(2s1 + 1
2 − ξ) ∂
∂ξ+
(2s2 + 1
2 − η) ∂
∂η+ Q
]Φ = EΦ (1.8)
whereQ = s1(s1 − 1/2)
ξ+ s2(s2 − 1/2)
η+ ξ + η
4− 1
4α2 V(√
ξ/α,√
η/α)
(1.9)
stands for the modified potential and
E = E(s1, s2, α) = s1 + s2 + 12
− E
4α2 (1.10)
the shifted eigenvalues. Notice from (1.7) that the new dependent variable Φ does nothave to satisfy any boundary condition as long as it is a bounded function of ξ and η.
Then, to get rid of the unwelcome terms that are proportional to ξ−1 and η−1 appearingin the modified potential, we are free to choose any one of the elements from the set
(0, 0), (0, 12), (1
2 , 0), (12 , 1
2)
for (s1, s2). Moreover, setting
2si + 12
= γi + 1, i = 1, 2 (1.11)
the last equation reads as[ξ
∂2
∂ξ2 + η∂2
∂η2 + (γ1 + 1 − ξ) ∂
∂ξ+ (γ2 + 1 − η) ∂
∂η+ Q
]Φ = EΦ (1.12)
where the modified potential
Q = Q(ξ, η; α) = 14
[ξ + η − 1
α2 V(√
ξ/α,√
η/α)]
(1.13)
is now free of the parameters γi, and hence si, for i = 1, 2. Meanwhile, the shiftedeigenvalues becomes
E = E(γ1, γ2, α) = 12
(γ1 + γ2) + 1 − E
4α2 . (1.14)
Notice from (1.7) that the (γ1, γ2) ∈
(−12 , −1
2), (−12 , 1
2), (12 , −1
2), (12 , 1
2)
values obtainedfrom (1.11) lead to the states E2m,2n, E2m,2n+1, E2m+1,2n and E2m+1,2n+1 respectively.This can be seen on returning back to the original variables (x, y) via (1.4).
The Laguerre pseudospectral method for the two-dimensional Schrödinger equation 541
Note that the Schrödinger equation in (1.1) is defined over the whole real plane whereasthe transformed equation (1.12) of a reflection symmetric system is defined over the firstquadrant. Thus, the direct application of the HPM used in [1] is not possible anymore.Instead, Laguerre pseudospectral method (LPM) seems more feasible since the Laguerrepolynomials constitute a complete orthogonal set on the half line (0, ∞).
Therefore, in section 2 we construct the pseudospectral formulation of the transformedequation in (1.12) based on the Laguerre polynomials. Section 3 includes numerical ex-amples and some implementation notes. The last section concludes the paper with someremarks.
2. Pseudospectral formulation of the problemSince a pseudospectral method is based on a polynomial interpolation in the Lagrange
form, we propose an approximate solution of the form
Φ(ξ, η) =N∑
n=0
N∑j=0
ℓn(ξ)ℓj(η)Φnj (2.1)
whereℓk(ξ) = ϕN+1(ξ)
(ξ − ξk)ϕ′N+1(ξk)
, k = 0, 1, . . . , N (2.2)
are the set of N -th degree Lagrange interpolating polynomials in which
ϕN+1(ξ) = kN+1
N∏i=0
(ξ − ξi), kN+1 ∈ R (2.3)
is the (N + 1)-st degree polynomial having real and distinct zeros. Here, Φnj = Φ(ξn, ηj)are the exact values of Φ(ξ, η) at the specified grid points (ξn, ηj) for n, j = 0, 1, . . . , N .Now to approximate the eigenvalues of the Schrödinger equation with symmetric potentialswe set
ϕγN+1(t) =
LγN+1(t)hN+1
, h2k = Γ(k + γ + 1)
k!, t ∈ (0, ∞) (2.4)
in which LγN+1(t) is the Laguerre polynomial of degree N + 1 and order γ = ±1
2 . Ap-proximating the solutions of differential equations by Laguerre polynomials are usuallynot stable for large N due to their wild behaviors at infinity, and hence, one usually workswith the Laguerre functions Lγ
N (t) = e−t/2LγN (t) instead. This situation is theoretically
investigated, for example in [5, 7–9], and it is shown that the Laguerre functions havebetter stability properties than Laguerre polynomials. In this study, we search for thesquare integrable solutions of (1.1) which vanish exponentially at infinity. Therefore, theLaguerre functions are suitable for the numerical treatment of (1.1). But, since we factoroff the term e−(ξ+η)/2 from the solution by means of the transformation in (1.7), the useof Laguerre polynomials for equation (1.12) is equivalent to the use of Laguerre functionsfor the original problem in (1.1). Thus, we continue with the normalized Laguerre poly-nomials in (2.4) to approximate the eigenvalues of (1.12) and hence, those of the originalequation in (1.1).
By (2.4) the grid points (ξn, ηj) are set to be the cartesian product of the zeros ξn andηj of the Laguerre polynomials Lγ1
N+1(ξ) and Lγ2N+1(η), respectively. That is, we have four
set of grid points according to the choice of the parameters
(γ1, γ2) ∈
(−12
, −12
), (−12
,12
), (12
, −12
), (12
,12
)
which will be used to approximate the states E2m,2n, E2m,2n+1, E2m+1,2n and E2m+1,2n+1respectively.
542 H. Alıcı
On the other hand, zeros of the normalized Laguerre polynomials may be determinedto a desired accuracy as the eigenvalues of the tridiagonal matrix
R =
B0 A1 0A1 B1 A2
A2 B2. . .
. . . . . . AN
0 AN BN
(2.5)
where the An = −√
n(n + γ) and Bn = 2n + γ + 1 are the coefficients in the three termrecursion
An+1ϕγn+1(ξ) + (Bn − ξ)ϕγ
n(ξ) + Anϕγn−1(ξ) = 0, n = 0, 1, . . . (2.6)
of the normalized Laguerre polynomials [2, 12]. Actually, the matrix R is obtained byrunning the recursion from n = 0 to n = N and setting ϕN+1(ξ) to zero. Thus, themth computed eigenvector rm = [r0,m r1,m . . . rN−1,m rN,m]T of R corresponding tothe m−th eigenvalue (or root) ξm is a constant multiple of the vector of orthonormalpolynomial values ϕγ m = [ϕγ
0(ξm) ϕγ1(ξm) . . . ϕγ
N−1(ξm) ϕγN (ξm)]T at the point ξm. That
is, rm = aϕm. This constant may be determined by comparing the first elements r0,m andϕγ
0(ξm) of these two vectors since ϕγ0(ξm) = 1/h0 = 1/
√Γ(γ + 1) is a constant. Therefore,
for n = 0, 1, . . . , N , the values ϕγn(ξm) of the normalized classical orthogonal polynomials
at the zeros of ϕγN+1(ξ) may be computed as
[ϕγ0(ξm) ϕγ
1(ξm) . . . ϕγN−1(ξm) ϕγ
N (ξm)]T =1√
Γ(γ + 1)r0,m[r0,m r1,m . . . rN−1,m rN,m]T (2.7)
in terms of the computed eigenvector rm of tridiagonal symmetric matrix R of size N + 1.It is known as the Golub-Welsch algorithm whose details may be found, for example, in
[2,4]. This procedure, unfortunately, suffers from computing the full set of eigenvalues witha uniform accuracy especially for large values of N . Therefore, in this case, alternativeroot finding algorithms, for example, Newton method may be used to compute the roots.Nevertheless, for moderate N values the Golub-Welsch algorithm can be used without anyhesitation.
After determining the mesh points (ξn, ηj), we insert the approximate solution in (2.1)into the equation (1.12) and require its satisfaction at the grid points (ξm, ηi) to obtainthe set of (N + 1)2 equations
N∑n=0
N∑j=0
[Ln(ξm)ℓj(ηi) + Lj(ηi)ℓn(ξm) + Q(ξm, ηi; α)ℓn(ξm)ℓj(ηi)
]Φnj
= E(γ1, γ2, α)N∑
n=0
N∑j=0
ℓn(ξm)ℓj(ηi)Φnj (2.8)
for m, i = 0, 1, . . . , N where
Lr(ζp) := Lγpr = ζpℓ′′
r(ζp) + (γ + 1 − ζp)ℓ′r(ζp). (2.9)
Here, notice the dependence of the roots to the order γ of the Laguerre polynomials. Thatis, the points ζp := ζγ
p are the roots of the (N + 1)st degree Laguerre polynomials of orderγ.
The Laguerre pseudospectral method for the two-dimensional Schrödinger equation 543
Therefore, by using (2.9) and the well-known property ℓr(ζp) = δpr of the Lagrangepolynomials, the algebraic equations in (2.8) reads as
N∑n=0
N∑j=0
(Lγ1
mnδij + Lγ2ij δmn + Qmiδmnδij
)Φnj =
E(γ1, γ2, α)N∑
n=0
N∑j=0
δmnδijΦnj (2.10)
where the Qmi = Q(ξm, ηi; α) are the known values of the modified potential in (1.13)at the mesh points. The last equation may be written in the matrix-vector form
BΦ = E(γ1, γ2, α)Φ (2.11)
whereB(γ1, γ2) := B = Lγ1 ⊗ I + I ⊗ Lγ2 + Q1 (2.12)
is a matrix of dimension (N + 1)2 × (N + 1)2. Here Φ is an (N + 1)2 × 1 vector containingthe vectorized unknown wavefunction values at the grid points in the order
Φ =[Φ00 . . . Φ0N Φ10 . . . Φ1N . . . . . . ΦN0 . . . ΦNN
]T (2.13)
where Φij stand for the values Φ(ξi, ηj) of the wavefunction at the nodal points (ξi, ηj).Moreover, Lγi for i = 1, 2 stands for the kinetic energy matrix with entries (2.9), I the(N + 1) × (N + 1) identity matrix, Q the diagonal potential matrix whose diagonal entriescontain the values Q(ξm, ηi; α) in the order specified in (2.13) and M1 ⊗M2 the Kroneckerproduct of the matrices M1 and M2.
The explicit entries [12]
Lγpr = −1
6
12ζp
(ζp − ζr)2ϕγ ′
N+1(ζp)ϕγ ′
N+1(ζr)if p = r
2N + 1ζr
[(γ − ζr)2 − 1
]if p = r
(2.14)
of the matrix Lγ in (2.9) reveal that it is, and hence B, is not symmetric. However,fortunately, the similarity transformation
Lγ = (Sγ)−1LγSγ (2.15)
in whichSγ = spδpr =
√ζpϕγ ′
N+1(ζp)δpr (2.16)
is a diagonal matrix symmetrizes the matrix Lγ . In this case, the entries of the symmetricmatrix Lγ reads as
Lγpr = −1
6
12
√ζpζr
(ζp − ζr)2 if p = r
2N + 1ζr
[(γ − ζr)2 − 1
]if p = r
(2.17)
with Kγpr = Kγ
rp. Therefore, we may state the following proposition.
Proposition 2.1. The matrixT = Sγ1 ⊗ Sγ2 (2.18)
in which Sγ is given by (2.16), symmetrizes the matrix B in (2.12).
544 H. Alıcı
Proof. By using the basic linear algebra tools
(A ⊗ B)−1 = A−1 ⊗ B−1 and (A ⊗ B)(C ⊗ D) = AC ⊗ BD (2.19)
and after a little algebra we end up with
B = T−1BT = (Sγ1 ⊗ Sγ2)−1(Lγ1 ⊗ I + I ⊗ Lγ2 + Q
)(Sγ1 ⊗ Sγ2)
= (Sγ1)−1Lγ1Sγ1 ⊗ I + I ⊗ (Sγ2)−1Lγ2Sγ2
+ (Sγ1 ⊗ Sγ2)−1 Q (Sγ1 ⊗ Sγ2).
Now using (2.15) for the first two term and keeping in mind that T = Sγ1 ⊗ Sγ2 and Qare diagonal matrices the last equation reads as
B = Lγ1 ⊗ I + I ⊗ Lγ2 + Q. (2.20)
Hence, being the addition of symmetric matrices, B is symmetric.
Thus we may replace the unsymmetrical system in (2.11) with the simple symmetricone
Bu = Eu (2.21)since the similar matrices participate of the same eigenvalue set. It is simple in thesense that the unpleasant term ϕ′
N+1(ζp)/ϕ′N+1(ζr) in (2.14) is removed by the similarity
transformation in (2.18). Notice that the eigenvectors of the unsymmetrical and symmetricsystems are linked as
Φ = Tu (2.22)
since Bu = T−1BTu = Eu implies B[Tu] = E[Tu]. The last equation may be written innodal form as
Φ(ξm, ηi) =√
ξmηiϕγ1 ′N+1(ξm)ϕγ2 ′
N+1(ηi)umi (2.23)
with the help of (2.16) and (2.18). Then, (1.7) together with (1.11) and (2.23) leads tothe values of the original wavefunction
Ψ(ξm, ηi) = ϕγ1 ′N+1(ξm)ϕγ2 ′
N+1(ηi)ξ12 (γ1+ 3
2 )m η
12 (γ2+ 3
2 )i e− 1
2 (ξ+η)umi (2.24)
at the mesh point (ξm, ηi) in terms of an eigenvector u of (2.21). On the other hand,setting n = N + 1 and x = ξm, ηi in the differential difference relation
xϕγ ′n (x) = nϕγ ′
n (x) −√
n(n + γ)ϕγ ′n−1(x) (2.25)
of the normalized Laguerre polynomials we obtain
ξmϕγ1 ′N+1(ξm) = −
√(N + 1)(N + γ1 + 1)ϕγ1
N (ξm)
ηiϕγ2 ′N+1(ηi) = −
√(N + 1)(N + γ2 + 1)ϕγ2
N (ηi)(2.26)
since ϕγ1N+1(ξm) = ϕγ2
N+1(ηi) = 0. Now, using (2.7) and (2.26), equation (2.24) reads as
Ψ(ξm, ηi) = C(N, γ1, γ2)rN,m
r0,m
rN,i
r0,iξ
12 (γ1− 1
2 )m η
12 (γ2− 1
2 )i e− 1
2 (ξm+ηi)umi (2.27)
where
C(N, γ1, γ2) = (N + 1)√
(N + γ1 + 1)(N + γ2 + 1)Γ(γ1 + 1)Γ(γ2 + 1)
. (2.28)
Now, we are in a position to state the following theorem:
The Laguerre pseudospectral method for the two-dimensional Schrödinger equation 545
Theorem 2.2. The approximate eigenvalues Ek of the two-dimensional Schrödinger equa-tion in (1.1) together with the boundary condition (1.2) are related with the eigenvaluesEk of the matrix B(γ1, γ2) in (2.21) by the formula
Ek = 4α2[1
2(γ1 + γ2) + 1 − Ek(γ1, γ2, α)
], k = 0, 1, . . . (2.29)
and the values of the corresponding normalized eigenfunctions Ψk(xn, yj) (in L2 sense) atthe mesh points in the first quadrant (xn, yj) = (
√ξn/α,
√ηj/α) are given by
Ψk(xm, yi) = αC(N, γ1, γ2)rN,m
r0,m
rN,i
r0,iξ
12 (γ1− 1
2 )m η
12 (γ2− 1
2 )i e− 1
2 (ξm+ηi)ukmi (2.30)
provided that uk is the k−th normalized (in Euclidean 2-norm) eigenvector of (2.21).Finally, the complete picture of the normalized eigenfunction Ψk(x, y) in the whole plane,at the original mesh points (xm, yi) = (±
√ξn/α, ±√
ηj/α) is obtained by appropriately(even and/or odd) extending the values in (2.30) to the other quadrants.
Proof. The first part easily follows from (1.10) and (1.11). For the second part, keepingin mind that for symmetric potentials the squares of the eigenfunctions are equal in allquadrants and then using the transformations in (1.4) and (1.7) we write∥∥∥Ψk
∥∥∥2
L2=
∫ ∫R2
Ψ2k(x, y)dxdy = 4
∫ ∫R+2
Ψ2k(x, y)dxdy
= 4α2∫ ∫
R+2Ψ2
k(x, y)dxdy = 4α2∫ ∫
R+2Ψ2
k(ξ, η) 14α2√
ξηdξdη
=∫ ∫
R+2Φ2
k(ξ, η)ξγ1ηγ2e−(ξ+η)dξdη
(2.31)
Now, applying the N+1 point Gauss-Laguerre quadrature to the last integral we have∥∥∥Ψk
∥∥∥2
L2= lim
N→∞
N∑m=0
N∑i=0
Φ2k(ξm, ξi)ωγ1
m ωγ2i (2.32)
whereωγ1
m = 1(N + 1)(N + γ1 + 1)
ξm
ϕγ1 2N (ξm)
, m = 0, 1, . . . , N
ωγ2i = 1
(N + 1)(N + γ2 + 1)ηi
ϕγ2 2N (ηi)
, i = 0, 1, . . . , N
(2.33)
are known as the Christofell numbers of the Laguerre-Gauss quadrature in terms of thenormalized Laguerre polynomials [6]. Now, plugging (2.26) into (2.23) we obtain
Φ(ξm, ηi) = (N + 1)√
(N + γ1 + 1)(N + γ2 + 1)ϕγ1N (ξm)ϕγ2
N (ηi)√ξmηi
. (2.34)
Finally, with the help of (2.34) and (2.33), (2.32) takes the form∥∥∥Ψk
∥∥∥2
L2= lim
N→∞
N∑n=0
N∑j=0
(uknj)2 (2.35)
in which the term with double sum is the squared Euclidean 2-norm of the vector uk(N+1)2×1.
Thus we have ∥∥∥Ψk
∥∥∥2
L2= lim
N→∞
∥∥∥uk∥∥∥2
2= lim
N→∞1 = 1 (2.36)
since∥∥∥uk
∥∥∥2
= 1 by hypothesis which completes the proof.
546 H. Alıcı
yx
-0.4
-0.2
z 0
0.2
0.4
-10 -10-5
05
10
-5
0
5
10
(a) Ψ00(x, y), E00 = 2.2629597964443
-5y0
5
10
-10 -10
-0.4
z 0
0.2
0.4
-0.2
x
105
-50
(b) Ψ10(x, y), E10 = 2.2629597981908
yx
-0.4
-0.2
z 0
0.2
0.4
5
-10 -10-5
0
10
-5
0
10
5
(c) Ψ01(x, y), E01 = 3.9992895301372
-5y0
5
10
-10 -10
-0.4
x
-0.2
z 0
0.2
0.4
50
-5
10
(d) Ψ11(x, y), E11 = 3.9992895313331
Figure 1. First four normalized wavefunctions of the symmetric double-well po-tential in (3.1) when µ = 0.01.
3. Numerical results and discussionIn this section, we apply the methods described in the previous section to several sym-
metric two dimensional quantum mechanical potentials. As a first example we considerthe symmetric double-well potential
V (x, y) = x2 − y2 + µy2(2y2 − x2) + 18µ
(3.1)
where µ is a positive real parameter. In all tables, n stands for the eigenvalue index,N the truncation order for which the desired accuracy of the corresponding eigenvalue isobtained, and αopt denotes the optimum value of the scaling or an optimization parameterα for which the desired accuracy is obtained with the smallest possible truncation orderN . A method to determine the optimum value of α for Gaussian type functions is givenin [10] which can also be applied in our problem since the eigenfunctions of (1.1) are ofGaussian type.
The accuracy of the results in all tables reported here has been checked by inspectingthe number of stable digits between two consecutive truncation orders. Therefore, onlythe last digits might be incorrect because of rounding.
Table 1 presents the accuracy improvement of the ground state eigenvalue of the sym-metric double well potential by systematically increasing the truncation size N . It is clearfrom Table 1 that the increase in N by one results in an accuracy gain of one digit for
The Laguerre pseudospectral method for the two-dimensional Schrödinger equation 547
Table 1. Accuracy improvement of the ground state eigenvalue of the symmetricdouble well potential in (3.1).
µ αopt N E0,00.01 1 12 2.261
15 2.262 95618 2.262 959 79021 2.262 959 796 4624 2.262 959 796 444 6927 2.262 959 796 444 74028 2.262 959 796 444 740
this potential. This is typical for all potentials considered here. In fact, for some specificpotentials the accuracy gain is more than one digit when the truncation size is increasedby one.
Table 2. Several nearly degenerate eigenvalues of the symmetric double-well po-tential in (3.1). For comparison, we include the corresponding results from [1]which uses the HPM. For both methods αopt = 1.
µ NLP M NHP M E2m,2n LPM/HPM E2m,2n+1 LPM/HPM0.01 28 56 2.262 959 796 444 3 / 43 2.262 959 798 190 7 / 08
4.967 130 049 336 0 / 68 4.967 130 461 753 9 / 385.734 657 066 616 3 / 67 5.734 657 071 515 1 / 517.523 965 294 560 8 / 11 7.524 006 494 536 1 / 688.458 313 203 321 6 / 21 8.458 314 106 626 5 / 689.203 671 170 577 8 / 78 9.203 671 208 017 7 / 809.898 300 048 693 6 / 40 9.900 487 409 690 5 / 10
11.045 853 380 364 7 / 47 11.045 937 822 097 3 / 7311.936 552 764 608 0 / 86 11.936 627 549 541 6 / 1211.994 455 823 810 8 / 09 12.054 502 192 145 4 / 5112.672 301 933 093 4 / 32 12.672 302 141 021 5 / 18E2m+1,2n LPM/HPM E2m+1,2n+1 LPM/HPM3.999 289 530 137 4 / 72 3.999 289 531 333 6 / 316.714 422 914 369 3 / 92 6.714 423 198 940 9 / 157.469 353 731 555 9 / 53 7.469 353 740 847 8 / 729.287 707 580 307 3 / 73 9.287 736 596 968 9 / 88
10.198 992 794 453 7 / 37 10.198 994 433 172 2 / 2310.937 895 736 016 2 / 57 10.937 895 835 513 0 / 3111.685 001 492 931 3 / 05 11.686 593 204 436 4 / 59
Table 2 presents some nearly degenerate states of the symmetric double-well potential.For comparison we include the results of [1], which uses Hermite pseudospectral methodsin both direction. It is clear from Table 2 that the use of Laguerre instead of Hermitepolynomials halved the truncation size in each direction that is necessary to obtain theaccuracy quoted. More clearly, by using the HPM we need to diagonalize a matrix ofsize (2N)2 × (2N)2 while with the use of LPM we only need to diagonalize four separateN2 × N2 matrices to get the same number of eigenvalues to the same accuracy.
As a second example, we take into account the Gaussian type potential
V (x, y) = −e−β(x4+y4), β > 0 (3.2)
548 H. Alıcı
yx
-0.2
-0.1
z 0
0.1
0.2
20
10
-10
0
-20 -20-10
010
20
(a) Ψ00(x, y), E00 = −0.8002386220129
yx
-0.1
z 0
0.1
0.2
-0.2
10
-10
0
10
20
-20 -20-10
0
20
(b) Ψ10(x, y), E10 = −0.5544027490889
-10y0
10
20
-20 -20
-0.2
x
-0.1
z 0
0.1
0.2
2010
-100
(c) Ψ01(x, y), E01 = −0.5544027490889
xy
-0.2
-0.1
z 0
0.1
0.2
20
-10
0
10
-20 -20-10
010
20
(d) Ψ11(x, y), E11 = −0.3214069787951
Figure 2. First four normalized wavefunctions of the Gaussian type potential in(3.2) when β = 0.001.
Table 3. Comparison of the truncation sizes of the LPM and HPM for the discretestates of the Gaussian potential in (3.2) when β = 0.001.
NLP M NHP M αopt E2m,2n
30 60 0.5 -0.800 238 622 012 90.3 -0.270 627 665 0
-0.252 579 345 5E2m,2n+1 and E2m+1,2n
30 60 0.5 -0.554 402 749 0880.3 -0.064 141 0400.2 -0.004 59
E2m+1,2n+130 60 0.5 -0.321 406 978 7
which has a finite number of discrete states between −1 < Em,n < 0 together with acontinuous spectrum over the entire positive real axis for small values of the real parameterβ.
Next, as a third example, we consider the hyperbolic secant potential
V (x, y) = −m(m + 1) sech2[(x4 + y4)1/2
](3.3)
The Laguerre pseudospectral method for the two-dimensional Schrödinger equation 549
Table 4. Improved results for the discrete states of the Gaussian potential in(3.2) when β = 0.001 by using LPM.
NLP M αopt E2m,2n
42 0.5 -0.800 238 622 012 90.3 -0.270 627 664 825 0
-0.252 579 345 455 4E2m,2n+1 and E2m+1,2n
35 0.5 -0.554 402 749 088 948 0.3 -0.064 141 043 4960 0.2 -0.004 589 7
E2m+1,2n+148 0.5 -0.321 406 978 795 1
where m > 0. Like the Gaussian potential, there exist finitely many discrete states locatedon the negative real axis between −m(m + 1) < E < 0 together with the continuousspectrum on the whole positive real axis. Tables 3 and 5 presents the computed eigenvaluesof the Gaussian type and hyperbolic secant potential by using the LPM and HPM.
Table 5. Comparison of the truncation sizes of the LPM and HPM for the discretestates of the hyperbolic secant potential in (3.3) when m = 5.
NLP M NHP M αopt E2m,2n
30 60 2.5 -23.904 156 425 7342.0 -8.276 916 534 5
-7.607 559 120 2E2m,2n+1 and E2m+1,2n
30 60 2.5 -16.505 099 700 871.5 -2.343 009
-0.532 94E2m+1,2n+1
30 60 2.0 -9.599 644 620 3
For the last two potentials again we see that NHP M ≈ 2NLP M = 60. Therefore, byincreasing the truncation size NLP M of the present algorithm it is possible to obtainmore accurate results than the HPM which are presented in Tables 4 and 6. However, aremarkable slowing down of the convergence is met for the discrete states at the borderof the continuum which is a common drawback of almost all methods even in the one-dimensional case. This is because of the nonexistence of the contributions coming fromthe continuous spectrum eigenfunctions in the basis functions [3, 11].
Finally, we take into account the quartic anharmonic oscillatorV (x, y) = x2 + y2 + c4(x4 + 2ax2y2 + y4) (3.4)
with c4 > 0, and −1 ≤ a ≤ 1. Table 7 present the states E2m,2n, E2m,2n+1, E2m+1,2n andE2m+1,2n+1 when c4 = 1000 and a = 1. For comparison we also tabulate the results from[13] that employs the certain trigonometric basis set in a Rayleigh-Ritz variational scheme.We have executed our computer program in gfortran–4.8 by using quadruple precisionarithmetic so that the comparison of the numerical results with those of [13] becomes moremeaningful and informative. Clearly, our results are slightly better than those of [13] where17−18 decimal points of accuracy was obtained with the truncation size of N = 22. On theother hand, with the same truncation size, we obtain 18 − 22 decimal points of accuracy.Moreover, the algorithm of [1] is also implemented in Fortran programming language and
550 H. Alıcı
Table 6. Improved results for the discrete states of the hyperbolic secant potentialin (3.3) when m = 5 by using LPM.
NLP M αopt E2m,2n
30 2.5 -23.904 156 425 73433 2.0 -8.276 916 534 56
-7.607 559 120 19E2m,2n+1 and E2m+1,2n
30 2.5 -16.505 099 700 872 260 1.5 -2.343 009 097 47
-0.532 963 25E2m+1,2n+1
60 2.0 -9.599 644 620 311
executed in quadruple precision arithmetic. Again, the doubling NHP M ≈ 2NLP M = 44in the truncation sizes of the HPM and LPM is clear. Note that the HPM produces thefull spectrum at once, however, while reporting in Table 7 we split it into four states tocompare the results easily.
Table 7. First few energy eigenvalues of the quartic anharmonic oscillator in (3.4)when c4 = 1000, a = 1.
NLP M NHP M E2m,2n (αopt = 6) E2m,2n (αcr = 1.5)LPM (N = 22)/HPM (N = 44) Reference [13] (N = 22)
22 44 23.513 389 183 129 853 963 236 1 23.513 389 183 129 853 96389.433 434 033 749 367 276 85 89.433 434 033 749 367 27795.437 449 804 059 634 223 44 95.437 449 804 059 634 223
170.997 778 280 937 441 267 170.997 778 280 937 441 27183.306 338 107 785 976 427 183.306 338 107 785 976 43187.549 037 142 026 458 968 187.549 037 142 026 458 97
E2m,2n+1 and E2m+1,2n E2m,2n+1 and E2m+1,2n
54.054 855 795 519 439 394 244 4 54.054 855 795 519 439 394128.619 616 180 914 730 350 60 128.619 616 180 914 730 35138.283 038 429 442 399 890 09/11 138.283 038 429 442 399 89216.151 947 586 633 607 188 0/1 216.151 947 586 633 607 19
E2m+1,2n+1 E2m+1,2n+189.433 434 033 749 367 276 845 8 89.433 434 033 749 367 277
170.997 778 280 937 441 267 36 170.997 778 280 937 441 27183.306 338 107 785 976 426 61 183.306 338 107 785 976 43
Closeness in the numerical results with the same truncation size is not too much surpris-ing since both the trigonometric and Laguerre basis sets are somehow the exact solutionsof the problem in (1.1). More specifically, the former is the exact solution of (1.1) overa finite rectangle (x, y) ∈ (α, α) × (β, β) with Dirichlet boundary conditions in which thepotential is in the form of a rectangular box with impenetrable walls [13]. The latter is thesolution of the system (1.1)-(1.2) under the two-dimensional harmonic oscillator potentialV (x, y) = x2 + y2 (see the transformed equation (1.12) together with (1.13)-(1.14) whenα = 1).
The Laguerre pseudospectral method for the two-dimensional Schrödinger equation 551
4. ConclusionIn this article, the two-dimensional Schrödinger equation over the whole real plane with
symmetric nonseparable potentials is solved numerically by using the LPM. Transforma-tion of the problem over the first quadrant enabled us to treat the states E2m,2n, E2m,2n+1,E2m+1,2n and E2m+1,2n+1 separately. By this way, instead of diagonalizing a matrix ofsize 4N2 × 4N2 we compute the eigenvalues of four matrices of size N2 × N2 to obtain thesame number of eigenvalues to a certain accuracy.
It is known that the numerical eigenvalue problems suffer from the problem of computingthe full set of eigenvalues with a uniform accuracy [14]. Only the portion of the eigenvaluescan be obtained with a desired accuracy for a fixed truncation order N . Therefore, thetreatment of the states E2m,2n, E2m,2n+1, E2m+1,2n and E2m+1,2n+1 by separate basis setsquadruples the number of high accurate eigenvalues for the same fixed truncation size N .
On the other hand, comparison of the present numerical results with those obtained byapplying the HPM [1] reveals evidently that NHP M ≈ 2NLP M . Clearly, the use of LPMinstead of HPM halves the number of points and reduces the cost by a factor of two ineach direction which strongly supports the main argument of the present study. Note that,we do not claim that the LPM is superior to the HPM from the efficiency or accuracypoint of view in general. In fact, even the direct comparison of these two methods is notmeaningful since the Hermite and Laguerre polynomials are defined over the whole realline and half line, respectively. However, here we have shown that if the system (1.1)-(1.2) has the reflection symmetries, that is, both the differential equation in (1.1) andthe boundary conditions in (1.2) are invariant under the replacement of the independentvariables x and/or y by their negatives, then the transformation of the problem over thefirst quadrant and the use of LPM with γ = ±1
2 separate the states E2m,2n, E2m,2n+1,E2m+1,2n and E2m+1,2n+1, and hence, halve the truncation size N in each direction. Thisreduction is important since in two dimension the size of the resulting discrete systemincreases as the square of the truncation size. Alternatively, without transforming theequation, one may use the even or odd indexed Hermite polynomials as basis sets toseparate the above four states. Actually, both approaches can be regarded as equivalentif we remember the interrelations
H2n(x) = (−1)n22nn!L−1/2n (x2) (4.1)
andH2n+1(x) = (−1)n22n+1n!xL1/2
n (x2) (4.2)between the Hermite and Laguerre polynomials. Unfortunately, direct use of H2n(ξ) orH2n+1(ξ) in (2.2) does not allow the separation of the above states. Thus, in this studywe have presented the use of Lγ
n(ξ) with γ = ±12 after the transformation of the problem
over the first quadrant in order to reduce the total cost by a factor of four. As a result,the same accuracy of the HPM for the numerical eigenvalues is obtained with considerablylow cost which is in accordance with the main aim of this study. Remember that our claimwas to present a pseudospectral method for the numerical solution of the two dimensionalSchrödinger equation with symmetric nonseparable potentials which is more efficient thanthe HPM from a computational point of view.
References[1] H. Alıcı, The Hermite pseudospectral method for the two-dimensional Schrödinger
equation with nonseparable potentials, Comput. Math. Appl. 69 (6), 466–477, 2015.[2] H. Alıcı and H. Taşeli, Pseudospectral methods for an equation of hypergeometric type
with a perturbation, J. Comput. Appl. Math. 234, 1140–1152, 2010.[3] M. Demiralp, N.A. Baykara, and H. Taşeli, A basis set comparison in a variational
scheme for Yukawa potential, J. Math. Chem. 11, 311–323, 1992.
552 H. Alıcı
[4] G.H. Golub and J.H. Welsch, Calculation of Gauss quadrature rules, Math. Comput.23, 221–230 s1–s10, 1969.
[5] B.-Y. Guo, L.-L. Wang, and Z.-Q. Wang, Generalized Laguerre interpolation andpseudospectral method for unbounded domains, SIAM J. Numer. Anal. 43, 2567–2589,2006.
[6] F.B. Hildebrand, Method of Applied Mathematics, McGraw-Hill, NewYork, 1956, pp.319–323.
[7] G. Mastroianni and D. Occorsio, Lagrange interpolation at Laguerre zeros in someweighted uniform spaces, Acta Math. Hungar. 91 (1-2), 27–52, 2001.
[8] J. Shen, Stable and efficient spectral methods in unbounded domains using Laguerrefunctions, SIAM J. Numer. Anal. 38 (4), 1113–1133, 2000.
[9] J. Shen and L.-L. Wang, Some recent advances on spectral methods for unboundeddomains, Commun. Comput. Phys. 5, 195–241, 2009.
[10] T. Tang, The Hermite spectral method for Gaussian-type functions, SIAM J. Sci.Comput. 14 (3), 594–606, 1993.
[11] H. Taşeli, Modified Laguerre basis for hydrogen-like systems, Int. J. Quantum Chem.63, 949–959, 1997.
[12] H. Taşeli and H. Alıcı, The Laguerre pseudospectral method for the reflection sym-metric Hamiltonians on the real line, J. Math. Chem. 41, 407–416, 2007.
[13] H. Taşeli and R. Eid, Eigenvalues of the two-dimensional Schrödinger equation withnonseparable potentials, Int. J. Quantum Chem. 59, 183–201, 1996.
[14] J.A.C. Weideman and L.N. Trefethen, Eigenvalues of second-order spectral differen-tiation matrices, SIAM J. Numer. Anal. 25, 1279–1298, 1988.
Hacettepe Journal ofMathematics & Statistics
Hacet. J. Math. Stat.Volume 49 (2) (2020), 553 – 564
DOI : 10.15672/hujms.460975
Research Article
Lucas polynomial solution of nonlinear differentialequations with variable delays
Sevin Gümgüm∗1, Nurcan Baykuş Savaşaneril2, Ömür Kıvanç Kürkçü1,Mehmet Sezer3
1Izmir University of Economics, Department of Mathematics, Izmir, 35330, Turkey2Izmir Vocational School, Dokuz Eylül University, Izmir, Turkey
3Department of Mathematics, Manisa Celal Bayar University, Manisa, Turkey
AbstractIn this study, a novel matrix method based on Lucas series and collocation points hasbeen used to solve nonlinear differential equations with variable delays. The applicationof the method converts the nonlinear equation to a matrix equation which corresponds toa system of nonlinear algebraic equations with unknown Lucas coefficients. The method istested on three problems to show that it allows both analytical and approximate solutions.
Mathematics Subject Classification (2010). 34A34, 65L60, 40C05
Keywords. nonlinear delay differential equations, variable delays, matrix andcollocation methods, Lucas polynomials and series
1. IntroductionOrdinary differential equations fail to model many physical phenomena when the model
is not only determined by its present state but also by a certain past state. Consequently,differential equations with time delays are used in modeling of real life situations such ashuman body control and multibody control systems, electric circuits, dynamical behaviourof a system in fluid mechanics, chemical engineering [21], spread of bacteriophage infection[38], stage structured populations [37], epidemic model in biology and dynamic diseasesmodel in physiology [35].
Some numerical methods have been developed to solve nonlinear differential equations(NDE) with proportional and constant delays; among them, one can mention Aboodhtransformation method [5], Adomian decomposition method [12,31], Power series method[11], Decomposition method [39], Differential transform method [28], Hermite waveletbased method [36], Variational iteration method [25,29,41], Power and Padé series basedmethod [24], Spectral method [6], Variable multistep methods [27], Quasilinearizationtechnique [34], Runge-Kutta-Fehlberg methods [30], Polynomial least squares method [13],Homotopy perturbation method [35], and First Boubaker polynomial approach [16].
On the other hand, there are few studies about nonlinear differential equations withvariable delays. A study on the existence of positive ω-periodic solutions has been carried∗Corresponding Author.
Email addresses: [email protected] (S. Gümgüm), [email protected] (N.B.Savaşaneril), [email protected] (Ö.K. Kürkçü), [email protected] (M. Sezer)Received: 18.09.2018; Accepted: 22.01.2019
554 S. Gümgüm , N.B. Savaşaneril, Ö.K. Kürkçü, M. Sezer
out by Dorociaková and Olach [19]. Chen et.al. [14] presented new criteria for asymptoticstability. Asymptotic behaviour of solutions is studied by Dix [18]. Fixed points andstability are studied [7, 17, 23, 44]. Only a few numerical techniques have been applied tosolve such kind of equations: A new multi-step technique [10], Legendre-Gauss collocationmethod [40] and Runge-Kutta method using Hermite interpolation [22].
Numerical solutions of ODEs, fractional differential equations (FDE) and integro-differential equations are of great interest. Recently, methods based on Lucas, Fibonacciand Fermat polynomials have been proposed to solve FDEs [1–4,8,42,43]. In these studies,they derived the operational matrix of fractional derivatives and observed that the numer-ical solutions have smaller errors than those obtained by using orthogonal polynomials.
In the present study, we consider the NDE with variable delays of the form
2∑k=0
1∑j=0
Pkj(t) y(k)(t − τkj(t)) +1∑
p=0
p∑q=0
Rpq(t) y(p)(t)y(q)(t) = g(t) (1.1)
or precisely
P00(t)y(t − τ00(t)) + P10(t)y′(t − τ10(t)) + P01(t)y(t − τ01(t)) + P11(t)y′(t − τ11(t))+ P20(t)y′′(t − τ20(t)) + P21(t)y′′(t − τ21(t)) + R00(t)y2(t) + R10(t)y′(t)y(t)+ R11(t)(y′(t))2 = g(t)
with the initial conditions y(a) = λ1 and y′(a) = λ2.Here, Pkj(t), Rpq(t), g(t), and the variable delays τkj(t) are given continuous functions
defined on 0 ≤ a ≤ t ≤ b, where τkj(t) ≥ 0.We propose a new matrix technique, developed by Sezer et.al. [9, 15, 20, 32], to solve
Eq. (1.1) with the initial conditions, in the finite Lucas series of the form
y(t) ∼= yN (t) =N∑
n=0an Ln(t), a ≤ t ≤ b (1.2)
where an, n = 0, 1, ..., N are unknown coefficients and Ln(t), n = 0, 1, ..., N ; N ≥ mare the Lucas polynomials [26]. These polynomials are constructed from the recurrencerelation
L0(t) = 2, L1(t) = t,Ln+2(t) = tLn+1(t) + Ln(t), n ≥ 0
The Binet and power form representations of Lucas polynomials can be seen in [2, 3].
2. Operational matrix relationsIn this section, we derive the operational matrix relations of Eq. (1.1) and (1.2). For
this purpose, we write the series in Eq. (1.2) as a matrix equation as follows
y(t) ∼= yN (t) = L(t)A = T(t)MA (2.1)
where
L(t) =[
L0(t) L1(t) · · · LN (t)], T(t) =
[1 t · · · tN
],
A =[
a0 a1 · · · aN]T
Lucas polynomial solution of nonlinear differential equations with variable delays 555
and M has the form
MT =
2 0 0 · · · 0
0 11
(10
)0 · · · 0
21
(11
)0 2
2
(20
)0
0 32
(21
)0 · · · 0
......
... . . . ...(n−1)( n−1
2 )
(n−1
2n−1
2
)0 n−1
( n+12 )
(n+1
2n−3
2
)· · · 0
0 n
( n+12 )
(n+1
2n−1
2
)0 · · · n
n
(n0
)
when N is odd, and
MT =
2 0 0 · · · 0
0 11
(10
)0 · · · 0
21
(11
)0 2
2
(20
)0
0 32
(21
)0 · · · 0
......
... . . . ...
0 n−1( n
2 )
( n2
n−22
)0 · · · 0
n
( n2 )
( n2n2
)0 n
( n+22 )
(n+2
2n−2
2
)· · · n
n
(n0
)
when N is even.
One can write the relation between T(t) and its derivatives T′(t) and T′′(t) as follows
T′(t) = T(t)B and T′′(t) = T(t)B2 (2.2)
where
B =
0 1 0 · · · 00 0 2 · · · 0...
...... . . . ...
0 0 0 · · · N0 0 0 · · · 0
and B0 =
1 0 0 · · · 00 1 0 · · · 0...
...... . . . ...
0 0 0 1 00 0 0 · · · 1
.
In a similar way, the approximate solution and its derivatives can be expressed by usingEq. (2.1) and (2.2) as
y(t) ∼= yN (t) = L(t)A = T(t)MA
y′(t) ∼= y′N (t) = T′(t)MA = T(t)BMA
y′′(t) ∼= y′′N (t) = T′′(t)MA = T(t)B2MA.
(2.3)
556 S. Gümgüm , N.B. Savaşaneril, Ö.K. Kürkçü, M. Sezer
Replacing t by t − τkj(t) in each equation in (2.3) yields the recurrence relation,
y(t − τkj(t)) ∼= yN (t − τkj(t)) = T(t − τkj(t))MA = T(t)S(−τkj(t))MA
y′(t − τkj(t)) ∼= y′N (t − τkj(t)) = T(t − τkj(t))BMA = T(t)S(−τkj(t))BMA
y′′(t − τkj(t)) ∼= y′′N (t − τkj(t)) = T(t − τkj(t))B2MA = T(t)S(−τkj(t))B2MA.
(2.4)
Note that, T(t − τkj(t)) = T(t)S(−τkj(t)) and
S(−τkj(t)) =
(00
)(−τkj(t))0
(10
)(−τkj(t))1 · · ·
(N0
)(−τkj(t))N
0(
11
)(−τkj(t))0 · · ·
(N1
)(−τkj(t))N−1
0 0 · · ·(
N2
)(−τkj(t))N−2
......
.... . .
...
0 0 · · ·(
NN
)(−τkj(t))0
.
In addition, we can obtain the matrix forms of (y(0)(t))2, y(1)(t)y(0)(t) and (y(1)(t))2
which appears in the nonlinear part of Eq. (1.1), by using Eq. (2.3) as
(y(0)(t)
)2= T(t)MT(t) M A
y(1)(t) y(0)(t) = T(t) BMT(t) M A
(y(1)(t)
)2= T(t)BMT(t) B M A
(2.5)
where
T(t) = diag [T(t)](N+1)×(N+1)2 , M = diag [M](N+1)2×(N+1)2 ,
B = diag [B](N+1)2×(N+1)2 , A =[
a0A a1A · · · aN A]T
Substituting the collocation points ( ti = a+(b−a)i/N, i = 0, 1, · · · , N) into Eq. (1.1),gives the system of equations
2∑k=0
1∑j=0
Pkj(ti) y(k)(ti − τkj(ti)) +1∑
p=0
p∑q=0
Rpq(ti) y(p)(ti)y(q)(ti) = g(ti),
which can be expressed with the aid of Eqs. (2.4) and (2.5) as
2∑k=0
1∑j=0
PkjT SkjBkMA +1∑
p=0
p∑q=0
RpqTpqA = G (2.6)
Lucas polynomial solution of nonlinear differential equations with variable delays 557
wherePkj = diag
[Pkj(t0) Pkj(t1) · · · Pkj(tN )
],
T =
T (t0)T (t1)
...T (tN )
=
1 t0 · · · tN
01 t1 · · · tN
1...
... . . . ...1 tN · · · tN
N
,
Skj =
S(−τkj(t0))S(−τkj(t1))
...S(−τkj(tN ))
, G =
g(t0)g(t1)
...g(tN )
,
Rpq = diag[
Rpq(t0) Rpq(t1) · · · Rpq(tN )],
Tpq =
T (t0)BpM T (t0) Bq MT (t1)BpM T (t1) Bq M
...T (tN )BpM T (tN ) Bq M
; p, q = 0, 1.
The fundamental matrix equation (2.6) can be briefly expressed in the form
WA + ZA = G (2.7)
where
W =2∑
k=0
1∑j=0
Pkj TSkjBkM = [wij ] ; i, j = 0, 1, · · · , N
Z =1∑
p=0
p∑q=0
RpqTpq = [zmn] ; m = 0, 1, · · · , N, n = 0, 1, · · · , (N + 1)2
G =[
g(t0) g(t1) · · · g(tN )]T
.
Also we can write the matrix equation (2.7) in the augmented form as
[W; Z; G] =
w00 w01 · · · w0N ; z00 z01 · · · z0(N+1)2 ; g(t0)w10 w11 · · · w1N ; z10 z11 · · · z1(N+1)2 ; g(t1)
......
. . .... ;
......
. . ....
. . ....
wN0 wN1 · · · wNN ; zN0 zN1 · · · zN(N+1)2 ; g(tN )
. (2.8)
Now, let us write the initial conditions y(a) = λ1 and y′(a) = λ2 in the matrix form byusing the relations in Eq. (2.3)
T(a)MA = λ1 and T(a)BMA = λ2
or brieflyU1A + O∗A = λ1 and U2A + O∗A = λ2, (2.9)
whereU1 =
[u00 u01 · · · u0N
]= T(a)M,
U2 =[
u10 u11 · · · u1N]
= T(a)BM,
O∗ =[
0 0 · · · 0].
558 S. Gümgüm , N.B. Savaşaneril, Ö.K. Kürkçü, M. Sezer
In order to find the unknown Lucas coefficients an, (n = 0, 1, · · · , N), related to theapproximate solution Eq. (1.2), we replace the row matrices in Eq. (2.9) by any rows ofthe augmented matrix in Eq. (2.8). Consequently, we obtain a new augmented matrix[
W ; Z ; G]
related to the matrix equation WA + Z A = G.
We solve this nonlinear algebraic system using NSolve routine in Mathematica, andobtain the unknown coefficients. Then, substitute them in Eq. (2.1) to obtain the approx-imate solution. A detailed theoretical convergence and error analysis of Lucas expansionof a function is given in [2,3]. Thus, rate of convergence is investigated numerically in thisstudy.
3. Examples and discussionIn this section, we apply the method to three problems to demonstrate the validity and
accuracy of the method. In the first problem, the application of the method yields theexact solution. In order to show the efficiency of the method for the next two problems,we compute the absolute errors for each collocation point ti as follows
EN (ti) = |y(ti) − yN (ti)|.
3.1. Example 1:Consider the first order nonlinear differential equation with variable delay t2:
y′(t) + ty(t − t2) + ty2(t) = 1 + t2, 0 ≤ t ≤ 1y(0) = 0
The exact solution of the above problem is y(t) = t. We aim to show that the exactsolution could be found using the present method.
First, we approximate the solution y(t) by the Lucas polynomial yN (t) =∑N
k=0 ak Lk(t),and formulate the problem in the form of Eq. (1.1). Here,
P10(t) = 1, P11(t) = 0, P00(t) = t, P01(t) = 0,
τ10(t) = 0, τ11(t) = 0, τ00(t) = t2, τ01(t) = 0,
R00(t) = t, R10(t) = 0, R11(t) = 0,
g(t) = 1 − t2
The collocation points are computed as
t0 = 0, t1 = 12 , t2 = 1
by taking N = 2.
Then, we write the fundamental matrix equation of the given problem as[P10T S10B1M + P00T S00B0M
]︸ ︷︷ ︸
W
A + R00T00︸ ︷︷ ︸Z
A = G.
where
Lucas polynomial solution of nonlinear differential equations with variable delays 559
P10 =
1 0 20 1 00 0 1
, T =
1 0 0 0 0 0 0 0 00 0 0 1 1
214 0 0 0
0 0 0 0 0 0 1 1 1
,
S10 =
1 0 00 1 00 0 11 0 00 1 00 0 11 0 00 1 00 0 1
, B1 =
0 1 00 0 20 0 0
, M =
2 0 20 1 00 0 1
,
P00 =
0 0 00 1
2 00 0 1
, S00 =
1 0 00 1 00 0 10 − 1
21116
0 1 −12
0 0 11 −1 10 1 −10 0 1
, R00 =
0 0 00 1
2 00 0 1
,
T00 =
4 0 4 0 0 0 4 2 64 0 4 1 1
498
92
94
274
4 0 4 2 12
94 6 3 9
, G =
1542
.
Now, we can calculate W and Z
W = P10T S10B1M + P00T S00B0M =
0 1 01 1.125 2.031252 1 4
,
Z = R00T00 =
0 0 0 0 0 0 0 0 02 0 2 1
218
916
94
98
278
4 0 4 2 12
94 6 3 9
.
Hence the augmented matrix [W; Z; G] can be written as
0 1 0 ; 0 0 0 0 0 0 0 0 0 ; 1
1 1.125 2.03125 ; 2 0 2 12
18
916
94
98
278 ; 5
4
2 1 4 ; 4 0 4 2 12
94 6 3 9 ; 2
.
The initial condition matrix is calculated as
U =[
2 0 2 ; 0 0 0 0 0 0 0 0 0 ; 0].
Replacing this row by the third row of the augmented matrix gives
560 S. Gümgüm , N.B. Savaşaneril, Ö.K. Kürkçü, M. Sezer
0 1 0 ; 0 0 0 0 0 0 0 0 0 ; 1
1 1.125 2.03125 ; 2 0 2 12
18
916
94
98
278 ; 5
4
2 0 2 ; 0 0 0 0 0 0 0 0 0 ; 0
.
Once we solve this system, we get the unknown coefficients as A =[
0 1 0]T ,
and hence we obtain the analytical solution
y(t) = L(t)A =[
2 t t2 + 2] 0
10
= t.
The present method is said to be accurate and efficient.
3.2. Example 2:The next example is a second order NDE with variable delays t2, − t
2y′′(t) + y′(t − t2) − t2 y(t + t
2) + (y′(t))2 − y′(t)y(t) = et + et−t2 − t2e3t/2
y(0) = y′(0) = 1, t ∈ [0, 1]
The analytical solution of this problem is y(t) = et. We solve the problem for severalvalues of N . Table 1 shows the absolute errors for N = 4, 5, 7 and 9. One can see thateven N = 4 yields an accuracy up to four decimal places. Increasing N decreases theabsolute error for each collocation point.
Table 1. Absolute errors for several values of N
ti E4(ti) E5(ti) E7(ti) E9(ti)0.2 5.12e − 06 2.05e − 06 2.79e − 09 1.86e − 120.4 2.44e − 05 6.22e − 06 5.53e − 09 3.56e − 120.6 1.77e − 04 1.37e − 05 8.75e − 09 4.95e − 120.8 2.08e − 04 1.85e − 05 4.84e − 09 8.55e − 111.0 9.90e − 04 9.09e − 05 5.35e − 07 4.78e − 09
Table 2 presents the convergence rate calculated by, [33]
RN = log[ ||y(t) − yN (t)||
||y(t) − yN+1(t)||
] 1log 2
, t ∈ [a, b].
We observe a cubic convergence for several values of N at the point t = 1.
Table 2. Convergence rate of the present method at t = 1.
N 4 5 6 7 8RN 3.4446 3.5060 3.9020 3.7218 3.0843
Figure 1 presents the analytical and approximate solution when N = 4. One can seethat the approximate solution agrees very well with the exact solution.
Lucas polynomial solution of nonlinear differential equations with variable delays 561
0.0 0.2 0.4 0.6 0.8 1.00.5
1.0
1.5
2.0
2.5
3.0
t
y
Exact solution y4(t)
Figure 1. Exact solution and approximate solution for N = 4.
3.3. Example 3:For the third example, we consider a second order NDE with variable delay t − t3/8,
[10] y′′(t) + 2y(t) − y2(t) + y( t3
8 ) = sin t − sin2 t + sin (t3/8), 0 ≤ t ≤ 1y(0) = 0, y′(0) = 1
The analytical solution of this problem is y(t) = sin t. Figure 2 presents the the exactand approximate solutions for N = 9, 10 and 11. One can see that the numerical solutionwith N = 11 agrees well with the exact solution.
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
t
y
Exact solution y11(t) y10(t) y9(t)
Figure 2. Comparison of analytical and numerical solutions for several values of N .
4. ConclusionIn this paper, nonlinear differential equations with variable delays are solved by Lucas
polynomial approach. The main advantage of the method is to convert the nonlinear
562 S. Gümgüm , N.B. Savaşaneril, Ö.K. Kürkçü, M. Sezer
equations to a system of nonlinear algebraic equations. The efficiency of the proposedmethod is tested on three problems. The results are presented in terms of absolute errorscalculated at each collocation point. It is observed that the method enables high accuracynumerical solutions or even analytical solution. Thus, we can say that this is an effectiveand convenient approach to solve the indicated type of problems.
References[1] W.M. Abd-Elhameed and Y.H. Youssri, A novel operational matrix of Caputo frac-
tional derivatives of Fibonacci polynomials: Spectral solutions of fractional differentialequations, Entropy, 18 (345), 2016.
[2] W.M. Abd-Elhameed and Y.H. Youssri, Spectral solutions for fractional differentialequations via a novel Lucas operational matrix of fractional derivatives, Rom. J. Phys.61 (5-6), 795–813, 2016.
[3] W.M. Abd-Elhameed and Y.H. Youssri, Generalized Lucas polynomial sequence ap-proach for fractional differential equations, Nonlinear Dyn. 89, 1341–1355, 2017.
[4] W.M. Abd-Elhameed and Y.H. Youssri, Spectral tau algorithm for certain coupled sys-tem of fractional differential equations via generalized Fibonacci polynomial sequence,Iran J. Sci. Technol. A, doi.org/10.1007/s40995-017-0420-9, 2017.
[5] K.S. Aboodh, R.A. Farah, I.A. Almardy and A.K. Osman, Solving delay differentialequations by Aboodh transformation method, Int. J. Appl. Math. Stats. Sci. 7 (1),21–30, 2018.
[6] I. Ali, H. Brunner, and T. Tang, A spectral method for pantograph-type delay dif-ferential equations and its convergence analysis, J. Comp. Math. 27 (2-3), 254–265,2009.
[7] A. Ardjouni and A. Djoudi, Fixed points and stability in linear neutral differentialequations with variable delays, Nonlinear Anal. 74, 2062–2070, 2011.
[8] A.G. Atta, G.M. Moatimid, and Y.H. Youssri, Generalized Fibonacci operational col-location approach for fractional initial value problems, Int. J. Appl. Comput. Math 5(9), 2019.
[9] N. Baykuş-Savaşaneril and M. Sezer, Hybrid Taylor-Lucas collocation method for nu-merical solution of high-order Pantograph type delay differential equations with vari-ables delays, Appl. Math. Inf. Sci. 11 (6), 1795–1801, 2017.
[10] B. Benhammouda and H. Vazquez-Leal, A new multi-step technique with differentialtransform method for analytical solution of some nonlinear variable delay differentialequations, SpringerPlus 5 (1723), 2016.
[11] B. Benhammouda, H. Vazquez-Leal and L. Hernandez-Martinez, Procedure for exactsolutions of nonlinear pantograph delay differential equations, British J. Math. Comp.Sci. 40 (19), 2738–2751, 2014.
[12] L. Blanco-Cocom, A.G. Estrella and E. Avila-Vales, Solving delay differential systemswith history functions by the Adomian decomposition method, Appl. Math. Comput.218, 5994–6011 2012.
[13] B. Caruntu and C. Bota, Analytical approximate solutions for a general class of non-linear delay differential equations, Scientific World J. 2014, (6 pp), 2014.
[14] G. Chen, L. Dingshi, O.V. Gaans and S.V. Lunel, Stability of nonlinear Neutral delaydifferential equations with variable delays, Electron. J. Differ. Eq. 2017 (118), 1–14,2017.
[15] M. Çetin, M. Sezer and C. Güler, Lucas polynomial approach for system of high-orderlinear differential equations and residual error estimation, Math. Prob. Eng. 2015,(14 pp), 2015.
[16] S. Davaeifar and J. Rashidinia, Solution of a system of delay differential equations ofmultipantograph type, J. Taibah Univ. Sci. 11, 1141–1157, 2017.
Lucas polynomial solution of nonlinear differential equations with variable delays 563
[17] L. Ding, X. Li, and Z. Li, Fixed points and stability in nonlinear equations withvariable delays, Fixed Point Theory Appl. 2010, (14pp), 2010.
[18] J.G. Dix, Asymptotic behavior of solutions to a first-order differential equation withvariable delays, Comp. Math. Appl. 50, 1791–1800, 2005.
[19] B. Dorociaková and R. Olach, Some notes to existence and stability of the positiveperiodic solutions for a delayed nonlinear differential equations, Open Math. 14, 361–369, 2016.
[20] S. Gümgüm, N. Baykuş-Savaşaneril, Ö.,K. Kürkçü and M. Sezer, A numerical tech-nique based on Lucas polynomials together with standard and Chebyshev-Lobatto col-location points for solving functional integro-differential equations involving variabledelays, Sakarya Univ. J. Sci. 22 (6), 2018.
[21] P. Ha, Analysis and numerical solutions of delay differential-slgebraic equations, PhDThesis, Berlin, 4-9, 2015.
[22] F. Ismail, R.A. Al-Khasawneh, A.S. Lwin and M. Suleiman, Numerical treatmentof delay differential equations by Runge-Kutta method using Hermite interpolation,Matematika 18 (2), 79–90, 2002.
[23] C. Jin and J. Luo, Fixed points and stability in neutral differential equations withvariable delay, Proc. Amer. Math. Soc. 136, 909–918, 2008.
[24] S. Karimi-Vanani and A. Aminataei, On the numerical solution of nonlinear delaydifferential equations, J. Concrete Applicable Math. 8 (4), 568–576, 2010.
[25] M.M. Khader, Numerical and theoretical treatment for solving linear and nonlineardelay differential equations using variational iteration method, Arab J. Math. Sci. 19(2), 243–256, 2013.
[26] E. Lucas, Theorie de fonctions numeriques simplement periodiques, Amer. J. Math.1, 184–240; 289–321, 1878.
[27] J.A. Martín and O. García, Variable multistep methods for delay differential equations,Math. Comput. Model. 35, 241–257, 2002.
[28] F. Mirzaee and L. Latifi, Numerical solution of delay differential equations by differ-ential transform method, J. Sci. I. A. U. 20 (78/2), 83–88, 2011.
[29] S.T. Mohyud-Dina and A. Yıldırım, Variational iteration method for delay differentialequations using He’s polynomials, Z. Naturforsch. 65a, 1045–1048, 2010.
[30] H.J. Oberle and H.J. Pesch, Numerical treatment of delay differential equations byHermite interpolation, Numer. Math. 37, 235–255, 1981.
[31] F.O. Ogunfiditimi, Numerical solution of delay differential equations using the Ado-mian decomposition method, Int. J. Eng. Sci. 4 (5), 18–23, 2015.
[32] M. Özel, M. Tarakçı and M. Sezer, A numerical approach for a nonhomogeneousdifferential equation with variable delays, Math. Sci. 12, 145–155, 2018.
[33] L. Pezza and F. Pitolli, A multiscale collocation method for fractional differentialproblems, Math. Comp. Simul. 174, 210–219, 2018.
[34] A.S.V. Ravi-Kanth and P. Murali-Mohan Kumar, A numerical technique for solvingnonlinear singularly perturbed delay differential equations, Math. Model. Anal. 23 (1),64–78, 2018.
[35] F. Shakeri and M. Dehghan, Solution of delay differential equations via a homotopyperturbation method, Math. Comput. Model. 48, 486–498, 2008.
[36] S.C. Shiralashetti, B.S. Hoogar and S. Kumbinarasaiah, Hermite wavelet based methodfor the numerical solution of linear and nonlinear delay differential equations, Int. J.Eng. Sci. Math. 6 (8), 71–79, 2017.
[37] A.G. Stephen and Y. Kuang, A stage structured predator-prey model and its depen-dence on maturation delay and death rate, J. Math. Biol. 49, 188-200, 2004.
[38] A.G. Stephen and Y. Kuang, A delay reaction-diffusion model of the spread of bacte-riophage infection, SIAM J. Appl. Math. 65 (2), 550–566, 2005.
564 S. Gümgüm , N.B. Savaşaneril, Ö.K. Kürkçü, M. Sezer
[39] O.A. Taiwo and O.S. Odetunde, On the numerical approximation of delay differentialequations by a decomposition method, Asian J. Math. Stat. 3 (4), 237–243, 2010.
[40] Z.Q. Wang and L.L. Wang, A Legendre-Gauss collocation method for nonlinear delaydifferential equations, Discrete Continuous Dyn. Syst. Ser. B 13 (3), 685–708, 2010.
[41] A. Yıldırım, H. Kocak, and S. Tutkun, Reliable analysis for delay differential equationsarising in mathematical biology, J. King Saud Univ. Sci. 24, 359–365, 2012.
[42] H.Y. Youssri, A new operational matrix of Caputo fractional derivatives of Fermatpolynomials: an application for solving the Bagley-Torvik equation, Adv. Diff. Eq.2017 (73), 2017.
[43] H.Y. Youssri and W.M. Abd-Elhameed, Spectral solutions for multi-term fractionalinitial value problems using a new Fibonacci operational matrix of fractional integra-tion, Progr. Fract. Differ. Appl. 2 (2), 141–151, 2016.
[44] B. Zhang, Fixed points and stability in differential equations with variable delays,Nonlinear Anal. 63, 233–242, 2005.
Hacettepe Journal ofMathematics & Statistics
Hacet. J. Math. Stat.Volume 49 (2) (2020), 565 – 577
DOI : 10.15672/hujms.568340
Research Article
Convolutions of the bi-periodic Fibonaccinumbers
Takao Komatsu1, José L. Ramírez∗2
1Department of Mathematics, School of Science, Zhejiang Sci-Tech University, Hangzhou, 310018, China2Departamento de Matemáticas, Universidad Nacional de Colombia, Bogotá, Colombia
AbstractLet qn be the bi-periodic Fibonacci numbers, defined by qn = c(n)qn−1 + qn−2 (n ≥ 2)with q0 = 0 and q1 = 1, where c(n) = a if n is even, c(n) = b if n is odd, where a andb are nonzero real numbers. When c(n) = a = b = 1, qn = Fn are Fibonacci numbers.In this paper, the convolution identities of order 2, 3 and 4 for the bi-periodic Fibonaccinumbers qn are given with binomial (or multinomial) coefficients, by using the symmetricformulas.Mathematics Subject Classification (2010). 11B39, 05A15, 05A19
Keywords. bi-periodic Fibonacci numbers, convolutions, symmetric formulas
1. IntroductionConvolution identities for various types of numbers (or polynomials) have been studied,
with or without binomial (or multinomial) coefficients, including Bernoulli, Euler, Genoc-chi, Cauchy, Stirling and balancing numbers (cf. [1–3,6,9,10,15,16,19]). A typical formulais due to Euler, given by
n∑k=0
(n
k
)BkBn−k = −nBn−1 − (n − 1)Bn (n ≥ 0) ,
where Bn are Bernoulli numbers, defined byx
ex − 1=
∞∑n=0
Bnxn
n!(|x| < 2π) .
On the other hand, many kinds of generalizations of Fibonacci numbers have beenpresented in the literature. A typical one is a generalized Fibonacci sequence Wn∞
n=0,defined by Wn = pWn−1 + qWn−2 (n ≥ 2) with W0 = a and W1 = b. In [5] some newidentities involving differences of products of generalized Fibonacci numbers are shown.One of different types is the bi-periodic Fibonacci sequence [7]. For any two nonzero realnumbers a and b, the bi-periodic Fibonacci sequence, say qn∞
n=0, is determined by:
q0 = 0, q1 = 1, qn =
aqn−1 + qn−2, if n ≡ 0 (mod 2);bqn−1 + qn−2, if n ≡ 1 (mod 2);
n > 2. (1.1)
∗Corresponding Author.Email addresses: [email protected] (T. Komatsu), [email protected] (J.L. Ramírez)Received: 16.08.2017; Accepted: 22.01.2019
566 T. Komatsu and J. L. Ramírez
When a = b = 1, qn = Fn are Fibonacci numbers. The explicit expression of the bi-periodic Fibonacci numbers can be expressed explicitly as
q2n = an−1∑k=0
(2n − k − 1
k
)(ab)n−k−1 ,
q2n+1 =n∑
k=0
(2n − k
k
)(ab)n−k .
Moreover, the ordinary generating function of the bi-periodic Fibonacci numbers isgiven by
F (x) :=∞∑
n=0qnxn = x(1 + ax − x2)
1 − (ab + 2)x2 + x4 .
For more properties about this sequence see for example [4, 7, 8, 18,20].Recently, in [13], the convolution identities of two Fibonacci numbers Fn are explicitly
given:n∑
k=0FkFn−k =
n−1∑m=0
mFm cos (n − m − 1)π2
as special cases of higher-order identities. In [14], this result is generalized by using a moregeneral form:
Fr + (−1)rFk−rx
1 − Lkx + (−1)kx2 =∞∑
n=0Fkn+rxn ,
with k > r ≥ 0, where Ln are Lucas numbers. In [11, 12, 17], convolution identities forFibonacci numbers are generalized as Tribonacci numbers and Tetranacci numbers. Inparticular, in [12,17], symmetric formulas are used to yield the results.
In this paper, motivated by the previous results, the convolution identities for the bi-periodic Fibonacci numbers qn are given with binomial (or multinomial) coefficients. In[15] the so-called exponential generating functions of generalized Fibonacci-type numbersun and Lucas-type numbers vn are considered:
eαx − eβx
√a2 + 4b
=∞∑
n=0un
xn
n!and eαx + eβx =
∞∑n=0
vnxn
n!.
Here, un = aun−1 +bun−2 (n ≥ 2) with u0 = 0 and u1 = 1, and vn = avn−1 +bvn−2 (n ≥ 2)with v0 = 2 and v1 = a. α and β are the roots of the quadratic equation x2 − ax − b = 0,given by
α = a +√
a2 + 4b
2and β = a −
√a2 + 4b
2.
Then the higher-order convolution identities with multinomial coefficients
∑k1+···+kr=n
k1,...,kr≥0
(n
k1, . . . , kr
)uk1 · · · ukr and
∑k1+···+kr=n
k1,...,kr≥0
(n
k1, . . . , kr
)vk1 · · · vkr
are given in the linear combinations of un and vn. We consider this kind of convolutionidentities for bi-periodic Fibonacci numbers.
This paper is organized as follows. In Section 2, convolution identities for two bi-periodic Fibonacci numbers with binomial coefficients are shown. In Section 3, convolutionidentities for three and four bi-periodic Fibonacci numbers with multinomial coefficientsare shown. The main tools are symmetric formulas which are often used in [12,17].
Convolutions of the bi-periodic Fibonacci numbers 567
2. Convolution identities with binomial coefficientsConsider the exponential generating function
f(x) :=∞∑
n=0qn
xn
n!.
We introduce two supplementary functions
f1(x) =∞∑
n=0q2n
x2n
(2n)!and f2(x) =
∞∑n=0
q2n+1x2n+1
(2n + 1)!
so that f(x) = f1(x) + f2(x). By using the recurrence relations (1.1), we have the systemof the differential equations:
f ′′1 (x) − af ′
2(x) − f1(x) = 0 , (2.1)f ′′
2 (x) − bf ′1(x) − f2(x) = 0 . (2.2)
Therefore, we get two 4-th order differential equations:
f(4)1 (x) − (ab + 2)f ′′
1 (x) + f1(x) = 0 , (2.3)
f(4)2 (x) − (ab + 2)f ′′
2 (x) + f2(x)) = 0 . (2.4)
Since the roots of x4 − (ab + 2)x2 + 1 = 0 are given by
±α = ±
√ab + 2 +
√ab(ab + 4)
2and ± β = ±
√ab + 2 −
√ab(ab + 4)
2,
the generating function f1(x) can be expressed as
f1(x) = c1eαx + c2e−αx + c3eβx + c4e−βx ,
where
c1 + c2 + c3 + c4 = q0 = 0 ,
c1α − c2α + c3β − c4β = 0 ,
c1α2 + c2α2 + c3β2 + c4β2 = q2 = a ,
c1α3 − c2α3 + c3β3 − c4β3 = 0 .
Solving this system, we get
c1 = c2 = a
2√
ab(ab + 4)and c3 = c4 = − a
2√
ab(ab + 4).
Note that αβ = 1, α2 + β2 = ab + 2 and α2 − β2 =√
ab(ab + 4).Similarly, the generating function f2(x) can be expressed as
f2(x) = d1eαx + d2e−αx + d3eβx + d4e−βx ,
where
d1 + d2 + d3 + d4 = 0 ,
d1α − d2α + d3β − d4β = q1 = 1 ,
d1α2 + d2α2 + d3β2 + d4β2 = 0 ,
d1α3 − d2α3 + d3β3 − d4β3 = q3 = ab + 1 .
568 T. Komatsu and J. L. Ramírez
Solving this system, we get
d1 = −d2 = ab + 4 +√
ab(ab + 4)
2√
2(ab + 4)√
ab + 2 +√
ab(ab + 4),
d3 = −d4 = ab + 4 −√
ab(ab + 4)
2√
2(ab + 4)√
ab + 2 −√
ab(ab + 4).
Therefore, we obtain thatf(x) = r1eαx + r2e−αx + r3eβx + r4e−βx ,
where
r1 = c1 + d1 = a
2√
ab(ab + 4)+ ab + 4 +
√ab(ab + 4)
2√
2(ab + 4)√
ab + 2 +√
ab(ab + 4),
r2 = c2 + d2 = a
2√
ab(ab + 4)− ab + 4 +
√ab(ab + 4)
2√
2(ab + 4)√
ab + 2 +√
ab(ab + 4),
r3 = c3 + d3 = − a
2√
ab(ab + 4)+ ab + 4 −
√ab(ab + 4)
2√
2(ab + 4)√
ab + 2 −√
ab(ab + 4),
r4 = c4 + d4 = − a
2√
ab(ab + 4)− ab + 4 −
√ab(ab + 4)
2√
2(ab + 4)√
ab + 2 −√
ab(ab + 4).
Now, we shall consider the sum of the product of two bi-periodic Fibonacci numbers.We need three Lemmas to get the main result.
Lemma 2.1.
r21e2αx + r2
2e−2αx + r23e2βx + r2
4e−2βx = 1b(ab + 4)
∞∑n=0
Qnxn
n!,
where Qn are numbers, satisfying for n ≥ 1Q2n = (a + b)Q2n−1 + 4Q2n−2 ,
Q2n+1 = 4ab
a + bQ2n + 4Q2n−1,
with Q0 = a + b and Q1 = 2ab.
Remark 2.2. We have explicit expressions: for n ≥ 0
Q2n = 22n−1(a + b)n∑
k=0
2n
2n − k
(2n − k
k
)(ab)n−k ,
Q2n+1 = 22n+1abn∑
k=0
2n + 12n − k + 1
(2n − k + 1
k
)(ab)n−k .
Proof of Lemma 2.1. Assume that the exponential generating functionr2
1e2αx + r22e−2αx + r2
3e2βx + r24e−2βx
determines the sequence Qn∞n=0. By αβ = 1 and α2 + β2 = ab + 2, we have the
characteristic equation(x + 2α)(x − 2α)(x + 2β)(x − 2β)= x4 − 4(α2 + β2)x2 + 16α2β2
= x4 − 4(ab + 2)x2 + 16 .
Convolutions of the bi-periodic Fibonacci numbers 569
Thus, Qn satisfies the recurrence relation
Qn = 4(ab + 2)Qn−2 − 16Qn−4 (n ≥ 4) . (2.5)
Now, we get that
Q0 = r21 + r2
2 + r23 + r2
4 = a + b
b(ab + 4),
Q1 = 2r21α − 2r2
2α + 2r23β − 2r2
4β = 2a
ab + 4,
Q2 = 4r21α2 + 4r2
2α2 + 4r23β2 + 4r2
4β2 = 2(a + b)(ab + 2)b(ab + 4)
,
Q3 = 8r21α3 − 8r2
2α3 + 8r23β3 − 8r2
4β3 = 8a(ab + 3)ab + 4
.
Using the recurrence relation (2.5), by induction, we have the recurrence relations: forn ≥ 1
Q2n = (a + b)Q2n−1 + 4Q2n−2 ,
Q2n+1 = 4ab
a + bQ2n + 4Q2n−1 .
Putting Qn = b(ab + 4)Qn, we get the desired result.
Lemma 2.3.
r1r3e(α+β)x + r2r4e−(α+β)x + r1r4e(α−β)x + r2r3e−(α−β)x = − 1b(ab + 4)
∞∑n=0
QQnxn
n!,
where for n ≥ 0
QQ2n = a − b
2((ab + 4)n − (ab)n)+ an+1bn ,
QQ2n+1 = (ab)n+1 .
Proof. Assume that the exponential generating function
r1r3e(α+β)x + r2r4e−(α+β)x + r1r4e(α−β)x + r2r3e−(α−β)x
determines the sequence QQn∞n=0. By α2 + β2 = ab + 2 and α2 − β2 =
√ab(ab + 4), we
have the characteristic equation(x + (α + β)
)(x − (α + β)
)(x + (α − β)
)(x − (α − β)
)= x4 − 2(α2 + β2)x2 + (α2 − β2)2
= x4 − 2(ab + 2)x2 + (ab)(ab + 4) .
Thus, QQn satisfies the recurrence relation
QQn = 2(ab + 2)QQn−2 − (ab)(ab + 4)QQn−4 (n ≥ 4) . (2.6)
570 T. Komatsu and J. L. Ramírez
Now, we get that
QQ0 = r1r3 + r2r4 + r1r4 + r2r3 = − a
b(ab + 4),
QQ1 = r1r3(α + β) − r2r4(α + β) + r1r4(α − β) − r2r3(α − β) = − a
ab + 4,
QQ2 = r1r3(α + β)2 + r2r4(α + β)2 + r1r4(α − β)2 + r2r3(α − β)2 = −2(a − b) + a2b
b(ab + 4),
QQ3 = r1r3(α + β)3 − r2r4(α + β)3 + r1r4(α − β)3 − r2r3(α − β)3 = − a2b
ab + 4.
Using the recurrence relation (2.6), by induction, we have for n ≥ 0
QQ2n = (b − a)(ab + 4)n − (a + b)(ab)n
2b(ab + 4),
QQ2n+1 = −an+1bn
ab + 4.
Putting QQn = −b(ab + 4)QQn, we get the desired result. Lemma 2.4.
r1r2 + r3r4 = a − b
2b(ab + 4).
Proof. Sincer1r2 = r3r4 = a
4b(ab + 4)− 1
4(ab + 4)= a − b
4b(ab + 4),
we get the result. Theorem 2.5. For n ≥ 1, we have
n∑k=0
(n
k
)qkqn−k = Qn − 2QQn
b(ab + 4). (2.7)
Proof. Since (a + b + c + d)2 = (a2 + b2 + c2 + d2) + 2(ab + ac + ad + bc + bd + cd), byLemmas 2.1, 2.3 and 2.4 we have
(r1eαx + r2e−αx + r3eβx + r4e−βx)2
= (r21e2αx + r2
2e−2αx + r23e2βx + r2
4e−2βx)
+ 2(r1r3e(α+β)x + r2r4e−(α+β)x + r1r4e(α−β)x + r2r3e−(α−β)x)+ 2(r1r2 + r3r4)
= 1b(ab + 4)
∞∑n=0
Qnxn
n!− 2
b(ab + 4)
∞∑n=0
QQnxn
n!+ a − b
b(ab + 4)
= 1b(ab + 4)
( ∞∑n=0
(Qn − 2QQn)xn
n!+ (a − b)
).
Since
f(x)2 =∞∑
n=0
n∑k=0
(n
k
)qkqn−k
xn
n!,
by comparing the coefficients on both sides, we get the desired result. Examples. For n = 1, 2, 3, 4 in Theorem 2.5, both sides of (2.7) equal to 0, 2, 6a,6a2 + 8ab + 8, respectively.
Convolutions of the bi-periodic Fibonacci numbers 571
3. Higher-order identitiesNext, we shall consider the convolution identities for three bi-periodic Fibonacci num-
bers ∑k1+k2+k3=n
k1,k2,k3≥0
(n
k1, k2, k3
)qk1qk2qk3 .
We use the following symmetric formula.
Lemma 3.1 ([17]). The following equality holds:
(a + b + c + d)3
= A(a3 + b3 + c3 + d3) + B(abc + abd + acd + bcd)
+ C(a2 + b2 + c2 + d2)(a + b + c + d)
+ D(ab + ac + ad + bc + bd + cd)(a + b + c + d),
where A = D − 2, B = −3D + 6 and C = −D + 3.
Lemma 3.2.
r31e3αx + r3
2e−3αx + r33e3βx + r3
4e−3βx = 14b(ab + 4)
∞∑n=0
Pnxn
n!,
where the numbers Pn satisfy for n ≥ 1
P2n = 3a(a + 3b)3a + b
P2n−1 + 9P2n−2 ,
P2n+1 = 3b(3a + b)a + 3b
P2n + 9P2n−1
with P0 = 0 and P1 = 3(3a + b).
Remark 3.3. We have explicit expressions: for n ≥ 0
P2n = 32na(a + 3b)n−1∑k=0
(2n − k − 1
k
)(ab)n−k−1 = 32n(a + 3b)q2n ,
P2n+1 = 32n+1(3a + b)n∑
k=0
(2n − k
k
)(ab)n−k = 32n+1(3a + b)q2n+1 .
Proof of Lemma 3.2. Assume that the exponential generating function
r31e3αx + r3
2e−3αx + r33e3βx + r3
4e−3βx
determines the sequence Pn∞n=0. By αβ = 1 and α2 + β2 = ab + 2, we have the
characteristic equation
(x + 3α)(x − 3α)(x + 3β)(x − 3β)= x4 − 9(α2 + β2)x2 + 81α2β2
= x4 − 9(ab + 2)x2 + 81 .
Thus, Pn satisfies the recurrence relation
Pn = 9(ab + 2)Pn−2 − 81Pn−4 (n ≥ 4) . (3.1)
572 T. Komatsu and J. L. Ramírez
Now, we get that
P0 = r31 + r3
2 + r33 + r3
4 = 0 ,
P1 = 3r31α − 3r3
2α + 3r33β − 3r3
4β = 3(3a + b)4b(ab + 4)
,
P2 = 9r31α2 + 9r3
2α2 + 9r33β2 + 9r3
4β2 = 9a(a + 3b)4b(ab + 4)
,
P3 = 27r31α3 − 27r3
2α3 + 27r33β3 − 27r3
4β3 = 27(3a + b)(ab + 1)4b(ab + 4)
.
Using the recurrence relation (3.1), by induction, we have the recurrence relations: forn ≥ 1
P2n = 3a(a + 3b)3a + b
P2n−1 + 9P2n−2 ,
P2n+1 = 3b(3a + b)a + 3b
P2n + 9P2n−1 .
Putting Pn = 4b(ab + 4)Pn, we get the desired result.
Lemma 3.4.
r1r3r4eαx + r2r3r4e−αx + r1r2r3eβx + r1r2r4e−βx = a − b
4b(ab + 4)
∞∑n=0
qnxn
n!,
Proof. Since
r1r2 = r3r4 = a − b
4b(ab + 4),
we get the result.
Theorem 3.5. For n ≥ 1, we have∑k1+k2+k3=n
k1,k2,k3≥0
(n
k1, k2, k3
)qk1qk2qk3
= 14b(ab + 4)
(A · Pn + B(a − b)qn + 4C
n∑k=0
(n
k
)Qkqn−k
−4Dn∑
k=0
(n
k
)QQkqn−k + 2(a − b)Dqn
), (3.2)
where the numbers A, B, C and D satisfy the condition in Lemma 3.1.
Remark 3.6. It is clear that this value is 0 when n = 0, 1, 2. Assume that n ≥ 3. WhenD = 2, by A = B = 0 and C = 1, we have a simpler form:∑
k1+k2+k3=nk1,k2,k3≥0
(n
k1, k2, k3
)qk1qk2qk3
= 1b(ab + 4)
(n∑
k=0
(n
k
)(Qk − 2QQk)qn−k + (a − b)qn
)
= 1b(ab + 4)
n−1∑k=2
(n
k
)(Qk − 2QQk)qn−k .
Notice that Q0 − 2QQ0 = −(a − b), Q1 − 2QQ1 = 0 and q0 = 0.
Convolutions of the bi-periodic Fibonacci numbers 573
Proof of Theorem 3.5. By Lemma 3.1 together with Lemmas 2.1, 2.3, 2.4, 3.2, 3.4, wehave
(r1eαx + r2e−αx + r3eβx + r4e−βx)3
= A(r31e3αx + r3
2e−3αx + r33e3βx + r3
4e−3βx)
+ B(r1r2r3eβx + r1r2r4e−βx + r1r3r4eαx + r2r3r4e−αx)
+ C(r21e2αx + r2
2e−2αx + r23e2βx + r2
4e−2βx)(r1eαx + r2e−αx + r3eβx + r4e−βx)
+ D(r1r2 + r1r3e(α+β)x + r1r4e(α−β)x + r2r3e−(α−β)x + r2r4e−(α+β)x + r3r4)
× (r1eαx + r2e−αx + r3eβx + r4e−βx)
= A1
4b(ab + 4)
∞∑n=0
Pnxn
n!+ B
a − b
4b(ab + 4)
∞∑n=0
qnxn
n!
+ C
(1
b(ab + 4)
∞∑n=0
Qnxn
n!
)( ∞∑n=0
qnxn
n!
)
+ D
(− 1
b(ab + 4)
∞∑n=0
QQnxn
n!+ a − b
2b(ab + 4)
)( ∞∑n=0
qnxn
n!
)
= 14b(ab + 4)
∞∑n=0
(A · Pn + B(a − b)qn + 4C
n∑k=0
(n
k
)Qkqn−k
−4Dn∑
k=0
(n
k
)QQkqn−k + 2(a − b)Dqn
)xn
n!.
On the other hand,( ∞∑n=0
qnxn
n!
)3
=∞∑
n=0
∑k1+k2+k3=n
k1,k2,k3≥0
(n
k1, k2, k3
)qk1qk2qk3
xn
n!.
Comparing the coefficients on both sides, we get the desired result.
Next, we shall consider the convolution identities for four bi-periodic Fibonacci numbers∑k1+k2+k3+k4=n
k1,k2,k3,k4≥0
(n
k1, k2, k3, k4
)qk1qk2qk3qk4 .
We need the following symmetric formula.
Lemma 3.7 ([17]). The following equality holds:
(a + b + c + d)4
= A(a4 + b4 + c4 + d4) + Babcd + C(a3 + b3 + c3 + d3)(a + b + c + d)
+ D(a2 + b2 + c2 + d2)2 + E(a2 + b2 + c2 + d2)(ab + ac + ad + bc + bd + cd)
+ F (ab + ac + ad + bc + bd + cd)2 + G(a2 + b2 + c2 + d2)(a + b + c + d)2
+ H(ab + ac + ad + bc + bd + cd)(a + b + c + d)2
+ I(abc(a + b + c) + abd(a + b + d) + bcd(b + c + d) + acd(a + c + d))
+ J(abc + abd + bcd + acd)(a + b + c + d) ,
where A = −D + E + G + H − 3, B = 12D + 12G − 4J − 12, C = −E − 2G − H + 4,F = −2D − 2G − 2H + 6 and I = 4D − E + 2G − H − J .
574 T. Komatsu and J. L. Ramírez
Lemma 3.8.
r41e4αx + r4
2e−4αx + r43e4βx + r4
4e−4βx = 14b2(ab + 4)2
∞∑n=0
Rnxn
n!,
where the numbers Rn satisfy for n ≥ 1
R2n = a2 + 6ab + b2
a + bR2n−1 + 24R2n−2 ,
R2n+1 = 24ab(a + b)a2 + 6ab + b2 R2n + 24R2n−1
with R0 = a2 + 6ab + b2 and R1 = 8ab(a + b).
Remark 3.9. We have explicit expressions: for n ≥ 0
R2n = 24n−1(a2 + 6ab + b2)n∑
k=0
2n
2n − k
(2n − k
k
)(ab)n−k ,
P2n+1 = 24n+3ab(a + b)n∑
k=0
2n + 12n − k + 1
(2n − k + 1
k
)(ab)n−k .
Theorem 3.10. For n ≥ 1∑k1+k2+k3+k4=n
k1,k2,k3,k4≥0
(n
k1, k2, k3, k4
)qk1qk2qk3qk4
= A
4b2(ab + 4)2 Rn + C
4b(ab + 4)
n∑k=0
(n
k
)Pkqn−k + D
b2(ab + 4)2
n∑k=0
(n
k
)QkQn−k
+ E
(1
b2(ab + 4)2
n∑k=0
(n
k
)QkQn−k + a − b
2b2(ab + 4)2 Qn
)
+ F
(1
b2(ab + 4)2
n∑k=0
(n
k
)QQkQQn−k + a − b
b2(ab + 4)2 QQn
)
+ G
b(ab + 4)∑
k1+k2+k3=nk1,k2,k3≥0
(n
k1, k2, k3
)Qk1qk2qk3
+ H
1b(ab + 4)
∑k1+k2+k3=n
k1,k2,k3≥0
(n
k1, k2, k3
)QQk1qk2qk3 + a − b
2b(ab + 4)
(n
k
)QkQn−k
+ I
a − b
4b2(ab + 4)2 (Qn − QQn) + J
(1
b(ab + 4)
n∑k=0
QQkqn−k + a − b
2b(ab + 4)qn
),
where the numbers A, C, D, E, F , G, H, I and J are given in Lemma 3.7.
Remark 3.11. The above form becomes much simpler for some specific values of thenumbers A to J . For example, when A = C = G = H = I = 0, by B = J = 0, D = 1 andE = F = 4, we have∑
k1+k2+k3+k4=nk1,k2,k3,k4≥0
(n
k1, k2, k3, k4
)qk1qk2qk3qk4
= 1b2(ab + 4)2
(2(a − b)(Qn + 2QQn) +
n∑k=0
(n
k
)(5QkQn−k + 4QQkQQn−k)
).
Convolutions of the bi-periodic Fibonacci numbers 575
Proof of Theorem 3.10. We apply Lemma 3.7 as a = r1eαx, b = r2e−αx, c = r3eβx andd = r4e−βx. By Lemma 3.8, we have
A(a4 + b4 + c4 + d4) = A1
4b2(ab + 4)2
∞∑n=0
Rnxn
n!.
By r1r2 = r3r4 = B(a − b)/4b(ab + 4),
Babcd = (a − b)2
16b2(ab + 4)2 .
By Lemma 3.2,
C(a3 + b3 + c3 + d3)(a + b + c + d)
= C
(1
4b(ab + 4)
∞∑n=0
Pnxn
n!
)( ∞∑n=0
qnxn
n!
)
= C1
4b(ab + 4)
∞∑n=0
n∑k=0
(n
k
)Pkqn−k
xn
n!.
By Lemma 2.1
D(a2 + b2 + c2 + d2)2
= D
(1
b(ab + 4)
∞∑n=0
Qnxn
n!
)2
= D1
b2(ab + 4)2
∞∑n=0
n∑k=0
(n
k
)QkQn−k
xn
n!.
By Lemma 2.3 and Lemma 2.4,
E(a2 + b2 + c2 + d2)(ab + ac + ad + bc + bd + cd)
= E
(1
b(ab + 4)
∞∑n=0
Qnxn
n!
)(1
b(ab + 4)
∞∑n=0
QQnxn
n!+ a − b
2b(ab + 4)
)
= E
(1
b2(ab + 4)2
∞∑n=0
n∑k=0
(n
k
)QkQn−k
xn
n!+ a − b
2b2(ab + 4)2
∞∑n=0
Qnxn
n!
).
By Lemma 2.3 and Lemma 2.4,
F (ab + ac + ad + bc + bd + cd)2
= F
(1
b(ab + 4)
∞∑n=0
QQnxn
n!+ a − b
2b(ab + 4)
)2
= F
(1
b2(ab + 4)2
∞∑n=0
n∑k=0
(n
k
)QQkQQn−k
xn
n!+ a − b
b2(ab + 4)2
∞∑n=0
QQnxn
n!+ (a − b)2
4b2(ab + 4)2
).
By Lemma 2.1,
G(a2 + b2 + c2 + d2)(a + b + c + d)2
= G
(1
b(ab + 4)
∞∑n=0
Qnxn
n!
)( ∞∑n=0
qnxn
n!
)2
= G1
b(ab + 4)
∞∑n=0
∑k1+k2+k3=n
k1,k2,k3≥0
(n
k1, k2, k3
)Qk1qk2qk3
xn
n!.
576 T. Komatsu and J. L. Ramírez
By Lemma 2.3 and Lemma 2.4,
H(ab + ac + ad + bc + bd + cd)(a + b + c + d)2
= H
(1
b(ab + 4)
∞∑n=0
QQnxn
n!+ a − b
2b(ab + 4)
)( ∞∑n=0
qnxn
n!
)2
= H
1b(ab + 4)
∞∑n=0
∑k1+k2+k3=n
k1,k2,k3≥0
(n
k1, k2, k3
)QQk1qk2qk3
xn
n!
+ a − b
2b(ab + 4)
∞∑n=0
n∑k=0
(n
k
)QkQn−k
xn
n!
).
By Lemma 2.1 and Lemma 2.3 with r1r2 = r3r4 = (a − b)/4b(ab + 4),
I(abc(a + b + c) + abd(a + b + d) + bcd(b + c + d) + acd(a + c + d))
= Ia − b
4b(ab + 4)(r1r3e(α+β)x + r2r4e−(α+β)x + r1r4e(α−β)x + r2r3e−(α−β)x)
+ I(r21e2αx + r2
2e−2αx + r23e2βx + r2
4e−2βx)
= −Ia − b
4b(ab + 4)1
b(ab + 4)
∞∑n=0
QQnxn
n!+ I
a − b
4b(ab + 4)1
b(ab + 4)
∞∑n=0
Qnxn
n!
= Ia − b
4b2(ab + 4)2
∞∑n=0
(Qn − QQn)xn
n!.
By Lemma 2.3 and Lemma 2.4,
J(abc + abd + bcd + acd)(a + b + c + d)
= J
(1
b(ab + 4)
∞∑n=0
QQnxn
n!+ a − b
2b(ab + 4)
)( ∞∑n=0
qnxn
n!
)
= J
(1
b(ab + 4)
∞∑n=0
n∑k=0
QQkqn−kxn
n!+ a − b
2b(ab + 4)
∞∑n=0
qnxn
n!
).
We combine all the relations to get the main result.
4. Final remarksOne can continue to get the convolution identities of five and more bi-periodic Fibonacci
numbers. The situation becomes more complicated. Even in the case of five bi-periodicFibonacci numbers, we need the symmetric formula for (a + b + c + d)5 (see [12,17]).
Acknowledgment. The authors thank the anonymous referee for careful reading ofthe manuscript and helpful comments and suggestions. The research of José L. Ramírezwas partially supported by Universidad Nacional de Colombia, Project No. 37805.
References[1] T. Agoh and K. Dilcher, Convolution identities and lacunary recurrences for Bernoulli
numbers, J. Number Theory, 124, 105–122, 2007.[2] T. Agoh and K. Dilcher, Higher-order recurrences for Bernoulli numbers, J. Number
Theory, 129, 1837–1847, 2009.[3] T. Agoh and K. Dilcher, Higher-order convolutions for Bernoulli and Euler polyno-
mials, J. Math. Anal. Appl. 419, 1235–1247, 2014.
Convolutions of the bi-periodic Fibonacci numbers 577
[4] M. Alp, N. Irmak and L. Szalay, Two-Periodic ternary recurrences and their Binet-formula, Acta Math. Univ. Comenianae 2, 227–232, 2012.
[5] C. Cooper, Some identities involving differences of products of generalized Fibonaccinumbers, Colloq. Math. 141 (1), 45–49, 2015.
[6] K. Dilcher and C. Vignat, General convolution identities for Bernoulli and Eulerpolynomials, J. Math. Anal. Appl. 435, 1478–1498, 2016.
[7] M. Edson and O. Yayenie, A new generalization of Fibonacci sequences and extendedBinet’s Formula, Integers, 9 (A48), 639-654, 2009.
[8] N. Irmak and L. Szalay, On k-periodic binary recurrences, Ann. Math. Inform. 40,25–35, 2012.
[9] T. Komatsu, Higher-order convolution identities for Cauchy numbers of the secondkind, Proc. Jangjeon Math. Soc. 18, 369–383, 2015.
[10] T. Komatsu, Higher-order convolution identities for Cauchy numbers, Tokyo J. Math.39, 225–239, 2016.
[11] T. Komatsu, Convolution identities for Tribonacci numbers, Ars Combin. 136, 199–210, 2018.
[12] T. Komatsu and R. Li, Convolution identities for Tribonacci numbers with symmetricformulae, Math. Rep. (Bucur.) 21 (1), 27-47, 2019, arXiv:1610.02559.
[13] T. Komatsu, Z. Masakova and E. Pelantova, Higher-order identities for Fibonaccinumbers, Fibonacci Quart. 52 (5), 150-163, 2014.
[14] T. Komatsu and G.K. Panda, On several kinds of sums involving balancing and Lucas-balancing numbers, Ars Combin. (to appear). arXiv:1608.05918.
[15] T. Komatsu and P.K. Ray, Higher-order identities for balancing numbers,arXiv:1608.05925, 2016.
[16] T. Komatsu and Y. Simsek, Third and higher order convolution identities for Cauchynumbers, Filomat 30, 1053–1060, 2016.
[17] R. Li, Convolution identities for Tetranacci numbers, arXiv:1609.05272.[18] J.L. Ramírez, Bi-periodic incomplete Fibonacci sequences, Ann. Math. Inform. 42,
83–92, 2013.[19] W. Wang, Some results on sums of products of Bernoulli polynomials and Euler
polynomials, Ramanujan J. 32, 159–186, 2013.[20] O. Yayenie, A note on generalized Fibonacci sequence, Applied. Math. Comp. 217
(12), 5603–5611, 2011.
Hacettepe Journal ofMathematics & Statistics
Hacet. J. Math. Stat.Volume 49 (2) (2020), 578 – 585
DOI : 10.15672/hujms.568378
Research Article
On centrally-extended multiplicative(generalized)-(α, β)-derivations in semiprime rings
Najat Muthana1,2, Zakeiah Alkhamisi∗2
1Department of Mathematics, Faculty of Science, University of Jeddah, Jeddah, Saudia Arabia2Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudia Arabia
AbstractLet R be a ring with center Z and α, β and d mappings of R. A mapping F of R iscalled a centrally-extended multiplicative (generalized)-(α, β)-derivation associated withd if F (xy) − F (x)α(y) − β(x)d(y) ∈ Z for all x, y ∈ R. The objective of the present paperis to study the following conditions: (i) F (xy) ± β(x)G(y) ∈ Z, (ii) F (xy) ± g(x)α(y) ∈ Zand (iii) F (xy) ± g(y)α(x) ∈ Z for all x, y in some appropriate subsets of R, where G is amultiplicative (generalized)-(α, β)-derivation of R associated with the map g on R.
Mathematics Subject Classification (2010). 16N60,16W10
Keywords. semiprime ring, left ideal, multiplicative (generalized)-derivation,multiplicative (generalized)-(α, β)-derivation, centrally-extended generalized(α, β)-derivation, centrally-extended multiplicative (generalized)-(α, β)-derivation,generalized (α, β)-derivation
1. IntroductionThroughout this work R will be a ring with center Z. Recall that a ring R is said to
be semiprime if aRa = 0 then a = 0. For x, y ∈ R, the commutator xy − yx and the anti-commutator xy + yx will be written as [x, y] and (x y) respectively. For given x, y ∈ R,put [x, y]0 = x, then [x, y]k = [[x, y]k−1, y] for integer k ≥ 1. Let S be a nonempty subsetof R and α a mapping of R. If α(xy) = α(x)α(y) or α(xy) = α(y)α(x) for all x, y ∈ S, thenwe say that α acts as homomorphism or anti-homomorphism on S, respectively. A map f :S → R is said to be α-commuting on S in case [α(x), f(x)] = 0 satisfies for all x ∈ S. Wewill make some extensive use of the basic commutator identities [x, yz] = [x, y]z + y[x, z]and [xy, z] = [x, z]y + x[y, z].
Let α and β be mappings of R. A map D on R is called an (α, β)-derivation of R ifit is additive and satisfying D(xy) = D(x)α(y) + β(x)D(y), for all x, y ∈ R. Let D bean (α, β)-derivation of R, a map F on R is called a generalized (α, β)-derivation if it isadditive and satisfying F (xy) = F (x)α(y) + β(x)D(y) for all x, y ∈ R.
Recently, Bell and Daif [2] introduced the notion of centrally-extended derivations (CE-derivation) on rings. A CE-derivation D of R is a mapping of R such that D(x+y)−D(x)−D(y) ∈ Z and D(xy)−D(x)y−xD(y) ∈ Z, for all x, y in R. Tammam et al. [5] generalizedthis notion to the concepts CE-(α, β)-derivation and CE-generalized (α, β)-derivation.∗Corresponding Author.Email addresses: [email protected] (N. Muthana), [email protected] (Z. Alkhamisi)Received: 14.01.2017; Accepted: 23.01.2019
On centrally-extended multiplicative (generalized)-(α, β)-derivations in semiprime rings 579
A CE-(α, β)-derivation D of R is a mapping of R such that D(x+y)−D(x)−D(y) ∈ Z andD(xy)−D(x)α(y)−β(x)D(y) ∈ Z hold for all x, y ∈ R. Let D be a CE-(α, β)-derivation ofR, a map F on R is called a CE-generalized (α, β)-derivation if F (x+y)−F (x)−F (y) ∈ Zand F (xy) − F (x)α(y) − β(x)D(y) ∈ Z are fulfilled for all x, y ∈ R.
A map F on R is said to be a multiplicative (generalized)-(α, β)-derivation (M-(generalized)-(α, β)-derivation) associated with a map d on R if F (xy) = F (x)α(y) + β(x)d(y) holdsfor all x, y ∈ R. According to [3], an M-(generalized)-(I, I)-derivation is simply called anM-(generalized)-derivation, where I is the identity map on R.
We begin by the following definitions.
Definition 1.1. Let R be a ring and α be a mapping of R. A map T on R is calleda centrally-extended multiplicative left α-centralizer (CEM-left α-centralizer) if T (xy) −T (x)α(y) ∈ Z holds for all x, y ∈ R.
Definition 1.2. Let R be a ring and α, β be mappings of R. A map D on R is called aCEM-(α, β)-derivation if D(xy) − D(x)α(y) − β(x)D(y) ∈ Z holds for all x, y ∈ R.
Definition 1.3. Let R be a ring and α, β and d be mappings of R. A map F on R is calleda CEM-(generalized)-(α, β)-derivation associated with d if F (xy)−F (x)α(y)−β(x)d(y) ∈Z holds for all x, y ∈ R.
Hence the concept of CEM-(generalized)-(α, β)-derivation covers both the concept ofCEM-(α, β)-derivation and the concept of CEM-left α-centralizer. Moreover, every CE-generalized (α, β)-derivation is a CEM-(generalized)-(α, β)-derivation and every M-(generalized)-(α, β)-derivation is a CEM-(generalized)-(α, β)-derivation. Also, every generalized (α, β)-derivation is an M-(generalized)-(α, β)-derivation.
In this paper, our aim is to investigate certain identities involving CEM-(generalized)-(α, β)-derivations on some appropriate subsets of the ring R.
2. PreliminariesWe shall require throughout this paper to the following results.
Lemma 2.1. Let R be a semiprime ring, U a left ideal of R and α, β mappings of R suchthat β(U) ⊆ U . If either [xyα(z), β(z)] = 0 or x[yα(z), β(z)] = 0 holds for all x, y, z ∈ U ,then U [α(z), β(z)] = (0) for all z ∈ U .
Proof. First assume that[xyα(z), β(z)] = 0 for all x, y, z ∈ U. (2.1)
Substituting rx for x in (2.1), where r ∈ R, and then using (2.1), we obtain[r, β(z)]xyα(z) = 0 for all x, y, z ∈ U, r ∈ R. (2.2)
Replacing x by α(z)sx, where s ∈ R, we get [r, β(z)]α(z)sxyα(z) = 0, which implies[r, β(z)]α(z)Rxyα(z) = (0) for all x, y, z ∈ U, r ∈ R. (2.3)
Interchanging x and y then subtracting one from the other, we get[r, β(z)]α(z)R[x, y]α(z) = (0) for all x, y, z ∈ U, r ∈ R. (2.4)
In particular,[x, β(z)]α(z)R[x, β(z)]α(z) = (0) for all x, z ∈ U. (2.5)
The semiprimeness of R yields that[x, β(z)]α(z) = 0 for all x, z ∈ U. (2.6)
Right multiplying (2.6) by β(z), we get[x, β(z)]α(z)β(z) = 0 for all x, z ∈ U. (2.7)
580 N. Muthana, Z. Alkhamisi
Replace x by xβ(z) in (2.6) to get[x, β(z)]β(z)α(z) = 0 for all x, z ∈ U. (2.8)
Now (2.7) and (2.8) together imply that[x, β(z)][α(z), β(z)] = 0 for all x, z ∈ U. (2.9)
Replacing x by α(z)x in the last expression, we obtain[α(z), β(z)]x[α(z), β(z)] = 0 for all x, z ∈ U, (2.10)
that isU [α(z), β(z)]RU [α(z), β(z)] = (0) for all z ∈ U. (2.11)
Hence, since R is a semiprime ring,U [α(z), β(z)] = (0) for all z ∈ U. (2.12)
Now suppose thatx[yα(z), β(z)] = 0 for all x, y, z ∈ U. (2.13)
Replacing y with α(z)y in (2.13), we getx[α(z)yα(z), β(z)] = 0 for all x, y, z ∈ U. (2.14)
Now replacing y by yα(z)u, where u ∈ U , we havex[α(z)yα(z)uα(z), β(z)] = 0 for all x, y, z, u ∈ U. (2.15)
This impliesx[α(z)yα(z), β(z)]uα(z) + xα(z)yα(z)[uα(z), β(z)] = 0 for all x, y, z, u ∈ U. (2.16)
Then using (2.14), we obtainxα(z)yα(z)[uα(z), β(z)] = 0 for all x, y, z, u ∈ U, (2.17)
and this is equivalent toxα(z)y([α(z)uα(z), β(z)] − [α(z), β(z)]uα(z)) = 0 for all x, y, z, u ∈ U. (2.18)
Again, using (2.14), we getxα(z)y[α(z), β(z)]uα(z) = 0 for all x, y, z, u ∈ U. (2.19)
Replacing y with β(z)y in (2.19), we obtainxα(z)β(z)y[α(z), β(z)]uα(z) = 0 for all x, y, z, u ∈ U. (2.20)
Now replace x by xβ(z) in (2.19) to getxβ(z)α(z)y[α(z), β(z)]uα(z) = 0 for all x, y, z, u ∈ U. (2.21)
Subtracting (2.21) from (2.20), we havex[α(z), β(z)]y[α(z), β(z)]uα(z) = 0 for all x, y, z, u ∈ U. (2.22)
Right multiplying (2.22) by β(z), we getx[α(z), β(z)]y[α(z), β(z)]uα(z)β(z) = 0 for all x, y, z, u ∈ U. (2.23)
Replacing u by uβ(z) in (2.22), we obtainx[α(z), β(z)]y[α(z), β(z)]uβ(z)α(z) = 0 for all x, y, z, u ∈ U. (2.24)
Now (2.23) and (2,24) together imply thatx[α(z), β(z)]y[α(z), β(z)]u[α(z), β(z)] = 0 for all x, y, z, u ∈ U, (2.25)
that is(U [α(z), β(z)])3 = (0) for all z ∈ U. (2.26)
Since a semiprime ring contains no nonzero nilpotent left ideals, it follows thatU [α(z), β(z)] = (0) for all z ∈ U. (2.27)
On centrally-extended multiplicative (generalized)-(α, β)-derivations in semiprime rings 581
Lemma 2.2. Let R be a semiprime ring and U a left ideal of R. If [y[x, z]2, z] = 0 for allx, y, z ∈ U , then U [U, U ] = (0).
Proof. By the hypothesis, we have[y[x, z]2, z] = 0 for all x, y, z ∈ U. (2.28)
Substituting y by xy in (2.28) and then using (2.28), we obtain[x, z]y[x, z]2 = 0 for all x, y, z ∈ U, (2.29)
that isU [x, z]2RU [x, z]2 = (0) for all x, z ∈ U. (2.30)
The semiprimeness of R forces thatU [x, z]2 = (0) for all x, z ∈ U. (2.31)
Linearizing (2.31) with respect to z, we haveU([[x, u], v] + [[x, v], u]) = (0) for all x, u, v ∈ U. (2.32)
Replacing u with uv in (2.32), then using (2.31) and (2.32) to getU [u, v][x, v] = (0) for all x, u, v ∈ U. (2.33)
Now substituting x by xu, we obtainU [u, v]x[u, v] = (0) for all x, u, v ∈ U, (2.34)
that isU [u, v]RU [u, v] = (0) for all u, v ∈ U, (2.35)
Hence, the semiprimeness of R yields that U [U, U ] = (0). Lemma 2.3 ([4], Theorem 2). Let R be a semiprime ring and U a nonzero left ideal ofR. For integers n, k ≥ 1, and some a ∈ R, if [a, xk]n = 0 for all x ∈ U , then [a, U ] = (0).
3. The resultsTheorem 3.1. Let R be a semiprime ring, U a left ideal of R, α, β, d and g mappings of R,F a CEM-(generalized)-(α, β)-derivation of R associated with d and G an M-(generalized)-(α, β)-derivation of R associated with g, where α(U) ⊆ U , β(U) = U and β acts ashomomorphism on U . If F (xy)±β(x)G(y) ∈ Z for all x, y ∈ U , then U [(d±g)(x), α(x)] =(0) for all x ∈ U . Moreover, if U is an ideal of R, d ± g is an α−commuting map on U .
Proof. By the hypothesis, we haveF (xy) ± β(x)G(y) ∈ Z for all x, y ∈ U. (3.1)
Replacing y with yz in (3.1), where z ∈ U , and then we getF (xyz) ± β(x)G(yz) = F (xy)α(z) + β(xy)d(z) + a ± β(x)G(y)α(z)
± β(x)β(y)g(z)= (F (xy) ± β(x)G(y))α(z) + β(x)β(y)(d ± g)(z) + a, (3.2)
where a ∈ Z. Applying (3.1) and (3.2) yields[β(x)β(y)(d ± g)(z), α(z)] = 0 for all x, y, z ∈ U. (3.3)
Since β(U) = U , we get[xy(d ± g)(z), α(z)] = 0 for all x, y, z ∈ U. (3.4)
Hence, by Lemma 2.1, we obtainU [(d ± g)(z), α(z)] = (0) for all z ∈ U. (3.5)
582 N. Muthana, Z. Alkhamisi
Moreover, if U is an ideal of R, the semiprimeness of U yields that[(d ± g)(z), α(z)] = 0 for all z ∈ U. (3.6)
Theorem 3.2. Let R be a semiprime ring, U a left ideal of R, α, β, d and g mappingsof R and F a CEM-(generalized)-(α, β)-derivation of R associated with d, where α(U) ⊆U , β(U) = U , α acts as homomorphism on U , and β acts as homomorphism or anti-homomorphism on U . If F (xy) ± g(x)α(y) ∈ Z for all x, y ∈ U , then U [d(x), α(x)] = (0)for all x ∈ U . Moreover, if U is an ideal of R, d is an α−commuting map on U .
Proof. Assume thatF (xy) ± g(x)α(y) ∈ Z for all x, y ∈ U. (3.7)
Now we replace y with yz in (3.7), where z ∈ U , then we getF (xyz) ± g(x)α(yz) = F (xy)α(z) + β(xy)d(z) + a ± g(x)α(y)α(z)
= (F (xy) ± g(x)α(y))α(z) + β(xy)d(z) + a, (3.8)where a ∈ Z. Applying (3.7) and (3.8) yields
[β(xy)d(z), α(z)] = 0 for all x, y, z ∈ U. (3.9)Since β(U) = U , we get
[xyd(z), α(z)] = 0 for all x, y, z ∈ U. (3.10)Henceforth, by Lemma 2.1, we get the required result.
If we put α = β = g = I in Theorem 3.2, we get
Corollary 3.3 ([3], Theorem 2.9). Let R be a semiprime ring, U a nonzero left idealof R, d a mapping of R and F an M-(generalized)-derivation of R associated with d. IfF (xy) ± xy ∈ Z for all x, y ∈ U , then U [d(x), x] = (0) for all x ∈ U .
Theorem 3.4. Let R be a semiprime ring, U a nonzero left ideal of R, α, d and gmappings of R, and F a CEM-(generalized)-(α, α)-derivation of R associated with d, whereα(U) = U and α acts as anti-homomorphism on U . If F (xy) ± g(y)α(x) ∈ Z for allx, y ∈ U , then U [d(x), α(x)] = (0) for all x ∈ U .
Proof. By the hypothesis, we haveF (xy) ± g(y)α(x) ∈ Z for all x, y ∈ U. (3.11)
Replacing y with yz in (3.11), where z ∈ U , we getF (xyz) ± g(yz)α(x) = F (xy)α(z) + α(xy)d(z) + a ± g(yz)α(x), (3.12)
where a ∈ Z. Applying (3.11) and (3.12), we get[α(xy)d(z), α(z)] + [±g(yz)α(x) ∓ g(y)α(x)α(z), α(z)] = 0 for all x, y, z ∈ U. (3.13)
Now substituting zx for x in (3.13), we have for all x, y, z ∈ U
[α(zxy)d(z), α(z)] + [±g(yz)α(x) ∓ g(y)α(x)α(z), α(z)]α(z) = 0. (3.14)Right multiplying (3.13) by α(z) and then subtracting it from (3.14), we get
[α(zxy)d(z), α(z)] − [α(xy)d(z), α(z)]α(z) = 0 for all x, y, z ∈ U. (3.15)Since α(U) = U , we obtain
[yx[d(z), α(z)], α(z)] = 0 for all x, y, z ∈ U. (3.16)Replacing y with d(z)y, in the above relation and then using (3.16), we get
[d(z), α(z)]yx[d(z), α(z)] = 0 for all x, y, z ∈ U, (3.17)
On centrally-extended multiplicative (generalized)-(α, β)-derivations in semiprime rings 583
that isyx[d(z), α(z)]Ryx[d(z), α(z)] = (0) for all x, y, z ∈ U. (3.18)
The semiprimeness of R yields thatyx[d(z), α(z)] = 0 for all x, y, z ∈ U, (3.19)
Since U is a left ideal, [d(z), α(z)]rx ∈ U for all x, z ∈ U, r ∈ R. In equation (3.19), replacey by x and replace x by [d(z), α(z)]rx to get
x[d(z), α(z)]rx[d(z), α(z)] = 0 for all x, z ∈ U, r ∈ R, (3.20)that is
x[d(z), α(z)]Rx[d(z), α(z)] = (0) for all x, z ∈ U. (3.21)Therefore we have
U [d(z), α(z)] = (0) for all z ∈ U. (3.22)
The following theorem is an extension and generalization to [3, Theorem 2.11].
Theorem 3.5. Let R be a semiprime ring, U a nonzero left ideal of R, α, d and gmappings of R, and F a CEM-(generalized)-(α, α)-derivation of R associated with d, whereα(U) = U and α acts as homomorphism on U . If F (xy) ± g(y)α(x) ∈ Z for all x, y ∈ U ,then U [d(x), α(x)] = (0) for all x ∈ U . Moreover, if α = g and α is homomorphism onU , then U ⊆ Z, Ud(R) ⊆ Z and UF (R) ⊆ Z.
Proof. Suppose thatF (xy) ± g(y)α(x) ∈ Z for all x, y ∈ U. (3.23)
In the above relation, replacing y with yz, where z ∈ U , we getF (xyz) ± g(yz)α(x) = F (xy)α(z) + α(xy)d(z) + a ± g(yz)α(x), (3.24)
where a ∈ Z. Applying (3.23) and (3.24), we get[α(xy)d(z), α(z)] + [±g(yz)α(x) ∓ g(y)α(x)α(z), α(z)] = 0 for all x, y, z ∈ U. (3.25)
Now substituting xz for x in (3.25), we have for all x, y, z ∈ U
[α(xzy)d(z), α(z)] + [±g(yz)α(x) ∓ g(y)α(x)α(z), α(z)]α(z) = 0. (3.26)Right multiplying (3.25) by α(z) and then subtracting it from (3.26), we get
[α(xzy)d(z), α(z)] − [α(xy)d(z), α(z)]α(z) = 0 for all x, y, z ∈ U. (3.27)Since α(U) = U , we have
[x[yd(z), α(z)], α(z)] = 0 for all x, y, z ∈ U. (3.28)Replacing x with yd(z)x, in the above relation and then using (3.28), we get
[yd(z), α(z)]x[yd(z), α(z)] = 0 for all x, y, z ∈ U, (3.29)that is
x[yd(z), α(z)]Rx[yd(z), α(z)] = (0) for all x, y, z ∈ U. (3.30)The semiprimeness of R yields that
x[yd(z), α(z)] = 0 for all x, y, z ∈ U. (3.31)Therefore, by Lemma 2.1, we have
U [d(z), α(z)] = (0) for all z ∈ U. (3.32)Now, assume that α = g and α is additive on U , then (3.25) becomes
[α(xy)d(z), α(z)] + [±α(yz)α(x) ∓ α(y)α(x)α(z), α(z)] = 0 for all x, y, z ∈ U. (3.33)
584 N. Muthana, Z. Alkhamisi
Replacing y with yz in (3.33), we have for all x, y, z ∈ U
[α(xyz)d(z), α(z)] + [±α(yz2)α(x) ∓ α(yz)α(x)α(z), α(z)] = 0. (3.34)Right multiplying (3.33) by α(z) and then subtracting it from (3.34), we get
[α(xy)[α(z), d(z)], α(z)] + [±α(yz2)α(x) ∓ α(yz)α(x)α(z), α(z)]−[±α(yz)α(x) ∓ α(y)α(x)α(z), α(z)]α(z) = 0 for all x, y, z ∈ U. (3.35)
By using (3.32), we have
[±α(yz2)α(x) ∓ α(yz)α(x)α(z), α(z)] − [±α(yz)α(x)∓α(y)α(x)α(z), α(z)]α(z) = 0 for all x, y, z ∈ U. (3.36)
Since α is epimorphism on U , then we have[±yz2x ∓ yzxz, z] − [±yzx ∓ yxz, z]z = 0 for all x, y, z ∈ U. (3.37)
That is[yz[x, z], z] − [y[x, z], z]z = 0 for all x, y, z ∈ U, (3.38)
which is equivalent to[y[x, z]z, z] − [yz[x, z], z] = 0 for all x, y, z ∈ U, (3.39)
that is[y[x, z]2, z] = 0 for all x, y, z ∈ U. (3.40)
Thus Lemma 2.2 get us U [U, U ] = (0). Replacing y with [y, z] in (3.23), we haveF (x[y, z]) ± α([y, z])α(x) ∈ Z for all x, y, z ∈ U, (3.41)
that isF (0) ± [α(y), α(z)]α(x) ∈ Z for all x, y, z ∈ U. (3.42)
Now we replace x by [x, z] in (3.23) to getF ([x, z]y) ± α(y)α([x, z]) ∈ Z for all x, y, z ∈ U. (3.43)
Thus F ([x, z]y) ∈ Z for all x, y, z ∈ U. Then F (0) ∈ Z and from (3.42) we have [U, U ]U ⊆Z. Therefore we get [x, y]2 ∈ Z for all x, y ∈ U . In particular, [x, y]3 = 0 for all x, y ∈ U .Then by Lemma 2.3, we get U ⊆ Z. Thus (3.23) gives F (xy) ∈ Z for all x, y ∈ U, and sowe have
F (x)α(y) + α(x)d(y) ∈ Z for all x, y ∈ U. (3.44)Replacing x with xz in the last expression, we get
F (xz)α(y) + α(xz)d(y) ∈ Z for all x, y, z ∈ U, (3.45)which implies that xzd(y) ∈ Z for all x, y, z ∈ U, and then, for r ∈ R, we have xz[d(y), r] =0 for all x, y, z ∈ U, that is xRz[d(y), r] = (0) for all x, y, z ∈ U, r ∈ R. In particular,x[d(y), r]Rx[d(y), r] = (0) for all x, y ∈ U, r ∈ R. Since R is semiprime, we get Ud(U) ⊆ Z.Then (3.44) gives us F (U)U ⊆ Z, and so F (xr)y ∈ Z for all x, y ∈ U, r ∈ R. Then we getb, c ∈ Z such that
F (x)α(r)y + α(x)d(r)y + by ∈ Z (3.46)and
F (r)α(x)y + α(r)d(x)y + cy ∈ Z (3.47)for all x, y ∈ U, r ∈ R. Thus, we have xyd(r) ∈ Z and xyF (r) ∈ Z for all x, y ∈ U, r ∈ R.Hence, since R is semiprime, we obtain Ud(R) ⊆ Z and UF (R) ⊆ Z.
Corollary 3.6. Let R be a semiprime ring, U a nonzero left ideal of R, α a mapping of R,and T a CEM-left α-centralizer of R, where α is an epimorphism of U . If T (xy)±α(yx) ∈Z for all x, y ∈ U , then U ⊆ Z and UT (R) ⊆ Z.
On centrally-extended multiplicative (generalized)-(α, β)-derivations in semiprime rings 585
Note that if F is a CEM-(generalized)-(α, β)-derivation of a ring R associated with amap d on R, where α and β are mappings of R such that α acts as homomorphism on R,then F ± α is a CEM-(generalized)-(α, β)-derivation of R associated with d.
Theorem 3.7. Let R be a semiprime ring, U a nonzero left ideal of R, α and d mappings ofR, and F a CEM-(generalized)-(α, α)-derivation of R associated with d, where α(U) = Uand α acts as homomorphism on R. If one of the following conditions:
(1) F (xy) ± [α(x), α(y)] ∈ Z(2) F (xy) ± (α(x) α(y)) ∈ Z
is satisfied for all x, y ∈ U , then U [d(x), α(x)] = (0) for all x ∈ U . Moreover, if α ishomomorphism on U , then U ⊆ Z, Ud(R) ⊆ Z and UF (R) ⊆ Z.
Proof. Replacing F with F ∓ α, then by Theorem 3.5 we get the desired result. Corollary 3.8 ([1], Theorem 2.18 and Theorem 2.19). Let R be a semiprime ring, Ua nonzero left ideal of R, d a mapping on R, and F an M-(generalized)-derivation of Rassociated with d. If one of the following conditions:
(1) F (xy) ± [x, y] ∈ Z(2) F (xy) ± (x y) ∈ Z
is satisfied for all x, y ∈ U , then U ⊆ Z and F (xy) ∈ Z for all x, y ∈ U .
The following corollary is an immediate consequence of Theorems 3.5 and 3.7.
Corollary 3.9. Let R be a semiprime ring, α epimorphism of R, d map on R and Fa CEM-(generalized)-(α, α)-derivation of R associated with d. If one of the followingconditions:
(1) F (xy) ± α(yx) ∈ Z(2) F (xy) ± α([x, y]) ∈ Z(3) F (xy) ± α((x y)) ∈ Z
is satisfied for all x, y ∈ R, then R is commutative.
Acknowledgment. The authors are grateful to the referee for his/her valuable sugges-tions and comments.
References[1] A. Ali, B. Dhara, S. Khan and F. Ali, Multiplicative (generalized)-derivations and
left ideals in semiprime rings, Hacettepe J. Math. Stat. 44 (6), 1293–1306, 2015.[2] H.E. Bell and M.N. Daif, On centrally-extended maps on rings, Beitrage Algebra
Geom. Article No. 244, 1–8, 2015.[3] B. Dhara and S. Ali, On multiplicative (generalized)-derivations in prime and
semiprime rings, Aequat. Math. 86 (1-2), 65–79, 2013.[4] C. Lanski, An Engel condition with derivation for left ideals, Proc. Amer. Math. Soc.
125 (2), 339–345, 1997.[5] M.S. Tammam El-Sayiad, N.M. Muthana and Z.S. Alkhamisi, On rings with some
kinds of centrally-extended maps, Beitrage Algebra Geom. Article No. 274, 1–10,2015.
Hacettepe Journal ofMathematics & Statistics
Hacet. J. Math. Stat.Volume 49 (2) (2020), 586 – 598
DOI : 10.15672/hujms.544489
Research Article
Euler sums and non-integerness of harmonic typesums
Haydar Göral1, Doga Can Sertbaş∗2
1Department of Mathematics, Faculty of Sciences, Dokuz Eylül University, Tınaztepe Yerleşkesi, 35390Buca/İzmir, TURKEY
2Department of Mathematics, Faculty of Sciences, Sivas Cumhuriyet University, 58140, Sivas, TURKEY
AbstractWe show that Euler sums of generalized hyperharmonic numbers can be evaluated in termsof Euler sums of generalized harmonic numbers and special values of the Riemann zetafunction. Then we focus on the non-integerness of generalized hyperharmonic numbers.We prove that almost all generalized hyperharmonic numbers are not integers and ourerror term is sharp and the best possible. Finally, we analyze generalized hyperharmonicnumbers in terms of topology and relate this to non-integerness.
Mathematics Subject Classification (2010). 11B83, 5A10, 11B75
Keywords. Harmonic numbers, hyperharmonic numbers, Euler sums
1. IntroductionThe goal of this paper is to study Euler sums and non-integerness of generalized hyper-
harmonic numbers. In order to achieve this, we make use of a recurrence relation of thesenumbers and the distribution of prime numbers. Before stating our results, we first givethe necessary definitions. The sequence of partial sums of the harmonic series is calledharmonic numbers, namely
hn =n∑
k=1
1k
for n ≥ 1. These numbers have been studied recurrently and exponentially. It is well-known that
hn = log n + γ + O
( 1n
)(1.1)
and a finer one is
hn ∼ log n + γ + 12n
−∞∑
k=1
B2k
2kn2k(1.2)
= log n + γ + 12n
− 112n2 + 1
120n4 − · · · (1.3)
∗Corresponding Author.Email addresses: [email protected] (H. Göral), [email protected] (D.C. Sertbaş)Received: 24.07.2018; Accepted: 30.01.2019
Euler sums and non-integerness of harmonic type sums 587
as n tends to infinity, where γ is Euler’s constant and Bm is the m-th Bernoulli number.Recall that the Bernoulli numbers are defined as
z
ez − 1=
∞∑n=1
Bn
n!zn.
Linking the sum of the first n positive integers with the sum of their reciprocals, that isto say with harmonic numbers, Ramanujan gave the striking asymptotic
hn ∼ 12
log 2m + γ + 112m
− 1120m2 + 1
630m3 − 11680m4 (1.4)
+ 12310m5 − 191
360360m6 + 2930030m7
− 28331166880m8 + 140051
17459442m9 − · · ·
as n goes to infinity where
m = n(n + 1)2
= 1 + 2 + · · · + n
is the n-th triangular number. For the details, we refer the reader to page 531 of [6]. Then-th generalized harmonic number of order m, namely H
(m)n , is defined as
H(m)n =
n∑k=1
1km
. (1.5)
The generating function∑∞
n=1 H(m)n zn of H
(m)n is equal to
Lim(z)1 − z
where
Lim(z) =∞∑
n=1
zn
nm
is the polylogarithm function. Special values of the zeta function
Em(s) =∞∑
k=1
H(m)n
ns, ℜ(s) > 1
of H(m)n are called Euler sums. In other words, for any integers m ≥ 1 and k ≥ 2, the sum
Em(k) is called an Euler sum. For m = 1 and k ≥ 2 an integer, Euler proved that
E1(k) =∞∑
k=1
hn
nk= 1
2(k + 2)ζ(k + 1) − 1
2
k−2∑j=1
ζ(k − j)ζ(j + 1), (1.6)
where
ζ(s) =∞∑
n=1
1ns
is the Riemann zeta function defined in ℜ(s) > 1. Euler sums have received continuousattention and been reevaluated since then. For instance if m+k is odd, then as in equation(1.6), the corresponding Euler sum can be evaluated in terms of the special values of theRiemann zeta function, see [7]. For more on the analytic properties of harmonic numbersand approximation type results, we refer the reader to [1]. Next, we define hyperharmonicnumbers. Hyperharmonic numbers were first defined in the book of Conway and Guy
588 H. Göral, D.C. Sertbaş
[10] and they are another generalization of harmonic numbers. The n-th hyperharmonicnumber of order r is defined recursively by
h(r)n :=
n∑k=1
h(r−1)k , (1.7)
where h(1)n = hn. By [10], one has that h
(r)n can be expressed in terms of binomial coeffi-
cients and harmonic numbers with the formula
h(r)n =
(n + r − 1
r − 1
)(hn+r−1 − hr−1). (1.8)
Mező [15] conjectured that hyperharmonic numbers are never integers except 1, thatis to say if n ≥ 2 then h
(r)n cannot be an integer. The case r = 1 was already proved by
Theisinger [17]. Based on three different approaches, namely analytic, combinatorial andalgebraic, the authors [11] proved that almost all hyperharmonic numbers are not integers.This yields an almost answer to Mező’s problem [15]. Moreover, in the same paper [11],it was deduced that if n is even or a prime power, or r is odd then the correspondinghyperharmonic number is not integer.
Extending the definition of h(r)n and H
(m)n simultaneously, generalized hyperharmonic
numbers (see [9]) are defined by
H(m,r)n :=
n∑k=1
H(m,r−1)k , r ≥ 2, (1.9)
where H(m,1)n = H
(m)n as defined by (1.5). In [9], Dirichlet series with generalized hyper-
harmonic numbers were computed in terms of the values of the Hurwitz zeta function.In the same paper, they also gave a combinatorial identity for generalized hyperharmonicnumbers given by
H(m,r)n =
n∑j=1
(n − j + r − 1
r − 1
)1
jm, (1.10)
and this was generalized in [12, Proposition 2.1]. Unlike the hyperharmonic case m = 1,we do not have such a formula as given in (1.8). Note also that we have
∞∑n=1
H(m,r)n zn = Lim(z)
(1 − z)r. (1.11)
Now we explain our results. Throughout the article, we let n, m, r, k to be positive in-tegers. Our first result states that Euler sums of generalized hyperharmonic numbers canbe computed in terms of ordinary Euler sums and special values of the Riemann zeta func-tion. For complex numbers z1, . . . , zℓ, let ⟨z1, . . . , zℓ⟩Q denote the vector space generatedby z1, . . . , zℓ over Q. Euler sums of hyperharmonic numbers and analytic continuationof their Dirichlet series were studied in [8, 14]. The next result is a structural theoremon Euler sums of generalized hyperharmonic numbers and generalizes the correspondingresults of [8, 14].Theorem 1.1. Let
ζH(m,r)(s) =∑n≥1
H(m,r)n
ns,
where m, r ≥ 1 and ℜ(s) > r. For any integer k ≥ r + 1, the function ζH(m,r)(k) can bewritten as a finite Q-linear combinations of ordinary Euler sums and special values of theRiemann zeta function. Moreover, we have
ζH(m,r)(k) ∈⟨Ei(j), ζ(j) : 1 ≤ i ≤ m, 2 ≤ j ≤ k
⟩Q
Euler sums and non-integerness of harmonic type sums 589
and all rational coefficients are effectively computable in this expression. Furthermore, thefunction ζH(m,r)(s) has a meromorphic continuation to the whole complex plane.
Some remarks are as follows: even in the case r = 2 and m + k is odd, it does not seemthat the corresponding Euler sum ζH(m,r)(k) can be expressed as a Q-linear combinationsof at most two products of special values of the Riemann zeta functions as in equation(1.6), see also [7]. Therefore the representation given in Theorem 1.1 seems to be the bestpossible one.
Our second theorem generalizes the result in [9] and it reveals the existence of plentifulnon-integer generalized hyperharmonic numbers.
Theorem 1.2. Let n, m, r be positive integers where n ≥ 2.
(1) For any prime p and k ≥ 1, if n = pk then H(m,r)n is not an integer.
(2) Let p be the maximum prime that is less than n. If for all c ∈ N, we have thatcpm /∈ [r, n − p + r − 1], then H
(m,r)n is not an integer. In fact, for sufficiently large
n, if cpm /∈ [r, r + n0.525 − 1] for all c ∈ N, then H(m,r)n /∈ Z.
(3) If n is sufficiently large and r ≤(n − n0.525)m − n0.525 + 1, then H
(m,r)n /∈ Z. In
particular for a given r, if n is large enough, then the corresponding H(m,r)n is not
an integer.(4) If
(n − 1)(r − 1)!
r−3∏i=−1
(n + i) < 2m
holds, then H(m,r)n /∈ Z.
LetS(x) =
∣∣∣(n, r) ∈ [0, x] × [0, x] : h(r)n /∈ Z
∣∣∣ .In other words, S(x) counts the number of pairs (n, r) in the rectangle [0, x] × [0, x] wherethe corresponding hyperharmonic number h
(r)n is not an integer. In [2,11], it was obtained
thatS(x) ∼ x2,
which means that non-integer hyperharmonics have the full asymptotic in the first quadru-ple. This result is based on primes in short intervals. Our third theorem extends this resultto generalized hyperharmonic numbers.
Theorem 1.3.(1) For sufficiently large n, the set of integers r with H
(m,r)n /∈ Z contains a set of
density
1 − n0.525
(n − n0.525)m.
In particular, this density goes to 1, if one of n or m tends to infinity.(2) For a fixed m ≥ 2, let
Sm(x) =∣∣∣(n, r) ∈ [0, x] × [0, x] H(m,r)
n /∈ Z∣∣∣ .
ThenSm(x) = x2 + Om
(x1+ 1
m
).
In other words for a fixed m ≥ 2, almost all generalized hyperharmonic numbersare not integers.
590 H. Göral, D.C. Sertbaş
Now we consider the case where m is also not fixed. We obtain that almost all gener-alized hyperharmonic numbers are not integers as well. The following result is sharp andthe best possible as for n = 1, H
(m,r)n = 1 which is independent from the choice of m and
r. That is to say, the error term O(x2) in the following theorem is inevitable and one
cannot give a better error term.
Theorem 1.4. LetT (x) =
∣∣∣(n, m, r) ∈ [0, x]3 H(m,r)n /∈ Z
∣∣∣ .Then T (x) = x3 +O
(x2) . This means that almost all generalized hyperharmonic numbers
are not integers.
Next, we analyze generalized hyperharmonic numbers in terms of topology, and thisleads to non-integerness of these numbers. Also the first part of the following theoremgives a characterization of Mező’s problem [15].
Theorem 1.5.(1) The set H =
h
(r)n n, r ≥ 1
is closed in R in the usual topology. In particular,
there exists an integer a in H if and only if a is a limit point of the set H.(2) The topological closure of the set G =
H
(m,r)n m, n, r ≥ 1
is
G ∪ ζ(k) k ≥ 2 ∪ 1, 2, . . . .
(3) Given n, r ≥ 1, there exists M = M(n, r) such that if m ≥ M then H(m,r)n /∈ Z.
Note that one can obtain Theorem 1.5 (3) from Theorem 1.2 (4) in an effective way.However, from topological point of view, this part follows immediately.
Now we fix our notations in this paper: we denote p as a prime, unless it is statedotherwise. Also for a given a ∈ Z, we define
νp(a) :=
m if pm ∥ a∞ if a = 0
as the p-adic valuation of a. Here pm ∥ a means pm | a but pm+1 - a. We extend thisnotation to a rational number q = a/b ∈ Q by νp(q) = νp(a) − νp(b) where a, b ∈ Z. Notethat for any q1, q2 ∈ Q, we have
νp (q1q2) = νp (q1) + νp (q2)νp (q1 + q2) ≥ min νp (q2) , νp (q2) , (1.12)
and the last property is called the non-Archimedian property of the p-adic valuation. More-over we have equality in (1.12) if νp (q1) = νp (q2) .
Next we define the big-O notation as given in [3]. Let g(x) be a function from R toitself and suppose that g(x) > 0 for x ≥ a, where a is a real number. We use the notation
f(x) = O (g(x))to mean that there exists a constant c > 0 such that
|f(x)| ≤ cg(x), for all x ≥ a.
We writef(x) = Oℓ (g(x))
to indicate that the big-O constant may depend on ℓ. We say that f(x) is asymptotic tog(x), denoted by f(x) ∼ g(x), if
limx→∞
f(x)g(x)
= 1.
Euler sums and non-integerness of harmonic type sums 591
2. ProofsProof of Theorem 1.1. By (1.11), we know that
∞∑n=1
H(m,r)n zn = Lim(z)
(1 − z)r, |z| < 1.
Taking derivatives of both sides and using the equality
(Lim(z))′ = Lim−1(z)z
,
we get that∞∑
n=1nH(m,r)
n zn−1 =Lim−1(z)
z (1 − z)r + r(1 − z)r−1Lim(z)(1 − z)2r
= Lim−1(z)z(1 − z)r
+ rLim(z)(1 − z)r+1 .
This yields by (1.11) again that∞∑
n=1nH(m,r)
n zn =∞∑
n=1H(m−1,r)
n zn +∞∑
n=1rH(m,r+1)
n zn+1.
By adjusting the index and comparing the coefficients, we obtain the following recurrencerelation:
rH(m,r+1)n = (n + 1)H(m,r)
n+1 − H(m−1,r)n+1 . (2.1)
To prove the first part of the theorem, we proceed by induction on r and we apply equation(2.1). If r = 1, then it is clear. Moreover if m = 1 then the theorem follows from [8, 14].Now suppose the theorem for 1 ≤ j ≤ r, and we will show it for r + 1 where m ≥ 2. Bythe recursion formula (1.9) for any p ≥ 1, we know that
H(p,r)n+1 = H(p,r)
n + H(p,r−1)n+1
= H(p,r)n + · · · + H(p,2)
n + H(p)n + 1
(n + 1)p.
By (2.1), we get that
rH(m,r+1)n = (n + 1)H(m,r)
n+1 − H(m−1,r)n+1
= (n + 1)r∑
j=1H(m,j)
n −r∑
j=1H(m−1,j)
n .
So the corresponding zeta function can be found as
ζH(m,r+1)(s) = 1r
r∑j=1
∑n≥1
H(m,j)n
ns−1 +r∑
j=1
∑n≥1
H(m,j)n
ns−
r∑j=1
∑n≥1
H(m−1,j)n
ns
= 1
r
r∑j=1
(ζH(m,j)(s − 1) + ζH(m,j)(s) − ζH(m−1,j)(s)) . (2.2)
By induction, we know that each of the summands on the right hand side of (2.2) can bewritten as a finite Q-linear combinations of ordinary Euler sums and special values of theRiemann zeta function, when s = k ≥ r + 1. Moreover by equation (2.2) and inductionon r, we deduce that
ζH(m,r)(k) ∈⟨Ei(j), ζ(j) : 1 ≤ i ≤ m, 2 ≤ j ≤ k
⟩Q
.
592 H. Göral, D.C. Sertbaş
For the last part of the theorem, when r = 1, the meromorphic continuation follows from[4]. If m = 1, then by [14] we finish the proof again. So suppose that m, r ≥ 2. Now byinduction on r and the functional equation (2.2), we obtain the meromorphic continuationof ζH(m,r)(s) to the whole complex plane.
Proof of Theorem 1.2 (1). Let n = pk. By equation (1.10), we have
H(m,r)n =
n∑j=1
(n − j + r − 1
r − 1
)1
jm= 1
pkm+
pk−1∑j=1
(pk − j + r − 1
r − 1
)1
jm.
Note that νp (j) < k, for any j ∈ 1, . . . , n − 1 . Therefore
νp
((pk − j + r − 1
r − 1
)1
jm
)≥ (1 − k)m ≥ 1 − km.
So by the non-Archimedian property of the p-adic valuation, we deduce that
νp
(H(m,r)
n
)= νp
(p−km
)= −km.
As k, m ≥ 1, the non-integerness follows by the previous equation.
Proof of Theorem 1.2 (2). Let p be the largest prime that is less than n. From the firstpart of the theorem, we may suppose that n is not a prime. By the Bertrand’s Postulate,we know that n
2 < p < n < 2p. Therefore, we get
H(m,r)n =
n∑j=1j =p
(n − j + r − 1
r − 1
)1
jm+(
n − p + r − 1r − 1
)1
pm. (2.3)
Notice that the first summand in equation (2.3) has a positive p-adic valuation as there isonly one multiple of p in [1, n]. Now we analyze the second summand. Observe that(
n − p + r − 1r − 1
)= r(r + 1) · · · (n − p + r − 1)
(n − p)!.
Note that there exists at most one multiple of p in [r, n − p + r − 1], since the length of thecorresponding interval is n − p and n − p < p. As cpm /∈ [r, n − p + r − 1] for any c ∈ N,we deduce that pm - r(r + 1) · · · (n − p + r − 1). Therefore
νp
((n − p + r − 1
r − 1
))≤ νp (r(r + 1) · · · (n − p + r − 1))
≤ m − 1. (2.4)
Combining (2.3), (2.4) and the non-Archimedian property of the p-adic valuation, we getthat νp
(H
(m,r)n
)≤ −1, which yields the non-integerness of H
(m,r)n . Now suppose that n is
sufficiently large. By [5], we know that the prime p lies in the interval (n−n0.525, n). Hencen − p < n0.525, which implies that [r, n − p + r − 1] ⊆ [r, r + n0.525 − 1]. By the argumentabove, we see that if cpm /∈ [r, r + n0.525 − 1] for any c ∈ N, then the non-integernessfollows.
Proof of Theorem 1.2 (3). Suppose that r ≤(n − n0.525)m − n0.525 + 1. Again we may
assume by the first part of Theorem 1.2 that n is not a prime. Since n is sufficiently large,we obtain that the largest prime p that is less than n satisfies the inequality n − n0.525 <p < n. Therefore n − p < n0.525, which implies that
r ≤(n − n0.525
)m− n0.525 + 1 < pm − (n − p) + 1. (2.5)
Euler sums and non-integerness of harmonic type sums 593
Observe that this inequality holds if and only if n − p + r − 1 < pm. Note by this fact thatcpm /∈ [r, n−p+r −1] for any c ∈ N, as 0 < r ≤ n−p+r −1 < pm and 1 ≤ n−p. By part(2) of Theorem 1.2, we conclude that H
(m,r)n /∈ Z. The last statement follows directly.
Proof of Theorem 1.2 (4). Again by equation (1.10), we know that
H(m,r)n =
(n − 1 + r − 1
r − 1
)+
n∑j=2
(n − j + r − 1
r − 1
)1
jm. (2.6)
If the second summand in equation (2.6) is less than 1, then H(m,r)n cannot be integer. We
may assume that r ≥ 2, because if r = 1 then non-integerness follows by [17] and the factthat
1 < H(m)n < ζ(m) ≤ ζ(2) < 2
for n, m ≥ 2. Suppose that the inequality
(n − 1)(r − 1)!
r−3∏i=−1
(n + i) < 2m (2.7)
holds. Note that the product (n − 1) · · · (n + r − 3) is greater than or equal to the product(n − j + 1) · · · (n − j + r − 1) for any j ≥ 2. Therefore, we have
n∑j=2
(n − j + r − 1
r − 1
)≤ (n − 1) ·
(n − 1 + r − 1
r − 1
).
For j ≥ 2, we see thatn∑
j=2
(n − j + r − 1
r − 1
)1
jm≤ 1
2m
n∑j=2
(n − j + r − 1
r − 1
)
≤ n − 12m
·(
n − 1 + r − 1r − 1
). (2.8)
By (2.7) and (2.8), we get that
0 <n∑
j=2
(n − j + r − 1
r − 1
)1
jm< 1,
and hence the non-integerness follows.
Proof of Theorem 1.3 (1). Let n be sufficiently large, and p denote the biggest primethat is less than n. Observe by Theorem 1.2 (2) that if cpm /∈ [r, r + n0.525 − 1] for anyc ∈ N, then H
(m,r)n /∈ Z. Thus the only possibility for H
(m,r)n to be an integer comes from
the condition r ∈ [cpm − n0.525 + 1, cpm]. When we consider the numbers modulo pm, theobservation above leads to the fact that the set of integers r with H
(m,r)n /∈ Z contains a
set of density
1 − n0.525
pm> 1 − n0.525
(n − n0.525)m, (2.9)
as p > n − n0.525. This completes the proof.
Proof of Theorem 1.3 (2). Let m ≥ 2 be fixed and put
Sm(x) =∣∣∣(n, r) ∈ [0, x] × [0, x] H(m,r)
n /∈ Z∣∣∣ .
By Theorem 1.2 (3), we know that if n is sufficiently large and r ≤(n − n0.525)m−n0.525+1
then H(m,r)n /∈ Z. Note that r = Om (nm) . Consider the following Figure 1.
594 H. Göral, D.C. Sertbaş
n
r
nm x1m
cm1m
r = cmnm
x
x
Figure 1. The graph of r = cmnm, for a fixed m ≥ 2 and for some cm > 0. Theshaded area indicates the lattice points (n, r) where the corresponding generalizedhyperharmonic number H
(m,r)n is not an integer.
Observe that the function intersects the line r = x at(
xcm
) 1m . Therefore there are
at most Om
(x · x
1m
)possibly many tuples (n, r) where the corresponding generalized
hyperharmonic H(m,r)n may be an integer. Thus we get that Sm(x) = x2 +Om
(x1+ 1
m
), as
desired. Thus for a fixed m ≥ 2, almost all generalized hyperharmonics are not integers.
Proof of Theorem 1.4. We will use the part (2) of Theorem 1.3. Note that we cannotapply this part directly, as the big-O term depends on m as well. So our method to obtainthe theorem will be more delicate. Observe that
T (x) =∑
1≤m≤x
Sm(x). (2.10)
Now we split equation (2.10) into three parts, i.e.
T (x) = T1(x) + T2(x) + T3(x),
where
T1(x) =∑
1≤m<4Sm(x), (2.11)
T2(x) =∑
4≤m<2√
x+1Sm(x), (2.12)
T3(x) =∑
2√
x+1≤m≤x
Sm(x). (2.13)
ClearlyT1(x) = O
(x2)
. (2.14)
Now we estimate the proper subsums given in (2.12) and (2.13). There exists an absoluteconstant n0 which does not depend on m such that if n ≥ n0, then
(n − n0.525)m − n0.525 + 1 ≥ nm2 , (2.15)
if m ≥ 4. Now consider the following Figure 2.
Euler sums and non-integerness of harmonic type sums 595
n
r
n0 x2m
r = nm2
r = (n − n0.525)m − n0.525 + 1
r = nm
x
x
Figure 2. For a fixed m ≥ 4, the shaded area represents the lattice points (n, r)where the corresponding generalized hyperharmonic number H
(m,r)n is not an in-
teger.
By the argument in Theorem 1.3 (2) and equation (2.15), we obtain that
Sm(x) = x2 + O(x1+ 2
m
), (2.16)
when m ≥ 4. To estimate the sum T2(x), observe by (2.16) that
T2(x) =∑
4≤m<2√
x+1Sm(x) =
∑4≤m<2
√x+1
(x2 + O
(x1+ 2
m
))
=∑
4≤m<2√
x+1x2 + O
∑4≤m<2
√x+1
x1+ 2m
=
∑4≤m<2
√x+1
x2 + O
x1+ 24
∑4≤m<2
√x+1
1
=
∑4≤m<2
√x+1
x2 + O(x
32 · 2
√x)
=∑
4≤m<2√
x+1x2 + O
(x2)
(2.17)
as m ∈ [4, 2√
x + 1]. For the last sum, note that
T3(x) =∑
2√
x+1≤m≤x
Sm(x)
=∑
2√
x+1≤m≤x
x2 + O
∑2√
x+1≤m≤x
x1+ 2m
=
∑2√
x+1≤m≤x
x2 + O
(x · x
1+ 1√x
)=
∑2√
x+1≤m≤x
x2 + O(x2)
, (2.18)
as x1√x = e
log x√x = O (1) . Now combining equations (2.14), (2.17) and (2.18) yields to
T (x) =∑m≤x
x2 + O(x2)
= x3 + O(x2)
,
and this concludes the theorem.
596 H. Göral, D.C. Sertbaş
Proof of Theorem 1.5 (1). For the first part, let α be a limit point of the set H. As 1is in H, we may suppose that α is different from 1. Thus, there exists a sequence h
(rk)nk
such thatlim
k→∞h(rk)
nk= α.
Since h(r)1 = 1 for any r, we can say that nk > 1 for k ≥ 1. Assume that the sequence
nk is unbounded. As h(rk)nk ≥ hnk
by the recursion formula (1.7) of hyperharmonics, wededuce that the sequence h
(rk)nk is also unbounded since the harmonic series diverges. As
a consequence, we see that the sequence nk is finite and does not contain 1. Next, assumethat the sequence rk is unbounded. By the recursion formula (1.7) of hyperharmonicsagain, we observe that if n ≥ m then h
(r)n ≥ h
(r)m . Therefore, we get that
h(rk)nk
≥ h(rk)2 = 2rk + 1
2.
In other words, the sequence h(rk)nk is also unbounded. Thus, the sequence rk is finite.
Hence h(rk)nk is finite and this yields that α belongs to H.
Proof of Theorem 1.5 (2). Let β be a limit point of G. We may assume that β = 1and β = ζ(k) for any k ≥ 2, as they are the limit points of the set. Write
limk→∞
H(mk,rk)nk
= β.
For k is large enough, we may assume that nk, rk ≥ 2, because if r = 1 then the limit isalready contained in G, as
limn→∞
H(m)n = ζ(m), lim
m→∞H(m)
n = 1.
Suppose that nk tends to infinity. By the recursion formula (1.9), we have that
H(mk,rk)nk
=nk∑i=1
H(mk,rk−1)i ≥ nk,
as rk ≥ 2. Therefore, if nk is unbounded, then so is the set
H(mk,rk)nk
. Thus the set
nk k ≥ 1 must be finite. Clearly we also have H(mk,rk)nk ≥ H
(mk,rk)2 . Now suppose that
rk tends to infinity, as k tends to infinity. Since
H(mk,rk)2 = H
(mk,rk−1)1 + H
(mk,rk−1)2 , (2.19)
we see by induction that H(mk,rk)2 ≥ rk. So again, we get that
H
(mk,rk)nk
is not bounded.
Hence (nk, rk) k ≥ 1 is finite. As a result of this, we may assume that (nk, rk) = (n, r)for some fixed n, r ≥ 2. If the set mk k ≥ 1 is also finite, we see that β ∈ G. Nowsuppose that mk tends to infinity, as k goes to infinity. We know by recursion that
H(m,r)n = H
(m,r−1)1 + H
(m,r−1)2 + · · · + H(m,r−1)
n .
By induction on n + r, this indicates that limm→∞
H(m,r)n is a positive integer. Finally by
equation (2.19), we get thatlim
m→∞H
(m,r)2 = r.
This concludes the proof of the second part of Theorem 1.5. Proof of Theorem 1.5 (3). By the second part of Theorem 1.5, we know that for afixed n, r ∈ N, where n ≥ 2, the limit lim
m→∞H(m,r)
n is an integer. But we also know that
the sequence
H(m,r)n
m≥1
is decreasing as n ≥ 2. Thus the third part of Theorem 1.5follows.
Euler sums and non-integerness of harmonic type sums 597
3. Concluding remarksAs we mentioned earlier, currently we do not have any such fundamental equation like
(1.8) for generalized hyperharmonic numbers when m ≥ 2. Therefore, it is hard to adaptcombinatorial and algebraic approaches which were given in [11], since there does notexist any fixed binomial term that we can compute its p-adic valuation for a given primep. Moreover, it can be shown by using [16] that there are some generalized hyperharmonicnumbers whose 2-adic valuation is greater than 0. The following Table 1 contains suchexamples.
n m r H(m,r)n ν2
(H
(m,r)n
)2 2 2 9/4 −25 2 3 22073/1200 −45 3 3 1189951/72000 −65 3 32 119538143/2250 −17 2 64 270270407641/2205 014 2 11 83741315760829/61486425 015 2 11 433932992403542/184459275 115 2 12 1653452996692253/312161850 -115 2 64 12505713644625510548024/14189175 3
Table 1. The generalized hyperharmonic numbers and their 2-adic valuations fordifferent values of n, m and r.
As it can be seen from these values, one cannot give the upper bound 0 for the 2-adicvaluation of H
(m,r)n unlike the hyperharmonic case which is obtained in [13, Corollary
3.7]. Moreover, it was proved in [11, Theorem 2] that if n is even or r is odd, then thecorresponding hyperharmonic number h
(r)n is not an integer, as ν2
(h
(r)n
)≤ −1. However,
this does not work for generalized hyperharmonic numbers, as we see from the table above.
References[1] E. Alkan, Approximation by special values of harmonic zeta function and log-sine
integrals, Commun. Number Theory Phys. (10) 7, 515–550, 2013.[2] E. Alkan, H. Göral and D.C. Sertbaş, Hyperharmonic Numbers can Rarely be Integers,
Integers, 18 Paper No. A43, 1–16, 2018.[3] T.M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, New York,
First edition, 1976.[4] T.M. Apostol and T.H. Vu, Dirichlet Series Related to the Riemann Zeta Function,
J. Number Theory, 19, 85–102, 1984.[5] R.C. Baker, G. Harman and J. Pintz, The difference between consecutive primes, II,
Proc. Lond. Math. Soc. (3), 83, 532–562, 2001.[6] B.C. Berndt, Ramanujan’s Notebooks, Part V, Springer-Verlag, New York, 1998.[7] D. Borwein, J.M. Borwein and R. Girgensohn, Explicit evaluation of Euler sums,
Proc. Edinb. Math. Soc. (2), 38, 277–294, 1995.[8] K. Boyadzhiev and A. Dil, Euler sums of hyperharmonic numbers, J. Number Theory,
147, 490–498, 2015.[9] M. Cenkçi, A. Dil and I. Mező, Evaluation of Euler-like sums via Hurwitz zeta values,
Turkish J. Math. 41, 1640–1655, 2017.[10] J.H. Conway and R.K. Guy, The Book of Numbers, Springer-Verlag, New York, 1996.
598 H. Göral, D.C. Sertbaş
[11] H. Göral and D.C. Sertbaş, Almost all Hyperharmonic Numbers are not Integers, J.Number Theory, 171, 495–526, 2017.
[12] H. Göral and D.C. Sertbaş, A congruence for some generalized harmonic type sums,Int. J. Number Theory, 14 (4), 1033–1046, 2018.
[13] H. Göral and D.C. Sertbaş, Divisibility Properties of Hyperharmonic Numbers, ActaMath. Hungar. 154 (1), 147–186, 2018.
[14] K. Kamano, Dirichlet series associated with hyperharmonic numbers, Mem. OsakaInst. Tech. Ser. A, 56 (2), 11–15, 2011.
[15] I. Mező, About the non-integer property of hyperharmonic numbers, Ann. Univ. Sci.Budapest. Sect. Math. 50, 13–20, 2007.
[16] W.A. Stein et. al., Sage Mathematics Software (Version 6.10.rc2), The Sage Devel-opment Team, 2015, http://www.sagemath.org.
[17] L. Theisinger, Bemerkung über die harmonische reihe, Monatsh. Math. Phys. 26,132–134, 1915.
Hacettepe Journal ofMathematics & Statistics
Hacet. J. Math. Stat.Volume 49 (2) (2020), 599 – 607
DOI : 10.15672/hujms.568386
Research Article
Almost L-Dunford-Pettis sets in Banach latticesand its applications
Abderrahman RetbiIbn Tofail University, Faculty of Sciences, Department of Mathematics, B.P. 133, Kenitra, Morocco
AbstractWe introduce and study the notion of almost L-Dunford-Pettis sets in Banach lattices andwe give some characterizations of it in terms of sequences. As an application, we establishnew properties of almost Dunford-Pettis completely continuous operators. Finally, byintroducing the concept of aL-Dunford-Pettis property in Banach lattices, we investigatethe weak compactness of almost Dunford-Pettis completely continuous operator.
Mathematics Subject Classification (2010). 46A40, 46B40
Keywords. Banach lattice, Dunford-Pettis set, relatively compact Dunford-Pettisproperty, Dunford-Pettis completely continuous operator
1. Introduction and notationA norm bounded subset A of a Banach space X is said to be Dunford-Pettis set,
if every weakly null sequence (fn) in X ′ converges uniformly to zero on A, that is,lim
n→∞supx∈A fn(x) = 0. Recall from [6] that a norm bounded subset A of a topological
dual Banach space X ′ is an L-Dunford-Pettis if every weakly null sequence (xn), which is aDunford-Pettis subset of X converges uniformly to zero on A, that is lim
n→∞supf∈A f(xn) =
0.A Banach space X has
- the relatively compact Dunford-Pettis property (DPrcP for short) if every weakly nullsequence, which is a Dunford-Pettis set in X, is norm null [7].- the L-Dunford-Pettis property if every L-Dunford-Pettis set in X ′ is relatively weaklycompact [6].
A Banach lattice E has the positive relatively compact Dunford-Pettis property (PDPrcPfor short) if every disjoint weakly null sequence, which is a Dunford-Pettis set in X, isnorm null [4]. Note that if a Banach lattice E has the DPrcP then, it has PDPrcP butthe converse is not true in general (see Example 3.4 of [4]).
An operator T from a Banach space X into a another Banach space Y is called Dunford-Pettis completely continuous (DPcc for short) if each weakly null sequence (xn), which isa Dunford-Pettis set in X, we have ∥T (xn)∥Y → 0, as n → ∞ [7]. Recall from [4] that anoperator T from a Banach lattice E into a Banach space Y is called almost Dunford-Pettiscompletely continuous (aDPcc for short) if each disjoint weakly null sequence (xn), whichis a Dunford-Pettis set in E, we have ∥T (xn)∥Y → 0, as n → ∞.
Email address: [email protected]: 23.12.2016; Accepted: 30.01.2019
600 A. Retbi
Dunford-Pettis sets definition is given firstly by K.T. Andrews [2] as a norm boundedsubset A of a Banach space X is a Dunford-Pettis set whenever every weakly compactoperator from X to an arbitrary Banach space Y carries A to a norm totally boundedset. Then Andrew characterized the Dunford-Pettis sets by using sequences (fn) in X ′.Recently in [3], Bouras considered the disjoint version of the Dunford-Pettis sets andintroduced the almost Dunford-Pettis sets in Banach lattices. Following Bouras, a boundedsubset A of a Banach lattice E is said to be an almost Dunford-Pettis set if every disjointweakly null sequence (fn) in E′ converges uniformly to zero on A. In this paper, using thedisjoint sequence techniques we consider the disjoint version of L-Dunford-Pettis sets, thatwe call almost L-Dunford-Pettis sets in Banach lattices (Definition 2.1). In addition, weintroduce the aL-Dunford-Pettis property which is shared by those Banach lattice whoseevery almost L-Dunford-Pettis subset of his topological dual is relatively weakly compact(Definition 4.1).
The article is organized as follows. In Section 2 we establish some characterizations ofalmost L-Dunford-Pettis set in terms of sequences (Proposition 2.2), and we show thateach order interval in a dual Banach lattice is an almost L-Dunford-Pettis set (Proposition2.4). Also, we give some equivalent condition for T ′(A) to be almost L-Dunford-Pettis setwhere A is a norm bounded solid subset of E and T : E → F is an order boundedoperator between two Banach lattices (Theorem 2.7). In Section 3, using the notion ofalmost L-Dunford-Pettis set, we give characterizations of aDPcc operator and PDPrcP(Theorem 3.1 and Corollary 3.2). After that, we characterize Banach lattice E such thateach almost L-Dunford-Pettis set of E′ is L-Dunford-Pettis (Theorem 3.12), and we derivesome sufficient conditions such that the PDPrcP coincide with the DPrcP (Corollary 3.13).In Section 4, we prove that a Banach lattice E has the aL-Dunford-Pettis property if andonly if each aDPcc operator from a Banach lattice E into any Banach space Y is weaklycompact (Theorem 4.2), and we deduce an important result about the reflexive space(Corollary 4.3).
To state our results, we need to fix some notations and recall some definitions. A Banachlattice is a Banach space (E, ∥ · ∥) such that E is a vector lattice and its norm satisfiesthe following property: for each x, y ∈ E such that |x| ≤ |y|, we have ∥x∥ ≤ ∥y∥. If E isa Banach lattice, its topological dual E′, endowed with the dual norm, is also a Banachlattice. The sequence (xn) of a Banach lattice E is disjoint if |xn| ∧ |xm| = 0, n = m (wedenote by xn⊥xm).
Recall that a nonzero element x of a vector lattice G is discrete if the order idealgenerated by x equals the subspace generated by x. The vector lattice G is discrete, if itadmits a complete disjoint system of discrete elements. The lattice operations of a Banachlattice E are weakly sequentially continuous, whenever xn → 0 for σ(E, E′) as n → ∞imply |xn| → 0 for σ(E, E′), as n → ∞. We will use the term operator T : X −→ Ybetween two Banach space to mean a bounded linear mapping, its dual operator T ′ isdefined from Y ′ into X ′ by T ′(f)(x) = f(T (x)) for each f ∈ Y ′ and for each x ∈ X. Werefer the reader to [1] for unexplained terminology of Banach lattice theory and operators.
2. Almost L-Dunford-Pettis set in a topological dual of Banach latticeWe start this work by a definition of almost L-Dunford-Pettis set, which is a disjoint
version of L-Dunford-Pettis set.
Definition 2.1. Let E be a Banach lattice. A norm bounded subset A of E′ is called analmost L-Dunford-Pettis set, if every disjoint weakly null sequence (xn), which is a DP setin E converge uniformly to zero on A, that is, lim
n→∞supf∈A |f(xn)| = 0.
Now, for a norm bounded subset of a topological dual Banach lattice, we give a char-acterization of an almost L-Dunford-Pettis sets.
Almost L-Dunford-Pettis sets in Banach lattices and its applications 601
Proposition 2.2. Let E be a Banach lattice and let A be a norm bounded subset of E′.The following statements are equivalent:
(1) A is an almost L-Dunford-Pettis set in E′.(2) For every sequence (fn) in A and every disjoint weakly null sequence (xn), which
is a Dunford-Pettis set in E, we have fn(xn) → 0 as n → ∞.
Proof. (2) ⇒ (1) Assume by way of contradiction that A is not an almost L-Dunford-Pettis set in E′. Then, there exists a disjoint weakly null sequence (xn), which is aDunford-Pettis subset of E such that supf∈A |f(xn)| > ϵ > 0 for some ϵ > 0 and each n.Hence, for every n there exists some fn in A such that |fn(xn)| > ϵ, which is impossiblefrom our hypothesis (2). This prove that A is an almost L-Dunford-Pettis set in E′.
(1) ⇒ (2) Let (fn) be a sequence in A and (xn) be a disjoint weakly null sequence,which is a Dunford-Pettis set in E. Since
|fn(xn)| ≤ supf∈A |f(xn)|,
for every n, and A is an almost L-Dunford-Pettis set in E′ then, fn(xn) → 0 as n → ∞.This completes the proof.
As a consequence of Proposition 2.2, we obtain the following result.
Proposition 2.3. Let E be a Banach lattice and let (fn) be a norm bounded sequence inE′. The following statements are equivalent:
(1) The subset fn, n ∈ N is an almost L-Dunford-Pettis set in E′.(2) For every disjoint weakly null sequence (xn), which is a Dunford-Pettis set in E,
we have fn(xn) → 0 as n → ∞.
The following proposition shows that every order interval in a topological dual Banachlattice is an almost L-Dunford-Pettis set.
Proposition 2.4. Let E be a Banach lattice. Then, for every f ∈ (E′)+, [−f, f ] is analmost L-Dunford-Pettis set in E′.
Proof. Let (xn) be a disjoint weakly null sequence, which is a Dunford-Pettis set in E,and put W = xn : n ∈ N. Then, W is a relatively weakly compact set of E and (|xn|)is a disjoint sequence in the solid hull of W . Now, by Theorem 4.34 of [1], we see that(|xn|) is a weakly null sequence of E. Since
f(|xn|) = sup |g(xn)| : g ∈ [−f, f ] → 0
as n → ∞ for all f ∈ (E′)+, it follows that [−f, f ] is an almost L-Dunford-Pettis set in E′
for all f ∈ (E′)+, and this ends the proof.
From Proposition 2.4 and Theorem 1.73 of [1], we get
Corollary 2.5. Let T be an order bounded operator from a Banach lattice E into anotherBanach lattice F . Then, T ′ ([−f, f ]) is an almost L-Dunford-Pettis set in E′ for everyf ∈ (F ′)+.
Proof. Since T be an order bounded operator from a Banach lattice E into anotherBanach lattice F , by Theorem 1.73 of [1], we obtain that T ′ : F ′ → E′ is also orderbounded. Thus, T ′([−f, f ]) is an order bounded subset of E′ for all f ∈ (F ′)+, and sothere exists g ∈ (E′)+ such that T ′([−f, f ]) ⊂ [−g, g]. Now, from Proposition 2.4, weconclude that T ′([−f, f ]) is an almost L-Dunford-Pettis set in E′ for every f ∈ (F ′)+, asdesired.
In order to prove the next theorem, we need the following lemma.
602 A. Retbi
Lemma 2.6. Let E be a Banach lattice, and let (gn) be a norm bounded sequence in E+.Then the sequence defined for n ≥ 2 by
fn =(
gn − 4nn−1∑i=1
gi − 2−n∞∑
i=12−igi
)+
,
is a disjoint sequence of E+.Proof. Let n > m ≥ 2, then
0 ≤ fn ≤ (gn − 4ngm)+,
and0 ≤ 4nfm ≤ 4n(gm − 4−ngn)+
= (4ngm − gn)+
= (gn − 4ngm)−.
Since (gn − 4ngm)+⊥(gn − 4ngm)−, we deduce that fn⊥fm, as desired. Theorem 2.7. Let T be an order bounded operator from a Banach lattice E into anotherBanach lattice F , and let A be a norm bounded solid subset of F ′. The following statementsare equivalent:
(1) T ′(A) is an almost L-Dunford-Pettis set in E′.(2) T ′(fn), n ∈ N is an almost L-Dunford-Pettis set in E′, for each disjoint sequence
(fn) ⊂ A+ = A ∩ (F ′)+.Proof. (1) ⇒ (2) Obvious.
(2) ⇒ (1) Let (xn) be a disjoint weakly null sequence, which is a Dunford-Pettis setin E. To finish the proof, we have to prove that supg∈A |T ′(g)(xn)| → 0 as n → ∞.Assume by way of contradiction that supg∈A |T ′(g)(xn)| does not converge to 0 as n → ∞.So there exists some ϵ > 0 such that supg∈A |T ′(g)(xn)| > ϵ for each n. Hence, thereexists gn ∈ A+ such that gn(|T (xn)|) > ϵ for all natural number n. Let g ∈ A+. Thenfrom Corollary 2.5, we see that T ′([g, g]) is an almost L-Dunford-Pettis sets in E′, andwe have g(|T (xn)|) → 0 as n → ∞. Let n1 = 1. Since gn1(T (xn)) → 0 as n → ∞, thereexists some natural number n2 such that n2 > n1 = 1 and gn1(|T (xn2)|) < ϵ
22×2+2 . Also,because
∑2k=1 gnk
(|T (xn)|) → 0 as n → ∞, there exists some natural number n3 suchthat n3 > n2 > n1 = 1 and
∑2k=1 gnk
(|T (xn3)|) < ϵ22×3+2 . By induction, we get a strictly
increasing subsequence (nk) of N such that
(∑m−1
k=1 gnk)(|T (xnm)|) < ϵ
22m+2 for all m ≥ 2.
Now, let
h =∑∞
k=1 2−kgnk
and
fm = (gnm − 4m∑m−1k=1 gnk
− 2−mh)+ for all m ≥ 2.
So by Lemma 2.6, we see that (fm) is a disjoint sequence in (F ′)+, as 0 ≤ fm ≤ gnm ,gnm ∈ A and A is a solid subset of F ′ then, fm ∈ A+. Hence, we have
fm(|T (xnm)|) = (gnm − 4mm−1∑k=1
gnk− 2−mh)+(|T (xnm)|)
≥ (gnm − 4mm−1∑k=1
gnk− 2−mh)(|T (xnm)|)
> ϵ − ϵ
4− 2−mh(|T (xnm)|).
Almost L-Dunford-Pettis sets in Banach lattices and its applications 603
This prove that fm(|T (xnm)|) > ϵ2 for m sufficiently large (because 2−mh(|T (xnm)|) → 0).
Since fm(|T (xnm)|) = sup |T ′(y)(xnm)| , |y| ≤ fm, for m sufficiently large there existssome ym ∈ F ′ such that |ym| ≤ fm and |T ′(ym)(xnm)| > ϵ
2 . It is clear that (y+m) and (y−
m)are norm bounded disjoint sequences in A+ and so, by our hypothesis we obtain
ϵ
2<
∣∣T ′(ym)(xnm)∣∣
≤∣∣∣T ′(y+
m)(xnm)∣∣∣+ ∣∣T ′(y−
m)(xnm)∣∣
≤ supk∈N
∣∣∣T ′(y+k )(xnm)
∣∣∣+ supk∈N
∣∣∣T ′(y−k )(xnm)
∣∣∣ → 0,
as m → ∞. This leads to a contradiction, and we are done. As a consequence of Theorem 2.7, we obtain the following result.
Corollary 2.8. Let T be an order bounded operator from a Banach lattice E into anotherBanach lattice F , and let A be a norm bounded solid subset of F ′. The following statementsare equivalent:
(1) T ′(A) is an almost L-Dunford-Pettis set in E′.(2) fn(T (xn)) → 0 as n → ∞, for every disjoint weakly null sequence (xn), which is a
Dunford-Pettis set in E+ and for each disjoint sequence (fn) in A+.
Next, we derive another consequence of Theorem 2.7.
Corollary 2.9. Let E be a Banach lattice and let A be a norm bounded solid subset ofE′. The following statements are equivalent:
(1) A is an almost L-Dunford-Pettis set in E′.(2) fn, n ∈ N is an almost L-Dunford-Pettis set in E′, for each disjoint sequence
(fn) ⊂ A+ = A ∩ (F ′)+.
3. Almost L-Dunford-Pettis set, aDPcc operator and PDPrcPThe following theorem gives a new characterization of order bounded aDPcc operator
from a Banach lattice E into another F in term of almost L-Dunford-Pettis sets in E′.
Theorem 3.1. For an order bounded operator T from a Banach lattice E into anotherF . The following statements are equivalent:
(1) T is an aDPcc operator.(2) T ′(BF ′) is an almost L-Dunford-Pettis set in E′.(3) T ′(fn), n ∈ N is an almost L-Dunford-Pettis set in E′, for each disjoint sequence
(fn) ⊂ B+F ′.
(4) fn(T (xn)) → 0 as n → ∞, for every disjoint weakly null sequence (xn), which is aDunford-Pettis set in E+ and for each disjoint sequence (fn) ⊂ B+
F ′.
Proof. (1) ⇔ (2) Let (xn) be a disjoint weakly null sequence, which is a Dunford-Pettissubset of E′. Since
∥T (xn)∥ = supf∈T ′(BF ′ ) |f(xn)|,
then, it is clear that T is an aDPcc operator if and only if T ′(BF ′) is an almost L-Dunford-Pettis in E′.
(2) ⇔ (3) Follows from Theorem 2.7.(3) ⇔ (4) Follows from Proposition 2.3. As a simple consequence of Theorem 3.1, we get a characterization of PDPrcP in Banach
lattices.
604 A. Retbi
Corollary 3.2. Let E be a Banach lattice. The following statements are equivalent:(1) E has the PDPrcP.(2) BE′ is an almost L-Dunford-Pettis set.(3) fn, n ∈ N is an almost L-Dunford-Pettis set in E′, for each disjoint sequence
(fn) ⊂ B+E′.
(4) fn(xn) → 0 as n → ∞, for every disjoint weakly null sequence (xn), which is aDunford-Pettis set in E+ and for each disjoint sequence (fn) ⊂ B+
E′.
In the next result, we obtain a new characterization of PDPrcP in Banach lattices interm of almost L-Dunford-Pettis sets.
Theorem 3.3. A Banach lattice E has the PDPrcP if and only if every bounded subsetof E′ is an almost L-Dunford-Pettis set.
Proof. For the "if" part, since BE′ is an almost L-Dunford-Pettis set, by Corollary 3.2 weconclude that E has the PDPrcP.
For the "only if" part, assume by way of contradiction that there exists a bounded subsetA, which is not an almost L-Dunford-Pettis set of E′. Then, there exists a disjoint weaklynull sequence (xn), which is a Dunford-Pettis set of E such that supf∈A |f(xn)| > ϵ > 0 forsome ϵ > 0 and each n. Hence, for every n there exists some fn in A such that |fn(xn)| > ϵ.
On the other hand, since (fn) ⊂ A, there exists some K > 0 such that ∥fn∥E′ ≤ K forall n. Thus,
|fn(xn)| ≤ K ∥xn∥,
for each n, so by our hypothesis, |fn(xn)| → 0 as n → ∞, which is impossible. Thiscompletes the proof.
Let us define the following.
Definition 3.4. Let E be a Banach lattice, E has the property (a) if for every weaklynull sequence (xn), which is a Dunford-Pettis set in E we have |xn| → 0 for σ(E, E′) asn → ∞.
Remark 3.5. Let E be a Banach lattice. Note that E is discrete with order continuousnorm ⇒ the lattice operations of E are weakly sequentially continuous (see Proposition2.5.23 of [5]) ⇒ E has the property (a).
We need to recall of the following characterization of aDPcc operators, which is estab-lished in Theorem 3.9 of [4].
Theorem 3.6. An operator T from a Banach lattice E into a Banach space Y is aDPccif and only if ∥T (xn)∥ → 0 as n → ∞ for every weakly null sequence (xn), which is aDunford-Pettis set in E+.
In the following result, we establish a sufficient condition such that the class of aDPccoperators and the class of DPcc operators coincide.
Theorem 3.7. Let E be a Banach lattice and Y be a Banach space such that E has theproperty (a), then each aDPcc operator from E into Y is DPcc.
Proof. Let T be an aDPcc operator from E into Y . We prove that T is DPcc, let (xn) bea weakly null sequence, which is a Dunford-Pettis set in E. Since E has the property (a)then (x+
n ) and (x−n ) be weakly null sequences in E+, and it is clear that are Dunford-Pettis
sets. Now, it follows from Theorem 3.6 that∥∥T (x+
n )∥∥ → 0 and ∥T (x−
n )∥ → 0 as n → ∞.Thus,
∥T (xn)∥ =∥∥T (x+
n ) − T (x−n )∥∥ ≤
∥∥T (x+n )∥∥+ ∥T (x−
n )∥ → 0 as n → ∞,
and we are done.
Almost L-Dunford-Pettis sets in Banach lattices and its applications 605
Now, from Theorem 3.7 and Corollary 3.20 of [4], we derive
Corollary 3.8. Let E and F be two Banach lattices such that E has the property (a) orF is discrete with order continuous norm, then each positive aDPcc operator from E intoF is DPcc.
The following result give a necessary and sufficient condition such that each order in-terval in a topological dual Banach lattice is an L-Dunford-Pettis set.
Proposition 3.9. Let E be a Banach lattice. The following statements are equivalent:(1) For every f ∈ (E′)+, [−f, f ] is an L-Dunford-Pettis set in E′.(2) E has the property (a).
Proof. Let (xn) be a weakly null sequence, which is a Dunford-Pettis set of E, then theresult follows from the equality:
f(|xn|) = sup |g(xn)| : g ∈ [−f, f ],
for every f ∈ (E′)+ and every n. We need the following proposition.
Proposition 3.10. A Banach space X has the DPrcP if and only if the closed unit ballBX′ of X ′ is L-Dunford-Pettis.
Proof. Let (xn) be a weakly null sequence, which is a Dunford-Pettis set of X, then theresult follows from the equality:
∥xn∥ = supf∈BX′ |f(xn)|,
for every n. Remark 3.11. It is clear that every L-Dunford-Pettis set in a dual Banach lattice isalmost L-Dunford-Pettis, but the converse is not true in general. In fact, if we put E =L1 [0, 1] ⊕ L2 [0, 1] then, E has the PDPrcP but does not have the DPrcP (see Example3.4 of [4]), hence from Corollary 3.2 and Proposition 3.10, we see that the closed unit ballBE′ is an almost L-Dunford-Pettis set but it is not L-Dunford-Pettis.
Now, we are in a position to give our major result, and we characterize Banach latticeE such that each almost L-Dunford-Pettis set of E′ is L-Dunford-Pettis.
Theorem 3.12. Let E be a Banach lattice. The following statements are equivalent:(1) Each almost L-Dunford-Pettis set of E′ is L-Dunford-Pettis.(2) E has the property (a).(3) Each aDPcc operator from E to any Banach lattice F is DPcc.(4) Each aDPcc operator from E to ℓ∞ is DPcc.
Proof. (1) ⇒ (2) Let f ∈ (E′)+ then, [−f, f ] is an almost L-Dunford-Pettis set in E′ (seeProposition 2.4), and by our hypothesis, we have that [−f, f ] is an L-Dunford-Pettis setin E′. Now, from Proposition 3.9, we see that E has the propery (a).
(2) ⇒ (3) Let T be an aDPcc operator from E to any Banach lattice F , since E hasthe property (a) then, by Theorem 3.7, T is DPcc operator.
(3) ⇒ (4) Obvious.(4) ⇒ (1) Suppose by way of contradiction that there exist an almost L-Dunford-Pettis
set A in E′ which is not L-Dunford-Pettis. As A is not L-Dunford-Pettis subst of E′, sothere exists a weakly null sequence (xn), which is a Dunford-Pettis subset of E such thatsupf∈A |f(xn)| > ϵ > 0 for some ϵ > 0 and each n. Hence, for every n there exists somefn in A such that |fn(xn)| > ϵ.
On the other hand, consider the operator T : E → ℓ∞ defined by
606 A. Retbi
T (x) = (fn(x))∞n=0 for all x ∈ E.
We show that T is aDPcc operator. Since A is almost L-Dunford-Pettis subst of E′, thenfor every disjoint weakly null sequence (ym), which is a Dunford-Pettis of E, we obtain
∥T (ym)∥∞ = ∥(fn(ym))∞n=0∥∞
= supn∈N
|fn(ym)|
≤ supf∈A
|f(ym)| → 0,
as m → ∞, this prove that T is aDPcc, and by our hypothesis we see that T is DPcc.Now, we have
ϵ < |fn(xn)| ≤ ∥T (xn)∥∞ → 0, as n → ∞,
which is impossible, and this ends the proof. Consequently, we obtain some sufficient conditions such that the PDPrcP and DPrcP
in Banach lattice conicide.
Corollary 3.13. Let E be a Banach lattice. Suppose that one of the following assertionsis valid:
(1) Each almost L-Dunford-Pettis set of E′ is L-Dunford-Pettis.(2) E has the property (a).(3) The lattice operations of E are weakly sequentially continuous.(4) E is discrete.(5) Each aDPcc operator from E to ℓ∞ is DPcc.
Then, E has the PDPrcP if and only if E has the DPrcP.
Proof. (1), (2) and (5) Follows from Theorem 3.12, in particular, we put in assertion (3)of this Theorem F = E and T = IdE : E → E the identity operator.
(3) Follows from Remark 3.5 and (2).(4) If E has the PDPrcP, then, its norm is order continuous, and as E is discrete so by
Remark 3.5 and assertion (3), we deduce that E has the DPrcP, and this completes theproof.
4. aL-Dunford-Pettis property in Banach latticesLet E be a Banach lattice, note that each relatively weakly compact subset A of a dual
topological Banach lattice E′ is L-Dunford-Pettis (see Proposition 2.3 of [6]), and henceA is almost L-Dunford-Pettis. The converse of this property is not true in general, in fact,the closed unit ball Bℓ∞ of ℓ∞ is almost L-Dunford-Pettis set (see Corollary 3.2), but itis not relatively weakly compact.
Now, we give the following definition.
Definition 4.1. A Banach lattice E has the aL-Dunford-Pettis property, if every almostL-Dunford-Pettis set in E′ is relatively weakly compact.
Note that an aDPcc operator is not weakly compact in general. In fact, Idℓ1 is aDPcc,but it is not weakly compact.
Used the idea of aL-Dunford-Pettis property in Banach lattice, we establish the weakcompactness of aDPcc operators.
Theorem 4.2. Let E be a Banach lattice, then, the following assertions are equivalent:(1) E has the aL-Dunford-Pettis property,(2) for each Banach space Y, every aDPcc operator from E into Y is weakly compact,(3) every aDPcc operator from E into ℓ∞ is weakly compact.
Almost L-Dunford-Pettis sets in Banach lattices and its applications 607
Proof. (1) ⇒ (2) Suppose that E has the aL-Dunford-Pettis property and T : E → Y isaDPcc operator. Thus T ′(BY ′) is an almost L-Dunford-Pettis set in E′. So by hypothesis,it is relatively weakly compact and T is a weakly compact operator.
(2) ⇒ (3) Obvious.(3) ⇒ (1) If E does not have the aL-Dunford-Pettis property, there exists an almost
L-Dunford-Pettis subset A of E′ which is not relatively weakly compact. So there isa sequence (fn) ⊆ A with no weakly convergent subsequence. Now, we show that theoperator T : E → ℓ∞ defined by T (x) = (fn(x)) for all x ∈ E is aDPcc but it is notweakly compact. As (fn) ⊆ A is almost L-Dunford-Pettis set, then for every disjointweakly null sequence (xm), which is a Dunford-Pettis set in E we have
∥T (xm)∥ = supn |fn(xm)| → 0, as m → ∞,
so T is aDPcc operator. Hence T ′((λn)∞n=1) =
∑∞n=1 λnfn for every (λn)∞
n=1 ∈ ℓ1 ⊂ (ℓ∞)′.If e′
n is the usual basis element in ℓ1 then T ′(e′n) = fn, for all n ∈ N . Thus, T ′ is not a
weakly compact operator and neither is T . This finishes the proof. As a consequence of Theorem 4.2, we derive the following result.
Corollary 4.3. A PDPrc space has the aL-Dunford-Pettis property if and only if it isreflexive.
Proof. (⇒) If a Banach lattice E has the PDPrcP, then the identity operator IdE on Eis aDPcc. As E has the aL-Dunford-Pettis property, it follows from Theorem 4.2 that IdE
is weakly compact, and hence E is reflexive.(⇐) Obvious.
Remark 4.4. Note that the Banach lattice ℓ1 is not reflexive and has the PDPrcP, thenfrom Corollary 4.3, we conclude that ℓ1 does not have the aL-Dunford-Pettis property.
Acknowledgment. The author would like to thank the referee for his comments whichhave improved this paper.
References[1] C.D. Aliprantis and O. Burkinshaw, Positive operators, Reprint of the 1985 original.
Springer, Dordrecht, 2006.[2] K.T. Andrews, Dunford-Pettis sets in the space of Bochner integrable functions, Math.
Ann. 241, 35-41, 1979.[3] K. Bouras, Almost Dunford-Pettis sets in Banach lattices, Rend. Circ. Mat. Palermo.
62, 227-236, 2013.[4] K. El Fahri, N. Machrafi and M. Moussa, Banach Lattices with the Positive Dunford-
Pettis Relatively Compact Property, Extracta Math. 30 (2), 161-179, 2015.[5] P. Meyer-Nieberg, Banach lattices, Universitext, Springer-Verlag, Berlin, 1991.[6] A. Retbi and B. El Wahbi, L-Dunford-Pettis property in Banach spaces, Methods
Funct. Anal. Topology, accepted.[7] Y. Wen and J. Chen, Characterizations of Banach Spaces With Relatively Compact
Dunford-Pettis Sets, Advances in Mathematics (China), 45 (1), 122-132, 2016.
Hacettepe Journal ofMathematics & Statistics
Hacet. J. Math. Stat.Volume 49 (2) (2020), 608 – 616
DOI : 10.15672/hujms.478373
Research Article
Ideal based trace graph of matricesT. Tamizh Chelvam∗, M. Sivagami
Department of Mathematics, Manonmaniam Sundaranar University, Abishekapatti, Tirunelveli 627012,Tamil Nadu, India.
AbstractLet R be a commutative ring and Mn(R) be the set of all n × n matrices over R wheren ≥ 2. The trace graph of the matrix ring Mn(R) with respect to an ideal I of R, denotedby ΓIt(Mn(R)), is the simple undirected graph with vertex set Mn(R) \ Mn(I) and twodistinct vertices A and B are adjacent if and only if Tr(AB) ∈ I. Here Tr(A) representsthe trace of the matrix A. In this paper, we exhibit some properties and structure ofΓIt(Mn(R)).
Mathematics Subject Classification (2010). 16S50, 05C25, 13A15, 05C69
Keywords. trace graph, matrix ring, ideal-based, clique number
1. IntroductionThe concept of associating graphs to commutative rings was first introduced by Beck [3].
He introduced the concept of zero-divisor graph of a commutative ring R as an undirectedgraph whose vertices are the elements of R with two distinct vertices x and y joined by anedge if and only if xy = 0. Later on, Anderson and Livingston [2] modified the definitionwith vertex set, the set of all nonzero zero divisors of R and introduced the zero-divisorgraph Γ(R) corresponding to a commutative ring R. In [9], Redmond introduced the notionof the zero-divisor graph with respect to an ideal I of a commutative ring R, denoted byΓI(R), as the graph with vertex set x ∈ R \ I : xy ∈ I for some y ∈ R \ I, and twodistinct vertices x and y are adjacent if and only if xy ∈ I. The concept of trace graphof a matrix ring over a commutative ring was introduced by Almahdi, Louartiti, andTamekkante [1]. Several authors have extensively studied about zero-divisor graph withrespectxzs to an ideal. For example one may refer [8]. Let R be a commutative ring andn be a positive integer. Let Mn(R) denote the set of all n × n matrices over R, Mn(R)∗
denotes the set of all n × n non-zero matrices over R and let Tr(A) be the trace of thematrix A ∈ Mn(R). The trace graph of the matrix ring Mn(R), denoted by Γt(Mn(R)),is the simple undirected graph with vertex set A ∈ Mn(R)∗ : there exists B ∈ Mn(R)∗
such that Tr(AB) = 0 and two distinct vertices A and B are adjacent if and only ifTr(AB) = 0. Further study on the trace graph of matrices was done by authors [10].
In this paper, as a parallel approach of generalization of Γ(R) to ΓI(R), we generalizethe notion of the trace graph Γt(Mn(R)) of a matrix ring Mn(R) to the trace graphΓIt(Mn(R)) with respect to an ideal I of R. Actually ΓIt(Mn(R)) is the simple undirected∗Corresponding Author.Email addresses: [email protected] (T. Tamizh Chelvam), [email protected] (M. Sivagami)Received: 04.11.2018; Accepted: 04.02.2019
Ideal based trace graph of matrices 609
graph with vertex set Mn(R) \ Mn(I) and two distinct vertices A and B are adjacent ifand only if Tr(AB) ∈ I. Note that if A ∈ Mn(I), then Tr(AB) ∈ I for every B ∈ Mn(R).Due to this, matrices in Mn(I) are not considered for the vertex set of ΓIt(Mn(R)). Asusual, Eij denotes the matrix whose ijth entry is 1 and 0 elsewhere. For a set X, |X|denotes the cardinality of X, X \ Y denotes the set of elements that belong to X and notto set Y . For basic definitions on rings, one may refer [6] and for noncommutative ringssee [5, 7].
Let G be a graph. For distinct vertices x and y of G, let d(x, y) be the length of theshortest path between x and y (d(x, y) = ∞ if there is no such path). The diameter of Gis diam(G) = supd(x, y) : x and y are distinct vertices of G. The girth of G, denotedby gr(G), is defined as the length of the shortest cycle in G (gr(G) = ∞ if G contains nocycles). For a graph G and a vertex v ∈ V (G), the eccentricity e(v) of v is the maximumdistance to any vertex in the graph, i.e., e(v) = max
u∈V (G)d(v, u). The radius rad(G) of a G
is the minimum eccentricity among all vertices in G and a vertex of G is a central vertexif e(v) = rad(G). G is self-centered if every vertex is in the center i.e., e(v) = rad(G) forevery vertex v ∈ V (G). A subset Ω of V (G) is called a clique if the induced subgraph ofΩ is complete. The order of the largest clique in G is its clique number, which is denotedby ω(G). The chromatic number of a graph G, denoted by χ(G), is the smallest numberof colors needed to color the vertices of G so that no two adjacent vertices share the samecolor. An independent set or stable set is a set of vertices in a graph G such that no two ofthem are adjacent. A maximum independent set is an independent set of largest possiblesize for the given graph G. This size is called the independence number of G and denotedby α(G).
If the edges of G are partitioned into subgraphs H1, . . . , Hk, . . . Hn, then we write G ∼=H1 ⊕· · ·⊕Hn, and if Hi
∼= Hj for all 1 ≤ i, j ≤ k, then we write G ∼= kH ⊕Hk+1 ⊕· · ·⊕Hn,where H ∼= Hi, (1 ≤ i ≤ k). For general reference of graph theoretical terms and results,we refer [11].
Remark 1.1. Let R be a commutative ring and n be a positive integer.1. The graph ΓIt(M1(R)) coincides with ΓI(R) (ideal based zero-divisor graph of the
ring R).2. If I = (0), then ΓIt(Mn(R)) = Γt(Mn(R)) for all n ≥ 1.3. If I = R, then ΓIt(Mn(R)) is the null graph.
Throughout this paper, unless otherwise specified, R is a commutative ring with identity,n ≥ 2 is an integer, and I is a non-trivial ideal of R. If A = [aij ] ∈ Mn(R) \ Mn(I) thecorresponding matrix in Mn(R/I) is [aij +I]. If A = [aij ] ∈ Mn(I), then the correspondingmatrix in Mn(R/I) is the zero matrix in Mn(R/I). For convenience, we denote the matrix[aij + I] ∈ Mn(R/I) as A corresponding to the matrix A = (aij). In Section 2, we provethat for n ≥ 2, ΓIt(Mn(R)) is a connected graph of diameter 2 and of girth 3. In Section 3,we study the structure of ΓIt(Mn(R)) through the relationship between ΓIt(Mn(R)) andΓt(Mn(R/I)). In Section 4, we discuss the clique, chromatic, and independence numbersof ΓIt(Mn(R)).
2. Girth and diameterIn this section, we list some properties of the trace graph of matrix ring with re-
spect to an ideal I of R that can be proved by similar arguments as in the case of thetrace graph of matrix rings over commutative rings. For A = [aij ] ∈ Mn(R), we setJI(A) =
∑1≤i,j≤n
(R/I)(aij + I) ∈ (R/I); the sum of the ideals of R/I generated by all
entries of A = [aij + I] over R/I. Note that JI(A) is an ideal of R/I.
610 T. Tamizh Chelvam, M. Sivagami
Proposition 2.1. For a non-zero ideal I of R and an integer n ≥ 2, ΓIt(Mn(R))) containsno isolated vertex.
Proof. Let A = [aij ] ∈ Mn(R) \ Mn(I).
Case 1. If A = In and Tr(A) ∈ I, then A is adjacent to the identity matrix In.
Case 2. Assume that Tr(A) /∈ I.
Case 2.1. Suppose A has exactly one entry akℓ such that akℓ /∈ I. Choose B = [bij ]such that bℓk ∈ I and bij /∈ I otherwise. Then B ∈ Mn(R) \ Mn(I) and Tr(AB) =a11b11 + · · · + a1nbn1 + a21b12 + · · · + a2nbn2 + · · · + an1b1n + · · · + annbnn. Note that ineach term of Tr(AB) either aij ∈ I or bij ∈ I. Since I is an ideal of R, aijbij ∈ I for every1 ≤ i, j ≤ n and hence their sum belongs to I. Thus Tr(AB) ∈ I.
Case 2.2. Suppose that A has at least two entries akℓ, ak1ℓ1 which are not elementsof I. Then choose B = [bij ] such that bℓk = −ak1ℓ1 , bℓ1k1 = akℓ, bij ∈ I elsewhere. ThusB ∈ Mn(R)\Mn(I) and Tr(AB) = akℓbℓk +ak1ℓ1bℓ1k1 + elements of I. Hence Tr(AB) ∈ I.
Thus in all the cases for every A ∈ Mn(R) \ Mn(I), there exists B ∈ Mn(R) \ Mn(I) suchthat Tr(AB) ∈ I. Hence, ΓIt(Mn(R))) contains no isolated vertex.
In the following, we prove that no vertex in ΓIt(Mn(R))) is adjacent to all other vertices.
Proposition 2.2. For a non-zero ideal I of R and an integer n ≥ 2, no vertex ofΓIt(Mn(R)) is adjacent to every other vertex of ΓIt(Mn(R)).
Proof. Given a matrix A = [aij ] ∈ Mn(R) \ Mn(I). There exists at least one entry akℓ
such that akℓ /∈ I. Choose B = [bij ] such that bℓk = 1 and bij = 0 elsewhere. ThusB ∈ V (ΓIt(Mn(R))) and Tr(AB) = akℓ /∈ I. If A = B, then Tr(AIn) /∈ I.
Now we obtain, the degree of vertices in ΓIt(Mn(R)).
Proposition 2.3. Let R be a finite commutative ring and n ≥ 2 be an integer.1. For any vertex A of ΓIt(Mn(R)), we have:
a. deg(A) = |R|n2
|JI(A)| − 1 if Tr(A2) /∈ I, and
b. deg(A) = |R|n2
|JI(A)| − 2 if Tr(A2) ∈ I.
2. δ(Γt(Mn(R))) = |R|n2−1|I| − 2.
Proof. 1. Let A ∈ Mn(R). Consider fA : Mn(R) → R defined by fA(B) =Tr(AB) andnatural homomorphism φ : R → R/I by φ(x) = x + I. Clearly φ fA : Mn(R) → R/Iis a surjective homomorphism with (φ fA)(B) =Tr(AB) + I, Im(φ fA) = JI(A) andker(φ fA) = B ∈ Mn(R)| Tr(AB) ∈ I.
By the isomorphism theorem, Mn(R)ker(φfA)
∼= JI(A) and so |ker(φfA)| = |Mn(R)||JI(A)| = |R|n2
|JI(A)| .
When Tr(A2) /∈ I, ker(φ fA) contains exactly the vertices adjacent to A and the zeromatrix. When Tr(A2) ∈ I, ker(φ fA) contains additionally A. Hence (a) and (b) hold.
2. Consider the matrix A = [aij ] ∈ Mn(R)\Mn(I) with aii ∈ I for every 1 ≤ i ≤ n, aij /∈ Iimplies aji ∈ I for every i = j and aij is a unit for some i and j. Clearly JI(A) = R/I andTr(A2) ∈ I. Thus by 1(b), we have deg(A) = |R|n2−1|I| − 2 and so δ ≤ |R|n2−1|I| − 2.
Since |JI(A)| ≤ |R/I| for every ideal JI(A) of R/I, |R|n2
|JI(A)| ≥ |R|n2−1|I|. From this
|R|n2−1|I| − 2 ≤ |R|n2
|JI(A)| − 2 ≤ deg(A) for every A ∈ Mn(R). Thus, δ = |R|n2−1|I| − 2.
Ideal based trace graph of matrices 611
From Proposition 2.3, for a finite commutative ring R, ΓIt(Mn(R)) can never be anEulerian graph. For, consider the matrices E11 and E1n where n = 1. Tr(E2
11) = 1 /∈ Iand Tr(E2
1n) = 0 ∈ I. Hence, by the Proposition 2.3, either of E11 and E1n must have odddegree.
Proposition 2.4. Let R be a commutative ring, n ≥ 2 be an integer and I be a non trivialideal of R. Then ΓIt(Mn(R)) is connected with diam(ΓIt(Mn(R))) = 2 and gr(ΓIt(Mn(R)))= 3.
Proof. Let A = [aij ] and B = [bij ] be two distinct elements of Mn(R)\Mn(I). If Tr(AB) ∈I, then d(A, B) = 1. Assume that Tr(AB) /∈ I. By Proposition 2.2, diam(ΓIt(Mn(R))) > 1.Now let us consider two cases:
Case 1. Suppose aijbkℓ − akℓbij ∈ I for each (i, j), (k, l) ∈ 1, . . . , n2.Let (i0, j0) and (i1, j1) be two distinct elements of 1, . . . , n2 such that ai0j0 /∈ I. Considerthe matrix C = [cij ] with cj0i0 = −ai1j1 , cj1i1 = ai0j0 , and ckℓ ∈ I elsewhere. ThenC ∈ Mn(R) \ Mn(I) and
Tr(AC) = ai0j0cj0i0 + ai1j1cj1i1 + elements of I
= −ai0j0ai1j1 + ai1j1ai0j0 + elements of I ∈ I
andTr(BC) = bi0j0cj0i0 + bi1j1cj1i1 + elements of I
= −bi0j0ai1j1 + bi1j1ai0j0 + elements of I ∈ I.
Case 2. Suppose there exist (i0, j0), (i1, j1) ∈ 1, . . . , n2 such that ai0j0bi1j1 − ai1j1bi0j0 /∈I.Let (i2, j2) ∈ 1, . . . , n2 \ (i0, j0), (i1, j1) and consider the matrix C = [cij ] where
cj0i0 = ai1j1bi2j2 − ai2j2bi1j1 ,
cj1i1 = ai2j2bi0j0 − ai0j0bi2j2 ,
cj2i2 = ai0j0bi1j1 − ai1j1bi0j0 andckℓ ∈ I elsewhere.
Then C ∈ Mn(R) \ Mn(I) andTr(AC) = ai0j0cj0i0 + ai1j1cj1i1 + ai2j2cj2i2
= ai0j0ai1j1bi2j2 − ai0j0ai2j2bi1j1 + ai1j1ai2j2bi0j0
− ai1j1ai0j0bi2j2 + ai2j2ai0j0bi1j1 − ai2j2ai1j1bi0j0 + elements of I ∈ I,
and Tr(BC) = bi0j0cj0i0 + bi1j1cj1i1 + bi2j2cj2i2
= bi0j0ai1j1bi2j2 − bi0j0ai2j2bi1j1 + bi1j1ai2j2bi0j0
− bi1j1ai0j0bi2j2 + bi2j2ai0j0bi1j1 − bi2j2ai1j1bi0j0 + elements of I ∈ I.
In both cases, A = C and B = C (otherwise Tr(AB) ∈ I) and hence d(A, B) = 2.Consequently, ΓIt(Mn(R)) is connected and diam(ΓIt(Mn(R))) = 2.Consider nonzero distinct matrices A = [aij ] with a11 = 1 and aij ∈ I elsewhere, B = [bij ]with bnn = 1 and bij ∈ I elsewhere and C = [cij ] with c1n = 1 and cij ∈ I elsewhere. Bythe choice of A, B, C, we have Tr(AB), Tr(BC), Tr(AC) ∈ I. Thus A − B − C − A is acycle, and so gr(ΓIt(Mn(R))) = 3. Remark 2.5. (i). By Propositions 2.2 and 2.4, the eccentricity of every vertex in
ΓIt(Mn(R)) is 2 and hence the radius of ΓIt(Mn(R)) is 2. i.e., the graph ΓIt(Mn(R))is self-centered.
(ii). By Proposition 2.4 ΓIt(Mn(R)) contains an odd cycle, and so ΓIt(Mn(R)) cannever be a bipartite graph.
612 T. Tamizh Chelvam, M. Sivagami
3. Relationship between ΓIt(Mn(R)) and Γt(Mn(R/I))In this section, we study the graph ΓIt(Mn(R)) through Γt(Mn(R/I)). The following
theorem is useful in the further discussion of this paper.
Theorem 3.1. Let R be a ring and I be an ideal of R. Then Mn(R)/Mn(I) ∼= Mn (R/I) .
Proof. The map φ : Mn(R)/Mn(I) → Mn (R/I) by [aij ] + Mn(I) = [aij + I] defines anisomorphism between Mn(R)/Mn(I) and Mn (R/I) . Note 3.2. From the isomorphism defined in Theorem 3.1, given an ideal I of R and amatrix A ∈ Mn(R), we can view the trace of the coset A + Mn(I) in Mn(R)/Mn(I) asthe trace of A in Mn(R/I). Thus, the trace graph of Mn(R)/Mn(I) is the trace graph ofMn(R/I).
Theorem 3.3. Let I be an ideal of a commutative ring R, n ≥ 2 be a positive integer andA = [aij ], B = [bij ] ∈ Mn(R) \ Mn(I) Then the following are true:
1. If A is adjacent to B in Γt(Mn(R/I)), then A and B are adjacent in ΓIt(Mn(R)).2. If A is adjacent to B in ΓIt(Mn(R)) and A = B, then A is adjacent to B in
Γt(Mn(R/I)).3. If A is adjacent to B in ΓIt(Mn(R)) and A = B, then Tr(A2), Tr(B2) ∈ I.4. If Tr(A2) ∈ I and A = B, then A is adjacent to B in ΓIt(Mn(R)) and Tr(B2) ∈ I.5. If A and B are (distinct) adjacent vertices in ΓIt(Mn(R)), then all (distinct) el-
ements of A are adjacent to all elements of B in ΓIt(Mn(R)). In particular, ifTr(A2) ∈ I, then all the distinct elements of A are adjacent in ΓIt(Mn(R)).
Proof. 1. In view of the fact mentioned in Note 3.2, it is enough to prove that A+Mn(I)is adjacent to B + Mn(I) in Γt (Mn(R)/Mn(I)) implies A is adjacent to B in ΓIt(Mn(R)).When A + Mn(I) is adjacent to B + Mn(I) in Γt (Mn(R)/Mn(I)) , we have Tr(AB +Mn(I)) = Mn(I) and so Tr(AB) ∈ I. Thus A is adjacent to B in ΓIt(Mn(R)).2. If A is adjacent to B in ΓIt(Mn(R)), then Tr(AB) ∈ I. This gives that Tr(AB) + I = Iand hence Tr(AB + Mn(I)) = Mn(I). Thus, Tr((A + Mn(I))(B + Mn(I))) = Mn(I), andso A + Mn(I) is adjacent to B + Mn(I) in Γt (Mn(R)/Mn(I)) .
3. If A is adjacent to B in ΓIt(Mn(R)), by (2) above Tr((A + Mn(I))(B + Mn(I))) =Mn(I). Since A + Mn(I) = B + Mn(I), Tr((A + Mn(I))(A + Mn(I))) = Mn(I). i.e.,Tr(A2+Mn(I)) = Mn(I) giving Tr(A2)+I = I. Thus Tr(A2) ∈ I and similarly Tr(B2) ∈ I.4. If Tr(A2) ∈ I, then Tr(A2) + I = I and so Tr(A2 + Mn(I)) = Mn(I). Thus Tr((A +Mn(I))(B+Mn(I))) = Mn(I) giving Tr(AB)+I = I. Thus Tr(AB) ∈ I. i.e., A is adjacentto B in ΓIt(Mn(R)). By (3), Tr(B2) ∈ I.
5. It is enough to prove that if A and B are (distinct) adjacent vertices in ΓIt(Mn(R)),then all (distinct) elements of A + Mn(I) are adjacent to all elements of B + Mn(I) inΓIt(Mn(R)). In particular, if Tr(A2) ∈ I, then all the distinct elements of A + Mn(I) areadjacent in ΓIt(Mn(R)).
By (1) and (2), if A and B are adjacent vertices in ΓIt(Mn(R)), then all (distinct)elements of A + Mn(I) and B + Mn(I) are adjacent in ΓIt(Mn(R)). As a particular case,taking B = A, we get if Tr(A2) ∈ I, then all the distinct elements of A + Mn(I) areadjacent in ΓIt(Mn(R)). Corollary 3.4. Let I be an ideal of a commutative ring R and n ≥ 2 be a positive integer.Then ΓIt(Mn(R)) contains |Mn(I)| disjoint subgraphs each isomorphic to Γt (Mn(R/I)) .
Proof. Let Aii∈Λ be distinct coset representatives of elements in the quotient ringMn(R)/Mn(I). Then the vertex set of Γt(Mn(R)/Mn(I)) is partitioned into Ai+Mn(I)i∈Λ.
Ideal based trace graph of matrices 613
Note that Ai +Mn(I) = Aj +Mn(I) for i = j. Fix X ∈ Mn(I). Consider the subgraph HX
with vertex set Ai + X : i ∈ Λ ⊆ V (ΓIt(Mn(R))) and two vertices Ai + X and Aj + Xare adjacent in HX if Ai + Mn(I) and Aj + Mn(I) are adjacent in Γt (Mn(R)/Mn(I)) .Clearly, HX is isomorphic to Γt (Mn(R)/Mn(I)) .
Assume that Ai + X and Aj + X are adjacent in HX . By the definition of HX , Ai +Mn(I) is adjacent to Aj + Mn(I) in Γt (Mn(R)/Mn(I)) . By Theorem 3.3(1), Ai and Aj
are adjacent in ΓIt(Mn(R)). By Theorem 3.3(4), Ai + X and Aj + X are adjacent inΓIt(Mn(R)). Hence HX is a subgraph of ΓIt(Mn(R)).
Also, for any Y (= X) ∈ Mn(I), V (HX) ∩ V (HY ) = ϕ. Thus, ΓIt(Mn(R)) contains|Mn(I)| disjoint subgraphs each isomorphic to Γt (Mn(R)/Mn(I)) and so contains |Mn(I)|disjoint subgraphs isomorphic to Γt (Mn(R/I)) . Remark 3.5. The following are true:
1. Γt (Mn(R)/Mn(I)) is a graph with |Mn(R/I)| − 1 vertices.2. ΓIt(Mn(R)) is a graph with |Mn(R)| − |Mn(I)| vertices.3. Let R be a finite commutative ring. Note that Corollary 3.4 exhibits a partition
of ΓIt(Mn(R)) into vertex disjoint subgraphs. Thus|Mn(I)| |V (Γt (Mn(R/I)))| = |V (ΓIt(Mn(R)))| .
The following theorem puts forth a partition of ΓIt(Mn(R)) into edge disjoint subgraphs.In view of Proposition 3.3(4), if Tr(A2) ∈ I and A = B, then A is adjacent to B inΓIt(Mn(R)) and Tr(B2) ∈ I. This means that if Tr(A2) ∈ I for a matrix A, then the sameis true for all matrices in the coset of A.
Theorem 3.6. Let R be a commutative ring with identity, I be a non trivial ideal of R,n ≥ 2 be an integer and
λ = |A ∈ V (Γt(Mn(R/I))) : Tr(A2) ∈ I and A is a coset representative of A|.Then ΓIt(Mn(R)) ∼= |Mn(I)|2Γt(Mn(R/I)) ⊕ λK|Mn(I)|.
Proof. Consider the partition of edges of Γt(Mn(R/I)) given below:E1 = e = (A, B) : Tr(A2),Tr(B2) /∈ IE2 = e = (A, B) : Tr(A2),Tr(B2) ∈ IE3 = e = (A, B) : either Tr(A2) or Tr(B2) ∈ I.
Let e = (A, B) ∈ E(Γt(Mn(R/I))). By Theorem 3.3(1) and (4), the subgraph induced bythe set Ve = A + N1, B + N2 : N1, N2 ∈ Mn(I) in ΓIt(Mn(R)) is
⟨Ve⟩ =
K|Mn(I)|,|Mn(I)| if Tr(A2),Tr(B2) /∈ I;K|Mn(I)|,|Mn(I)| ⊕ 2K|Mn(I)| if Tr(A2),Tr(B2) ∈ I;K|Mn(I)|,|Mn(I)| ⊕ K|Mn(I)| if either Tr(A2) or Tr(B2) ∈ I.
By [4, p.192], we have K|Mn(I)|,|Mn(I)| ∼= M(e)1 ⊕ · · · ⊕ M
(e)|Mn(I)|, where each of M
(e)i is a
perfect matching of K|Mn(I)|,|Mn(I)|. Thus,
⟨Ve⟩ =
M
(e)1 ⊕ · · · ⊕ M
(e)|Mn(I)| if Tr(A2),Tr(B2) /∈ I;
M(e)1 ⊕ · · · ⊕ M
(e)|Mn(I)| ⊕ 2K|Mn(I)| if Tr(A2),Tr(B2) ∈ I;
M(e)1 ⊕ · · · ⊕ M
(e)|Mn(I)| ⊕ K|Mn(I)| if either Tr(A2) or Tr(B2) ∈ I.
Note that Hi =⊕
e∈E(ΓtMn(R/I))M
(e)i is a subgraph of ΓIt(Mn(R)) and Hi can be di-
vided into |Mn(I)| edge disjoint subgraphs each isomorphic to Γt(Mn(R/I)), i.e., Hi∼=
|Mn(I)|Γt(Mn(R/I)).Clearly H = H1 ⊕ · · · ⊕ H|Mn(I)| is a subgraph with vertex set Mn(R) \ Mn(I) andH ∼= |Mn(I)|2Γt(Mn(R/I)). Thus ΓIt(Mn(R)) ∼= |Mn(I)|2Γt(Mn(R/I)) ⊕ λK|Mn(I)| where
λ = |A ∈ V (Γt(Mn(R/I)) : Tr(A2) ∈ I and A is a coset representative of A|.
614 T. Tamizh Chelvam, M. Sivagami
4. Chromatic, clique and independence numbers of ΓIt(Mn(R))In this section, we obtain bounds for the clique, chromatic and independence numbers
of ΓIt(Mn(R)) and obtain a condition for the chromatic and clique numbers of ΓIt(Mn(R))to be equal.
Theorem 4.1. Let n ≥ 2 be an integer, R be a commutative ring and I be a non trivialideal of R. Then the following hold:
1. ω(Γt(Mn(R/I))) ≤ ω(ΓIt(Mn(R))) ≤ |Mn(I)|ω(Γt(Mn(R/I))). Moreover, theequality ω(ΓIt(Mn(R))) = |Mn(I)|ω(Γt(Mn(R/I))) holds if there exists a clique ofmaximum order in Γt(Mn(R/I)) such that Tr(A2) ∈ I for every vertex A in theclique.
2. χ(Γt(Mn(R/I))) ≤ χ(ΓIt(Mn(R))) ≤ |Mn(I)|χ(Γt(Mn(R/I))).
Proof. 1. The first inequality follows from the fact that Γt(Mn(R/I)) is a subgraphof ΓIt(Mn(R)). Let ω(Γt(Mn(R/I))) = k. To conclude the proof, it is enough to provethat ω(ΓIt(Mn(R))) ≤ k|Mn(I)|. Since Γt(Mn(R/I)) ∼= Γt(Mn(R)/Mn(I)), we haveω(Γt(Mn(R)/Mn(I))) = k.
Suppose there exists a clique of order k|Mn(I)| + 1 in ΓIt(Mn(R)). Let B1, . . . ,Bk|Mn(I)|+1 be a clique in ΓIt(Mn(R)). Consider the set
X = B1 + Mn(I), . . . , Bk|Mn(I)|+1 + Mn(I) ⊆ V (Γt(Mn(R)/Mn(I))).Since Bi is adjacent to Bj in ΓIt(Mn(R)), for i = j, either Bi + Mn(I) = Bj + Mn(I) orBi +Mn(I) is adjacent to Bj +Mn(I) in Γt(Mn(R)/Mn(I)). Since |Bi +Mn(I)| = |Mn(I)|we have at least k + 1 distinct elements in X such that the k + 1 elements are adjacent toeach other in Γt(Mn(R)/Mn(I)). Thus ω(Γt(Mn(R)/Mn(I))) ≥ k + 1, which is a contra-diction. Hence ω(ΓIt(Mn(R))) ≤ k|Mn(I)|. The moreover case is clear from the precedingarguments and Theorem 3.3(1) and (5).
2. The first inequality is clear since Γt(Mn(R/I)) is a subgraph of ΓIt(Mn(R)). Letχ(Γt(Mn(R/I))) = k and C1, . . . , Ck be the color classes of Γt(Mn(R/I)). Consider A ∈Γt(Mn(R/I)) belongs to the color class C1 and the set XA = [aij ] ∈ ΓIt(Mn(R)) : [aij +I] = A. Note that |XA| = |Mn(I)|. Assign |Mn(I)| distinct colors C11, . . . , C1|Mn(I)| to thevertices of XA. Assign the same colors C11, . . . , C1|Mn(I)| for the vertices arising out of othervertices B ∈ Γt(Mn(R/I)) belonging to the color class C1. Since A is not adjacent to B novertex of XA is adjacent to XB. Similarly for 2 ≤ i ≤ k assigning colors, Ci1, . . . , Ci|Mn(I)|to the vertices of ΓIt(Mn(R)) arising out of the vertices of the color class Ci we havek|Mn(I)| colors and the coloring is proper. Thus χ(ΓIt(Mn(R))) ≤ k|Mn(I)|.
The following theorem is a generalization of the moreover case of Theorem 4.1(1).
Theorem 4.2. Let n ≥ 2 be an integer, R be a commutative ring and I be a non triv-ial ideal in R. Let S be a clique of maximum order in Γt(Mn(R/I)) and S have thelargest number of elements A with Tr(A2) ∈ I. Let X = A ∈ S : Tr(A2) /∈ I. Thenω(ΓIt(Mn(R))) = |X| + |Mn(I)|(ω(Γt(Mn(R/I))) − |X|).
Proof. Let |X| = |A ∈ S : Tr(A2) /∈ I| = k1, |A ∈ S : Tr(A2) ∈ I| = k2 andω(Γt(Mn(R/I))) = |S| = k. Then k1 + k2 = k. In view of Note 3.2, Γt(Mn(R/I)) ∼=Γt(Mn(R)/Mn(I)) and so ω(Γt(Mn(R)/Mn(I))) = k.
Further by our assumption on S, any maximal clique of Γt(Mn(R/I)) and hence ofΓt(Mn(R)/Mn(I)) can have at most k2 number of vertices A with Tr(A2) ∈ I.
Hence one can take the clique corresponding to S of Γt(Mn(R/I)) as a clique < A1 +Mn(I), . . . , Ak + Mn(I) > of Γt(Mn(R)/Mn(I)) with Tr(A2
i ) ∈ I for 1 ≤ i ≤ k2 andTr(A2
i ) /∈ I for k2 + 1 ≤ i ≤ k. Clearly the set Aij : Aij ∈ Ai + Mn(I), 1 ≤ i ≤
Ideal based trace graph of matrices 615
k2 and 1 ≤ j ≤ |Mn(I)| ∪ Ai2 = Ai : k2 + 1 ≤ i ≤ k is a clique of size |Mn(I)|k2 + k1in ω(ΓIt(Mn(R))).
Hence ω(ΓIt(Mn(R))) ≥ k1 + |Mn(I)|k2.
To prove our result, it is enough to prove that ω(ΓIt(Mn(R))) ≤ k1 + |Mn(I)|k2. SupposeΓIt(Mn(R)) has a clique S′ of order k1 + |Mn(I)|k2 + 1. Without loss of generality, wemay assume that S′ is a maximal clique of order ≥ k1 + |Mn(I)|k2 + 1 in ΓIt(Mn(R)). LetA ∈ S′ with Tr(A2) /∈ I.
By Theorem 3.3 (3), no vertex in the set A + B : B ∈ Mn(I)∗ is adjacent to A. HenceA + B : B ∈ Mn(I)∗ has no intersection with S′. Also note that due to the maximalityof the clique S′, if A ∈ S′ with Tr(A2) ∈ I then by Theorem 3.3 (2) and (4), the setA + B : B ∈ Mn(I) ⊂ S′.
If S′ contains at least k2|Mn(I)| + 1 vertices with Tr(A2) ∈ I, then by Theorem 3.3 (2),the clique S′
I of Γt(Mn(R)/Mn(I)) with respect to S′ contains at least k2 + 1 verticeswith Tr(A2) ∈ I which is a contradiction to our assumption that among the cliques ofΓt(Mn(R/I)), S has the largest number of elements A with Tr(A2) ∈ I.
Hence the number of vertices in S′ with Tr(A2) ∈ I is less than or equal to k2|Mn(I)|. i.e.,S′ contains at least k1+1 vertices with Tr(A2) /∈ I. Now, the clique S′
I of Γt(Mn(R)/Mn(I))corresponding to S′ contains at least k2 + k1 + 1 vertices which is a contradiction toω(Γt(Mn(R/I))) = k. Thus ω(ΓIt(Mn(R))) ≤ k1 + |Mn(I)|k2 and hence ω(ΓIt(Mn(R))) =k1 + |Mn(I)|k2. Theorem 4.3. Let n ≥ 2 be an integer, R be a commutative ring and I be a non trivialideal of R. Let Γt(Mn(R/I)) contain a clique of maximum order such that Tr(A2) ∈ Ifor every A in the clique. If χ(Γt(Mn(R/I))) = ω(Γt(Mn(R/I))), then χ(ΓIt(Mn(R))) =ω(ΓIt(Mn(R))).
Proof. Firstly, let us assume that χ(Γt(Mn(R/I))) = ω(Γt(Mn(R/I))) = k. From this wehave χ(Γt(Mn(R)/Mn(I))) = ω(Γt(Mn(R)/Mn(I))) = k.
Let A1 + Mn(I), . . . , Ak + Mn(I) be a clique of order k in Γt(Mn(R)/Mn(I)) suchthat Tr(A2
i ) ∈ I, 1 ≤ i ≤ k. Let c1, . . . , ck be a set of minimum colors required for aproper coloring of the graph Γt(Mn(R)/Mn(I)). Without loss of generality assume thatAi + Mn(I) is colored by the color ci. Since A2
i ∈ I, the set X = A ∈ ΓIt(Mn(R)) : A ∈Ai + Mn(I) for some i ∈ 1, . . . , k forms a clique of order |Mn(I)|k in ΓIt(Mn(R)).
By Theorem 4.1(1), this clique is maximum and ω(ΓIt(Mn(R))) = k|Mn(I)|. Assignk|Mn(I)| distinct colors c′
1, . . . , c′k|Mn(I)| to the vertices in the set X.
For a vertex B ∈ V (ΓIt(Mn(R))) \ X, there exists M ∈ Mn(I) such that B = Bℓ + M ∈Bℓ + Mn(I) for some Bℓ + Mn(I) /∈ A1 + Mn(I), . . . , Ak + Mn(I). Let cj be the color ofBℓ + Mn(I) in Γt(Mn(R)/Mn(I)). Note that Aj + Mn(I) belongs to the color class cj inΓt(Mn(R)/Mn(I)).
Assign the color of Aj + M in ΓIt(Mn(R)) to B = Bℓ + M in ΓIt(Mn(R)). Let C ∈V (ΓIt(Mn(R))) \ X be adjacent to B ∈ V (ΓIt(Mn(R))) \ X. Then Bℓ + Mn(I) is adjacentto C + Mn(I) in Γt(Mn(R)/Mn(I)) and hence they belong to different color classes inΓt(Mn(R)/Mn(I)) and so B and C belong to different color classes in ΓIt(Mn(R)).
Thus we have given a proper coloring for the graph ΓIt(Mn(R)) with k|Mn(I)| colorsand so χ(ΓIt(Mn(R))) ≤ k|Mn(I)|. Since k|Mn(I)| = ω(ΓIt(Mn(R))) ≤ χ(ΓIt(Mn(R))),χ(ΓIt(Mn(R))) = k|Mn(I)|.
616 T. Tamizh Chelvam, M. Sivagami
Theorem 4.4. Let n ≥ 2 be an integer, R be a commutative ring and I be a non trivialideal of R. Then
1. α(Γt(Mn(R/I))) ≤ α(ΓIt(Mn(R))) ≤ |Mn(I)|α(Γt(Mn(R/I)));2. In particular, if there exists an independent set of maximum order in Γt(Mn(R/I))
such that Tr(A2) /∈ I for every vertex A in the independent set, then α(ΓIt(Mn(R)))= |Mn(I)|α(Γt(Mn(R/I))).
Proof. 1. Let α(Γt(Mn(R/I))) = k and X be the corresponding maximum independentset of Γt(Mn(R/I)). Consider the set X1 = A : A ∈ X ⊆ V (ΓIt(Mn(R))). By theTheorem 3.3(2), we have that X1 is an independent set of order k in ΓIt(Mn(R)). Henceα(Γt(Mn(R/I))) ≤ α(ΓIt(Mn(R))). By Note 3.2, we have α(Γt(Mn(R)/Mn(I))) = k.
Suppose that there exists an independent set of order k|Mn(I)| + 1 in ΓIt(Mn(R)).Let B1, . . . , Bk|Mn(I)|+1 be an independent set in ΓIt(Mn(R)). Consider the set X =B1 + Mn(I), . . . , Bk|Mn(I)|+1 + Mn(I) ⊆ V (Γt(Mn(R)/Mn(I))).
Note that for i = j, Bi + Mn(I) = Bj + Mn(I) or Bi + Mn(I) is not adjacent toBj + Mn(I) in Γt(Mn(R)/Mn(I)). Since |Bi + Mn(I)| = |Mn(I)| we have at least k + 1distinct elements in X such that the k + 1 elements are not adjacent to each other inΓt(Mn(R)/Mn(I)),i.e., α(Γt(Mn(R)/Mn(I))) ≥ k + 1 which is a contradiction. Henceα(ΓIt(Mn(R))) ≤ k|Mn(I)|.2. From Theorem 3.3(2) and (3), we have α(ΓIt(Mn(R))) ≥ |Mn(I)|α(Γt(Mn(R/I))). Bythe previous part, we have α(ΓIt(Mn(R))) = |Mn(I)|α(Γt(Mn(R/I))). Acknowledgment. This work is supported by the INSPIRE programme (IF 160672) ofDepartment of Science and Technology, Government of India, India for the second author.Further this work is supported by UGC-SAP(DRS-II) programme through the first author.
References[1] F.A.A. Almahdi, K. Louartiti, and M. Tamekkante, The trace graph of the matrix
ring over a finite commutative ring, Acta Math. Hungar. 156 (1), 132–144, 2018.[2] D.F. Anderson and P.S. Livingston, The zero divisor graph of a commutative ring, J.
Algebra 217, 434–447, 1999.[3] I. Beck, Coloring of commutative rings, J. Algebra 116, 208–226, 1988.[4] G. Chartrand, O.R. Oellermann, Applied and algorithmic graph theory, McGraw-Hill,
Inc., New York, 1993.[5] I.N. Herstein, Noncommutative rings, Carus Monographs in Mathematics, Math. As-
soc. of America, 1968.[6] I. Kaplansky, Commutative Rings, University of Chicago Press, Chicago, 1974.[7] T.Y. Lam, A first course in noncommutative rings, Graduate Texts in Mathematics,
Springer-Verlag, New York 2001.[8] H.R. Maimani, M.R. Pournaki, and S. Yassemi, Zero-divisor graph with respect to an
ideal, Comm. Algebra 34 (3), 923–929, 2006.[9] S.P. Redmond, An ideal-based zero-divisor graph of a commutative ring, Comm. Al-
gebra 31, 4425–4443, 2003.[10] M. Sivagami and T. Tamizh Chelvam, On the trace graph of matrices, Acta Math.
Hungar. 158 (1), 235–250, 2019, https://doi.org/10.1007/s10474-019-00918-5.[11] D.B. West, Introduction to graph theory, Prentice Hall, 2001.
Hacettepe Journal ofMathematics & Statistics
Hacet. J. Math. Stat.Volume 49 (2) (2020), 617 – 637
DOI : 10.15672/hujms.568393
Research Article
A note for some parabolic multilinearcommutators generated by a class of parabolic
maximal and linear operators with rough kernelon the parabolic generalized local Morrey spaces
Ferit Gürbüz
Hakkari University, Faculty of Education, Department of Mathematics Education, Hakkari 30000, Turkey
AbstractIn this paper, we give the boundedness of some parabolic multilinear commutators gener-ated by a class of parabolic maximal and linear operators with rough kernel and paraboliclocal Campanato functions on the parabolic generalized local Morrey spaces, respectively.Indeed, the results in this paper are extensions of some known results.
Mathematics Subject Classification (2010). 42B20, 42B25
Keywords. parabolic multilinear commutators, rough kernel, parabolic generalizedlocal Morrey space, parabolic local Campanato space
1. Introduction and main resultsLet Sn−1 = x ∈ Rn : |x| = 1 denote the unit sphere on Rn (n ≥ 2) equipped with the
normalized Lebesgue measure dσ (x′), where x′ denotes the unit vector in the direction ofx and αn ≥ αn−1 ≥ · · · ≥ α1 ≥ 1 be fixed real numbers.
Note that for each fixed x = (x1, . . . , xn) ∈ Rn, the function
F (x, ρ) =n∑
i=1
x2i
ρ2αi
is a strictly decreasing function of ρ > 0. Hence, there exists a unique ρ = ρ (x) such thatF (x, ρ) = 1. It is clear that for each fixed x ∈ Rn, the function F (x, ρ) is a decreasingfunction in ρ > 0. Fabes and Riviére [5] showed that (Rn,ρ) is a metric space which isoften called the mixed homogeneity space related to αin
i=1. For t > 0, we let At be thediagonal n × n matrix
At = diag [tα1 , . . . , tαn ] =
tα1 0. . .
0 tαn
.
Email address: [email protected]: 26.07.2017; Accepted: 06.02.2019
618 F. Gürbüz
Let ρ ∈ (0, ∞) and 0 ≤ φn−1 ≤ 2π, 0 ≤ φi ≤ π, i = 1, . . . , n − 2. For any x =(x1, x2, . . . , xn) ∈ Rn, set
x1 = ρα1 cos φ1 . . . cos φn−2 cos φn−1,
x2 = ρα2 cos φ1 . . . cos φn−2 sin φn−1,
...xn−1 = ραn−1 cos φ1 sin φ2,
xn = ραn sin φ1.
Thus dx = ρα−1J (x′) dρdσ(x′), where α =n∑
i=1αi, x′ ∈ Sn−1, J (x′) =
n∑i=1
αi (x′i)
2, dσ is
the element of area of Sn−1 and ρα−1J (x′) is the Jacobian of the above transform. It iseasy to see that J (x′) ∈ C∞ (
Sn−1) with 1 ≤ J (x′) ≤ M and x′ ∈ Sn−1 for some M ≥ 1.Let P be a real n × n matrix, whose all the eigenvalues have positive real part. Let
At = tP (t > 0), and set γ = trP . Then, there exists a quasi-distance ρ associated with Psuch that (see [3])
(1 − 1) ρ (Atx) = tρ (x), t > 0, for every x ∈ Rn,(1 − 2) ρ (0) = 0, ρ (x − y) = ρ (y − x) ≥ 0, and ρ (x − y) ≤ k (ρ (x − z) + ρ (y − z)),(1 − 3) dx = ργ−1dσ (w) dρ, where ρ = ρ (x), w = Aρ−1x and dσ (w) is a measure on
the unit ellipsoid w : ρ (w) = 1.Then, Rn, ρ, dx becomes a space of homogeneous type in the sense of Coifman-Weiss
(see [3]) and a homogeneous group in the sense of Folland-Stein (see [6]).Denote by E (x, r) the ellipsoid with center at x and radius r, more precisely, E (x, r) =
y ∈ Rn : ρ (x − y) < r. Moreover, by the property of ρ and the polar coordinates trans-form above, we have
|E (x, r)| =∫
ρ(x−y)<r
dy = υρrα1+···+αn = υρrγ ,
where |E(x, r)| stands for the Lebesgue measure of E(x, r) and υρ is the volume of theunit ellipsoid on Rn. By EC(x, r) = Rn\ E (x, r), we denote the complement of E (x, r).
If we take α1 = · · · = αn = 1 and P = I, then obviously ρ (x) = |x| =(
n∑i=1
x2i
) 12
, γ = n,
(Rn, ρ) = (Rn, |·|), EI(x, r) = B (x, r), At = tI and J (x′) ≡ 1. Moreover, in the standardparabolic case P0 = diag [1, . . . , 1, 2] we have
ρ (x) =
√√√√ |x′|2 +√
|x′|4 + x2n
2, x =
(x′, xn
).
Note that we deal not exactly with the parabolic metric, but with a general anisotropicmetric ρ of generalized homogeneity, the parabolic metric being its particular case, butwe keep the term parabolic in the title and text of the paper, the above existing tradition,see for instance [2].
Suppose that Ω (x) is a real-valued and measurable function defined on Rn. Supposethat Sn−1 is the unit sphere on Rn (n ≥ 2) equipped with the normalized Lebesguesurface measure dσ. Let Ω ∈ Ls(Sn−1) with 1 < s ≤ ∞ be homogeneous of degree zerowith respect to At (Ω (x) is At-homogeneous of degree zero), that is, Ω(Atx) = Ω(x), forany t > 0, x ∈ Rn. We define s′ = s
s−1 for any s > 1. One of the important problems onparabolic homogeneous spaces investigates the boundedness of parabolic linear operatorssatisfying the following size conditions ((1.1) and (1.2)). Therefore, in this paper, weconsider parabolic linear operators T P
Ω and T PΩ,α, α ∈ (0, γ) satisfying the size conditions
Some parabolic multilinear commutators 619
for any f ∈ L1(Rn) with compact support and x /∈ suppf , respectively
|T PΩ f(x)| ≤ c0
∫Rn
|Ω(x − y)|ρ (x − y)γ |f(y)| dy, (1.1)
|T PΩ,αf(x)| ≤ c0
∫Rn
|Ω(x − y)|ρ (x − y)γ−α |f(y)| dy, (1.2)
where c0 is independent of f and x.We point out that the conditions (1.1) and (1.2) in the case Ω ≡ 1, α = 0 and P = I
was first introduced by Soria and Weiss in [11]. Indeed, in 1944, Soria and Weiss devel-oped Stein’s result [12] in the above shape. The conditions (1.1) and (1.2) are satisfied bymany interesting operators in harmonic analysis, such as the parabolic Calderón–Zygmundoperators, parabolic Carleson’s maximal operator, parabolic Hardy–Littlewood maximaloperator, parabolic C. Fefferman’s singular multipliers, parabolic R. Fefferman’s singularintegrals, parabolic Ricci–Stein’s oscillatory singular integrals, parabolic the Bochner–Riesz means, the parabolic fractional integral operator(parabolic Riesz potential), para-bolic fractional maximal operator, parabolic fractional Marcinkiewicz operator and so on(see [7, 8, 11] for details).
The parabolic fractional maximal function MPΩ,αf and T p
Ω,αf by with rough kernels,0 < α < γ, of a function f ∈ Lloc (Rn) are defined by
MPΩ,αf(x) = sup
t>0|E(x, t)|−1+ α
γ
∫E(x,t)
|Ω (x − y)| |f(y)|dy,
T PΩ,αf(x) =
∫Rn
Ω(x − y)ρ (x − y)γ−α f(y)dy,
satisfy condition (1.2). It is obvious that when Ω ≡ 1, MP1,α ≡ Mp
α and T P1,α ≡ T P
α arethe parabolic fractional maximal operator and the parabolic fractional integral operator,respectively. If P = I, then M I
Ω,α ≡ MΩ,α and T IΩ,α ≡ TΩ,α are the fractional maximal
operator with rough kernel and fractional integral operator with rough kernel, respectively.It is well known that the parabolic fractional maximal and integral operators play animportant role in harmonic analysis (see [2, 6, 8]).
We notice that when α = 0, the above operators become the parabolic Calderón–Zygmund singular integral operator with rough kernel T P
Ω = T PΩ,0 and the corresponding
parabolic maximal operator with rough kernel MPΩ,0 ≡ MP
Ω :
T PΩ f(x) = p.v.
∫Rn
Ω(x − y)ρ (x − y)γ f(y)dy,
MPΩ f(x) = sup
t>0|E(x, t)|−1
∫E(x,t)
|Ω (x − y)| |f(y)|dy,
satisfy condition (1.1). It is obvious that when Ω ≡ 1, T PΩ ≡ T P and MP
Ω ≡ MP are theparabolic singular operator and the parabolic maximal operator, respectively. If P = I,then M I
Ω ≡ MΩ is the Hardy-Littlewood maximal operator with rough kernel, and T IΩ ≡ TΩ
is the homogeneous singular integral operator. It is well known that the parabolic maximaland singular operators play an important role in harmonic analysis (see [2, 6, 7, 14]).
On the other hand let b be a locally integrable function on Rn, then for 0 < α < γ,we define commutators generated by parabolic fractional maximal and integral operators
620 F. Gürbüz
with rough kernel and b as follows, respectively.
MPΩ,b,α (f) (x) = sup
t>0|E(x, t)|−1+ α
γ
∫E(x,t)
|b (x) − b (y)| |Ω (x − y)| |f(y)|dy,
[b, T PΩ,α]f(x) ≡ b(x)T P
Ω,αf(x) − T PΩ,α(bf)(x) =
∫Rn
[b(x) − b(y)] Ω(x − y)ρ (x − y)γ−α f(y)dy.
Similarly, for α = 0, we define commutators generated by parabolic maximal and sin-gular integral operators by with rough kernels and b as follows, respectively.
MPΩ,b (f) (x) = sup
t>0|E(x, t)|−1
∫E(x,t)
|b (x) − b (y)| |Ω (x − y)| |f(y)|dy,
[b, T PΩ ]f(x) ≡ b(x)T P
Ω f(x) − T PΩ (bf)(x) = p.v.
∫Rn
[b(x) − b(y)] Ω(x − y)ρ (x − y)γ f(y)dy.
Because of the need for the study of partial differential equations (PDEs), Morrey [10]introduced Morrey spaces Mp,λ which naturally are generalizations of Lebesgue spaces.We also refer to [1] for the latest research on the theory of Morrey spaces associated withharmonic analysis.
A measurable function f ∈ Lp (Rn), p ∈ (1, ∞), belongs to the parabolic Morrey spacesMp,λ,P (Rn) with λ ∈ [0, γ) if the following norm is finite:
∥f∥Mp,λ,P=
supx∈Rn,r>0
1rλ
∫E(x,r)
|f (y)|p dy
1/p
,
where E(x, r) stands for any ellipsoid with center at x and radius r. When λ = 0,Mp,λ,P (Rn) coincides with the parabolic Lebesgue space Lp,P (Rn).
If P = I, then Mp,λ,I(Rn) ≡ Mp,λ(Rn) and Lp,I (Rn) ≡ Lp (Rn) are the classical Morreyand the Lebesgue spaces, respectively.
We now recall the definition of parabolic generalized local (central) Morrey spaceLM
x0p,φ,P in the following.
Definition 1.1 (parabolic generalized local (central) Morrey space, [7, 8]). Letφ(x, r) be a positive measurable function on Rn × (0, ∞) and 1 ≤ p < ∞. For any fixedx0 ∈ Rn we denote by LM
x0p,φ,P ≡ LM
x0p,φ,P (Rn) the parabolic generalized local Morrey
space, the space of all functions f ∈ Llocp (Rn) with finite quasinorm
∥f∥LM
x0p,φ,P
= supr>0
φ(x0, r)−1|E(x0, r)|−1p ∥f∥Lp(E(x0,r)) < ∞.
According to this definition, we recover the local parabolic Morrey space LMx0p,λ,P and
weak local parabolic Morrey space WLMx0p,λ,P under the choice φ(x0, r) = r
λ−γp :
LMx0p,λ,P = LM
x0p,φ,P |
φ(x0,r)=rλ−γ
p, WLM
x0p,λ,P = WLM
x0p,φ,P |
φ(x0,r)=rλ−γ
p.
Now, let us recall the defination of the space of LCx0p,λ,P (parabolic local Campanato
space).
Some parabolic multilinear commutators 621
Definition 1.2 ([7, 8]). Let 1 ≤ p < ∞ and 0 ≤ λ < 1γ . A parabolic local Campanato
function b ∈ Llocp (Rn) is said to belong to the LC
x0p,λ,P (Rn), if
∥b∥LC
x0p,λ,P
= supr>0
1|E (x0, r)|1+λp
∫E(x0,r)
∣∣∣b (y) − bE(x0,r)
∣∣∣p dy
1p
< ∞,
wherebE(x0,r) = 1
|E (x0, r)|
∫E(x0,r)
b (y) dy.
Define
LCx0p,λ,P (Rn) =
b ∈ Lloc
p (Rn) : ∥b∥LC
x0p,λ,P
< ∞
.
Let bi (i = 1, . . . , m) be locally integrable functions on Rn, then the fractional typeparabolic multilinear commutators generated by parabolic fractional maximal and integraloperators with rough kernel and
−→b = (b1, . . . , bm) (parabolic local Campanato functions)
are given as follows, respectively:
[−→b , T P
Ω,α]f (x) =∫Rn
m∏i=1
[bi (x) − bi (y)] Ω(x − y)ρ (x − y)γ−α f (y) dy, 0 < α < γ,
MP
Ω,−→b ,α
f (x) = supt>0
|E(x, t)|−1+ αγ
∫E(x,t)
m∏i=1
[|bi (x) − bi (y)|] |Ω(x − y)| |f(y)|dy, 0 < α < γ.
We notice that when α = 0, the above operators become the parabolic multilinear commu-tators generated by parabolic singular integral operators and the corresponding parabolicmaximal operators with rough kernel and
−→b = (b1, . . . , bm) as follows, respectively:
[−→b , T P
Ω ]f (x) =∫Rn
m∏i=1
[bi (x) − bi (y)] Ω(x − y)ρ (x − y)γ f (y) dy,
MP
Ω,−→b
f (x) = supt>0
|E(x, t)|−1∫
E(x,t)
m∏i=1
[|bi (x) − bi (y)|] |Ω(x − y)| |f(y)|dy.
In [7, 8] the boundedness of a class of parabolic sublinear operators with rough kerneland their commutators on the parabolic generalized local Morrey spaces under genericsize conditions which are satisfied by most of the operators in harmonic analysis has beeninvestigated, respectively.
Inspired by [7, 8], our main purpose in this paper is to consider the boundedness ofabove operators ([
−→b , TΩ], MΩ,
−→b
, [−→b , TΩ,α], MΩ,
−→b ,α
) on the parabolic generalized localMorrey spaces, respectively. But, the techniques and non-trivial estimates which havebeen used in the proofs of our main results are quite different from [7, 8]. For example,using inequality about the weighted Hardy operator Hw in [7,8], in this paper we will onlyuse the following relationship between essential supremum and essential infimum(
essinfx∈E
f (x))−1
= esssupx∈E
1f (x)
, (1.3)
where f is any real-valued nonnegative function and measurable on E (see [13], page 143).Our main results can be formulated as follows.
622 F. Gürbüz
Theorem 1.3. Let x0 ∈ Rn, Ω ∈ Ls(Sn−1), 1 < s ≤ ∞, be At-homogeneous of degreezero. Let also, 1 < q, pi, p < ∞ with 1
q =m∑
i=11pi
+ 1p and
−→b ∈ LC
x0pi,λi,P
(Rn) for 0 ≤ λi < 1γ ,
i = 1, . . . , m.For s′ ≤ q, if functions φ1,φ2 : Rn × R+→ R+ satisfy the inequality
∞∫r
(1 + ln t
r
)m essinft<τ<∞
φ1(x0, τ)τγp
t
γ
(1p
−m∑
i=1λi
)+1
≤ C φ2(x0, r), (1.4)
and for p < s, if functions φ1,φ2 : Rn × R+→ R+ satisfy the inequality∞∫r
(1 + ln t
r
)m essinft<τ<∞
φ1(x0, τ)τγp
t
γ
(1p
− 1s
−m∑
i=1λi
)+1
dt ≤ C φ2(x0, r)rγs ,
where C does not depend on r, then the commutators [−→b , T P
Ω ] and MP
Ω,−→b
are bounded from
LMx0p,φ1,P to LM
x0q,φ2,P . Moreover,∥∥∥[−→b , T P
Ω ]f∥∥∥
LMx0q,φ2,P
.m∏
i=1∥−→b ∥
LCx0pi,λi,P
∥f∥LM
x0p,φ1,P
, (1.5)
∥∥∥[−→b , MP
Ω,−→b
]f∥∥∥
LMx0q,φ2,P
.m∏
i=1∥−→b ∥
LCx0pi,λi,P
∥f∥LM
x0p,φ1,P
. (1.6)
Corollary 1.4 ([7]). Let x0 ∈ Rn, Ω ∈ Ls(Sn−1), 1 < s ≤ ∞, be At-homogeneous ofdegree zero. Let T P
Ω be a parabolic linear operator satisfying condition (1.1), bounded onLp(Rn) for p > 1, and bounded from L1(Rn) to WL1(Rn). Let also, b ∈ LC
x0p2,λ,P (Rn),
0 ≤ λ < 1γ and 1
p = 1p1
+ 1p2
.For s′ ≤ p, if functions φ1,φ2 : Rn × R+→ R+ satisfy the inequality
∞∫r
(1 + ln t
r
) essinft<τ<∞
φ1(x0, τ)τγ
p1
tγ
p1+1−γλ
dt ≤ C φ2(x0, r),
and for p1 < s, if functions φ1,φ2 : Rn × R+→ R+ satisfy the inequality∞∫r
(1 + ln t
r
) essinft<τ<∞
φ1(x0, τ)τγ
p1
tγ
p1− γ
s+1−γλ
dt ≤ C φ2(x0, r)rγs ,
where C does not depend on r, then the commutators [b, T PΩ ] and MP
Ω,b are bounded fromLM
x0p1,φ1,P to LM
x0p,φ2,P . Moreover,∥∥∥[b, T P
Ω ]f∥∥∥
LMx0p,φ2,P
. ∥b∥LC
x0p2,λ,P
∥f∥LM
x0p1,φ1,P
,
∥∥∥MPΩ,bf
∥∥∥LM
x0p,φ2,P
. ∥b∥LC
x0p2,λ,P
∥f∥LM
x0p1,φ1,P
.
Theorem 1.5. Let x0 ∈ Rn, Ω ∈ Ls(Sn−1), 1 < s ≤ ∞, be At-homogeneous of degreezero. Let also, 0 < α < γ and 1 < q, q1, pi, p < γ
α with 1q =
m∑i=1
1pi
+ 1p , 1
q1= 1
q − αγ and
−→b ∈ LC
x0pi,λi,P
(Rn) for 0 ≤ λi < 1γ , i = 1, . . . , m.
Some parabolic multilinear commutators 623
For s′ ≤ q, if functions φ1,φ2 : Rn × R+→ R+ satisfy the inequality
∞∫r
(1 + ln t
r
)m essinft<τ<∞
φ1(x0, τ)τγp
t
γ
(1
q1−
( m∑i=1
λi+m∑
i=1
1pi
))+1
≤ C φ2(x0, r), (1.7)
and for q1 < s, if functions φ1,φ2 : Rn × R+→ R+ satisfy the inequality
∞∫r
(1 + ln t
r
)m essinft<τ<∞
φ1(x0, τ)τγp
t
γ
(1
q1−
(1s
+m∑
i=1λi+
m∑i=1
1pi
))+1
dt ≤ C φ2(x0, r)rγs ,
where C does not depend on r, then the commutators [−→b , T P
Ω,α] and MP
Ω,−→b ,α
are bounded
from LMx0p,φ1,P to LM
x0q1,φ2,P . Moreover,
∥∥∥[−→b , T PΩ,α]f
∥∥∥LM
x0q1,φ2,P
.m∏
i=1∥−→b ∥
LCx0pi,λi,P
∥f∥LM
x0p,φ1,P
, (1.8)
∥∥∥MP
Ω,−→b ,α
f∥∥∥
LMx0q1,φ2,P
.m∏
i=1∥−→b ∥
LCx0pi,λi,P
∥f∥LM
x0p,φ1,P
. (1.9)
Corollary 1.6 ([8]). Let x0 ∈ Rn, Ω ∈ Ls(Sn−1), 1 < s ≤ ∞, be At-homogeneous ofdegree zero. Let T P
Ω,α be a parabolic linear operator satisfying condition (1.2) and boundedfrom Lp(Rn) to Lq(Rn). Let also, 0 < α < γ, 1 < p < γ
α , b ∈ LCx0p2,λ,P (Rn), 0 ≤ λ < 1
γ ,1p = 1
p1+ 1
p2, 1
q = 1p − α
γ , 1q1
= 1p1
− αγ .
For s′ ≤ p, if functions φ1,φ2 : Rn × R+→ R+ satisfy the inequality
∞∫r
(1 + ln t
r
) essinft<τ<∞
φ1(x0, τ)τγ
p1
tγq1
+1−γλdt ≤ C φ2(x0, r),
and for q1 < s, if functions φ1,φ2 : Rn × R+→ R+ satisfy the inequality
∞∫r
(1 + ln t
r
) essinft<τ<∞
φ1(x0, τ)τγ
p1
tγq1
− γs
+1−γλdt ≤ C φ2(x0, r)r
γs ,
where C does not depend on r, then the commutators [b, T PΩ,α] and MP
Ω,b,α are boundedfrom LM
x0p1,φ1,P to LM
x0q,φ2,P . Moreover,∥∥∥[b, T P
Ω,α]f∥∥∥
LMx0q,φ2,P
. ∥b∥LC
x0p2,λ,P
∥f∥LM
x0p1,φ1,P
,
∥∥∥MPΩ,b,αf
∥∥∥LM
x0q,φ2,P
. ∥b∥LC
x0p2,λ,P
∥f∥LM
x0p1,φ1,P
.
At last, let F, G ≥ 0. Here and henceforth, the symbol F ≈ G means that F . G andG . F happen simultaneously; while F . G and G . F mean that there exists a constantC > 0 such that F ≤ CG.
624 F. Gürbüz
2. Some lemmasTo prove the main results (Theorems 1.3 and 1.5), we need the following lemmas.
Lemma 2.1 ([7, 8]). Let b be a parabolic local Campanato function in LCx0p,λ,P (Rn), 1 <
p < ∞, 0 ≤ λ < 1γ and 0 < r2 < r1. Then
1|E (x0, r1)|1+λp
∫E(x0,r1)
∣∣∣b (y) − bE(x0,r2)
∣∣∣p dy
1p
≤ C
(1 + ln r1
r2
)∥b∥
LCx0p,λ,P
, (2.1)
where C > 0 is independent of b, r1 and r2.From this inequality (2.1), we have∣∣∣bE(x0,r1) − bE(x0,r2)
∣∣∣ ≤ C
(1 + ln r1
r2
)|E (x0, r1)|λ ∥b∥
LCx0p,λ,P
, (2.2)
and it is easy to see that
∥b − bE∥Lp(E) ≤ C
(1 + ln r1
r2
)r
γp
+γλ ∥b∥LC
x0p,λ,P
. (2.3)
Lemma 2.2. Suppose that x0 ∈ Rn, Ω ∈ Ls(Sn−1), 1 < s ≤ ∞, is At-homogeneousof degree zero. Let T P
Ω be a parabolic linear operator satisfying condition (1.1). Let also1 < q, pi, p < ∞ with 1
q =m∑
i=11pi
+ 1p and
−→b ∈ LC
x0pi,λi,P
(Rn) for 0 ≤ λi < 1γ , i = 1, . . . , m.
Then, for s′ ≤ q the inequality
∥[−→b , T P
Ω ]f∥Lq(E(x0,r)) .m∏
i=1∥−→b ∥
LCx0pi,λi,P
rγq
∞∫2kr
(1 + ln t
r
)m ∥f∥Lp(E(x0,t))
t
γ
(1p
−m∑
i=1λi
)+1
dt (2.4)
holds for any ellipsoid E (x0, r) and for all f ∈ Llocp (Rn). Also, for p < s the inequality
∥[−→b , T P
Ω ]f∥Lq(E(x0,r)) .m∏
i=1∥−→b ∥
LCx0pi,λi,P
rγq
− γs
∞∫2kr
(1 + ln t
r
)m ∥f∥Lp(E(x0,t))
t
γ
(1p
− 1s
−m∑
i=1λi
)+1
dt
holds for any ellipsoid E (x0, r) and for all f ∈ Llocp (Rn).
Proof. Without loss of generality, it is sufficient to show that the conclusion holds for[−→b , T P
Ω ]f = [(b1, b2) , T PΩ ]f . Let 1 < q, pi, p < ∞ with 1
q =m∑
i=11pi
+ 1p and
−→b ∈ LC
x0pi,λi,P
(Rn)
for 0 ≤ λi < 1γ , i = 1, . . . , m. Set E = E (x0, r) for the parabolic ball (ellipsoid) centered at
x0 and of radius r and for k > 0, we denote 2kE = E (x0, 2kr) = y ∈ Rn : ρ (x − y) < 2kr.We represent f as
f = f1 + f2, f1 (y) = f (y) χ2kE (y) , f2 (y) = f (y) χ(2kE)C (y) , r > 0 (2.5)
and thus have∥∥∥[(b1, b2) , T PΩ ]f
∥∥∥Lq(E)
≤∥∥∥[(b1, b2) , T P
Ω ]f1∥∥∥
Lq(E)+∥∥∥[(b1, b2) , T P
Ω ]f2∥∥∥
Lq(E)=: F + G.
Let us estimate F + G, respectively.
Some parabolic multilinear commutators 625
For [(b1, b2) , T PΩ ]f1 (x), we have the following decomposition,
[(b1, b2) , T PΩ ]f1 (x) = (b1 (x) − (b1)E) (b2 (x) − (b2)E) T P
Ω f1 (x)− (b1 (·) − (b1)E) T P
Ω ((b2 (·) − (b2)E) f1) (x)+ (b2 (x) − (b2)E) T P
Ω ((b1 (x) − (b1)E) f1) (x)− T P
Ω ((b1 (·) − (b1)E) (b2 (·) − (b2)E) f1) (x) .
Hence, we get
F =∥∥∥[(b1, b2) , T P
Ω ]f1∥∥∥
Lq(E).∥∥∥(b1 − (b1)E) (b2 (x) − (b2)E) T P
Ω f1∥∥∥
Lq(E)
+∥∥∥(b1 − (b1)E) T P
Ω ((b2 − (b2)E) f1)∥∥∥
Lq(E)
+∥∥∥(b2 − (b2)E) T P
Ω ((b1 − (b1)E) f1)∥∥∥
Lq(E)
+∥∥∥T P
Ω ((b1 − (b1)E) (b2 − (b2)E) f1)∥∥∥
Lq(E)
≡F1 + F2 + F3 + F4. (2.6)
One observes that the estimate of F2 is analogous to that of F3. Thus, we will onlyestimate F1, F2 and F4.
To estimate F1, let 1 < q, τ < ∞, such that 1q = 1
τ + 1p , 1
τ = 1p1
+ 1p2
. Then, usingHölder’s inequality and by Theorem 1.2. in [7] it follows that:
F1 =∥∥∥(b1 − (b1)E) (b2 (x) − (b2)E) T P
Ω f1∥∥∥
Lq(E)
. ∥(b1 − (b1)E) (b2 (x) − (b2)E)∥Lτ (E)
∥∥∥T PΩ f1
∥∥∥Lp(E)
. ∥b1 − (b1)E∥Lp1 (E) ∥b2 − (b2)E∥Lp2 (E) ∥f∥Lp(2kE)
. ∥b1 − (b1)E∥Lp1 (E) ∥b2 − (b2)E∥Lp2 (E) rγp
∞∫2kr
∥f∥Lp(E(x0,t))dt
tγp
+1 .
From Lemma 2.1, it is easy to see that
∥bi − (bi)E∥Lpi (E) ≤ Crγpi
+γλi ∥bi∥LCx0pi,λi,P
, (2.7)
and
∥bi − (bi)E∥Lpi (2kE) ≤ ∥bi − (bi)2kE∥Lpi (2kE) + ∥(bi)E − (bi)2kE∥Lpi (2kE)
. rγpi
+γλi ∥bi∥LCx0pi,λi,P
, (2.8)
for i = 1, 2. Hence, by (2.7) we get
F1 . ∥b1∥LC
x0p1,λ1,P
∥b2∥LC
x0p2,λ2,P
rγ
(1
p1+ 1
p2+ 1
p
) ∞∫2kr
(1 + ln t
r
)2t− γ
p+γ(λ1+λ2)−1∥f∥Lp(E(x0,t))dt
. ∥b1∥LC
x0p1,λ1,P
∥b2∥LC
x0p2,λ2,P
rγq
∞∫2kr
(1 + ln t
r
)2 ∥f∥Lp(E(x0,t))
tγp
−γ(λ1+λ2)+1 dt.
626 F. Gürbüz
To estimate F2, let 1 < τ < ∞, such that 1q = 1
p1+ 1
τ . Then, similar to the estimates forF1, we have
F2 . ∥b1 − (b1)E∥Lp1 (E)
∥∥∥T PΩ ((b2 (·) − (b2)E) f1)
∥∥∥Lτ (E)
. ∥b1 − (b1)E∥Lp1 (E) ∥(b2 (·) − (b2)E) f1∥Lk(E)
. ∥b1 − (b1)E∥Lp1 (E) ∥b2 − (b2)E∥Lp2 (2kE) ∥f∥Lp(2kE) ,
where 1 < k < ∞, such that 1k = 1
p2+ 1
p = 1τ . By (2.7) and (2.8), we get
F2 . ∥b1∥LC
x0p1,λ1,P
∥b2∥LC
x0p2,λ2,P
rγq
∞∫2kr
(1 + ln t
r
)2 ∥f∥Lp(E(x0,t))
tγp
−γ(λ1+λ2)+1 dt.
In a similar way, F3 has the same estimate as above, so we omit the details. Then we havethat
F3 . ∥b1∥LC
x0p1,λ1,P
∥b2∥LC
x0p2,λ2,P
rγq
∞∫2kr
(1 + ln t
r
)2 ∥f∥Lp(E(x0,t))
tγp
−γ(λ1+λ2)+1 dt.
Now let us consider the term F4. Let 1 < q, τ < ∞, such that 1q = 1
τ + 1p , 1
τ = 1p1
+ 1p2
.Then by Theorem 1.2. in [7], Hölder’s inequality and (2.8), we obtain
F4 =∥∥∥T P
Ω ((b1 − (b1)E) (b2 − (b2)E) f1)∥∥∥
Lq(E)
. ∥(b1 − (b1)E) (b2 − (b2)E) f1∥Lq(E)
. ∥(b1 − (b1)E) (b2 − (b2)E)∥Lτ (E) ∥f1∥Lp(E)
. ∥b1 − (b1)E∥Lp1 (2kE) ∥b2 − (b2)E∥Lp2 (2kE) ∥f∥Lp(2kE)
. ∥b1∥LC
x0p1,λ1,P
∥b2∥LC
x0p2,λ2,P
rγq
∞∫2kr
(1 + ln t
r
)2 ∥f∥Lp(E(x0,t))
tγp
−γ(λ1+λ2)+1 dt.
Combining all the estimates of F1, F2, F3, F4; we get
F =∥∥∥[(b1, b2) , T P
Ω ]f1∥∥∥
Lq(E). ∥b1∥
LCx0p1,λ1,P
∥b2∥LC
x0p2,λ2,P
rγq
∞∫2kr
(1 + ln t
r
)2 ∥f∥Lp(E(x0,t))
tγp
−γ(λ1+λ2)+1 dt.
Now, let us estimate G =∥∥∥[(b1, b2) , T P
Ω ]f2∥∥∥
Lq(E). For G, it’s similar to (2.6) we also
write
G =∥∥∥[(b1, b2) , T P
Ω ]f2∥∥∥
Lq(E).∥∥∥(b1 − (b1)E) (b2 (x) − (b2)E) T P
Ω f2∥∥∥
Lq(E)
+∥∥∥(b1 − (b1)E) T P
Ω ((b2 − (b2)E) f2)∥∥∥
Lq(E)
+∥∥∥(b2 − (b2)E) T P
Ω ((b1 − (b1)E) f2)∥∥∥
Lq(E)
+∥∥∥T P
Ω ((b1 − (b1)E) (b2 − (b2)E) f2)∥∥∥
Lq(E)
≡G1 + G2 + G3 + G4.
Some parabolic multilinear commutators 627
To estimate G1, let 1 < q, τ < ∞, such that 1q = 1
τ + 1p , 1
τ = 1p1
+ 1p2
. Then, using Hölder’sinequality and by (12) in [7] and (2.7), we have
G1 =∥∥∥(b1 − (b1)E) (b2 (x) − (b2)E) T P
Ω f2∥∥∥
Lq(E)
. ∥(b1 − (b1)E) (b2 − (b2)E)∥Lτ (E)
∥∥∥T PΩ f2
∥∥∥Lp(E)
. ∥b1 − (b1)E∥Lp1 (E) ∥b2 − (b2)E∥Lp2 (E) rγ
p
∞∫2kr
∥f∥Lp(E(x0,t))t− γ
p−1
dt
. ∥b1∥LC
x0p1,λ1,P
∥b2∥LC
x0p2,λ2,P
rγ
(1
p1+ 1
p2+ 1
p
)
×∞∫
2kr
(1 + ln t
r
)2t− γ
p+γ(λ1+λ2)−1∥f∥Lp(E(x0,t))dt
. ∥b1∥LC
x0p1,λ1,P
∥b2∥LC
x0p2,λ2,P
rγq
∞∫2kr
(1 + ln t
r
)2 ∥f∥Lp(E(x0,t))
tγp
−γ(λ1+λ2)+1 dt.
On the other hand, for the estimates used in G2, G3, we have to prove the belowinequality:
∣∣∣T PΩ ((b2 − (b2)E) f2) (x)
∣∣∣ . ∥b2∥LC
x0p2,λ2,P
∞∫2kr
(1 + ln t
r
)t− γ
p+γλ2−1 ∥f∥Lp(E(x0,t)) dt. (2.9)
Indeed, when s′ ≤ q, for x ∈ E, by Fubini’s theorem and applying Hölder’s inequality andfrom (2.2), (2.3), 0 < r < t, (11) in [7] we have∣∣∣T P
Ω ((b2 (·) − (b2)E) f2) (x)∣∣∣
.∫
(2kE)C
|b2 (y) − (b2)E | |Ω (x − y)| |f(y)|ρ(x0−y)γ dy
≈∞∫
2kr
∫2kr<ρ(x0−y)<t
|b2 (y) − (b2)E | |Ω (x − y)| |f (y)| dy dttγ+1
.∞∫
2kr
∫E(x0,t)
∣∣∣b2 (y) − (b2)E(x0,t)
∣∣∣ |Ω (x − y)| |f (y)| dy dttγ+1
+∞∫
2kr
∣∣∣(b2)E(x0,t) − (b2)E(x0,r)
∣∣∣ ∫E(x0,t)
|Ω (x − y)| |f (y)| dy dttγ+1
.∞∫
2kr
∥∥∥b2 (·) − (b2)E(x0,t)
∥∥∥Lp2 (E(x0,t))
∥Ω (· − y)∥Ls(E(x0,t)) ∥f∥Lp(E(x0,t))
× |E (x0, t)|1− 1p2
− 1s
− 1p dt
tγ+1
+∞∫
2kr
∣∣∣(b2)E(x0,t) − (b2)E(x0,r)
∣∣∣ ∥f∥Lp(E(x0,t)) ∥Ω (· − y)∥Ls(E(x0,t)) |E (x0, t)|1− 1p
− 1s dt
tγ+1
.∞∫
2kr
∥∥∥b2 (·) − (b2)E(x0,t)
∥∥∥Lp2 (E(x0,t))
∥f∥Lp(E(x0,t)) t−1− γ
p2− γ
p dt
628 F. Gürbüz
+ ∥b2∥LC
x0p2,λ2,P
∞∫2kr
(1 + ln t
r
)∥f∥Lp(E(x0,t)) t
−1− γp
+γλ2dt
. ∥b2∥LC
x0p2,λ2,P
∞∫2kr
(1 + ln t
r
)t− γ
p+γλ2−1 ∥f∥Lp(E(x0,t)) dt.
This completes the proof of inequality (2.9).Let 1 < τ < ∞, such that 1
q = 1p1
+ 1τ . Then, using Hölder’s inequality and from (2.9),
(2.3) and 0 < r < t, we get
G2 =∥∥∥(b1 − (b1)E) T P
Ω ((b2 − (b2)E) f2)∥∥∥
Lq(E)
. ∥b1 − (b1)E∥Lp1 (E)
∥∥∥T PΩ ((b2 (·) − (b2)E) f2)
∥∥∥Lτ (E)
. ∥b1∥LC
x0p1,λ1,P
∥b2∥LC
x0p2,λ2,P
rγq
∞∫2kr
(1 + ln t
r
)2 ∥f∥Lp(E(x0,t))
tγp
−γ(λ1+λ2)+1 dt.
Similarly, G3 has the same estimate above, so here we omit the details. Then the inequality
G3 =∥∥∥(b2 − (b2)E) T P
Ω ((b1 − (b1)E) f2)∥∥∥
Lq(E)
. ∥b1∥LC
x0p1,λ1,P
∥b2∥LC
x0p2,λ2,P
rγq
∞∫2kr
(1 + ln t
r
)2 ∥f∥Lp(E(x0,t))
tγp
−γ(λ1+λ2)+1 dt
is valid.Now, let us estimate G4 =
∥∥∥T PΩ ((b1 − (b1)E) (b2 − (b2)E) f2)
∥∥∥Lq(E)
. It’s similar to theestimate of (2.9), for any x ∈ E, we also write∣∣∣T P
Ω ((b1 − (b1)E) (b2 − (b2)E) f2) (x)∣∣∣
.∞∫
2kr
∫E(x0,t)
∣∣∣b1 (y) − (b1)E(x0,t)
∣∣∣ ∣∣∣b2 (y) − (b2)E(x0,t)
∣∣∣ |Ω(x − y)| |f (y)| dy dttγ+1
+∞∫
2kr
∫E(x0,t)
∣∣∣b1 (y) − (b1)E(x0,t)
∣∣∣ ∣∣∣(b2)E(x0,t) − (b2)E(x0,r)
∣∣∣ |Ω(x − y)| |f (y)| dy dttγ+1
+∞∫
2kr
∫E(x0,t)
∣∣∣(b1)E(x0,t) − (b2)E(x0,r)
∣∣∣ ∣∣∣b2 (y) − (b2)E(x0,t)
∣∣∣ |Ω(x − y)| |f (y)| dy dttγ+1
+∞∫
2kr
∫E(x0,t)
∣∣∣(b1)E(x0,t) − (b2)E(x0,r)
∣∣∣ ∣∣∣(b2)E(x0,t) − (b2)E(x0,r)
∣∣∣ |Ω(x − y)| |f (y)| dy dttγ+1
≡ G41 + G42 + G43 + G44.Let us estimate G41, G42, G43, G44, respectively.Firstly, to estimate G41, similar to the estimate of (2.9), we get
G41 . ∥b1∥LC
x0p1,λ1,P
∥b2∥LC
x0p2,λ2,P
∞∫2kr
(1 + ln t
r
)2 ∥f∥Lp(E(x0,t))
tγp
−γ(λ1+λ2)+1 dt.
Secondly, to estimate G42 and G43, from (2.9), (2.2), (2.3) and 0 < r < t, it follows that
G42 . ∥b1∥LC
x0p1,λ1,P
∥b2∥LC
x0p2,λ2,P
∞∫2kr
(1 + ln t
r
)2 ∥f∥Lp(E(x0,t))
tγp
−γ(λ1+λ2)+1 dt,
Some parabolic multilinear commutators 629
and
G43 . ∥b1∥LC
x0p1,λ1,P
∥b2∥LC
x0p2,λ2,P
∞∫2kr
(1 + ln t
r
)2 ∥f∥Lp(E(x0,t))
tγp
−γ(λ1+λ2)+1 dt.
Finally, to estimate G44, similar to the estimate of (2.9), from (2.2), (2.3) and 0 < r < t,we have
G44 . ∥b1∥LC
x0p1,λ1,P
∥b2∥LC
x0p2,λ2,P
∞∫2kr
(1 + ln t
r
)2 ∥f∥Lp(E(x0,t))
tγp
−γ(λ1+λ2)+1 dt.
By the estimates of G4j above, where j = 1, 2, 3, we know that∣∣∣T PΩ ((b1 − (b1)E) (b2 − (b2)E) f2) (x)
∣∣∣. ∥b1∥
LCx0p1,λ1,P
∥b2∥LC
x0p2,λ2,P
∞∫2kr
(1 + ln t
r
)2 ∥f∥Lp(E(x0,t))
tγp
−γ(λ1+λ2)+1 dt.
Then, we have
G4 =∥∥∥T P
Ω ((b1 − (b1)E) (b2 − (b2)E) f2)∥∥∥
Lq(E)
. ∥b1∥LC
x0p1,λ1,P
∥b2∥LC
x0p2,λ2,P
rγq
∞∫2kr
(1 + ln t
r
)2 ∥f∥Lp(E(x0,t))
tγp
−γ(λ1+λ2)+1 dt.
So, combining all the estimates for G1, G2, G3, G4, we get
G =∥∥∥[(b1, b2) , T P
Ω ]f2∥∥∥
Lq(E). ∥b1∥
LCx0p1,λ1,P
∥b2∥LC
x0p2,λ2,P
rγq
∞∫2kr
(1 + ln t
r
)2 ∥f∥Lp(E(x0,t))
tγp
−γ(λ1+λ2)+1 dt.
Thus, putting estimates F and G together, we get the desired conclusion∥∥∥[(b1, b2) , T PΩ ]f
∥∥∥Lq(E)
. ∥b1∥LC
x0p1,λ1,P
∥b2∥LC
x0p2,λ2,P
rγq
∞∫2kr
(1 + ln t
r
)2 ∥f∥Lp(E(x0,t))
tγp
−γ(λ1+λ2)+1 dt.
For the case of p < s, we can also use the same method, so we omit the details. Thiscompletes the proof of Lemma 2.2. Lemma 2.3. Suppose that x0 ∈ Rn, Ω ∈ Ls(Sn−1), 1 < s ≤ ∞, is At-homogeneous ofdegree zero. Let T P
Ω,α be a parabolic linear operator satisfying condition (1.2). Let also
0 < α < γ and 1 < q, q1, pi, p < γα with 1
q =m∑
i=11pi
+ 1p , 1
q1= 1
q − αγ and
−→b ∈ LC
x0pi,λi,P
(Rn)
for 0 ≤ λi < 1γ , i = 1, . . . , m.
Then, for s′ ≤ q the inequality
∥[−→b , T P
Ω,α]f∥Lq1 (E(x0,r)) .m∏
i=1∥−→b ∥
LCx0pi,λi,P
rγq1
∞∫2kr
(1 + ln t
r
)m ∥f∥Lp(E(x0,t))
t
γ
(1
q1−
( m∑i=1
λi+m∑
i=1
1pi
))+1
dt
(2.10)holds for any ellipsoid E (x0, r) and for all f ∈ Lloc
p (Rn). Also, for q1 < s the inequality
∥[−→b , T P
Ω,α]f∥Lq1 (E(x0,r))
.m∏
i=1∥−→b ∥
LCx0pi,λi,P
rγq1
− γs
∞∫2kr
(1 + ln t
r
)m ∥f∥Lp(E(x0,t))
t
γ
(1
q1−
(1s
+m∑
i=1λi+
m∑i=1
1pi
))+1
dt
630 F. Gürbüz
holds for any ellipsoid E (x0, r) and for all f ∈ Llocp (Rn).
Proof. Similar to the proof of Lemma 2.2, it is sufficient to show that the conclusionholds for m = 2. As in the proof of Lemma 2.2, we split f = f1 + f2 in form (2.5) andhave∥∥∥[(b1, b2) , T P
Ω,α]f∥∥∥
Lq1 (E)≤∥∥∥[(b1, b2) , T P
Ω,α]f1∥∥∥
Lq1 (E)+∥∥∥[(b1, b2) , T P
Ω,α]f2∥∥∥
Lq1 (E)=: A + B.
Let us estimate A + B, respectively.For [(b1, b2) , T P
Ω,α]f1 (x), it is easy to see that
A =∥∥∥[(b1, b2) , T P
Ω,α]f1∥∥∥
Lq1 (E).∥∥∥(b1 − (b1)E) (b2 (x) − (b2)E) T P
Ω,αf1∥∥∥
Lq1 (E)
+∥∥∥(b1 − (b1)E) T P
Ω,α ((b2 − (b2)E) f1)∥∥∥
Lq1 (E)
+∥∥∥(b2 − (b2)E) T P
Ω,α ((b1 − (b1)E) f1)∥∥∥
Lq1 (E)
+∥∥∥T P
Ω,α ((b1 − (b1)E) (b2 − (b2)E) f1)∥∥∥
Lq1 (E)
≡A1 + A2 + A3 + A4. (2.11)
Let 1 < q < ∞. Since 1q = 1
p − αγ , it is obvious that 1
q1= 1
p1+ 1
p2+ 1
q . Thus, using Hölder’sinequality and by Theorem 0.1 in [8] and (2.7) it follows that:
A1 . ∥(b1 − (b1)E) (b2 − (b2)E)∥Lr(E)
∥∥∥T PΩ,αf1
∥∥∥Lq(E)
. ∥b1 − (b1)B∥Lp1 (E) ∥b2 − (b2)B∥Lp2 (E) ∥f∥Lp(2kE)
. ∥b1 − (b1)E∥Lp1 (E) ∥b2 − (b2)E∥Lp2 (E) rγq
∞∫2kr
∥f∥Lp(E(x0,t))dt
tγq
+1
. ∥b1∥LC
x0p1,λ1,P
∥b2∥LC
x0p2,λ2,P
rγq1
∞∫2kr
(1 + ln t
r
)2 ∥f∥Lp(E(x0,t))
tγq1
−γ(λ1+λ2)−γ
(1
p1+ 1
p2
)+1
dt.
To estimate A2, let 1 < τ < ∞, such that 1q1
= 1p1
+ 1τ . Then, similar to the estimates for
A1, we have
A2 . ∥b1 − (b1)E∥Lp1 (E)
∥∥∥T PΩ,α ((b2 (·) − (b2)E) f1)
∥∥∥Lτ (E)
. ∥b1 − (b1)E∥Lp1 (E) ∥(b2 (·) − (b2)E) f1∥Lk(E)
. ∥b1 − (b1)E∥Lp1 (E) ∥b2 − (b2)E∥Lp2 (2kE) ∥f∥Lp(2kE) ,
where 1 < k < 2γα , such that 1
k = 1p2
+ 1p = 1
τ + αγ . By (2.7) and (2.8), we get
A2 . ∥b1∥LC
x0p1,λ1,P
∥b2∥LC
x0p2,λ2,P
rγq1
∞∫2kr
(1 + ln t
r
)2 ∥f∥Lp(E(x0,t))
tγq1
−γ(λ1+λ2)−γ
(1
p1+ 1
p2
)+1
dt.
In a similar way, A3 has the same estimate as above, so we omit the details. Then we havethat
A3 . ∥b1∥LC
x0p1,λ1,P
∥b2∥LC
x0p2,λ2,P
rγq1
∞∫2kr
(1 + ln t
r
)2 ∥f∥Lp(E(x0,t))
tγq1
−γ(λ1+λ2)−γ
(1
p1+ 1
p2
)+1
dt.
Now let us consider the term A4. Let 1 < q < 2γα such that 1
q1= 1
q − αγ . It is easy to
see that 1q = 1
p1+ 1
p2+ 1
p . Thus, by Theorem 0.1 in [8], Hölder’s inequality and (2.8), we
Some parabolic multilinear commutators 631
obtain
A4 . ∥(b1 − (b1)E) (b2 − (b2)E) f1∥Lq(E)
. ∥b1 − (b1)E∥Lp1 (2kE) ∥b2 − (b2)E∥Lp2 (2kE) ∥f∥Lp(2kE)
. ∥b1∥LC
x0p1,λ1,P
∥b2∥LC
x0p2,λ2,P
rγq1
∞∫2kr
(1 + ln t
r
)2 ∥f∥Lp(E(x0,t))
tγq1
−γ(λ1+λ2)−γ
(1
p1+ 1
p2
)+1
dt.
Combining all the estimates of A1, A2, A3, A4; we get
A =∥∥∥[(b1, b2) , T P
Ω,α]f1∥∥∥
Lq1 (E)
. ∥b1∥LC
x0p1,λ1,P
∥b2∥LC
x0p2,λ2,P
rγq1
∞∫2kr
(1 + ln t
r
)2 ∥f∥Lp(E(x0,t))
tγq1
−γ(λ1+λ2)−γ
(1
p1+ 1
p2
)+1
dt.
Now, let us estimate B =∥∥∥[(b1, b2) , T P
Ω,α]f2∥∥∥
Lq1 (E). For B, it’s similar to (2.11) we also
write
B =∥∥∥[(b1, b2) , T P
Ω,α]f2∥∥∥
Lq1 (E).∥∥∥(b1 − (b1)E) (b2 (x) − (b2)E) T P
Ω,αf2∥∥∥
Lq1 (E)
+∥∥∥(b1 − (b1)E) T P
Ω,α ((b2 − (b2)E) f2)∥∥∥
Lq1 (E)
+∥∥∥(b2 − (b2)E) T P
Ω,α ((b1 − (b1)E) f2)∥∥∥
Lq1 (E)
+∥∥∥T P
Ω,α ((b1 − (b1)E) (b2 − (b2)E) f2)∥∥∥
Lq1 (E)
≡B1 + B2 + B3 + B4.
Let 1 < p1, p2 < 2γα . Since 1
q = 1p − α
γ , it is easy to see that 1q1
= 1p1
+ 1p2
+ 1q . Thus, using
Hölder’s inequality and by (2.6) in [8] and (2.7), we have
B1 . ∥(b1 − (b1)E) (b2 − (b2)E)∥Lp(E)
∥∥∥T PΩ,αf2
∥∥∥Lq(E)
. ∥b1 − (b1)E∥Lp1 (E) ∥b2 − (b2)E∥Lp2 (E) rγ
q
∞∫2kr
∥f∥Lp(E(x0,t))t− γ
q−1
dt
. ∥b1∥LC
x0p1,λ1,P
∥b2∥LC
x0p2,λ2,P
rγq1
∞∫2kr
(1 + ln t
r
)2 ∥f∥Lp(E(x0,t))
tγq1
−γ(λ1+λ2)−γ
(1
p1+ 1
p2
)+1
dt.
On the other hand, for the estimates used in B2, B3, we have to prove the belowinequality:
∣∣∣T PΩ,α ((b2 − (b2)E) f2) (x)
∣∣∣ . ∥b2∥LC
x0p2,λ2,P
∞∫2kr
(1 + ln t
r
)t− γ
p+γλ2−1+α ∥f∥Lp(E(x0,t)) dt.
(2.12)Indeed, when s′ ≤ q, for x ∈ E, by Fubini’s theorem and applying Hölder’s inequality andfrom (2.2), (2.3), 0 < r < t and (11) in [7] we have∣∣∣T P
Ω,α ((b2 (·) − (b2)E) f2) (x)∣∣∣
.∫
(2kE)C
|b2 (y) − (b2)E | |Ω (x − y)| |f(y)|ρ(x0−y)γ−α dy
632 F. Gürbüz
≈∞∫
2kr
∫2kr<ρ(x0−y)<t
|b2 (y) − (b2)E | |Ω (x − y)| |f (y)| dy dttγ−α+1
.∞∫
2kr
∫E(x0,t)
∣∣∣b2 (y) − (b2)E(x0,t)
∣∣∣ |Ω (x − y)| |f (y)| dy dttγ−α+1
+∞∫
2kr
∣∣∣(b2)E(x0,t) − (b2)E(x0,r)
∣∣∣ ∫E(x0,t)
|Ω (x − y)| |f (y)| dy dttγ−α+1
.∞∫
2kr
∥∥∥b2 (·) − (b2)E(x0,t)
∥∥∥Lp2 (E(x0,t))
∥Ω (· − y)∥Ls(E(x0,t)) ∥f∥Lp(E(x0,t))
× |E (x0, t)|1− 1p2
− 1s
− 1p dt
tγ−α+1
+∞∫
2kr
∣∣∣(b2)E(x0,t) − (b2)E(x0,r)
∣∣∣ ∥f∥Lp(E(x0,t)) ∥Ω (· − y)∥Ls(E(x0,t)) |E (x0, t)|1− 1p
− 1s dt
tγ−α+1
. ∥b2∥LC
x0p2,λ2,P
∞∫2kr
(1 + ln t
r
)t− γ
p+γλ2−1+α ∥f∥Lp(E(x0,t)) dt.
This completes the proof of inequality (2.12).Let 1 < τ < ∞, such that 1
q1= 1
p1+ 1
τ and 1τ = 1
p2+ 1
p − αγ . Then, using Hölder’s
inequality and from (2.12), (2.3) and 0 < r < t, we get
B2 . ∥b1 − (b1)E∥Lp1 (E)
∥∥∥T PΩ,α ((b2 (·) − (b2)E) f2)
∥∥∥Lτ (E)
. ∥b1∥LC
x0p1,λ1,P
∥b2∥LC
x0p2,λ2,P
rγq1
∞∫2kr
(1 + ln t
r
)2 ∥f∥Lp(E(x0,t))
tγq1
−γ(λ1+λ2)−γ
(1
p1+ 1
p2
)+1
dt.
Similarly, B3 has the same estimate above, so here we omit the details. Then the inequality
B3 =∥∥∥(b2 − (b2)E) T P
Ω,α ((b1 − (b1)E) f2)∥∥∥
Lq1 (E)
. ∥b1∥LC
x0p1,λ1,P
∥b2∥LC
x0p2,λ2,P
rγq1
∞∫2kr
(1 + ln t
r
)2 ∥f∥Lp(E(x0,t))
tγq1
−γ(λ1+λ2)−γ
(1
p1+ 1
p2
)+1
dt
is valid.Now, let us estimate B4 =
∥∥∥T PΩ,α ((b1 − (b1)E) (b2 − (b2)E) f2)
∥∥∥Lq1 (E)
. It’s similar to the
estimate of (2.12), for any x ∈ E, we also write∣∣∣T PΩ,α ((b1 − (b1)E) (b2 − (b2)E) f2) (x)
∣∣∣.
∞∫2kr
∫E(x0,t)
∣∣∣b1 (y) − (b1)E(x0,t)
∣∣∣ ∣∣∣b2 (y) − (b2)E(x0,t)
∣∣∣ |Ω(x − y)| |f (y)| dy dttγ−α+1
+∞∫
2kr
∫E(x0,t)
∣∣∣b1 (y) − (b1)E(x0,t)
∣∣∣ ∣∣∣(b2)E(x0,t) − (b2)E(x0,r)
∣∣∣ |Ω(x − y)| |f (y)| dy dttγ−α+1
+∞∫
2kr
∫E(x0,t)
∣∣∣(b1)E(x0,t) − (b2)E(x0,r)
∣∣∣ ∣∣∣b2 (y) − (b2)E(x0,t)
∣∣∣ |Ω(x − y)| |f (y)| dy dttγ−α+1
+∞∫
2kr
∫E(x0,t)
∣∣∣(b1)E(x0,t) − (b2)E(x0,r)
∣∣∣ ∣∣∣(b2)E(x0,t) − (b2)E(x0,r)
∣∣∣ |Ω(x − y)| |f (y)| dy dttγ−α+1
Some parabolic multilinear commutators 633
≡ B41 + B42 + B43 + B44.Let us estimate B41, B42, B43, B44, respectively.Firstly, to estimate B41, similar to the estimate of (2.12), we get
B41 . ∥b1∥LC
x0p1,λ1,P
∥b2∥LC
x0p2,λ2,P
∞∫2kr
(1 + ln t
r
)2 ∥f∥Lp(E(x0,t))
tγq1
−γ(λ1+λ2)−γ
(1
p1+ 1
p2
)+1
dt.
Secondly, to estimate B42 and B43, from (2.12), (2.2), (2.3) and 0 < r < t, it follows that
B42 . ∥b1∥LC
x0p1,λ1,P
∥b2∥LC
x0p2,λ2,P
∞∫2kr
(1 + ln t
r
)2 ∥f∥Lp(E(x0,t))
tγq1
−γ(λ1+λ2)−γ
(1
p1+ 1
p2
)+1
dt,
and
B43 . ∥b1∥LC
x0p1,λ1,P
∥b2∥LC
x0p2,λ2,P
∞∫2kr
(1 + ln t
r
)2 ∥f∥Lp(E(x0,t))
tγq1
−γ(λ1+λ2)−γ
(1
p1+ 1
p2
)+1
dt.
Finally, to estimate B44, similar to the estimate of (2.12) and from (2.2), (2.3) and0 < r < t, we have
B44 . ∥b1∥LC
x0p1,λ1,P
∥b2∥LC
x0p2,λ2,P
∞∫2kr
(1 + ln t
r
)2 ∥f∥Lp(E(x0,t))
tγq1
−γ(λ1+λ2)−γ
(1
p1+ 1
p2
)+1
dt.
By the estimates of B4j above, where j = 1, 2, 3, we know that
∣∣∣T PΩ,α ((b1 − (b1)E) (b2 − (b2)E) f2) (x)
∣∣∣ . ∥b1∥LC
x0p1,λ1,P
∥b2∥LC
x0p2,λ2,P
∞∫2kr
(1 + ln t
r
)2
×∥f∥Lp(E(x0,t))
tγq1
−γ(λ1+λ2)−γ
(1
p1+ 1
p2
)+1
dt.
Then, we have
B4 . ∥b1∥LC
x0p1,λ1,P
∥b2∥LC
x0p2,λ2,P
rγq1
∞∫2kr
(1 + ln t
r
)2 ∥f∥Lp(E(x0,t))
tγq1
−γ(λ1+λ2)−γ
(1
p1+ 1
p2
)+1
dt.
So, combining all the estimates for B1, B2, B3, B4, we get
B =∥∥∥[(b1, b2) , T P
Ω,α]f2∥∥∥
Lq1 (E)
. ∥b1∥LC
x0p1,λ1,P
∥b2∥LC
x0p2,λ2,P
rγq1
∞∫2kr
(1 + ln t
r
)2 ∥f∥Lp(E(x0,t))
tγq1
−γ(λ1+λ2)−γ
(1
p1+ 1
p2
)+1
dt.
Thus, putting estimates A and B together, we get the desired conclusion∥∥∥[(b1, b2) , T PΩ,α]f
∥∥∥Lq1 (E)
. ∥b1∥LC
x0p1,λ1,P
∥b2∥LC
x0p2,λ2,P
rγq1
∞∫2kr
(1 + ln t
r
)2 ∥f∥Lp(E(x0,t))
tγq1
−γ(λ1+λ2)−γ
(1
p1+ 1
p2
)+1
dt.
For the case of q1 < s, we can also use the same method, so we omit the details. Thus,we complete the the proof of Lemma 2.3.
634 F. Gürbüz
3. Proofs of the main results3.1. Proof of Theorem 1.3.
We consider (1.5) firstly. Since f ∈ LMx0p,φ1,P , by (1.3) and it is also non-decreasing,
with respect to t, of the norm ∥f∥Lp(E(x0,t)), we get
∥f∥Lp(E(x0,t))
essinf0<t<τ<∞
φ1(x0, τ)τγp
≤ esssup0<t<τ<∞
∥f∥Lp(E(x0,t))
φ1(x0, τ)τγp
≤ esssup0<τ<∞
∥f∥Lp(E(x0,τ))
φ1(x0, τ)τγp
≤ ∥f∥LM
x0p,φ,P
. (3.1)
For s′ ≤ q < ∞, since (φ1, φ2) satisfies (1.4), we have∞∫r
(1 + ln t
r
)m ∥f∥Lp(E(x0,t))
t
γ
(1p
−m∑
i=1λi
)+1
dt
≤∞∫r
(1 + ln t
r
)m ∥f∥Lp(E(x0,t))
essinft<τ<∞
φ1(x0, τ)τγp
essinft<τ<∞
φ1(x0, τ)τγp
t
γ
(1p
−m∑
i=1λi
)+1
dt
≤ C∥f∥LM
x0p,φ,P
∞∫r
(1 + ln t
r
)m essinft<τ<∞
φ1(x0, τ)τγp
t
γ
(1p
−m∑
i=1λi
)+1
dt
≤ C∥f∥LM
x0p,φ,P
φ2(x0, r). (3.2)
Then by (2.4) and (3.2), we get∥∥∥[−→b , T PΩ ]f
∥∥∥LM
x0q,φ2,P
= supr>0
φ2 (x0, r)−1 |E(x0, r)|−1q
∥∥∥[−→b , T PΩ ]f
∥∥∥Lq(E(x0,r))
≤ Cm∏
i=1∥−→b ∥
LCx0pi,λi,P
supr>0
φ2 (x0, r)−1
×∞∫r
(1 + ln t
r
)m ∥f∥Lp(E(x0,t))
t
γ
(1p
−m∑
i=1λi
)+1
dt
≤ Cm∏
i=1∥−→b ∥
LCx0pi,λi,P
∥f∥LM
x0p,φ,P
.
For the case of p < s, we can also use the same method, so we omit the details. Thus, wefinish the proof of (1.5).
We are now in a place of proving (1.6) in Theorem 1.3.
Remark 3.1. The conclusion of (1.6) is a direct consequence of the following Lemma 3.2and (1.5). In order to do this, we need to define an operator by
[−→b , T P
|Ω|] (|f |) (x) =∫Rn
m∏i=1
[|bi (x) − bi (y)|] |Ω(x − y)|ρ (x − y)γ |f(y)| dy,
where Ω ∈ Ls(Sn−1), 1 < s ≤ ∞, is At-homogeneous of degree zero in Rn.
Some parabolic multilinear commutators 635
Using the idea of proving Lemma 2 in [4] (see also [9]), we can obtain the followingpointwise relation:
Lemma 3.2. Let Ω ∈ Ls(Sn−1), 1 < s ≤ ∞, be At-homogeneous of degree zero. Then wehave
MP
Ω,−→b
f(x) ≤ [−→b , T P
|Ω|] (|f |) (x) for x ∈ Rn.
In fact, for any t > 0, we have
[−→b , T P
|Ω|] (|f |) (x) ≥∫
ρ(x−y)<t
m∏i=1
[|bi (x) − bi (y)|] |Ω(x − y)|ρ (x − y)γ |f(y)| dy
≥ 1tγ
∫E(x,t)
m∏i=1
[|bi (x) − bi (y)|] |Ω(x − y)| |f(y)| dy.
Taking the supremum for t > 0 on the inequality above, we get
[−→b , T P
|Ω|] (|f |) (x) ≥ MP
Ω,−→b
f(x) for x ∈ Rn.
From the process proving (1.5), it is easy to see that the conclusions of (1.5) also holdfor [
−→b , T P
|Ω|]. Combining this with Lemma 3.2, we can immediately obtain (1.6), whichcompletes the proof.
3.2. Proof of Theorem 1.5.Similar to the proof of Theorem 1.3, We consider (1.8) firstly.For s′ ≤ q < ∞, since (φ1, φ2) satisfies (1.7) and by (3.1), we have
∞∫r
(1 + ln t
r
)m ∥f∥Lp(E(x0,t))
t
γ
(1
q1−
( m∑i=1
λi+m∑
i=1
1pi
))+1
dt
≤∞∫r
(1 + ln t
r
)m ∥f∥Lp(E(x0,t))
essinft<τ<∞
φ1(x0, τ)τγp
essinft<τ<∞
φ1(x0, τ)τγp
t
γ
(1
q1−
( m∑i=1
λi+m∑
i=1
1pi
))+1
dt
≤ C∥f∥LM
x0p,φ,P
∞∫r
(1 + ln t
r
)m essinft<τ<∞
φ1(x0, τ)τγp
t
γ
(1
q1−
( m∑i=1
λi+m∑
i=1
1pi
))+1
dt
≤ C∥f∥LM
x0p,φ,P
φ2(x0, r). (3.3)
Then by (2.10) and (3.3), we get∥∥∥[−→b , T PΩ,α]f
∥∥∥LM
x0q1,φ2,P
= supr>0
φ2 (x0, r)−1 |E(x0, r)|−1
q1
∥∥∥[−→b , T PΩ,α]f
∥∥∥Lq1 (E(x0,r))
≤ Cm∏
i=1∥−→b ∥
LCx0pi,λi,P
supr>0
φ2 (x0, r)−1
×∞∫r
(1 + ln t
r
)m ∥f∥Lp(E(x0,t))
t
γ
(1
q1−
( m∑i=1
λi+m∑
i=1
1pi
))+1
dt
636 F. Gürbüz
≤ Cm∏
i=1∥−→b ∥
LCx0pi,λi,P
∥f∥LM
x0p,φ,P
.
For the case of q1 < s, we can also use the same method, so we omit the details. Thus,we finish the proof of (1.8).
We are now in a place of proving (1.9) in Theorem 1.5.
Remark 3.3. The conclusion of (1.9) is a direct consequence of the following Lemma 3.4and (1.8). In order to do this, we need to define an operator by
[−→b , T P
|Ω|,α] (|f |) (x) =∫Rn
m∏i=1
[|bi (x) − bi (y)|] |Ω(x − y)|ρ (x − y)γ−α |f(y)| dy 0 < α < γ,
where Ω ∈ Ls(Sn−1), 1 < s ≤ ∞, is At-homogeneous of degree zero in Rn.
Using the idea of proving Lemma 2 in [4] (see also [9]), we can obtain the followingpointwise relation:
Lemma 3.4. Let 0 < α < γ and Ω ∈ Ls(Sn−1), 1 < s ≤ ∞, be At-homogeneous of degreezero. Then we have
MP
Ω,−→b ,α
f(x) ≤ [−→b , T P
|Ω|,α] (|f |) (x) for x ∈ Rn.
In fact, for any t > 0, we have
[−→b , T P
|Ω|,α] (|f |) (x) ≥∫
ρ(x−y)<t
m∏i=1
[|bi (x) − bi (y)|] |Ω(x − y)|ρ (x − y)γ−α |f(y)| dy
≥ 1tγ−α
∫E(x,t)
m∏i=1
[|bi (x) − bi (y)|] |Ω(x − y)| |f(y)| dy.
Taking the supremum for t > 0 on the inequality above, we get
[−→b , T P
|Ω|,α] (|f |) (x) ≥ MP
Ω,−→b ,α
f(x) for x ∈ Rn.
From the process proving (1.8), it is easy to see that the conclusions of (1.8) also holdfor [
−→b , T P
|Ω|,α]. Combining this with Lemma 3.4, we can immediately obtain (1.9), whichcompletes the proof.
References[1] D.R. Adams, Morrey spaces, Lecture Notes in Applied and Numerical Harmonic Anal-
ysis, Birkhäuser/Springer, Cham, 2015.[2] A.P. Calderón and A. Torchinsky, Parabolic maximal functions associated with a dis-
tribution, Adv. Math. 16, 1-64, 1975.[3] R.R. Coifman and G. Weiss, Analyse harmonique non-commutative sur certains es-
paces homogènes, Étude de certaines intégrales singuliàres, Lecture Notes in Mathe-matics, 242, Berlin-New York: Springer-Verlag, 1971, (in French).
[4] Y. Ding, Weak type bounds for a class of rough operators with power weights, Proc.Amer. Math. Soc. 125, 2939-2942, 1997.
[5] E. Fabes and N. Riviére, Singular integrals with mixed homogeneity, Stud. Math. 27,19-38, 1966.
[6] G.B. Folland and E.M. Stein, Hardy Spaces on homogeneous groups, Math. Notes 28,Princeton Univ. Press, Princeton, 1982.
Some parabolic multilinear commutators 637
[7] F. Gürbüz, Parabolic sublinear operators with rough kernel generated by parabolicCalderón-Zygmund operators and parabolic local Campanato space estimates for theircommutators on the parabolic generalized local Morrey spaces, Open Math. 14 (1),300-323, 2016.
[8] F. Gürbüz Parabolic generalized local Morrey space estimates of rough parabolic sub-linear operators and commutators, Adv. Math. (China) 46 (5), 765-792, 2017.
[9] F. Gürbüz, Some estimates for generalized commutators of rough fractional maximaland integral operators on generalized weighted Morrey spaces, Canad. Math. Bull. 60(1), 131-145, 2017.
[10] C.B. Morrey, On the solutions of quasi-linear elliptic partial differential equations,Trans. Amer. Math. Soc. 43, 126-166, 1938.
[11] F. Soria and G. Weiss, A remark on singular integrals and power weights, IndianaUniv. Math. J. 43, 187-204, 1994.
[12] E.M. Stein, Note on singular integrals, Proc. Amer. Soc. Math. 8, 250-254, 1957.[13] R.L. Wheeden and A. Zygmund, Measure and Integral: An Introduction to Real
Analysis, vol. 43 of Pure and Applied Mathematics, Marcel Dekker, New York, NY,USA, 1977.
[14] Q.Y. Xue, Y. Ding and K. Yabuta, Parabolic Littlewood-Paley g-function with roughkernel, Acta Math. Sin. (Engl. Ser.) 24 (12), 2049-2060, 2008.
Hacettepe Journal ofMathematics & Statistics
Hacet. J. Math. Stat.Volume 49 (2) (2020), 638 – 647
DOI : 10.15672/hujms.460348
Research Article
Construction of arithmetic secret sharing schemesby using torsion limits
Seher Tutdere∗1, Osmanbey Uzunkol2,3
1Balıkesir University, Department of Mathematics, Altıeylül, 10145, Balıkesir2FernUniversität in Hagen, Faculty of Mathematics and Computer Science, Germany
3Mathematical and Computational Sciences, TÜBİTAK BİLGEM, Turkey
AbstractCascudo, Cramer, and Xing [Torsion limits and Riemann-Roch systems for function fieldsand applications, IEEE Trans. Inf. Theory, 2014] gave a construction of arithmetic secretsharing schemes by using the torsion limits of algebraic function fields and Riemann-Rochsystems. In this work, we give some new conditions for the construction of arithmeticsecret sharing schemes. Furthermore, we give new bounds on the torsion limits of certaintowers of function fields over finite fields.
Mathematics Subject Classification (2010). 94A62, 11R58, 11T71
Keywords. algebraic function fields, torsion limits, arithmetic secret sharing schemes
1. IntroductionSecret sharing is a cryptographic mechanism allowing to distribute secret shares among
different parties. This is achieved by a trusted dealer in such a way that only authorizedsubsets of the parties can determine the secret [3]. Secret sharing schemes have theadvantage of enabling the user to eliminate the root of trust problem [3,21]. Furthermore,secret sharing has plenty of privacy preserving real-life applications ranging from accesscontrols [20], oblivious transfers [23] to biometric authentication schemes [13].
The set of all subsets for a group of users authorized to access to some resources withina system is called its access structure. If the authorized subsets of a secret sharing schemeare exactly those sets whose cardinality is larger than a predetermined lower bound, thenthe secret sharing scheme is said to have a threshold access structure [10]. Moreover, asecret sharing scheme is called ideal if the shares have the same size as the secrets [3].Shamir’s secret sharing scheme is a classical example of an ideal secret sharing schemehaving threshold access structure. Since the shares are computed and reconstructed byusing only linear functions [18], it is also an example of a linear secret sharing scheme(LSSS). Moreover, an LSSS can be constructed for any access structure [17] following thenotion of general access structures introduced in Ito et al. [16]. However, the shares growexponentially in the number of parties, and the optimization of secret sharing schemes forarbitrary access structures is a difficult problem [3].
∗Corresponding Author.Email addresses: [email protected] (S. Tutdere), [email protected] (O. Uzunkol)Received: 16.09.2018; Accepted: 08.02.2019
Construction of arithmetic secret sharing schemes by using torsion limits 639
Chen and Cramer [8] introduced an LSSS defined over a finite field using algebraic-geometry codes (AG-codes). Unlike the general case, this scheme has the advantage thatshares are much smaller than the number of parties since one uses algebraic curves withmany rational points. Therefore, this achieves larger information rate by generalizingShamir’s secret sharing scheme into an algebra-geometric setting. One inevitable disad-vantage (due to the bounds on MDS codes [8]) is that this scheme is an ideal ramp secretsharing scheme, i.e. a quasi-threshold scheme. In particular, one has the property thatthe scheme has t-rejecting and t + 1 + 2g-accepting structure, where g is the genus of theunderlying algebraic curve. For LSSS over finite fields, in [5], an upper bound on the limitof t is given in some cases.
Cascudo, Cramer, and Xing [4, 6] introduced the notion of arithmetic secret sharingschemes based on AG-codes which are special quasi-threshold Fq-linear secret sharingschemes. They can be used as the main algorithmic primitives in realizing information the-oretically secure multi-party computation schemes (in particular, communication-efficientsecure two-party computations) and verifiable secret sharing schemes [7,9]. More precisely,it is shown in [8] that asymptotically good arithmetic secret sharing schemes can be usedto achieve constant-rate communication in secure two-party communication by removinglogarithmic terms which appear if one instead uses Shamir’s secret sharing scheme [21].As argued in [6], as an important primitive, these schemes can also be used in plenty ofother useful applications in cryptography including zero-knowledge for circuit satisfiability[14] and efficient oblivious transfer [15].
Constructing asymptotically good arithmetic secret sharing schemes is based on somespecial sequences of algebraic function fields. Besides the well-known notion of Iharalimits for constructing asymptotically good towers of function fields, the notion torsionlimits for algebraic function fields is introduced in [6]. Geometrically, in order to con-struct arithmetic secret sharing schemes with asymptotically good properties, we neednot only to have algebraic curves with many rational points but also to have jacobians(of corresponding algebraic curves) having comparably small d-torsion subgroups. On thealgebraic side, the torsion limit for a tower of function fields with a given Ihara limit givesinformation on the size of d−torsion subgroups of the corresponding degree-zero divisorclass groups. In [6], the authors give asymptotical results improving the classical boundsof Weil [24] on the size of torsion subgroups of abelian varieties over finite fields. For thispurpose, the existence of solutions for certain Riemann-Roch systems of equations is inves-tigated. The authors further give new bounds on the torsion limits of certain sequences offunction fields. Consequently, they use these bounds in constructing asymptotically goodarithmetic secret-sharing schemes by weakening the lower bound condition on the Iharaconstant A(q).
Following the lines of [6], the contributions of this work are given as follows:• We give a necessary condition on the asymptotic constructions of arithmetic se-
cret sharings which is helpful for the security of the construction (i.e. how manyadversaries it can tolerate) by using an important class of towers of function fieldsintroduced by Bassa et al. depending on the Ihara limit given in [2].
• We then give a simplification of Theorem 2.2 [6, Corollary 4.12] in Theorems 4 and5 under some conditions which eliminate not only the requirements of computingthe class number h, but also the number of effective divisors with degree r1, r2(Ar1 , Ar2 , resp.). These new conditions are much easier to verify for a givenfunction field though making the results less general than [6]. For this purpose,we mainly use the bound on class number [19] and the bound on the number ofeffective divisors [1, Theorem 3.5]. In Theorem 2.2, one needs to know both theclass number h and the values Ar1 , Ar2 of a given function field. In particular,our improvements imply that it is enough to know the genus g and the number of
640 S. Tutdere, O. Uzunkol
places of degree n ≤ g−1 (which are much easier to estimate) to obtain a sufficientcondition.
In Section 2, we first introduce the preliminaries and notations. We then give definitionsand some results regarding arithmetic-secret sharing schemes (based on AG-codes). Wefurther investigate the bounds on the torsion limits in this section. In Section 3, we givean application of these bounds for a family of towers of function fields over finite fieldsintroduced by Bassa et al. [2]. Finally, Section 4 concludes the paper with new conditionsfor the construction of arithmetic secret sharing schemes.
2. PreliminariesLet F/Fq be a function field over the finite field Fq with q elements, where q is a power
of a prime number p. We denote by g := g(F ) its genus, by Bi(F ) its number of places ofdegree i for any i ∈ N, and by P(F ) its set of rational places.
An asymptotically exact sequence of algebraic function fields F = Fii≥0 over a finitefield Fq is a sequence of function fields with gi := g(Fi) → ∞ such that for all m ≥ 1 thefollowing limit exists:
βm(F) = limi→∞
Bm(Fi)gi
.
It is well-known that any tower of function fields over any finite field is an asymptoticallyexact sequence, see for instance [12].
Throughout this paper, we will use the following notations frequently:• F/Fq: A function field with full constant field Fq.• An = An(F ): The number of effective divisors of F with degree n, for n ≥ 1. Set
An := 0 for n < 0.• P(k)(F ): The set of places of F with degree k ∈ N.• log := ln.• Div(F ): The group of divisors of F with Div(F ) ⊃ Div0(F ) ⊃ Princ(F ), where
Div0(F ) denotes the group of divisors of F with degree zero and Princ(F ) denotesthe group of principal divisors of F .
• JF = Div0(F )/Prin(F ): The zero divisor class group of F with cardinality |JF | =h(F ), which is called the class number.
For a positive integer r, letJF [r] := [D] ∈ JF : r · [D] = O
be the r-torsion subgroup of JF , where O denotes the identity element of JF . For eachfamily F = F/Fq of function fields with g(F ) → ∞, the limit
Jr(F) := lim infF ∈F
logq |JF [r]|g(F )
is called the r-torsion limit of the family F. Let a ∈ R and F be the set of sequences Fof function fields over Fq such that in each family genus tends to infinity and the Iharalimit
A(F) = limg(F )→∞
B1(F )g(F )
≥ a for every F ∈ F.
Then the asymptotic quantity Jr(q, a) is defined byJr(q, a) := lim inf
F∈FJr(F).
It is well-known that the Ihara constant is given by A(q) = lim supF
A(F), where F runs
over all infinite families of function fields over Fq. We note that we here only considerasymptotically exact sequences of function fields over finite fields.
Construction of arithmetic secret sharing schemes by using torsion limits 641
An (n, t, d, r)-arithmetic secret sharing scheme for Fkq over Fq is an n-code C for Fk
q suchthat t ≥ 1, d ≥ 2, C is t-disconnected, the d powering C∗d is an n-code for Fk
q , and C∗d
is r-reconstructing. This means that the secret sharing scheme is linear with the secret inFk
q for every share in Fq such that• no set of ≤ t parties has any information about the secret,• if d secrets are shared with the scheme, then for any set of r parties, the product
of the d secrets is a linear function of the vector containing the products of the dshares which correspond to each party.
These schemes are secret sharing schemes with additional properties regarding the recon-struction of the product of d secrets given the local products of the respective shares. Forfurther details and how such schemes may be constructed using function fields with manyplaces of degree one, see [6]. The results of [6] can be divided into two main categories;results related to the asymptotic existence of arithmetic secret sharing schemes, and theconditions for the existence of arithmetic secret sharing schemes.
Firstly, we investigate the bounds on torsion limits, which are only related to the resultsin [6] on asymptotically good arithmetic secret sharing and will be revisited in Section 3,in the following theorem by combining the bounds in Theorems 2.3 and 2.4 of [6]:
Theorem 2.1. Let Fq be a finite field of characteristic p. For any integer r ≥ 2, setJr := Jr(q, A(q)). Write r as r = plr′ for some l ≥ 0 and a positive integer r′ coprime top. Let c := gcd(r′, q − 1) and γ := l
√q√
q+1 .
(i) For any r one has Jr ≤ 2 logq r.(ii) If r | q and q is a square, then Jr ≤ 1√
q+1 logq r.(iii) If r - (q − 1) and, q is non-square or c > pγ, then Jr ≤ logq r.(iv) If r - q, r - (q − 1), q is a square, and c ≤ pγ, then
Jr ≤ l√
q + 1logq p + logq(cr′).
Proof. We give a complete proof by comparing the results of [6]:(i) It is well-known from a result of Weil [24] that for any function field F/Fq with
genus g one has |JF [r]| ≤ r2g, and hence assertion (i) always holds.(ii) Applying [6, Theorem 2.4(ii)] with r = pl and r′ = c = 1 we obtain the inequality
Jr ≤ l√
q + 1logq p = 1
√q + 1
logq r.
(iii) and (iv) When r - (q − 1), [6, Theorem 2.3(ii)] yields to Jr ≤ logq r. Furthermore,when q is a square, we obtain
Jr ≤ l√
q + 1logqr, (2.1)
by [6, Theorem 2.3(iii)]. Using [6, Theorem 2.4(ii)], also the following inequalityholds:
Jr ≤ l√
q + 1logq p + logq(cr′). (2.2)
Hence, by inequalities (2.1), (2.2), and substituting the value r = plr′, we get
A := l√
q + 1logq p + logq(cr′) − logq r =
−l√
q√
q + 1logq p + logq c.
Since A ≥ 0 if and only if c ≥ pγ , assertion (iv) follows.
642 S. Tutdere, O. Uzunkol
We remark that for Theorem 2.1(iv) with c < pγ , [6, Theorem 2.4] gives a better upperbound on Jr than [6, Theorem 2.3].
Secondly, we revisit the following theorem from [6] which is about the general conditionsrequired for the existence of arithmetic secret sharing which will be improved in Section4, under some conditions.
Theorem 2.2 ([6, Corollary 4.12]). Let F/Fq be an algebraic function field. Let d, k, t, n ∈Z with d ≥ 2, n > 1 and 1 ≤ t < n. Suppose Q1, . . . , Qk, P1, . . . , Pn ∈ P(k)(F ) are pairwisedistinct. If there is an integer s such that
h(F ) >
(n
t
)(Ar1 + Ar2 |JF [d]|)
where r1 := 2g−s+t+k−2 and r2 := ds−n+t, then there exists an (n, t, d, n−t)-arithmeticsecret sharing scheme for Fk
q over Fq with uniformity.
3. Torsion-limits of towersFor some cryptographic applications [6], one is interested in the families of function
fields F with positive limit A(F) and small torsion limit Jr(F). To determine the torsionlimit seems to be much harder than determining the Ihara limit. In [6, Theorem 2.6], itis proved that for all q ≥ 8 except perhaps for q = 11 or 13, A(F) > 1 + J2(F). We heregive a discussion on these limits. We begin with an application of Theorem 2.1 when q isa square:
Proposition 3.1. Suppose that q = pm is a square (with m ≥ 1 and p prime) andr = plr′ where gcd(r′, p) = 1. We set c := gcd(r′, q − 1) and γ := l
√q√
q+1 . Then there existsa recursive tower of function fields F over Fq such that one has
A(F) ≥ √q − 1 − B + Jr(F),
where
B =
1√q+1 logq r if r | q
2 logq r if r - q but r | (q − 1)logq r if r - q, r - (q − 1), c ≥ pγ
l√q+1 logq p + log(cr′) otherwise.
Proof. We know from [11] that there exists a recursive tower of function fields F over Fq
with A(F) = √q − 1. As q is a square, the proof follows easily from Theorem 2.1.
We now need the following result of Bassa et al. [2]:
Theorem 3.2 ([2, Theorem 1.2]). Let n = 2m + 1 ≥ 3 be an integer and q = pn with aprime p. There exists a recursive tower of function fields F over Fq such that
A(F) ≥ 2(pm+1 − 1)p + 1 + ϵ
, where ϵ = p − 1pm − 1
.
Next, the torsion limit of the tower given in Theorem 3.2 can be estimated by using thelower bound on the Ihara limit A(F):
Construction of arithmetic secret sharing schemes by using torsion limits 643
Proposition 3.3. Let n and q be given as in Theorem 3.2. There exists a recursive towerof function fields F over Fq with the following properties:
(i) If p is odd, then A(F) ≥ A + J2(F), where
A = 2(pm+1 − 1)p + 1 + ϵ
− 2 logq 2 with ϵ = p − 1pm − 1
. (3.1)
(ii) If p is even, then A(F) ≥ A + logq 2 + J2(F), where A is given as in Eqn. (3.1).
The proof of Proposition 3.3 is obvious; it follows from Theorems 2.1 and 3.2. Alterna-tively, the proposition follows from Theorem 3.2 and the fact that J2(F ) ≤ logq 2 if 2 | qand J2(F ) ≤ 2 logq 2 if 2 - q, which is immediate since for any function field F of genus g
one has JF [2] ≤ 22g in general, and JF [2] ≤ 2g in case char(F ) = 2.
Remark 3.4. More concretely, [6, Theorem 4.16] implies that parameters of the as-ymptotic constructions of arithmetic secret sharings improve depending on the ratioA(F)/(1 + J(F)). In particular, a larger κ corresponding to the length of the secretand the τ corresponding to the security of the construction (how many adversaries it cantolerate) can be obtained by using Proposition 3.3.
4. New conditions for the construction of arithmetic secret sharing schemesIn this part, we give an improvement of [6, Corollary 4.12] under some conditions.
Before this, we need the following: For an algebraic function field F/Fq with genus g, weset
∆ := i : 1 ≤ i ≤ g − 1 and Bi ≥ 1 with δ := |∆|, (4.1)fix an integer n ≥ 0, and further set
Un := b = (bi)i∈∆ : bi ≥ 0 and∑i∈∆
i · bi = n. (4.2)
It is well-known that the number of effective divisors of degree n of an algebraic functionfield F/Fq is given as follows:
An =∑
b∈Un
[ ∏i∈∆
(Bi + bi − 1
bi
)], (4.3)
see for instance [1]. By combining this formula for An with some results of [6] and thebound on class number given in [19] we obtain the following theorem. This improves thesufficient conditions on the existence of arithmetic secret sharing schemes with uniformity:
Theorem 4.1. Let F/Fq be a function field of genus g ≥ 2, d, k, t, n ∈ N with d ≥ 2 and1 ≤ t < n. Set
M := max(Bi + ⌊g−1
i ⌋⌊g−1
i ⌋
)| i ∈ ∆
. (4.4)
Suppose that Q1, . . . , Qk, P1, . . . , Pn ∈ P(1)(F ) are pairwise distinct rational places and
H >
(n
t
)(2g
√q + q + 1 + M δ · d2g), (4.5)
whereH := qg−1(q − 1)2
(q + 1)(g + 1)and δ is given as in (4.1). Assume further that
1 ≤ ds − n + t ≤ g − 1, (4.6)where s = 2g + t + k − 3. Then there exists an (n, t, d, n − t)-arithmetic secret sharingscheme for Fk
q over Fq with uniformity.
644 S. Tutdere, O. Uzunkol
Proof. We first note that |JF [d]| ≤ d2g by Theorem 2.1.Choose an s ∈ Z such that
r1 := 2g − s + t + k − 2 = 1, r2 := ds − n + t.
Note that r1 = 1 implies Ar1 = A1 = B1. Using the Hasse-Weil bound [22, Theorem 5.2.3]on B1 and the bound on Ar2 given in [1, Theorem 3.5] one obtains(
n
t
)(Ar1 + Ar2 |JF [d]|) ≤
(n
t
)(B1 +
∏i∈∆
(Bi + ⌊g−1
i ⌋⌊g−1
i ⌋
)|JF [d]|
)
≤(
n
t
)(B1 + M δ|JF [d]| ≤ B1 + M δd2g)
≤(
n
t
)(2g
√q + q + 1 + M δd2g)
< H ≤ h,
by assumption (4.5) and the bound H ≤ h shown in [19]. Now by Theorem 2.2, the prooffollows.
We now give an example satisfying the conditions of Theorem 4.1. Note that theparameters in the following example satisfy the conditions in [6, Proposition 4.8].
Example 4.2. Let q := 38, n = 9, d = t = 2, k = 1, and F := Fq(x) be the rationalfunction field over Fq. Consider the extension field E := F (y) of F where y2+x6+x+1 = 0.It has genus g(E) = 2 and ∆ = 1, so δ = 1. Using Magma†, one obtains B1(F ) = 6481,hence M = 6482, and
H = 29 · 37 · 52 · 412
17 · 193> 14342347.
Hence, H satisfies condition (4.5):
H ≥(
n
t
)(2g
√q + q + 1 + M δ · d2g) = 3981528.
The condition (4.6) is clearly satisfied. Thus, by Theorem 4.1, we obtain an (9, 2, 2, 7)-arithmetic secret sharing scheme over F38 with uniformity. Note that E/Fq is a hyperel-liptic function field.
Next, we give an estimation for the cardinality of Un (see (4.2)), which will be used inTheorem 4.5. For this, We know that the partitions of a number n is correspond to theset of solutions (j1, j2, ..., jn) to the Diophantine equation
1j1 + 2j2 + 3j3 + ... + njn = n.
For example, two distinct partitions of 4 in summands can be given by (1, 1, 1, 1), (1, 1, 2)corresponding to the solutions (j1, j2, j3, j4) = (4, 0, 0, 0), (2, 1, 0, 0), respectively. Thecardinalities of the summands in the partition (1, 1, 2) are j1 = 2 and j2 = 1. We now fixδ = |∆|, as in (4.1). We need to count the number of partitions of n in summands whosecardinalities are in ∆. We choose the values j′
is for the δ − 1 largest indices i in ∆. Thoseindices are all at least 2 (notice that if 1 ∈ ∆, then it is necessarily the smallest index in∆). Thus, each ji is at most n/2, i.e., it is within the range [0, n
2 ]. This means, we haven2 + 1 choices for each ji. Therefore, we have the following lemma:
Lemma 4.3. |Un| ≤(
n2 + 1
)δ−1.
†Magma Computational Algebra System: Magma Online Calculator,available under http://magma.maths.usyd.edu.au/calc/
Construction of arithmetic secret sharing schemes by using torsion limits 645
Remark 4.4. For the applications in arithmetic secret sharing schemes, it is highly desiredto construct other examples with q < n (i.e., improving Shamir’s secret sharing scheme). Inorder to find such examples we need algebraic function fields for which B1 is large, however,almost all Bi, 2 ≤ i ≤ g − 1, are zero so that Conditions (4.5) and (4.6) of Theorem 4.1simultaneously hold. However, finding such examples may not be easy. For example, thefunction field F3(x, y) ⊃ F3(x), with y3 − y − x4 + x2 = 0, over F3 has genus g = 3 andB2 = 0. Similarly, the function field F5(x, y) ⊃ F5(x), with (x4−1)y4+x3y3+3xy−x4 = 0,over F5 has g = 4 and B2 = 0 (but B3 = 40). When Be = 0 with e a prime number wouldimply that the corresponding curve attains no new points over the extension Fq of degreee. For a fixed genus g and e prime, assuming Be = 0 and comparing a Hasse-Weil lowerbound over Fqe to an upper bound over Fq yields
qe − 2g√
qe ≤ q + 2g√
q.
For instance, for g = 3 (which makes e = 2 the only relevant case to consider) this leadsto q ≤ 9.
Theorem 4.5. Let F/Fq be a function field, d, k, t, n ∈ N with d ≥ 2 and 1 ≤ t < n.Let 1 ≤ m ≤ g − 1, be such that Bm ≥ Bi for all i ∈ 1, . . . , g − 1. Suppose thatQ1, Q2, . . . , Qk, P1, P2, . . . , Pn ∈ P(1)(F ) are pairwise distinct rational places and
H >
(n
t
)(B1 +
(n
2+ 1
)δ−1(e · (Bm + n − 1)n
)nδ
d2g)
(4.7)
where H is as in Theorem 4.1, δ is as in (4.1), and e is Euler’s constant. Assume furtherthat
ds − n + t ≥ 1,
where s = 2g + t + k − 3. Then there exists an (n, t, d, n − t)-arithmetic secret sharingscheme for Fk
q over Fq with uniformity.
Proof. The proof is similar to that of Theorem 4.1. The main difference is that insteadof M, we use the assumption that Bm ≥ Bi and the bound (4.8) for binomial coefficients.Note that bi ≤ n for all i ∈ ∆. By applying induction on n the following inequality canbe proven: (
Bm + n − 1n
)=
(Bm + n − 1
Bm − 1
)(4.8)
≤(
e · (Bm + n − 1)n
)n
.
Hence, by applying Lemma 4.3 with n = r2, using (4.3) and (4.8), we obtain that
A1 + Ar2 |JF [d]| = B1 +∑
b∈Un
∏i∈∆
(Bi + bi − 1
Bi − 1
)|JF [d]|
≤ B1 +∑
b∈Un
(Bm + n − 1
Bm − 1
)δ
|JF [d]|
≤ B1 +(
n
2+ 1
)δ−1(Bm + n − 1Bm − 1
)δ
d2g
≤ B1 +(
n
2+ 1
)δ−1(e(Bm + n − 1)n
)nδ
d2g.
Thus, the desired results follows by the assumption (4.7).
646 S. Tutdere, O. Uzunkol
Acknowledgment. We thank the referee for providing constructive suggestions whichimprove the presentation of the paper. Uzunkol’s work was supported by the project(114C027) funded by EU FP7-The Marie Curie Action and TÜBİTAK (2236-CO-FUNDEDBrain Circulation Scheme).
References[1] S. Ballet, R. Rolland, and S. Tutdere, Lower bounds on the number of rational points
of Jacobians over finite fields and application to algebraic function fields in towers,Moscow Math. J. 15 (3), 1–9, 2015.
[2] A. Bassa, P. Beelen, A. Garcia, and H. Stichtenoth, Towers of function fields overnon-prime finite fields, Moscow Math. J. 15 (1), 1–29, 2015.
[3] A. Beimel, Secret-sharing schemes: A survey, IWCC 2011: LNCS 6639 SpringerVerlag: 11–46, 2011.
[4] I. Cascudo, R. Cramer, and C. Xing, The torsion-limit for algebraic function fieldsand its application to arithmetic secret sharing, CRYPTO 2011: LNCS 6841 SpringerVerlag: 685–705, 2011.
[5] I. Cascudo, R. Cramer, and C. Xing, Bounds on the threshold gap in secret sharingand its applications, IEEE Trans. Inf. Theory 59 (9), 5600–5612, 2013.
[6] I. Cascudo, R. Cramer, and C. Xing, Torsion limits and Riemann-Roch systems forfunction fields and applications, IEEE Trans. Inf. Theory 60 (7), 3871–3888, 2014.
[7] D. Chaum, C. Crépeau, and I. Damgaard, Multi-Party unconditionally secure proto-cols, Proceedings of STOC 1988: ACM Press, New York, 11–19, 1988.
[8] H. Chen and R. Cramer, Algebraic geometric secret sharing schemes and secure multi-party computations over small fields, CRYPTO 2006: LNCS 4117 Springer Verlag:516–531, 2006.
[9] R. Cramer, I. Damgaard, and U. Maurer, General secure multi-party computationfrom any linear secret sharing scheme, EUROCRYPT 2000: LNCS 1807 SpringerVerlag: 316–334, 2000.
[10] O. Farràs, C. Padró, C. Xing, and A. Yang, Natural generalizations of threshold secretsharing IEEE Trans. Inf. Theory 60 (3), 1652–1664, 2014.
[11] A. Garcia and H. Stichtenoth, A tower of Artin-Schreier extensions of function fieldsattaining the Drinfeld-Vladut bound, Invent. Math. 121, 211–222, 1995.
[12] F. Hess, H. Stichtenoth, and S. Tutdere, On invariants of towers of function fieldsover finite fields, J. Algebra Appl. 12 (4), 1250190, 2013.
[13] T. Ignatenko and F.M.J. Willems, Biometric systems: Privacy and secrecy aspects,IEEE Trans. Inf. Forensics Secur. 4 (4), 956–973, 2009.
[14] Y. Ishai, E. Kushilevitz, R. Ostrovsky, and A. Sahai, Zero-knowledge from securemulti-party computation, Proceedings of 39th STOC: San Diego, Ca., USA: 21–30,2007.
[15] Y. Ishai, M. Prabhakaran, and A. Sahai, Founding cryptography on oblivious transfer-efficiently CRYPTO 2008: LNCS 157 Springer Verlag: 572–591, 2008.
[16] M. Ito, A. Saito, and T. Nishizeki, Multiple assignment scheme for sharing secret, J.Cryptol. 6 (1), 15–20, 1993.
[17] M. Ito, A. Saito, and T. Nishizeki, Secret sharing scheme realizing any access struc-ture, Proc IEEE Globecom: 99–102, 1987.
[18] E.D. Karnin, J.W. Greene, and M.E. Hellman, On secret sharing systems, IEEETrans. Inf. Theory 29 (1), 35–41, 1983.
[19] G. Lachaud and M. Martin-Deschamps, Nombre de points des jacobiennes sur uncorps finis, Acta Arith. 56 (4), 329–340, 1990.
[20] M. Naor and A. Wool, Access control and signatures via quorum secret sharing IEEETrans. Parallel Distrib. Syst. 9(1), 909–922, 1998.
Construction of arithmetic secret sharing schemes by using torsion limits 647
[21] A. Shamir, How to share a secret, Comm. ACM. 22 (11), 612–613, 1979.[22] H. Stichtenoth, Algebraic function fields and codes, 2nd Ed. Springer-Verlag 254,
2009.[23] T. Tassa, Generalized oblivious transfer by secret sharing, Design Code Cryptogr. 58
(1), 11–21, 2011.[24] A. Weil, Varietes abeliennes et courbes algebriques, Hermann, Paris, 1948.
Hacettepe Journal ofMathematics & Statistics
Hacet. J. Math. Stat.Volume 49 (2) (2020), 648 – 675
DOI : 10.15672/hujms.588684
Research Article
Representations and T ∗-extensions ofδ-Bihom-Jordan-Lie algebras
Abdelkader Ben Hassine1,2, Liangyun Chen∗3, Juan Li31Department of Mathematics, Faculty of Science and Arts at Belqarn, University of Bisha, Kingdom of
Saudi Arabia2Faculty of Sciences, University of Sfax, BP 1171, 3000 Sfax, Tunisia
3School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China
AbstractThe purpose of this article is to study representations of δ-Bihom-Jordan-Lie algebras. Inparticular, adjoint representations, trivial representations, deformations, T ∗-extensions ofδ-Bihom-Jordan-Lie algebras are studied in detail. Derivations and central extensions ofδ-Bihom-Jordan-Lie algebras are also discussed as an application.
Mathematics Subject Classification (2010). 17A99, 17B56, 16S80
Keywords. δ-Bihom-Jordan-Lie algebras, representations, derivations, deformations,T ∗-extensions
1. IntroductionThe notion of Jordan-Lie algebras was introduced in [7], which is closely related to both
Lie and Jordan superalgebras. Engel’s theorem of Jordan-Lie algebras was proved, andsome properties of Cartan subalgebras of Jordan-Lie algebras were given in [8].
Recently, the definition of δ-hom-Jordan-Lie algebras were introduced in [10], and theirrepresentations and T ∗-extensions were studied in detail.
A Bihom-algebra is an algebra in such a way that the identities defining the structureare twisted by two homomorphisms α, β. This class of algebras was introduced from acategorical approach in [4] as an extension of the class of Hom-algebras. The origin ofHom-structures can be found in the physics literature around 1900, appearing in the studyof quasi deformations of Lie algebras of vector fields, in particular q-deformations of Wittand Virasoro algebras in [5]. Since then, many authors have been interested in the studyof Hom-algebras, mainly motivated by their applications in mathematical physics (seefor instance the recent references [1, 6]). The fundamental for getting the basic notions,motivations, and results on Bihom-algebras is the reference [4].
More applications of the Bihom-Lie algebras, Bihom-algebras, Bihom-Lie superalgebrasand Bihom-Lie admissible superalgebras can be found in [3, 9].
The notion of derivations, representations, and T ∗-extensions of δ-Bihom-Jordan Liealgebras are not so well developed.∗Corresponding Author.Email addresses: [email protected] (A. Ben Hassine), [email protected] (L. Chen),
[email protected] (J. Li)Received: 11.03.2018; Accepted: 12.02.2019
Representations and T ∗-extensions of δ-Bihom-Jordan-Lie algebras 649
The paper is organized as follows. In Section 2 we give the definition of δ-Bihom-Jordan-Lie algebras, and show that the direct sum of two δ-Bihom-Jordan-Lie algebras isstill a δ-Bihom-Jordan-Lie algebra. A linear map between δ-Bihom-Jordan-Lie algebras isa morphism if and only if its graph is a Bihom subalgebra. In Section 3 we study deriva-tions of multiplicative δ-Bihom-Jordan-Lie algebras. For any nonnegative integers k and l,we define αkβl-derivations of multiplicative δ-Bihom-Jordan-Lie algebras. Considering thedirect sum of the space of αkβl-derivations, we prove that it is a Lie algebra. In particular,any α0β1-derivation gives rise to a derivation extension of the multiplicative δ-hom-Jordan-Lie algebra (L, [·, ·]L, α, β) (Theorem 3.3). In Section 4 we give the definition of representa-tions of multiplicative δ-Bihom-Jordan-Lie algebras. We can obtain the semidirect productmultiplicative δ-Bihom-Jordan-Lie algebra (L⊕M, [·, ·]ρ, α+αM , β+βM ) associated to anyrepresentation ρ on M of the multiplicative δ-Bihom-Jordan-Lie algebra (L, [·, ·]L, α, β).In Section 5 we study trivial representations of multiplicative δ-Bihom-Jordan-Lie alge-bras. We show that central extensions of a multiplicative δ-Bihom-Jordan-Lie algebra arecontrolled by the second cohomology with coefficients in the trivial representation. InSection 6 we study the adjoint representation of a regular δ-Bihom-Jordan-Lie algebra(L, [·, ·]L, α, β). For any integers s, t, we define the αsβt-derivations. We show that a 1-cocycle associated to the αsβt-derivation is exactly an αs+2βt−1-derivation of the regularδ-Bihom-Jordan-Lie algebra (L, [·, ·]L, α, β) in some conditions. We also give the definitionof Bihom-Nijienhuis operators of regular δ-Bihom-Jordan-Lie algebras. We show that thedeformation generated by a Bihom-Nijienhuis operator is trivial. In Section 7 we studyT ∗-extensions of δ-Bihom-Jordan-Lie algebras, show that T ∗-extensions preserve manyproperties such as nilpotency, solvability and decomposition in some sense.
2. Definitions and proprieties of δ-Bihom-Jordan-Lie algebrasDefinition 2.1 ([7]). A δ-Jordan Lie algebra is a couple (L, [·, ·]L) consisting of a vectorspace L and a bilinear map (bracket) [·, ·]L : L× L → L satisfying
[x, y] = −δ[y, x], δ = ±1,[x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0, ∀x, y, z ∈ L.
Definition 2.2 ([10]). A δ-hom-Jordan Lie algebra is a triple (L, [·, ·]L, α) consisting of avector space L, a bilinear map (bracket) [·, ·]L : L ⊗ L → L and a linear map α : L → Lsatisfying
[x, y] = −δ[y, x], δ = ±1,[α(x), [y, z]] + [α(y), [z, x]] + [α(z), [x, y]] = 0, ∀x, y, z ∈ L.
Especially, for δ = 1 one has a hom-Lie algebra and for δ = −1 a hom-Jordan Lie algebra.
Definition 2.3 ([3]). A Bihom-Lie algebra is a 4-tuple (L, [·, ·]L, α, β) consisting of vectorspace L, a bilinear map [·, ·] : L×L → L and two homomorphisms α, β : L → L such thatfor all elements x, y, z ∈ L we have
α β = β α,[β(x), α(y)] = −[β(y), α(x)],[β2(x), [β(y), α(z)]] + [β2(y), [β(z), α(x)]] + [β2(z), [β(x), α(y)]] = 0
(Bihom-Jacobi equation).
Definition 2.4. A δ-Bihom-Jordan Lie algebra is a 4-tuple (L, [·, ·]L, α, β) consisting ofa vector space L, a bilinear map (bracket) [·, ·]L : L ⊗ L → L and two linear maps
650 A. Ben Hassine, L. Chen, J. Li
α, β : L → L satisfying
α β = β α, (2.1)[β(x), α(y)] = −δ[β(y), α(x)], δ = ±1, (2.2)[β2(x), [β(y), α(z)]]+[β2(y), [β(z), α(x)]]+[β2(z), [β(x), α(y)]]=0,∀x, y, z ∈ L.(2.3)
Especially, for δ = 1 one has a Bihom-Lie algebra and for δ = −1 a Bihom-Jordan Liealgebra.
Definition 2.5. 1) A δ-Bihom-Jordan Lie algebra (L, [·, ·]L, α, β) is multiplicative ifα and β are algebra morphisms, i.e., for any x, y ∈ L, we have
α([x, y]L) = [α(x), α(y)]L and β([x, y]L) = [β(x), β(y)]L.
2) A δ-Bihom-Jordan Lie algebra (L, [·, ·]L, α, β) is regular if α and β are algebraautomorphisms.
3) A subvector space η ∈ L is a Bihom subalgebra of (L, [·, ·]L, α, β) if α(η) ∈ η,β(η) ∈ η and
[x, y]L ∈ η, ∀x, y ∈ η.
4) A subvector space η ∈ L is a Bihom ideal of (L, [·, ·]L, α, β) if α(η) ∈ η, β(η) ∈ ηand
[x, y]L ∈ η, ∀x ∈ η, y ∈ L.
Definition 2.6. A δ-Bihom associative algebra is a triple (L,α, β) consisting of a vectorspace L, a bilinear map on L, and two linear commuting maps α, β : L → L satisfying
α(x)(yz) = δ(xy)β(z), ∀x, y, z ∈ L. (2.4)
Proposition 2.7. Let (L,α, β) be a multiplicative δ-Bihom associative algebra. Define abilinear map (bracket) [·, ·]L : L× L → L satisfying
[x, y]L = xy − δα−1(β(y))β−1(α(x)),∀x, y ∈ L. (2.5)Then (L, [·, ·]L, α, β) is a δ-Bihom-Jordan-Lie algebra.
Proof. First we check that the bracket product [·, ·] is compatible with the structure mapsα and β. For any x, y ∈ L, we have
[α(x), α(y)] = α(x)α(y) − δ(α−1β(α(y)))(αβ−1(α(x)))= α(x)α(y) − δβ(y)(α2β−1(x))= α([x, y]).
Similarly, one can prove that β([x, y]) = [β(x), β(y)].And
[β(x), α(y)] = β(x)α(y) − δ(α−1β(α(y)))(αβ−1(β(x)))= β(x)α(y) − δβ(y)(α(x))= −δ[β(y), α(x)].
Now we prove the Bihom-Jacobi condition. For any elements x, y ∈ L, we have
[β2(x), [β(y), α(z)]] =[β2(x), β(y)α(z) − δα−1β(α(z))αβ−1(β(y))]=[β2(x), β(y)α(z)] − δ[β2(x), β(z)α(y)]
=(β2(x)(β(y)α(z)) − δ(α−1(β2(y))β(z))α(β(x))
)− δ
(β2(x)(β(z)α(y)) − δ(α−1(β2(z))β(y))α(β(x))
).
Representations and T ∗-extensions of δ-Bihom-Jordan-Lie algebras 651
Similarly, we have
[β2(y), [β(z), α(x)]] =(β2(y)(β(z)α(x)) − δ(α−1(β2(z))β(x))α(β(y))
)− δ
(β2(y)(β(x)α(z)) − δ(α−1(β2(x))β(z))α(β(y))
).
[β2(z), [β(x), α(y)]] =(β2(z)(β(x)α(y)) − δ(α−1(β2(x))β(y))α(β(z))
)− δ
(β2(z)(β(y)α(x)) − δ(α−1(β2(y))β(x))α(β(z))
).
Note that
β2(x)(β(y)α(z)) = δ(α−1(β2(x))β(y))α(β(z)),β2(y)(β(x)α(z)) = δ(α−1(β2(y))β(x))α(β(z)),β2(x)(β(z)α(y)) = δ(α−1(β2(x))β(z))α(β(y)),β2(y)(β(z)α(x)) = δ(α−1(β2(y))β(z))α(β(x)),β2(z)(β(x)α(y)) = δ(α−1(β2(z))β(x))α(β(y)),β2(z)(β(y)α(x)) = δ(α−1(β2(z))β(y))α(β(x)).
Then we obtain [β2(x), [β(y), α(z)]] + [β2(y), [β(z), α(x)]] + [β2(z), [β(x), α(y)]] = 0.
Proposition 2.8. Given two δ-Bihom-Jordan-Lie algebras (L, [·, ·]L, α1, β1) and (L′, [·, ·]L′ ,α2, β2), there is a δ-Bihom-Jordan-Lie algebra (L ⊕ L′, [·, ·]L⊕L′ , α1 + α2, β1 + β2), wherethe bilinear map [·, ·]L⊕L′ : L⊕ L′ × L⊕ L′ → L⊕ L′ is given by
[u1 + v1, u2 + v2]L⊕L′ = [u1, v1]L + [u2, v2]L′ , ∀u1, u2 ∈ L, v1, v2 ∈ L′,
and the two linear maps α1 + α2, β1 + β2 : L⊕ L′ → L⊕ L′ defined by
(α1 + α2)(u1 + v1) = α1(u1) + α2(v1),(β1 + β2)(u1 + v1) = β1(u1) + β2(v1).
Proof. For any u1, u2, u3 ∈ L and v1, v2, v3 ∈ L′ we have:
[(β1 + β2)(u1 + v1), (α1 + α2)(u2 + v2)]L⊕L′
= [β1(u1), α1(u2)]L + [β2(v1), α2(v2)]L′ = −δ[β1(u2), α1(u1)]L − δ[β2(v2), α2(v1)]L′
= −δ([β1(u2), α1(u1)]L + [β2(v2), α2(v1)]L′)= −δ[(β1 + β2)(u2 + v2), (α1 + α2)(u1 + v1)]L⊕L′ .
(α1 + α2) (β1 + β2)(u1 + v1)= (α1 + α2)(β1(u1) + β2(v1)) = α1 β1(u1) + α2 β2(v1)= β1 α1(u1) + β2 α2(v1)= (β1 + β2) (α1 + α2)(u1 + v1).
Then, we have (α1 + α2) (β1 + β2) = (β1 + β2) (α1 + α2).By a direct computation, we have
(u1+v1),(u2+v2),(u3+v3) [(β1 + β2)2(u1 + v1), [(β1 + β2)(u2 + v2), (α1 + α2)(u3 + v3)]L⊕L′ ]L⊕L′
=(u1+v1),(u2+v2),(u3+v3) [β21(u1) + β2
2(v1), [β1(u2), α1(u3)]L + [β2(v2), α2(v3)]L′ ]L⊕L′
=u1,u2,u3 [β21(u1), [β1(u2), α1(u3)]L]L+ v1,v2,v3 [β2
1(v1), [β1(v2), α1(v3)]L′ ]L′
= 0,
where x,y,z denotes summation over the cyclic permutation on x, y, z.
652 A. Ben Hassine, L. Chen, J. Li
Definition 2.9. Let (L, [·, ·]L, α1, β1) and (L′, [·, ·]L′ , α2, β2) be two δ-Bihom-Jordan-Liealgebras. A linear map ϕ : L → L′ is said to be a morphism of δ-Bihom-Jordan-Liealgebras if
ϕ[u, v]L = [ϕ(u), ϕ(v)]L′ , ∀u, v ∈ L, (2.6)ϕ α1 = β1 ϕ, (2.7)ϕ α2 = β2 ϕ. (2.8)
Denote by Gϕ ∈ L⊕ L′ is the graph of a linear map ϕ : L → L′.Proposition 2.10. A map ϕ : (L, [·, ·]L, α1, β1) → (L′, [·, ·]L′ , α2, β2) is a morphism ofδ-Bihom-Jordan-Lie algebras if and only if the graph Gϕ ∈ L ⊕ L′ is a Bihom subalgebraof (L⊕ L′, [·, ·]L⊕L′ , α1 + α2, β1 + β2).Proof. Let ϕ : (L, [·, ·]L, α1, β1) → (L′, [·, ·]L′ , α2, β2) be a morphism of δ-Bihom-Jordan-Lie algebras, then for any u, v ∈ L, we have
[u+ ϕ(u), v + ϕ(v)]L⊕L′ = [u, v]L + [ϕ(u), ϕ(v)]L′ = [u, v]L + ϕ[u, v]L.Then the graph Gϕ is closed under the bracket operation [·, ·]L⊕L′ . So, we obtain
(α1 + α2)(u+ ϕ(u)) = α1(u) + α2 ϕ(u) = α1(u) + ϕ α2(u),and
(β1 + β2)(u+ ϕ(u)) = β1(u) + β2 ϕ(u) = β1(u) + ϕ β2(u),which implies that (α1 + α2)(Gϕ) ⊂ Gϕ and (β1 + β2)(Gϕ) ⊂ Gϕ. Then Gϕ is a Bihomsubalgebra of (L⊕ L′, [·, ·]L⊕L′ , α1 + α2, β1 + β2).
Now, suppose that the graph Gϕ ⊂ L⊕L′ is a Bihom subalgebra of (L⊕L′, [·, ·]L⊕L′ , α1+α2, β1 + β2), then we have
[u+ ϕ(u), v + ϕ(v)]L⊕L′ = [u, v]L + [ϕ(u), ϕ(v)]L′ ∈ Gϕ,
which implies that[ϕ(u), ϕ(v)]L′ = ϕ[u, v]L.
Furthermore, (α1 + α2)(Gϕ) ⊂ Gϕ and (β1 + β2)(Gϕ) ⊂ Gϕ implies(α1+α2)(u+ϕ(u)) = α1(u)+α2ϕ(u) ∈ Gϕ and (β1+β2)(u+ϕ(u)) = β1(u)+β2ϕ(u) ∈ Gϕ.
Which is equivalent to the condition α1 ϕ(u) = ϕ β1(u), and α2 ϕ(u) = ϕ β2(u) i.e.α1 ϕ = ϕ β1
and α2 ϕ = ϕ β2.
Therefore, ϕ is a morphism of δ-Bihom-Jordan-Lie algebras. Example 2.11. Let (L, [·, ·]) be a δ-Jordan-Lie algebra and α, β : L → L two commutinglinear maps such that α([x, y]) = [α(x), α(y)] and β([x, y]) = [β(x), β(y)], for all x, y ∈L. Then (L, [·, ·]L, α, β), where [x, y]L = [α(x), β(y)], is a δ-Bihom-Jordan-Lie algebra.Moreover, suppose that (L′, [·, ·]) is another δ-Jordan-Lie algebra and α′, β′ : L′ → L′ betwo algebra endomorphisms. If f : L → L′ is a δ-Jordan-Lie algebra homomorphism thatsatisfies f α = α′ f and f β = β′ f , then f : (L, [·, ·]L, α, β) → (L′, [·, ·]L′ , α′, β′) isalso a homomorphism of δ-Bihom-Jordan-Lie algebras.Proof. It is easy to show that (L, [·, ·]L, α, β) satisfies [β(x), α(y)]L = [αβ(x), βα(y)]) =αβ([x, y]) = αβ(−δ[y, x]) = −δ[αβ(y), αβ(x)]) = −δ[β(y), α(x)]L, and[β2(x), [β(y), α(z)]L]L + [β2(y), [β(z), α(x)]L]L + [β2(z), [β(x), α(y)]L]L= [β2(x), [αβ(y), βα(z)]]L + [β2(y), [αβ(z), βα(x)]]L + [β2(z), [αβ(x), βα(y)]L]L= [αβ2(x), β[αβ(y), βα(z)]]L + [αβ2(y), β[αβ(z), βα(x)]]L + [αβ2(z), β[αβ(x), βα(y)]L]L= αβ2([x, [y, z]] + [y, [z, x]] + [z, [x, y]])= 0.
Representations and T ∗-extensions of δ-Bihom-Jordan-Lie algebras 653
Then (L, [·, ·]L, α, β) is a δ-Bihom-Jordan-Lie algebra.The second assertion follows fromf([x, y]L) = f([α(x), β(y)]) = [f(α(x)), f(β(y))]) = [α(f(x)), β(f(y))] = [f(x), f(y)]L′ .
Then f : (L, [·, ·]L, α, β) → (L′, [·, ·]L′ , α′, β′, ) is also a homomorphism of δ-Bihom-Jordan-Lie algebras. Example 2.12. A three dimensional linear space L has a basis
e1 =
0 0 10 0 00 0 0
, e2 =
0 1 00 0 00 0 0
, e3 =
0 0 00 0 10 0 0
.
Then (L, [·, ·]) is a δ-Jordan-Lie algebra with respect to the product: 0 a b0 0 c0 0 0
,
0 a′ b′
0 0 c′
0 0 0
= δ
0 0 ac′
0 0 00 0 0
−
0 0 a′c0 0 00 0 0
If we define two algebra endomorphisms α and β by
α(e1) = δe1, α(e2) = e3, α(e3) = e2,
andβ(e1) = δe1, β(e2) = e3, β(e3) = e2.
Then (L,α⊗ β([·, ·]L) = [α(.), β(.)], α, β) is a δ-Bihom-Jordan-Lie algebra.
3. Derivations of δ-Bihom-Jordan-Lie algebrasIn this section, we will study derivations of δ-Bihom-Jordan-Lie algebras. Let (L, [·, ·]L, α,
β) be a multiplicative δ-Bihom-Jordan-Lie algebra. For any nonnegative integers k, l, de-note by αk the k-times composition of α and βl the l-times composition of β, i.e.
αk = α · · · α︸ ︷︷ ︸(k−times)
, βl = β · · · β︸ ︷︷ ︸(l−times)
.
Since the maps α, β commute, we denote byαkβl = α · · · α︸ ︷︷ ︸
(k−times)
β · · · β︸ ︷︷ ︸(l−times)
.
In particular, α0β0 = Id, α1β1 = αβ, α−kβ−l is the inverse of αkβl. If (L, [·, ·]L, α, β) isa regular δ-Bihom-Jordan-Lie algebra, we denote by α−k the k-times composition of α−1,the inverse of α.
Definition 3.1. For any nonnegative integers k, l, a linear map D : L → L is called anαkβl-derivation of the multiplicative δ-Bihom-Jordan-Lie algebra (L, [·, ·]L, α, β), if
[D,α] = 0, i.e. D α = α D, (3.1)
[D,β] = 0, i.e. D β = β D, (3.2)and
D[u, v]L = δk([D(u), αkβl(v)]L + [αkβl(u), D(v)]L),∀u, v ∈ L. (3.3)For a regular δ-Bihom-Jordan-Lie algebra, α−kβ−l-derivations can be defined similarly.
Note first that if α and β are bijective, the skew-symmetry condition (2.3) implies
[u, v] = −δ[α−1β(v), αβ−1(u)]L, ∀u, v ∈ L. (3.4)Denote by Derαsβl(L) is the set of αsβl-derivations of the multiplicative δ-Bihom-
Jordan-Lie algebra (L, [·, ·]L, α, β). For any u ∈ L satisfying α(u) = u, and β(u) = u,
654 A. Ben Hassine, L. Chen, J. Li
define Dk,l(u) : L → L by
Dk,l(u)(v) = −δ[αkβl(v), u]L, δk = 1, ∀v ∈ L.
By Equation (3.4),Dk,l(u)(v) = −δ[αkβl(v), u]L
= δ[α−1β(u), αβ−1(αkβl(v))]L= δ[u, αk+1βl−1(v)]L.
Then Dk,l(u) is an αk+1βl-derivation. We call an inner αk+1βl-derivation. In fact, wehave
Dk,l(u)(α(v)) = −δ[αk+1βl(v), u]L = −α(δ[αkβl(v), u]L) = α Dk,l(u)(v).
Dk,l(u)(β(v)) = −δ[αkβl+1(v), u]L = −β(δ[αkβl(v), u]L) = β Dk,l(u)(v).On the other hand, we have
Dk,l(u)([v, w]L)
= −δ[αkβl([v, w]L), u]L= −δ[[βαkβl−1(v), ααkβl−1(w)]L, β2(u)]L= δ[β2(u), [βαkβl−1(v), ααkβl−1(w)]L]
= −δ([αk+1βl(v), [αkβl(w), α(u)]L]L + [αkβl+1(w), [β(u), αk+2βl−2(v)]L]L)
= −δ[αk+1βl(v), [αkβl(w), α(u)]L]L − δ[αkβl+1(w), [β(u), αk+2βl−2(v)]L]L= −δk+1[αk+1βl(v), δ[αkβl(w), u]L]L − δk+1[δ[u, αk+1βl−1(v)]L, αk+1βl(w)]L= δk+1[αk+1βl(v), Dk,l(u)(w)]L + [Dk,l(u)(v), αk+1βl(v)]L.
Therefore, Dk,l(u) is an αk+1βl-derivation. Denote by Innαkβl(L) the set of inner αkβl-derivations, i.e.
Innαk,βl(L) = −δ[αk−1βl(·), u]L|u ∈ L,α(u) = u, β(u) = u, δk = 1. (3.5)
For any D ∈ Derαkβl(L) and D′ ∈ Derαsβt(L), define their commutator [D,D′ ] as usual:
[D,D′ ] = D D′ −D′ D. (3.6)
Lemma 3.2. For any D ∈ Derαk,βl(L) and D′ ∈ Derαs,βt(L), we have
[D,D′ ] ∈ Derαk+s,βl+t(L).
Proof. For any u, v ∈ L, we have[D,D′ ]([u, v]L) = D D′([u, v]L) −D
′ D([u, v]L)= δsD([D′(u), αsβt(v)]L + [αsβt(u), D′(v)]L)
−δkD′([D(u), αkβl(v)]L + [αkβl(u), D(v)]L)
= δsD([D′(u), αsβt(v)]L) + δsD([αsβt(u), D′(v)]L)−δkD
′([D(u), αkβl(v)]L) − δkD′([αkβl(u), D(v)]L)
= δk+s([D D′(u), αk+sβl+t(v)]L + [αkβl D′(u), D αsβt(v)]L+[D αsβt(u), αkβl D′(v)]L + [αk+s D(u), D D′(v)]L−[D′ D(u), αk+s(v)]L − [αs D(u), D′ αk(v)]L−[D′ αk(u), αs D(v)]L − [αk+sβl+t(u), D′ D(v)]L).
Representations and T ∗-extensions of δ-Bihom-Jordan-Lie algebras 655
Since any two of maps D,D′ , α, β commute, we have
D αs = αs D , D′ αk = αk D′
,
D βt = βt D , D′ βl = βl D′
.
Therefore, we have
[D,D′ ]([u, v]L) = δk+s([D D′(u) −D′ D(u), αk+sβl+t(v)]L
+[αk+sβl+t(u), D D′(u) −D′ D(v)]L)
= δk+s([[D,D′ ](u), αk+sβl+t(v)]L + [αk+sβl+t(u), [D,D′ ](v), ]L).Furthermore, it is straightforward to see that
[D,D′ ] α = D D′ α−D D′ α= α D D′ − α D D′
= α [D,D′ ],and
[D,D′ ] β = D D′ β −D D′ β= β D D′ − β D D′
= β [D,D′ ].
Therefore, [D,D′ ] ∈ Derαk+sβl+t(L).
For any integer k, l, denote by Der(L) = ⊕k≥0,l≥0Derαkβl(L). Obviously, Der(L) is aLie algebra, in which the Lie bracket is given by equation (3.6).
In the end, we consider the derivation extension of the regular δ-Bihom-Jordan-Liealgebra (L, [·, ·]L, α, β) and give an application of the α0β1-derivation Derα0β1(L).
For any linear map D,α, β : L → L, where α and β are inverse, consider the vectorspace L⊕RD. Define a skew-symmetric bilinear bracket operation [·, ·]D on L⊕RD by
[u, v]D = [u, v]L, [D,u]D = −δ[α−1β(u), αβ−1D]D = D(u),∀u, v ∈ L.
Define two linear maps by αD, βD : L⊕RD → L⊕RD byαD(u,D) = (α(u), D), and βD(u,D) = (β(u), D).
And the linear maps α, β involved in the definition of the bracket operation [·, ·]D arerequired to be multiplicative, that is
α [D,u]D = [α D,α(u)]D, β [D,u]D = [β D,β(u)]D.Then, we have
[u,D]D = −δ[α−1βD,αβ−1(u)]D= −δα−1β[D,α2β−2(u)]D= −δα−1βD(α2β−2(u))= −δαβ−1D(u).
Theorem 3.3. With the above notations, (L ⊕ RD, [·, ·]D, αD, βD) is a multiplicative δ-Bihom-Jordan-Lie algebra if and only if D is an α0β1-derivation of the multiplicativeδ-Bihom-Jordan-Lie algebra (L, [·, ·]L, α, β).
Proof. For any u, v ∈ L,m, n ∈ R, we haveαD βD(u,mD) = αD(β(u),mD) = (α β(u),mD),
andβD αD(u,mD) = βD(α(u),mD) = (β α(u),mD).
656 A. Ben Hassine, L. Chen, J. Li
Hence, we have
αD βD = βD αD ⇐⇒ α β = β α.
On the other hand,
αD[(u,mD), (v, nD)]D = αD([u, v]L + [u, nD]D + [mD, v]D)= αD([u, v]L − δnD αβ−1(u) +mD(v))= α([u, v]L) − δnα D αβ−1(u) +mα D(v)),
[αD(u,mD), αD(v, nD)]D = [(α(u),mD), (α(v), nD)]D= [α(u), α(v)]L + [α(u), nD]D + [mD,α(v)]D= [α(u), α(v)]L − δnD αβ−1(α(u)) +mD(α(v)).
Since α([u, v]L) = [α(u), α(v)]L,
αD[(u,mD), (v, nD)]D = [αD(u,mD), αD(v, nD)]D
if and only if
D α = α D, D β = β D.
Similarly
βD[(u,mD), (v, nD)]D = [βD(u,mD), βD(v, nD)]D
if and only if
D α = α D, D β = β D.
Next, we have
[βD(v, nD), αD(u,mD)]D = [(β(v), nD), (α(u),mD)]D= [β(v), α(u)]L + [β(v),mD]D + [nD,α(u)]D= [β(v), α(u)]L − δmαβ−1 D (β(v)) + nD(α(u))= −δ([β(u), α(v)]L +mαβ−1 D (β(v)) − δnD(α(u))),
[βD(u,mD), αD(v, nD)]D = [(β(u),mD), (α(v), nD)]D= [β(u), α(v)]L + [β(u), nD]D + [mD,α(v)]D= [β(u), α(v)]L − δnαβ−1 D (β(u)) +mD(α(v)),
thus
[βD(v, nD), αD(u,mD)]D = −δ[βD(u,mD), αD(v, nD)]D
if and only if
D α = α D, D β = β D.
Representations and T ∗-extensions of δ-Bihom-Jordan-Lie algebras 657
On the other hand, we have[β2
D(u,mD), [βD(v, nD), α(w, lD)]D]D + [β2D(v, nD), [βD(w, lD), αD(u,mD)]D]D
+ [β2D(w, lD), [βD(u,mD), αD(v, nD)]D]D
= [(β2(u),mD), [(β(v), nD), (α(w), lD)]D]D +[(β2(v), nD), [(β(w), lD), (α(u),mD)]D]D+ [(β2(w), lD), [(β(u),mD), (α(v), nD)]D]D
= [(β2(u),mD), ([β(v), α(w)] − δlα D(v) + nD α(w))]D+ [(β2(v), nD), ([β(w), α(u)] − δnα D(w) +mD α(u))]D+ [(β2(w), lD), ([β(u), α(v)] − δmα D(u) +mD α(v))]D
= [β2(u), [β(v), α(w)]] − δ[β2(u), lα D(v)] + [β2(u), nD α(w)]+ [mD, [β(v), α(w)]] − δ[mD, lα D(v)] + [mD,nD α(w)]+ [β2(v), [β(w), α(u)]] − δ[β2(v),mα D(w)] + [β2(v), lD α(u)]+ [nD, [β(w), α(u)]] − δ[nD,mα D(w)] + [nD, lD α(u)]+ [β2(w), [β(u), α(v)]] − δ[β2(w), nα D(u)] + [β2(w),mD α(v)]+ [lD, [β(u), α(w)]] − δ[lD, nα D(u)] + [lD,mD α(v)]
= [β2(u), [β(v), α(w)]] − δ[mD, lα D(v)] + [β2(u), nD α(w)]+ [mD, [β(v), α(w)]] − δmlα D2(v) +mnD2 α(w)+ [β2(v), [β(w), α(u)]] − δ[β2(v),mα D(w)] + [β2(v), lD α(u)]+ [nD, [β(w), α(v)]] − δmnα D2(w) + nlD2 α(w)]+ [β2(w), [β(u), α(v)]] − δ[β2(w), nα D(u)] + [β2(w),mD α(v)]+ [lD, [β(u), α(w)]] − δlnα D2(u) + lmD2 α(v)].
IfD is an α0β1-derivation of the multiplicative δ-Bihom-Jordan-Lie algebra (L, [·, ·]L, α, β),then
[mD, [β(v), α(w)]]D = mD[β(v), α(w)]= δ[mD β(v), α0β1(α(w))] + [α0β2(v),mD α(w)]= −δ[α0β2(w),mD α(v)] + [α0β2(v),mD α(w)]= −δ[β2(w),mD α(v)] + [β2(v),mD α(w)].
Similarly[nD, [β(w), α(u)]]D = −δ[β2(u), nD α(w)] + [β2(w), nα D(u)].
And[lD, [β(u), α(v)]]D = −δ[β2(v), lD α(u)] + [β2(w), lα D(w)].
Therefore, the δ-Bihom-Jacobi identity is satisfied if and only if D is an α0β1-derivationof (L, [·, ·]L, α, β). Thus (L ⊕ RD, [·, ·]D, αD, βD) is a multiplicative δ-Bihom-Jordan-Liealgebra if and only if D is an α0β1-derivation of (L, [·, ·]L, α, β).
4. Representations of δ-Bihom-Jordan-Lie algebrasIn this section we study representations of δ-Bihom-Jordan-Lie algebras and give the
corresponding coboundary operators. We can also construct the semidirect product of δ-Bihom-Jordan-Lie algebras. Let A ∈ End(V ) be an arbitrary linear transformation fromV to V .
658 A. Ben Hassine, L. Chen, J. Li
Definition 4.1. Let (L, [·, ·]L, α, β) be a multiplicative δ-Bihom-Jordan-Lie algebra. Arepresentation of L is a 4-tuple (M,ρ, αM , βM ), where M is a linear space, αM , βM : M →M are two commuting linear maps and ρ : L → End(M) is a linear map such that, for allu, v ∈ L, we have
ρ(α(u)) αM = αM ρ(u), (4.1)ρ(β(u)) βM = βM ρ(u), (4.2)ρ([β(u), v]L) βM = ρ(αβ(u)) ρ(v) − δρ(β(v)) ρ(α(u)). (4.3)
Let (L, [·, ·]L, α, β) be a regular δ-Bihom-Jordan-Lie algebra. The set of k-cochains on Lwith values in M , which we denote by Ck(L;M), is the set of k-linear maps from L×· · ·×L(k-times) to M :
Ck(L;M) , f : L× · · · × L(k − times) → M is a linear map.A k-Bihom-cochain on L with values in M is defined to be a k-cochain f ∈ Ck(L;M)
such that it is compatible with α, β and αM , βM in the sense that αM f = f α,βM f = f β, i.e.
αM (f(u1, . . . , uk)) = f(α(u1), . . . , α(uk)),βM (f(u1, . . . , uk)) = f(β(u1), . . . , β(uk)).
Denote by Ck(α,αM )(β,βM )
(L,M) the set of k-Bihom-cochains:
Ck(α,αM )(β,βM )
(L,M) , f ∈ Ck(L,M)|αM f = f α, βM f = f β.
Define the linear map dkρ : Ck
(α,αM )(β,βM )
(L,M) → Ck+1(L,M)(k = 1, 2) as follows: we set
d1ρf(u1, u2) = ρ(α(u1))f(u2) − δρα(u2))f(u1) − δf([α−1β(u1), u2]L),
d2ρf(u1, u2, u3) = ρ(αβ(u1))f(u2, u3) − δρ(αβ(u2))f(u1, u3) + ρ(αβ(u3))f(u1, u2)
−f([α−1β(u1), u2]L, β(u3)) + δf([α−1β(u1), u3]L, β(u2))−f([α−1β(u2), u3]L, β(u1)).
Lemma 4.2. With the above notations, for any f ∈ Ck(α,αM )(β,βM )
(L,M), we have
(dkρ f) α = αM dk
ρf,
(dkρ f) β = βM dk
ρf.
Thus we obtain a well-defined mapdk
ρ : Ck(α,αM )(β,βM )
(L,M) → Ck+1(α,αM )(β,βM )
(L,M)
with k = 1, 2.
Proposition 4.3. With the above notations, we have d2ρ d1
ρ = 0.
Proof. By straightforward computations, we haved2
ρ d1ρf(u1, u2, u3)
= ρ(αβ(u1))d1ρf(u2, u3) − δρ(αβ(u2))d1
ρf(u1, u3) + ρ(αβ(u3))d1ρf(u1, u2)
− d1ρf([α−1β(u1), u2]L, β(u3)) + δd1
ρf([α−1β(u1), u3]L, β(u2))− d1
ρf([α−1β(u2), u3]L, β(u1))= ρ(αβ(u1))(ρ(α(u2))f(u3) − δρα(u3))f(u2) − δf([α−1β(u2), u3]L))
Representations and T ∗-extensions of δ-Bihom-Jordan-Lie algebras 659
− δρ(αβ(u2))(ρ(α(u1))f(u3) − δρα(u3))f(u1) − δf([α−1β(u1), u3]L))+ ρ(αβ(u3))(ρ(α(u1))f(u2) − δρα(u2)f(u1) − δf([α−1β(u1), u2]L))− ρ(α([α−1β(u1), u2]L))f(β(u3)) + δρα(β(u3))f(u1)+ δf([α−1β([α−1β(u1), u2]L), β(u3)]L)− δρ(α([α−1β(u1), u3]L))f(β(u2)) + ρα(β(u2))f(u1)+ f([α−1β([α−1β(u1), u3]L), β(u2)]L)+ δρ(α([α−1β(u2), u3]L))f(β(u1)) − ρα(β(u1))f(u2)− f([α−1β([α−1β(u2), u3]L), β(u1)]L)
= ρ(αβ(u1))ρ(α(u2))f(u3) − δρ(αβ(u1))ρα(u3))f(u2) − δρ(αβ(u1))f([α−1β(u2), u3]L)− δρ(αβ(u2))ρ(α(u1))f(u3) + ρ(αβ(u2))ρα(u3))f(u1) + ρ(αβ(u2))f([α−1β(u1), u3]L)+ ρ(αβ(u3))ρ(α(u1))f(u2)−δρ(αβ(u3))ρα(u2)f(u1)−δρ(αβ(u3))f([α−1β(u1), u2]L))− ρ([β(u1), α(u2)]L)f(β(u3)) + δρ(α(β(u3)))f(u1)+ δf([([α−2β2(u1), α−1β(u2)]L), β(u3)]L)− δρ([β(u1), α(u3)]L))f(β(u2)) + ρ(α(β(u2)))f(u1)+ f([([α−2β2(u1), α−1β(u3)]L), β(u2)]L)+ δρ([β(u2), α(u3)]L)f(β(u1)) − ρ(α(β(u1)))f(u2)− f([([α−2β2(u2), α−1β(u3)]L), β(u1)]L)
= 0.
Then d2ρ d1
ρf(u1, u2, u3) = 0.
Associated to the representation ρ, we obtain the complex (Ck(α,αM )(β,βM )
(L,M), dρ). Denote
the set of closed k-Bihom-cochains by Zkα,β(L; ρ) and the set of exact k-Bihom-cochains
by Bkα,β(L, ρ), k = 1, 2.
Denote the corresponding cohomology byHk
α,β(L, ρ) = Zkα,β(L; ρ)/Bk
α,β(L, ρ),where
Zkα,β(L; ρ) = f ∈ Ck
(α,αM )(β,βM )
(L,M) | dkρf = 0,
Bkα,β(L, ρ) = dk
ρg | g ∈ Ck−1(α,αM )(β,βM )
(L,M).
In the case of Lie algebras, we can form semidirect products when given representations.Similarly, we have
Proposition 4.4. Let (L, [·, ·]L, α, β) be a multiplicative δ-Bihom-Jordan-Lie algebra and(M,ρ, αM , βM ) a representation of L. Assume that the maps αM and βM are bijective.Then L nM = (L ⊕ M, [·, ·]ρ, α ⊕ αM , β ⊕ βM ) is a δ-Bihom-Jordan-Lie algebra, whereα⊕ αM , β ⊕ βM : L⊕M → L⊕M are defined by (α⊕ αM )(u+ x) = α(u) + αM (x) and(β⊕βM )(u+x) = β(u)+βM (x), for all u, v ∈ L and x, y ∈ M , the bracket [·, ·]ρ is definedby
[u+ x, v + y]ρ = [u, v]L + δρ(u)(y) − ρ(α−1β(v))(αMβ−1M (x)). (4.4)
We call L nM the semidirect product of the multiplicative δ-Bihom-Jordan-Lie algebra(L, [·, ·]L, α, β) and M .
660 A. Ben Hassine, L. Chen, J. Li
Proof. First we show that [·, ·]ρ satisfies antisymmetry,
[(β ⊕ βM )(v + y), (α⊕ αM )(u+ x)]ρ= [β(v) + βM (y), α(u) + αM (x)]ρ= [β(v), α(u)]L + δρ(β(v))(αM (x)) − ρ(α−1β(α(u)))(αMβ−1
M (βM (v)))= [β(v), α(u)]L + δρ(β(v))(αM (x)) − ρ(β(u))(αM (v))= −δ([β(u), α(v)]L + δρ(β(u))(αM (y)) − ρ(β(v))(αM (u)))= −δ[(β ⊕ βM )(u+ x), (α⊕ αM )(v + y)]ρ.
Next we show that (α⊕ αM ) and (β ⊕ βM ) are algebra morphisms. On the one hand, wehave
(α⊕ αM )[u+ x, v + y]ρ= ((α⊕ αM )([u, v]L + δρ(u)(y) − ρ(α−1β(v))(αMβ−1
M (x)))= α([u, v]L) + δαM ρ(u)(y) − αM ρ(α−1β(v))(αMβ−1
M (x)))= [α(u), α(v)]L + δρ(α(u))(αM (y)) − ρ(α(α−1β(v)))(αM (αMβ−1
M (x))))= [α(u), α(v)]L + δρ(α(u))(αM (y)) − ρ(β(v))(α2
Mβ−1M (x))
= [(α⊕ αM )(u+ x), (α⊕ αM )(v + y)]ρ.
Similarly, we obtain
(β ⊕ βM )[u+ x, v + y]ρ = [(β ⊕ βM )(u+ x), (β ⊕ βM )(v + y)]ρ.
Furthermore
[(β ⊕ βM )2(u+ x), [(β ⊕ βM )(v + y), (α⊕ αM )(w + z)]ρ]ρ= [β2(u) + β2
M (x), [β(v) + βM (y), α(w) + αM (z)]ρ]ρ= [β2(u) + β2
M (x), [β(v), α(w)]L + δρ(β(v))(αM (z)) − ρ(α−1β(α(w)))(αMβ−1M (βM (y)))]ρ
= [β2(u) + β2M (x), [β(v), α(w)]L + δρ(β(v))(αM (z)) − ρ(β(w))(αM (y))]ρ
= [β2(u), [β(v), α(w)]L]L + δ2ρ(β2(u))ρ(β(v))(αM (z)) − δρ(β2(u))ρ(β(w))(αM (y))− ρ(α−1β([β(v), α(w)])(αMβ−1
M (β2M (x))))
= [β2(u), [β(v), α(w)]L]L + ρ(β2(u))ρ(β(v))(αM (z)) − δρ(β2(u))ρ(β(w))(αM (y))− ρ([α−1β2(v), β(w)])(αMβM (x)).
Similarly,
[(β ⊕ βM )2(v + y), [(β ⊕ βM )(w + z), (α⊕ αM )(u+ x)]ρ]ρ= [β2(v), [β(w), α(u)]L]L + ρ(β2(v))ρ(β(w))(αM (x)) − δρ(β2(v))ρ(β(u))(αM (z))
− ρ([α−1β2(w), β(u)])(αMβM (y)).
And
[(β ⊕ βM )2(w + z), [(β ⊕ βM )(u+ x), (α⊕ αM )(v + y)]ρ]ρ= [β2(w), [β(u), α(v)]L]L + ρ(β2(w))ρ(β(u))(αM (y)) − δρ(β2(w))ρ(β(v))(αM (x))
− ρ([α−1β2(u), β(v)])(αMβM (z)).
By (4.3), the δ-Bihom-Jacobi identity is satisfied. Thus, (L ⊕ M, [·, ·]ρ, α ⊕ αM , β ⊕ βM )is a multiplicative δ-Bihom-Jordan-Lie algebra.
Representations and T ∗-extensions of δ-Bihom-Jordan-Lie algebras 661
5. The trivial representation of δ-Bihom-Jordan-Lie algebrasIn this section, we study the trivial representation of multiplicative δ-hom-Jordan-Lie
algebras. As an application, we show that the central extension of a multiplicative δ-Bihom-Jordan-Lie algebra (L, [·, ·]L, α, β) is controlled by the second cohomology of Lwith coefficients in the trivial representation.
Now let M = R, Then we have End(M) = R. Any αM , βM ∈ End(M) is exactly tworeal numbers, which we denote by r1, r2 respectively. Let ρ : L → End(M) = R be the zeromap. Obviously, ρ is a representation of the multiplicative δ-Bihom-Jordan-Lie algebra(L, [·, ·]L, α, β) with respect to any r1, r2 ∈ R. We will always assume that r1 = r2 = 1. Wecall this representation the trivial representation of the multiplicative δ-Bihom-Jordan-Liealgebra (L, [·, ·]L, α, β).
Associated to the trivial representation, the set of k-cochains on L, which we denote byCk(V ) = ∧kL∗, is the set of skew-symmetric k-linear maps from V × · · · × V to R.The set of k-Bihom-cochains is given by
Ckα,β(L) = f ∈ Ck(L)|f α = f, f β = f.
The corresponding coboundary operator dT : Ckα,β(L) → Ck+1
α,β (L)(k = 1, 2) is given by
d1T f(u1, u2) = −δf([α−1β(u1), u2]L), (5.1)
d2T f(u1, u2, u3) = −f([α−1β(u1), u2]L, β(u3)) + δf([α−1β(u1), u3]L, β(u2))
−f([α−1β(u2), u3]L, β(u1)).
Denote Zkα,β(L) and Bk
α,β(L)(k = 1, 2) similarly.In the following we consider central extensions of the multiplicative δ-Bihom-Jordan-Lie
algebra (L, [·, ·]L, α, β). Obviously, (R, 0, 1, 1) is an abelian multiplicative δ-Bihom-Jordan-Lie algebra with the trivial bracket and the identity morphism. Let θ ∈ C2
α,β(L), we haveθ α = θ, θ β = θ and θ(u, v) = −δθ(v, u), ∀u, v ∈ L. We consider the direct sumg = L⊕ R with the following bracket
[u+ s, v + t]θ = [u, v]L + θ(αβ−1(u), v), ∀u, v ∈ L, s, t ∈ R. (5.2)
Define αg, βg : g → g by αg(u+ s) = α(u) + s, and βg(u+ s) = β(u) + s.
Theorem 5.1. With the above notations, the 4-tuple (g, [·, ·]g, αg, βg) is a multiplicativeδ-Bihom-Jordan-Lie algebra if and only if θ ∈ C2
α,β(L) is a 2-cocycle associated to thetrivial representation, i.e.
dT θ = 0.We call the multiplicative δ-Bihom-Jordan-Lie algebra (g, [·, ·]g, αg, βg) the central ex-
tension of (L, [·, ·]L, α, β) by the abelian δ-Bihom-Jordan-Lie algebra (R, 0, 1, 1).
Proof. Obviously, since α β = β α, we have αg βg = βg αg. Then we show that αg
is an algebra morphism with the respect to the bracket [·, ·]θ. On one hand, we have
αg([u+ s, v + t]θ = αg([u, v]L + θ(αβ−1(u), v))= α([u, v]L) + θ(αβ−1(u), v)).
On the other hand, we have
[αg(u+ s), αg(v + t)]θ = [α(u) + s, α(v) + t]θ= [α(u), α(v)]L + θ(αβ−1(α(u)), α(v)).
Since α is an algebra morphism and θ(αβ−1(α(u)), α(v)) = θα(αβ−1(u), v) = θ(αβ−1(u), v).Then αg is an algebra morphism.
Similarly, we have βg is also an algebra morphism.
662 A. Ben Hassine, L. Chen, J. Li
Furthermore, we have
[βg(u+ s), αg(v + t)]θ = [β(u) + s, α(v) + t]θ= [β(u), α(v)]L + θ(αβ−1(β(u)), α(v))= [β(u), α(v)]L + θ(α(u), α(v))= [β(u), α(v)]L + θ(u, v)
and
[βg(v + t), αg(u+ s)]θ = [β(v) + t, α(u) + s]θ= [β(v), α(u)]L + θ(αβ−1(β(v)), α(u))= [β(v), α(u)]L + θ(α(v), α(u))= [β(v), α(u)]L + θ(v, u)= −δ([β(u), α(v)]L + θ(u, v)).
Then [βg(u+ s), αg(v + t)]θ = −δ[βg(v + t), αg(u+ s)]θ.By direct computations, we have
[β2g (u+ s), [βg(v + t), αg(w + r)]θ]θ + [β2
g (v + t), [βg(w + r), αg(u+ s)]θ]θ+ [β2
g (w + r), [βg(u+ s), αg(v + t)]θ]θ= [β2(u) + s, [β(v) + t, α(w) + r]θ]θ + [β2(v) + t, [β(w) + r, α(u) + s]θ]θ
+ [β2(w) + r, [β(u) + s, α(v) + t]θ]θ= [β2(u) + s, [β(v)α(w)]L + θ(αβ−1(β(v)), α(w))]θ
+ [β2(v) + t, [β(w)α(u)]L + θ(αβ−1(β(w)), α(u))]θ+ [β2(w) + r, [β(u)α(v)]L + θ(αβ−1(β(u)), α(v))]θ)
= [β2(u), [β(v)α(w)]L]L + θ(αβ−1(β2(u))), [β(v), α(w)]L+ [β2(v), [β(w)α(u)]L]L + θ(αβ−1(β2(v))), [β(w), α(u)]L+ [β2(w), [β(u)α(v)]L]L + θ(αβ−1(β2(w))), [β(u), α(v)]L
= [β2(u), [β(v)α(w)]L]L + θ(αβ(u), [β(v), α(w)]L)+ [β2(v), [β(w)α(u)]L]L + θ(αβ(v), [β(w), α(u)]L)+ [β2(w), [β(u)α(v)]L]L + θ(αβ(w), [β(u), α(v)]L).
Thus by the Bihom-Jacobi identity of L, [·, ·]θ satisfies the δ-Bihom-Jacobi identity if andonly if
θ(αβ(u)), [β(v), α(w)]L + θ(αβ(v)), [β(w), α(u)]L + θ(αβ(w)), [β(u), α(v)]L = 0.
Namely,
θ(β(u), [α−1β(v), w]L) + θ(β(v), [α−1β(w), u]L) + θ(β(w), [α−1β(u), v]L) = 0.
On the other hand,
dT θ(u, v, w)= δ3(−δθ([α−1β(u), v]L, β(w)) + θ([α−1β(w), u]L, β(v)) − δθ([α−1β(v), w]L, β(u)))= −(θ([α−1β(u), v]L, β(w)) + θ([α−1β(w), u]L, β(v)) + θ([α−1β(v), w]L, β(u)))= δ([β2
g (u+ s), [βg(v + t), αg(w + r)]θ]θ + [β2g (v + t), [βg(w + r), αg(u+ s)]θ]θ)
= 0.
Representations and T ∗-extensions of δ-Bihom-Jordan-Lie algebras 663
Then the 4-tuple (g, [·, ·]θ, αg, βg) is a multiplicative δ-Bihom-Jordan-Lie algebra if andonly if θ ∈ C2
α,β(L) satisfies dT θ = 0.
Proposition 5.2. For θ1, θ2 ∈ Z2(V ), if δ(θ1 −θ2) is exact, the corresponding two centralextensions (g, [·, ·]θ1 , αg, βg) and (g, [·, ·]θ2 , αg, βg) are isomorphic.
Proof. Assume that θ1 − θ2 = δdT f , f ∈ C1α,β(L). Thus we have
θ1(αβ−1(u), v) − θ2(u, v) = δd1T f(αβ−1(u), v) = −f([α−1β αβ−1(u), v]) = −f([u, v]).
Define φg : g → g byφg(u+ s) = u+ s+ f(u).
Obviously, φg is an isomorphism of vector spaces. The fact that φg is a morphism of theδ-Bihom-Jordan-Lie algebra follows from the fact θ α = θ, θ β = θ. More precisely, wehave
φg αg(u+ s) = φg(α(u) + s) = α(u) + s+ f(α(u)) = α(u) + s+ f(u).On the other hand, we have
αg φg(u+ s) = αg(u+ s+ f(u)) = α(u) + s+ f(u).Thus, we obtain that φg αg = αg φg. Similarly
φg βg = βg φg.
We also haveφg[u+ s, v + t]θ1 = φg([u, v]L + θ1(αβ−1(u), v))= [u, v]L + θ1(αβ−1(u), v) + f([u, v]L) = ([u, v]L, θ2(αβ−1(u), v)= [φg(u+ s), φg(v + t)]θ2 .
Therefore, φg is also an isomorphism of multiplicative δ-Bihom-Jordan-Lie algebras.
6. The adjoint representation of δ-Bihom-Jordan-Lie algebrasLet (L, [·, ·]L, α, β) be a regular δ-Bihom-Jordan-Lie algebra. We consider that L rep-
resents on itself via the bracket with respect to the morphisms α, β. A very interestingphenomenon is that the adjoint representation of a δ-Bihom-Jordan-Lie algebra is notunique as one will see in sequel.
Definition 6.1. For any integer s, t, the αsβt-adjoint representation of the regular δ-Bihom-Jordan-Lie algebra (L, [·, ·]L, α, β), which we denote by ads,t, is defined by
ads,t(u)(v) = δ[αsβt(u), v]L, ∀u, v ∈ L.
Lemma 6.2. With the above notations, we haveads,t(α(u)) α = α ads,t(u);ads,t(β(u)) β = β ads,t(u);
ads,t([β(u), v]L) β = ads,t(αβ(u)) ads,t(v) − δads,t(α(v)) ads,t(β(u)).
Thus the definition of αsβt-adjoint representation is well defined.
Proof. For any u, v, w ∈ L, first we show that ads,t(α(u)) α = α ads,t(u)
ads,t(α(u))(α(v)) = δ[αs+1βt(u), α(v)]L= α(δ[αsβt(u), v]L) = α ads,t(u)(v).
Similarly, we haveads,t(β(u)) β = β ads,t(u).
664 A. Ben Hassine, L. Chen, J. Li
Note that the skew-symmetry condition impliesads(u)(v) = δ[αsβt(u), v]L
= δ[β(αsβt−1(u)), α(α−1(v))]L= −δ2[α−1β(v), αs+1βt−1(u)]L= −[α−1β(v), αs+1βt−1(u)]L,∀u, v ∈ L.
On one hand, we haveads,t([β(u), v]L) β(w) = ads,t([β(u), v]L)(β(w))
= −[α−1β(β(w)), αs+1βt−1([β(u), v]L)]L= −[α−1β2(w), [αs+1βt(u), αs+1βt−1(v)]L]L.
On the other hand, we haveads,t(αβ(u)) ads,t(v)(w) − δads,t(α(v)) ads,t(β(u))(w)= ads,t(αβ(u))(−[α−1β(w), αs+1βt−1(v)]L)
− δads,t(α(v))(−[α−1β(w), αs+2βt−1(u)]L)= [α−1β([α−1β(w), αs+1βt−1(v)]L), αs+1βt−1(αβ(u))]L)
− δ[α−1β([α−1β(w), αs+2βt−1(u)]L), αs+1βt−1(β(v))]L)= [β([α−2β(w), αsβt−1(v)]L), αs+2βt(u)]L
− δ[β([α−2β(w), αs+1βt−1(u)]L), αs+1βt(v)]L)= −δ[β(αs+1βt(u)), α[α−2β(w), αsβt−1(v)]L]L
+ [β(αsβt(v)), α[α−2β(w), αs+1βt−1(u)]L]L)= −δ[β(αs+1βt(u)), [α−1β(w), αs+1βt−1(v)]L]L
+ [αsβt+1(v), [α−1β(w), αs+2βt−1(u)]L]L)= [αs+1βt+1(u)), [αsβt(v), w]L]L
+ [αsβt+1(v), [α−1β(w), αs+2βt−1(u)]L]L)= [β2(αs+1βt−1(u)), [β(αsβt−1(v)), α(α−1(w))]L]L
+ [β2(αsβt−1(v)), [β(α−1(w)), α(αs+1βt−1(u))]L]L)= −[β2(α−1(w)), [β(αs+1βt−1(u)), α(αsβt−1(v))]L]L= −[α−1β2(w), [αs+1βt(u), αs+1βt−1(v)]L]L.
Thus, the definition of αsβt-adjoint representation is well defined. The proof is completed.
The set of k-Bihom-cochains on L with coefficients in L, which we denote by Ckα,β(L;L),
is given byCk
α,β(L;L) = f ∈ Ck(L;L)|α f = f α, β f = f β.In particular, the set of 0-Bihom-cochains is given by:
C0α,β(L;L) = u ∈ L|α(u) = u, β(u) = u.
Associated to the αsβt-adjoint representation, the corresponding operatords,t : Ck
α,β(L;L) → Ck+1α,β (L;L)(k = 1, 2)
is given byds,tf(u1, u2) = δ[α1+sβt(u1), f(u2)] − [α1+sβt(u2), f(u1)] − δf([α−1β(u1), u2]); (6.1)
Representations and T ∗-extensions of δ-Bihom-Jordan-Lie algebras 665
ds,tf(u1, u2, u3) = δ[α1+sβt+1(u1), f(u2, u3)] − [α1+sβt+1(u2), f(u1, u3)]+ δ[α1+sβt+1(u3), f(u1, u2)] − f([α−1β(u1), u2], β(u3))+ δf([α−1β(u1), u3], β(u2)) − f([α−1β(u2), u3], β(u1)).
For the αsβt-adjoint representation ads,t, we obtain the αsβt-adjoint complex(Ck
α,β(L;L), ds,t).We have known that a 1-cocycle associated to the adjoint representation is a derivation
for Lie algebras and Hom-Lie algebras. Similarly, we have
Proposition 6.3. Associated to the αsβt-adjoint representations ads,t of the regular δ-Bihom-Jordan-Lie algebra (L, [·, ·]L, α, β), it satisfies δs+1 = 1 , D ∈ C1
α,β(L,L) is a1-cocycle if and only if D is an αs+2βt−1-derivation, i.e. D ∈ Derαs+2βt−1(L).
Proof. The conclusion follows directly from the definition of the operator ds,t. D is closedif and only if
ds,t(D)(u, v) = δ[αs+1βt(u), D(v)]L − [αs+1βt(v), D(u)]L − δD[α−1β(u), v]L = 0.D is an αs+2βt−1-derivation if and only if
D[α−1β(u), v]L = −δ[αs+2βt−1α−1β(v), D(u)]L + [αs+2βt−1α−1β(u), D(v)]L= δs+1([D(u), αs+1βt(v)]L + [αs+1βt(u), D(v)]L).
Then, D ∈ C1α,β(L,L) is a 1-cocycle if and only if D is an αs+2βt−1-derivation, i.e. D ∈
Derαs+2βt−1(L).
Let ψ ∈ C2α,β(L;L) be a bilinear operator commuting with α and β, also ψ(u, v) =
−δψ(v, u). Consider a t-parameterized family of bilinear operations[u, v]t = [u, v]L + tψ(u, v). (6.2)
Since ψ commutes with α, β, then α, β are morphisms with respect to the bracket [·, ·]tfor every t. If all the brackets [·, ·]t endow (L, [·, ·]t, α, β) with regular δ-Bihom-Jordan-Liealgebra structures, we say that ψ generates a deformation of the regular δ-Bihom-Jordan-Lie algebra (L, [·, ·]L, α, β). The anti-symmetry of [·, ·]t means that
[β(v), α(u)]t = [β(v), α(u)]L + tψ(β(v), α(u))and [β(u), α(v)]t = [β(u), α(v)]L + tψ(β(u), α(v)).
Then [β(v), α(u)]t = −δ[β(u), α(v)]t if and only ifψ(β(v), α(u)) = −δψ(β(u), α(v)). (6.3)
By computing the Bihom-Jacobi identity of [·, ·]t[β2(u), [β(v), α(w)]t]t + [β2(v), [β(w), α(u)]t]t + [β2(w), [β(u), α(v)]t]t= [β2(u), [β(v), α(w)]L + tψ(β(v), α(w))]t
+ [β2(v), [β(w), α(u)]L + tψ(β(w), α(u))]t+ [β2(w), [β(u), α(v)]L + tψ(β(u), α(v))]t
= [β2(u), [β(v), α(w)]L]t + [β2(u), tψ(β(v), α(w))]t+ [β2(v), [β(w), α(u)]L]t + [β2(v), tψ(β(w), α(u))]t+ [β2(w), [β(u), α(v)]L]t + [β2(w), tψ(β(u), α(v))]t
= [β2(u), [β(v), α(w)]L]L + tψ(β2(u), [β(v), α(w)]L)+ [β2(u), tψ(β(v), α(w))]L + tψ(β2(u), tψ(β(v), α(w)))+ [β2(v), [β(w), α(u)]L]L + tψ(β2(v), [β(w), α(u)]L)
666 A. Ben Hassine, L. Chen, J. Li
+ [β2(v), tψ(β(w), α(u))]L + tψ(β2(v), tψ(β(w), α(u)))+ [β2(w), [β(u), α(v)]L]L + tψ(β2(w), [β(u), α(v)]L)+ [β2(w), tψ(β(u), α(v))]L + tψ(β2(w), tψ(β(u), α(v))).
This is equivalent to the conditions
ψ(β2(u), ψ(β(v), α(w))) + ψ(β2(v), ψ(β(w), α(u))) + ψ(β2(w), ψ(β(u), α(v))) = 0, (6.4)
ψ(β2(u), [β(v), α(w)]L) + [β2(u), ψ(β(v), α(w))]L+ψ(β2(v), [β(w), α(u)]L) + [β2(v), ψ(β(w), α(u))]L+ψ(β2(w), [β(u), α(v)]L) + [β2(w), ψ(β(u), α(v))]L = 0. (6.5)
Obviously, (6.4) and (6.3) means that ψ must itself define a δ-Bihom-Jordan-Lie algebrastructure on L. Furthermore, (6.5) means that ψ is closed with respect to the α−1β-adjointrepresentation ad−1,1, i.e. d−1,1ψ = 0.
d−1,1ψ(u, v, w)= δ[β2(u), ψ(v, w)]L − [β2(v), ψ(u,w)]L + δ[β2(w), ψ(u, v)]L
− ψ([α−1β(u), v]L, β(w)) + δψ([α−1β(u), w]L, β(v)) − ψ([α−1β(v), w]L, β(u))= δ[β2(u), ψ(v, w)]L + δ[β2(v), ψ(w, u)]L + δ[β2(w), ψ(u, v)]L
+ δψ(β(w), [α−1β(u), v]L) + δψ(β(v), [α−1β(w), u]L) + δψ(β(u), [α−1β(v), w]L)= 0.
A deformation is said to be trivial if there is a linear operator N ∈ C1α,β(L;L) such that
for Tt = id + tN , there holds
Tt[u, v]t = [Tt(u), Tt(v)]L. (6.6)
Definition 6.4. A linear operator N ∈ C1α,β(L,L) is called a Bihom-Nijienhuis operator
if we have
[Nu,Nv]L = N [u, v]N , (6.7)
where the bracket [·, ·]N is defined by
[u, v]N , [Nu, v]L + [u,Nv]L −N [u, v]L. (6.8)
Theorem 6.5. Let N ∈ C1α(L,L) be a Bihom-Nijienhuis operator. Then a deformation
of the regular δ-Bihom-Jordan-Lie algebra (L, [·, ·]L, α, β) can be obtained by putting
ψ(u, v) = δd−1,1N(u, v) = [u, v]N .
Furthermore, this deformation is trivial.
Proof. Since ψ = δd−1,1N , d−1,1ψ = 0 is valid. To see that ψ generates a deformation,we need to check the Bihom-Jacobi identity for ψ. Using the explicit expression of ψ, andwe denote u,v,w the summation over the cyclic permutation on u, v, w. We have
u,v,w ψ(β2(u), ψ(β(v), α(w))=u,v,w ψ(β2(u), [Nβ(v), α(w)] + [β(v), Nα(u)] −N [β(v), α(w)])=u,v,w ψ(β2(u), [Nβ(v), α(w)]) + ψ(β2(u), [β(v), Nα(u)]) − ψ(β2(u), N [β(v), α(w)])=u,v,w [Nβ2(u), [Nβ(v), α(w)]] + [Nβ2(v), [β(w), Nβ(u)]] + [β2(w), N [β(u), α(v)]N ]
+ u,v,w N [β2(v), N [β(w), α(u)]] − [Nβ2(v), N [β(w), α(u)]]u,v,w −N [β2(u), [Nβ(v), α(w)]] +N [β2(w), [β(u), Nα(v)]]
Representations and T ∗-extensions of δ-Bihom-Jordan-Lie algebras 667
Since N commutes with α and β, by the Bihom-Jacobi identity of L, we have
[Nβ2(u), [Nβ(v), α(w)]] + [Nβ2(v), [β(w), Nα(u)]] = [[Nβ(u), Nα(v)], β2(w)].
Since N is a Bihom-Nijenhuis operator, the last equation becomes
u,v,w [Nβ2(u), [Nβ(v), α(w)]] + [Nβ2(v), [β(w), Nα(u)]] + [β2(w), N [β(u), α(v)]N ] = 0.
Furthermore, also by the fact that N is a Bihom-Nijenhuis operator and we take in(6.7) and (6.8), u = β2(v) and v = [β(w), α(u)], we have N [β2(v), N [β(w), α(u)]] −[Nβ2(v), N [β(w), α(u)]] = −N [Nβ2(v), [β(w), α(u)]] +N2[β2(v), [β(w), α(u)]].
By the Bihom-Jacobi identity of L, we have
u,v,w N [β2(v), N [β(w), α(u)]] − [Nβ2(v), N [β(w), α(u)]]=u,v,w N [Nβ2(v), [β(w), α(u)]].
Then,
u,v,w ψ(β2(u), ψ(β(v), α(w))= −N [Nβ2(v), [β(w), α(u)]] −N [β2(u), [Nβ(v), α(w)]] + [β2(w), [β(u), Nβ(v)]]= −N [β2(Nv), [β(w), α(u)]] + [β2(u), [Nβ(v), α(w)]] + [β2(w), [β(u), Nβ(v)]]= 0.
Thus ψ generates a deformation of the δ-Bihom-Jordan-Lie algebra (L, [·, ·]L, α, β).Let Tt = id + tN , then we have
Tt[u, v]t = (id + tN)([u, v]L + tψ(u, v))= (id + tN)([u, v]L + t[u, v]N )= [u, v]L + t([u, v]N +N [u, v]L) + t2N [u, v]N .
On the other hand, we have
[Tt(u), Tt(v)]L = [u+ tNu, v + tNv]L= [u, v]L + t([Nu, v]L + [u,Nv]L) + t2[Nu,Nv]L.
By the equations (6.7) and (6.8), we have
Tt[u, v]t = [Tt(u), Tt(v)]L,
which implies that the deformation is trivial.
7. T*-extensions of δ-Bihom-Jordan-Lie algebrasThe last part deals with T*-extension. We provide in this section, for δ-Bihom-Jordan-
Lie algebras, characterizations of T*-extensions and observations about T*-extensions ofnilpotent and solvable δ-Bihom-Jordan-Lie algebras. This method was introduced byMartin Bordemann in [2].
Definition 7.1. Let (L, [·, ·]L, α, β) be a δ-Bihom-Jordan-Lie algebra. A bilinear form fon L is said to be nondegenerate if
L⊥ = x ∈ L|f(x, y) = 0,∀y ∈ L = 0;
αβ-invariant if
f([β(x), α(y)], α(z)) = f(α(x), [β(y), α(z)]),∀x, y, z ∈ L;
symmetric iff(x, y) = f(y, x).
A subspace I of L is called isotropic if I ⊆ I⊥.
668 A. Ben Hassine, L. Chen, J. Li
Definition 7.2. Let (L, [·, ·]L, α, β) be a δ-Bihom-Jordan-Lie algebra over a field K. IfL admits a nondegenerate invariant symmetric bilinear form f , then we call (L, f, α, β)a quadratic δ-Bihom-Jordan-Lie algebra. In particular, a quadratic vector space V is avector space admitting a nondegenerate symmetric bilinear form.
Let (L′, [·, ·]′L, α1, β1) be another δ-Bihom-Jordan-Lie algebra. Two quadratic δ-Bihom-
Jordan-Lie algebras (L, f, α, β) and (L′, f ′, α1, β1) are said to be isometric if there exists
a δ-Bihom-Jordan-Lie algebra isomorphism ϕ : L → L′ such that
f(x, y) = f ′(ϕ(x), ϕ(y)),∀x, y ∈ L.
Lemma 7.3. Let ad be the adjoint representation of a δ-Bihom-Jordan-Lie algebra(L, [·, ·]L, α, β). Let us consider L∗ the dual space of L, α, β : L∗ → L∗ two homomorphismsdefined by
α(f) = f α, β(f) = f β, ∀f ∈ L∗.
Then the linear map π : L → End(L∗) defined by, π(x)(f)(y) = −δf ad(x)(y), ∀x, y ∈ L,is a representation of L on (L∗, α, β) if and only if
α adα(x) = adx α; (7.1)
β adβ(x) = adx β; (7.2)
ad(α(x)) adβ(y) − δady ad(αβ(x)) = β ad[β(x), y]L. (7.3)We call the representation π the coadjoint representation of L.
Proof. Firstly, we have
(π(α(x)) α)(f) = −δα(f) adα(x) = −δf α adα(x),
and
α(π(x))(f) = −δα(f adx) = −δf adx α.
Similarly,
(π(β(x)) β)(f) = −δβ(f) adβ(x) = −δf β adβ(x),
and
β(π(x))(f) = −δβ(f adx) = −δf adx β.
Therefore,
(π([β(x), y]) β)(f) = −δf β ad[β(x), y];
(π(αβ(x)) π(y) − δπ(β(y)) π(α(x)))(f)= −δπ(αβ(x))(f ady) + π(β(y))(f adα(x))= f ady adαβ(x) − δf adα(x) adβ(y)= −δf (adα(x) adβ(y) − δady adαβ(x)).
Then we have
π(α(x)) α = α(π(x));π(β(x)) β = β(π(x));
π([β(x), y]) β = π(αβ(x)) π(y) − δπ(β(y)) π(α(x)).
Then π is a representation of L on (L∗, α, β).
Representations and T ∗-extensions of δ-Bihom-Jordan-Lie algebras 669
Lemma 7.4. Under the above notations, let (L, [·, ·]L, α, β) be a δ-Bihom-Jordan-Lie al-gebra, and ω : L × L → L∗ be a bilinear map. Assume that the coadjoint representationexists. The space L ⊕ L∗, provided with the following bracket and a linear map definedrespectively by
[x+ f, y + g]L⊕L∗ = [x, y]L + ω(x, y) + δπ(x)g − π(α−1β(y)αβ−1(f), (7.4)
α′(x+ f) = α(x) + f α, (7.5)β
′(x+ f) = β(x) + f β. (7.6)Then (L ⊕ L∗, [·, ·]L⊕L∗ , α
′, β
′) is a δ-Bihom-Jordan-Lie algebra if and only if ω is a 2-cocycle: L× L → L∗, i.e. ω ∈ Z2(L,L∗).
Proof. For any elements x+ f, y + g, z + h ∈ L⊕ L∗. We have
[β′(x+ f), α′(y + g)]= [β(x) + f β, α(y) + g α]= [β(x), α(y)]L + w(β(x), α(y)) + δπ(β(x))(g α) − π(α−1β(α(y)))αβ−1(f β)= [β(x), α(y)]L + w(β(x), α(y)) + δπ(β(x))(g α) − π(β(y))(f α).
Similarly, we have
[β′(y + g), α′(x+ f)] = [β(y), α(x)]L + w(β(y), α(x)) + δπ(β(y))(f α) − π(β(x))(g α).
Then, we have [β′(x+ f), α′(y + g)] = −δ[β′(y + g), α′(x+ f)] if and only ifw(β(x), α(y)) = −δw(β(y), α(x)).
Therefore,
[β′ 2(x+ f), [β′(y + g), α′(z + h)]]= [β2(x) + f β2, [β(y) + g β, α(z) + h α]]= [β2(x) + f β2, [β(y), α(z)]L + w(β(y), α(z))
+ δπ(β(y))(h α) − π(α−1β(α(z)))(αβ−1(g β))]= [β2(x) + f β2, [β(y), α(z)]L + w(β(y), α(z)) + δπ(β(y))(h α) − π(β(z))(g α)]= [β2(x), [β(y), α(z)]L]L + w(β2(x), [β(y), α(z)]L)
+ δπ(β2(x))w(β(y), α(z)) + π(β2(x))π(β(y))(h α)− δπ(β2(x))π(β(z))(g α) − π(α−1β[β(y), α(z)])(f β α)).
And[β′ 2(y + g), [β′(z + h), α′(x+ f)]]= [β2(y), [β(z), α(x)]L]L + w(β2(y), [β(z), α(x)]L)
+ δπ(β2(y))w(β(z), α(x)) + π(β2(y))π(β(z))(f α)− δπ(β2(y))π(β(x))(h α) − π(α−1β[β(z), α(x)])(g β α),
[β′ 2(z + h), [β′(x+ f), α′(y + g)]]= [β2(z), [β(x), α(y)]L]L + w(β2(z), [β(x), α(y)]L)
+ δπ(β2(z))w(β(x), α(y)) + π(β2(z))π(β(x))(g α)− δπ(β2(z))π(β(y))(f α) − π(α−1β[β(x), α(y)])(h β α).
Since π is the coadjoint representation of L, we haveπ(α−1β[β(x), α(y)]L)h β α
670 A. Ben Hassine, L. Chen, J. Li
= π([β(α−1β(x)), β(y)]L) β(h α)= π(αβ(α−1β(x)))π(β(y))(h α) − δπ(β(β(y)))π(α(α−1β(x)))(h α)= π(β2(x))π(β(y))(h α) − δπ(β2(y)))π(β(x))(h α).
Similarly,π(α−1β[β(y), α(z)]L)f β α = π(β2(y))π(β(z))(f α) − δπ(β2(z)))π(β(y))(f α),
andπ(α−1β[β(z), α(x)]L)g β α = π(β2(z))π(β(x))(g α) − δπ(β2(x)))π(β(z))(g α).
Consequently, [β′ 2(x+f), [β′(y+g), α′(z+h)]]+[β′ 2(y+g), [β′(z+h), α′(x+f)]]+[β′ 2(z+h), [β′(x+ f), α′(y + g)]] = 0 if and only if
0 = w(β2(x), [β(y), α(z)]L) + δπ(β2(x))w(β(y), α(z))+w(β2(y), [β(z), α(x)]L) − δπ(β2(y))w(β(z), α(x))+δw(β2(z), [β(x), α(y)]L) + δπ(β2(z))w(β(x), α(y))
= [β2(x), w(β(y), α(z))] − δ[β2(y), w(β(z), α(x))] + δ[β2(z), w(β(x), α(y))]−δw([β(y), α(z)]L, β2(x)) + w(β2(y), [β(x), α(y)]L) − δw([β(x), α(y)]L, β2(z))
= δd−1,1ω(x, y, z).That is ω ∈ Z2
α,β(L,L∗). Then confirmation holds if and only if ω ∈ Z2(L,L∗). Conse-quently, we prove the lemma.
Clearly, L∗ is an abelian Bihom-ideal of (L ⊕ L∗, [·, ·], α′, β
′) and L is isomorphic tothe factor δ-Bihom-Jordan-Lie algebra (L ⊕ L∗)/L∗. Moreover, consider the followingsymmetric bilinear form qL on L⊕ L∗ for all x+ f, y + g ∈ L⊕ L∗,
qL(x+ f, y + g) = f(y) + g(x).Then we have the following lemma.
Lemma 7.5. Let L, L∗, ω and qL be as above. Then the 4-tuple (L ⊕ L∗, qL, α′, β
′) isa quadratic δ-Bihom-Jordan-Lie algebra if and only if ω is Jordancyclic in the followingsense:
ω(β(x), α(y))(α(z)) = ω(β(y), α(z))(α(x)) for all x, y, z ∈ L.
Proof. If x+f is orthogonal to all elements of L⊕L∗, then f(y) = 0 and g(x) = 0, whichimplies that x = 0 and f = 0. So the symmetric bilinear form qL is nondegenerate.
Now suppose that x+ f, y + g, z + h ∈ L⊕ L∗, thenqL([β′(x+ f), α′(y + g)]L⊕L∗ , α
′(z + h))= qL([β(x) + f β, α(y) + g α]L⊕L∗ , α(z) + h α)= qL([β(x), α(y)]L + ω(β(x), α(y)) + δπ(β(x))g α− π(α−1βα(y))αβ−1(f β), α(z)
+ h α)= qL([β(x), α(y)]L + ω(β(x), α(y)) + δπ(β(x))g α− π(β(y))(f α), α(z) + h α)= ω(β(x), α(y))(α(z)) + δ(π(β(x))g α)(α(z)) − π(β(y))(f α)(α(z))
+ h α([β(x), α(y)]L)= ω(β(x), α(y))(α(z)) − δg α([β(x), α(z)]L) + f α([β(y), α(z)]L) + h α([β(x), α(y)]L)= ω(β(x), α(y))(α(z)) + g α([β(z), α(x)]L) + f α([β(y), α(z)]L) − δh α([β(y), α(x)]L).On the other hand,qL(α′(x+ f), [β′(y + g), α′(z + h)]L⊕L∗)= qL(α(x) + f α, [β(y) + g β, α(z) + h α]L⊕L∗)
Representations and T ∗-extensions of δ-Bihom-Jordan-Lie algebras 671
= qL(α(x) + f α, [β(y), α(z)]L + ω(β(y), α(z)) + δπ(β(y))h α− π(α−1βα(z))αβ−1(g β))
= qL(α(x) + f α, [β(y), α(z)]L + ω(β(y), α(z)) + δπ(β(y))h α− π(β(z))(g α))= f α([β(y), α(z)]L) + ω(β(y), α(z))(α(x)) + δπ(β(y))h α(α(x))
− π(β(z))(g α))(α(x))= ω(β(y), α(z))(α(x)) + g α([β(z), α(x)]L) + f α([β(y).α(z)]L) − δh α([β(y), α(x)]L).Hence the lemma follows.
Now, for a Jordancyclic 2-cocycle ω we shall call the quadratic δ-Bihom-Jordan-Liealgebra (L⊕ L∗, qL, α
′, β
′) the T ∗-extension of L (by ω) and denote the δ-Bihom-Jordan-Lie algebra (L⊕ L∗, [·, ·], α′
, β′) by T ∗
ωL.
Definition 7.6. Let L be a δ-Bihom-Jordan-Lie algebra over a field K. We inductivelydefine a derived series
(L(n))n≥0 : L(0) = L, L(n+1) = [L(n), L(n)],and a central descending series
(Ln)n≥0 : L0 = L, Ln+1 = [Ln, L].L is called solvable and nilpotent(of length k) if and only if there is a (smallest) integer
k such that L(k) = 0 and Lk = 0, respectively.
In the following theorem we discuss some properties of T ∗ωL.
Theorem 7.7. Let (L, [·, ·]L, α, β) be a δ-Bihom-Jordan-Lie algebra over a field K.(1) If L is solvable (nilpotent) of length k, then the T ∗-extension T ∗
ωL is solvable (nilpo-tent) of length r, where k ≤ r ≤ k + 1 (k ≤ r ≤ 2k − 1).
(2) If L is decomposed into a direct sum of two Bihom-ideals of L, so is the trivialT ∗-extension T ∗
0L.
Proof. (1) Firstly we suppose that L is solvable of length k. Since (T ∗ωL)(n)/L∗ ∼= L(n)
and L(k) = 0, we have (T ∗ωL)(k) ⊆ L∗, which implies (T ∗
ωL)(k+1) = 0 because L∗ is abelian,and it follows that T ∗
ωL is solvable of length k or k + 1.Suppose now that L is nilpotent of length k. Since (T ∗
ωL)n/L∗ ∼= Ln and Lk = 0,we have (T ∗
ωL)k ⊆ L∗. Let g ∈ (T ∗ωL)k ⊆ L∗, b ∈ L, x1 + f1, · · · , xk−1 + fk−1 ∈ T ∗
ωL,1 ≤ i ≤ k − 1, we have
[[· · · [g, x1 + f1]L⊕L∗ , · · · ]L⊕L∗ , xk−1 + fk−1]L⊕L∗(b)
= δk−1gad(x1)ad(β−1α(x2)) · · · ad(xk−1)β−(k−1)αk−1(b)
= g([x1, [β−1α(x2), [· · · , [β−(k−2)αk−2(xk−1), β−(k−1)αk−1(b)]L · · · ]L]L]L)
∈ g(Lk) = 0.
This proves that (T ∗ωL)2k−1 = 0. Hence T ∗
wL is nilpotent of length at least k and at most2k − 1.
(2) Suppose that 0 6= L = I ⊕ J , where I and J are two nonzero Bihom-ideals of(L[·, ·]L, α, β). Let I∗ (resp. J∗) denote the subspace of all linear forms in L∗ vanishingon J (resp. I). Clearly, I∗ (resp. J∗) can canonically be identified with the dual space ofI (resp. J) and L∗ ∼= I∗ ⊕ J∗.
Since [I∗, L]L⊕L∗(J) = I∗([L, β−1α(J)]L) ⊆ I∗([L, J ]L) ⊆ I∗(J) = 0 and [I, L∗]L⊕L∗(J) =L∗([I, J ]L) ⊆ L∗(I ∩ J) = 0, we have [I∗, L]L⊕L∗ ⊆ I∗ and [I, L∗]L⊕L∗ ⊆ I∗. Then
[T ∗0 I, T
∗0L]L⊕L∗ = [I ⊕ I∗, L⊕ L∗]L⊕L∗
= [I, L]L + [I, L∗]L⊕L∗ + [I∗, L]L⊕L∗ + [I∗, L∗]L⊕L∗ ⊆ I ⊕ I∗ = T ∗0 I.
672 A. Ben Hassine, L. Chen, J. Li
T ∗0 I is a Bihom-ideal of L and so is T ∗
0 J in the same way. Hence T ∗0L can be decomposed
into the direct sum T ∗0 I ⊕ T ∗
0 J of two nonzero Bihom-ideals of T ∗0L.
In the proof of a criterion for recognizing T ∗-extensions of a δ-Bihom-Jordan-Lie algebra,we will need the following result.
Lemma 7.8. Let (L, qL, α, β) be a quadratic δ-Bihom-Jordan-Lie algebra of even dimen-sion n over a field K and I be an isotropic n/2-dimensional subspace of L. If I is aBihom-ideal of (L, [·, ·]L, α, β), then [β(I), α(I)] = 0.
Proof. Since dimI+dimI⊥ = n/2 + dimI⊥ = n and I ⊆ I⊥, we have I = I⊥. IfI is a ideal of (L, [·, ·]L, α, β), then qL(α(L), [β(I), α(I⊥)]) = qL([β(L), α(I)], α(I⊥)) ⊆qL([β(L), I], α(I⊥)) ⊆ qL(I, I⊥) = 0, which implies [β(I), α(I)] = [β(I), α(I⊥)] ⊆ α(L)⊥ =0.
Theorem 7.9. Let (L, qL, α, β) be a quadratic regular δ-Bihom-Jordan-Lie algebra of evendimension n over a field K of characteristic not equal to two. Then (L, qL, α, β) is isometricto a T ∗-extension (T ∗
ωB, qB, α′, β
′) if and only if n is even and (L, [·, ·]L, α, β) contains anisotropic Bihom-ideal I of dimension n/2. In particular, B ∼= L/I, with B∗ satisfyingα(B∗) ⊆ B∗and β(B∗) ⊆ B∗.
Proof. (=⇒) Since dimB=dimB∗, dimT ∗ωB is even. Moreover, it is clear that B∗ is a
Bihom-ideal of half the dimension of T ∗ωB and by the definition of qB, we have qB(B∗, B∗) =
0, i.e., B∗ ⊆ (B∗)⊥ and so B∗ is isotropic.(⇐=) Suppose that I is an n/2-dimensional isotropic Bihom-ideal of L. By Lemma
7.8, [β(I), α(I)] = 0. Let B = L/I and p : L → B be the canonical projection. SincechK 6= 2, we can choose an isotropic complement subspace B0 to I in L, i.e., L = B0 u Iand B0 ⊆ B⊥
0 . Then B⊥0 = B0 since dimB0 = n/2.
Denote by p0 (resp. p1) the projection L → B0 (resp. L → I) and let q∗L denote the
homogeneous linear map I → B∗ : i 7→ q∗L(i), where q∗
L(i)(p(x)) := qL(i, x), ∀x ∈ L. Weclaim that q∗
L is a linear isomorphism. In fact, if p(x) = p(y), then x − y ∈ I, henceqL(i, x − y) ∈ qL(I, I) = 0 and so qL(i, x) = qL(i, y), which implies q∗
L is well-defined andit is easily seen that q∗
L is linear. If q∗L(i) = q∗
L(j), then q∗L(i)(p(x)) = q∗
L(j)(p(x)), ∀x ∈ L,i.e., qL(i, x) = qL(j, x), which implies i − j ∈ L⊥ = 0, hence q∗
L is injective. Note thatdimI = dimB∗, then q∗
L is surjective.In addition, q∗
L has the following property:
q∗L([β(x), α(i)])(p(α(y)))= qL([β(x), α(i)]L, α(y)) = −δqL([β(i), α(x)]L, α(y))= −δqL(α(i), [β(x), α(y)]L) = −δq∗
L(α(i))p([β(x), α(y)]L)= −δq∗
L(α(i))[p(β(x)), p(α(y))]L = −q∗L(α(i))(adp(β(x))(p(α(y))))
= δ(π(p(β(x)))q∗L(α(i)))(p(α(y))) = [p(β(x)), q∗
L(α(i))]L⊕L∗(p(α(y))),
where x, y ∈ L, i ∈ I. A similar computation shows that
q∗L([β(x), α(i)]) = [p(β(x)), q∗
L(α(i))]L⊕L∗ , q∗L([β(i), α(x)]) = [q∗
L(β(i)), p(β(x))]L⊕L∗ .
Define a homogeneous bilinear map
ω : B ×B −→ B∗
(p(b0), p(b′0)) 7−→ q∗
L(p1([b0, b′0])),
where b0, b′0 ∈ B0. Then w is well-defined since the restriction of the projection p to B0 is
a linear isomorphism.Let φ be the linear map L → B ⊕ B∗ defined by φ(b0 + i) = p(b0) + q∗
L(i),∀b0 + i ∈B0 u I = L. Since the restriction of p to B0 and q∗
L are linear isomorphisms, φ is also a
Representations and T ∗-extensions of δ-Bihom-Jordan-Lie algebras 673
linear isomorphism. Note thatφ([β(b0 + i), α(b′
0 + i′)]L)= φ([β(b0), α(b′
0)]L + [β(b0), α(i′)]L + [β(i), α(b′0)]L)
= φ(p0(β(b0), α(b′0)]L) + p1([β(b0), α(b′
0)]L) + [β(b0), α(i′)]L + [β(i), α(b′0)]L)
= p(p0([β(b0), α(b′0)]L)) + q∗
L(p1([β(b0), α(b′0)]L) + [β(b0), α(i′)]L + [β(i), α(b′
0)]L)= [p(β(b0)), p(α(b′
0))]L + ω(p(β(b0)), p(α(b′0))) + [p(β(b0)), q∗
L(α(i′))]L+ [q∗
L(β(i)), p(α(b′0))]L
= [p(β(b0)), p(α(b′0))]L + ω(p(β(b0)), p(α(b′
0))) + δπ(p(β(b0))(q∗L(α(i′)))
− π(p(β(b′0))(q∗
L(α(i)))= [p(β(b0)) + q∗
L(β(i)), p(α(b′0)) + q∗
L(α(i′))]B⊕B∗
= [φβ((b0 + i)), φα((b′0 + i′))]L⊕L∗ .
Then φ is an isomorphism of algebras, and so (B⊕B∗, [·, ·]B⊕B∗ , α, β) is a δ-Bihom-Jordan-Lie algebra. Furthermore, we have
qB(φ(b0 + i), φ(b′0 + i′)) = qB(p(b0) + q∗
L(i), p(b′0) + q∗
L(i′))= q∗
L(i)(p(b′0)) + q∗
L(i′)(p(b0))= qL(i, b′
0) + qL(i′, b0)= qL(b0 + i, b′
0 + i′),then φ is isometric. The relation
qB([β′(φ((x)), α′(φ(α(y)))], α′(φ(α(z))))= qB([φ(β(x)), φ(α(y))], φ(α(z))) = qB(φ([β(x), α(y)]), φ(α(z))) = qL([β(x), α(y)], α(z))= qL(α(x), [β(y), α(z)]) = qB(φ(α(x)), [φ(β(y)), φ(α(z))])
= qB(α′(φ(x)), [β′(φ(y)), α′(φ(z))])which implies that qB is a nondegenerate invariant symmetric bilinear form, and so(B ⊕ B∗, qB, α
′, β′) is a quadratic δ-Bihom-Jordan-Lie algebra. In this way, we get a
T ∗-extension T ∗ωB of B and consequently, (L, qL, α, β) and (T ∗
ωB, qB, α′, β
′) are isometricas required.
Let(L, [·, ·]L, α, β) be a δ-Bihom-Jordan-Lie algebra over a field K, and let ω1 : L×L →L∗ and ω2 : L×L → L∗ be two different Jordancyclic 2-cocycles. The T ∗-extensions T ∗
ω1Land T ∗
w2L of L are said to be equivalent if there exists an isomorphism of δ-Bihom-Jordan-Lie algebras ϕ : T ∗
ω1L → T ∗ω2L which is the identity on the Bihom-ideal L∗ and which
induces the identity on the factor δ-Bihom-Jordan-Lie algebra T ∗ω1L/L
∗ ∼= L ∼= T ∗ω2L/L
∗.The two T ∗-extensions T ∗
ω1L and T ∗ω2L are said to be isometrically equivalent if they are
equivalent and ϕ is an isometry.
Proposition 7.10. Let L be a δ-Bihom-Jordan-Lie algebra over a field K of characteristicnot equal to 2, and ω1, ω2 be two Jordan cyclic 2-cocycles L× L → L∗. Then we have
(i) T ∗ω1L is equivalent to T ∗
ω2L if and only if there is z ∈ C1(L,L∗) such that
ω1(x, y) − ω2(x, y) = δπ(x)z(y) − π(α−1β(y)αβ−1z(x) − z([x, y]L), ∀x, y ∈ L. (7.7)If this is the case, then the symmetric part zs of z, defined by zs(x)(y) :=
12(z(x)(y) + z(y)(x)), for all x, y ∈ L, induces a symmetric invariant bilinear formon L.
(ii) T ∗ω1L is isometrically equivalent to T ∗
ω2L if and only if there is z ∈ C1(L,L∗) suchthat (29) holds for all x, y ∈ L and the symmetric part zs of z vanishes.
674 A. Ben Hassine, L. Chen, J. Li
Proof. (i) T ∗ω1L is equivalent to T ∗
ω2L if and only if there is an isomorphism of δ-Bihom-Jordan-Lie algebras Φ : T ∗
ω1L → T ∗ω2L satisfying Φ|L∗ = 1L∗ and x− Φ(x) ∈ L∗,∀x ∈ L.
Suppose that Φ : T ∗ω1L → T ∗
ω2L is an isomorphism of δ-hom-Jordan-Lie algebra anddefine a linear map z : L → L∗ by z(x) := Φ(x) − x, then z ∈ C1(L,L∗) and for allx+ f, y + g ∈ T ∗
ω1L, we haveΦ([x+ f, y + g]Ω)= Φ([x, y]L + ω1(x, y) + δπ(x)g − π(α−1β(y)αβ−1(f)= [x, y]L + z([x, y]L) + ω1(x, y) + δπ(x)g − π(α−1β(y)αβ−1(f).
On the other hand,[Φ(x+ f),Φ(y + g)]= [x+ z(x) + f, y + z(y) + g]= [x, y]L + ω2(x, y) + δπ(x)g + δπ(x)z(y) − π(α−1β(y)αβ−1z(x) − π(α−1β(y)αβ−1(f).
Since Φ is an isomorphism, (7.7) holds.Conversely, if there exists z ∈ C1(L,L∗) satisfying (7.7), then we can define Φ : T ∗
ω1L →T ∗
ω2L by Φ(x + f) := x + z(x) + f . It is easy to prove that Φ is an isomorphism of δ-Bihom-Jordan-Lie algebras such that Φ|L∗ = idL∗ and x− Φ(L) ∈ L∗, ∀x ∈ L, i.e. T ∗
ω1L isequivalent to T ∗
ω2L.Consider the symmetric bilinear form qL : L×L → K, (x, y) 7→ zs(x)(y) induced by zs.
Note thatω1(β(x), α(y))(α(m)) − ω2(β(x), α(y))(α(m))= δπ(β(x))z(α(y))(α(m)) − π(α−1β(α(y)))αβ−1z(β(x))(α(m)) − z([β(x), α(y)]L)(α(m))= δπ(β(x))z(α(y))(α(m)) − π(α(y))z(α(x))(α(m)) − z([β(x), α(y)]L)(α(m))= −δz(α(y))([β(x), α(m)]L) + z(α(x))([β(y), α(m)]L) − z([β(x), α(y)]L)(α(m)),
andω1(β(y), α(m))(α(x)) − ω2(β(y), α(m))(α(x))= δπ(β(y))z(α(m))(α(x)) − π(α(m))z(α(y))(α(x)) − z([β(y), α(m)]L)(α(x))= −δz(α(m))([β(y), α(x)]L) + z(α(y))([β(m), α(x)]L) − z([β(y), α(m)]L)(α(x))= z(α(m))([β(x), α(y)]L) − δz(α(y))([β(x), α(m)]L) − z([β(y), α(m)]L)(α(x)).
Since both ω1 and ω2 are Jordancyclic, the right hand sides of above two equations areequal. Hence
− δz(α(y))([β(x), α(m)]L) + z(α(x))([β(y), α(m)]L) − z([β(x), α(y)]L)(α(m))= z(α(m))([β(x), α(y)]L) − δz(α(y))([β(x), α(m)]L) − z([β(y), α(m)]L)(α(x)).
That isz(α(x))([β(y), α(m)]L) + z([β(y), α(m)]L)(α(x))= z([β(x), α(y)]L)(α(m)) + z(α(m))([β(x), α(y)]L).
Since chK 6= 2, qL(α(x), [β(y), α(m)]) = qL([β(x), α(y)], α(m)), which proves the invari-ance of the symmetric bilinear form qL induced by zs.
(ii) Let the isomorphism Φ be defined as in (i). Then for all x+ f, y + g ∈ L⊕ L∗, wehave
qB(Φ(x+ f),Φ(y + g)) = qB(x+ z(x) + f, y + z(y) + g)= z(x)(y) + f(y) + z(y)(x) + g(x)= z(x)(y) + z(y)(x) + f(y) + g(x)
Representations and T ∗-extensions of δ-Bihom-Jordan-Lie algebras 675
= 2zs(x)(y) + qB(x+ f, y + g).Thus, Φ is an isometry if and only if zs = 0. Acknowledgment. The authors would like to thank the referee for valuable com-ments and suggestions on this article. This work was supported by NNSF of China (No.11771069), NSF of Jilin province (No. 20170101048JC) and the project of Jilin provincedepartment of education (No. JJKH20180005K).
References[1] S. Benayadi and A. Makhlouf, Hom-Lie algebras with symmetric invariant nondegen-
erate bilinear forms, J. Geom. Phys. 76, 38–60, 2014.[2] M. Bordemann, Nondegenerate invariant bilinear forms on nonassociative algebras,
Acta Math. Univ. Comenian. (N.S.) 66 (2), 151–201, 1997.[3] Y. Cheng and H. Qi, Representations of Bihom-Lie algebras, arXiv:1610.04302.[4] G. Graziani, A. Makhlouf, C. Menini, and F. Panaite, Bihom-associative algebras,
Bihom-Lie algebras and Bihom-bialgebras, SIGMA Symmetry Integrability Geom.Methods Appl. (11), Paper 086, 34 pp, 2015.
[5] J. Hartwig, D. Larsson and S. Silvestrov, Deformations of Lie algebras using σ-derivations, J. Algebra 295 (2), 314–361, 2006.
[6] A. Makhlouf and S. Silvestrov, Hom-algebra structures, J. Gen. Lie Theory Appl. 2(2), 51–64, 2008.
[7] S. Okubo and N. Kamiya, Jordan Lie superalgebra and Jordan Lie triple system, J.Algebra 198 (2), 388–411, 1997.
[8] L. Qian, J. Zhou, and L. Chen, Engel’s theorem for Jordan-Lie algebras and itsapplications, Chinese Ann. Math. Ser. A 33 (5), 517–526, 2012 (in Chinese).
[9] S. Wang and S. Guo, Bihom-Lie superalgebra structures, arXiv:1610.02290.[10] J. Zhao, L. Chen, and L. Ma, Representations and T ∗-extensions of Hom-Jordan-Lie
Algebras, Comm. Algebra 44 (7), 2786–2812, 2016.
Hacettepe Journal ofMathematics & Statistics
Hacet. J. Math. Stat.Volume 49 (2) (2020), 676 – 683
DOI : 10.15672/hujms.588700
Research Article
Complex fuzzy soft matrices with applicationsMadad Khan∗1, Saima Anis1, Seok-Zun Song2,3, Young Bae Jun4
1 Department of Mathematics, COMSATS University Islamabad, Abbottabad Campus, Pakistan2Department of Mathematics, Jeju National University, Jeju 63243, Korea
3School of Computational Sciences, Korean Institute for Advanced Study, Seoul, 02455, Korea4Department of Mathematics Education, Gyeongsang National University, Jinju, 52828, Korea
AbstractIn this paper, we introduce complex fuzzy soft matrices and define some new operationson these matrices. Moreover we develop an algorithm using complex fuzzy soft matricesand apply it to a decision making problem in signal processing.
Mathematics Subject Classification (2010). 03E72, 15B15
Keywords. complex fuzzy sets, complex fuzzy matrices
1. IntroductionFuzzy set has been introduced by Zadeh in 1965 [14], which is nowadays used in almost
all the branches of science. Basically, it is a suitable tool for modeling as the crisp modelslack clarity while handling problems in different fields of science like artificial intelligence,computer science, control engineering, decision theory, expert system, logic managementscience, operations research, robotics, and many others. In other words this concept offuzzy set is used to address those uncertain problems which arise in models representingreal life phenomenon. This concept is not only limited to the above mentioned problems,rather it is also been used in business, medical, and related health sciences successfully.Maiers and Sherif [5] in their index of fuzzy set applications identified more then twelvesubject areas including decision making, economics, engineering, and operations researchon the basis of literature available on the subject.
In 2002, Ramot et al. [13] introduced a new concept of a complex fuzzy set which isthe generalization of a fuzzy set whose range is not restricted to a closed interval [0, 1] butexpanded to a unit circle in complex plane. It is defined as:
µS(x) = γS(x)eιωS(x),
where γS(x) is called amplitude of grade of membership belongs to [0, 1] and ωS(x) is areal valued function. Thus µS(x) is a complex valued function that maps each input valueto a value in unit circle in complex plane. The authors in this paper added one moredimension to the degree of membership. Thus the complex fuzzy set has a phase term inaddition to its amplitude term. The phase term represents wave type characteristic of this∗Corresponding Author.Email addresses: [email protected] (M. Khan), [email protected] (S. Anis),
[email protected] (S.Z. Song), [email protected] (Y.B. Jun)Received: 15.01.2018; Accepted: 14.02.2019
Complex fuzzy soft matrices with applications 677
set which makes it different from the basic concept of the fuzzy set. The idea of complexfuzzy set is the extended form of the fuzzy set.
In [13], Ramot et al., defined union, intersection, and complement on this set andexplored several interesting properties and they gave real life application of this concept.
Complex fuzzy sets solved the issues raised in the problems that are very difficult orimpossible to be represented by the traditional concept of the fuzzy sets.
N. Çağman and S. Enginoğlu introduced soft matrices and explore several propertiesof these matrices. They developed a soft max-min decision making algorithm which canbe used as a powerful tool in the problems involving uncertainties [3]. Moreover theyintroduced fuzzy soft matrices along with a fs-max-min decision making method whichcan be used successfully in decision making problems that contain ambiguities [4]. In factthey gave a new direction for fundamental work as well as applications in the theory ofsoft sets and fuzzy soft sets. For more details about soft sets, complex fuzzy sets and fuzzysoft matrices see [1], [6–12], and [15].
In this paper we introduce complex fuzzy soft matrices with some new operations. Wedevelop an algorithm using these matrices and apply it to a problem in signals and systems.
2. PreliminariesSuppose that U is an initial universe set and E is a set of parameters, let P (U) denote
the power set of U . A pair (F, E) is called a soft set over U where F is a mapping givenby F : E → P (U). Clearly a soft set is a mapping from parameters to P (U) and it is nota set, but a parameterized family of subsets of the universe.
Let I = [0, 1] and let us denote all the subintervals of I by [I]A function A : X → [I] is called an interval-valued fuzzy set (briefly, an IVF set) in anon-empty set X.
Definition 2.1. A complex fuzzy set is defined as [13]:
µS(x) = γS(x)eιωS(x),
where γS(x) is called amplitude of grade of membership belongs to [0, 1] and ωS(x) is areal valued function.
Definition 2.2. Let U be an initial universe set, E be the set of parameters and A ⊂ E.Then a pair (Φµ, A) is called fuzzy soft set over U , where Φµ is a mapping given byΦµ : A → P (µU ), where P (µU ) denote the set of all complex fuzzy subsets of U .
3. Complex fuzzy soft matrix theoryIn this section we introduce a new concept of a complex fuzzy soft matrix.
Complex Fuzzy Soft Matrix: Let U = c1, c2, c3, . . . , cn be the universal set andE be the set of parameters given by E = e1, e2, e3, . . . , em. If A ⊂ E, then (Φµ, A)is a fuzzy soft matrix over U , where Φµ is a mapping given by Φµ : A → P (µU ) andηΞ : A → CU where CU denotes the set of all fuzzy subsets and P (CU ) denotes thepower set of CU . Then complex fuzzy soft set (Φµ, A) can be expressed in matrix form as[Am×n] = [|aij |]m×n, for i = 1, 2, . . . , m and j = 1, 2, 3, . . . , n, where
|aij | =
|η (ei) |j if ei ∈ A0 otherwise
|η (ei) |j ( η (ei)j is a complex fuzzy set ) represents the element of A correspondsto element cj of U , for j = 1, 2, 3, . . . , n, where |η (ei) |j = αRC such that α ∈ [0, 1] andR = 1, 2, 3, . . . , m and C = 1, 2, 3, . . . , n.
Following is an example to illustrate the newly defined complex fuzzy soft matrix.
678 M. Khan, S. Anis, S.Z. Song, Y.B. Jun
Example 3.1. Suppose that there are three houses under consideration, namely the uni-verses U = h1, h2, h3, and the parameter set E = e1, e2, e3, where ei stands for “nearto downtown”, “green”, and “cheap ” respectively. Consider the mapping Φµ from pa-rameter set A = e1, e2 ⊂ E to the set of all fuzzy subsets of power set U . Consider afuzzy soft set (Φµ, A) which describes the “attractiveness of houses” that is consideringfor purchase. Then fuzzy soft set (Φµ, A) is given as.
[Φµ, A] = [|η (ei) |j ]m×n , where
Φµ(e1) = (h1, |0.2ei π2 |), (h2, |0.3ei2π|, (h3, |0.4ei π
4 |)Φµ(e2) = (h1, |0.6ei π
6 |), (h2, |0.9ei(0.5)|), (h3, |0.1ei(0)|),
where
0.2ei π2 = 0.2(cos π
2+ i sin π
2) = 0.2(0 + i) = 0.2i
|0.2i| =√
0.04 = 0.2
0.3ei2π = 0.3(cos 2π + i sin 2π) = 0.3(1 + 0) = 0.3|0.3| =
√0.09 = 0.3.
0.4ei π4 = 0.4(cos π
4+ i sin π
4) = 0.4( 1√
2+ i
1√2
)
= 0.4(0.707 + i0.707) = 0.28 + i0.28|0.28 + i0.28| =
√0.078 + 0.078 = 0.16
0.6ei π6 = 0.6(cos π
6+ i sin π
6) = 0.6(0.87 + 0.5i)
= 0.52 + 0.3i
|0.52 + 0.3i| =√
0.27 + 0.09 = 0.6
0.9ei(0.5) = 0.9(cos 0.5 + i sin 0.5) = 0.9(0.99 + 0.009i)= 0.891 + 0.081i
|0.891 + 0.081i| =√
0.793 + 0.006 = 0.89
0.1ei(0) = 0.1(cos 0 + i sin 0) = 0.1|0.1ei(0)| = 0.1
We would represent this complex fuzzy soft set in matrix form as: 0.2 0.6 00.3 0.89 00.16 0.1 0
Definition 3.2. Let [aij ] be a complex fuzzy soft matrix. Then
[aA
ij
]is called complex
zero soft matrix if (aij , rij) = (0, 0) for all i and j and denoted by[aA
ij
]= [0].
Am×n = [|aij |]m×n, for i = 1, 2, . . . , m and j = 1, 2, 3, . . . , n, where
|aij | =
|η (ei) |j if ei ∈ A0 otherwise
|η (ei) |j represents the element of A corresponds to the element cj of U , for j = 1, 2, 3, . . . , n,where |η (ei) |j = αRC such that α ∈ [0, 1] and R = 1, 2, 3, . . . , m and C = 1, 2, 3, . . . , n.
Complex fuzzy soft matrices with applications 679
Definition 3.3. Let [Am×n] and [Bm×n] be complex fuzzy soft matrices. Then(i) [Am×n] is a complex fuzzy soft submatrices of [Bm×n], denoted by [Am×n] ⊑ [Bm×n],
if (|aij | ≤ |bij | for all |aij | ∈ [Am×n] and |bij | ∈ [Bm×n].(ii) [Am×n] is a proper complex fuzzy soft submatrices of [Bm×n], denoted by [Am×n] ⊂
[Bm×n], if (|aij | ≤ |bij | for all |aij | ∈ [Am×n] and |bij | ∈ [Bm×n], n, and for at least oneterm |aij | < |bij |.
(iii) [Am×n] is an equal complex fuzzy soft matrix of [Bm×n], denoted by [Am×n] =[Bm×n], if (|aij | = |bij | for all |aij | ∈ [Am×n] and |bij | ∈ [Bm×n].
Definition 3.4. Let [Am×n] and [Bm×n] be complex fuzzy soft matrices. Then thecomplex fuzzy soft matrices [Cm×n] are called
(i) union of [Am×n] and [Bm×n], denoted [Am×n] ∪ [Bm×n] if [Cm×n] = max|aij |, |bij |for all |aij | ∈ [Am×n] and |bij | ∈ [Bm×n].
(ii) intersection of [Am×n] and [Bm×n], denoted [Am×n]∩[Bm×n] if [Cm×n] = min|aij |, |bij |for all |aij | ∈ [Am×n] and |bij | ∈ [Bm×n].
(iii) complement of [Am×n] denoted by [Am×n], if Cm×n = 1− Am×n for all m and n.
Example 3.5. Assume that
[Am×n] =
0.2 0.6 0 0.30.3 0.9 0 0.20.9 0.1 0 0.40.16 0.4 0 0.8
and [Bm×n] =
0.1 0.5 0 0.30.3 0.8 0 0.20.9 0.7 0 0.50.4 0.4 0 0.6
.
Then
[Am×n] ∪ [Bm×n] =
0.2 0.6 0 0.30.3 0.9 0 0.20.9 0.7 0 0.50.16 0.4 0 0.8
, [Am×n] ∩ [Bm×n] =
0.1 0.5 0 0.30.3 0.8 0 0.20.9 0.1 0 0.40.4 0.4 0 0.6
and
[Am×n] =
0.8 0.4 1 0.70.7 0.1 1 0.80.1 0.9 1 0.60.84 0.6 1 0.2
.
Definition 3.6. Let [Am×n] and [Bm×n] be complex fuzzy soft matrices. Then [Am×n]and [Bm×n] are disjoint if [Am×n] ∩ [Bm×n] = [0] for all m and n.
Proposition 3.7. Let [Am×n] be a complex fuzzy soft matrix. Then(i) ([Am×n]) = [Am×n],(ii) [0] = 1.
Proposition 3.8. If [Am×n], [Bm×n] and [Cm×n] are complex fuzzy soft matrices then(i) [Am×n] = [Bm×n] and [Bm×n] = [Cm×n] ⇔ [Am×n] = [Cm×n].(ii) [Am×n] ⊑ [Bm×n] and [Bm×n] ⊑ [ Am×n] ⇔ [Am×n] = [Bm×n].
Proposition 3.9. If [Am×n] and [Bm×n] are complex fuzzy soft matrices then(i) [Am×n] ⊑ [Bm×n] and [Bm×n] ⊑ [Cm×n] ⇒ [Am×n] ⊑ [Cm×n].(ii) [Am×n] ⊑ [Bm×n] ⇔ [Am×n] ∩ [Bm×n] = [Am×n] ⇔ [Am×n] ∪ [Bm×n] = [Bm×n].
680 M. Khan, S. Anis, S.Z. Song, Y.B. Jun
Proposition 3.10. If [Am×n] and [Bm×n] are complex fuzzy soft matrices then
(i) [ Am×n] ∪ [Bm×n] = [Bm×n] ∪ [Am×n].(ii) [ Am×n] ∩ [Bm×n] = [Bm×n] ∩ [Am×n].
(iii)([ Am×n] ∪ [Bm×n]
)∪ [Cm×n] = [Am×n] ∪
([Bm×n] ∪ [Cm×n]
).
(iv)([Am×n] ∩ [Bm×n]
)∩ [Cm×n] = [Am×n] ∩
([Bm×n] ∩ [Cm×n]
).
(v) [Am×n] ∪([Bm×n] ∩ [Cm×n]
)=
([Am×n] ∪ [Bm×n]
)∩
([Am×n] ∪ [Cm×n]
).
(vi) [Am×n] ∩([Bm×n] ∪ [Cm×n]
)=
([ Am×n] ∩ [Bm×n]
)∪
([Am×n] ∩ [Cm×n]
).
Proposition 3.11. Let [Am×n] and [Bm×n] be complex fuzzy soft matrices. Then DeMorgan law are valid
(i)([Am×n] ∪ [Bm×n]
)=
([Am×n]
)∩
([Bm×n]
).
(ii)([Am×n] ∩ [Bm×n]
)=
([Am×n]
)∪
([Bm×n]
).
Proof. For all m and n
(i)([Am×n] ∪ [Bm×n]
)
=[max
([Am×n], [Bm×n]
)]
=[1 − max
(Am×n, Bm×n
)]=
[min
(1 − Am×n, 1 − Am×n
)]= [Am×n] ∩ [Bm×n].
(ii) It can be proved similarly.
Note. It is worth mentioning to add here that a complex fuzzy soft matrix is moregeneral than a fuzzy soft matrix since that both the degree of membership function andphase terms are added in each entry of the matrix yielding better choice in decision makingproblems.
4. Decision making algorithmWe are going to discuss a real life application of newly defined complex fuzzy soft matrix.
In fact we will show that how our theoretical results have real life applications. Specificallythe complex fuzzy soft matrix explains how to get better and clear signal for identificationwith given reference signal.
Definition 4.1. [2] Let U = u1, u2, u3, ...un be initial universal set and Mm (Cki) =(Di1). Then subset of U can be obtained by using [Di1] = opt
[(di1, di1)
], where
opt(di1
)(U) = di1/ui : ui ∈ U and opt (di1) (U) = di1/ui : ui ∈ U,
opt(di1
), opt (di1) (U) is called an optimum fuzzy set on U .
Step 1. If a receiver gets various signals S1(g), S2(g), S3(g), ..., Sm(g) from any source.Each signal is sampled N times by the receiver. Then Si(g) (i varies from 1 to m) signalscan be recognized with respect to R, where R is given known signal. Assume that bothSi(g) and R are considered as n times.
Complex fuzzy soft matrices with applications 681
Assume that Sj(t) is i-th signal, where 1 ≤ t ≤ m. Then the absolute value of each Sj(t)in terms of discrete complex fuzzy transform given as follows:
where gj,n is the complex Fourier coefficients of signals Sj , and Sj(t) varies from 1 tom. The above expression with alternate complex Fourier coefficients can be written as:
where gj,s = uj,s · eiβj,s such that uj,s and βj,s are real valued and uj,s ≥ 1 for all s(1 ≤ s ≤ m).Step 2. Expressed in matrix form as Am×n = [|Sj(t)|]N×m, that is, in the matrix take allthe signals in columns and each column contains N samples of every signal, so we get
A =
|S1(1)| |S2(1)| ... |Sm(1)||S1(2)| |S2(2)| ... |Sm(2)|
. . . .|S1(N)| |S2(N)| ... |Sm(N)|
Step 3. In same way make another matrix by the signals S
/j (t) (1 ≤ t ≤ m and 1 ≤ j ≤ N).
B =
|S/
1(1)| |S/2(1)| ... |S/
m(1)||S/
1(2)| |S/2(2)| ... |S/
m(2)|. . . .
|S/1(N)| |S/
2(N)| ... |S/m(N)|
Step 4. Find usual product of the matrices.Step 5. Find complex fuzzy soft max-min decision making matrix (CSMmDM).Step 6. Find optimum fuzzy set on U .
4.1. ApplicationLet us assume that the set of four signals U = η1, η2, η3, η4. Now each of these signals
is sampled four times. Let R be the given known reference signal. Each signal is comparedwith the reference signal in order to get the high degree of resemblance with the referencesignal R. First use the (1), then we obatin the matrix A by setting the signals alongcolumn and their four times sampling along row. Similarly we will obtain the matrix B.
Now on the basis of steps algorithm (1 − 3) defined above we discuss an example.
A =
0 0.1 0.2 0.20 0.3 0.3 0.30 0.2 0.2 0.40 0.3 0.2 0.1
B =
0.1 0 0.1 0.20.2 0 0.3 0.10.3 0 0.2 0.40.4 0 0.1 0.3
.
Step 4. Now the product of fuzzy soft matrices A and B is given as:
A ∗ B =
0.16 0.0 0.09 0.150.27 0.0 0.18 0.240.18 0.0 0.14 0.220.16 0.0 0.14 0.14
.
It is simple usual matrix multiplication.
682 M. Khan, S. Anis, S.Z. Song, Y.B. Jun
Step 5. To calculate Mm[A ∗ B
]= [Di1] = [di1)], we have to find [Di1] for all i ∈
1, 2, 3, 4, such that
di1 = min tk1 = min t11, t21, t31, t41 , for k ∈ 1, 2, 3, 4.
Now let us calculate [D11] = (d11) for fixed j = 1.
d11 = min tk1 = min t11, t21, t31, t41 .
Here we have to find tk1 for all k ∈ 1, 2, 3, 4 and tk1 for all k ∈ 1, 2, 3, 4. Let us findtk1 for k ∈ 1, 2, 3, 4 as:
t11 = 0.16, t21 = 0.27, t31 = 0.18, t41 = 0.16.
Nowd11 = min 0.16, 0.27, 0.18, 0.16 = 0.16
So (D11) = (d11) = (0.16) .Similarly we can find (D21) = (d21) , (D31) = (d31) and (D41) = (d41), where
d21 = min t12, t22, t32, t42 = min 0, 0, 0, 0 = 0.
So (D21) = (d21) = 0.
d31 = min 0.09, 0.18, 0.14, 0.14 = 0.09.
Therefore D31 = d31 = 0.09.Also
d41 = min 0.15, 0.24, 0.22, 0.14 = 0.14So (D41) = d41 = 0.14.Finally we can obtain fuzzy soft max-min decision fuzzy soft matrix as:
mM(A ∗ B
)= (Di1) = (di1)
=
(D11) = (d11)(D21) = (d21)(D31) = (d31)(D41) = (d41)
=
0.160.000.090.14
.
Step 6. Finally we can find an optimum fuzzy set on [U ] = (u) .
optMm (A ∗ B) (η) = 0.16/η1, 0.09/η3, 0.14/η4 .
Hence identify signal η1 as R.
5. ConclusionIn this paper, we introduced the concept of a complex fuzzy soft matrix and defined
different types of matrices in fuzzy soft set theory along with examples. Then we intro-duced some new operations on these matrices and explored related properties. Further weconstructed a complex fuzzy soft decision making model, designed an algorithm with thehelp of these matrices, and used it in decision making problems. We hope that our findingwill help enhancing the study on fuzzy soft set theory and will open a new direction forapplications especially in decision analysis.
Acknowledgment. The first two authors are grateful to Higher Education Commissionof Pakistan for financial support. The third author (S.Z. Song) was supported by BasicScience Research Program through the National Research Foundation of Korea(NRF)funded by the Ministry of Education (No. 2016R1D1A1B02006812).
Complex fuzzy soft matrices with applications 683
References[1] B. Chetia and P.K. Das, Some results of intuitionistic fuzzy soft matrix theory, Adv.
Appl. Sci. Res. 3 (1), 412–423, 2012.[2] N. Çağman and S. Enginoğlu, Soft set theory and uni-int decision making, Eur. J.
Oper. Res. 207 (2), 848–855, 2010.[3] N. Çağman and S. Enginoğlu, Soft matrix theory and its decision making, Comput.
Math. Appl. 59, 3308–3314, 2010.[4] N. Çağman and S. Enginoğlu, Fuzzy soft matrix theory and its applications in decision
making, Iran J. Fuzzy Syst. 9 (1), 109–119, 2012.[5] J. Maiers and Y.S. Sherif, Applications of fuzzy set theory, IEEE Trans. Syst. Man
Cybern. 15 (1), 175–189, 1985.[6] D. Molodstov, Soft set theory first-results, Comput. Math. Appl. 37 (4-5), 190–31,
1999.[7] S. Mondal and M. Pal, Soft matrices, Afr. J. Math. Comput. Sci. Res. 4 (13), 379–388,
2011.[8] P.K. Maji, R. Biswas, and A.R. Roy, Fuzzy soft sets, J. Fuzzy Math. 9 (3), 589–502,
2001.[9] S. Miodrag and L.D. Petković, Complex interval arithmetic and its applications, John
Wiley & Sons, 1998.[10] P. Rajarajeswari and P. Dhanalakshmi, An application of interval valued intuition-
istic fuzzy soft matrix theory in medical diagnosis, to appear in Ann. Fuzzy Math.Inform.
[11] A.K. Shyamal and M. Pal, Interval-valued fuzzy matrices, J. Fuzzy Math. 14 (3),583–604, 2006.
[12] Y. Yang and J. Chenli, Fuzzy soft matrices and their applications, Lect. Notes Com-put. Sc. (Part I) 7002, 618–627, 2011.
[13] D. Ramot, R. Milo, M. Friedman, and A. Kandel, Complex fuzzy sets, IEEE Trans.Fuzzy Syst. 10 (2), 171–186, 2012.
[14] L.A. Zadeh, Fuzzy sets, Inform. Control 8, 338–353, 1965.[15] G. Zhang, T.S. Dillon, K.-Y. Cai, J. Ma, and J. Lu, Operation properties and δ-
equalities of complex fuzzy sets. Int. J. Approx. Reason. 50, 1227–1249, 2009.
Hacettepe Journal ofMathematics & Statistics
Hacet. J. Math. Stat.Volume 49 (2) (2020), 684 – 694
DOI : 10.15672/hujms.473495
Research Article
New analogues of the Filbert and Lilbert matricesvia products of two k-tuples asymmetric entries
Emrah Kılıç1, Neşe Ömür2, Sibel Koparal∗2
1TOBB University of Economics and Technology, Department of Mathematics, 06560 Ankara, Turkey2Kocaeli University, Department of Mathematics, 41380 Kocaeli, Turkey
AbstractIn this paper, we present new analogues of the Filbert and Lilbert matrices via productsof two k-tuples asymmetric entries consist of the Fibonacci and Lucas numbers. We shallderive explicit formulæ for their LU -decompositions and inverses. To prove the claimedresults, we write all the identities to be proven in q-word and then use the celebratedZeilberger algorithm to prove required q-identities.
Mathematics Subject Classification (2010). 15A23, 05A30, 11B39
Keywords. generalized Filbert matrix, q-analogues, LU-decomposition, Zeilberger’salgorithm, computer algebra system (CAS)
1. IntroductionLet Un and Vn be the generalized Fibonacci and Lucas sequences, respectively,
whose the Binet forms are
Un = αn − βn
α − β= αn−1 1 − qn
1 − qand Vn = αn + βn = αn (1 + qn) ,
where q = β/α = −α−2, so that α = i/√q. When α = 1+
√5
2 (or equivalently q =(1 −
√5)/(1 +
√5)), the sequences Un, Vn are reduced to the Fibonacci sequence Fn
and the Lucas sequence Ln.Throughout this paper we shall use the q-Pochhammer symbol
(x; q)n = (1 − x) (1 − xq) ...(1 − xqn−1
).
In the literature, there are many combinatorial matrices constructed by terms of spe-cial integer sequences or their functional analogues. For example, they are constructed viathe binomial coefficients, the Gaussian q-binomial coefficients or the well-known integersequences such as natural numbers, the Fibonacci and Lucas numbers. For these combi-natorial matrices and their properties, we refer to the works [2–6, 14–17, 21, 22]. Now werecall some well-known combinatorial matrices from the current literature:∗Corresponding Author.Email addresses: [email protected] (E. Kılıç), [email protected] (N. Ömür),
[email protected] (S. Koparal)Received: 22.10.2018; Accepted: 12.03.2019
New analogues of the Filbert and Lilbert matrices 685
• Chu and Di Claudio [5] studied the matrix[
(a)j+λi(c)j+λi
]0≤i,j≤n
, where a and c are
complex numbers, λini=0 are integers and (x)n is the shifted factorial of order n.
They also presented some variants of the above matrix.• For nonnegative integer g, Zhou and Zhaolin [21] studied the g-circulant matrices
whose elements consist of the Fibonacci and Lucas numbers, separately.• Hilbert matrix H = [hij ] is defined with entries
hij = 1i + j − 1
.
• As an analogue of the well known Hilbert matrix, Richardson [20] defined andstudied the Filbert matrix Hn =
[hij
]ni,j=1
with entries
hij = 1Fi+j−1
,
where Fn is the nth Fibonacci number.• The Filbert matrix has been extended by Berg [1] and Ismail [7].• Also several generalizations and analogues of it have been investigated and studied.
For example, Kılıç and Prodinger [8] gave a generalization of the Filbert matrixdenoted by F with entries
fij = 1Fi+j+r
,
where r ≥ −1 is an integer parameter.• Kılıç and Prodinger [10] introduced two new variations of the Filbert matrix F
denoted by G and L with entries, respectively
gij =Fλ(i+j)+r
Fλ(i+j)+sand lij =
Lλ(i+j)+r
Lλ(i+j)+s,
where s, r, and λ are integer parameters such that s = r, s ≥ −1, and λ ≥ 1. Thiswas the first nontrivial example where the numerator of the entries is not equal tozero.
• Kılıç and Prodinger [9] gave a further generalization of the generalized Filbertmatrix F by defining the matrix Q with entries
Qij = 1Fi+j+rFi+j+r+1...Fi+j+r+k−1
,
where r ≥ −1 is an integer parameter and k ≥ 0 is an integer parameter.• Some authors generalized and extended the concept in a series of papers [8–13,19]
to matrices with entries1
Fλ(i+j)+rFλ(i+j+1)+r...Fλ(i+j+k−1)+r
and1
Lλ(i+j)+rLλ(i+j+1)+r...Lλ(i+j+k−1)+r,
where r ≥ −1 and λ, k ≥ 1 are integer parameters.• Kılıç and Prodinger [11] went one step further, by allowing an asymmetric growth
of indices. They, however, confined themselves to k = 1; for this instance, theinverse matrix also enjoys nice closed form entries, which is no longer true for
686 E. Kılıç, N. Ömür, S. Koparal
k ≥ 2. To be more specific, they introduced four generalizations of the Filbertmatrix F, and defined the matrices T, M, H, and Z with entries by
tij = 1Fλi+µj+r
, mij = Fλi+µj+r
Fλi+µj+s, hij = 1
Lλi+µj+rand zij = Lλi+µj+r
Lλi+µj+s,
respectively, where s, r, λ, and µ are integer parameters such that s = r, s, r ≥ −1,and λ, µ ≥ 1. To prove their results, the authors could not use the q-Zeilbergeralgorithm because the summand they needed, are not q-hypergeometric. So theyused the backward induction to prove their claims.
In these summarized works, the authors derived explicit formulæ for the LU -decompositionsfor the matrices mentioned above. Also they derived explicit formulæ for their inverses.
In this paper, inspiring by the all works mentioned above, we shall present new analoguesof the Filbert and Lilbert matrices. We study two matrices A and B with entries
Aij = 1(Fi+j+r+1Fi+j+r+2...Fi+j+r+k) · (Li−j+s+1Li−j+s+2...Li−j+s+k)
andBij = 1
(Li+j+r+1Li+j+r+2...Li+j+r+k) · (Li−j+s+1Li−j+s+2...Li−j+s+k),
where r, s, λ, and µ are integer parameters such that s = r, r, s ≥ −1, and λ, µ ≥ 1. Weshall derive explicit formulæ for the LU -decompositions and their inverses.
Thus we denote q-forms of the matrices A and B by A and B, respectively. In thatcase, they are
An,d = (1 − q)k (−1)k(n+r+s+1)+1 ik(k+s+r)q12 k(k+r+s)+nk
× 1k∏
t=1[(1 − qn+d+r+t) (1 + qn−d+s+k)]
and
Bn,d = (−1)k(n+r+s)+1 ik(k+r+s+1)q12 k(r+s+k+1)+kn
× 1k∏
t=1[(1 + qn+d+r+t) (1 + qn−d+s+k)]
,
respectively. After this, we will present all our results for the matrices A and B becauseall our identities hold for a general q. Thus one can obtain the results related with thematrices A and B by taking q =
(1 −
√5)
/(1 +
√5)
.
We briefly clarify what will be done throughout this paper. We will derive explicitformulæ for LU -decompositions and inverses of the matrices A and B. Here we only provesome of the claimed results rather than all of them. We will use the celebrated q-Zeilbergeralgorithm [19] to prove the claimed results. In detail, all the results related with the matrixA will be listed in Section 2 without proof. All the results related with the matrix B willbe listed in Section 3 without proof. We also give Fibonacci-Lucas corollaries of our resultsafter each result by choosing a special value of q, q =
(1 −
√5)
/(1 +
√5)
in Sections 2and 3. Then, in Section 4.1, we will give the proofs related with the matrix A. For thematrix A, we prove the claim about the LU -decompositions of the matrix A as well as theclaim about the matrix L and its inverse matrix L−1. In Section 4.2, for the matrix B, weprove the claim about the LU -decompositions of the matrix B as well as the claim aboutthe matrix U and its inverse matrix U−1. While proving the claimed identities mentionedjust above, we use the q-Zeilberger algorithm.
New analogues of the Filbert and Lilbert matrices 687
2. The main resultsIn this section, we will present all our results related to matrix A. We start with giving
the matrices L and U yielding from the LU -decomposition of the matrix A of the formA = L.U in the following next two theorems.
Theorem 2.1. For 1 ≤ d ≤ n,
Ln,d = (−q)k(n−d) (q; q)n−1(q; q)n−d (q; q)d−1
×
(qd+r+2; q
)d+k−1
(−qs+1; q
)d+k−1
(−q2d+k+s+r+1; q
)n−d
(qn+r+2; q)d+k−1 (−qn−d+s+1; q)d+k−1 (−qd+k+s+r+2; q)n−d
.
Fibonacci-Lucas corollary for k = 3, r = 4, and s = 2 :
Corollary 2.2. For 1 ≤ d ≤ n,
Ln,d =(−1)n−d
(n−1∏t=1
Ft
)(d+2∏t=1
Ft+d+5
)(d+2∏t=1
Lt+2
)(n−d∏t=1
Lt+2d+9
)(
n−d∏t=1
Ft
)(d−1∏t=1
Ft
)(d+2∏t=1
Ft+n+5
)(d+2∏t=1
Lt+n−d+2
)(n−d∏t=1
Lt+d+10
) .
Theorem 2.3. For 1 ≤ d ≤ n,
Ud,n = (−1)k(d−1) ik(k−r−s)
× qd+k−1−n(d−1)+ k(k+r+s)2 (1 − q)k (q; q)n−1
(q; q)n−d
×
(qk; q
)d−1
(−qd+k+r+s+2; q
)d−1
(−qn+r−s+1; q
)d−1
(−q−d−k−s+1; q)d−1 (qd+k+r+1; q)d−1 (qn+r+2; q)d+k−1 (−qs−n+2; q)d+k−1.
Fibonacci-Lucas corollary for k = 3, r = 4, and s = 2 :
Corollary 2.4. For 1 ≤ d ≤ n,
Ud,n = (−1)−n(d−1)+ 32 53/2
×
(d−1∏t=1
Ft+2
)(n−1∏t=1
Ft
)(d−1∏t=1
Lt+d+10
)(d−1∏t=1
Lt+n+2
)(
n−d∏t=1
Ft
)(d−1∏t=1
Lt−d−5
)(d−1∏t=1
Ft+d+7
)(d+2∏t=1
Ft+n+5
)(d+2∏t=1
Lt−n+3
) .
Now we shall formulate determinant of the matrix A in the following theorem.
Theorem 2.5. For 1 ≤ n, d ≤ N,
detAN = (−1)kN ik(k−r−s)N q(k−1+n+ k(k+r+s)2 )N
× (1 − q)kNN∏
d=1(−1)kd qd(1−n)
×
(−qd+k+r+s+2; q
)d−1
(qk; q
)d−1
(q; q)d−1
(−qd+r−s+1; q
)d−1
(−q−d−k−s+1; q)d−1 (qd+k+r+1; q)d−1 (qd+r+2; q)d+k−1 (−qs−d+2; q)d+k−1.
Now we present the inverse matrices L−1 and U−1 in the following next two theorems.
688 E. Kılıç, N. Ömür, S. Koparal
Theorem 2.6. For 1 ≤ d ≤ n,
L−1n,d = (−1)(k−1)(d−n) q(n−d+1
2 )+(k−1)(n−d)
×(1 + q2d+k+r+s+1
) (q; q)n−1(q; q)n−d (q; q)d−1
×
(qd+r+2; q
)n−d
(−qd−n+s+2; q
)n−d
(−qn+k+s+r+2; q
)n−1
(qn+d+k+r; q)n−d (−qd+k+s; q)n−d (−qd+k+s+r+2; q)n
.
Fibonacci-Lucas corollary for k = 3, r = 4, and s = 2 :
Corollary 2.7. For 1 ≤ d ≤ n,
L−1n,d = (−1)(
d2)+(n+1
2 )−dn L2(d+5)
×
(n−1∏t=1
Ft
)(n−d∏t=1
Ft+d+5
)(n−d∏t=1
Lt+d−n+3
)(n−1∏t=1
Lt+n+10
)(
n−d∏t=1
Ft
)(d−1∏t=1
Ft
)(n−d∏t=1
Ft+n+d+6
)(n−d∏t=1
Lt+d+4
)(n∏
t=1Lt+d+10
) .
Theorem 2.8. For 1 ≤ d ≤ n,
U−1d,n = (−1)d+n(k−1) ik(k+r+s)q(d−1
2 )+(n2)+s−k k+r+s
2
(1 + q2d+r−s
)(1 − q)k
×
(qd+r+2; q
)n+k−2
(−qs+2−d; q
)n+k−2
(qn+k+r+1; q
)n
(−q−n−k−s+1; q
)n
(qk; q)n−1 (q; q)n−d (q; q)d−1 (−qd+r−s+1; q)n (−qn+k+r+s+2; q)n−1.
Fibonacci-Lucas corollary for k = 3, r = 4, and s = 2 :
Corollary 2.9. For 1 ≤ d ≤ n,
U−1d,n = (−1)(
d2)+(n+2
2 ) L2(d+1)5−3/2
×
(n+1∏t=1
Ft+d+5
)(n+1∏t=1
Lt−d+3
)(n∏
t=1Ft+n+7
)(n∏
t=1Lt−n−5
)(
n−1∏t=1
Ft+2
)(n−d∏t=1
Ft
)(d−1∏t=1
Ft
)(n∏
t=1Lt+d+2
)(n−1∏t=1
Lt+n+10
) .
Now let us consider the inverse matrix again. Since A−1 = U−1L−1 and by the definitionsof the matrices U−1 and L−1, we have the following result without proof.
Theorem 2.10. For 1 ≤ i, j ≤ n, we have((An)−1
)i,j
= ik(k+r+s) (−1)jk−j+i
× q(j+12 )− 1
2 k(r+s+k)+s−jk− 12 (3i−1)
(1 + q2j+k+r+s+1
) (1 + q2i+r−s
)(1 − q)k (q; q)i−1 (q; q)j−1
×∑
maxi,j≤t≤n
qt(t−j+k−1)
(qi+r+2; q
)t+k−2
(−qs+2−i; q
)t+k−2
(qt+k+r+1; q
)t
(qk; q)t−1 (−qi+r−s+1; q)t (−qt+k+r+s+2; q)t−1
×(q; q)t−1
(−q−t−k−s+1; q
)t
(qj+r+2; q
)t−j
(−qj−t+s+2; q
)t−j
(−qt+k+s+r+2; q
)t−1
(q; q)t−i (q; q)t−j (qt+j+k+r; q)t−j (−qj+k+s; q)t−j (−qj+k+s+r+2; q)t
.
New analogues of the Filbert and Lilbert matrices 689
The final formula given in the last theorem follows from some straightforward simplifica-tions. Unfortunately, the sum cannot be evaluated in closed form as we saw.
3. The matrix B
Now we present all our results related to the matrix B. For convenience, we use thesame letters L and U, but with a different meaning. We present the matrices L and Uyielding from the LU -decomposition B = L.U in the following next two theorems:
Theorem 3.1. For 1 ≤ d ≤ n,
Ln,d = (−q)k(n−d)
(qn−d+1; q
)d−1
(q; q)d−1
×
(−qd+r+2; q
)d+k−1
(−qs+1; q
)d+k−1
(qn+k+r+s+2; q
)d−1
(−qn+r+2; q)d+k−1 (−qn−d+s+1; q)d+k−1 (qd+k+r+s+2; q)d−1.
Fibonacci-Lucas corollary for k = 3, r = 4, and s = 2 :
Corollary 3.2. For 1 ≤ d ≤ n,
Ln,d = (−1)n−d
×
(d−1∏t=1
Ft+n−d
)(d+2∏t=1
Lt+s
)(d+2∏t=1
L−t+d+2
)(d−1∏t=1
Ft+n+10
)(
d−1∏t=1
Ft
)(d+2∏t=1
Lt+n+5
)(d+2∏t=1
Lt+n−d+2
)(d−1∏t=1
Ft+d+10
) .
Theorem 3.3. For 1 ≤ d ≤ n,
Ud,n = (−1)k(d−1) ik(k−r−s−1)q(n−1)(1−d)+k( r+s+22 )+(k+1
2 )
×(qn+r−s+1; q
)d−1
(−qn+r+2; q)d+k−1 (−q−n+s+2; q)d+k−1
×
(qk; q
)d−1
(q; q)n−1
(qd+k+r+s+2; q
)d−1
(q; q)n−d (−qd+k+r+1; q)d−1 (−q−d−k−s+1; q)d−1.
Fibonacci-Lucas corollary for k = 3, r = 4, and s = 2 :
Corollary 3.4. For 1 ≤ d ≤ n,
Ud,n = (−1)n(1−d)
×
(d−1∏t=1
Ft+2
)(n−1∏t=1
Ft
)(d−1∏t=1
Ft+n+2
)(d−1∏t=1
Ft+d+10
)(
n−d∏t=1
Ft
)(d+2∏t=1
Lt+n+5
)(d−1∏t=1
Lt+d+7
)(d+2∏t=1
Lt−n+3
)(d−1∏t=1
Lt−d−5
) .
We can give determinant of the matrix B since it is simply evaluated as product of themain diagonal elements of the matrix U yielding from the LU -decomposition B = L.U inthe following theorem.
690 E. Kılıç, N. Ömür, S. Koparal
Theorem 3.5. For 1 ≤ n, d ≤ N,
detBN
= (−1)kN ik(k−r−s−1)N q(
n−1+k( r+s+22 )+(k+1
2 ))
NN∏
d=1(−1)kd q−d(n−1)
×
(qk; q
)d−1
(q; q)d−1
(qd+r−s+1; q
)d−1
(qd+k+r+s+2; q
)d−1
(−qd+k+r+1; q)d−1 (−q−d−k−s+1; q)d−1 (−qd+r+2; q)d+k−1 (−q−d+s+2; q)d+k−1.
Now we present the inverse matrices L−1 and U−1 in the following next two theorems.
Theorem 3.6. For 1 ≤ d ≤ n,
L−1n,d = (−1)(k+1)(n−d) q(n−d+1
2 )+(k−1)(n−d)
×(1 − q2d+r+s+k+1
) (q; q)n−1(q; q)n−d (q; q)d−1
×
(−qd+r+2; q
)n−d
(−qd−n+s+2; q
)n−d
(qn+k+r+s+2; q
)n−1
(−qn+d+k+r; q)n−d (−qd+k+s; q)n−d (qd+k+r+s+2; q)n
.
Fibonacci-Lucas corollary for k = 3, r = 4, and s = 2 :
Corollary 3.7. For 1 ≤ d ≤ n,
L−1n,d = (−1)(
d2)+(n+1
2 )−dn−d+1 L2(d+5)
×
(n−1∏t=1
Ft
)(n−d∏t=1
Lt+d+5
)(n−d∏t=1
Lt+d−n+3
)(n−1∏t=1
Ft+n+10
)(
n−d∏t=1
Ft
)(d−1∏t=1
Ft
)(n−d∏t=1
Lt+n+d+6
)(n−d∏t=1
Lt+d+4
)(n∏
t=1Ft+d+10
) .
Theorem 3.8. For 1 ≤ d ≤ n,
U−1d,n = (−1)d+n(k−1) ik(k+r+s+1)q(d−1
2 )+(n2)− k(k+r+s+1)
2 +s
×(1 − q2d+r−s
) 1(q; q)n−d (q; q)d−1
×
(−qd+r+2; q
)n+k−2
(−q−d+s+2; q
)n+k−2
(−qn+k+r+1; q
)n
(−q−n−k−s+1; q
)n
(qk; q)n−1 (qd+r−s+1; q)n (qn+k+r+s+2; q)n−1.
Fibonacci-Lucas corollary for k = 3, r = 4, and s = 2 :
Corollary 3.9. For 1 ≤ d ≤ n,
U−1d,n = (−1)(
d2)+(n
2)+1 F2(d+1)
×
(n+1∏t=1
Lt+d+5
)(n+1∏t=1
Lt−d+3
)(n∏
t=1Lt+n+7
)(n∏
t=1Lt−n−5
)(
n−1∏t=1
Ft+2
)(n−d∏t=1
Ft
)(d−1∏t=1
Ft
)(n∏
t=1Ft+d+2
)(n−1∏t=1
Ft+n+10
) .
Now we consider the inverse matrix. Since B−1 = U−1L−1 and by the definitions of thematrices U−1 and L−1, we have the following result without proof.
New analogues of the Filbert and Lilbert matrices 691
Theorem 3.10. For n > 0,((Bn)−1
)i,j
= (−1)−j−jk+i ik(k+r+s+1)
× q(j+12 )−jk+s− 1
2 k(r+s+k)− 12 (3i+k−1)
(1 − q2i+r−s
) (1 − q2j+r+s+k+1
)(q; q)i−1 (q; q)j−1
×∑
t
qt(t+k−j−1)
(−qi+r+2; q
)t+k−2
(−q−i+s+2; q
)t+k−2
(−qt+k+r+1; q
)t
(qk; q)t−1 (q; q)t−i (qi+r−s+1; q)t (qt+k+r+s+2; q)t−1
×(q; q)t−1
(−qj+r+2; q
)t−j
(−qj−t+s+2; q
)t−j
(qt+k+r+s+2; q
)t−1
(−q−t−k−s+1; q
)t
(q; q)t−j (−qt+j+k+r; q)t−j (−qj+k+s; q)t−j (qj+k+r+s+2; q)t
.
The final formula as given in the theorem just above follows from some straightforwardsimplifications. Unfortunately, the sum again cannot be evaluated in closed form as wesaw.
4. ProofsHere we shall only prove four claimed identities related with matrices A and B in the
following subsections separately. We use the q-Zeilberger algorithm for all the identitieswill be proven, our experiments indicate that they are Gosper-summable. The entries inour examples, qualify for the q-Zeilberger algorithm that we used in our earlier papers.Nowadays, such identities are a routine verification using the q-Zeilberger algorithm, asdescribed in the book [18].
4.1. Proofs related with the matrix A
We shall present the proofs related to the matrix A. For LU -decomposition of A, wehave to prove that ∑
1≤t≤mind,nLd,tUt,n = Ad,n.
Thus, consider∑1≤t≤mind,n
Ld,tUt,n
= (−1)k(d−1) ik(k−r−s)qk(d+1)+n+ k(k+r+s)2 −1 (1 − q)k (q; q)d−1 (q; q)n−1
×∑
1≤t≤mind,nqt(1−n−k) 1
(q; q)n−t (q; q)d−t
×
(qk; q
)t−1
(−q2t+k+s+r+1; q
)d−t
(−qt+k+r+s+2; q
)t−1
(−qn+r−s+1; q
)t−1
(q; q)t−1 (−qt+k+s+r+2; q)d−t (−q−t−k−s+1; q)t−1 (qt+k+r+1; q)t−1
×(qt+r+2; q
)t+k−1
(−qs+1; q
)t+k−1
(qd+r+2; q)t+k−1 (−qd−t+s+1; q)t+k−1 (qn+r+2; q)t+k−1 (−qs−n+2; q)t+k−1.
Denote the last sum in the above equation by SUMn. The Mathematica version of theq-Zeilberger algorithm produces the recursion
SUMn = (1 − qd+n+r)(1 + qd−n+k+s+1)q(1 − qn−1)(1 − qd+n+k+r)(1 + qd−n+s+1)
SUMn−1.
SinceSUM1 = qs
(q; q)d−1(qd+r+2; q)k(−qd+s; q)k,
692 E. Kılıç, N. Ömür, S. Koparal
we obtain
SUMn = 1qn−1−s(q; q)d−1(q2; q2)k(q; q)n−1
[n + d + k + r
n + d + r
]−1
q
[d − n + k + s
d − n + s
]−1
−q
,
as claimed.
Now we prove ∑d≤t≤n
Ln,tL−1t,d = δn,d,
where δn,d is the Kronecker delta. By lower triangular matrices L and L−1, we need tolook at the entries indexed by (n, d):
∑d≤t≤n
Ln,tL−1t,d = (−1)−d+dk+kn q(d+1
2 ) (1 + q2d+k+r+s+1) (q; q)n−1
(q; q)d−1
×∑
d≤t≤n
(−1)tq(t
2)−dt(−qs+1; q
)t+k−1
(−q2t+k+s+r+1; q
)n−t
(qn+r+2; q)t+k−1 (−qn−t+s+1; q)t+k−1 (−qt+k+s+r+2; q)n−t
×
(qt+r+2; q
)t+k−1
(qd+r+2; q
)t−d
(−qd−t+s+2; q
)t−d
(−qt+k+s+r+2; q
)t−1
(q; q)n−t (q; q)t−d (qt+d+k+r; q)t−d (−qd+k+s; q)t−d (−qd+k+s+r+2; q)t
.
For the sum in the last expression, that is,
∑d≤t≤n
(−1)tq(t
2)−dt(−qs+1; q
)t+k−1
(−q2t+k+s+r+1; q
)n−t
(qn+r+2; q)t+k−1 (−qn−t+s+1; q)t+k−1 (−qt+k+s+r+2; q)n−t
×
(qt+r+2; q
)t+k−1
(qd+r+2; q
)t−d
(−qd−t+s+2; q
)t−d
(−qt+k+s+r+2; q
)t−1
(q; q)n−t (q; q)t−d (qt+d+k+r; q)t−d (−qd+k+s; q)t−d (−qd+k+s+r+2; q)t
.
the q-Zeilberger algorithm evaluates it as 0 provided that i = j. If i = j, it obvious thatthe sum is equal to 1. Thus ∑
d≤t≤n
Ln,tL−1t,d = δn,d,
as claimed.
Similarly, using the q-Zeilberger algorithm, one could prove the result∑d≤t≤n
Ud,tU−1t,n = δd,n.
4.2. Proofs related with the matrix B
In this part we shall give the proofs related to the matrix B. For LU -decomposition ofB, we have to prove that ∑
1≤t≤mind,nLd,tUt,n = Bd,n.
New analogues of the Filbert and Lilbert matrices 693
Thus, consider∑1≤t≤mind,n
Ld,tUt,n
= (−1)k(d−1) ik(k−r−s−1)q(k+12 )+ 1
2 k(r+s+2n)+k+n−1
× (q; q)n−1∑
1≤t≤mind,nqt(1−k−n)
(qd−t+1; q
)t−1
(q; q)t−1 (q; q)n−t
×(−qt+r+2; q
)t+k−1
(−qs+1; q
)t+k−1
(−qd+r+2; q)t+k−1 (−qd−t+s+1; q)t+k−1 (qt+k+r+s+2; q)t−1
×
(qk; q
)t−1
(qn+r−s+1; q
)t−1
(qt+k+r+s+2; q
)t−1
(qd+k+r+s+2; q
)t−1
(−qt+k+r+1; q)t−1 (−qn+r+2; q)t+k−1 (−q−n+s+2; q)t+k−1 (−q−t−k−s+1; q)t−1.
Denote the last sum in the above equation by SUMn. The algorithm produces the recursion
SUMn = q−1(1 + qd+n+r)(1 + qd−n+k+s+1)(1 − qn−1)(1 + qd+n+k+r)(1 + qd−n+s+1)
SUMn−1.
SinceSUM1 = qs
(−qd+k+r+1; q−1)k(−qd+s; q)k,
we obtain
SUMn = qs−n+1
(−q; q)k(−q; q)k(q; q)n−1
[n + d + k + r
n + d + r
]−1
−q
[d − n + k + s
d − n + s
]−1
−q
,
as claimed.
We continue with proving ∑d≤t≤n
Ud,tU−1t,n = δn,d,
where δn,d is the Kronecker delta. By the lower triangular matrices U and U−1, we needto look at the entries indexed by (n, d):∑
d≤t≤n
Ud,tU−1t,n = (−1)−n+k(d+n) q(n
2)+d+k+s
×
(qk; q
)d−1
(−qn+k+r+1; q
)n
(−q−n−k−s+1; q
)n
(qd+k+r+s+2; q
)d−1
(qk; q)n−1 (−q−d−k−s+1; q)d−1 (qn+k+r+s+2; q)n−1 (−qd+k+r+1; q)d−1
×∑
d≤t≤n
(−1)t q(t2)−dt
(1 − q2t+r−s
)(q; q)t−d (q; q)n−t
×(qt+r−s+1; q
)d−1
(−qt+r+2; q
)n+k−2
(−q−t+s+2; q
)n+k−2
(−qt+r+2; q)d+k−1 (−q−t+s+2; q)d+k−1 (qt+r−s+1; q)n
.
For the sum in the last expression, that is,∑d≤t≤n
(−1)t q(t2)−dt
(1 − q2t+r−s
)(q; q)t−d (q; q)n−t
×(qt+r−s+1; q
)d−1
(−qt+r+2; q
)n+k−2
(−q−t+s+2; q
)n+k−2
(−qt+r+2; q)d+k−1 (−q−t+s+2; q)d+k−1 (qt+r−s+1; q)n
694 E. Kılıç, N. Ömür, S. Koparal
the q-Zeilberger algorithm evaluates it as 0 for i = j. If i = j, it is obvious that the sumis equal to 1. Thus ∑
d≤t≤n
Ud,tU−1t,n = δd,n,
as claimed. Similarly, using the q-Zeilberger algorithm, we have∑d≤t≤n
Ln,tL−1t,d = δn,d.
References[1] C. Berg, Fibonacci numbers and orthogonal polynomials, Arab. J. Math. Sci. 17, 75–
88, 2011.[2] L. Carlitz, Some determinants of q-binomial coefficients, J. Reine Angew. Math. 226,
216–220, 1967.[3] W. Chu, On the evaluation of some determinants with q-binomial coefficients, J.
Systems Sci. Math. Science 8 (4), 361–366, 1988.[4] W. Chu, Generalizations of the Cauchy determinant, Publ. Math. Debrecen 58 (3),
353–365, 2001.[5] W. Chu and L. Di Claudio, Binomial determinant evaluations, Ann. Comb. 9 (4),
363–377, 2005.[6] W. Chu, Finite differences and determinant identities, Linear Algebra Appl. 430,
215–228, 2009.[7] M.E.H. Ismail, One parameter generalizations of the Fibonacci and Lucas numbers,
The Fibonacci Quart. 46/47, 167–180, 2008/2009.[8] E. Kılıç and H. Prodinger, A generalized Filbert matrix, The Fibonacci Quart. 48,
29–33, 2010.[9] E. Kılıç and H. Prodinger, The q-Pilbert matrix, Int. J. Comput. Math. 89, 1370–
1377, 2012.[10] E. Kılıç and H. Prodinger, Variants of the Filbert matrix, The Fibonacci Quart. 51,
153–162, 2013.[11] E. Kılıç and H. Prodinger, Asymmetric generalizations of the Filbert matrix and vari-
ants, Publ. Inst. Math. (Belgrad) (N.S) 95 (109), 267–280, 2014.[12] E. Kılıç and H. Prodinger, The generalized q-Pilbert matrix, Math. Slovaca 64, 1083–
1092, 2014.[13] E. Kılıç and H. Prodinger, The generalized Lilbert matrix, Periodica Math. Hungar.
73, 62–72, 2016.[14] G.Y. Lee, S.G. Lee, and H.G. Shin, On the k-generalized Fibonacci matrix Q∗
K , LinearAlgebra Appl. 251, 73–88, 1997.
[15] G.Y. Lee and S.H. Cho, The generalized Pascal matrix via the generalized Fibonaccimatrix and the generalized Pell matrix, J. Korean Math. Soc. 45 (2), 479–491, 2008.
[16] M. Merca, A note on the determinant of a Toeplitz-Hessenberg matrix, Spec. Matrices1, 10–16, 2013.
[17] A.M. Ostrowski, On some determinants with combinatorial numbers, J. Reine Angew.Math. 216, 25–30, 1964.
[18] M. Petkovsek, H. Wilf, and D. Zeilberger, A=B, A.K. Peters, Wellesley, MA, 1996.[19] H. Prodinger, A generalization of a Filbert matrix with 3 additional parameters, Trans.
Roy. Soc. South Afr. 65, 169–172, 2010.[20] T.M. Richardson, The Filbert matrix, The Fibonacci Quart. 39 (3), 268–275, 2001.[21] J. Zhou and J. Zhaolin, The spectral norms of g-circulant matrices with classical
Fibonacci and Lucas numbers entries, Appl. Math. Comput. 233, 582–587, 2014.[22] J. Zhou and J. Zhaolin, Spectral norms of circulant-type matrices with binomial coef-
ficients and Harmonic numbers, Int. J. Comput. Math. 11 (5), 1350076, 2014.
Hacettepe Journal ofMathematics & Statistics
Hacet. J. Math. Stat.Volume 49 (2) (2020), 695 – 707
DOI : 10.15672/hujms.467966
Research Article
Graphical calculus of Hopf crossed modules
Kadir Emir
Department of Mathematics and Statistics, Masaryk University, Brno, Czech Republic
AbstractWe give the graphical notion of crossed modules of Hopf algebras-will be called Hopfcrossed modules for short- in a symmetric monoidal category. We use the web proof assis-tant Globular to visualize our (colored) string diagrams. As an application, we introducethe homotopy of Hopf crossed module maps via Globular, and give some of its functorialrelations.
Mathematics Subject Classification (2010). 16T05, 16S40, 18D05, 18D10
Keywords. Globular, Hopf crossed module, symmetric monoidal category, homotopy
1. IntroductionA crossed module of groups is given by a group homomorphism ∂ : E → G together
with an action B of G on E satisfying the following relations for all e, f ∈ E, and g ∈ G:
∂(g B e) = g ∂(e) g−1 and ∂(e) B f = e f e−1 .
The essential example of a crossed module comes from a normal subgroup N E G withan inclusion map where the action is defined by conjugation. Thus crossed modules canbe considered as a generalization of normal subgroups. Crossed modules are introducedby Whitehead [24] as an algebraic model for homotopy 2-types. Another result is that,the category of crossed modules is also naturally equivalent to that of cat1-groups provenin [14]. This property leads to a groupoid structure as an example of strict 2-groups.Crossed modules are also appear in the context of simplicial homotopy theory, since theyare equivalent to the simplicial groups with Moore complex of length one. An essentialresult of this equivalence is that the homotopy category of n-types is equivalent to thehomotopy category of simplicial groups with Moore complex of length n − 1, which arealso the algebraic models for n-types.
A braided monoidal category [13] (or braided tensor category) is a monoidal category[21] with an isomorphism τx,y : A ⊗ B → B ⊗ A called "braiding" satisfying the certainhexagon diagrams. Graphically this braiding will be denoted by:
(1.1)
Email address: [email protected]: 06.10.2018; Accepted: 12.03.2019
696 K. Emir
where we have made the convention that the diagrams are to be read from top to bottom.Furthermore, a braided monoidal category will be called a symmetric monoidal categoryif the braiding (1.1) has the following property:
=
The well-known examples of symmetric monoidal categories are Vect, Set, and Cat.
A Hopf algebra [22] can be considered as an abstraction of the group algebra (of agroup) and the universal enveloping algebra (of a Lie algebra). There exists a wide varietyof variations of Hopf algebras by relaxing its properties or adding some extra structures, forinstance, quasi-Hopf algebras [2, 6], quasi-triangular Hopf algebras [17], quantum groups[17, 18], the Leibniz-Hopf algebras and its dual [3, 4, 12], the Steenrod algebras [5, 19],Hopfish algebras [23], etc. Most of these notions make sense via Tannaka duality by theproperties and structures on the corresponding categories. Hopf algebras are first definedover vector spaces. However, as a consequence of the relation between the category ofvector spaces and monoidal categories, any Hopf algebra can also be defined over bothsymmetric monoidal categories and (arbitrary) braided monoidal categories; see [16] fordetails. The notion of Hopf crossed modules is introduced by Majid in [15] which is givenby a Hopf algebra morphism ∂ : I → H where I is an H-module algebra and coalgebrasatisfying two Peiffer relations with an additional compatibility law that is less restrictedcase than being cocommutative. For any Hopf algebra H, an element x ∈ H is said to be
• Group-like, if ∆(x) = x ⊗ x,• Primitive, if ∆(x) = x ⊗ 1 + 1 ⊗ x.
Therefore, we have the functors
( )∗gl : Hopf Algebras → Groups
Prim: Hopf Algebras → Lie Algebraswhich preserve the crossed module structure [11] and can be extended to the correspondingfunctors
( )∗gl : Hopf Crossed Modules → Group Crossed Modules
Prim: Hopf Crossed Modules → Lie Algebra Crossed Modules.
Globular [1] is an online proof assistant for finitely-presented semistrict globular highercategories which currently operates up to the level of 4-categories. It allows one to for-malize higher-categorical proofs and visualize them as string diagrams, and share them inpublic.
In this paper, we present the graphical calculus of Hopf crossed modules over anysymmetric monoidal category, by using (colored) string diagrams via Globular. As anapplication, we introduce the homotopy of Hopf crossed module morphisms and prove thatthe functors ( )∗
gl and Prim preserve the homotopy, from which we get the correspondinggroupoid functors.
2. Hopf algebraic conventionsWe recall some notions from [15, 16] in a graphical point of view. From now on, C will
be a fixed symmetric monoidal category and all Hopf algebras will be defined over it. Weuse the convention appears in [8].
Graphical calculus of Hopf crossed modules 697
Definition 2.1. We picture any Hopf algebra by and denote its product (·), co-
product (∆), unit (1), counit (ϵ), and antipode (S) by
respectively, such that the Hopf algebra axioms [16] are satisfied. Any Hopf algebra is saidto be cocommutative if and only if
=
Moreover, if a Hopf algebra is cocommutative (or commutative), then we have
=
2.1. Hopf algebra actionSuppose that I is a bialgebra (not necessarily a Hopf algebra) and H is a Hopf algebra.
We say that I is an H-module algebra if there exists a left action ρ : H ⊗ I → I of H on I
such that the following conditions hold:•
= and =
•
=
698 K. Emir
•
=
Moreover ρ makes I an H-module coalgebra if it further satisfies
•
= ,
•
= .
In the above context, H-module algebra means a monoid in the category of H-modules.This requires the multiplication I ⊗ I → I to be an H-module morphism which yields thecorresponding conditions above. Similarly, an H-module coalgebra means a comonoid inthe same category, equivalently the action ρ : H ⊗I → I needs to be a coalgebra morphism.
Definition 2.2 (Adjoint Action). If H is a cocommutative Hopf algebra, then H itselfhas a natural H-module algebra and coalgebra structure which is given by the "adjoint
Graphical calculus of Hopf crossed modules 699
action"
= (2.1)
Remark 2.3. In the cocommutative case, we have:
=
which is proven in [7]. However, it is not true in a general context because of the followingequality:
=
700 K. Emir
Definition 2.4 (Smash Product). If I is an H-module algebra and coalgebra with actionρ : H ⊗ I → I satisying the following compatibility condition:
= (2.2)
then we have the smash product Hopf algebra I ⊗ρ H with the underlying tensor productI ⊗ H, where
• Product:
=
• Coproduct:
=
• Antipode:
=
Remark that, we need the compatibility condition to make the coproduct an algebramorphism.
Graphical calculus of Hopf crossed modules 701
3. Hopf Crossed ModulesThe non-diagrammatic version of the following notions appear in [11,15].
Definition 3.1. A Hopf crossed module is a Hopf algebra morphism ∂ : I → H:
where I is an H-module algebra and coalgebra such that the followings hold:• Compatibility condition:
= ,
• Pre-crossed module condition:
= ,
• Peiffer identity (crossed module condition):
= .
Let ∂ : I → H and ∂′ : I ′ → H ′ be two Hopf crossed modules
and
respectively. A crossed module morphism is a pair
f1 = , f0 =
702 K. Emir
of Hopf algebra morphisms f1 : I → I ′ and f0 : H → H ′ such that•
= (3.1)
•
= (3.2)
Therefore we have the category of Hopf crossed modules.
3.1. Strict Quantum 2-GroupsA strict quantum 2-group is a pair of Hopf algebras (H1, H0) with the Hopf algebra
morphisms s, t : H1 → H0 and e : H0 → H1 respectively,
, ,
which satisfies
= =
together with an associative product which forms an embedded quantum groupoid; see[15] for details.
Remark 3.2. For a given Hopf crossed module ∂ : I → H, there exists a strict quantum2-group (H1, H) where H1 is the smash product. However the converse is not true; namelywe cannot get a Hopf crossed module structure via a strict quantum 2-group. At this pointthe braided crossed module notion appears, see [15,20] for more details.
Graphical calculus of Hopf crossed modules 703
4. Globular Application: Homotopy TheoryFrom now on, all Hopf algebras will be colored black. Moreover, we just consider
the category of cocommutative Hopf algebras over a symmetric monoidal category. Thisrestriction will lead us to generalize the homotopy theory of crossed modules of groupsand of Lie algebras in the sense of the functors:
( )∗gl : Hopf Algebras → Groups , Prim: Hopf Algebras → Lie Algebras.
4.1. Derivation and HomotopyLet A .=
(∂ : I → H
)and A′ .=
(∂′ : I ′ → H ′) be two arbitrary but fixed cocommutative
Hopf crossed modules. Both of them denoted are by:
.
Definition 4.1. Let f0 : H → H ′ be a cocommutative Hopf algebra morphism:
An f0-derivation is a coalgebra morphism Γ: I → H ′ denoted by such that
= (4.1)
where denotes the action in the corresponding Hopf crossed module.
704 K. Emir
Lemma 4.2. If Γ is an f0-derivation, then
= , = (4.2)
and, recalling the adjoint action (2.1), we have
= (4.3)
Graphical calculus of Hopf crossed modules 705
Proof. Follows from (4.1) by graphical calculation.
Definition 4.3. Let A.=
(∂ : I → H
)and A′ .=
(∂′ : I ′ → H ′) be two arbitrary but fixed
cocommutative Hopf crossed modules, and denote both of them by:
Suppose that we have Hopf crossed module morphism f = (f1, f0) : A → A′ with
f1 = , f0 =
Define g = (g1, g0) by the components
g1 = , g0 =
where is the f0-derivation Γ: I → H ′.
Theorem 4.4. g0 and g1 define Hopf algebra morphisms.
Proof. It is clear that g0, g1 are coalgebra morphisms since Γ is a coalgebra morphism.Also they are algebra morphisms [7]. Therefore they are bialgebra morphisms; and fur-thermore Hopf algebra morphisms from (4.2).
Theorem 4.5. g = (g1, g0) : A → A′ is a Hopf crossed module morphism.
Proof. It is already proved in [7] that the condition (3.1) holds. Also by using (4.3), theaction is preserved, namely the condition (3.2) is satisfied.
Definition 4.6 (Homotopy). In the condition of the previous theorem, we write f(f0,Γ)−−−→ g
or shortly f ≃ g, and say that (f0, Γ) is a "homotopy (or derivation)" connecting f to g.
As a consequence of this homotopy definition, we can give the following:
Let A,A′ be Hopf crossed modules. If there exist Hopf crossed module morphismsf : A → A′ and g : A′ → A such that f g ≃ idA′ and g f ≃ idA; we say that the Hopfcrossed modules A and A′ are "homotopy equivalent", which is denoted by A ≃ A′.
706 K. Emir
4.2. A groupoid structureLet f = (f1, f0) : A → A′ be a Hopf crossed module morphism. By letting Γ be the
zero morphism, we get a derivation which connects f to itself. Also, the antipode ofa derivation which connects f to g is again a derivation connecting g to f . Moreover,let f, g, k be Hopf crossed module morphisms between A → A′, where Γ is a derivationconnecting f to g, and Γ′ is a derivation connecting g to k. Then the map ·(Γ ⊗ Γ′)∆defines a derivation which connects f to k. These yield the following theorem.
Theorem 4.7. Let A,A′ be two arbitrary but fixed cocommutative Hopf crossed modules.We have a groupoid HOMHA whose objects are the Hopf crossed module morphisms betweenA → A′, the morphisms being their homotopies.
5. Review by Sweedler’s notationIn Sweedler’s [22] notation, we denote the coproduct of a Hopf algebra by:
∆(x) =∑(x)
x′ ⊗ x′′.
We can easily obtain some applications and categorical properties of the previous notionsby using this notation. Hence the derivation (4.1) can also be expressed by the formula
Γ(ab) =∑(b)
(S(f0(b′) Bρ Γ(a)
)Γ(b′′), (5.1)
that connects Hopf crossed module morphism f = (f0, g0) to g = (f0, g0) where:
g0(a) =∑(a)
f0(a′) (∂′ Γ)(a′′), g1(x) =∑(x)
f1(x′) (Γ ∂)(x′′)
are diagrammatically given in Definition 4.3.
Theorem 5.1. The functors ( )∗gl and Prim preserve the homotopy and homotopy equiv-
alence.
Proof. The functor ( )∗gl yields S(x) = x−1 and also ϵ(x) = e. Therefore the formula (5.1)
will be turned intoΓ(ab) =
((f0(b))−1 B Γ(a)
)Γ(b),
which is given in [10] for the case of groups.On the other hand, by using the properties of the functor Prim, the formula (5.1) will
be turned intoΓ
([a, b]
)= f0(a) B Γ(b) + f0(b) B Γ(a) + [Γ(a), Γ(b)],
which is given in [9] for the case of Lie algebras.
Recall Theorem 4.7; and the groupoid structures in [9, 10] for the case of groups andLie algebras. As a result of the previous theorem we have the following.
Corollary 5.2. The functors ( )∗gl and Prim can be seen as groupoid functors such as:
( )∗gl : HOMHA → HOMGrp and Prim: HOMHA → HOMLie.
Acknowledgment. The author is thankful to Jamie Vicary for the Globular lecturesduring AARMS 2016 Summer School in Halifax, Canada. The author supported by theprojects MUNI/A/1186/2018 from Masaryk University, and also BAP/2018/2244 fromEskişehir Osmangazi University.
Graphical calculus of Hopf crossed modules 707
References[1] K. Bar, A. Kissinger, and J. Vicary, Globular: an online proof assistant for higher-
dimensional rewriting, Log. Methods Comput. Sci. 14 (1), 2018.[2] D. Bulacu, S. Caenepeel, F. Panaite, and F. Van Oystaeyen, Quasi-Hopf algebras. A
categorical approach, Cambridge University Press, 2019.[3] M. Crossley and N. Turgay, Conjugation invariants in the Leibniz-Hopf algebra, J.
Pure Appl. Algebra, 217 (12), 2247–2254, 2013.[4] M. Crossley and N. Turgay, Conjugation invariants in the mod 2 dual Leibniz-Hopf
algebra, Comm. Algebra 41 (9), 3261–3266, 2013.[5] M. Crossley and S. Whitehouse, On conjugation invariants in the dual Steenrod
algebra, Proc. Amer. Math. Soc. 128 (9), 2809–2818, 2000.[6] V. Drinfel’d, Quasi-Hopf algebras, Leningr. Math. J. 1 (6), 1419–1457, 1990.[7] K. Emir, Globular: Homotopy of Hopf crossed module maps, available at:
http://globular.science/1610.001v2.[8] K. Emir, Globular: Hopf crossed modules, available at:
http://globular.science/1611.002v1.[9] K. Emir and S. Çetin, From simplicial homotopy to crossed module homotopy in
modified categories of interest, Georgian Math. J. doi: 10.1515/gmj-2018-0069.[10] J. Faria Martins, The fundamental 2-crossed complex of a reduced CW-complex, Ho-
mology Homotopy Appl. 13 (2), 129–157, 2011.[11] J. Faria Martins, Crossed modules of Hopf algebras and of associative algebras and
two-dimensional holonomy, J. Geom. Phys. 99, 68–110, 2016.[12] M. Hazewinkel, The Leibniz-Hopf algebra and Lyndon words, CWI Report: AM-R
9612, January 1996.[13] A. Joyal and R. Street, Braided monoidal categories, Macquarie Math Reports 860081,
November 1986.[14] J. Loday, Spaces with finitely many non-trivial homotopy groups, J. Pure Appl. Al-
gebra 24 (2), 179–202, 1982.[15] S. Majid, Strict quantum 2-groups, arXiv:1208.6265.[16] S. Majid, Algebras and Hopf algebras in braided categories, in: Advances in Hopf
algebras, 55–105, New York: Marcel Dekker, 1994.[17] S. Majid, Foundations of quantum group theory, Cambridge University Press, 1995.[18] S. Majid, What is .. a quantum group?, Notices Amer. Math. Soc. 53 (1), 30–31,
2006.[19] J. Milnor, The Steenrod algebra and its dual, Ann. Math. (2) 67, 150–171, 1958.[20] D. Radford, The structure of hopf algebras with a projection, J. Algebra 92 (2), 322–
347, 1985.[21] P. Selinger, A survey of graphical languages for monoidal categories. in: New struc-
tures for physics, 289–355. Berlin: Springer, 2011.[22] M. Sweedler, Hopf algebras, Mathematics lecture note series. W. A. Benjamin, 1969.[23] X. Tang, A. Weinstein, and C. Zhu, Hopfish algebras, Pacific J. Math. 231 (1),
193–216, 2007.[24] J. Whitehead, On adding relations to homotopy groups, Ann. of Math. (2) 42, 409–
428, 1941.
Hacettepe Journal ofMathematics & Statistics
Hacet. J. Math. Stat.Volume 49 (2) (2020), 708 – 715
DOI : 10.15672/hujms.588726
Research Article
On ∗-differential identities in prime rings withinvolution
Shakir Ali∗1, Ali N.A. Koam2, Moin A. Ansari21Department of Mathematics, Faculty of Science, Aligarh Muslim University, Aligarh, India
2Department of Mathematics, Faculty of Science, Jazan University, Kingdom of Saudi Arabia
AbstractLet R be a ring. An additive map x 7→ x∗ of R into itself is called an involution if (i)(xy)∗ = y∗x∗ and (ii) (x∗)∗ = x hold for all x, y ∈ R. In this paper, we study the effect ofinvolution ” ∗ ” on prime rings that satisfying certain differential identities. The identitiesconsidered in this manuscript are new and interesting. As the applications, many knowntheorems can be either generalized or deduced. In particular, a classical theorem due toHerstein [A note on derivation II, Canad. Math. Bull., 1979] is deduced.
Mathematics Subject Classification (2010). 16N60, 16W10, 16W25
Keywords. prime ring, commutativity, involution, derivation, ∗-differential identities
1. Notations and introductionIn all that follows, unless specially stated, R always denotes an associative ring with
centre Z(R). As usual the symbols s t and [s, t] will denote the anti-commutator st + tsand commutator st − ts, respectively. Given an integer n ≥ 2, a ring R is said to ben-torsion free if nx = 0 (where x ∈ R) implies that x = 0. A ring R is called primeif aRb = (0) (where a, b ∈ R) implies a = 0 or b = 0, and is called semiprime ring ifaRa = (0) (where a ∈ R) implies a = 0. An additive map x 7→ x∗ of R into itself is calledan involution if (i) (xy)∗ = y∗x∗ and (ii) (x∗)∗ = x hold for all x, y ∈ R. A ring equippedwith an involution is called a ring with involution or ∗-ring. An element x in a ring withinvolution is said to be hermitian if x∗ = x and skew-hermitian if x∗ = −x. The setsof all hermitian and skew-hermitian elements of R will be denoted by H(R) and S(R),respectively. The involution is called the first kind if Z(R) ⊆ H(R), otherwise it is said tobe of the second kind. In the later case S(R) ∩ Z(R) 6= (0). Notice that x is normal i.e.,xx∗ = x∗x, if and only if h and k commute. If all elements in R are normal, then R iscalled a normal ring (see [15] for more details).
An additive mapping δ : R → R is said to be a derivation of R if δ(st) = δ(s)t+sδ(t) forall s, t ∈ R. A derivation δ is said to be inner if there exists a ∈ R such that δ(s) = as−safor all s ∈ R. Over the last some decades, several authors have investigated the relationshipbetween the commutativity of the ring R and certain special types of maps like derivationsand automorphisms of R. The criteria to discuss the commutativity of prime rings via∗Corresponding Author.Email addresses: [email protected] (S. Ali), [email protected] (A.N.A. Koam),
[email protected] (M.A. Ansari)Received: 26.04.2018; Accepted: 18.03.2019
On ∗-differential identities in prime rings with involution 709
derivations has been given for the first time by Posner [19]. In fact, he proved that theexistence of a nonzero centralizing derivation( i.e., δ(x)x − xδ(x) ∈ Z(R) for all x ∈ Z(R)on a prime ring forces the ring to be commutative. Since then many algebraists establishedthe commutativity of prime and semiprime rings via derivations and automorphisms thatsatisfying certain differential identities (see [1,3,6–10,13,14,17,18] and references therein).
In this paper, our intent is to continue to investigate and discuss the commutativityof prime rings with involution ‘∗” satisfying certain ∗- differential identities. In fact, ourresults generalize and unify several well known and classical theorems proved in [4], [12],and [16].
2. PreliminariesWe shall do a great deal of calculation with commutators and anti-commutators, rou-
tinely using the following basic identities:
For all s, t, w ∈ R;[st, w] = s[t, w] + [s, w]t and [s, tw] = t[s, w] + [s, t]w
so(tw) = (sot)w − t[s, w] = t(sow) + [s, t]w(st)ow = s(tow) − [s, w]t = (sow)t + s[t, w].
We start our investigation with some known facts and results about rings which will beused frequently throughout the discussions.
Fact 2.1 ([2, Lemma 2.1 ]). Let R be a prime ring with involution ”∗” of second kind suchthat char(R) 6= 2. If R is normal i.e., [x, x∗] = 0 for all x ∈ R, then R is commutative.
Fact 2.2. The center of a prime ring is free from zero divisors.
Fact 2.3. Let R be a 2-torsion free ring with involution ” ∗ ” . Then every x ∈ R can beuniquely represented as 2x = h + k, where h ∈ H(R) and k ∈ S(R).
In view of the Fact 2.1 and Theorem 2.4 of [2], we have the following.
Fact 2.4. Let R be a prime ring with involution ”∗” of second kind such that char(R) 6= 2.Let δ be a nonzero derivation of R such that [δ(x), x∗] = 0 for all x ∈ R. Then, R is acommutative integral domain.
Lemma 2.5. Let R be a prime ring with involution ” ∗ ” of the second kind such thatchar(R) 6= 2. Let δ be a derivation of R such that δ(h) = 0 for all h ∈ H(R) ∩Z(R). Thenδ(x) = 0 for all x ∈ Z(R).
Proof. Suppose that we have δ(h) = 0 for all h ∈ S(R) ∩ Z(R). Substituting k2 (wherek ∈ S(R) ∩ Z(R)) for h and using the fact that δ(k) ∈ Z, we obtain 2δ(k)k = 0 for allk ∈ S(R) ∩ Z(R). This implies that δ(k)k = 0 for all k ∈ S(R) ∩ Z(R). Application of theFact 2.2 yields δ(k) = 0 for all k ∈ S(R) ∩Z(R). In view of the Fact 2.3, we conclude that2δ(x) = δ(2x) = δ(h + k) = δ(h) + δ(k) = 0 and hence δ(x) = 0 for all x ∈ Z(R). Lemma 2.6. Let R be a prime ring with involution ”∗” of second kind such that char(R) 6=2. If x x∗ = 0 for all x ∈ R or xx∗ = 0 for all x ∈ R, then R is a commutative integraldomain.
Proof. First we assume that x x∗ = 0 for all x ∈ R. Direct linearization of the aboverelation gives
xy∗ + yx∗ + x∗y + y∗x = 0 (2.1)for all x, y ∈ R. Substituting sy for y (where s ∈ Z(R) ∩ S(R) in (2.1), we get
−sxy∗ + syx∗ + sx∗y − sy∗x = 0 (2.2)
710 S. Ali, A.N.A. Koam, M.A. Ansari
for all x, y ∈ R and s ∈ Z(R) ∩ S(R). Multiplying (2.1) by s and then combining with theobtained relation, we arrive at s(yx∗ + x∗y) = 0 for all x, y ∈ R and s ∈ Z(R) ∩ S(R).Invoking the primeness of R, we get yx∗ + x∗y = 0 for all x, y ∈ R. This implies thatx y = 0 for all x, y ∈ R. Replacing x by xz and using the second anti-commutatoridentity, we get y[x, z] = 0 for all x, y, z ∈ R. The primeness of R furnishes the requiredresult. On the other hand, we consider the case xx∗ = 0 for all x ∈ R. This implies thatx x∗ = 0 for all x ∈ R and therefore the result follows by above discussion. Hence, R iscommutative. This proves the lemma.
3. The resultsIn [16], Herstein proved that a prime ring R of characteristic different from two with
a nonzero derivation δ satisfying the differential identity [δ(x), δ(y)] = 0 for all x, y ∈ R,must be commutative. Further, Daif [11] showed that for a 2-torsion free semiprime ringR admitting a derivation δ such that [δ(x), δ(y)] = 0 for all x, y ∈ I , where I is a nonzeroideal of R and δ is nonzero on I, then R contains a nonzero central ideal. Further, this resultwas extended by first author together with Dar in [[12], Theorem 3.1] for prime rings withinvolution. In fact, they proved that if R is prime ring with involution ” ∗ ” of the secondkind such that char(R) 6= 2 and satisfying the ∗-differential identity [δ(x), δ(x∗)] = 0 forall x ∈ R, then R must be commutative. This result motivated us to prove the followingtheorem.
Theorem 3.1. Let R be a prime ring with involution ” ∗ ” of the second kind such thatchar(R) 6= 2. Let δ1 and δ2 be derivations of R such that at least one of them is nonzeroand satisfying the identity [δ1(x), δ1(x∗)] + δ2(x x∗) = 0 for all x ∈ R. Then R is acommutative integral domain.
Proof. We are given that δ1, δ2 : R → R derivations such that[δ1(x), δ1(x∗)] + δ2(x x∗) = 0 (3.1)
for all x ∈ R. We divide the proof in three cases.Case (i): First we assume that δ1 6= 0 and δ2 = 0. Then, the relation (3.1) reduces to
[δ1(x), δ1(x∗)] = 0 (3.2)for all x ∈ R. Henceforth, the proof follows by [[12], Theorem 3.1]. But the proof ofTheorem 3.1. given in [12] is very complicated and technical. Therefore, we presenthere short and elegant proof that may be considered as an alternative and brief proof ofTheorem 3.1. given in [12]. Polarizing the equation (3.2), we obtain
[δ1(x), δ1(y∗)] + [δ1(y), δ1(x∗)] = 0 (3.3)for all x, y ∈ R. Substituting yh for y (where h ∈ Z(R) ∩ H(R)) in (3.3) and using the factthat δ1(h) ∈ Z(R), we arrive at
[δ1(x), δ1(y∗)] + [δ1(y), δ1(x∗)]h + [δ1(x), y∗] + [y, δ1(x∗)]δ1(h) = 0 (3.4)for all x, y ∈ R. In view of (3.3), the above relation reduces to
[δ1(x), y∗] + [y, δ1(x∗)]δ1(h) = 0 (3.5)for all x, y ∈ R and h ∈ Z(R) ∩ H(R). The primeness of R yields that either δ1(h) = 0for all h ∈ Z(R) ∩ H(R) or [δ1(x), y∗] + [y, δ1(x∗)] = 0 for all x, y ∈ R. If δ1(h) = 0 forall h ∈ Z(R) ∩ H(R), then by the Fact 2.5 we conclude that δ1(x) = 0 for all x ∈ Z(R).Substituting ky for y(where k ∈ Z(R)∩S(R)) in (3.3) and then combining it with (3.3), weobtain [δ1(y), δ1(x∗)] = 0 for all x, y ∈ R. This implies that [δ1(x2), δ1(x)] = 0 for all x ∈ R.In view of [[5], Theorem 3.1], we conclude the result. Finally, we have the remaining case[δ1(x), y∗] + [y, δ1(x∗)] = 0 for all x, y ∈ R. Replace y by ys(where s ∈ Z(R) ∩ S(R)) in thelast expression to get s[δ1(x), y∗] − s[y, δ1(x∗)] = 0 for all x, y ∈ R and s ∈ Z(R) ∩ S(R).
On ∗-differential identities in prime rings with involution 711
Multiplying the above relation by s from left side and combine with last relation, we findthat 2s[δ1(x), x∗] = 0 for all x ∈ R. Since R is prime ring and char (R) 6= 2, so the lastidentity reduces to [δ1(x), x∗] = 0 for all x ∈ R. In view of the Fact 2.4, we conclude therequired conclusion. Hence, R is commutative.Case (ii): Now we assume that δ1 = 0 and δ2 6= 0 Then, the relation (3.1) reduces to
δ2(x x∗) = 0 (3.6)
for all x ∈ R. Substituting x + y for x in (3.6), we obtain
δ2(x y∗) + δ2(y x∗) = 0 (3.7)
for all x, y ∈ R. Replacing x by hx (where h ∈ Z(R) ∩ H(R)) in (3.7) and using theanti-commutator identities, we get
δ2(h(x y∗)) + δ2(h(y x∗)) = 0 (3.8)
for all x, y ∈ R and h ∈ Z(R) ∩ H(R). Since h ∈ Z(R) ∩ H(R), so δ2(h) ∈ Z(R) andconsequently equation (3.8) gives
δ2(h)(x y∗) + (y x∗) + hδ2((x y∗) + δ2((y x∗) = 0 (3.9)
for all x, y ∈ R and h ∈ Z(R) ∩ H(R)). Application of relation (3.7) yields
δ2(h)(x y∗) + (y x∗) = 0 (3.10)
for all x, y ∈ R and h ∈ Z(R) ∩ H(R). Taking x = y in 3.10) and using the fact that charR 6= 2, we obtain δ2(h)(x x∗) = 0 for all x ∈ R. Invoking the primeness of R, it followsthat either x x∗ = 0 for all x ∈ R or δ2(h) = 0. In case δ2(h) = 0 for all h ∈ Z(R) ∩ H(R),application of the Fact 2.5 implies that δ2(x) = 0 for all x ∈ Z. Substituting ky for y(wherek ∈ Z(R) ∩ S(R)) in (3.7) and then combining it with (3.7), we find that δ1(x y) = 0 forall x, y ∈ R. This implies that R is commutative. In consequence, we have x x∗ = 0 forall x ∈ R. Hence, Lemma 2.6 yields the required result.Case (iii): Finally, we assume that both δ1 and δ2 are nonzero. Interchanging the role ofx and x∗ in equation (3.1) and using the fact that [x, x∗] = −[x∗, x] and x x∗ = x∗ x,gives
−[δ1(x∗), δ1(x)] + δ2(x∗ x) = 0 (3.11)
for all x ∈ R. This implies that
[δ1(x), δ1(x∗)] − δ2(x x∗) = 0 (3.12)
for all x ∈ R. Combining (3.11) and (3.14) and using the fact that char(R) 6= 2, we get
[δ1(x), δ1(x∗)] = 0 (3.13)
for all x ∈ R. Therefore, the result follows by Case (i). Hence, R is a commutative Integraldomain. This completes the proof of the theorem.
Using a similar technique with necessary variations, we can prove the following result.
Theorem 3.2. Let R be a prime ring with involution ” ∗ ” of the second kind such thatchar(R) 6= 2. Let δ1 and δ2 be derivations of R such that at least one of them is nonzeroand satisfying the identity [δ1(x), δ1(x∗)] − δ2(x x∗) = 0 for all x ∈ R. Then R is acommutative integral domain.
Theorem 3.3. Let R be a prime ring with involution ” ∗ ” of the second kind such thatchar(R) 6= 2. Let δ1 and δ2 be derivations of R such that at least one of them is nonzeroand satisfying the identity δ1(x) δ1(x∗) + δ2([x, x∗]) = 0 for all x ∈ R. Then R is acommutative integral domain.
712 S. Ali, A.N.A. Koam, M.A. Ansari
Proof. By the assumption, we have
δ1(x) δ1(x∗) + δ2([x, x∗]) = 0 (3.14)
for all x ∈ R. Substituting x∗ for x in (3.14) and using the fact that x x∗ = x∗ x, weobtain
δ1(x) δ1(x∗) − δ2([x, x∗]) = 0 (3.15)
for all x ∈ R. From relations (3.14) and (3.15), we conclude that
δ1(x) δ1(x∗) = 0 (3.16)
for all x ∈ R. Henceforward, the result is follows by [[12], Theorem 3.3]. This proves thetheorem.
Theorem 3.4. Let R be a prime ring with involution ” ∗ ” of the second kind such thatchar(R) 6= 2. Let δ1 and δ2 be derivations of R such that at least one of them is nonzeroand satisfying the identity δ1(x) δ1(x∗) − δ2([x, x∗]) = 0 for all x ∈ R. Then R is acommutative integral domain.
Theorem 3.5. Let R be a prime ring with involution ” ∗ ” of the second kind such thatchar(R) 6= 2. Let δ be a nonzero derivation of R such that δ([x, x∗]) + [δ(x), δ(x∗)] = 0 forall x ∈ R. Then R is a commutative integral domain.
Proof. By the assumption, we have
δ([x, x∗]) + [δ(x), δ(x∗)] = 0 (3.17)
for all x ∈ R. Polarizing the relation (3.17), we obtain
δ([x, y∗]) + δ([y, x∗]) + [δ(x), δ(y∗)] + [δ(y), δ(x∗)] = 0 (3.18)
for all x, y ∈ R. Substituting yh for y (where h ∈ Z(R) ∩ H(R)) in (3.18) and using thefact δ(h) ∈ Z(R) (where h ∈ Z(R) ∩ H(R)), we arrive at
[x, y∗] + [y, x∗]δ(h) + hδ([x, y∗)] + δ([y, x∗]) + δ(h)[δ(x), y∗]
+h[δ(x), δ(y∗)] + [δ(y), δ(x∗)]h + [y, δ(x∗)]δ(h) = 0for all x, y ∈ R. This implies that
[x, y∗] + [y, x∗]δ(h) + δ([x, y∗)] + δ([y, x∗]) + [δ(x), δ(y∗)] + [δ(y), δ(x∗)]h
+[δ(x), y∗] + [y, δ(x∗)]δ(h) = 0Application of the relation (3.18) yields
[x, y∗] + [y, x∗]δ(h) + [δ(x), y∗] + [y, δ(x∗)]δ(h) = 0 (3.19)
for all x ∈ R. Since δ(h) ∈ Z(R), so the above expression can be written as
[x, y∗] + [y, x∗] + [δ(x), y∗] + [y, δ(x∗)]δ(h) = 0 (3.20)
for all x ∈ R. The primeness of R yields that either δ(h) = 0 or [x, y∗]+[y, x∗]+[δ(x), y∗]+[y, δ(x∗)]) = 0 for all x, y ∈ R. If δ(h) = 0, then δ(x) = 0 for all x ∈ Z(R) by the Fact 2.5.Replacing y by ky (where s ∈ Z(R) ∩ S(R)) in (3.18) and combining it with the obtainedresult, we get δ[(x, y]) + [δ(x), δ(y)] = 0 for all x, y ∈ R. Substituting x2 for y in the lastrelation, we find that [δ(x), δ(x2)] = 0 for all x ∈ R. Hence, R is commutative by Theorem3.1 of [5]. On the other hand, we have
[x, y∗] + [y, x∗] + [δ(x), y∗] + [y, δ(x∗)] = 0 (3.21)
for all x, y ∈ R. Replace x by xs (where s ∈ Z(R) ∩ S((R)) in (3.21) to get
[x, y∗]s − [y, x∗]s + [δ(x), y∗]s + [x, y∗]δ(s) − [y, x∗]δ(s) − [y, δ(x∗)]s = 0 (3.22)
On ∗-differential identities in prime rings with involution 713
for all x, y ∈ R and s ∈ Z(R) ∩ S(R). Multiplying by s to (3.21) from right and combiningwith (3.22), we arrive at
2[x, y∗]s + 2[δ(x), y∗]s + 2[x, y∗]δ(s) = 0 (3.23)for all x, y ∈ R. Taking y = x∗ in (3.23) and using the fact that char(R) 6= 2, we obtain[δ(x), x]s = 0 for all x ∈ R and s ∈ Z(R)∩S(R). The primeness of R gives that [δ(x), x] = 0for all x ∈ R. Since δ 6= 0, Posner’s theorem [19] yields the desired conclusion. This provesthe theorem completely. Theorem 3.6. Let R be a prime ring with involution ” ∗ ” of the second kind such thatchar(R) 6= 2. Let δ be a nonzero derivation of R such that δ(x x∗) + δ(x) δ(x∗) = 0 forall x ∈ R. Then R is a commutative integral domain.In view of above discussions, results, and as the applications of the main theorems, weobtain the following corollaries.Corollary 3.7. Let R be a prime ring with involution ” ∗ ” of the second kind such thatchar(R) 6= 2. Let δ1 and δ2 be derivations of R such that at least one of them is nonzeroand satisfying the identity [δ1(x), δ1(y)] ± δ2(x y) = 0 for all x, y ∈ R. Then R is acommutative integral domain.Corollary 3.8. Let R be a prime ring with involution ” ∗ ” of the second kind such thatchar(R) 6= 2. Let δ1 and δ2 be derivations of R such that at least one of them is nonzeroand satisfying the identity δ1(x) δ1(y) ± δ2([x, y]) = 0 for all x, y ∈ R. Then R is acommutative integral domain.Corollary 3.9 ([16], Theorem). Let R be a prime ring with involution ” ∗ ” of the secondkind such that char(R) 6= 2. Let δ be a nonzero derivation of R such that [δ(x), δ(y)] = 0for all x, y ∈ R. Then R is commutative.The next corollary is the ∗-version of Herstein classical theorem proved in [16].Corollary 3.10 ([12], Theorem 3.1). Let R be a prime ring with involution ” ∗ ” ofthe second kind such that char(R) 6= 2. Let δ be a nonzero derivation of R such that[δ(x), δ(x∗)] = 0 for all x ∈ R. Then R is commutative.Corollary 3.11 ([12], Theorem 3.2). Let R be a prime ring with involution ” ∗ ” ofthe second kind such that char(R) 6= 2. Let δ be a nonzero derivation of R such thatδ(x) δ(x∗) = 0 for all x ∈ R. Then R is commutative.Corollary 3.12 ([4], Theorem 2.2). Let R be a prime ring with involution ”∗” of the secondkind such that char(R) 6= 2. Let δ be a nonzero derivation of R such that δ([x, x∗]) = 0for all x ∈ R. Then R is commutative.Corollary 3.13 ([4], Theorem 2.3). Let R be a prime ring with involution ”∗” of the secondkind such that char(R) 6= 2. Let δ be a nonzero derivation of R such that δ(x x∗) = 0for all x ∈ R. Then R is commutative.Corollary 3.14. Let R be a prime ring with involution ” ∗ ” of the second kind such thatchar(R) 6= 2. Let δ be a nonzero derivation of R such that δ(xx∗) = 0 for all x ∈ R. ThenR is a commutative integral domain.Proof. We are given that δ is a nonzero derivation of R such that δ(xx∗) = 0 for all x ∈ R.For any x ∈ R, x∗ also is an element of R. Substitution x∗ for x in the given assertion, weobtain δ(x∗x) = 0 for all x ∈ R. This implies that δ(x x∗) = 0 for all x ∈ R. Hence R iscommutative by Corollary 3.13. Corollary 3.15. Let R be a prime ring with involution ” ∗ ” of the second kind such thatchar(R) 6= 2. Let δ be a nonzero derivation of R such that δ(xx∗) + δ(x)δ(x∗) = 0 for allx ∈ R. Then R is a commutative integral domain.
714 S. Ali, A.N.A. Koam, M.A. Ansari
Proof. By the assumption, we have δ(xx∗)+δ(x)δ(x∗) = 0 for all x ∈ R. Replace x by x∗
in the last expression to get δ(x∗x) + δ(x∗)δ(x) = 0 for all x ∈ R. By combining the lasttwo relations, we obtain δ([x, x∗]+[δ(x), δ(x∗)] = 0 for all x ∈ R. Hence, R is commutativeby Theorem 2.5. This proves the corollary.
4. Some examplesThe first example shows that the restriction of the second kind involution in Theo-
rems 3.1 and 3.5 is not superfluous.
Example 4.1. Let R =(
a bc d
)|a, b, c, d ∈ Z
. Obviously, R is prime ring. Define the
maps δ1, δ2, ∗ : R −→ R such that(
a bc d
)∗=
(d − b−c a
), δ1
(a bc d
)=
(0 − b
c 0
)Then, it is straightforward to check that δ1 is a derivation of R. It is easy to see that
Z(R) =(
a 00 b
)|a ∈ Z
. Then, x∗ = x for all x ∈ R and hence Z(R) ⊆ H(R),
which shows that the involution ∗ is of the first kind. Moreover, for δ1 = δ2 the followingconditions: (i) [δ1(x), δ1(x∗)]+δ2(xx∗) = 0 for all x ∈ R, (ii) δ([x, x∗])+[δ(x), δ(x∗)] = 0for all x ∈ R are satisfied. However, R is not commutative.
The next example demonstrates that Theorem 3.5 cannot be extended for semiprimerings.
Example 4.2. Let R be a ring with involution ” ∗ ” same as in Example 4.1. Next,let C be the field of complex numbers with the conjugation involution. Consider the setL = R×C. Then, it is obvious to see that (L, σ) a semiprime ring with involution ∗ of thesecond kind, where σ(r, z) = (r∗, z) for all (r, z) ∈ R × C. Define a derivation δ : L −→ L
by δ(r, z) = (δ1(r), 0) for all (r, z) ∈ R × C (where δ1 is a derivation on R). Then, it isstraightforward to check that δ is a derivation of R × C satisfying the conditions of thementioned theorems, but R is not commutative. Hence, in Theorem 3.5, the hypothesisof primeness is crucial.
Remark 4.3. At the end, let us also point out that we do not know yet whether Theorems3.1, 3.3, and 3.5 true for automorphisms of semi(prime)rings. Hence, these are openproblems for automorphisms of semi(prime)rings.
Acknowledgment. The authors are grateful to the referee(s) for their helpful com-ments. The final form was prepared when the first author was on a short visit at theDepartment of Mathematics, Faculty of Science, Jazan University, KSA, during Decem-ber 6-9, 2017. The first author appreciates the gracious hospitality he received at JazanUniversity during his visit.
References[1] S. Ali and H. Alhazmi, Some commutativity theorems in prime rings with involution
and derivations, J. Adv. Math. Comput. Sci. 24 (5), 1–6, 2017.[2] S. Ali and N.A. Dar, On ∗-centralizing mappings in rings with involution, Georgian
Math. J. 1, 25–28, 2014.[3] S. Ali and S. Huang, On derivations in semiprime rings, Algebr. Represent. Theory
15 (6), 1023–1033, 2012.[4] S. Ali, N.A. Dar, and M. Asci, On derivations and commutativity of prime rings with
involution, Georgian Math. J. 23 (1), 9–14, 2016.[5] S. Ali, M.S. Khan, and M. Al-Shomrani, Generalization of Herstein theorem and its
applications to range inclusion problems, J. Egyptian Math. Soc. 22, 322–326, 2014.
On ∗-differential identities in prime rings with involution 715
[6] N. Argac, On prime and semiprime rings with derivations, Algebra Colloq. 13 (3),371–380, 2006.
[7] M. Ashraf and M.A. Siddeeque, On ∗− n-derivations in prime rings with involution,Georgian Math. J. 21 (1), 9–18, 2014.
[8] M. Ashraf and N. Rehman, On commutativity of rings with derivations, Results Math.42 (1-2), 3–8, 2002.
[9] H.E. Bell, On the commutativity of prime rings with derivation, Quaest. Math. 22,329-333, 1991.
[10] H.E. Bell and M.N. Daif, On derivations and commutativity in prime rings, ActaMath. Hungar. 66, 337–343, 1995.
[11] M.N. Daif, Commutativity results for semiprime rings with derivation, Int. J. Math.Math. Sci. 21 (3), 471–474, 1998.
[12] N.A. Dar and S. Ali, On ∗-commuting mappings and derivations in rings with invo-lution, Turk. J. Math. 40, 884–894, 2016.
[13] V.De. Filippis, On derivation and commutativity in prime rings, Int. J. Math. Math.Sci. 69-72, 3859–3865, 2004.
[14] A. Fosner and J. Vukman, Some results concerning additive mappings and derivationson semiprime rings, Pul. Math. Debrecen, 78 (3-4), 575–581, 2011.
[15] I.N. Herstein, Rings with Involution, University of Chicago Press, Chicago, 1976.[16] I.N. Herstein, A note on derivation II, Canad. Math. Bull. 22, 509–511, 1979.[17] J. Mayne, Centralizing automorphisms of prime rings, Canad. Math. Bull. 19, 113–
117, 1976.[18] L. Oukhtite, Posner’s second theorem for Jordan ideals in ring with involution, Expo.
Math. 4 (29), 415–419, 2011.[19] E.C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc. 8, 1093–1100, 1957.
Hacettepe Journal ofMathematics & Statistics
Hacet. J. Math. Stat.Volume 49 (2) (2020), 716 – 726
DOI : 10.15672/hujms.588741
Research Article
On some subclasses k-uniformly Janowski starlikeand convex functions associated with t-symmetric
pointsKhalida Inayat Noor1, Nasir Khan2, Muhammad Arif3, Janusz Sokół∗4
1Department of Mathematics COMSATS Institute of Information Technology, Park Road, Islamabad,Pakistan
2Department of Mathematics FATA University F.R. Kohat, Pakistan3Department of Mathematics, Abdul Wali Khan University Mardan, Mardan 23200, Pakistan
4Faculty of Mathematics and Natural Sciences, University of Rzeszów, ul. Prof. Pigonia 1, 35-310Rzeszów, Poland
AbstractIn this paper, we define new subclasses of k-uniformly Janowski starlike and k-uniformlyJanowski convex functions associated with t-symmetric points. The integral representa-tions, convolution properties and coefficient bounds for these classes are studied.
Mathematics Subject Classification (2010). 30C45, 30C50
Keywords. subordination, convolution, t-symmetric points
1. IntroductionLet A denote the class of functions f(z) of the form
f(z) = z +∞∑
n=2anzn, (1.1)
which are analytic in the open unit disk U = z : z ∈ C and |z| < 1. Furthermore, Srepresents the class of all functions in A which are univalent in U .
Sakaguchi [10], introduced a class S∗s of functions starlike with respect to symmetric
points, it consists of functions f(z) ∈ S, satisfying the inequality
Re
f ′(z)
f(z) − f (−z)
> 0, (z ∈ U). (1.2)
Following him, many authors studied this class and its subclasses see [1, 9, 12,14].Motivated by S∗
s, we can easily obtain the following class Cs of functions convex withrespect to symmetric points. Let Cs denote the class of functions in S, satisfying theinequality
Re
(zf ′(z))′
f ′(z) − f ′(−z)
> 0, (z ∈ U).
∗Corresponding Author.Email addresses: [email protected] (K.I. Noor), [email protected] (N. Khan),
[email protected] (M. Arif), [email protected] (J. Sokół)Received: 15.04.2015; Accepted: 18.03.2019
On some subclasses 717
In [2], Chand and Singh considered a class Sts of functions starlike with respect to
t-symmetric points, which consists of functions f(z) ∈ S, satisfying the inequality
Re
zf ′(z)ft (z)
> 0, (z ∈ U),
where
ft(z) = 1t
t−1∑µ=0
ϵ−µf (ϵµz) ,(ϵt = 1 : t ∈ N
). (1.3)
From (1.1) equation (1.3) we can rewrite as
ft(z) = 1t
t−1∑µ=0
ϵ−µf (ϵµz) = 1t
t−1∑µ=0
ϵ−µ[ϵµz +∞∑
n=2an (ϵµz)n]
= z +∞∑
n=2bnanzn, (1.4)
where
bn = 1t
t−1∑µ=0
ϵ(n−1)µ =
1, n = lt + 1,0, n = lt + 1,
(1.5)
where l, t ∈ N; n ≥ 2; ϵt = 1.Notice that
ft (ϵµz) = ϵµft(z), (1.6)
f ′t (ϵµz) = ft(z) = 1
t
t−1∑µ=0
f ′ (ϵµz) , (z ∈ U) . (1.7)
Definition 1.1. For f(z) ∈ A, given by (1.1) and g(z) ∈ A of the form
g(z) = z +∞∑
n=2cnzn, (z ∈ U) ,
the Hadmard product (or convolution) of f(z) and g(z) is given by
(f ∗ g) (z) = z +∞∑
n=2ancnzn = (g ∗ f) (z), (z ∈ U).
For two functions f(z) and g(z) analytic in U , we say that f(z) is subordinate tog(z), denoted by f ≺ g or f(z) ≺ g(z), if there exists an analytic function w(z) with|w(z)| < |z| such that f(z) = g (w(z)). If g(z) is univalent in U then f(z) ≺ g(z) if andonly if f (0) = g (0) and f (U) ⊂ g (U). The idea of subordination was widely presentedby Miller and Mocanu [7].
Definition 1.2. A function p(z) is said to be in the class P [A, B], −1 ≤ B < A ≤ 1, if itis analytic in U with p (0) = 1 and
p(z) ≺ 1 + Az
1 + Bz, (z ∈ U).
Geometrically, if a function p belongs to P [A, B], then it maps the open unit disc Uonto the disk
Ω [A, B] =
w :∣∣∣∣w − 1 − AB
1 − B2
∣∣∣∣ <A − B
1 − B2
.
The class P [A, B] is connected with the class P of functions with positive real part bythe relation
p(z) ∈ P, if and only if (A + 1) p(z) − (A − 1)(B + 1) p (z) − (B − 1)
∈ P [A, B] .
718 K.I. Noor, N. Khan, M. Arif, J. Sokół
This class P [A, B] was presented by Janowski [3] and explored by a few creators. Kanasand Wiśniowska [4, 5] presented and examined the class k − ST of k−starlike functionsand the related class k − UCV of k−uniformly convex functions. These classes werecharacterized subject to the conic region Ωk, k ≥ 0, as
Ωk =
u + iv : u > k√
(u − 1)2 + v2
.
This domain represents the right half plane, a parabola, a hyperbola and an ellipse fork = 0, k = 1, 0 < k < 1 and k > 1 respectively. The functions such that pk(U) = Ωk are
pk(z) =
1+z1−z , k = 0,
1 + 2π2
(log 1+
√z
1−√
z
)2, k = 1,
1 + 21−k2 sinh2
(2π arccos k
)arctanh
√z
, 0 < k < 1,
1 + 2k2−1 sin
(π
2R(t)∫ u(z)√
t
0dx√
1−x2√
1−(tx)2
)+ 1
k2−1 , k > 1,
(1.8)
where
u(z) = z −√
t
1 −√
tx, (z ∈ U) ,
and t ∈ (0, 1) and z is chosen such that k = cosh(
πR′(t)4R(t)
). Here R(t) is Legendre’s complete
elliptic integral of first kind and R′(t) is the complementary integral of R (t).Following are the definitions of classes k − ST and k − UCV .
Definition 1.3. A function f(z) ∈ A is said to be in the class k − ST , k ≥ 0, if and onlyif,
zf ′(z)f(z)
≺ pk(z), (z ∈ U).
Definition 1.4. A function f(z) ∈ A is said to be in the class k − UCV , k ≥ 0, if andonly if,
(zf ′(z))′
f ′(z)≺ pk(z), (z ∈ U).
The classes k − ST and k − UCV were further generalized by Shams et al, [11], to theKD (k, β) and SD (k, β), respectively, with respect to the conic domain G (k, β), k ≥ 0and 0 ≤ β < 1 which is
G (k, β) = w : Rew > k |w − 1| + β .
Now using the concepts of Janowski functions and the conic regions, we define thefollowing class of functions.
Definition 1.5. A function p(z) is said to be in the class k − P [A, B], k ≥ 0, −1 ≤ B <A ≤ 1, if and only if
p(z) ≺ (A + 1) pk(z) − (A − 1)(B + 1) pk(z) − (B − 1)
, (z ∈ U),
where pk(z) is defined in (1.8).
Geometrically, the function p(z) ∈ k − P [A, B], takes all values from the domainΩk[A, B], −1 ≤ B < A ≤ 1, k ≥ 0 which is defined as
Ωk[A, B] =
w : Re
((B − 1) w(z) − (A − 1)(B + 1) w(z) − (A + 1)
)> k
∣∣∣∣(B − 1) w (z) − (A − 1)(B + 1) w(z) − (A + 1)
− 1∣∣∣∣ ,
On some subclasses 719
or equivalently as
Ωk[A, B] =
u + iv :[(
B2 − 1) (
u2 + v2)− 2 (AB − 1) u +(A2 − 1
)]2
> k2
(−2 (B + 1)
(u2 + v2)+ 2 (A + B + 2) u − 2 (A + 1)
)2+4 (A − B)2 v2
.
The domain Ωk[A, B] retains the conic domain Ωk inside the circular region defined byΩ[A, B]. The impact of Ω[A, B], on the conic domain Ωk, changes the original shape of theconic regions. The ends of hyperbola and parabola gets closer to one another but nevermeet anywhere and the ellipse gets the oval shape. When A → 1, B → −1 the radius ofthe circular disk define by Ω[A, B] tends to infinity, consequently the arm of the hyperbolaand parabola expand to the oval turns into ellipse. We see that Ωk[1, −1] = Ωk, the conicdomain defined by Kanas and Wiśniowska [4].
Definition 1.6. A function f(z) ∈ A is said to be in the class k − S(t)s T [A, B], k ≥
0, −1 ≤ B < A ≤ 1, if and only if
Re
(B − 1) zf ′(z)ft(z) − (A − 1)
(B + 1) zf ′(z)ft(z) − (A + 1)
> k
∣∣∣∣∣∣(B − 1) zf ′(z)
ft(z) − (A − 1)
(B + 1) zf ′(z)ft(z) − (A + 1)
− 1
∣∣∣∣∣∣ , (z ∈ U)
or equivalentlyzf ′(z)ft(z)
∈ k − P [A, B] ,
where ft(z) is defined by (1.3).
In some special cases we have the well known classes presented and studied earlier:(i) k − S
(1)s T [A, B] = k − ST [A, B], [8].
(ii) 0 − S(t)s T [A, B] = S
(t)s [A, B], [6].
(iii) k − S(1)s T [1, −1] = k − ST , [5].
(iv) k − S(1)s T [1 − 2β, −1] = SD (k, β), [11].
(v) 0 − S(1)s T [A, B] = S∗ [A, B], [3]
Definition 1.7. A function f(z) ∈ A is said to be in the class k − UC(t)s V [A, B], k ≥
0, −1 ≤ B < A ≤ 1, if and only if
Re
(B − 1) (zf ′(z))′
f ′t(z) − (A − 1)
(B + 1) (zf ′(z))′
f ′t(z) − (A + 1)
> k
∣∣∣∣∣∣∣(B − 1) (zf ′(z))′
f ′t(z) − (A − 1)
(B + 1) (zf ′(z))′
f ′t(z) − (A + 1)
− 1
∣∣∣∣∣∣∣ , (z ∈ U)
or equivalently(zf ′(z))′
f ′t (z)
∈ k − P [A, B] , (1.9)
where ft(z) is defined by (1.3).
In some special cases we have the well known classes presented and studied earlier:(i) k − UC
(1)s V [A, B] = k − UCV [A, B], [8].
(ii) k − UC(1)s V [1, −1] = k − UCV , [4].
(iii) k − UC(1)s V [A, B] = KD (k, β) , [11].
(iv) 0 − UC(1)s V [A, B] = C [A, B] , [3].
It is easy to see that
f(z) ∈ k − UC(t)s V [A, B] ⇐⇒ zf ′(z) ∈ k − S(t)
s T [A, B] .
720 K.I. Noor, N. Khan, M. Arif, J. Sokół
2. Main results2.1. Integral representation
First we give two meaningful conclusions about the classes k − S(t)s T [A, B] and k −
UC(t)s V [A, B].
Theorem 2.1. Let f(z) ∈ k − S(t)s T [A, B]. Then ft(z) ∈ k − S(1)T [A, B] ∈ S.
Proof. For f(z) ∈ k − S(t)s T [A, B], we can obtain
zf ′(z)ft(z)
≺ (A + 1) pk(z) − (A − 1)(B + 1) pk(z) − (B − 1)
, (z ∈ U). (2.1)
Substituting z by ϵµz respectively (µ = 0, 1, 2, 3, . . . , t − 1), we have
ϵµzf ′ (ϵµz)ft (ϵµz)
≺ (A + 1) pk(ϵµz) − (A − 1)(B + 1) pk(ϵµz) − (B − 1)
≺ (A + 1) pk(z) − (A − 1)(B + 1) pk(z) − (B − 1)
, (z ∈ U) . (2.2)
By definition of ft(z) and ϵ = exp(
2πit
), we know ϵ−µft (ϵµz) = ft(z). Then equation
(2.2) becomeszf ′ (ϵµz)
ft(z)≺ (A + 1) pk(z) − (A − 1)
(B + 1) pk(z) − (B − 1), (z ∈ U) . (2.3)
Let (µ = 0, 1, 2, 3, . . . , t − 1) in (2.3), respectively. Making the convex combination ofthem, we can get
zf ′t(z)
ft(z)= 1
t
t−1∑µ=0
zf ′ (ϵµz)ft(z)
≺ (A + 1) pk(z) − (A − 1)(B + 1) pk(z) − (B − 1)
, (z ∈ U) ,
because the function on right-hand site of (2.3) is convex univalent. That is, ft(z) ∈k − S
(1)s T [A, B] = k − ST [A, B] ⊂ S, [8] .
Putting k = 0 in Theorem 2.1, we can obtain Corollary 2.2, below which is comparableto the result obtained by Kwon and Sim [6] .
Corollary 2.2. Let f(z) ∈ S(t)s T [A, B]. Then ft(z) ∈ S∗ [A, B] ⊂ S.
Theorem 2.3. Let f(z) ∈ k − UC(t)s V [A, B]. Then ft(z) ∈ UCV [A, B] ⊂ S.
Proof. The proof of Theorem 2.3 is similar to that of Theorem 2.1 so the details areomitted.
Now we give the integral representations of the functions belonging to the classes k −S
(t)s T [A, B] and k − UC
(t)s V [A, B].
Theorem 2.4. Let f(z) ∈ k − S(t)s T [A, B]. Then
ft(z) = z ·
exp (A − B) 1t
t−1∑µ=0
∫ ϵµz
0
(pk(w(ς)) − 1)t (B + 1) pk(w(ς)) − (B − 1)
dς
, (2.4)
where ω(z) is analytic in U , ω (0) = 0 and |ω(z)| < 1.
Proof. For f(z) ∈ k − S(t)s T [A, B], from definition of the subordination we can have
zf ′(z)ft(z)
= (A + 1) pk(w(z)) − (A − 1)(B + 1) pk(w(z)) − (B − 1)
, (z ∈ U) , (2.5)
On some subclasses 721
where w(z) analytic in U , with w(0) = 0 and |w(z)| < 1. Substituting z by ϵµz respectively(µ = 0, 1, 2, 3, . . . , t − 1), we have
zf ′ (ϵµz)ϵ−µft (ϵµz)
= (A + 1) pk(w(ϵµz)) − (A − 1)(B + 1) pk(w(ϵµz)) − (B − 1)
. (2.6)
For (µ = 0, 1, 2, 3, . . . , t − 1) and z ∈ U . Using the equalities (1.6) and (1.7) we have
zf ′t(z)
ft(z)= 1
t
t−1∑µ=0
(A + 1) pk(w(ϵµz)) − (A − 1)(B + 1) pk(w(ϵµz)) − (B − 1)
, (2.7)
or equivalently,
f ′t(z)
ft(z)− 1
z= 1
t
t−1∑µ=0
(A − B) (pk(w(ϵµz)) − 1)z ((B + 1) pk(w(ϵµz)) − (B − 1))
. (2.8)
Integrating equality (2.8) , we have
log ft(z)z
= (A − B) 1t
t−1∑µ=0
∫ z
0
(pk(w(ϵµς)) − 1)ζ ((B + 1) pk(w(ϵµς)) − (B − 1))
dς
= (A − B) 1t
t−1∑µ=0
∫ ϵµz
0
(pk(w(ς)) − 1)t (B + 1) pk(w(ς)) − (B − 1)
dς. (2.9)
Therefore arranging equality (2.9) for ft(z), we can obtain
ft(z) = z ·
exp (A − B) 1t
t−1∑µ=0
∫ ϵµz
0
(pk(w(ς)) − 1)t (B + 1) pk(w(ς)) − (B − 1)
dς
,
and so the proof of Theorem 2.4 is complete.
Putting t = 1, in Theorem 2.4, we can obtain Corollary 2.5.
Corollary 2.5. Let f(z) ∈ k − ST [A, B]. Then
f(z) = z ·
exp (A − B)∫ z
0
(pk(w(ς)) − 1)t (B + 1) pk(w(ς)) − (B − 1)
dς
,
where ω(z) is analytic in U , ω (0) = 0 and |ω(z)| < 1.
Putting k = 0, in Theorem 2.4, we can obtain Corollary 2.6, below which is comparableto the result obtained by Kwon and Sim [6].
Corollary 2.6. Let f(z) ∈ k − S(t)s [A, B]. Then
ft(z) = z ·
exp (A − B) 1t
t−1∑µ=0
∫ z
0
w(ς)t (1 + Bw (ς))
dς
,
where ω(z) is analytic in U , ω (0) = 0 and |ω(z)| < 1.
Putting t = 1, A = 1 and B = −1 in Theorem 2.4, we can obtain Corollary 2.7.
Corollary 2.7. Let f(z) ∈ k − US. Then
f(z) = z ·
exp∫ z
0(pk(w(ς)) − 1) dς
,
where ω(z) is analytic in U , ω (0) = 0 and |ω(z)| < 1.
722 K.I. Noor, N. Khan, M. Arif, J. Sokół
Theorem 2.8. Let f(z) ∈ k − UC(t)s V [A, B]. Then
ft(z) =∫ z
0exp
(A − B) 1t
t−1∑µ=0
∫ ϵµς
0
(pk(w(τ)) − 1)t (B + 1) pk(w(τ)) − (B − 1)
dτ
dς,
where ω(z) is analytic in U , ω (0) = 0 and |ω(z)| < 1.
Proof. The proof of Theorem 2.8 is similar to that of Theorem 2.4 so the details areomitted. Theorem 2.9. Let f(z) ∈ k − S
(t)s T [A, B]. Then
f(z) =∫ z
0exp
(A − B) 1t
t−1∑µ=0
∫ ϵµζ
0
(pk(w(τ)) − 1)t (B + 1) pk(w(τ)) − (B − 1)
dτ
×( (A + 1) pk(w(ς)) − (A − 1)
(B + 1) pk(w(ς)) − (B − 1)
)dς,
where ω(z) is analytic in U , ω (0) = 0 and |ω(z)| < 1.
Proof. Let f(z) ∈ k − S(t)s T [A, B], then from equalities (2.4) and (2.5) we have
f ′(z) = ft(z)z
( (A + 1) pk(w(z)) − (A − 1)(B + 1) pk(w(z)) − (B − 1)
)
= exp (A − B) 1t
t−1∑µ=0
∫ ϵµz
0
(pk(w(τ)) − 1)t (B + 1) pk(w(τ)) − (B − 1)
dτ
×( (A + 1) pk(w(z)) − (A − 1)
(B + 1) pk(w(z)) − (B − 1)
). (2.10)
Integrating the equality (2.10) , we have
f(z) =∫ z
0exp
(A − B) 1t
t−1∑µ=0
∫ ϵµς
0
(pk(w(τ)) − 1)t (B + 1) pk(w(τ)) − (B − 1)
dτ
×( (A + 1) pk(w(ς)) − (A − 1)
(B + 1) pk(w(ς)) − (B − 1)
)dς.
and so the proof of Theorem 2.9 is complete. Putting k = 0, in Theorem 2.9, we can obtain Corollary 2.10, below which is comparable
to the result obtained by Kwon and Sim [6].
Corollary 2.10. Let f(z) ∈ S(t)s T [A, B]. Then
f(z) =∫ z
0exp
(A − B) 1t
t−1∑µ=0
∫ ϵµζ
0
w(τ)τ (1 + Bw(τ))
dτ
(1 + Aw(ς)1 + Bw(ς)
)dς.
where ω(z) is analytic in U , ω (0) = 0 and |ω(z)| < 1.
By applying similar method as in Theorem 2.9, we have
Theorem 2.11. Let f(z) ∈ k − UC(t)s V [A, B]. Then
f(z) =∫ z
0
1ξ
∫ ξ
0exp
(A − B) 1t
t−1∑µ=0
∫ ϵµς
0
(pk(w(τ)) − 1)t (B + 1) pk(w(τ)) − (B − 1)
dτ
×( (A + 1) pk(w(ς)) − (A − 1)
(B + 1) pk(w(ς)) − (B − 1)
)dςdξ,
where ω(z) is analytic in U , ω (0) = 0 and |ω(z)| < 1.
On some subclasses 723
2.2. Convolution conditionsIn this sections, we provide the convolutions conditions for the classes k − S
(t)s T [A, B]
and k − UC(t)s V [A, B].
Theorem 2.12. A function f(z) ∈ k − S(t)s T [A, B], if and only if
1z
f(z) ∗
(z
(1 − z)2
((B + 1)pk(eiθ) − (B − 1)
)−((A + 1)pk(eiθ) − (A − 1)
)h(z)
)= 0, (2.11)
for all z ∈ U and 0 ≤ θ < 2π. The coefficients of h(z) = z + b2z2 + · · · are given by (1.5).The function pk(z) is defined in (1.8).
Proof. Assume that f(z) ∈ k − S(t)s T [A, B], then we have
zf ′(z)ft(z)
≺ (A + 1) pk(z) − (A − 1)(B + 1) pk(z) − (B − 1)
, (z ∈ U)
if and only ifzf ′(z)ft(z)
= (A + 1) pk(eiθ) − (A − 1)(B + 1) pk(eiθ) − (B − 1)
, (2.12)
for all z ∈ U , and 0 ≤ θ < 2π. The condition (2.12), can be written as1z
zf ′(z)
[(B + 1) pk(eiθ) − (B − 1)
]− ft(z)
[((A + 1) pk(eiθ) − (A − 1)
)]= 0. (2.13)
On the other hand it is well known that
zf ′(z) = f(z) ∗ z
(1 − z)2 . (2.14)
And from (1.4), we have
ft(z) = z +∞∑
n=2anbnzn = (f ∗ h) (z), (2.15)
where
h(z) = z +∞∑
n=2bnzn, (2.16)
and where bn is given by (1.5). Substituting (2.14) and (2.15) in (2.13), we can get (2.11).This completes the proof of the Theorem 2.12.
Putting t = 1, in Theorem 2.12, we can obtain Corollary 2.13.
Corollary 2.13. A function f(z) ∈ k − ST [A, B], if and only if
1z
f(z) ∗
(z
(1 − z)2
(1 + B
(eiθ))
− z
(1 − z)
(1 + A
(eiθ)))
= 0,
for all z ∈ U .
Putting k = 0, in Theorem 2.12, we can obtain Corollary 2.14.
Corollary 2.14. A function f(z) ∈ S(t)s [A, B], if and only if
1z
f(z) ∗
(z
(1 − z)2
(1 + B
(eiθ))
− h(z)(1 + A
(eiθ)))
= 0,
for all z ∈ U and 0 ≤ θ < 2π, where h(z) is given by (2.16).
724 K.I. Noor, N. Khan, M. Arif, J. Sokół
Theorem 2.15. For k ≥ 0, −1 ≤ B < A ≤ 1. A function f(z) ∈ k − UC(t)s V [A, B], if
and only if
1z
f(z) ∗
(z (B + 1) pk(eiθ) − (B − 1)
(1 − z)2 −((A + 1) pk(eiθ) − (A − 1)
)h(z)
)′= 0,
where z ∈ U , 0 ≤ θ < 2π.
Proof. The proof of Theorem 2.15 , is similar to that of Theorem 2.12 , so the detail areomitted.
2.3. Coefficient inequalitiesFinally, we provide some coefficient inequalities which are sufficient for a function to be
in the class k − S(t)s T [A, B] or to be in the class k − UC
(t)s V [A, B].
Theorem 2.16. Assume that t is a positive integer, −1 ≤ B < A ≤ 1 and k ≥ 0. If afunction f(z) ∈ A of the form (1.1) satisfies the condition
∞∑n=1
2 (k + 1) tn + |(tn (B + 1) + (B − A))| |atn+1|
+∞∑
n=2,n =lt+12 (k + 1) n + |(n (B + 1))| |an| < |B − A| , (2.17)
then f(z) is in the class k − S(t)s T [A, B].
Proof. Assume that (2.17) holds, then it suffices to show that
k
∣∣∣∣∣∣(B − 1) zf ′(z)
ft(z) − (A − 1)
(B + 1) zf ′(z)ft(z) − (A + 1)
− 1
∣∣∣∣∣∣− Re
(B − 1) zf ′(z)ft(z) − (A − 1)
(B + 1) zf ′(z)ft(z) − (A + 1)
− 1
< 1.
We have
k
∣∣∣∣∣∣(B − 1) zf ′(z)
ft(z) − (A − 1)
(B + 1) zf ′(z)ft(z) − (A + 1)
− 1
∣∣∣∣∣∣− Re
(B − 1) zf ′(z)ft(z) − (A − 1)
(B + 1) zf ′(z)ft(z) − (A + 1)
− 1
≤ (k + 1)
∣∣∣∣ (B − 1)zf ′(z) − (A − 1)ft(z)(B + 1)zf ′(z) − (A + 1)ft (z)
− 1∣∣∣∣
= 2 (k + 1)∣∣∣∣ ft(z) − zf ′(z)(B + 1)zf ′(z) − (A + 1)ft(z)
∣∣∣∣≤ 2 (k + 1)
∑∞n=2 |bn − n| |an|
|B − A| −∑∞
n=2 |n (B + 1) − (A + 1) bn| |an|.
The last expression is bounded by 1, if∞∑
n=2(2 (k + 1) (n − bn) + |n (B + 1) − (A + 1) bn|) |an| < |B − A| . (2.18)
Using (1.5) in (2.18) we have∞∑
n=12 (k + 1) tn + |(tn (B + 1) + (B − A))| |atn+1|
+∞∑
n=2,n=lt+12 (k + 1) n + |(n (B + 1))| |an| < |B − A| ,
and this completes the proof of Theorem 2.16.
On some subclasses 725
Putting t = 1, in Theorem 2.16, we can obtain the following Corollary 2.17, which iscomparable to a result obtained by Noor and Malik in [8].
Corollary 2.17. A function f(z) ∈ A of the form (1.1) is in the class k − ST [A, B], if itsatisfies the condition
∞∑n=2
2 (k + 1) (n − 1) + |(n (B + 1) + (A + 1))| |an| < |B − A| ,
where k ≥ 0, −1 ≤ B < A ≤ 1.
Putting k = 0, in Theorem 2.16, we can obtain the following Corollary 2.18, which iscomparable to a result obtained by Kwon and Sim in [6].
Corollary 2.18. A function f(z) ∈ A of the form (1.1) is in the class S(t)s T [A, B], if it
satisfies the condition∞∑
n=1(tn + (A − B) (tn + 1)) |atn+1| +
∞∑n=2,n =lt+1
(1 + |B|) n |an| < |B − A| ,
where −1 ≤ B < A ≤ 1.
Putting t = 1, A = 1 and B = −1 in Theorem 2.16, we can obtain Corollary 2.19, whichis comparable to a result obtained by Kanas and Wiśniowska in [4].
Corollary 2.19. A function f(z) ∈ A of the form (1.1) is in the class k−ST , if it satisfiesthe condition ∞∑
n=2n + k (n − 1) |an| < 1, k > 0.
Putting t = 1, A = 1 − 2α, B = −1 with 0 ≤ β < 1 in Theorem 2.16, we can obtainCorollary 2.20, which is comparable to a result obtained by Shams et al. in [11].
Corollary 2.20. A function f(z) ∈ A of the form (1.1) is in the class SD (k, β), if itsatisfies the condition
∞∑n=2
n (k + 1) − (k + β) |an| < 1 − β,
where 0 ≤ β < 1. then k ≥ 0.
Putting t = 1, A = 1 − 2β, B = −1 with 0 ≤ β < 1 and k = 0 in Theorem 2.16, wecan obtain Corollary 2.21, below which is comparable to the known result obtained bySilverman in [13].
Corollary 2.21. A function f(z) ∈ A of the form (1.1) is in the class S∗ (β), if it satisfiesthe conditions ∞∑
n=2n − β |an| < 1 − β,
where 0 ≤ β < 1.
Theorem 2.22. A function f(z) ∈ A and of the form (1.1), is in the class k−UC(t)s V [A, B],
if it satisfies the condition∞∑
n=2[2 (k + 1) tn + |nt (B + 1) + (B − A)|] (tn + 1) |atn+1|
+∞∑
n=2,n =lt+1(2 (k + 1) n + n (B + 1)) |nan| < |B − A| ,
where −1 ≤ B < A ≤ 1 and k ≥ 0.
726 K.I. Noor, N. Khan, M. Arif, J. Sokół
Proof. The proof of Theorem 2.22, is similar to that of Theorem 2.16, so the detail areomitted.
References[1] M. Arif, K.I. Noor, and R. Khan, On subclasses of analytic functions with respect to
symmetrical points, Abst. Appl. Anal. 2012, 790689, 2012.[2] R. Chand and P. Singh, On certain schlicht mapping, Indian J. Pure App. Math. 10,
1167–1174, 1979.[3] W. Janowski, Some extremal problem for certain families of analytic functions, Ann.
Polon. Math. 28 (3), 298–326, 1973.[4] S. Kanas and A. Wiśniowska, Conic regions and k-uniform convexity, J. Comput.
Appl. Math. 105, 327–336, 1999.[5] S. Kanas and A. Wiśniowska, Conic domains and starlike functions, Rev. Roumaine
Math. Pure. Appl. 45 (4), 647–657, 2000.[6] O. Kwon and Y. Sim, A certain subclass of Janowski type functions associated with
k-symmetric points, Commun. Korean Math. Soc. 28, 143–154, 2013.[7] S.S. Miller and P.T. Mocanu, Differential Subordinations, Theory and Applications,
in: Series of Monographs and Textbooks in Pure and Applied Mathematics 225,Marcel Dekker Inc., New York, 2000.
[8] K.I. Noor and S.N. Malik, On coefficient inequalities of functions associated withconic domains, Comput. Math. Appl. 62, 2209–2217, 2011.
[9] K.I. Noor and S. Mustafa, Some classes of analytic functions related with functionsof bounded radius rotation eith respect to symmetrical points, J. Math. Ineq. 2 (3),267–276, 2009.
[10] K. Sakaguchi, On a certain univalent mapping, J. Math. Soc. Japan 11, 72–75, 1959.[11] S. Shams, S.R. Kulkarni, and J.M. Jahangiri, Classes of uniformly starlike and
convex functions, Int. J. Math. Math. Sci. 55, 2959–2961, 2004.[12] T.N. Shanmugam, C. Ramachandran, and V. Ravichandran, Fekete-Szegö problem
for subclass of starlike functions with respect to symmetric points, Bull. Korean Math.Soc. 43 (3), 589–598, 2006.
[13] H. Silverman, Univalent functions with negative coefficients, Proc. Amr. Math. Soc.51, 109–116, 1975.
[14] J. Sokół, Some remarks on the class of functions starlike with respect to symmetricpoints, Folia Sci. Univ. Tech. Resov. 73, 79–91, 1990.
Hacettepe Journal ofMathematics & Statistics
Hacet. J. Math. Stat.Volume 49 (2) (2020), 727 – 739
DOI : 10.15672/hujms.621536
Research Article
On total mean curvatures of foliated half-lightlikesubmanifolds in semi-Riemannian manifolds
Fortuné Massamba∗, Samuel SsekajjaSchool of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Private Bag X01,
Scottsville 3209 South Africa
AbstractWe derive total mean curvature integration formulas of a three co-dimensional foliationFn on a screen integrable half-lightlike submanifold, Mn+1 in a semi-Riemannian manifoldM
n+3. We give generalized differential equations relating to mean curvatures of a totallyumbilical half-lightlike submanifold admitting a totally umbilical screen distribution, andshow that they are generalizations of those given by [K. L. Duggal and B. Sahin, Differentialgeometry of lightlike submanifolds, Frontiers in Mathematics, Birkhäuser Verlag, Basel,2010].
Mathematics Subject Classification (2010). 53C25, 53C40, 53C50
Keywords. half-lightlike submanifold, Newton transformation, foliation and meancurvature
1. IntroductionThe rapidly growing importance of lightlike submanifolds in semi-Riemannian geome-
try, particularly Lorentzian geometry, and their applications to mathematical physics–likein general relativity and electromagnetism motivated the study of lightlike geometry insemi-Riemannian manifolds. More precisely, lightlike submanifolds have been shown torepresent different black hole horizons (see [4] and [6] for details). Among other motiva-tions for investing in lightlike geometry by many physicists is the idea that the universewe are living in can be viewed as a 4-dimensional hypersurface embedded in (4 + m)-dimensional spacetime manifold, where m is any arbitrary integer. There are significantdifferences between lightlike geometry and Riemannian geometry as shown in [4] and [6],and many more references therein. Some of the pioneering work on this topic is due toDuggal-Bejancu [4], Duggal-Sahin [6] and Kupeli [7]. It is upon those books that manyother researchers, including but not limited to [3, 5, 8–11], have extended their theories.
Lightlike geometry rests on a number of operators, like shape and algebraic invariantsderived from them, such as trace, determinants, and in general the r-th mean curvatureSr. There is a great deal of work so far on the case r = 1 (see some in [4, 6] and manymore) and as far as we know, very little has been done for the case r > 1. This is partlydue to the non-linearity of Sr for r > 1, and hence very complicated to study. A great∗Corresponding Author.Email addresses: [email protected] and [email protected] (F. Massamba),
[email protected] (S. Ssekajja)Received: 07.03.2017; Accepted: 18.03.2019
728 F. Massamba, S. Ssekajja
deal of research on higher order mean curvatures Sr in Riemannian geometry has beendone with numerous applications, for instance see [2] and [1]. This gap has motivated ourintroduction of lightlike geometry of Sr for r > 1. In this paper we have considered a half-lightlike submanifold admitting an integrable screen distribution, of a semi-Riemannianmanifold. On it we have focused on a codimension 3 foliation of its screen distributionand thus derived integral formulas of its total mean curvatures (see Theorems 4.9 and4.10). Furthermore, we have considered totally umbilical half-lightlike submanifolds, witha totally umbilical screen distribution and generalized Theorem 4.3.7 of [6] (see Theorem5.2 and its Corollaries). The paper is organized as follows; In Section 2 we summarize thebasic notions on lightlike geometry necessary for other sections. In Section 3 we give somebasic information on Newton transformations of a foliation F of the screen distribution.Section 4 focuses on integration formulae of F and their consequences. In Section 5 wediscus screen umbilical half-lightlike submanifolds and generalizations of some well-knownresults of [6].
2. PreliminariesLet (Mn+1, g) be a two-co-dimensional submanifold of a semi-Riemannian manifold
(Mn+3, g), where g = g|T M . The submanifold (Mn+1, g) is called a half-lightlike if the
radical distribution Rad TM = TM ∩ TM⊥ is a vector subbundle of the tangent bundleTM and the normal bundle TM⊥ of M , with rank one. Let S(TM) be a screen distri-bution which is a semi-Riemannian complementary distribution of Rad TM in TM , andalso choose a screen transversal bundle S(TM⊥), which is semi-Riemannian and comple-mentary to Rad TM in TM⊥. Then,
TM = Rad TM ⊥ S(TM), TM⊥ = Rad TM ⊥ S(TM⊥). (2.1)We will denote by Γ(Ξ) the set of smooth sections of the vector bundle Ξ. It is well-knownfrom [4] and [6] that for any null section E of Rad TM , there exists a unique null sectionN of the orthogonal complement of S(TM⊥) in S(TM)⊥ such that g(E, N) = 1, it followsthat there exists a lightlike transversal vector bundle ltr(TM) locally spanned by N . LetW ∈ Γ(S(TM⊥)) be a unit vector field, then g(N, N) = g(N, Z) = g(N, W ) = 0, for anyZ ∈ Γ(S(TM)).
Let tr(TM) be complementary (but not orthogonal) vector bundle to TM in TM . Thenwe have the following decompositions of tr(TM) and TM
tr(TM) = ltr(TM) ⊥ S(TM⊥), (2.2)
TM = S(TM) ⊥ S(TM⊥) ⊥ Rad TM ⊕ ltr(TM). (2.3)It is important to note that the distribution S(TM) is not unique, and is canonicallyisomorphic to the factor vector bundle TM/Rad TM [4]. Let P be the projection of TMon to S(TM). Then the local Gauss-Weingarten equations of M are the following;
∇XY = ∇XY + B(X, Y )N + D(X, Y )W, (2.4)∇XN = −AN X + τ(X)N + ρ(X)W, (2.5)∇XW = −AW X + ϕ(X)N, (2.6)∇XPY = ∇∗
XPY + C(X, PY )E, (2.7)∇XE = −A∗
EX − τ(X)E, (2.8)
for all E ∈ Γ(Rad TM), N ∈ Γ(ltr(TM)) and W ∈ Γ(S(TM⊥)), where ∇ and ∇∗ areinduced linear connections on TM and S(TM), respectively, B and D are called thelocal second fundamental forms of M , C is the local second fundamental form on S(TM).Furthermore, AN , AW and A∗
E are the shape operators on TM and S(TM) respectively,and τ , ρ, ϕ and δ are differential 1-forms on TM . Notice that ∇∗ is a metric connection
On total mean curvatures of foliated half-lightlike submanifolds 729
on S(TM) while ∇ is generally not a metric connection. In fact, ∇ satisfies the followingrelation
(∇Xg)(Y, Z) = B(X, Y )λ(Z) + B(X, Z)λ(Y ), (2.9)
for all X, Y, Z ∈ Γ(TM), where λ is a 1-form on TM given λ(·) = g(·, N). It is well-knownfrom [4] and [6] that B and D are independent of the choice of S(TM) and they satisfy
B(X, E) = 0, D(X, E) = −ϕ(X), ∀ X ∈ Γ(TM). (2.10)
The local second fundamental forms B, D and C are related to their shape operators bythe following equations
g(A∗EX, Y ) = B(X, Y ), g(A∗
EX, N) = 0, (2.11)g(AW X, Y ) = εD(X, Y ) + ϕ(X)λ(Y ), (2.12)g(AN X, PY ) = C(X, PY ), g(AN X, N) = 0, (2.13)g(AW X, N) = ερ(X), where ε = g(W, W ), (2.14)
for all X, Y ∈ Γ(TM). From equations (2.11) we deduce that A∗E is S(TM)-valued,
self-adjoint and satisfies A∗EE = 0. Let R denote the curvature tensor of M , then
g(R(X, Y )PZ, N) = g((∇XAN )Y, PZ) − g((∇Y AN )X, PZ)+ τ(Y )C(X, PZ) − ετ(X)C(Y, PZ)ρ(Y )D(X, PZ)− ρ(X)D(Y, PZ), ∀ X, Y, Z ∈ Γ(TM). (2.15)
A half-lightlike submanifold (M, g) of a semi-Riemannian manifold M is said to be totallyumbilical [6] if on each coordinate neighborhood U there exist smooth functions H1 andH2 on ltr(TM) and S(TM⊥) respect such that
B(X, Y ) = H1g(X, Y ), D(X, Y ) = H2g(X, Y ), ∀ X, Y ∈ Γ(TM). (2.16)
Furthermore, when M is totally umbilical then the following relations follows by straight-forward calculations
A∗EX = H1PX, P (AW X) = εH2PX, D(X, E) = 0, ρ(E) = 0, (2.17)
for all X, Y ∈ Γ(TM).Next, we suppose that M is a half-lightlike submanifold of M , with an integrable screen
distribution S(TM). Let M ′ be a leaf of S(TM). Notice that for any screen integrablehalf-lightlike M , the leaf M ′ of S(TM) is a co-dimension 3 submanifold of M whose normalbundle is Rad TM ⊕ ltr(TM) ⊥ S(TM⊥). Now, using (2.4) and (2.7) we have
∇XY = ∇∗XY + C(X, PY )E + B(X, Y )N + D(X, Y )W, (2.18)
for all X, Y ∈ Γ(TM ′). Since S(TM) is integrable, then its leave is semi-Riemannian andhence we have
∇XY = ∇∗′XY + h′(X, Y ), ∀ X, Y ∈ Γ(TM ′), (2.19)
where h′ and ∇∗′ are second fundamental form and the Levi-Civita connection of M ′ inM . From (2.18) and (2.19) we can see that
h′(X, Y ) = C(X, PY )E + B(X, Y )N + D(X, Y )W, (2.20)
for all X, Y ∈ Γ(TM ′). Since S(TM) is integrable, then it is well-known from [6] thatC is symmetric on S(TM) and also AN is self-adjoint on S(TM) (see Theorem 4.1.2 fordetails). Thus, h′ given by (2.20) is symmetric on TM ′.
Let L ∈ Γ(Rad TM ⊕ ltr(TM) ⊥ S(TM⊥)), then we can decompose L as
L = aE + bN + cW, (2.21)
730 F. Massamba, S. Ssekajja
for non-vanishing smooth functions on M given by a = g(L, N), b = g(L, E) and c =εg(L, W ). Suppose that g(L, L) > 0, then using (2.21) we obtain a unit normal vector Wto M ′ given by
W = 1g(L, L)
(aE + bN + cW ) = 1g(L, L)
L. (2.22)
Next we define a (1,1) tensor AW
in terms of the operators A∗E , AN and AW by
AW
X = 1g(L, L)
(aA∗EX + bAN X + cAW X), (2.23)
for all X ∈ Γ(TM). Notice that AW
is self-adjoint on S(TM). Applying ∇X to W andusing equations (2.23) (2.4) and (2.11)-(2.13), we have
g(AW
X, PY ) = −g(∇XW , PY ), ∀ X, Y ∈ Γ(TM). (2.24)
Let ∇∗⊥ be the connection on the normal bundle Rad TM ⊕ ltr(TM) ⊥ S(TM⊥). Thenfrom (2.24) we have
∇XW = −AW
X + ∇∗⊥X W , ∀ X ∈ Γ(TM), (2.25)
where
∇∗⊥X W = − 1
g(L, L)X(g(L, L))W + 1
g(L, L)[X(a) − aτ(X)E
+X(b) + bτ(X) + cϕ(X)N + X(c) + aD(X, E) + bρ(X)W ] .
Example 2.1. Let M = (R51, g) be a semi-Riemannian manifold, where g is of sig-
nature (−, +, +, +, +) with respect to canonical basis (∂x1, ∂x2, ∂x3, ∂x4, ∂x5), where(x1, · · · , x5) are the usual coordinates on M . Let M be a submanifold of M and givenparametrically by the following equations
x1 =φ1, x2 = sin φ2 sin φ3, x3 = φ1, x4 = cos φ2 sin φ3,
x5 = cos φ3, where φ2 ∈ [0, 2π] and φ3 ∈ (0, π/2).
Then we have TM = spanE, Z1, Z2 and ltr(TM) = spanN, where
E = ∂x1 + ∂x3, Z1 = cos φ3∂x2 − sin φ2 sin φ3∂x5,
Z2 = cos φ3∂x4 − cos φ2 sin φ3∂x5 and N = 12
(−∂x1 + ∂x3).
Also, by straightforward calculations, we have
W = sin φ2 sin φ3∂x2 + cos φ2 sin φ3∂x4 + cos φ3∂x5.
Thus, S(TM⊥) = spanW and hence M is a half-lightlike submanifold of M . Fur-thermore we have [Z1, Z2] = cos φ2 sin φ3∂x2 − sin φ2 sin φ3∂x4, which leads to [Z1, Z2] =cos φ2 tan φ3Z1−sin φ2 tan φ3Z2 ∈ Γ(S(TM)). Thus, M is a screen integrable half-lightlikesubmanifold of M . Finally, it is easy to see that AN is self-adjoint operator on S(TM).
In the next sections we shall consider screen integrable half-lightlike submanifolds ofsemi-Riemannian manifold M and derive special integral formulas for a foliation of S(TM),whose normal vector is W and with shape operator A
W.
On total mean curvatures of foliated half-lightlike submanifolds 731
3. Newton transformations of AW
Let (Mm+3, g) be a semi-Riemannian manifold and let (Mn+1, g) be a screen integrable
half-lightlike submanifold of M . Then S(TM) admits a foliation and let F be a suchfoliation. Then, the leaves of F are co-dimension three submanifolds of M , whose normalbundle is S(TM)⊥. Let W be unit normal vector to F such that the orientation of M
coincides with that given by F and W . The Levi-Civita connection ∇ on the tangentbundle of M induces a metric connection ∇′ on F. Furthermore, h′ and A
Ware the
second fundamental form and shape operator of F. Notice that AW
is self-adjoint on TF
and at each point p ∈ F has n real eigenvalues (or principal curvatures) κ1(p), · · · , κn(p).Attached to the shape operator A
Ware n algebraic invariants
Sr = σr(κ1, · · · , κn), 1 ≤ r ≤ n,
where σr : M′n → R are symmetric functions given by
σr(κ1, · · · , κn) =∑
1≤i1<···<ir≤n
κi1 · · · κir . (3.1)
Then, the characteristic polynomial of AW
is given by
det(AW
− tI) =n∑
α=0(−1)αSrtn−α,
where I is the identity in Γ(TF). The normalized r-th mean curvature Hr of M ′ is definedby
Hr =(
n
r
)−1
Sr and H0 = 1. (a constant function 1).
In particular, when r = 1 then H1 = 1ntr(A
W) which is the mean curvature of a F. On
the other hand, H2 relates directly with the (intrinsic) scalar curvature of F. Moreover,the functions Sr (Hr respectively) are smooth on the whole M and, for any point p ∈ F,Sr coincides with the r-th mean curvature at p. In this paper, we shall use Sr instead ofHr.
Next, we introduce the Newton transformations with respect to the operator AW
. TheNewton transformations Tr : Γ(TF) → Γ(TF) of a foliation F of a screen integrable half-lightlike submanifold M of an (n + 3)-dimensional semi-Riemannian manifold M withrespect to A
Ware given by by the inductive formula
T0 = I, Tr = (−1)rSrI + AW
Tr−1, 1 ≤ r ≤ n. (3.2)
By Cayley-Hamiliton theorem, we have Tn = 0. Moreover, Tr are also self-adjoint andcommutes with A
W. Furthermore, the following algebraic properties of Tr are well-known
(see [2], [1] and references therein for details).tr(Tr) = (−1)r(n − r)Sr, (3.3)
tr(AW
Tr) = (−1)r(r + 1)Sr+1, (3.4)tr(A2
W Tr) = (−1)r+1(−S1Sr+1 + (r + 2)Sr+2), (3.5)
tr(Tr ∇′XA
W) = (−1)rX(Sr+1) = (−1)rg(∇′Sr+1, X), (3.6)
for all X ∈ Γ(TM). We will also need the following divergence formula for the operatorsTr
div∇′(Tr) = tr(∇′Tr) =n∑
β=1(∇′
ZβTr)Zβ, (3.7)
where Z1, · · · , Zn is a local orthonormal frame field of TF.
732 F. Massamba, S. Ssekajja
4. Integration formulas for F
This section is devoted to derivation of integral formulas of foliation F of S(TM) witha unit normal vector W given by (2.22). By the fact that ∇ is a metric connectionthen g(∇
WW , W ) = 0. This implies that the vector field ∇
WW is always tangent to F.
Our main goal will be to compute the divergence of the vectors Tr∇W
W and Tr∇W
W +(−1)rSr+1W . The following technical lemmas are fundamentally important to this paper.Let E, Zi, N, W, for i = 1, · · · , n be a quasi-orthonormal field of frame of TM , suchthat S(TM) = spanZi and ϵi = g(Zi, Zi).
Lemma 4.1. Let M be a screen integrable half-lightlike submanifold of Mn+3 and let M ′
be a foliation of S(TM). Let AW
be its shape operator, where W is a unit normal vectorto F. Then
g((∇′XA
W)Y, Z) = g(Y, (∇′
XAW
)Z),g((∇′
XTr)Y, Z) = g(Y, (∇′XTr)Z),
for all X, Y, Z ∈ Γ(TF).Proof. By simple calculations we have
g((∇′XA
W)Y, Z) = g(∇′
X(AW
Y ), Z) − g(∇′XY,A
WZ). (4.1)
Using the fact that ∇′ is a metric connection and the symmetry of AW
, (4.1) gives
g((∇′XA
W)Y, Z) = g(Y, ∇′
X(AW
Z)) − g(Y,AW
(∇′XZ)). (4.2)
Then, from (4.2) we deduce the first relation of the lemma. A proof of the second relationfollows in the same way, which completes the proof. Lemma 4.2. Let M be a screen integrable half-lightlike submanifold of M and let F be aco-dimension three foliation of S(TM). Let A
Wbe its shape operator, where W is a unit
normal vector to F. Denote by R the curvature tensor of M . Thendiv∇′(T0) = 0,
div∇′(Tr) = AW
div∇′(Tr−1) +n∑
i=1ϵi(R(W , Tr−1Zi)Zi)′,
where (R(W , X)Z)′ denotes the tangential component of R(W , X)Z for X, Z ∈ Γ(TF).Equivalently, for any Y ∈ Γ(TF) then
g(div∇′(Tr), Y ) =r∑
j=1
n∑i=1
ϵig(R(Tr−1Zi, W )(−AW
)j−1Y, Zi). (4.3)
Proof. The first equation of the lemma is obvious since T0 = I. We turn to the secondrelation. By direct calculations using the recurrence relation (3.2) we derive
div∇′(Tr) = (−1)rdiv∇′(SrI) + div∇′(AW
Tr−1)
= (−1)r∇′Sr + AW
div∇′(Tr−1) +n∑
i=1ϵi(∇′
ZiA
W)Tr−1Zi. (4.4)
Using Codazzi equationg(R(X, Y )Z, W ) = g((∇′
Y AW)X, Z) − g((∇′
XAW
)Y, Z),for any X, Y, Z ∈ Γ(TF) and Lemma 4.1, we have
g((∇′ZiA
W)Y,Tr−1Zi) = g((∇′
Y AW)Zi, Tr−1Zi) + g(R(Y, Zi)Tr−1Zi, W )
= g(Tr−1(∇′Y AW
)Zi, Zi) + g(R(W , Tr−1Zi)Zi, Y ), (4.5)
On total mean curvatures of foliated half-lightlike submanifolds 733
for any Y ∈ Γ(TF). Then applying (4.4) and (4.5) we get
g(div∇′(Tr), Y ) = (−1)rg(∇′Sr, Y ) + tr(Tr−1(∇′Y AW
))
+ g(div∇′(Tr−1), Y ) + g(Y,n∑
i=1ϵiR(W , Tr−1Zi)Zi). (4.6)
Then, applying (4.6) and (3.6) we get the second equation of the lemma. Finally, (4.3)follows immediately by an induction argument.
Notice that when the ambient manifold is a space form of constant sectional curvature,then (R(W , X)Y )′ = 0, for each X, Y ∈ Γ(TF). Hence, from Lemma (4.2) we havediv∇′(Tr) = 0.
Lemma 4.3. Let M be a screen integrable half-lightlike submanifold of M and let F be aco-dimension three foliation of S(TM). Let A
Wbe its shape operator, where W is a unit
normal vector to F. Let Zi be a local field such (∇′XZi)p = 0, for i = 1, · · · , n and any
vector field X ∈ Γ(TM). Then at p ∈ F we have
g(∇′Zi
∇W
W , Zj) = g(A2W
Zi, Zj) − g(R(Zi, W )Zj , W )
− g((∇′WA
W)Zi, Zj) + g(∇
WW , Zi)g(Zj , ∇
WW ).
Proof. Applying ∇Zi to g(∇W
W , Zj) and g(W , ∇W
Zj) in turn and then using the tworesulting equations, we have
−g(∇W
W , ∇ZiZj) = g(∇Zi∇WW , Zj) + g(∇ZiW , ∇
WZj)
+ g(W , ∇Zi∇WZj). (4.7)
Furthermore, by direct calculations using (∇′XZi)p = 0 we have
g((∇′WA
W)Zi, Zj) = g(∇
WW , ZiZj) + g(W , ∇
WZiZj),
and hence
g(A2W
Zi, Zj) − g(R(Zi, W )Zj , W ) − g((∇′WA
W)Zi, Zj)
= g(A2W
Zi, Zj) − g(R(Zi, W )Zj , W )
− g(∇W
W , ZiZj) − g(W , ∇W
ZiZj)
= g(A2W
Zi, Zj) − g(∇ZiZj , ∇W
W )
− g(∇Zi∇WZj , W ) + g(∇[Zi,W ]Zj , W ). (4.8)
Now, applying (4.7), the condition at p and the following relations
∇ZiW =n∑
k=1ϵkg(∇ZiW , Zk)Zk, ∇
WZj = g(∇
WZj , W )W ,
and g(A2W
Zi, Zj) = −∑n
k=1 ϵkg(∇ZiW , Zk)g(∇ZkZj , W ) to the last line of (4.8) and the
fact that S(TM) is integrable we get
g(A2W
Zi, Zj) − g(R(Zi, W )Zj , W ) − g((∇′WA
W)Zi, Zj)
= g(∇′Zi
∇W
W , Zj) − g(∇W
W , Zi)g(Zj , ∇W
W ),
from which the lemma follows by rearrangement.
Notice that, using parallel transport, we can always construct a frame field from theabove lemma.
734 F. Massamba, S. Ssekajja
Proposition 4.4. Let M be a screen integrable half-lightlike submanifold of an indefinitealmost contact manifold M and let F be a foliation of S(TM). Then
div∇′(Tr∇W
W ) = g(div∇′(Tr), ∇W
W ) + (−1)r+1W (Sr+1)
+ (−1)r+1(−S1Sr+1 + (r + 2)Sr+2) −n∑
i=1ϵig(R(Zi, W )TrZi, W )
+ g(∇W
W , Tr∇W
W ),where Zi is a field of frame tangent to the leaves of F.
Proof. From (3.7),we deduce that
div∇′(TrZ) = g(div∇′(Tr), Z) +n∑
i=1ϵig(∇′
ZiZ, TrZi), (4.9)
for all Z ∈ Γ(TF). Then using (4.9), Lemmas 4.2 and 4.3, we obtain the desired result.Hence the proof.
From Proposition 4.4 we have
Theorem 4.5. Let M be a screen integrable half-lightlike submanifold of an indefinitealmost contact manifold M and let F be a co-dimension three foliation of S(TM). Then
div∇(Tr∇W
W ) = g(div∇′(Tr), ∇W
W ) + (−1)r+1W (Sr+1)+ (−1)r+1(−S1Sr+1 + (r + 2)Sr+2)
−n∑
i=1ϵig(R(Zi, W )TrZi, W ).
Proof. A proof follows easily from Proposition 4.4 by recognizing the fact that
div∇(Tr∇W
W ) = div∇′(Tr∇W
W )
− g(∇W
W , Tr∇W
W ),which completes the proof. Theorem 4.6. Let M be a screen integrable half-lightlike submanifold of M and let F bea co-dimension three foliation of S(TM). Then,
div∇(Tr∇W
W + (−1)rSr+1W ) = g(div∇′(Tr), ∇W
W )
+ (−1)r+1(r + 2)Sr+2 −n∑
i=1ϵig(R(Zi, W )TrZi, W ).
Proof. By straightforward calculations we haveS1 = tr(A
W)
= −n∑
i=1ϵig(∇ZiW , Zi)
= −n+1∑i=1
ϵig(∇ZiW , Zi)
= −div∇(W ),
where Zn+1 = W . From this equation we deduce
div∇(Sr+1W ) = −S1Sr+1 + W (Sr+1). (4.10)Then from (4.10) and Theorem 4.5 we get our assertion, hence the proof.
On total mean curvatures of foliated half-lightlike submanifolds 735
Next, we let dV denote the volume form M . Then from Theorem 4.6 we have thefollowing
Corollary 4.7. Let M be a screen integrable half-lightlike submanifold of a compact semi-Riemannian manifold M and let F be a co-dimension three foliation of S(TM). Then∫
Mg(div∇′(Tr), ∇
WW )dV =
∫M
((−1)r(r + 2)Sr+2
+n∑
i=1ϵig(R(Zi, W )TrZi, W )dV.
Setting r = 0 in the above corollary we get
Corollary 4.8. Let M be a screen integrable half-lightlike submanifold of a compact semi-Riemannian manifold M and let F be a co-dimension three foliation of S(TM) with meancurvatures Sr. Then for r = 0 we have∫
M2S2dV =
∫M
Ric(W , W )dV,
where Ric(W , W ) =n∑
i=1ϵig(R(Zi, W )W , Zi).
Notice that the equation in Corollary 4.8 is the lightlike analogue of (3.5) in [2] forco-dimension one foliations on Riemannian manifolds.
Next, we will discuss some consequences of the integral formulas developed so far.A semi-Riemannian manifold M of constant sectional curvature c is called a semi-
Riemannian space form [4, 6] and is denoted by M(c). Then, the curvature tensor R ofM(c) is given by
R(X, Y )Z = cg(Y , Z)X − g(X, Z)Y , ∀ X, Y , Z ∈ Γ(TM). (4.11)
Theorem 4.9. Let M be a screen integrable half-lightlike submanifold of a compact semi-Riemannian space form M(c) of constant sectional curvature c. Let F be a co-dimensionthree foliation of its screen distribution S(TM). If V is the total volume of M , then
∫M
SrdV =
0, r = 2k + 1,
cr2
(n2r2
)V, r = 2k,
(4.12)
for positive integers k.
Proof. By setting X = Zi, Y = W and Z = TrZi in (4.11) we deduce that
R(Zi, W )TrZi = −cg(Zi, TrZi)W .
Then substituting this equation in Corollary 4.7 we obtain∫M
g(div∇′(Tr), ∇W
W )dV =∫
M((−1)r(r + 2)Sr+2 − ctr(Tr))dV.
Since M is of constant sectional curvature c, then Lemma 4.2 implies that Tr = 0 for anyr and hence the above equation simplifies to
(r + 2)∫
MSr+2dV = c(n − r)
∫M
SrdV. (4.13)
Since S1 = −div∇(W ) and that M is compact, then∫
M S1dV = 0. Using this fact togetherwith (4.13), mathematical induction gives
∫M SrdV = 0 for all r = 2k + 1 (i.e., r odd).
736 F. Massamba, S. Ssekajja
For r even we will consider r = 2m and n = 2l (i.e., both M and M are odd dimensional).With these conditions, (4.13) reduces to∫
MS2m+2dV = c
l − m
1 + m
∫M
S2mdV. (4.14)
Now setting m = 0, 1, · · · and S0 = 1 in (4.14) we obtain∫M
S2dV = clV,
∫M
S4dV = c2 (l − 1)l2
V,
and more generally∫M
S2kdV = ck (l − k + 1)(l − k + 2)(l − k + 3) · · · l
k!V. (4.15)
Hence, our assertion follows from 4.15, which completes the proof.
Next, when M is Einstein i.e., Ric = µg we have the following.
Theorem 4.10. Let M be a screen integrable half-lightlike submanifold of an Einsteincompact semi-Riemannian manifold M . Let F be a co-dimension three foliation of itsscreen distribution S(TM) with totally umbilical leaves. If V is the total volume of M ,then
∫M
SrdV =
0, r = 2k + 1,
(µn
)n2
(n2r2
)V, r = 2k,
(4.16)
for positive integers k.
Proof. Suppose that AW
= 1nSrI. Then by direct calculations using the formula for Tr
we derive Tr = (−1)r+1 (n−r)n SrI. Then, under the assumptions of the theorem we obtain
Ric(W , ∇W
W ) = 0 and Ric(W , W ) = µ and hence, Corollary 4.7 reduces to
n(r + 2)∫
MSr+2dV = λ(n − r)
∫M
SrdV. (4.17)
Notice that (4.17) is similar to (4.13) and hence following similar steps as in the previoustheorem we get
∫M SrdV = 0 for r odd and for r even we get∫M
S2kdV =(
µ
n
)k (l − k + 1)(l − k + 2)(l − k + 3) · · · l
k!V,
which complete the proof.
5. Screen umbilical half-lightlike submanifoldsIn this section we consider totally umbilical half-lightlike submanifolds of semi-Riemannian
manifold, with a totally umbilical screen distribution and thus, give a generalized versionof Theorem 4.3.7 of [6] and its Corollaries, via Newton transformations of the operatorAN .
A screen distribution S(TM) of a half-lightlike submanifold M of a semi-Riemannianmanifold M is said to be totally umbilical [6] if on any coordinate neighborhood U thereexist a function K such that
C(X, PY ) = Kg(X, PY ), ∀ X, Y ∈ Γ(TM). (5.1)
In case K = 0, we say that S(TM) is totally geodesic. Furthermore, if S(TM) is totallyumbilical then by straightforward calculations we have
AN X = PX, C(E, PX) = 0, ∀ X ∈ Γ(TM). (5.2)
On total mean curvatures of foliated half-lightlike submanifolds 737
Let E, Zi, for i = 1, · · · , n, be a quasi-orthonormal frame field of TM which diagonalizesAN . Let l0, l1, · · · , ln be the respective eigenvalues (or principal curvatures). Then asbefore, the r-th mean curvature Sr is given by
Sr = σr(l0, · · · , ln) and S0 = 1.
The characteristic polynomial of AN is given by
det(AN − tI) =n∑
α=0(−1)αSrtn−α,
where I is the identity in Γ(TM). The normalized r-th mean curvature Hr of M is defined
by(
n
r
)Hr = Sr and H0 = 1. The Newton transformations Tr : Γ(TM) → Γ(TM) of
AN are given by the inductive formula
T0 = I, Tr = (−1)rSrI + AN Tr−1, 1 ≤ r ≤ n. (5.3)
By Cayley-Hamiliton theorem, we have Tn+1 = 0. Also, Tr satisfies the following proper-ties.
tr(Tr) = (−1)r(n + 1 − r)Sr, (5.4)tr(AN Tr) = (−1)r(r + 1)Sr+1, (5.5)tr(A2
N Tr) = (−1)r+1(−S1Sr+1 + (r + 2)Sr+2), (5.6)tr(Tr ∇XAN ) = (−1)rX(Sr+1), (5.7)
for all X ∈ Γ(TM).
Proposition 5.1. Let (M, g) be a totally umbilical half-lightlike submanifold of a semi-Riemannian manifold M of constant sectional curvature c. Then
g(div∇(Tr), X) = (−1)r−1λ(X)E(Sr) − τ(X)tr(AN Tr−1)− cλ(X)tr(Tr−1) + g(div∇(Tr−1), AN X) + g((∇EAN )Tr−1E, X)
+n∑
i=1ϵi−λ(X)B(Zi, AN (Tr−1Zi))
+ ετ(Zi)C(X, Tr−1Zi)ρ(X)D(Zi, Tr−1Zi) − ρ(Zi)D(X, Tr−1Zi),
for any X ∈ Γ(TM).
Proof. From the recurrence relation (5.3), we derive
g(div∇(Tr), X) = (−1)rPX(Sr) + g((∇EAN )Tr−1E, X)
+ g(div∇(Tr−1), AN X) +n∑
i=1ϵig((∇ZiAN )Tr−1Zi, X), (5.8)
for any X ∈ Γ(TM). But
g((∇ZiAN )Tr−1Zi, X) = g(Tr−1Zi, (∇ZiAN )X) + g(∇ZiAN (Tr−1Zi), X)− g(∇Zi(AN X), Tr−1Zi) + g(AN (∇ZiX), Tr−1Zi)− g(AN (∇ZiTr−1Zi), X), (5.9)
for all X ∈ Γ(TM).
Then applying (2.9) to (5.9) while considering the fact that AN is screen-valued, we get
g((∇ZiAN )Tr−1Zi, X) = g(Tr−1Zi, (∇ZiAN )X) − λ(X)B(Zi, AN (Tr−1Zi)). (5.10)
738 F. Massamba, S. Ssekajja
Furthermore, using (2.15) and (4.11), the first term on the right hand side of (5.10) reducesto
g(Tr−1Zi,(∇ZiAN )X) = −cλ(X)g(Zi, Tr−1Zi) + g((∇XAN )Zi, Tr−1Zi)− τ(X)C(Zi, Tr−1Zi) + ετ(Zi)C(X, Tr−1Zi)ρ(X)D(Zi, Tr−1Zi)− ρ(X)D(X, Tr−1Zi), (5.11)
for any X ∈ Γ(TM). Finally, replacing (5.11) in (5.10) and then put the resulting equationin (5.8) we get the desired result.
Next, from Proposition 5.1 we have the following.
Theorem 5.2. Let (M, g) be a half-lightlike submanifold of a semi-Riemannian manifoldM(c) of constant curvature c, with a proper totally umbilical screen distribution S(TM).If M is also totally umbilical, then the r-th mean curvature Sr, for r = 0, 1, · · · , n, withrespect to AN are solution of the following differential equation
E(Sr+1) − τ(E)(r + 1)Sr+1 − c(−1)r(n + 1 − r)Sr = H1(r + 1)Sr+1.
Proof. Replacing X with E in the Proposition 5.1 and then using (2.16) and (5.2) weobtain, for all r = 0, 1, · · · , n,
E(Sr+1) − (−1)rτ(E)tr(AN Tr) − c(−1)rtr(Tr) = (−1)rH1tr(AN Tr),from which the result follows by applying (5.4) and (5.5). Corollary 5.3. Under the hypothesis of Theorem 5.2, the induced connection ∇ on Mis a metric connection, if and only if, the r-th mean curvature Sr with respect to AN aresolution of the following equation
E(Sr+1) − τ(E)(r + 1)Sr+1 − c(−1)r(n + 1 − r)Sr = 0.
Also the following holds.
Corollary 5.4. Under the hypothesis of Theorem 5.2, M(c) is a semi-Euclidean space, ifand only if, the r-th mean curvature Sr with respect to AN are solution of the followingequation
E(Sr+1) − τ(E)(r + 1)Sr+1 = H1(r + 1)Sr+1.
Notice that Theorem 5.2 and Corollary 5.3 are generalizations of Theorem 4.3.7 andCorollary 4.3.8, respectively, given in [6].
Acknowledgment. This work is based on the research supported wholly by the Na-tional Research Foundation of South Africa (Grant Numbers: 95931 and 106072).
References[1] K. Andrzejewski, W. Kozlowski and K. Niedzialomski, Generalized Newton transfor-
mation and its applications to extrinsic geometry, Asian J. Math. 20 (2), 293–322,2016.
[2] K. Andrzejewski and Pawel G. Walczak, The Newton transformation and new integralformulae for foliated manifolds, Ann. Glob. Anal. Geom. 37 (2), 103–111, 2010.
[3] C. Calin, Contributions to geometry of CR-submanifold, Ph.D. thesis, University ofIasi (Romania), 1998.
[4] K.L. Duggal and A. Bejancu, Lightlike submanifolds of semi-Riemannian manifoldsand applications, Mathematics and Its Applications, Kluwer Academic Publishers,1996.
[5] K.L. Duggal and B. Sahin, Screen conformal half-lightlike submanifolds, Int. J. Math.Math. Sci. 2004 (68), 3737–3753, 2004.
On total mean curvatures of foliated half-lightlike submanifolds 739
[6] K.L. Duggal and B. Sahin, Differential geometry of lightlike submanifolds. Frontiersin Mathematics, Birkhäuser Verlag, Basel, 2010.
[7] D.N. Kupeli, Singuler semi-Riemannian geometry, Mathematics and Its Applications,Vol. 366, Kluwer Academic Publishers, 1996.
[8] F. Massamba, Totally contact umbilical lightlike hypersurfaces of indefinite Sasakianmanifolds, Kodai Math. J. 31, 338–358, 2008.
[9] F. Massamba, On semi-parallel lightlike hypersurfaces of indefinite Kenmotsu mani-folds, J. Geom. 95, 73–89, 2009.
[10] F. Massamba and S. Ssekajja, Some remarks on quasi generalized CR-null geometryin indefinite nearly cosymplectic manifolds, Int. J. Math. Math. Sci. Art. ID 9613182,10 pp, 2016.
[11] E. Yasar, A.C. Coken, and A. Yucesan, Lightlike hypersurfaces in semi-Riemannianmanifold with semi-symmetric non-metric connection, Math. Scand. 102 (2), 253–264, 2008.
Hacettepe Journal ofMathematics & Statistics
Hacet. J. Math. Stat.Volume 49 (2) (2020), 740 – 753
DOI : 10.15672/hujms.588747
Research Article
Pair of generalized derivations acting onmultilinear polynomials in prime rings
Basudeb Dhara∗1, Sukhendu Kar2, Priyadwip Das2
1Department of Mathematics, Belda College, Belda, Paschim Medinipur, 721424, W.B., India2Department of Mathematics, Jadavpur University, Kolkata-700032, W.B., India
AbstractLet R be a noncommutative prime ring of characteristic different from 2 with Utumiquotient ring U and extended centroid C and f(r1, . . . , rn) be a multilinear polynomial overC, which is not central valued on R. Suppose that F and G are two nonzero generalizedderivations of R such that G = Id (identity map) and
F (f(r)2) = F (f(r))G(f(r)) + G(f(r))F (f(r))for all r = (r1, . . . , rn) ∈ Rn. Then one of the following holds:
(1) there exist λ ∈ C and µ ∈ C such that F (x) = λx and G(x) = µx for all x ∈ Rwith 2µ = 1;
(2) there exist λ ∈ C and p, q ∈ U such that F (x) = λx and G(x) = px + xq for allx ∈ R with p + q ∈ C, 2(p + q) = 1 and f(x1, . . . , xn)2 is central valued on R;
(3) there exist λ ∈ C and a ∈ U such that F (x) = [a, x] and G(x) = λx for all x ∈ Rwith f(x1, . . . , xn)2 is central valued on R;
(4) there exist λ ∈ C and a, b ∈ U such that F (x) = ax + xb and G(x) = λx for allx ∈ R with a + b ∈ C, 2λ = 1 and f(x1, . . . , xn)2 is central valued on R;
(5) there exist a, p ∈ U such that F (x) = xa and G(x) = px for all x ∈ R, with(p − 1)a = −ap ∈ C and f(x1, . . . , xn)2 is central valued on R;
(6) there exist a, q ∈ U such that F (x) = ax and G(x) = xq for all x ∈ R witha(q − 1) = −qa ∈ C and f(x1, . . . , xn)2 is central valued on R.
Mathematics Subject Classification (2010). 16N60, 16W25, 16R50
Keywords. prime ring, derivation, generalized derivation, extended centroid, Utumiquotient ring
1. IntroductionThroughout this paper R always denotes an associative prime ring with extended cen-
troid C and U its Utumi ring of quotients. By a derivation, we mean an additive mappingd : R → R such that d(xy) = d(x)y + xd(y) holds for all x, y ∈ R. By a generalizedderivation, we mean an additive mapping F : R → R such that F (xy) = F (x)y + xd(y)
∗Corresponding Author.Email addresses: [email protected] (B. Dhara), [email protected] (S. Kar),
[email protected] (P. Das)Received: 23.12.2017; Accepted: 18.03.2019
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holds for all x, y ∈ R, where d is a derivation of R. Thus any derivation is a generalizedderivation.
A famous result proved by Posner [17, Theorem 2] states that if a prime ring R hasa nonzero derivation d such that [d(x), x] ∈ Z(R) for all x ∈ R, then R is commutative.Brešar [2] studied the case d(x)x−xδ(x) ∈ Z(R) for all x ∈ R, where d and δ are two deriva-tions of a prime ring R and obtained that either d = δ = 0 or R is commutative. Afterthat in [13] Lee and Shiue extended the previous result considering multilinear polynomial.They proved that if d(f(x1, . . . , xn))f(x1, . . . , xn) − f(x1, . . . , xn)δ(f(x1, . . . , xn)) ∈ Z(R)for all x1, . . . , xn ∈ I, where I is a nonzero ideal of R and f(x1, . . . , xn) is a non-centralmultilinear polynomial over C, then either d = 0 = δ or d = −δ and f(x1, . . . , xn)2 iscentral valued on RC unless char(R) = 2 and dimCRC = 4.
Recently in [4], De Filippis et al. showed that if d and δ are nonzero derivations of Rand f(x1, . . . , xn) is a multilinear polynomial over C, non-central valued on R, such that[d(f(x1, . . . , xn)), δ(f(x1, . . . , xn))] ∈ Z(R) for all x1, . . . , xn ∈ R, then d, δ are lineardependent over C unless when char(R) = 2 and dimCRC = 4.
More recently, Fosner and Vukman [7] proved that if R is a prime ring of char(R) = 2,F1 and F2 are generalized derivations of R such that F1(x)F2(x) + F2(x)F1(x) = 0 for allx ∈ R then either F1 = 0 or F2 = 0. In [18], Rania and Scudo extended this result tothe case G(f(x1, . . . , xn))d(f(x1, . . . , xn)) + d(f(x1, . . . , xn))G(f(x1, . . . , xn)) = 0 for allx1, . . . , xn ∈ R, where G is a generalized derivation of R and d is any derivation of R, andproved that either G = 0 or d = 0, except when d is inner, there exists λ ∈ C such thatG(x) = λx, ∀x ∈ R and f(x1, . . . , xn)2 is central valued on R. Recently, in [19] Yarbiland De Filippis studied the same situation, when G and d are two skew derivations of Rassociated to the same automorphism α and obtained that either G = 0 or d = 0. Hereskew derivation means an additive mapping d : R → R such that d(xy) = d(x)y+α(x)d(y)for all x, y ∈ R, where α is an automorphism of R.
Recently, Dhara et al. [6] extended the above result by taking generalized derivation Fin the place of derivation d, that is,
F (f(x1, . . . , xn))G(f(x1, . . . , xn)) + G(f(x1, . . . , xn))F (f(x1, . . . , xn)) = 0,
where F, G are two generalized derivations of R. In the present paper, we consider thecase F (f(r)2) = F (f(r))G(f(r)) + G(f(r))F (f(r)) for all r = (r1, . . . , rn) ∈ Rn, where Fand G are two generalized derivations of R. If G = Id (identity map), then F becomesa derivation of R. So our interest is to study the case when G = Id. More precisely, weprove the following theorem.
Main Theorem. Let R be a noncommutative prime ring of characteristic different from2 with Utumi quotient ring U and extended centroid C, and f(r1, . . . , rn) be a multilinearpolynomial over C, which is not central valued on R. Suppose that F and G are twononzero generalized derivations of R such that G = Id (identity map) and
F (f(r)2) = F (f(r))G(f(r)) + G(f(r))F (f(r))
for all r = (r1, . . . , rn) ∈ Rn. Then one of the following holds:(1) there exist λ ∈ C and µ ∈ C such that F (x) = λx and G(x) = µx for all x ∈ R
with 2µ = 1;(2) there exist λ ∈ C and p, q ∈ U such that F (x) = λx and G(x) = px + xq for all
x ∈ R with p + q ∈ C, 2(p + q) = 1, and f(x1, . . . , xn)2 is central valued on R;(3) there exist λ ∈ C and a ∈ U such that F (x) = [a, x] and G(x) = λx for all x ∈ R
with f(x1, . . . , xn)2 is central valued on R;(4) there exist λ ∈ C and a, b ∈ U such that F (x) = ax + xb and G(x) = λx for all
x ∈ R with a + b ∈ C, 2λ = 1 and f(x1, . . . , xn)2 is central valued on R;
742 B. Dhara, S. Kar, P. Das
(5) there exist a, p ∈ U such that F (x) = xa and G(x) = px for all x ∈ R, with(p − 1)a = −ap ∈ C and f(x1, . . . , xn)2 is central valued on R;
(6) there exist a, q ∈ U such that F (x) = ax and G(x) = xq for all x ∈ R witha(q − 1) = −qa ∈ C and f(x1, . . . , xn)2 is central valued on R.
Following corollaries are straightforward.
Corollary 1.1. Let R be a noncommutative prime ring of characteristic different from2 with Utumi quotient ring U and extended centroid C, and f(r1, . . . , rn) be a multilin-ear polynomial over C, which is not central valued on R. Suppose that F is a nonzerogeneralized derivation of R and d is a nonzero derivation of R such that
F (f(r)2) = F (f(r))d(f(r) + d(f(r))F (f(r)for all r = (r1, . . . , rn) ∈ Rn. Then there exist λ ∈ C and p ∈ U such that F (x) = λx andd(x) = [p, x] for all x ∈ R with f(x1, . . . , xn)2 is central valued on R.
Corollary 1.2. Let R be a noncommutative prime ring of characteristic different from2 with Utumi quotient ring U and extended centroid C, and f(r1, . . . , rn) be a multilin-ear polynomial over C, which is not central valued on R. Suppose that G is a nonzerogeneralized derivation of R such that
G(f(r))f(r) + f(r)G(f(r)) = f(r)2
for all r = (r1, . . . , rn) ∈ Rn, then one of the following holds:(1) there exists µ ∈ C such that G(x) = µx for all x ∈ R with 2µ = 1;(2) there exist p, q ∈ U such that G(x) = px + xq for all x ∈ R with p + q ∈ C,
2(p + q) = 1 and f(x1, . . . , xn)2 is central valued on R.
2. Main resultsLemma 2.1. [1, Lemma 3] Let R be a noncommutative prime ring with Utumi quotientring U and extended centroid C, and f(x1, . . . , xn) be a multilinear polynomial over C,which is not central valued on R. Suppose that there exist a, b, c, q ∈ U such that (af(r) +f(r)b)f(r) − f(r)(cf(r) + f(r)q) = 0 for all r = (r1, . . . , rn) ∈ Rn. Then one of thefollowing holds:
(1) a, q ∈ C and q − a = b − c ∈ C;(2) f(x1, . . . , xn)2 is central valued on R and q − a = b − c ∈ C;(3) char(R) = 2 and R satisfies s4.
In particular, from above Lemma, we have the followings:
Lemma 2.2. Let R be a noncommutative prime ring of characteristic different from 2with Utumi quotient ring U and extended centroid C, and f(x1, . . . , xn) be a multilinearpolynomial over C, which is not central valued on R. Suppose that there exist a, b, q ∈ Usuch that af(r)2 + f(r)2q + f(r)bf(r) = 0 for all r = (r1, . . . , rn) ∈ Rn. Then one of thefollowing holds:
(1) a, q ∈ C and q + a = −b ∈ C;(2) f(x1, . . . , xn)2 is central valued on R and q + a = −b ∈ C;(3) char(R) = 2 and R satisfies s4.
Lemma 2.3. [6, Corollary 2.14] Let R be a prime ring of characteristic different from 2,with Utumi quotient ring U and extended centroid C, and f(x1, . . . , xn) be a multilinearpolynomial over C. Suppose that d and δ are two nonzero derivations of R such that
d(f(r))δ(f(r)) + δ(f(r))d(f(r)) = 0for all r = (r1, . . . , rn) ∈ Rn. Then f(x1, . . . , xn) is central valued on R.
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Lemma 2.4. [6, Lemma 2.10] Let R be a prime ring of characteristic different from 2,U its Utumi quotient ring, and C its extended centroid, and f(x1, . . . , xn) a multilinearpolynomial over C which is non-central valued on R. Suppose that a, b, p ∈ U such that
af(r)2b + f(r)pf(r) = 0
for all r = (r1, . . . , rn) ∈ Rn. Then one of the following holds:(1) a ∈ C and ab = −p ∈ C;(2) b ∈ C and ab = −p ∈ C;(3) f(x1, . . . , xn)2 is central valued on R and ab = −p ∈ C.
Lemma 2.5. [5, Lemma 1.5] Let C be an infinite field and m ≥ 2. If A1, . . . , Ak are notscalar matrices in Mm(C) then there exists some invertible matrix P ∈ Mm(C) such thatany matrices PA1P −1, . . . , PAkP −1 have all nonzero entries.
Proposition 2.6. Let R = Mm(C) be the ring of all m × m matrices over the infinitefield C, f(x1, . . . , xn) a non-central multilinear polynomial over C and a, b, p, q ∈ R. If
(af(r)2 + f(r)2b) = (af(r) + f(r)b)(pf(r) + f(r)q) + (pf(r) + f(r)q)(af(r) + f(r)b)
for all r = (r1, . . . , rn) ∈ Rn, then either a or p and either b or q are scalar matrices.
Proof. By our assumption, R satisfies the generalized polynomial identity
(af(r1, . . . , rn)2 + f(r1, . . . , rn)2b)= (af(r1, . . . , rn) + f(r1, . . . , rn)b)(pf(r1, . . . , rn) + f(r1, . . . , rn)q)+(pf(r1, . . . , rn) + f(r1, . . . , rn)q)(af(r1, . . . , rn) + f(r1, . . . , rn)b). (2.1)
We assume first that a /∈ Z(R) and p /∈ Z(R). Now we shall show that this case leads toa contradiction.Since a /∈ Z(R) and p /∈ Z(R), by Lemma 2.5 there exists a C-automorphism ϕ of Mm(C)such that a1 = ϕ(a), p1 = ϕ(p) have all nonzero entries. Clearly a1, p1, b1 = ϕ(b) andq1 = ϕ(q) must satisfy the condition (2.1). Without loss of generality we may replacea, b, p, q with a1, b1, p1, q1, respectively.Here ekl denotes the usual matrix unit with 1 in (k, l)-entry and zero elsewhere. Sincef(x1, . . . , xn) is not central, by [14] (see also [15]), there exist u1, . . . , un ∈ Mm(C) and γ ∈C −0 such that f(u1, . . . , un) = γekl, with k = l. Moreover, since the set f(r1, . . . , rn) :r1, . . . , rn ∈ Mm(C) is invariant under the action of all C-automorphisms of Mm(C), thenfor any i = j there exist r1, . . . , rn ∈ Mm(C) such that f(r1, . . . , rn) = eij . Hence from(2.1) we have
0 = (aeij + eijb)(peij + eijq) + (peij + eijq)(aeij + eijb) (2.2)
and then left multiplying by eij , it follows eijaeijpeij+eijpeijaeij = 0, which gives 2ajipji =0, that is a contradiction, since a and p have all nonzero entries. Thus we conclude thateither a or p is central.
Similarly, we can prove that b or q is central.Therefore we conclude that either a or p and either b or q are scalar matrices.
Proposition 2.7. Let R = Mm(C) be the ring of all matrices over the field C withchar(R) = 2 and f(x1, . . . , xn) a non-central multilinear polynomial over C and a, b, p, q ∈R. If
(af(r)2 + f(r)2b) = (af(r) + f(r)b)(pf(r) + f(r)q) + (pf(r) + f(r)q)(af(r) + f(r)b)
for all r = (r1, . . . , rn) ∈ Rn, then either a or p and either b or q are scalar matrices.
Proof. If one assumes that C is infinite, then the conclusions follow by Proposition 2.6.Now let C be finite and K be an infinite field which is an extension of the field C. Let
744 B. Dhara, S. Kar, P. Das
R = Mm(K) ∼= R ⊗C K. Notice that the multilinear polynomial f(r1, . . . , rn) is centralvalued on R if and only if it is central valued on R. Consider the generalized polynomial
P (r1, . . . , rn) = (af(r1, . . . , rn)2 + f(r1, . . . , rn)2b)−(af(r1, . . . , rn) + f(r1, . . . , rn)b)(pf(r1, . . . , rn) + f(r1, . . . , rn)q)−(pf(r1, . . . , rn) + f(r1, . . . , rn)q)(af(r1, . . . , rn) + f(r1, . . . , rn)b) (2.3)
which is a generalized polynomial identity for R.Moreover, it is a multi-homogeneous of multi-degree (2, . . . , 2) in r1, . . . , rn.Hence the complete linearization of P (r1, . . . , rn) is a multilinear generalized polynomialΘ(r1, . . . , rn, y1, . . . , yn) in 2n indeterminates, moreover
Θ(r1, . . . , rn, r1, . . . , rn) = 2nP (r1, . . . , rn).
Clearly the multilinear polynomial Θ(r1, . . . , rn, y1, . . . , yn) is a generalized polynomialidentity for R and R too. Since char(C) = 2 we obtain P (r1, . . . , rn) = 0 for all r1, . . . , rn ∈R and then conclusion follows from Proposition 2.6.
In the above Proposition, replacing bp = b′ and qa = q′, it is straightforward to provethe following:
Corollary 2.8. Let R = Mm(C) be the ring of all matrices over the field C with char(R) =2 and f(x1, . . . , xn) a non-central multilinear polynomial over C and a, b, p, q, b′, q′ ∈ R. If
(af(r)2 + f(r)2b) = af(r)(pf(r) + f(r)q) + f(r)b′f(r) + f(r)bf(r)q
+pf(r)(af(r) + f(r)b) + f(r)q′f(r) + f(r)qf(r)bfor all r = (r1, . . . , rn) ∈ Rn, then either a or p and either b or q are scalar matrices.
Lemma 2.9. Let R be a noncommutative prime ring of characteristic different from 2with Utumi quotient ring U and extended centroid C, and f(r1, . . . , rn) be a multilinearpolynomial over C, which is not central valued on R. Suppose that F and G ( = Id, identitymap) are two nonzero inner generalized derivations of R such that
F (f(r)2) = F (f(r))G(f(r)) + G(f(r))F (f(r))
for all r = (r1, . . . , rn) ∈ Rn. Then one of the following holds:(1) there exist λ ∈ C and µ ∈ C such that F (x) = λx and G(x) = µx for all x ∈ R
with 2µ = 1;(2) there exist λ ∈ C and p, q ∈ U such that F (x) = λx and G(x) = px + xq for all
x ∈ R with p + q ∈ C, 2(p + q) = 1 and f(x1, . . . , xn)2 is central valued on R;(3) there exist λ ∈ C and a ∈ U such that F (x) = [a, x] and G(x) = λx for all x ∈ R
with f(x1, . . . , xn)2 is central valued on R;(4) there exist λ ∈ C and a, b ∈ U such that F (x) = ax + xb and G(x) = λx for all
x ∈ R with a + b ∈ C, 2λ = 1 and f(x1, . . . , xn)2 is central valued on R;(5) there exist a, p ∈ U such that F (x) = xa and G(x) = px for all x ∈ R, with
(p − 1)a = −ap ∈ C and f(x1, . . . , xn)2 is central valued on R;(6) there exist a, p ∈ U such that F (x) = ax and G(x) = xq for all x ∈ R with
a(q − 1) = −qa ∈ C and f(x1, . . . , xn)2 is central valued on R.
Proof. Since F and G are inner generalized derivations of R, there exist a, b, p, q ∈ Usuch that F (x) = ax + xb and G(x) = px + xq for all x ∈ R. Then by hypothesis, we have
h(r1, . . . , rn) = (af(r1, . . . , rn)2 + f(r1, . . . , rn)2b)−(af(r1, . . . , rn) + f(r1, . . . , rn)b)(pf(r1, . . . , rn) + f(r1, . . . , rn)q)
−(pf(r1, . . . , rn) + f(r1, . . . , rn)q)(af(r1, . . . , rn) + f(r1, . . . , rn)b) = 0 (2.4)
745
for all r1, . . . , rn ∈ R. Since R and U satisfy the same generalized polynomial identities(GPI) (see [3]), U satisfies h(r1, . . . , rn) = 0 that is
h(r1, . . . , rn) = (af(r1, . . . , rn)2 + f(r1, . . . , rn)2b)−(af(r1, . . . , rn) + f(r1, . . . , rn)b)(pf(r1, . . . , rn) + f(r1, . . . , rn)q)
−(pf(r1, . . . , rn) + f(r1, . . . , rn)q)(af(r1, . . . , rn) + f(r1, . . . , rn)b) = 0 (2.5)
for all r1, . . . , rn ∈ U . Suppose that h(r1, . . . , rn) is a trivial GPI for U and Cr1, . . . , rn,the free C-algebra in noncommuting indeterminates r1, . . . , rn. Then, h(r1, . . . , rn) is zeroelement in T = U ∗C Cr1, . . . , rn. This implies that a, p, 1 is linearly independent overC. Let αp + βa + γ = 0, where α, β, γ ∈ C. If α = 0, then β = 0 and hence a ∈ C. Ifα = 0, then p = λa + µ for some λ, µ ∈ C. In this case our identity reduces to
(af(r1, . . . , rn)2 + f(r1, . . . , rn)2b)−(af(r1, . . . , rn) + f(r1, . . . , rn)b)((λa + µ)f(r1, . . . , rn) + f(r1, . . . , rn)q)
−((λa + µ)f(r1, . . . , rn) + f(r1, . . . , rn)q)(af(r1, . . . , rn)+f(r1, . . . , rn)b) = 0 (2.6)
in T . If a is not in C, then from above we have
af(r1, . . . , rn)((f(r1, . . . , rn) − 2λaf(r1, . . . , rn) − µf(r1, . . . , rn)−f(r1, . . . , rn)q − λf(r1, . . . , rn)b) = 0 (2.7)
in T , that is
af(r1, . . . , rn)(2λaf(r1, . . . , rn) + f(r1, . . . , rn)(µ + q + λb − 1)) = 0. (2.8)
This implies that λa ∈ C and hence p = (λa + µ) ∈ C. Thus we conclude that eithera ∈ C or p ∈ C. Similarly, we can prove that either b ∈ C or q ∈ C.
Next suppose that h(r1, . . . , rn) is a non-trivial GPI for U . In case C is infinite, we haveh(r1, . . . , rn) = 0 for all r1, . . . , rn ∈ U ⊗C C, where C is the algebraic closure of C. Sinceboth U and U ⊗C C are prime and centrally closed [8, Theorems 2.5 and 3.5], we mayreplace R by U or U ⊗C C according to C finite or infinite. Then R is centrally closed overC and h(r1, . . . , rn) = 0 for all r1, . . . , rn ∈ R. By Martindale’s theorem [16], R is then aprimitive ring with nonzero socle soc(R) and with C as its associated division ring. Then,by Jacobson’s theorem [10, p.75], R is isomorphic to a dense ring of linear transformationsof a vector space V over C. Assume first that V is finite dimensional over C, that is,dimCV = m. By density of R, we have R ∼= Mm(C). Since f(r1, . . . , rn) is not centralvalued on R, R must be noncommutative and so m ≥ 2. In this case, by Proposition 2.7,we get that either a or p and either b or q are in C. If V is infinite dimensional over C,then for any e2 = e ∈ soc(R) we have eRe ∼= Mt(C) with t =dimCV e. In this case weprove that either a or p are in C. To prove this, assume that a /∈ C and p /∈ C. Thenthere exist h1, h2 ∈ soc(R) such that [a, h1] = 0 and [p, h2] = 0. By Litoff’s Theorem [9],there exists idempotent e ∈ soc(R) such that ah1, h1a, ph2, h2p, h1, h2 ∈ eRe. We haveeRe ∼= Mk(C) with k =dimCV e. Since R satisfies generalized identity
eaf(er1e, . . . , erne)2 + f(er1e, . . . , erne)2be
= e(af(er1e, . . . , erne) + f(er1e, . . . , erne)b) (2.9).(pf(er1e, . . . , erne) + f(er1e, . . . , erne)q)+(pf(er1e, . . . , erne) + f(er1e, . . . , erne)q).(af(er1e, . . . , erne) + f(er1e, . . . , erne)b)e, (2.10)
746 B. Dhara, S. Kar, P. Das
the subring eRe satisfieseaef(r1, . . . , rn)2 + f(r1, . . . , rn)2ebe
= eaef(r1, . . . , rn)(epef(r1, . . . , rn) + f(r1, . . . , rn)eqe)+f(r1, . . . , rn)ebpef(r1, . . . , rn) + f(r1, . . . , rn)ebef(r1, . . . , rn)eqe
+epef(r1, . . . , rn)(eaef(r1, . . . , rn) + f(r1, . . . , rn)ebe)+f(r1, . . . , rn)eqaef(r1, . . . , rn) + f(r1, . . . , rn)eqef(r1, . . . , rn)ebe. (2.11)
Then by Corollary 2.8, either eae or epe are central elements of eRe. Thus eitherah1 = (eae)h1 = h1eae = h1a or ph2 = (epe)h2 = h2(epe) = h2p, a contradiction. Henceeither a or p are in C.
Similarly, we can prove that either b or q are in C.Thus we have the following cases:
Case 1: Let a, b ∈ C.In this case, by (2.5) U satisfies
(a + b)f(r1, . . . , rn)2 − (a + b)(f(r1, . . . , rn)(pf(r1, . . . , rn) + f(r1, . . . , rn)q) −(pf(r1, . . . , rn) + f(r1, . . . , rn)q)(a + b)f(r1, . . . , rn) = 0. (2.12)
Since F = 0, a + b = 0. Hence from abovef(r1, . . . , rn)2 − (f(r1, . . . , rn)pf(r1, . . . , rn) + f(r1, . . . , rn)2q))−(pf(r1, . . . , rn)2 + f(r1, . . . , rn)qf(r1, . . . , rn)) = 0. (2.13)
This impliespf(r1, . . . , rn)2 + f(r1, . . . , rn)2(q − 1) + f(r1, . . . , rn)(p + q)f(r1, . . . , rn) = 0,
for all r1, . . . , rn ∈ U. Then by Lemma 2.2, one of the following holds:(1) p, q − 1 ∈ C and p + q − 1 = −(p + q) ∈ C. In this case we have F (x) = ax + xb =
(a + b)x and G(x) = px + xq = (p + q)x for all x ∈ R, with 2(p + q) = 1 which isour conclusion (1).
(2) f(x1, . . . , xn)2 is central valued on R and p+q −1 = −(p+q) ∈ C. In this case, wehave F (x) = ax + xb = (a + b)x and G(x) = px + xq for all x ∈ R with p + q ∈ Cand 2(p + q) = 1, which is our conclusion (2).
Case 2: Let p ∈ C and q ∈ C.Then by (2.5), U satisfies
(af(r1, . . . , rn)2 + f(r1, . . . , rn)2b)−(af(r1, . . . , rn) + f(r1, . . . , rn)b)(p + q)f(r1, . . . , rn)
−(p + q)f(r1, . . . , rn)(af(r1, . . . , rn) + f(r1, . . . , rn)b) = 0. (2.14)This can be written as
a(1 − p − q)f(r1, . . . , rn)2 + f(r1, . . . , rn)2b(1 − p − q)−(f(r1, . . . , rn)(a + b)(p + q)f(r1, . . . , rn) = 0 (2.15)
for all r1, . . . , rn ∈ U . Then by Lemma 2.2, one of the following holds:(1) a(1 − p − q), b(1 − p − q) ∈ C and a(1 − p − q) + b(1 − p − q) = (a + b)(p + q) ∈ C.
Since G = Id, p + q = 1 and hence a, b ∈ C. Then conclusion follows by Case 1.
(2) f(x1, . . . , xn)2 is central valued on R and a(1−p−q)+b(1−p−q) = (a+b)(p+q) ∈ C.This implies 2(p+q)(a+b) = a+b. Since G = 0, 0 = p+q ∈ C. Hence (a+b)(p+q) ∈C yields a+b ∈ C. Thus 2(p+q)(a+b) = a+b gives (2(p+q)−1)(a+b) = 0. Thisimplies either a + b = 0 or 2(p + q) = 1. When a + b = 0, F (x) = ax + xb = [a, x]
747
for all x ∈ R, G(x) = px + xq = (p + q)x for all x ∈ R, which is conclusion (3).On the other hand when 2(p + q) = 1, then F (x) = ax + xb for all x ∈ R witha + b ∈ C and G(x) = px + xq = (p + q)x for all x ∈ R with 2(p + q) = 1, which isour conclusion (4).
Case 3: Let a ∈ C and q ∈ C.Then by (2.5), we have
f(r1, . . . , rn)2(a + b)= f(r1, . . . , rn)(a + b)(p + q)f(r1, . . . , rn) + (p + q)f(r1, . . . , rn)2(a + b),
for all r1, . . . , rn ∈ U .This can be written as
(p + q − 1)f(r1, . . . , rn)2(a + b) + f(r1, . . . , rn)(a + b)(p + q)f(r1, . . . , rn) = 0.
Then by Lemma 2.4, one of the following holds:(1) p + q − 1, (a + b)(p + q) ∈ C and (p + q − 1)(a + b) + (a + b)(p + q) = 0. This
implies p + q ∈ C. Since G = 0, p + q = 0 and hence 0 = a + b ∈ C. Hence(p + q − 1)(a + b) + (a + b)(p + q) = 0 yields 2(p + q) = 1. Thus in this case wehave F (x) = ax + xb = x(a + b) = (a + b)x and G(x) = px + xq = (p + q)x for allx ∈ R with 2(p + q) = 1, which is our conclusion (1).
(2) a + b, (a + b)(p + q) ∈ C and (p + q − 1)(a + b) + (a + b)(p + q) = 0. Since a ∈ C,a + b ∈ C yields b ∈ C. Since F = 0, a + b = 0 and thus (a + b)(p + q) ∈ C impliesp+q ∈ C. Hence, (p+q−1)(a+b)+(a+b)(p+q) = 0 yields 2(p+q) = 1. Thus in thiscase we have F (x) = ax + xb = x(a + b) = (a + b)x and G(x) = px + xq = (p + q)xfor all x ∈ R with 2(p + q) = 1, which is our conclusion (1).
(3) f(x1, . . . , xn)2 is central valued on R and (p + q − 1)(a + b) = −(a + b)(p + q) ∈ C.Thus in this case we have F (x) = ax + xb = x(a + b) for all x ∈ R and G(x) =px + xq = (p + q)x for all x ∈ R, which is our conclusion (5).
Case 4: Let b ∈ C and p ∈ C.Then by (2.5), we have
(a + b)f(r1, . . . , rn)2
= (a + b)f(r1, . . . , rn)2(p + q) + f(r1, . . . , rn)(p + q)(a + b)f(r1, . . . , rn),
for all r1, . . . , rn ∈ U . This can be written as
(a + b)f(r1, . . . , rn)2(p + q − 1) + f(r1, . . . , rn)(p + q)(a + b)f(r1, . . . , rn) = 0
for all r1, . . . , rn ∈ U . Then by Lemma 2.4, one of the following holds:(1) a + b, (p + q)(a + b) ∈ C and (a + b)(p + q − 1) + (p + q)(a + b) = 0. Since b ∈ C,
a + b ∈ C yields a ∈ C. Since F = 0, a + b = 0 and thus (p + q)(a + b) ∈ C impliesp + q ∈ C. Hence, (a + b)(p + q − 1) + (p + q)(a + b) = 0 yields 2(p + q) = 1.Thus in this case we have F (x) = (a + b)x and G(x) = (p + q)x for all x ∈ R with2(p + q) = 1, which is our conclusion (1).
(2) p + q − 1, (p + q)(a + b) ∈ C and (a + b)(p + q − 1) + (p + q)(a + b) = 0. Since p ∈ C,p + q − 1 ∈ C yields q ∈ C. Since G = 0, p + q = 0 and thus (p + q)(a + b) ∈ Cimplies a + b ∈ C. Hence, (a + b)(p + q − 1) + (p + q)(a + b) = 0 yields 2(p + q) = 1.Thus in this case we have F (x) = (a + b)x and G(x) = (p + q)x for all x ∈ R with2(p + q) = 1, which is our conclusion (1).
(3) f(x1, . . . , xn)2 is central valued on R and (a + b)(p + q − 1) = −(p + q)(a + b) ∈ C.Thus in this case we have F (x) = ax + xb = (a + b)x for all x ∈ R and G(x) =px + xq = x(p + q) for all x ∈ R, which is our conclusion (6).
748 B. Dhara, S. Kar, P. Das
Proof of the Main Theorem. In [12, Theorem 3], Lee proved that every generalizedderivation g on a dense right ideal of R can be uniquely extended to a generalized derivationof U and thus can be assumed to be defined on the whole U with the form g(x) =ax + d(x) for some a ∈ U and d is a derivation of U . In the light of this, we mayassume that there exist a, b ∈ U and derivations d, δ of U such that F (x) = ax + d(x) andG(x) = bx + δ(x). Since I, R, and U satisfy the same generalized polynomial identities(see [3]) as well as the same differential identities (see [14]), without loss of generality, toprove our results, we may assume F (f(x1, . . . , xn))2 = F (f(x1, . . . , xn))G(f(x1, . . . , xn))+G(f(x1, . . . , xn))F (f(x1, . . . , xn)) for all x1, . . . , xn ∈ U .
If F and G both are inner generalized derivations of R, then by Lemma 2.9 we obtainour conclusions. Thus we assume that not both of F and G are inner. Hence U satisfies
af(x1, . . . , xn)2 + d(f(x1, . . . , xn)2)= (af(x1, . . . , xn) + d(f(x1, . . . , xn)))(bf(x1, . . . , xn) + δ(f(x1, . . . , xn)))+(bf(x1, . . . , xn) + δ(f(x1, . . . , xn)))(af(x1, . . . , xn) + d(f(x1, . . . , xn))) (2.16)
for all (x1, . . . , xn ∈ U , where d, δ are two derivations on U not both are inner.
Case 1: Assume that d and δ are C-dependent modulo inner derivations of U i.e.,αd + βδ = adq, where α, β ∈ C.
Subcase 1.i: Suppose α = 0. Then δ(x) = [p, x], where p = β−1q. Obviously d is not aninner derivation of U . From (2.16) we obtain that U satisfies
af(x1, . . . , xn)2 + d(f(x1, . . . , xn))f(x1, . . . , xn) + f(x1, . . . , xn)d(f(x1, . . . , xn))= (af(x1, . . . , xn) + d(f(x1, . . . , xn)))(bf(x1, . . . , xn) + [p, f(x1, . . . , xn)])+(bf(x1, . . . , xn) + [p, f(x1, . . . , xn)])(af(x1, . . . , xn) + d(f(x1, . . . , xn)). (2.17)
Let fd(x1, . . . , xn) be the polynomials obtained from f(x1, . . . , xn) replacing each coeffi-cients ασ with d(ασ). Then we have
d(f(x1, . . . , xn)) = fd(x1, . . . , xn) +∑
i
f(x1, . . . , d(xi), . . . , xn).
Thus (2.17) gives
af(x1, . . . , xn)2 + (fd(x1, . . . , xn) +∑i
f(x1, . . . , d(xi), . . . , xn))f(x1, . . . , xn)
+f(x1, . . . , xn)(fd(x1, . . . , xn) +∑i
f(x1, . . . , d(xi), . . . , xn))
= (af(x1, . . . , xn) + fd(x1, . . . , xn) +∑i
f(x1, . . . , d(xi), . . . , xn))
·(bf(x1, . . . , xn) + [p, f(x1, . . . , xn)]) + (bf(x1, . . . , xn) + [p, f(x1, . . . , xn)])·(af(x1, . . . , xn) + fd(x1, . . . , xn) +
∑i
f(x1, . . . , d(xi), . . . , xn)). (2.18)
Since d is outer derivation, by Kharchenko’s theorem [11], we have that U satisfies
749
af(x1, . . . , xn)2 + (fd(x1, . . . , xn) +∑i
f(x1, . . . , yi, . . . , xn))f(x1, . . . , xn)
+f(x1, . . . , xn)((fd(x1, . . . , xn) +∑i
f(x1, . . . , yi, . . . , xn))
= (af(x1, . . . , xn) + fd(x1, . . . , xn) +∑i
f(x1, . . . , yi, . . . , xn))
·(bf(x1, . . . , xn) + [p, f(x1, . . . , xn)]) + (bf(x1, . . . , xn) + [p, f(x1, . . . , xn)])·(af(x1, . . . , xn) + fd(x1, . . . , xn) +
∑i
f(x1, . . . , yi, . . . , xn)). (2.19)
Particularly, U satisfies the blended component,
∑i
f(x1, . . . , yi, . . . , xn)f(x1, . . . , xn) + f(x1, . . . , xn)∑i
f(x1, . . . , yi, . . . , xn)
=∑i
f(x1, . . . , yi, . . . , xn)(bf(x1, . . . , xn) + [p, f(x1, . . . , xn)])
+(bf(x1, . . . , xn) + [p, f(x1, . . . , xn)])∑i
f(x1, . . . , yi, . . . , xn). (2.20)
In particular, for y1 = x1, y2 = y3 = . . . = yn = 0, we get from above
2f(x1, . . . , xn)2 = f(x1, . . . , xn)(bf(x1, . . . , xn) + [p, f(x1, . . . , xn)])+(bf(x1, . . . , xn) + [p, f(x1, . . . , xn)])f(x1, . . . , xn), (2.21)
which gives
(b + p)f(x1, . . . , xn)2 − f(x1, . . . , xn)2(p + 2) + f(x1, . . . , xn)bf(x1, . . . , xn) = 0.
Then by Lemma 2.2, one of the following holds:(1) b + p, p + 2, b ∈ C and (b + p) − (p + 2) = −b. This implies p ∈ C and b = 1. Thus
in this case we have G(x) = bx + [p, x] = x for all x ∈ R, a contradiction.(2) f(x1, . . . , xn)2 is central valued on R and (b + p) − (p + 2) = −b ∈ C. This gives
b = 1. In this case, we have from (2.17) that U satisfies
af(x1, . . . , xn)2 + d(f(x1, . . . , xn)2)= (af(x1, . . . , xn) + d(f(x1, . . . , xn)))(f(x1, . . . , xn) + [p, f(x1, . . . , xn)])
+(f(x1, . . . , xn) + [p, f(x1, . . . , xn)])(af(x1, . . . , xn)+d(f(x1, . . . , xn))). (2.22)
This implies
0 = af(x1, . . . , xn)[p, f(x1, . . . , xn)] + d(f(x1, . . . , xn))[p, f(x1, . . . , xn)]+f(x1, . . . , xn)af(x1, . . . , xn) + [p, f(x1, . . . , xn)]af(x1, . . . , xn)+[p, f(x1, . . . , xn)]d((f(x1, . . . , xn)).
It gives
0 = af(x1, . . . , xn)[p, f(x1, . . . , xn)]+(fd(x1, . . . , xn) +
∑i
f(x1, . . . , yi, . . . , xn))[p, f(x1, . . . , xn)]
+f(x1, . . . , xn)af(x1, . . . , xn) + [p, f(x1, . . . , xn)]af(x1, . . . , xn)+[p, f(x1, . . . , xn)](fd(x1, . . . , xn) +
∑i
f(x1, . . . , yi, . . . , xn)).
750 B. Dhara, S. Kar, P. Das
In particular, U satisfies the blended component
∑i
f(x1, . . . , yi, . . . , xn)[p, f(x1, . . . , xn)]
+[p, f(x1, . . . , xn)]∑
i
f(x1, . . . , yi, . . . , xn) = 0. (2.23)
Putting yi = [q, xi] in (2.23), where q /∈ C, we have that U satisfies
[q, f(x1, . . . , xn)][p, f(x1, . . . , xn)] + [p, f(x1, . . . , xn)][q, f(x1, . . . , xn)] = 0.(2.24)
Then by Lemma 2.3, p ∈ C. Thus G(x) = bx + [p, x] = x for all x ∈ R, a contradiction.
Subcase 1.ii: Suppose α = 0, then αd + βδ = adq gives d = µδ + adc for some µ ∈ C andc ∈ U . Then we can assume that δ is not an inner derivation, otherwise d and δ both willbe inner derivations, a contradiction. From (2.16), U satisfies
af(x1, . . . , xn)2 + µδ(f(x1, . . . , xn)2) + [c, f(x1, . . . , xn)2]
=(
af(x1, . . . , xn) + µδ(f(x1, . . . , xn)) + [c, f(x1, . . . , xn)])
·(
bf(x1, . . . , xn) + δ(f(x1, . . . , xn)))
+(
bf(x1, . . . , xn) + δ(f(x1, . . . , xn)))
·(
af(x1, . . . , xn) + µδ(f(x1, . . . , xn))
+[c, f(x1, . . . , xn)])
,
that is,
af(x1, . . . , xn)2 + µ(f δ(x1, . . . , xn) +∑i
f(x1, . . . , δ(xi), . . . , xn))f(x1, . . . , xn)
+µf(x1, . . . , xn)(f δ(x1, . . . , xn) +∑i
f(x1, . . . , δ(xi), . . . , xn)) + [c, f(x1, . . . , xn)2]
=(
af(x1, . . . , xn) + µ(f δ(x1, . . . , xn) +∑i
f(x1, . . . , δ(xi), . . . , xn)) + [c, f(x1, . . . , xn)])
·(
bf(x1, . . . , xn) + f δ(x1, . . . , xn) +∑i
f(x1, . . . , δ(xi), . . . , xn))
+(
bf(x1, . . . , xn) + (f δ(x1, . . . , xn) +∑i
f(x1, . . . , δ(xi), . . . , xn)))
·(
af(x1, . . . , xn) + µ(f δ(x1, . . . , xn) +∑i
f(x1, . . . , δ(xi), . . . , xn))
+[c, f(x1, . . . , xn)])
.
751
Then by Kharchenko’s theorem [11], we have that U satisfies
af(x1, . . . , xn)2 + µ(f δ(x1, . . . , xn) +∑i
f(x1, . . . , yi, . . . , xn))f(x1, . . . , xn)
+µf(x1, . . . , xn)(f δ(x1, . . . , xn) +∑i
f(x1, . . . , yi, . . . , xn)) + [c, f(x1, . . . , xn)2]
=(
af(x1, . . . , xn) + µ(f δ(x1, . . . , xn) +∑i
f(x1, . . . , yi, . . . , xn)) + [c, f(x1, . . . , xn)])
·(
bf(x1, . . . , xn) + f δ(x1, . . . , xn) +∑i
f(x1, . . . , yi, . . . , xn))
+(
bf(x1, . . . , xn) + (f δ(x1, . . . , xn) +∑i
f(x1, . . . , yi, . . . , xn)))
·(
af(x1, . . . , xn) + µ(f δ(x1, . . . , xn) +∑i
f(x1, . . . , yi, . . . , xn))
+[c, f(x1, . . . , xn)])
. (2.25)
In particular, for x1 = 0, U satisfies
0 = µf(x1, . . . , xn)2 + µf(x1, . . . , xn)2, (2.26)
that is, 2µf(x1, . . . , xn)2 = 0. Since char(R) = 2, U satisfies µf(x1, . . . , xn)2 = 0. Thisimplies that either µ = 0 or f(x1, . . . , xn)2 = 0. Now f(x1, . . . , xn)2 = 0, impliesf(x1, . . . , xn) = 0 for all x1, . . . , xn ∈ U , a contradiction. Hence we have µ = 0. Thus(2.25) reduces to
af(x1, . . . , xn)2 =(
af(x1, . . . , xn) + [c, f(x1, . . . , xn)])
·(
bf(x1, . . . , xn) + f δ(x1, . . . , xn) +∑i
f(x1, . . . , yi, . . . , xn))
+(
bf(x1, . . . , xn) + (f δ(x1, . . . , xn) +∑i
f(x1, . . . , yi, . . . , xn)))
·(
af(x1, . . . , xn) + [c, f(x1, . . . , xn)])
.
In particular, U satisfies blended components
(af(x1, . . . , xn) + [c, f(x1, . . . , xn)])∑i
f(x1, . . . , yi, . . . , xn)
+∑i
f(x1, . . . , yi, . . . , xn)(af(x1, . . . , xn) + [c, f(x1, . . . , xn)]) = 0. (2.27)
For y1 = x1 and y2 = y3 =, . . . , = yn = 0, U satisfies
(af(x1, . . . , xn) + [c, f(x1, . . . , xn)])f(x1, . . . , xn)+f(x1, . . . , xn)(af(x1, . . . , xn) + [c, f(x1, . . . , xn)]) = 0, (2.28)
that is
(a + c)f(x1, . . . , xn)2 − f(x1, . . . , xn)2c + f(x1, . . . , xn)af(x1, . . . , xn) = 0,
for all x1, . . . , xn ∈ U . Then by Lemma 2.2, we have one of the followings:(1) a + c, c, a ∈ C and 2a = 0. Thus a = 0. In this case F (x) = ax + [c, x] = 0 for all
x ∈ U , a contradiction.
752 B. Dhara, S. Kar, P. Das
(2) f(x1, . . . , xn)2 is central valued on R and a ∈ C with 2a = 0. This implies a = 0.Then by (2.27), U satisfies
[c, f(x1, . . . , xn)]∑
i
f(x1, . . . , yi, . . . , xn)
+∑
i
f(x1, . . . , yi, . . . , xn)[c, f(x1, . . . , xn)] = 0.
Replacing yi with [q, xi] for some q /∈ C, we get from above that U satisfies
[c, f(x1, . . . , xn)][q, f(x1, . . . , xn)] + [q, f(x1, . . . , xn)][c, f(x1, . . . , xn)] = 0. (2.29)
By Lemma 2.3, c ∈ C. Then F (x) = ax + [c, x] = 0 for all x ∈ R, a contradiction.
Case 2: Let d and δ be linearly C-independent modulo inner derivations of U . Then from(2.16), U satisfies
af(x1, . . . , xn)2 + (fd(x1, . . . , xn) +∑i
f(x1, . . . , d(xi), . . . , xn))f(x1, . . . , xn)
+f(x1, . . . , xn)(fd(x1, . . . , xn) +∑i
f(x1, . . . , d(xi), . . . , xn))
= (af(x1, . . . , xn) + fd(x1, . . . , xn) +∑i
f(x1, . . . , d(xi), . . . , xn))
·(bf(x1, . . . , xn) + f δ(x1, . . . , xn) +∑i
f(x1, . . . , δ(xi), . . . , xn))
+(bf(x1, . . . , xn) + f δ(x1, . . . , xn) +∑i
f(x1, . . . , δ(xi), . . . , xn))
·(af(x1, . . . , xn) + (fd(x1, . . . , xn) +∑i
f(x1, . . . , d(xi), . . . , xn)) (2.30)
for all x1, . . . , xn ∈ U . Since d and δ are not inner, by Kharchenko’s theorem [11], Usatisfies
af(x1, . . . , xn)2 + (fd(x1, . . . , xn) +∑
i
f(x1, . . . , yi, . . . , xn))f(x1, . . . , xn)
+f(x1, . . . , xn)(fd(x1, . . . , xn) +∑
i
f(x1, . . . , yi, . . . , xn))
= (af(x1, . . . , xn) + fd(x1, . . . , xn) +∑
i
f(x1, . . . , yi, . . . , xn))
·(bf(x1, . . . , xn) + f δ(x1, . . . , xn) +∑
i
f(x1, . . . , zi, . . . , xn))
+(bf(x1, . . . , xn) + f δ(x1, . . . , xn) +∑
i
f(x1, . . . , zi, . . . , xn))
·(af(x1, . . . , xn) + fd(x1, . . . , xn) +∑
i
f(x1, . . . , yi, . . . , xn)).
In particular, for x1 = 0, z1 = y1, we get 2f(y1, x2, . . . , xn)2 = 0 implying f(x1, . . . , xn)2 =0 for all x1, . . . , xn ∈ U . It yields f(x1, . . . , xn) = 0 for all x1, . . . , xn ∈ U , a contradiction.Thus the proof of the theorem is completed.
Acknowledgment. The authors would like to thank the referee for pointing out somemisprints and providing very helpful comments and suggestions. The third author ex-presses his thanks to the University Grants Commission, New Delhi for its JRF awardedto him under grant No. 424222 dated 22.03.2017.
753
References[1] N. Argac and V. De Filippis, Actions of generalized derivations on multilinear poly-
nomials in prime rings, Algebra Colloq. 18 (Spec 01), 955–964, 2011.[2] M. Brešar, Centralizing mappings and derivations in prime rings, J. Algebra, 156,
385–394, 1993.[3] C.L. Chuang, GPIs having coefficients in Utumi quotient rings, Proc. Amer. Math.
Soc. 103 (3), 723–728, 1988.[4] V. De Filippis, O.M. Di Vincenzo, and C.Y. Pan, Quadratic central differential iden-
tities on a multilinear polynomial, Comm. Algebra, 36 (10), 3671–3681, 2008.[5] V. De Filippis and O.M. Di Vincenzo, Vanishing derivations and centralizers of gener-
alized derivations on multilinear polynomials, Comm. Algebra, 40, 1918–1932, 2012.[6] B. Dhara, S. Kar, and K.G. Pradhan, Identities with generalized derivations on mul-
tilinear polynomials in prime rings, Afr. Mat. 27, 1347–1360, 2016.[7] M. Fosner and J. Vukman, Identities with generalized derivations in prime rings,
Mediter. J. Math. 9 (4), 847–863, 2012.[8] T.S. Erickson, W.S. Martindale III, and J.M. Osborn, Prime nonassociative algebras,
Pacific J. Math. 60, 49–63, 1975.[9] C. Faith and Y. Utumi, On a new proof of Litoff’s theorem, Acta Math. Acad. Sci.
Hung. 14, 369–371, 1963.[10] N. Jacobson, Structure of rings, Amer. Math. Soc. Colloq. Pub. 37, Amer. Math.
Soc., Providence, RI, 1964.[11] V.K. Kharchenko, Differential identity of prime rings, Algebra Logic, 17, 155–168,
1978.[12] T.K. Lee, Generalized derivations of left faithful rings, Comm. Algebra, 27 (8), 4057–
4073, 1999.[13] T.K. Lee and W.K. Shiue, Derivations co-centralizing polynomials, Taiwanese J.
Math. 2 (4), 457–467, 1998.[14] T.K. Lee, Semiprime rings with differential identities, Bull. Inst. Math. Acad. Sinica,
20 (1), 27–38, 1992.[15] U. Leron, Nil and power central polynomials in rings, Trans. Amer. Math. Soc. 202,
297–103, 1975.[16] W.S. Martindale III, Prime rings satisfying a generalized polynomial identity, J. Al-
gebra, 12, 576–584, 1969.[17] E.C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc. 8, 1093–1100, 1957.[18] F. Rania and G. Scudo, A quadratic differential identity with generalized derivations
on multilinear polynomials in prime rings, Mediterr. J. Math. 11, 273–285, 2014.[19] N.B. Yarbil and V. De Filippis, A quadratic differential identity with skew derivations,
Comm. Algebra, 46 (1), 205–216, 2018.
Hacettepe Journal ofMathematics & Statistics
Hacet. J. Math. Stat.Volume 49 (2) (2020), 754 – 765
DOI : 10.15672/hujms.623990
Research Article
Computation of Zagreb indices and Zagrebpolynomials of Sierpiński graphs
Hafiz Muhammad Afzal Siddiqui
Department of Mathematics, COMSATS University Islamabad Lahore Campus, Pakistan
AbstractThe Sierpiński fractal or Sierpiński gasket and generalized Sierpiński graphs are objectsof great interest in dynamical systems and probability. In this paper, we consider theSierpiński gasket graph Sn, the generalized Sierpiński graphs S(n, C3) and S(n, C4). Weprovide explicit computing formulae for Zagreb indices, multiple Zagreb indices and Zagrebpolynomials of Sierpiński graphs.
Mathematics Subject Classification (2010). 05C12, 05C90
Keywords. Sierpiński gasket graph, generalized Sierpiński graph, topological indices,Zagreb index, augmented Zagreb index, Zagreb polynomials
1. IntroductionLet G be a graph with vertex set V (G) and edge set E(G). We denote the order of G
by |V (G)| and size of G by |E(G)|. An edge in E(G) with end vertices u and v is denotedby uv. Two vertices u and v are called adjacent if there is an edge between them. Theneighborhood of u, denoted by N(u), is the set of all vertices adjacent to u. The degree ofu is denoted by du and equals |N(u)|.
Graphs of Sierpiński type appear naturally in many different areas of mathematics aswell as in several other scientific fields. One of the most important family of such graphs isthe Sierpiński graphs Sn, obtained after a finite number of iterations that in the limit givethe Sierpiński gasket [23]. More simply, Sn+1 consists of three attached copies of Sn whichare referred as the top, bottom left and bottom right components of Sn+1. These grapheshad been already introduced in 1944 by Scorer, Grundy and Smith [26]. They play animportant role in dynamic systems and probability [21], as well as in psychology [26]. Thegeneralized Sierpiński graph, S(n, G) is constructed by copying |G| times S(n − 1, G) andadding one edge between copy x and copy y of S(n − 1, G), whenever xy is an edge of G.
The Sierpiński graphs S(n, k) and S(n, G) are defined as follows:S(n,k) has vertex set 1, 2, · · · , kn, and there is an edge between two vertices u =
(u1, u2 · · · , un) and v = (v1, v2, · · · , vn) iff there is an h ∈ 1, 2, · · · , n such that:• uj = vj for j = 1, · · · , h − 1;• uh = vh; and• uj = vh; vj = uh for j = h + 1, · · · , n.
Email address: [email protected]: 31.08.2016; Accepted: 25.03.2019
Computation of Zagreb indices and Zagreb polynomials of Sierpiński graphs 755
The generalized Sierpiński graph of dimension n denoted by S(n; G) is the graph withvertex set 1, 2, · · · , k and edge set defined by: u, v is an edge if and only if there existsi ∈ 1, 2, · · · , n such that:
• uj = vj if j < i,• ui = vi and (ui, vi) ∈ E(G),• uj = vi and vj = ui if j > i.
The topological indices are the objects of great importance in quantitative structure-activity research (QSAR) and structure-property relationships research (QSPR) study.The graph invariants, known as the first and second Zagreb indices, were the first vertex-degree-based structure descriptors [18,19]. The terms,
∑v∈V (G)
d2v,
∑uv∈E(G)
dudv and∑
v∈V (G)d3
v
were first appeared in the topological formula for total π-energy of conjugated moleculesthat was derived in 1972 by Gutman and Trinajstić [18]. Ten years later, Balaban et al.included
M1(G) =∑
v∈V (G)d2
v =∑
uv∈E(G)(du + dv) (1.1)
andM2(G) =
∑uv∈E(G)
dudv (1.2)
among topological indices and named them “Zagreb group indices” [2]. The name “Zagrebgroup indices” was abbreviated to “Zagreb indices” and now we call M1(G), the first Zagrebindex and M2(G), the second Zagreb index. Afterwards these indices have been used asbranching indices [5]. Later on the Zagreb indices found applications in QSPR and QSARstudies [16, 29]. These indices have been used to study molecular complexity, chirality,ZE-isomorphism and hetero-systems. For further study on chemical applications andmathematical properties see the following papers [1,4,6,10,13–15,17,20,22,25,28,30,31].
The term,∑
v∈V (G)(dv)3 was ignored for more than forty years. Recently, Furtula and
Gutman proved that this term have a very promising application potential [8]. Theyproposed that this term should be named the forgotten topological index or shortly theF-index that is defined as
F (G) =∑
v∈V (G)(dv)3 =
∑uv∈E(G)
[(du)2 + (dv)2]. (1.3)
They discovered a remarkable fact that the linear combination M1+ λF yields a highlyaccurate mathematical model of certain physico-chemical properties of alkanes [8].
Another important graph invariant that is necessarily encountered within the studies ofdifference between two Zagreb indices [3], is the reduced second Zagreb index, defined as
RM2(G) =∑
uv∈E(G)(du − 1) × (dv − 1). (1.4)
The augmented Zagreb index of G proposed by Furtula et al. in 2010 [9] is defined as
AZI(G) =∑
uv∈E(G)( dudv
du + dv − 2)3. (1.5)
This graph invariant has proven to be a valuable predictive index in the study of the heatof formation in octanes and heptanes [9]. Noting that if instead of the exponent 3 wewould set −0.5, then we would arrive at the ordinary ABC index. Preliminary studies[9, 12,17] indicate that AZI has an even better correlation potential than ABC index.
756 H.M.A. Siddiqui
The third Zagreb index was introduced by Shirdel in 2013 [27], defined as
M3(G) =∑
uv∈E(G)(du + dv)2. (1.6)
Clearly, this index is a combination of F-index and the second Zagreb index i.e.
M3(G) = F (G) + 2M2(G). (1.7)
Because of the above mentioned relation, we studied the F-index with the indices of theZagreb family.
The degree product P (G) =∏
v∈V (G)dv of a graph G was introduced and studied by
Narumi and Katayama for the first time. The Narumi-Katayama index was proposed in1984, by Narumi and Katayama [24]. It is defined as
NK(G) =∏
v∈V (G)dv. (1.8)
The first and second multiple Zagreb indices were introduced by Ghorbani and Azimiin 2012 [11], defined as
PM1(G) =∏
uv∈V (G)(du + dv) =
∏v∈V (G)
(dv)2 (1.9)
andPM2(G) =
∏uv∈V (G)
(dudv). (1.10)
Clearly, the first multiple Zagreb index is the square of Narumi-Katayama index.In 2009, Fath-Tabar [7] put forward the first and the second Zagreb polynomials of the
graph G, defined respectively as
ZG1(G, x) =∑
uv∈E(G)xdu+dv (1.11)
andZG2(G, x) =
∑uv∈E(G)
xdudv , (1.12)
where x is a dummy variable.In this paper, we compute the above mentioned topological indices for Sn, S(n, C3) and
S(n, C4).
2. Zagreb indices and Zagreb polynomials for the Sierpiński gasket graph,Sn
The Sierpiński gasket graphs for n = 1, 2, 3 are given in Figure 1. The order of Sn is12(3n + 3) and the size of Sn is 3n. There are two kinds of edges corresponding to theirdegrees of end vertices for n > 1. The edge partition of edge set of Sn is shown in Table 1.
Table 1. Edge partition of edge set of Sn
(du, dv) (2, 4) (4, 4)Number of Edges 6 3n − 6
There are two kinds of vertices in the set V (G) corresponding to their degrees. Table 2shows such a partition of the set V (G) of Sn.
Computation of Zagreb indices and Zagreb polynomials of Sierpiński graphs 757
1
2
3
11
12 13
21 3323
111
112 113
121
123
133
223212 213
312 313 322 332 333
321
S1 S
2
S3
Figure 1. The graph of S1, S2 and S3
Table 2. The partition of V (G) of Sn
dv 2 4Number of Vertices 3 1
2(3n − 3)
The following theorems present the analytically closed formulae of Zagreb indices andZagreb polynomials for Sn .
Theorem 2.1. The first and second Zagreb indices for G = Sn are given byM1(G) = 8 × 3n − 12 and M2(G) = 16 × 3n − 48.
Proof. Using Equation 1.1 and Table 1, we haveM1(G) =
∑uv∈E(G)
(du + dv)
= 6(2 + 4) + (3n − 6)(4 + 4).After simplification, we get the required result, M1(G) = 8 × 3n − 12.
Similarly, Using Equation 1.2 and Table 1, we haveM2(G) =
∑uv∈E(G)
(du × dv)
= 6(2 × 4) + (3n − 6)(4 × 4) = 16 × 3n − 48.
Theorem 2.2. The reduced second Zagreb index for G = Sn is given by
RM2(G) = 9 × 3n − 36.
Proof. Using Equation 1.4 and Table 1, we findRM2(G) =
∑uv∈E(G)
(du − 1) × (dv − 1)
= 6(1 × 3) + (3n − 6)(3 × 3).After simplification, we get the required result, RM2(G) = 9 × 3n − 36. Theorem 2.3. The third Zagreb index for G = Sn is given by
M3(G) = 8(8 × 3n − 21).
758 H.M.A. Siddiqui
Proof. Using Equation 1.6 and Table 1, we get
M3(G) =∑
uv∈E(G)(du + dv)2 = 6(6)2 + (3n − 6)(8)2.
After simplification, we get the required result, M3(G) = 8(8 × 3n − 21). Theorem 2.4. The F-index for G = Sn is given by
F (G) = 32 × 3n − 72.
Proof. Using Equation 1.7, Theorem 2.1 and Theorem 2.3, we getF (G) = 8(8 × 3n − 21) − 2(16 × 3n − 48) = 32 × 3n − 72.
Theorem 2.5. The augmented Zagreb index for G = Sn is given by
AZI(G) = 16(32 × 3n−3 − 379 ).
Proof. Using Equation 1.5, Table 1 and simplifying, we have
AZI(G) =∑
uv∈E(G)[ dudv
du + dv − 2]3
= 6(2)3 + (3n − 6)(83
)3
= 16(32 × 3n−3 − 379
).
Theorem 2.6. The first and second multiple Zagreb indices for G = Sn are given by
PM1(G) = 32(3n+1 − 9) and PM2(G) = 44(3n+1 − 18).
Proof. Using Equation 1.9 and Table 2, we get
PM1(G) =∏
v∈V (G)(dv)2 = (2)2 × 3 × 8(3n − 3),
which after simplification gives PM1(G) = 32(3n+1 − 9).Similarly, using Equation 1.10 and Table 1 and after simplification, we have
PM2(G) =∏
uv∈E(G)dudv
= 8 × 6 × (16)(3n − 6) = 44(3n+1 − 18).
Corollary 2.7. The Narumi-Katayama index for G = Sn is given byNK(G) =
√PM1(G) = 12
√4(3n−1 − 1).
Theorem 2.8. The first Zagreb polynomial for G = Sn is given byZG1(G, x) = 6 × x6 + (3n − 6) × x8.
Proof. Using Equation 1.11 and Table 1, we have
ZG1(G, x) =∑
uv∈E(G)xdu+dv = 6 × x6 + (3n − 6) × x8
. Theorem 2.9. The second Zagreb polynomial for G = Sn is given by
ZG2(G, x) = 6 × x8 + (3n − 6) × x16.
Computation of Zagreb indices and Zagreb polynomials of Sierpiński graphs 759
Proof. Using Equation 1.12 and Table 1, we get
ZG2(G, x) =∑
uv∈E(G)xdudv = 6 × x8 + (3n − 6) × x16.
3. Zagreb indices and Zagreb polynomials for S(n, C3)The generalized Sierpiński graphs, S(1, C3), S(2, C3) and S(3, C3) are shown in Figure
2.
1
2
3
11
12
13
21
22
3323 31
32
111
112
113
121
122
123 131
132
133
221
222
223
212
211 213 311
232
233231
333
312
313
321
322
323 331
332
Figure 2. The graphs S(1, C3), S(2, C3) and S(3, C3)
The order and size of S(n, C3) are 3n and 32(3n − 1), respectively. There are two kinds
of edges corresponding to their degrees of end vertices for n > 1. The edge partition ofedge set of S(n, C3) is shown in Table 3.
Table 3. Edge partition of edge set of S(n, C3)
(du, dv) (2, 3) (3, 3)Number of Edges 6 3
2(3n − 5)
There are two kinds of vertices in the set V (G) corresponding to their degrees. Table 4shows such a partition of the set V (G) of S(n, C3).
Table 4. The partition of V (G) of S(n, C3)
dv 2 3Number of Vertices 3 3n − 3
The following theorems present the analytically closed formulae of Zagreb indices andZagreb polynomials for S(n, C3) for n > 1 .
Theorem 3.1. The first and second Zagreb indices for G = S(n, C3) are given byM1(G) = 9 × 3n − 15 and M2(G) = 9
2(3n+1 − 7).
760 H.M.A. Siddiqui
Proof. Using Equation 1.1 and Table 3, we have
M1(G) =∑
uv∈E(G)(du + dv)
= 6(2 + 3) + 32
(3n − 5)(3 + 3).
After simplification, we get the required result, M1(G) = 9 × 3n − 15.Similarly, Using Equation 1.2 and Table 3, we have
M2(G) =∑
uv∈E(G)(du × dv)
= 6(2 × 3) + 32
(3n − 5)(3 × 3) = 92
(3n+1 − 7).
Theorem 3.2. The reduced second Zagreb index for G = S(n, C3) is given by
RM2(G) = 6(3n − 3).
Proof. Using Equation 1.4 and Table 3, we find
RM2(G) =∑
uv∈E(G)(du − 1) × (dv − 1)
= 6(1 × 2) + 32
(3n − 5)(2 × 2).
After simplification, we get the required result, RM2(G) = 6(3n − 3). Theorem 3.3. The third Zagreb index for G = S(n, C3) is given by
M3(G) = 2(3n+3 − 20).
Proof. Using Equation 1.6 and Table 3, we get
M3(G) =∑
uv∈E(G)(du + dv)2 = 6(5)2 + 3
2(3n − 5)(6)2.
After simplification, we get the required result, M3(G) = 2(3n+3 − 20). Theorem 3.4. The F-index for G = S(n, C3) is given by
F (G) = 3(3n+2 − 23).
Proof. Using Equation 1.7, Theorem 3.1 and Theorem 3.3, we get
F (G) = 6(9 × 3n − 20) − 2 × 92
(3n+1 − 7) = 3(3n+2 − 23).
Theorem 3.5. The augmented Zagreb index for G = S(n, C3) is given by
AZI(G) = 127 (3n+7 + 4791).
Proof. Using Equation 1.5, Table 3 and simplifying, we have
AZI(G) =∑
uv∈E(G)[ dudv
du + dv − 2]3
= 6(2)3 + 32
(3n − 5)(94
)3
= 127 (3n+7 + 4791).
Computation of Zagreb indices and Zagreb polynomials of Sierpiński graphs 761
Theorem 3.6. The first and second multiple Zagreb indices for G = S(n, C3) are givenby
PM1(G) = 4(3n+3 − 81) and PM2(G) = 2(3n+5 − 1215).Proof. Using Equation 1.9 and Table 4, we get
PM1(G) =∏
v∈V (G)(dv)2 = 4 × 3 × 9(3n − 3),
which after simplification gives PM1(G) = 4(3n+3 − 81).Similarly, using Equation 1.10 and Table 3 and after simplification, we havePM2(G) =
∏uv∈E(G)
dudv
= 6 × 6 × 9(32(3n − 5)) = 2(3n+5 − 1215).
Corollary 3.7. The Narumi-Katayama index for G = S(n, C3) is given byNK(G) =
√PM1(G) = 2
√3n+3 − 81.
Theorem 3.8. The first Zagreb polynomial for G = S(n, C3) is given byZG1(G, x) = 6 × x5 + 1
2(3n+1 − 15)x6.
Proof. Using Equation 1.11 and Table 3, we have
ZG1(G, x) =∑
uv∈E(G)xdu+dv = 6 × x5 + 3
2(3n − 5) × x6
= 6 × x5 + 12
(3n+1 − 15)x6.
Theorem 3.9. The second Zagreb polynomial for G = S(n, C3) is given by
ZG2(G, x) = 6 × x6 + 12(3n+1 − 15)x9.
Proof. Using Equation 1.12 and Table 3, we get
ZG2(G, x) =∑
uv∈E(G)xdudv = 6 × x6 + 3
2(3n − 5) × x9
= 6 × x6 + 12
(3n+1 − 15)x9.
4. Zagreb indices and Zagreb polynomials for S(n, C4)The generalized Sierpiński graphs, S(1, C4), S(2, C4) and S(3, C4) are shown in Figure
3.The order and size of S(n, C4) are 4n and 4
3(4n − 1), respectively. There are two kindsof edges corresponding to their degrees of end vertices for n > 1. The edge partition ofedge set of S(n, C4) is shown in Table 5.
Table 5. Edge partition of edge set of S(n, C4)
(du, dv) (2, 3) (3, 3)Number of Edges 2
3(4n + 8) 23(4n − 10)
There are two kinds of vertices in the set V (G) corresponding to their degrees. Table 6shows such a partition of the set V (G) of S(n, C4).
The following theorems present the analytically closed formulae of Zagreb indices andZagreb polynomials for S(n, C4), for n > 1 .
762 H.M.A. Siddiqui
1
32
4
22
11
1213
14
21
23
2431
32 33
34
41
42 43
44
111
112113
114
222
221224 331
333
334
441
442443
444
copy 1
copy 2
333223
copy 3
copy 4
copy 1
copy 2 copy 3
copy 4
Figure 3. The graphs S(1, C4), S(2, C4) and S(3, C4)
Table 6. The partition of V (G) of S(n, C4)
dv 2 3Number of Vertices 1
3(4n + 8) 23(4n − 4)
Theorem 4.1. The first and second Zagreb indices for G = S(n, C4) are given byM1(G) = 2
3(11 × 4n − 20) and M2(G) = 2(5 × 4n − 14).
Proof. Using Equation 1.1 and Table 5, we have
M1(G) =∑
uv∈E(G)(du + dv)
= 23
(4n + 8)(2 + 3) + 23
(4n − 10)(3 + 3).
After simplification, we get the required result, M1(G) = 23(11 × 4n − 20).
Similarly, using Equation 1.2 and Table 5, we have
M2(G) =∑
uv∈E(G)(du × dv)
= 23
(4n + 8)(2 × 3) + 23
(4n − 10)(3 × 3) = 2(5 × 4n − 14).
Theorem 4.2. The reduced second Zagreb index for G = S(n, C4) is given by
RM2(G) = 4n+1 − 16.
Proof. Using Equation 1.4 and Table 5, we find
RM2(G) =∑
uv∈E(G)(du − 1) × (dv − 1)
= 23
(4n + 8)(1 × 2) + 23
(4n − 10)(2 × 2).
After simplification, we get the required result, RM2(G) = 4n+1 − 16. Theorem 4.3. The third Zagreb index for G = S(n, C4) is given by
Computation of Zagreb indices and Zagreb polynomials of Sierpiński graphs 763
M3(G) = 23(61 × 4n − 160).
Proof. Using Equation 1.6 and Table 5, we getM3(G) =
∑uv∈E(G)
(du + dv)2
= 0(4)2 + 23
(4n + 8)(5)2 + 23
(4n − 10)(6)2.
After simplification, we get the required result, M3(G) = 23(61 × 4n − 160).
Theorem 4.4. The F-index for G = S(n, C4) is given byF (G) = 2
3(31 × 4n − 76).
Proof. Using Equation 1.7, Theorem 4.1 and Theorem 4.3, we get
F (G) = 23
(61 × 4n − 160) − 2 × 2(5 × 4n − 14) = 23
(31 × 4n − 76).
Theorem 4.5. The augmented Zagreb index for G = S(n, C4) is given by
AZI(G) = 196(1241 × 4n − 3194).
Proof. Using Equation 1.5, Table 5 and simplifying, we have
AZI(G) =∑
uv∈E(G)[ dudv
du + dv − 2]3
= 23
(4n + 8)(2)3 + 23
(4n − 10)(94
)3
= 196
(1241 × 4n − 3194).
Theorem 4.6. The first and second multiple Zagreb indices for G = S(n, C4) are givenby
PM1(G) = 8(4n + 8)(4n − 4) and PM2(G) = 24(4n + 8)(4n − 10).
Proof. Using Equation 1.9 and Table 6, we get
PM1(G) =∏
v∈V (G)(dv)2 = 22×1
3(4n + 8) × 32×2
3(4n − 4),
which after simplification gives PM1(G) = 8(4n + 8)(4n − 4).Similarly, using Equation 1.10 and Table 5 and after simplification, we have
PM2(G) =∏
uv∈E(G)dudv
= 6 × 23
(4n + 8) × 9×23
(4n − 10)
= 24(4n + 8)(4n − 10).
As an immediate consequence of Theorem 4.6, we have the following result.
Corollary 4.7. The Narumi-Katayama index for G = S(n, C4) is given byNK(G) =
√PM1(G) = 2
√(4n + 8)(4n − 4).
Theorem 4.8. The first Zagreb polynomial for G = S(n, C4) is given byZG1(G, x) = 2
3 [(4n + 8) × x5 + (4n − 10) × x6].
764 H.M.A. Siddiqui
Proof. Using Equation 1.11 and Table 5, we have
ZG1(G, x) =∑
uv∈E(G)xdu+dv = 2
3(4n + 8) × x5 + 2
3(4n − 10) × x6
= 23
[(4n + 8) × x5 + (4n − 10) × x6].
Theorem 4.9. The second Zagreb polynomial for G = S(n, C4) is given byZG2(G, x) = 2
3 [(4n + 8) × x6 + (4n − 10) × x9].
Proof. Using Equation 1.12 and Table 5, we get
ZG2(G, x) =∑
uv∈E(G)xdudv = 2
3(4n + 8) × x6 + 2
3(4n − 10) × x9
= 23
[(4n + 8) × x6 + (4n − 10) × x9].
5. Conclusion and general remarksIn this paper, we have conducted the study of Zagreb indices and Zagreb polynomials
for the Sierpiński gasket graph Sn, the generalized Sierpiński graphs S(n, C3) and S(n, C4).We have computed the exact formulae of Zagreb indices and Zagreb polynomials for thesestructures. Various graph-theoretic parameters and certain distance based and count-ing related topological descriptors for the Sierpiński gasket graph Sn and the generalizedSierpiński graphs S(n, C3) and S(n, C4) can be considered for future study.
Acknowledgment. The author would like to thank the referee for his/her corrections,comments and useful criticism which improved the first version of this paper.
References[1] H. Ali, A.Q. Baig and M.K. Shafiq, On topological properties of hierarchical inter-
connection networks, J. Appl. Math. Comput. 55 (1-2), 313–334, doi:10.1007/s12190-016-1038-3, 2016.
[2] A.T. Balaban, I. Motoc, D. Bonchev and O. Makenyan, Topological indices forstructure-activity correlations, Topics Curr. Chem. 114, 21–55, 1983.
[3] G. Caporossi, P. Hansen and D. Vukičević, Comparing of Zagreb indices of cylicgraphs, MATCH Commun. Math. Comput. Chem. 63, 441–451, 2010.
[4] K.C. Das and I. Gutman, Some properties of the second Zagreb index, MATCHCommun. Math. Comput. Chem. 52, 103–112, 2004.
[5] M.V. Diudea, (Ed.), QSPR/QSAR Studies by molecular descriptors, NOVA, NewYork, 2001.
[6] M.R. Farahani, H.M.A. Siddiqui, Sh. Baby, M. Imran and M.K. Siddiqui, The Secondand Second Sum connectivity Indices of TUC4C8 Nanutubes, J. Optoelectron. Bio.Mater. , 8 (3), 107–111, 2016.
[7] H. Fath-tabar, Zagreb polynomials and PI indices of some nanostructures, Digest. J.Nanomater. Bios. 4, 189–191, 2009.
[8] B. Furtula and I. Gutman, A forgotten topological index, J. Math. Chem. 53, 1184-1190, 2015.
[9] B. Furtula, A. Graovac and D. Vukičević, Augmented Zagreb index, J. Math. Chem.48, 370–380, 2010.
Computation of Zagreb indices and Zagreb polynomials of Sierpiński graphs 765
[10] B. Furtula, I. Gutman and M. Dehmer, On structural-sensitivity of degree-based topo-logical indices, Appl.Math. Comput. 219, 8973–8978, 2013.
[11] M. Ghorbani and N. Azimi, Note on multiple Zagre indices, Iran. J. Math. Chem. 3,137–143, 2012.
[12] I. Gutman, Degree-based topological indices, Croat. Chem. Acta, 86, 351–361, 2013.[13] I. Gutman, An exceptional property of first Zagreb index, MATCH Commun. Math.
Comput. Chem. 72, 733–740, 2014.[14] I. Gutman and K.C. Das, The first Zagreb index 30 years after, MATCH Commun.
Math. Comput. Chem. 50, 83–92, 2004.[15] I. Gutman and B. Furtula, Ž. K. Vukićević and G. Popivoda, On Zagreb indices and
coindices, MATCH Commun. Math. Comput. Chem. 74, 5–16, 2015.[16] I. Gutman and O. Polansky, Mathematical Concepts in Organic Chemistry, Springer-
Verlag, Berlin, 1986.[17] I. Gutman and J. Tošović, Testing the quality of molecular structures descriptors.
Vertex-degree-based topological indices, J. Serb. Chem. Soc. 78, 805–810, 2013.[18] I. Gutman and N. Trinajstić, Graph theory and molecular orbitals. Total π-electron
energy of alternate hydrocarbons, Chem. Phy. Lett. 17, 535–538, 1972.[19] I. Gutman, B. Ruščić, N. Trinajstić and C.F. wilcox, Graph theory and molecular
orbitals. XII. Acyclic polyenes, J. Chem. Phys. 62, 1692–1704, 1975.[20] S. Hayat and H.M.A. Siddiqui, On bipartite edge frustration of carbon and boron
nanotubes, Studia UBB Chemia, LXI(1), 283–290, 2016.[21] A.M. Hinz and D. Parisse, Coloring Hanoi and Sierpiński graphs, Disc. Math. 312
(9), 1521–1535, 2012.[22] M. Imran, A.Q. Baig, H. Ali and S.U. Rehman, On topological properties of poly
honeycomb networks, Period. Math. Hungr. 73 (1), 100–119, 2016.[23] A. Jonsson, A trace theorem for the Drichlet form on the Sierpiński gasket, Math. Z.
250, 599–609, 2005.[24] H. Narumi and H. Katayama, Simple topological index, a newly devised index charac-
terizing the topological nature of structural isomers of saturated hydrocarbons, Mem.Fac. Engin. Hokkaido Univ. 16, 209–214, 1984.
[25] S. Nikolić, G. Kovačević, A. Milič and N. Trinajstić, The Zagreb indices 30 yearsafter, Croat. Chem. Acta, 76, 113–124, 2003.
[26] R.S. Scorer, P.M. Grundy and C.A.B. Smith, Some binary games, Math. Gaz. 28,96–103, 1944.
[27] G.H. Shirdel, H. Rezapour and A.M. Sayadi, The hyper-Zagreb index of graph opera-tions, Iran. J. Math. Chem. 4, 213–220, 213.
[28] S. Wang, M.R. Farahani and A.Q. Baig, The Sadhana polynomial and Sadhana indexof polycyclic aromatic hydrocarbon PAHk, J. Chem. Pharm. Res. 8 (6), 526–531, 2016.
[29] K. Xu and K.Ch. Das, Zagreb indices and polynomials of TUHRC4 and TUSC4C8nanotubes, MATCH Commun. Math. Comput. Chem. 68, 257–272, 2012.
[30] L. Yang, X. Ai and L. Zhang, The Zagreb coindices of a type of composite graph,Hacettepe J. Math. Stat. 45 (4), 1135–1145, 2016.
[31] B. Zhou, Zagreb indices, MATCH Commun. Math. Comput. Chem. 52, 113–118,2004.
Hacettepe Journal ofMathematics & Statistics
Hacet. J. Math. Stat.Volume 49 (2) (2020), 766 – 776
DOI : 10.15672/hujms.471023
Research Article
Oscillatory behavior of n-th order nonlinear delaydifferential equations with a nonpositive neutral
termS.R. Grace1, I. Jadlovská2, A. Zafer∗3
1Department of Engineering Mathematics, Faculty of Engineering, Cairo University, Orman, Giza 12221,Egypt
2Department of Mathematics and Theoretical Informatics, Faculty of Electrical Engineering andInformatics, Technical University of Košice, Letná 9, 042 00 Košice, Slovakia
3Department of Mathematics, College of Engineering and Technology, American University of the MiddleEast, Kuwait
AbstractWe study the oscillation problem for solutions of a class of n-th order nonlinear delaydifferential equations with nonpositive neutral terms. The obtained results improve andcorrelate many of the known oscillation criteria in the literature for neutral and non-neutralequations.
Mathematics Subject Classification (2010). 34K11, 34C10
Keywords. oscillation, second order differential equation, neutral term
1. IntroductionConsider the nonlinear n-th order delay differential equation of the form(
a(t)([x(t) − p(t)x(σ(t))](n−1)(t)
)α)′+ q(t)xβ(τ(t)) = 0, t ≥ t0, (1.1)
where n is even and t0 > 0 is fixed. It will be assumed that(i) α, β are the ratios of positive odd integers such that α ≥ β;
(ii) a ∈ C1([t0, ∞),R), a(t) > 0, a′(t) ≥ 0.(iii) p, q ∈ C([t0, ∞),R), 0 < p(t) ≤ p0 < 1, q(t) ≥ 0 and q(t) is not identically zero for
all large t;(iv) τ, σ ∈ C1([t0, ∞),R), τ(t) ≤ t, σ(t) ≤ t, τ ′(t) ≥ 0, σ′(t) > 0, and limt→∞ τ(t) =
limt→∞ σ(t) = ∞.By a solution of Eq. (1.1) we mean a function x(t) ∈ Cn−1([Tx, ∞),R), for some tx ≥
t0, which has the property a(t)([x(t) − p(t)x(σ(t))](n−1))α ∈ C1([tx, ∞),R) and satisfiesEq. (1.1) on [tx, ∞). We consider only those solutions x(t) of (1.1) which satisfy supx(t) :t ≥ T > 0 for all T ≥ tx. Such a solution of (1.1) is said to be oscillatory if it hasarbitrarily large zeros, and otherwise it is called nonoscillatory. Equation (1.1) is said
∗Corresponding Author.Email addresses: [email protected] (S.R. Grace), [email protected] (I. Jadlovská),
[email protected] (A. Zafer)Received: 16.10.2018; Accepted: 26.03.2019
Nonpositive neutral term 767
to be oscillatory if all its solutions are oscillatory. We note that the equation is calledhalf-linear when α = β, and sub-half-linear when α > β.
Recently, the oscillation of equations of the form (1.1) with linear and nonlinear neutralterm, has been considered in [1–8,11,13–15,17,20,21], where it is usually assumed that
−∞ < −p0 ≤ p(t) ≤ 0.
We note that there are only few results dealing with the oscillation of differential equationshaving a nonpositive neutral term. For an important initial contribution for such equationswe refer in particular to [20], where equation (1.1) was studied in the special case n = 2and α = 1 under the assumptions
0 ≤ p(t) ≤ p0 < 1, τ(t) = t − τ0, σ(t) = t − σ0.
Further contributions for (1.1) and its particular cases can be found in [5, 11, 15, 17, 21],where the authors established sufficient conditions ensuring that every solution x of (1.1)is either oscillatory or converges to zero as t → ∞. Unfortunately, these results cannotdistinguish solutions with different behaviors.
In this article, mainly motivated by the ideas [5, 8, 9, 19], we present new oscillationtheorems for n-th order nonlinear differential equations with a nonpositive neutral termof type (1.1). The obtained results improve and correlate many of the known results inthe literature even for the case p(t) = 0. The method we employ here in this work hasnaturally a partial resemblance for the second-order case [9], however the results and mostarguments are quite different due to higher-order nature of (1.1).
In the sequel, we let
A(v, u) =∫ v
u
1a1/α(s)
ds, v ≥ u ≥ t0,
and assume thatA(t, t0) → ∞ as t → ∞. (1.2)
It turns out that the improper integral∫ ∞
t0q(s) ds (1.3)
plays a key role in our study. In case it is convergent we define
Q(t) =∫ ∞
tq(s)ds, t ≥ t0.
The results of this paper are presented in a form which is essentially new. The paperis organized as follows. In Section 2 we provide some useful lemmas to be relied uponin the proofs of the theorems in Section 3. The last section is devoted to the illustrativeexamples. It may be of interest to study equation (1.1) with β > α.
2. LemmasAll the functional inequalities are assumed to hold eventually, that is, they are satisfied
for all t large enough.In what follows, we put
y(t) = x(t) − p(t)x(σ(t)). (2.1)
Lemma 2.1 (See [12]). Let u be a positive and k-times differentiable function on aninterval [ta, ∞) with its k-th derivative u(k) nonpositive on [ta, ∞) and not identically zeroon any subarray of [ta, ∞). Then there exists a tb ≥ ta and an integer l, 0 ≤ l ≤ k − 1,with k + l odd so that
(−1)l+ju(j) > 0 on [tb, ∞) (j = l, . . . , k − 1),u(i) > 0 on [tb, ∞) (i = 1, . . . , l − 1), when l > 1.
768 S.R. Grace, I. Jadlovská, A. Zafer
Lemma 2.2 (See [16]). Let u be as in Lemma 2.1 and tb ≥ ta be assigned to u by Lemma2.1. Moreover, let θ be a number with 0 < θ < 1. Then there exists a tc ≥ tb/θ such that
u(θt) ≥ [θ(1 − θ)]k−1
(k − 1)!tk−1u(k−1)(t), for all t ≥ tc. (2.2)
In addition, when limt→∞ u(t) = 0, for some tc ≥ ta we have
u(t) ≥ θ
(k − 1)!tk−1u(k−1)(t), for every t ≥ tc. (2.3)
Lemma 2.3 (See [18]). Let u(t) be a bounded k-times differentiable function on an interval[ta, ∞) with
u(t) > 0 (−1)ku(k)(t) ≥ 0 for t ≥ ta.
Then there exists a tb ≥ ta such that(−1)iu(i)(t) ≥ 0 for every t ≥ tb, i = 1, 2, . . . , k
and
u(ξ) ≥ (−1)k−1u(k−1)(η)(k − 1)!
(η − ξ)k−1 for every t ≥ tb, tb ≤ ξ ≤ η. (2.4)
Lemma 2.4. Assume that x(t) is a positive solution of (1.1) for t ≥ t1, t1 ∈ [t0, ∞).Then there exists t2 ∈ [t1, ∞) such that the corresponding function y(t) defined by (2.1)satisfies one of the following two cases:
y(t) > 0, y′(t) > 0, y(n−1)(t) > 0,(a(t)
(y(n−1)(t)
)α)′≤ 0, (C1)
y(t) < 0, (−1)i+1y(i)(t) > 0, i = 1, 2, . . . , n,(a(t)
(y(n−1)(t)
)α)′≤ 0, (C2)
for t ≥ t2.
Proof. Let x(t) be a positive solution of (1.1), say x(t), x(τ(t)) > 0, and x(σ(t)) > 0 fort ≥ t1. By Eq. (1.1), we have(
a(t)(y(n−1)(t)
)α)′
= − q(t)xβ(τ(t)) ≤ 0, t ≥ t1. (2.5)
Hence a(t)(y(n−1)(t)
)αis nonincreasing and of one sign eventually. That is, there exists
t2 ≥ t1 such that either y(n−1)(t) > 0 or y(n−1)(t) < 0 for t ≥ t2. We claim thaty(n−1)(t) > 0 for t ≥ t2. To see this, suppose on the contrary that y(n−1)(t) < 0 for t ≥ t2.Then
a(t)(y(n−1)(t)
)α≤ a(t2)
(y(n−1)(t2)
)α=: c < 0, t ≥ t2.
Integrating the above inequality, we see that
y(n−2)(t) ≤ y(n−2)(t2) + c1/α∫ t
t2a−1/α(s)ds.
By virtue of (1.2), we have limt→∞ y(t) = −∞. Since y(t) > −x(σ(t)), x(t) must beunbounded, and so there exists a sequence Tk∞
k=0 such that x(Tk) = maxx(s) : T0 ≤s ≤ Tk with limk→∞ Tk = ∞ and limk→∞ x(Tk) = ∞. Furthermore, since σ(Tk) > T0 forall k sufficiently large and σ(t) ≤ t, we see that
x(σ(Tk)) ≤ maxx(s) : T0 ≤ s ≤ Tk = x(Tk).Therefore, for all large k,
y(Tk) = x(Tk) − p(Tk)x(σ(Tk)) ≥ (1 − p(Tk))x(Tk) > 0which contradicts the fact that limt→∞ y(t) = −∞. Hence, we have proven the claim. Inview of (2.5) and (ii), we also have y(n)(t) < 0 for t ≥ t2. There are two possibilities to
Nonpositive neutral term 769
consider: either y(t) > 0 or y(t) < 0 for t ≥ t2. If y(t) > 0, then it follows from Lemma2.1 that y(t) satisfies (C1). If y(t) < 0, then we see that
x(t) ≤ p(t)x(σ(t)) ≤ x(σ(t)), (2.6)
which implies that x(t) and hence y(t) are bounded functions. Using Lemma 2.3 withu = −y, we obtain that y(t) satisfies (C2). The proof is complete.
Remark 2.1. For any positive solution x(t) of (1.1), the case (C2) is completely causedby presence of the neutral term. If p(t) = 0, such a case never occurs.
3. Oscillation of solutionsFor the sake of clarity, we put
k(t) =
1, when β = αc (tn−2A(t, t1))α−β, when β < α,
l(t) =
(
41−n
(n−1)!
)β, when β = α
c (tn−2A(t, t1))α−β, when β < α,
and
R(t) = τn−2(t)τ ′(t)(a(τ(t))k(t))1/α
, h(t) = σ−1(τ(t))
where c, c, t1 ∈ R.
We start with the following theorem.
Theorem 3.1. Let conditions (i)–(iv) and (1.2) hold, and let the integral (1.3) be con-vergent. If there exists a function ρ ∈ C1([t0, ∞), (0, ∞)) with ρ′(t) ≥ 0 such that, for allsufficiently large c, c, t1, and for some T > t1,
lim supt→∞
[ρ(t)Q(t) +
∫ t
T
[ρ(s)q(s) − µ
a(τ(s))k(s)(ρ′(s))α+1
(τn−2(s)τ ′(s)ρ(s))α
]ds
]= ∞, (3.1)
where
µ = αα
(1 + α)α+1
(2(n − 2)!β 42−n
)α
,
and
lim supt→∞
a−1(h(t))∫ t
h(t)
((h(t) − h(s))n−1
(n − 1)!
)βq(s)
pβ(h(s))ds
>
1 when β = α,
0 when β < α,(3.2)
then (1.1) is oscillatory.
Proof. Let x(t) be a nonoscillatory solution of (1.1), say x(t) > 0, x(τ(t)) > 0, x(σ(t)) > 0for t ≥ t1 for some t1 ≥ t0. It follows from Lemma 2.4 that there exists t2 ∈ [t1, ∞) suchthat the function y defined by (2.1) satisfies either (C1) or (C2) for t ≥ t2. We will considerboth cases separately.
At first, assume that (C1) holds. In view of (2.5) and x(t) ≥ y(t), we may write that(a(t)
(y(n−1)(t)
)α)′≤ −q(t)yβ(τ(t)) ≤ −q(t)yβ (τ(t)/2) . (3.3)
Define
w(t) := ρ(t)a(t)(y(n−1)(t))α
yβ(τ(t)/2), t ≥ t2. (3.4)
770 S.R. Grace, I. Jadlovská, A. Zafer
Therefore, w(t) > 0. By differentiating (3.4) and using (3.3), we get
w′(t) =(
ρ(t)yβ(τ(t)/2)
)′(a(t)(y(n−1)(t))α +
(a(t)(y(n−1)(t))α
)′ ρ(t)yβ(τ(t)/2)
≤ −ρ(t)q(t) +(
ρ′(t)ρ(t)
)w(t) − βρ(t)a(t)(y(n−1)(t))αy′(τ(t)/2)τ ′(t)
2yβ+1(τ(t)/2). (3.5)
Employing the inequality (2.2) in Lemma 2.2 with u = y′, it follows that there existst3 ≥ t2 such that
y′ (τ(t)/2) ≥ M1/2τn−2(t)y(n−1)(τ(t)), M1/2 = 42−n
(n − 2)!, for t ≥ t3. (3.6)
Using (3.5), (3.6) and the fact that a1/α(t)y(n−1)(t) is decreasing, we have
w′(t) ≤ −ρ(t)q(t) + ρ′(t)ρ(t)
w(t) −βM1/2
2τn−2(t)τ ′(t)ρ(t)
a1/α(τ(t))
(a1/α(t)y(n−1)(t)
)α+1
yβ+1(τ(t)/2), (3.7)
and hence
w′ ≤ −ρ(t)q(t) + ρ′(t)ρ(t)
w −βM1/2
2τn−2(t)τ ′(t)
(a(τ(t))ρ(t))1/αy(β−α)/α(τ(t)/2)w(α+1)/α(t).
If β = α, then y(β−α)/α(t) = 1 while for the case β < α and since a(t)(y(n−1)(t))α isdecreasing, there exists a constant c1 > 0 such that
a(t)(y(n−1)(t))α ≤ c1 for t ≥ t2,
which by integrating (n − 1)-times from t2 to t leads toy(t) ≤ c2tn−2A(t, t2) for t ≥ t4
for some constant c2 > 0 and t4 ≥ t2 . Then,
y(β−α)/α(τ(t)/2) ≥ y(β−α)/α(t) ≥ c(β−α)/α2 t(n−2)(β−α)/αA(β−α)/α(t, t2).
Using the two cases and the definition of k(t) in (3.7), we get
w′ ≤ −ρ(t)q(t) + ρ′(t)ρ(t)
w −βM1/2
2τn−2(t)τ ′(t)
(a(τ(t))k(t)ρ(t))1/αw(α+1)/α. (3.8)
Setting
B1 := ρ′(t)ρ(t)
, B2 :=βM1/2
2τn−2(t)τ ′(t)
(a(τ(t))k(t)ρ(t))1/α
and employing the inequality
B1u − B2u(1+α)/α ≤ αα
(1 + α)α+1 Bα+11 B−α
2 ,
(see [10]), we have from (3.8),
w′(t) ≤ −ρ(t)q(t) + µa(τ(t))k(t)
(τn−2(t)τ ′(t))α
(ρ′(t))α+1
ρα(t).
Integrating this inequality from t4 to t we get
w(t) ≤ w(t4) −∫ t
t4
[ρ(s)q(s) − µ
a(τ(s))k(s)(τn−2(s)τ ′(s))α
(ρ′(s))α+1
ρα(s)
]ds. (3.9)
On the other hand, it follows from (3.5) that
w′(t) ≤ −ρ(t)q(t) + ρ′(t)ρ(t)
w(t), (3.10)
Nonpositive neutral term 771
that is, (w(t)ρ(t)
)′≤ −q(t).
Integrating the above inequality from t to t′, we get
w(t′)ρ(t′)
≤ w(t)ρ(t)
−∫ t′
tq(s)ds,
and hencew(t) ≥ ρ(t)Q(t). (3.11)
By using (3.11) in (3.9), we find that
w(t4) ≥ ρ(t)Q(t) +∫ t
t4
[ρ(s)q(s) − µ
a(τ(s))k(s)(τn−2(s)τ ′(s))α
(ρ′(s))α+1
ρα(s)
]ds,
which clearly contradicts (3.1).Consider now case (C2). If we put z = −y, then Eq. (1.1) gives(
a(t)(z(n−1)(t)
)α)′≥ q(t)xβ(τ(t)).
Using the inequality z(t) ≤ p(t)x(σ(t)), we get(a(t)
(z(n−1)(t)
)α)′≥ q(t)
pβ(h(t))zβ(h(t)). (3.12)
In view of Lemma 2.3, we have
z(h(s)) ≥ (h(t) − h(s))n−1
(n − 1)!
(−z(n−1)(h(t))
), t ≥ s ≥ t2. (3.13)
Integrating (3.12) from h(t) to t and using (3.13) in the resulting inequality gives
(−z(n−1)(h(t))
)α≥
(−z(n−1)(h(t))
)β
a(h(t))
∫ t
h(t)
q(s)pβ(h(s))
((h(t) − h(s))n−1
(n − 1)!
)β
ds
or (−z(n−1)(h(t))
)α−β≥ a−1(h(t))
∫ t
h(t)
q(s)pβ(h(s))
((h(t) − h(s))n−1
(n − 1)!
)β
ds,
which contradicts (3.2). Note that z(n−1)(t) → 0 as t → ∞ is used when α > β.
Remark 3.1. As it will be shown in Example 4.1, the additional term ρ(t)Q(t) in (3.1)plays an important role in case
lim supt→∞
∫ t
T
[ρ(s)q(s) − αα
(1 + α)α+1
(2(n − 2)!β 42−n
)α a(τ(s))k(s)(ρ′(s))α+1
(τn−2(s)τ ′(s)ρ(s))α
]ds < ∞. (3.14)
Theorem 3.2. Let conditions (i)–(iv), (1.2), and (3.2) hold. If the integral (1.3) isdivergent, then (1.1) is oscillatory.
Proof. Let x(t) be a nonoscillatory solution of equation (1.1), say x(t) > 0, x(τ(t)) > 0,x(σ(t)) > 0 for t ≥ t1 for some t1 ≥ t0. It follows from Lemma 2.4 that there existst2 ∈ [t1, ∞) such that y satisfies either (C1) or (C2) for t ≥ t2.
If we assume that (C1) holds, then by letting t′ → ∞ in (3.10), we obtain a contradictionto the positivity of w(t). The rest of the proof is similar to that of Theorem 3.1 and henceis omitted.
In the following results we use different approaches to replace (3.1) in Theorem 3.1.
772 S.R. Grace, I. Jadlovská, A. Zafer
Theorem 3.3. Assume that α ≤ 1 and the hypotheses of Theorem 3.1 hold with (3.1)replaced by
lim supt→∞
[ρ(t)Q(t) +
∫ t
T
(ρ(s)q(s) − (n − 2)!
β42−n
(a(τ(s))k(s))1/α (ρ′(s))2
τn−2(s)τ ′(s)ρ(s)Q(1−α)/α(s)
)ds
]= ∞.
(3.15)Then (1.1) is oscillatory.
Proof. Let x(t) be a nonoscillatory solution of equation (1.1), say x(t) > 0, x(τ(t)) > 0,x(σ(t)) > 0 for t ≥ t1 for some t1 ≥ t0. It follows from Lemma 2.4 that there existst2 ∈ [t1, ∞) such that y satisfies either (C1) or (C2) for t ≥ t2. If (C1) holds, then as inthe proof of Theorem 3.1, we obtain (3.8). Thus, in view of (3.11), we have
w′(t) ≤ −ρ(t)q(t) + ρ′(t)ρ(t)
w(t) −βM1/2
2τn−2(t)τ ′(t)
(a(τ(t))k(t)ρ(t))1/αw(α+1)/α(t)
≤ −ρ(t)q(t) + ρ′(t)ρ(t)
w(t) −βM1/2
2τn−2(t)τ ′(t)
(a(τ(t))k(t))1/α ρ(t)Q(1−α)/α(t)w2(t)
≤ −ρ(t)q(t) + 1βM1/2
(a(τ(t))k(t))1/α (ρ′(t))2
τn−2(t)τ ′(t)ρ(t)Q(1−α)/α(t).
The rest of the proof is similar to that of Theorem 3.1 and hence is omitted. Theorem 3.4. Assume that the hypotheses of Theorem 3.1 hold with (3.1) replaced by
lim inft→∞
( 1Q(t)
∫ ∞
tR(s)Q(α+1)/α(s)ds
)>
α
(α + 1)(α+1)/α
2(n − 2)!β42−n
. (3.16)
Then (1.1) is oscillatory.
Proof. Let x(t) be a nonoscillatory solution of equation (1.1) , say x(t) > 0, x(τ(t)) > 0,x(σ(t)) > 0 for t ≥ t1 for some t1 ≥ t0. It follows from Lemma 2.4 that there existst2 ∈ [t1, ∞) such that y satisfies either (C1) or (C2) for t ≥ t2. We will consider both casesseparately.
At first, assume that (C1) holds. Define w(t) as in (3.4) with ρ(t) = 1, i.e.,
w(t) := a(t)(y(n−1)(t))α
yβ(τ(t)/2), t ≥ t2. (3.17)
Then as in proof of Theorem 3.1 we get
w′(t) ≤ −q(t) −βM1/2
2τn−2(t)τ ′(t)
(a(τ(t))k(t))1/αw(α+1)/α(t), (3.18)
Integrating (3.18) from t to t′, we see that
w(t′) ≤ w(t) −∫ t′
tq(s)ds −
βM1/22
∫ t′
t
τn−2(s)τ ′(s)(a(τ(s))k(s))1/α
w(α+1)/α(s)ds
= w(t) −∫ t′
tq(s)ds −
βM1/22
∫ t′
tR(s)w(α+1)/α(s)ds.
(3.19)
As in the proof of Theorem 3.1, we can show Q(t) < ∞ and∫∞
t R(s)w(α+1)/α(s)ds < ∞for t ≥ t3. Letting t′ → ∞ in (3.19), we get
w(t) ≥ Q(t) + β42−n
2(n − 2)!
∫ ∞
tR(s)w(α+1)/α(s)ds. (3.20)
Hence,
w(t)Q(t)
≥ 1 + β42−n
2(n − 2)!1
Q(t)
∫ ∞
tR(s)Q(α+1)/α(s)
(w(s)Q(s)
)(α+1)/α
ds. (3.21)
Nonpositive neutral term 773
Let λ = inft≥T
(w(t)/Q(t)). Then it is easy to see that λ ≥ 1 and, from (3.16) and (3.21),
λ ≥ 1 + α
(λ
α + 1
)(α+1)/α
,
which contradicts the admissible value of λ and α.Consider now case (C2). Similar to the proof of Theorem 3.1, one can get a contradiction
to (3.2). The proof is complete.
Let the integral (1.3) be convergent. We define the sequence un(t)∞n=0 by
u0(t) = Q(t),
un(t) =∫ ∞
tR(s)u(α+1)/α
n−1 (s)ds + u0(t), n = 1, 2, . . .
for t ≥ T ≥ t1 ≥ t0. By induction, it is easy to see that un(t) ≤ un+1(t), n = 0, 1, 2, . . ..
Theorem 3.5. Assume that the hypotheses of Theorem 3.1 except (3.1) hold. If thereexists any ui(t) such that
lim supt→∞
l(t)τβ(n−1)(t)aβ/α(τ(t))
ui(t) > 1, (3.22)
then (1.1) is oscillatory.
Proof. Let x(t) be a nonoscillatory solution of equation (1.1) , say x(t) > 0, x(τ(t)) > 0,x(σ(t)) > 0 for t ≥ t1 for some t1 ≥ t0. It follows from Lemma 2.4 that there existst2 ∈ [t1, ∞) such that y satisfies either (C1) or (C2) for t ≥ t2.
If y(t) satisfies (C1), then as in the proof of Theorem 3.4, we get that (3.20) holds forw(t) defined by (3.17) and some T ≥ t0 large enough, and thus, w(t) ≥ Q(t) = u0(t). Byinduction, we can see that
w(t) ≥ ui(t), t ≥ T, i = 1, 2, . . . . (3.23)
Since the sequence ui(t)∞i=0 is monotone increasing and bounded above, there exists a
function u(t) such that u(t) = limi→∞ ui(t). By Lebesgue monotone theorem,
u(t) = β42−n
2(n − 2)!
∫ ∞
tR(s)u(α+1)/α(s)ds + Q(t).
On the other hand, using (2.2) and the fact that a(t)(y(n−1)(t))α is decreasing in (3.17),we arrive at
1w(t)
= yβ(τ(t)/2)a(t)
(y(n−1)(t)
)α≥(
41−n
(n − 1)!τn−1(t)
)β(y(n−1)(τ(t))
)β
a(t)(y(n−1)(t)
)α≥(
41−n
(n − 1)!τn−1(t)
)β(a1/α(t)y(n−1)(t)
)β−α
aβ/α(τ(t)).
(3.24)
If α = β, then evidently1
w(t)≥ l(t)τβ(n−1)(t)
aβ/α(τ(t)). (3.25)
If α > β, then there exists a constant c > 0 such that
a1/α(t)y(n−1)(t) ≤ c for t ≥ T .
774 S.R. Grace, I. Jadlovská, A. Zafer
Thus, in view of (i) and (3.24), we also get (3.25). Combining (3.23) with (3.25), we seethat
l(t)τβ(n−1)(t)aβ/α(τ(t))
ui(t) ≤ 1,
which contradicts (3.22).Consider now case (C2). Similar to the proof of Theorem 3.1, one can get a contradiction
to (3.2). The proof is complete.
Theorem 3.6. Assume that the hypotheses of Theorem 3.1 except (3.1) hold. If
lim supt→∞
τ (n−1)β(t)a−1(τ(t))Q(t) > ((n − 1)!)β , β = α (3.26)
andlim sup
t→∞τ (n−1)β(t)a−β/α(τ(t))Q(t) = ∞, β < α, (3.27)
then (1.1) is oscillatory.
Proof. Let x(t) be a nonoscillatory solution of (1.1), say x(t) > 0, x(τ(t)) > 0, x(σ(t)) > 0for t ≥ t1 for some t1 ≥ t0. It follows from Lemma 2.4 that there exists t2 ∈ [t1, ∞) suchthat y satisfies either (C1) or (C2) for t ≥ t2. We will consider both cases separately.
At first, assume that (C1) holds. Now, set w(t) := a(t)(y(n−1)(t)
)α. Integrating (1.1)
from t to ∞ and using (iii), we have
w(t) ≥∫ ∞
tq(s)yβ(τ(s))ds ≥ Q(t)yβ(τ(t)). (3.28)
By virtue of Lemma 2.2, we get
y(τ(t)) ≥ θ
(n − 1)!τn−1(t)y(n−1)(τ(t)) (3.29)
for every θ ∈ (0, 1). Thus,
w(t) ≥ Q(t)(
θ
(n − 1)!
)β
τβ(n−1)(t)(y(n−1)(τ(t))
)β
= Q(t)(
θ
(n − 1)!
)β τβ(n−1)(t)aβ/α(τ(t))
wβ/α(τ(t)).(3.30)
Using the fact that w(t) is decreasing, we have
w(t) ≥ Q(t)(
θ
(n − 1)!
)β τβ(n−1)(t)aβ/α(τ(t))
wβ/α(t)
or
w1−β/α(t) ≥ Q(t)(
θ
(n − 1)!
)β τβ(n−1)(t)aβ/α(τ(t))
.
Taking lim sup of both sides of this inequality as t → ∞, we arrive at a contradiction to(3.26). when β = α and (3.27) when β < α.
Consider now case (C2). Similar to the proof of Theorem 3.1, one can get a contradictionto (3.2). The proof is complete.
If the equation is not of neutral type, then we can drop the condition (3.2). Withoutthis condition, a weaker result is still possible.
Theorem 3.7. Assume that excluding (3.2) all the assumptions of Theorems 3.1 or The-orems 3.3 or Theorems 3.4 or Theorems 3.5 or Theorems 3.6 hold. Then every solutionx(t) of (1.1) is oscillatory when p(t)=0, and is either oscillatory or approaches zero as ttends to infinity.
Nonpositive neutral term 775
Proof. It suffices to show that if x(t) is a positive solution of (1.1) and y(t) satisfies (C2),then lim
t→∞x(t) = 0. To see this we observe from y(t) < 0 and (2.6) that x(t) is bounded.
Therefore, we havelim sup
t→∞x(t) = a ≥ 0.
We claim that a = 0. If not, then there exists a sequence Tk∞k=0 such that limk→∞ Tk =
∞ and limk→∞ x(Tk) = a > 0. Let ϵ = a(1 − p0)/(2p0); then, for all large k, we havex(σ(Tk)) < a + ϵ. From this and the definition of y, we obtain
0 ≥ limk→∞
y(Tk) ≥ limk→∞
x(Tk) − p0(a + ϵ) = a(1 − p0)2
> 0,
a contradiction. Thus a = 0 and limt→∞ x(t) = 0. The proof is complete.
4. ExamplesThe following examples are illustrative.
Example 4.1. Consider the neutral equation(((x(t) − p0x(σ0t))′′′
)1/4)′
+ q0t7/4 x1/4
(45 t)
= 0, (4.1)
where q0 is a positive constant, σ0 ∈ (0, 1) and p0 ∈ [0, 1). If we set ρ(t) := t, thencondition (3.1) reduces to
q0 > 3√
10/5 ≈ 1.89737, (4.2)
while (3.14) gives only q0 > 4√
10/5 ≈ 2.52828. This improvement is due to the additionalterm ρ(t)Q(t) in (3.1). In view of Theorems 3.1 and 3.7, we conclude that Eq. (1.1) isoscillatory for p0 = 0. For p0 > 0 and, e.g., σ0 = 10/9, it is easy to see that h(t) =(8/9)t ≤ t, and by Theorem 3.1, we have that Eq. (4.1) is oscillatory if
q0 > 122.8072 p0.
Example 4.2. Consider the neutral equation(x(t) − 1
2x(t − π2 ))′′
+ 8x(t − π) = 0. (4.3)
Clearly, σ(t) = t − π2 and σ−1(t) = t + π
2 , τ(t) = t − π, and so h(t) := t − π2 . All conditions
of Theorem 3.2 are satisfied and hence the Eq. (4.3) is oscillatory. One such solution isx(t) = sin(4t).
Example 4.3. Consider the neutral equation(((x(t) − 1
2x(√
t))′)3)′
+ q0t5/4 + 1
x(t1/4) = 0, (4.4)
where q0 is a positive constant. Here, σ(t) =√
t and σ−1(t) = t2, τ(t) = t1/4, and soh(t) =
√t. All conditions of Theorem 3.1 are satisfied for every q0 and all large t and
hence the Eq. (4.4) is oscillatory.
Acknowledgment. We thank the anonymous reviewer whose comments have greatlyimproved this manuscript.
776 S.R. Grace, I. Jadlovská, A. Zafer
References[1] R.P. Agarwal, S.R. Grace, and D. O’Regan, Oscillation theory for second order linear,
half-linear, superlinear and sublinear dynamic equations, Springer Science & BusinessMedia, 2002.
[2] R.P. Agarwal, S.R. Grace, and P.J.Y. Wong, Oscillation theorems for certain higherorder nonlinear functional differential equations, Appl. Anal. Discr. Math. 2, 1–30,2008.
[3] R.P. Agarwal, M. Bohner, T. Li, and C. Zhang, A new approach in the study ofoscillatory behavior of even-order neutral delay differential equations, Appl. Math.Comput. 225, 787–794, 2013.
[4] R.P. Agarwal, M. Bohner, T. Li, and C. Zhang, Oscillation of second order differentialequations with a sublinear neutral term, Carpathian J. Math. 30 (1), 1–6, 2014.
[5] J.G. Dong, Oscillation behavior of second order nonlinear neutral differential equa-tions with deviating arguments, Comput. Math. Appl. 59, 3710 – 3717, 2010.
[6] S.R. Grace and B.S. Lalli, Oscillation of nonlinear second order neutral differentialequations, Rat. Math. 3, 77 – 84, 1987.
[7] S.R. Grace, J.R. Graef, and M.A. El-Beltagy, On the oscillation of third order neutraldelay dynamic equations on time scales, Comput. Math. Appl. 63 (4), 775–782, 2012.
[8] S.R. Grace and I. Jadlovská, Oscillation Criteria for second-order neutral dampeddifferential equations with delay argument, in: Dynamical Systems - Analytical andComputational Techniques, InTech, 2017.
[9] S.R. Grace,Oscillatory behavior of second-order nonlinear differential equations witha nonpositive neutral term, Mediterr. J. Math. 14 (6), Art. 229, 2017.
[10] G.H. Hardy, I.E. Littlewood, and G. Polya, Inequalities, University Press, Cambridge,1959.
[11] B. Karpuz, O. Ocalan, and S. Ozturk, Comparison theorems on the oscillation andasymptotic behaviour of higher-order neutral differential equations, Glasgow Math. J.52 (1), 107–114, 2010.
[12] I.T. Kiguradze, On the oscillation of solutions of the Eq. dmu/dtm +a(t)|u|nsgnu = 0,Mat. Sb. 65, 172–187, 1964 (in Russian).
[13] T. Li, Z. Han, C. Zhang, and H. Li, Oscillation criteria for second-order superlinearneutral differential equations, Abstr. Appl. Anal. 2011, 2011.
[14] T. Li, Yu.V. Rogovchenko, and C. Zhang, Oscillation results for second-order non-linear neutral differential equations, Adv. Differ. Equ. 2013, 1 – 13, 2013.
[15] Q. Li, R. Wang, F. Chen, and T. Li , Oscillation of second-order nonlinear delaydifferential equations with nonpositive neutral coefficients, Adv. Differ. Equ. 2015,1–15, 2015.
[16] Ch. G. Philos, A new criterion for the oscillatory and asymptotic behavior of delaydifferential equations, Bull. Acad. Pol. Sci. Ser. Sci. Mat. 39 (1), 61–64, 1981.
[17] H. Qin, N. Shang, and Y. Lu, A note on oscillation criteria of second order nonlinearneutral delay differential equations, Comput. Math. Appl. 56, 2987–299, 2008.
[18] V. Staikos and I. Stavroulakis, Bounded oscillations under the effect of retardationsfor differential equations of arbitrary order, P. Roy. Soc. Edinb. 77 (1), 129–136, 1977.
[19] H. Wu, L. Erbe, and A. Peterson, Oscillation of solution to second-order half-lineardelay dynamic equations on time scales, Electron. J. Differ. Eq. 2016 (71), 1–15,2016.
[20] J.S.W. Wong, Necessary and sufficient conditions for oscillation of second order neu-tral differential equations, J. Math. Anal. Appl. 252, 342–352, 2000.
[21] Q. Yang, l. Yang, and S. Zhu, Interval criteria for oscillation of second-order nonlinearneutral differential equations, Comput. Math. Appl. 46 (5), 903–918, 2003.
Hacettepe Journal ofMathematics & Statistics
Hacet. J. Math. Stat.Volume 49 (2) (2020), 777 – 783
DOI : 10.15672/hujms.499848
Research Article
Hopf algebra structure on superspace SP2|1q
Salih Celik
Department of Mathematics, Yıldız Technical University, Davutpasa-Esenler, 34220 Istanbul, Turkey
AbstractSuper-Hopf algebra structure on the function algebra on the extended quantum symplecticsuperspace SP2|1
q , denoted by F(SP2|1q ), is defined. A quantum Lie superalgebra derived
from F(SP2|1q ) is explicitly obtained.
Mathematics Subject Classification (2010). 16W30, 16T20, 17B37, 17B66, 20G42,58B32
Keywords. quantum symplectic superspace, super ⋆-algebra, super-Hopf algebra,quantum supergroup, quantum Lie superalgebra
1. IntroductionQuantum supergroups and quantum superalgebras are even richer mathematical sub-
jects as compared to Lie supergroups and Lie superalgebras. A quantum superspace is aspace that quantum supergroup acts with linear transformations and whose coordinatesbelong to a noncommutative associative superalgebra [7].
Some algebras have been considered which are covariant with respect to the quantumsupergroups in [4]. Using the corepresentation of the quantum supergroup OSPq(1|2),some non-commutative spaces covariant under its coaction have been constructed [2]. Inthe present work, we set up a super-Hopf algebra structure on an algebra which appearsin both paper. We denote this algebra by O(SP2|1
q ). As is known, the matrix elementsof the quantum supergroups OSPq(1|2) and OSPq(2|1) are the same and they act bothquantum superspaces SP1|2
q and SP2|1q . But these two quantum superspaces are not the
same. A study on SP1|2q was made in [3]. Here we will work on the quantum symplectic
superspace SP2|1q .
2. Review of quantum symplectic groupIn this section, we will give some information about the structures of quantum sym-
plectic groups as much as needed.The algebra O(OSPq(1|2)) is generated by the even elements a, b, c, d and odd elements
α, δ. Standard FRT construction [5] is obtained via the matrix R given in [6]. Using theRTT-relations and the q-orthosymplectic condition, all defining relations of O(OSPq(1|2))are explicitly obtained in [2]:
Email address: [email protected]: 20.12.2018; Accepted: 27.03.2019
778 S. Celik
Theorem 2.1. The generators of O(OSPq(1|2)) satisfy the relations
ab = q2ba, ac = q2ca, aα = qαa,
aδ = qδa + (q − q−1)αc, ad = da + (q − q−1)[(1 + q−1)bc + q−1/2αδ],bc = cb, bd = q2db, bα = q−1αb, bδ = qδb, (2.1)cd = q2dc, cα = q−1αc, cδ = qδc,
dα = q−1αd + (q−1 − q)δb, dδ = q−1δd,
αδ = −qδα + q−1/2(q2 − 1)bc, α2 = q1/2(q − 1)ba, δ2 = q1/2(q − 1)dc.
In (2.1), the relations involving the elements γ, e and β are not written. They can befound in [2]. Other relations that we need in this study are given below:
[e, α]q = q1/2(q − 1)(γb + βa), [e, β]q−1 = q−1/2(q−1 − 1)(δb + αd),
[e, γ]q = q1/2(1 − q)(δa + αc), [e, δ]q−1 = q−1/2(1 − q−1)(γd + βc), (2.2)
β2 = q1/2(q − 1)db, γ2 = q1/2(q − 1)ca,
e2 = 1 − q−1/2[α, δ]q = 1 + q1/2[β, γ]q−1
where [u, v]Q = uv − Qvu.
The quantum superdeterminant is defined by
Dq = ad − qbc − q1/2αδ
= da − q−1bc + q−1/2δα.
The element Dq is a central element of O(OSPq(2|1)).
If A and B are Z2-graded algebras, then their tensor product A ⊗ B is the Z2-gradedalgebra whose underlying space is Z2-graded tensor product of A and B. The followingdefinition gives the product rule for tensor product of algebras. Let us denote by τ(a) thegrade (or degree) of an element a ∈ A.
Definition 2.2. If A is a Z2-graded algebra, then the product rule in the Z2-gradedalgebra A ⊗ A is defined by
(a1 ⊗ a2)(a3 ⊗ a4) = (−1)τ(a2)τ(a3)a1a3 ⊗ a2a4
where ai’s are homogeneous elements in the algebra A.
Definition 2.3. A super-Hopf algebra is a vector space A over K with three linear maps∆ : A −→ A ⊗ A, called the coproduct, ϵ : A −→ K, called the counit, and S : A −→ A,called the coinverse, such that
(∆ ⊗ id) ∆ = (id ⊗ ∆) ∆, (2.3)m (ϵ ⊗ id) ∆ = id = m (id ⊗ ϵ) ∆, (2.4)m (S ⊗ id) ∆ = η ϵ = m (id ⊗ S) ∆, (2.5)
together with ∆(1) = 1 ⊗ 1, ϵ(1) = 1, S(1) = 1 and for any a, b ∈ A
∆(ab) = ∆(a)∆(b), ϵ(ab) = ϵ(a)ϵ(b), S(ab) = (−1)τ(a)τ(b)S(b)S(a) (2.6)
where m : A ⊗ A −→ A is the product map, id : A −→ A is the identity map andη : K −→ A.
Hopf algebra structure on superspace SP2|1q 779
3. Quantum symplectic superspace SP2|1q
In this section, we define a super-Hopf algebra structure on the extended functionalgebra of the quantum superspace SP2|1
q .
3.1. The algebra of polynomials on the superspace SP2|1q
The elements of the symplectic superspace are supervectors generated by two evenand an odd components. We define a Z2-graded symplectic space SP2|1 by dividing thesuperspace SP2|1 of 3x1 matrices into two parts SP2|1 = V0 ⊕ V1. A vector is an elementof V0 (resp. V1) and is of grade 0 (resp. 1) if it has the formx
0y
, resp.
0θ0
.
While the even elements commute to everyone, the odd element satisfies the relationθ2 = 0.
In [3], the quantum superspace SP2|1q is considered as the dual space of quantum super-
space SP1|2q and then relations (3.1) below are obtained by interpreting the coordinates as
differentiations.
Definition 3.1. Let K⟨x, θ, y⟩ be a free associative algebra generated by x, θ, y and Iq bea two-sided ideal generated by xθ − qθx, xy − q2yx, yθ − q−1θy, θ2 − q1/2(q − 1)yx. Thequantum superspace SP2|1
q with the function algebra
O(SP2|1q ) = K⟨x, θ, y⟩/Iq
is called Z2-graded quantum symplectic space (or quantum symplectic superspace).
Here the coordinates x and y with respect to the Z2-grading are of grade 0 (or even),the coordinate θ with respect to the Z2-grading is of grade 1 (or odd).
According to the above definition, if (x, θ, y)t ∈ SP2|1q then we have
xθ = qθx, θy = qyθ, yx = q−2xy, θ2 = q1/2(q − 1)yx (3.1)
where q is a non-zero complex number. This associative algebra over the complex numbersis known as the algebra of polynomials over quantum (2+1)-superspace.
It is easy to see the existence of representations that satisfy (3.1); for instance, thereexists a representation ρ : O(SP2|1
q ) → M(3,C) such that matrices
ρ(x) =
q 0 00 q2 00 0 1
, ρ(θ) =
0 q − 1 00 0 0
q1/2 0 0
, ρ(y) =
0 0 00 0 00 1 0
representing the coordinate functions satisfy the relations (3.1).
Note that the last two relations in (3.1) can be also written as a single relation. There-fore, we say that O(SP2|1
q ) is the superalgebra with generators x± and θ satisfying therelations [4]
x±θ = q±1θx±, [x+, x−] = q−1/2(q + 1)θ2. (3.2)where x+ = x and x− = y.
Definition 3.2 ([4]). The quantum supersphere on the quantum symplectic superspaceis defined by
r = q1/2x−x+ + θ2 − q−1/2x+x−.
780 S. Celik
3.2. A ⋆-structure on the algebra O(SP2|1q )
Here we define a Z2-graded involution on the algebra O(SP2|1q ).
Definition 3.3. Let A be an associative superalgebra. A Z2-graded linear map ⋆ : A −→ Ais called a superinvolution (or Z2-graded involution) if
(ab)⋆ = (−1)τ(a)τ(b)b⋆a⋆, (a⋆)⋆ = a
for any elements a, b ∈ A. The pair (A, ⋆) is called a Z2-graded ⋆-algebra.
If the parameter q is real, then the algebra O(SP2|1q ) becomes a ⋆-algebra with involution
determined by the following proposition.
Proposition 3.4. If q > 0 then the algebra O(SP2|1q ) supplied with the Z2-graded involu-
tion determined byx⋆
+ = q1/2 x−, θ⋆ = i θ, x⋆− = q−1/2 x+
becomes a super ⋆-algebra where i =√
−1.
Proof. We must show that the relations (3.2) are invariant under the star operation. Ifq is a positive number, we have
(x±θ − q±1θx±)⋆ = (i θ)(q±1/2x∓) − q±1(q±1/2x∓)(i θ) = q±1/2i (θx∓ − q±1θx∓)
and since [x+, x−]⋆ = [x+, x−]
[x+, x−] = [x+, x−]⋆ = q−1/2(q + 1)(−θ⋆θ⋆) = q−1/2(q + 1)θ2.
Hence the ideal (x±θ − q±1θx±, [x+, x−] − q−1/2(q + 1)θ2) is ⋆-invariant and the quotientalgebra K⟨x+, θ, x−⟩/(x±θ − q±1θx±, [x+, x−] − q−1/2(q + 1)θ2) becomes a ⋆-algebra.
3.3. The super-Hopf algebra structure on SP2|1q
We define the extended Z2-graded quantum symplectic space to be the algebra contain-ing SP2|1
q , the unit and x−1+ , the inverse of x+, which obeys x+x−1
+ = 1 = x−1+ x+. We will
denote the unital extension of O(SP2|1q ) by F(SP2|1
q ). The following theorem asserts thatthe superalgebra F(SP2|1
q ) is a super-Hopf algebra:
Theorem 3.5. The algebra F(SP2|1q ) is a Z2-graded Hopf algebra. The definitions of a
coproduct, a counit and a coinverse on the algebra F(SP2|1q ) are as follows
(i) the coproduct ∆ : F(SP2|1q ) −→ F(SP2|1
q ) ⊗ F(SP2|1q ) is defined by
∆(x+) = x+ ⊗ x+, ∆(θ) = θ ⊗ 1 + 1 ⊗ θ, ∆(x−) = x−1+ ⊗ x− + x− ⊗ x−1
+ , (3.3)
(ii) the counit ϵ : F(SP2|1q ) −→ C is given by
ϵ(x+) = 1, ϵ(θ) = 0, ϵ(x−) = 0,
(iii) the algebra F(SP2|1q ) admits a C-algebra antihomomorphism S : F(SP2|1
q ) −→ F(SP2|1q−1)
defined byS(x+) = x−1
+ , S(θ) = −θ, S(x−) = −x+x−x+.
Proof. The axioms (2.3)-(5) are satisfied automatically. It is also not difficult to showthat the co-maps preserve the relations (3.2). In fact, for instance,
∆([x+, x−]) = ∆(x+x− − x−x+) = 1 ⊗ [x+, x−] + [x+, x−] ⊗ 1
= q−1/2(q + 1)(1 ⊗ θ2 + θ2 ⊗ 1)∆(θ2) = 1 ⊗ θ2 + θ2 ⊗ 1,
Hopf algebra structure on superspace SP2|1q 781
andS([x+, x−]) = −[x+, x−], S(θ2) = −θ2.
Since S2(a) = id(a) for all a ∈ F(SP2|1q ), the coinverse S is of second order.
The set xkθlym : k, l, m ∈ N0 form a vector space basis of F(SP2|1q ). The formula (3.3)
gives the action of the coproduct ∆ only on the generators. The action of ∆ on producton generators can be calculated by taking into account that ∆ is a homomorphism.
Corollary 3.6. For the quantum supersphere r, we have∆(r) = r ⊗ 1 + 1 ⊗ r, ϵ(r) = 0, S(r) = −r.
Proof. Using the definition of ∆, as an algebra homomorphism, on the generators ofF(SP2|1
q ) in (3.3), it is easy to see that the element r ∈ F(SP2|1q ) is a primitive element,
that is,∆(r) = q1/2(x−1
+ ⊗ x− + x− ⊗ x−1+ )(x+ ⊗ x+) + (θ ⊗ 1 + 1 ⊗ θ)(θ ⊗ 1 + 1 ⊗ θ)
− q−1/2(x+ ⊗ x+)(x−1+ ⊗ x− + x− ⊗ x−1
+ )
= q1/2(1 ⊗ x−x+ + x−x+ ⊗ 1) + θ2 ⊗ 1 + 1 ⊗ θ2 − q−1/2(1 ⊗ x+x− + x+x− ⊗ 1)
= 1 ⊗ (q1/2x−x+ + θ2 − q−1/2x+x−) + (q1/2x−x+ + θ2 − q−1/2x+x−) ⊗ 1.
Since ϵ(1) = 1 andm(id ⊗ ϵ)∆(r) = rϵ(1) + ϵ(r)1 = r = m(ϵ ⊗ id)∆(r),
we obtain ϵ(r) = 0. Finally, using the fact that S is an anti-homomorphism we get
S(r) = q1/2x−1+ (−x+x−x+) − (−θ)(−θ) − q−1/2(−x+x−x+)x−1
+
= −(q1/2x−x+ + θ2 − q−1/2x+x−),as desired.
3.4. Coactions on the quantum symplectic superspaceLet a, b, c, d, e, γ, α, δ, β be elements of an algebra A. Assuming that the generators of
O(OSPq(2|1)) super-commute with the elements of O(SP2|1q ), define the components of
the vectors X ′ = (x′, θ′, y′)t and X ′′ = (x′′, θ′′, y′′)t using the following matrix equalitiesX ′ = T X and (X ′′)t = Xt T (3.4)
where X = (x, θ, y)t ∈ SP2|1q and T ∈ OSPq(2|1). If we assume that q = 1 then we have
the following theorem proving straightforward computations.
Theorem 3.7. If the transformations in (3.4) preserve the relations (3.1), then the entriesof T satisfy the relations (2.1) and then generate the algebra O(OSPq(2|1)) together withq-orthosymplectic condition.
A left quantum space (or left comodule algebra) for a Hopf algebra H is an algebra Xtogether with an algebra homomorphism (left coaction) δL : X −→ H ⊗ X such that
(id ⊗ δL) δL = (∆ ⊗ id) δL and (ϵ ⊗ id) δL = id.
Right comodule algebra can be defined in a similar way.
Theorem 3.8. (i) The algebra O(SP2|1q ) is a left and right comodule algebra of the Hopf
algebra O(OSPq(2|1)) with left coaction δL and right coaction δR such that
δL(Xi) =3∑
k=1tik ⊗ Xk, δR(Xi) =
3∑k=1
Xk ⊗ tki. (3.5)
782 S. Celik
(ii) The quantum supersphere r belongs to the center of O(SP2|1q ) and satisfies δL(r) = 1⊗r
and δR(r) = r ⊗ 1.
Proof. (i) These assertions are obtained from the relations in (2.1) and (2.2) togetherwith (3.1).
(ii) That r is a central element of O(SP2|1q ) is shown by using the relations in (3.1). To
show that δL(r) = 1 ⊗ r and δR(r) = r ⊗ 1 we use the definitions of δL and δR in (3.5)and the relations (2.1) and (2.2) with Dq = 1.
4. An h-deformation of the superspace SP2|1
In this section, we introduce an h-deformation of the superspace SP2|1 from the q-deformation via a contraction following the method of [1]. Consider the q-deformed algebraof functions on the quantum superspace SP2|1
q generated by x± and θ with the relations(3.2).
We introduce new coordinates X± and Θ by
x =
x+θ
x−
=
1 0 00 1 0h
q−1 0 1
X+
ΘX−
= g X.
When the relations (3.2) are used, taking the limit q → 1 we obtain the following exchangerelations, which define the h-superspace SP2|1
h :
Definition 4.1. Let O(SP2|1h ) be the algebra with the generators X± and Θ satisfying
the relationsX+Θ = ΘX+, X−Θ = ΘX− − 2hΘX+, X+X− = X−X+ + 2Θ2 (4.1)
where the coordinates X± are even and the coordinate Θ is odd. We call O(SP2|1h ) the
algebra of functions on the Z2-graded quantum space SP2|1h .
h-deformed supersphere on the symplectic h-superspace is given byrh = X−X+ + Θ2 + h X2
+ − X+X− = h X2+ − Θ2.
It is easily seen that the quantum supersphere rh belongs to the center of the superal-gebra O(SP2|1
h ).The definition of dual q-deformed symplectic superspace is given as follows [2].
Definition 4.2. Let Kφ+, z, φ− be a free associative algebra generated by z, φ+, φ−and Iq be a two-sided ideal generated by zφ± − q±1φ±z, φ−φ+ + q−2φ+φ− + q−2Qz2 andφ2
±. The quantum superspace SP1|2q with the function algebra
O(SP 1|2q ) = Kφ+, z, φ−/Iq
is called Z2-graded quantum symplectic space (or quantum symplectic superspace) whereQ = q1/2 − q3/2 and q = 0.
In case of exterior h-superspace, we use the transformationx = gX
with the components φ+, z and φ− of x. The definition is given below.
Definition 4.3. Let Λ(SP2|1h ) be the algebra with the generators Φ± and Z satisfying the
relationsΦ+Z = ZΦ+, ZΦ− = Φ−Z − 2hΦ+Z, Φ−Φ+ = −Φ+Φ−,
Φ2+ = 0, Φ2
− = h(2Φ−Φ+ − Z2)
Hopf algebra structure on superspace SP2|1q 783
where the coordinate Z is even and the coordinates Φ± are odd. We call Λ(SP2|1h ) the
quantum exterior algebra of the Z2-graded quantum space SP2|1h .
5. A Lie superalgebra derived from F(SP2|1q )
It is known that an element of a Lie group can be represented by exponential of anelement of its Lie algebra. By virtue of this fact, one can define the generators of thealgebra F(SP2|1
q ) asx+ := eu, θ := q−1/2 ξ, x− := e−uv. (5.1)
Then, the following lemma can be proved by direct calculations using the relations
xk±θ = q±k θxk
±, [xk+, x−] = q−1/2 q2k − 1
q − 1θ2 xk−1
+ , ∀k ≥ 1
whose the proof follows from induction on k.
Lemma 5.1. The generators u, ξ, v have the following commutation relations (Lie (anti-)brackets)
[u, ξ] = ~ ξ, [ξ, v] = 0, [u, v] = 2~1 − e−~ ξ2, (5.2)
where q = e~ and ~ ∈ R.
We denote the algebra for which the generators obey the relations (5.2) by L~ :=L(SP2|1
q ). The Z2-graded Hopf algebra structure of L~ can be read off from Theorem 3.5:
Theorem 5.2. The algebra L~ is a Z2-graded Hopf algebra. The definitions of a coproduct,a counit and a coinverse on the algebra L~ are as follows:
∆(ui) = ui ⊗ 1 + 1 ⊗ ui, ϵ(ui) = 0, S(ui) = −ui
for ui ∈ u, ξ, v.
The following proposition can be easily proved by using the Proposition 3.4 togetherwith (5.1).
Proposition 5.3. The algebra L~ supplied with the Z2-graded involution determined byu⋆ = 1
2 ~ + ln(e−uv), ξ⋆ = i ξ, v⋆ = v
becomes a super Lie ⋆-algebra.
References[1] A. Aghamohammadi, M. Khorrami, and A. Shariati, h-deformation as a contraction
of q-deformation, J. Phys. A: Math. Gen. 28, L225-L231, 1995.[2] N. Aizawa and R. Chakrabarti, Quantum Spheres for OSPq(1|2), J. Math. Phys. 46,
103510:1-25, 2005.[3] S. Celik, Covariant differential calculi on quantum symplectic superspace SP1|2
q , J.Math. Phys. 58, 023508:1-15, 2017.
[4] M. Chaichian and P.P. Kulish, Quantum group covariant systems, in From field theoryto quantum groups. World Sci. Publ., River Edge, NJ, 99-111, 1996.
[5] L.D. Faddeev, N.Yu. Reshetikhin, and L.A. Takhtajan, Quantization of Lie groupsand Lie algebras, Leningrad Math. J. 1, 193-225, 1990.
[6] P.P. Kulish and N.Yu Reshetikhin, Universal R-matrix of the quantum superalgebraosp(2|1), Lett. Math. Phys. 18, 143-149, 1989.
[7] Yu I. Manin, Multiparametric quantum deformation of the general linear supergroup,Commun. Math. Phys. 123, 163-175, 1989.
Hacettepe Journal ofMathematics & Statistics
Hacet. J. Math. Stat.Volume 49 (2) (2020), 784 – 792
DOI : 10.15672/hujms.456426
Research Article
On LPI rings
Rachida El Khalfaoui1, Najib Mahdou1, Abdeslam Mimouni∗2
1Department of Mathematics, Faculty of Science and Technology of Fez, Box 2202, University S.M. BenAbdellah Fez, Morocco
2Department of Mathematics and Statistics, King Fahd University of Petroleum & Minerals, Dhahran31261, Saudi Arabia
AbstractIn this paper, we extend the LPI property (that is, every locally principal ideal in anintegral domain is invertible) to rings with zero-divisors and we study the class of com-mutative rings in which every regular locally principal ideal is invertible called LPI rings.We investigate the stability of this property under homomorphic image, and its transferto various contexts of constructions such as direct products, amalgamation of rings andtrivial ring extensions. Our results generate examples which enrich the current literaturewith new and original families of rings that satisfy this property.
Mathematics Subject Classification (2010). 13A15, 13F05, 13F10
Keywords. locally principal ideals, regular ideals, trivial extension, pullback,amalgamation of rings, LPI-rings
1. IntroductionAll rings considered in this paper are assumed to be commutative with identity elements.
It is well-known that a finitely generated flat module over a domain is projective, and overan integral domain, the notion of projective ideal is equivalent to the one of invertibleideal. In general, an invertible ideal is projective but the converse is not true. The notionof domains with flat ideals invertible was first studied by Sally and Vasconcelos in 1975(see [21]) as domains with property P. They showed that if a domain D has the propertyP, then so does the polynomial ring D[X]. In 1977 Glas and Vasconcelos studied theinvertibility of faithfully flat ideals over an H-domain and conjectured that over an H-domain, a faithfully flat ideal is finitely generated (and hence invertible) see [14].
In [8], S. Bazzoni conjectured that Prüfer domains for which “an ideal is invertible ifand only if it is a locally principal" are exactly the ones with the finite character prop-erty, i.e. each nonzero element of the domain belongs to finitely many maximal ideals.This conjecture was first resolved in the affirmative by Holland, Martinez, McGovern andTesemma ([16]). Later, Halter-Koch stated and proved an analog of Bazzoni’s conjectureusing the language of ideal systems, that is, r-Prüfer monoids, which in the domain case
∗Corresponding Author.Email addresses: [email protected] (R. El Khalfaoui), [email protected] (N. Mahdou),
[email protected]; [email protected] (A. Mimouni)Received: 31.08.2018; Accepted: 27.03.2019
On LPI rings 785
are PV MD’s and include Prüfer domains ([15]). In 2010, Picozza and Tartarone intro-duced the notion of quasi-stable ideals as ideals I that are flat in their ring homomorphisms(I : I). They also studied domains in which every ideal is quasi-stable, and proved thatGlaz-Vascocncelos conjecture is false ([20]).
In 2011, D. D. Anderson and Muhammad Zafrullah [6] introduced and studied thenotion of LPI domains as integral domains in which every nonzero locally principal idealis invertible. They proved that a finite character intersection of LPI overrings is an LPI-domain and so if a domain D is a finite character intersection D = ∩DP for some set ofprime ideals of D, then D is an LPI domain. In 2013, Kui, Wang and Chen answeredpositively a question raised by Anderson and Zafrullah of whetter a polynomial ring overan LPI domain is an LPI domain ([17, Theorem 1.8]). In 2014, D. D. Anderson andA. Mimouni studied LPI domains in pullbacks ([3]). Very recently, Xing and Wang F.answered negatively a question by Anderson-Zafurllah by showing that if R is an LPIdomain and S is a multiplicatively closed set, then RS need not be an PLI domain (see[22]).
The purpose of the present work is to extend the notion of LPI domain to an arbitraryring with zero-divisors. A ring R is said to be an LPI ring if every regular locally principalideal of R is invertible. Noetherian rings are obviously LPI rings by [18, Lemma 18.1]. Ouraim is to give some simple methods in order to construct LPI rings outside the context ofintegral domains that are not Noetherian. For this, we investigate the stability of the LPIproperty under homomorphic image, and its transfer to various contexts of constructionssuch as direct products, amalgamation of rings and trivial ring extensions. Our resultsgenerate original examples which enrich the current literature with new families of ringssatisfying the LPI property. We denote Z(R) the set of zero-divisors of R and by Reg(R)the set of regular elements in R.
Let A be a ring and E an A-module. The trivial ring extension of A by E (also calledthe idealization of E over A) is the ring R = A n E whose underlying group is A × Ewith multiplication given by (a, e)(a′, e′) = (aa′, ae′ + a′e). Recall that if I is an ideal ofA and E′ is a submodule of E such that IE ⊆ E′, then J = I n E′ is an ideal of R.However, prime (resp., maximal) ideals of R have the form P n E, where P is a prime(resp., maximal) ideal of A [5, Theorem 3.2]. Suitable background on commutative trivialring extensions is [5, 7, 13].
Let A and B be two rings, let J be an ideal of B and let f : A −→ B be a ringhomomorphism. In this setting, we consider the following subring of A × B
A f J = (a, f(a) + j)|a ∈ A, j ∈ Jcalled the amalgamation of A and B along J with respect to f . Moreover, other classicalconstructions (such as the A + XB[X], A + XB[[X]], and the D + M constructions) canbe studied as particular cases of the amalgamation (see [9, Examples 2.5 and 2.6]). A par-ticular case of this construction is the amalgamated duplication of a ring along an ideal I(introduced and studied by D’Anna and Fontana in [9–11]). Let A be a ring, and let I bean ideal of A. A I := (a, a + i) : a ∈ A, i ∈ I is called the amalgamated duplicationof A along the ideal I. See for instance [9–12].
2. Main resultsNotice that a characterization of locally principal ideals in integral domains is given in
[6, Theorem 1]. The following proposition extends this characterization to regular locallyprincipal ideals in rings with zero-divisors. The proof is similar to that one in [6, Theorem1], and for the convenience of the reader we include it here. Recall that an ideal I in a ringR is called a cancellation ideal if IJ ⊆ IK for ideals J and K of R implies that J ⊆ K.
786 R. El Khafaoui, N. Mahdou, A. Mimouni
Proposition 2.1 ([6, Theorem 1]). Let R be a ring and let I be a regular ideal of R. Thefollowing conditions are equivalent:
(1) I is locally principal.(2) I is faithfully flat.(3) I is a cancellation ideal.
Proof. (1) ⇒ (2) Let I be a regular locally principal ideal of R. Then, IM is free for allmaximal ideals M of R. Indeed, IM is a regular ideal of RM for all M ∈ Max(R) since Iis regular. Therefore, IM is a regular principal ideal of RM for all M ∈ Max(R) and soIM is genereted by a regular element and so it is free. Hence, IM is faithfully flat for allM ∈ Max(R). Then, I is faithfully flat.(2) ⇒ (3) Let I be a regular faithfully flat ideal of R. Let IJ ⊆ IK for some ideals J andK of R. We concider the exact sequence,
0 → K → K + J → K+JK → 0.
Since I is faithfully flat, then we have
0 → I ⊗ K → I ⊗ (K + J) → I ⊗ K+JK → 0.
Therefore, I ⊗ K+JK = I⊗(K+J)
I⊗K = I(K+J)IK = IK
IK = 0. Since I is faithfully flat, thenK+J
K = 0 and so J ⊆ K.(3) ⇔ (1) By [4, Theorem].
Next, we study the transfer of the LPI property to direct products.Theorem 2.2. Let (Ri)i=1,··· ,n be a family of commutative rings. Then R =
∏ni=1 Ri is
an LPI ring if and only if so is Ri for each i = 1, · · · , n.The proof of this theorem needs the following lemmas. The proof of the first lemma is
straightforward and it is omitted, and the second lemma is a well-known result.Lemma 2.3. Let (Ri)i=1,2 be two rings and Ii be an ideal of Ri for i = 1, 2. Then, I1 × I2is a regular locally principal ideal of R1 × R2 if and only if I1 and I2 are regular locallyprincipal ideals of R1 respectively R2.Lemma 2.4 ([18, Lemma 18.1]). Let I be a regular locally principal ideal of a ring A.Then, I is invertible if and only if it is finitely generated.Proof of Theorem 2.2. The proof is done by induction on n and it suffices to check itfor n = 2. By Lemma 2.3, I1 × I2 is a regular locally principal ideal of R1 × R2 if andonly if I1 and I2 are locally principal ideals of R1 respectively R2 and it is easy to see thatI1 ×I2 is a finitely generated ideal of R1 ×R2 if and only if I1 and I2 are finitely generatedideals of R1 respectively R2 and we conclude by Lemma 2.4.
The next result shows that the LPI-property descends into a faithfully flat ring homo-morphism.Proposition 2.5. Let R and S be two rings and f : R → S be a ring homomorphismmaking S a faithfully flat R-module. Assume that f(Reg(R)) ⊆ Reg(S). If S is an LPIring, then so is R.In particular, if R and S are two domains and f : R → S is a ring homomorphism makingS a faithfully flat R-module then if S is an LPI domain, then so is R.Proof. Let I be a regular locally principal ideal of R. Then I is a faithfully flat R-moduleby Proposition 2.1. So I⊗S = IS is a faithfully flat S-module. Since f(Reg(R)) ⊆ Reg(S),IS is a regular locally principal ideal of S and so it is invertible since S is an LPI-ringwhich, in turn, is equivalent to IS is finitely generated. Therefore I ⊗ S = IS is finitelygenerated. Hence I is a finitely generated ideal of R (as S is a faithfully flat R-module),and therefore I is an invertible ideal of R. Thus R is an LPI ring as desired.
On LPI rings 787
As an immediate consequence, we recover [6, Teorem 5 (1)].
Corollary 2.6 ([6, Theorem 5 (1)]). Let R be a domain and let X be an indeterminateover R. If R[X] (resp.R[[X]]) is an LPI ring, then so is R.
Now, we will see the transfer of the LPI property to homomorphic image. Recall thatan ideal I of a ring R is a pure ideal if for every maximal ideal M ∈ Max(R), IM = 0M
or IM = RM .
Proposition 2.7. Let R be a ring.(1) Let P be a finitely generated non-regular prime ideal of R such that P is contained
in all regular locally principal ideals of R. If RP is an LPI ring, then so is R.
(2) Let I be a regular pure ideal of R. If R is an LPI ring, then RI is an LPI ring.
The proof of this proposition needs the following lemma.
Lemma 2.8. Let P be a non-regular prime ideal of a ring R.If J is a regular locally principal ideal of R such that P ⊆ J , then J
P is a regular locallyprincipal ideal of R
I .
Proof. Let J be a regular locally principal ideal of R such that P ⊆ J and let a be aregular element of J . Necessarily a ∈ P . Now, for every x ∈ R, ax = 0 in J
P implies thatax ∈ P . Hence x ∈ P since P is prime. Then, x = 0 and so a is a regular element in J
P .Therefore, J
P is a regular ideal of RI . Thus if J is locally principal, then the localization
of JP at any maximal ideal M
P of RP is isomorphic to JM
PM. Since JM is principal, JM
PMis
principal and therefore JP is locally principal.
Proof of Proposition 2.7. (1) Let P be a finitely generated non-regular prime ideal ofR contained in all regular locally principal ideals. Assume that R
P is an LPI ring. Let J
be a regular locally principal ideal of R. Then JP is a regular locally principal ideal of R
P
(Lemma 2.8) and so it is invertible. By Lemma 2.4, JP is finitely generated. Now, by the
exact sequence,
0 −→ P −→ J −→ JP −→ 0
J is finitely generated and by Lemma 2.4 we conclude that J is invertible. Hence R is anLPI ring.
(2) Let JI be a regular locally principal ideal of R
I . Then JI is a faithfully flat R
I -module.Since I is a regular ideal of R and I ⊆ J , then J is a regular ideal of R. Now, considerthe following exact sequence:
0 → I → J → JI → 0
Since I is a regular pure ideal of R, I is a regular locally principal ideal of R. So I isa faithfully flat ideal of R by Proposition 2.1. Thus J
I is a flat R-module (since RI is a
flat R-module). By the exact sequence, we conclude that J is a faithfully flat ideal ofR. Hence J is a regular locally principal ideal of R. So J is invertible and so finitelygenerated. Therefore J ⊗ R
I = JI is a finitely generated ideal of R
I and so it is invertibleby Lemma 2.4. Therefore R
I is an LPI ring. Example 2.9. Let K be a field and R = K n K and let P = 0 n K. Then P is afinitely generated non-regular prime ideal of R that is contained in all regular ideals of Rby Lemma 2.12. Since R
P = KnK0nK = K, K n K is an LPI ring by Proposition 2.7.
Our next theorem develops a result on the transfer of the LPI property to trivial ringextension. Recall that if E is an A-module, then Z(E) = a ∈ A such that ae = 0 forsome 0 = e ∈ E.
788 R. El Khafaoui, N. Mahdou, A. Mimouni
Theorem 2.10. Let A be a ring, E an A-module, R = A n E the trivial ring extensionof A by E and let S = A r (Z(A) ∪ Z(E)).
(1) Assume that for every regular locally principal ideal I of A, I ∩ S = ∅. If R is anLPI ring, then so is A.
(2) Assume that E is torsion-free and divisible. Then A is an LPI ring if and only ifso is R. In particular, if A is a domain and E is a flat divisible R-module, thenA is an LPI ring if and only if so is R.
(3) Assume E is finitely generated and E = S−1E. If A is an LPI ring, then so is R.If furthermore, for every regular locally principal ideal I of A, I ∩ S = ∅, then Ais an LPI ring if and only if so is R.
The proof of this theorem needs the following lemmas.
Lemma 2.11. (1) Let A be a ring, E an A-module and I an ideal of A and let S =A r (Z(A) ∪ Z(E)).(a) If InIE is a regular locally principal ideal of AnE, then I is a regular locally
principal ideal of A.(b) If I ∩ S = ∅, then I is a regular locally principal ideal of A if and only if
I n IE is a regular locally principal ideal of A n E.(c) If E is torsion free, then I is a regular locally principal ideal of A if and only
if I n IE is a regular locally principal ideal of A n E.(2) Let E′ is a submodule of E. If I n E′ is a regular locally principal ideal of A n E,
then so is I.
Proof. (1) (a) Clearly if (a, e) is a regular element of I n IE, then a is a regular elementof I by [1, Lemma 6]; and I is a locally principal ideal of A if and only if I n IE is alocally principal ideal of A n E by [1, Theorem 7].
(b) Assume that I∩S = ∅. Then there exists a regular element a in I such that a /∈ Z(E).Clearly (a, 0) is a regular element of I n IE. Indeed, let (0, 0) = (b, e) ∈ I n IE such that(a, 0)(b, e) = (0, 0). Then ab = 0 and ae = 0. Since a is regular, b = 0 and since a /∈ Z(E),e = 0. Thus (b, e) = (0, 0) and so (a, 0) is regular. Hence InIE is a regular ideal of AnE.
(c) Follows from [2, Lemma 10].(2) Let E′ is a submodule of E such that I n E′ is a regular locally principal ideal of
A n E. Then I is a regular ideal of A and for every maximal ideal M of R,(I n E′)MnE = IM n EM
= (AM n EM )(x, e)= AM x n (AM e + EM e)
Hence IM = AM x and therefore I is a locally principal ideal of A. Lemma 2.12 ([5, Theorem 3.9]). Let A be a ring, E an A-module and S = A r (Z(A) ∪Z(E)). Then the following conditions are equivalent.
(1) Every regular ideal of A n E has the form I n E where I is an ideal of A withI ∩ S = ∅.
(2) Every regular ideal of A n E is homogenous.(3) E = S−1E.
Lemma 2.13 ([1, Theorem 7]). Let A be a ring, I a nonzero ideal of A and E an A-module. If I n IE is an invertible ideal of A n E, then I is invertible.
Lemma 2.14 ([2, Theorem 11]). Let A be a ring, E be a torsion-free and divisible A-module and I n N a homogenous ideal of A n E. If I is an invertible ideal of A, thenI n N is invertible.
On LPI rings 789
Lemma 2.15 ([1, Theorem 9]). Let A be a ring, E an A-module, I nN an homogeneousideal of AnE. If I is a finitely generated ideal of A and N is a finitely generated submoduleof E, then I n N is finitely generated.
Proof of Theorem 2.10. (1) Assume that for every regular locally principal ideal I ofA, I ∩ S = ∅ and let I be a regular locally principal ideal of A. By Lemma 2.11 (1)(b),I n IE is a regular locally principal ideal of A n E. Since A n E is an LPI ring, thenI n IE is invertible. By Lemma 2.13, I is invertible and so A is an LPI ring.
(2) Assume that E is torsion-free and divisible. Then E = S−1E where S = Ar(Z(A)∪Z(E)) (since E is divisible).Let J be a regular locally principal ideal of R. By Lemma 2.12, J has the form I n Ewhere I is an ideal of A with I ∩ S = ∅. Thus I is a regular locally principal ideal of Aby Lemma 2.11 (2). Since A is an LPI ring, I is invertible. By Lemma 2.14, I n E is aninvertible ideal of A n E and therefore A n E is an LPI ring.
Conversely, assume that A n E is an LPI ring. Since E is torsion free, Z(E) ⊆ Z(A)and so for every regular locally principal ideal I of A, I ∩ S = ∅. Hence A is an LPI ringby (1).
(3) Assume that E is finitely generated, E = S−1E and A is an LPI ring. Let I n Ebe a regular locally principal ideal of R where I is an ideal of A with I ∩ S = ∅. Then Iis a regular locally principal ideal of A (Lemma 2.11 (2)) and thus I is invertible (and sois finitely generated). Since E is finitely generated, I n E is a finitely generated ideal ofA n E (Lemma 2.15) and therefore it is invertible by Lemma 2.4. Thus A n E is an LPIring. If furthermore, for every regular locally principal ideal I of A, I ∩ S = ∅, then theequivalence by (1).
Theorem 2.10 leads to the following result.
Corollary 2.16. Let A is a domain and E a K-vector space where K = qf(A). ThenA n E is an LPI ring if and only if so is A.
Example 2.17. Let A = Z(2) = ab | a, b ∈ Z and b is not divisible by 2 and E = Q
Z(2).
Then A n E is an LPI ring.
Proof. Z(2) , as a DV R, is an LPI ring.Q
Z(2)is a divisible (Z(2))-module since Q is a divisible (Z(2))-module. Also, Q
Z(2)= a
b |a, b ∈ Z and b is divisible by 2 is a flat (Z(2))-module since it is a free module generatedby 1
2 . So A n E is an LP -ring by Theorem 2.10 (2). By Theorem 2.10, we are able to give new examples of LPI rings that are not Noether-
ian.
Example 2.18. Let K be a field and E be an infinite dimensional vector space over K.Then :
(1) K n E is an LPI ring (by Corollary 2.16).(2) K n E is not Noetherian (since E is not finitely generated and by [5, Theorem
4.8]).
Next, we will see the transfer of the LPI property to the amalgamated duplication.Recall that an ideal J of A I is called homogeneous if J = K I for some ideal K ofA. If K is an ideal of A I such that 0× I ⊆ K, then K is homogeneous (see [19, Lemma2.9]).
790 R. El Khafaoui, N. Mahdou, A. Mimouni
Theorem 2.19. Let A be a ring and I be an ideal of A.(1) Assume that I is flat. If A I is an LPI ring, then so is A.(2) Assume that I is a non-regular finitely generated ideal of A and S−1I = I where
S = A \ Z(A). If A is an LPI ring, then so is A I. If furthermore I is flat,then A is an LPI ring if and only if so is A I.
The proof of this theorem needs the following lemmas. The proofs of the first and thirdlemma are elementary proofs and for convenience we include them here.
Lemma 2.20. Let A be a ring and I be an ideal of A. If (a, a + i) is a regular element ofA I, then a is a regular element of A.
Proof. Let (a, a + i) be a regular element of A I and b ∈ A such ab = 0. If there existsj ∈ I such that jb = 0, then (a, a + i)(bj, 0) = (0, 0), which is absurd since (a, a + i) isregular. Hence, for all j ∈ I, jb = 0. Thus (a, a + i)(b, b) = (0, 0) and so (b, b) = (0, 0).Hence a is a regular element of A. Lemma 2.21. Assume that I is a non-regular proper ideal of a ring A and S = A\(Z(A)).The following are equivalent:
(1) All regular ideals of A I are homogeneous.(2) For all a ∈ S we have I = aI which is equivalent to I = S−1I.
Proof. (1) ⇒ (2) Assume that all regular ideals of A I are homogeneous and leta ∈ S. Then (a, a) is a regular element of A I and so ⟨(a, a)⟩ is a homogeneous idealof A I. Thus 0 × I ⊆ ⟨(a, a)⟩ = J I for some ideal J of A. Let i ∈ I. Then,(0, i) = (α, α + k)(a, a) where α ∈ A and k ∈ I. Since a is a regular element of A, αa = 0implies that α = 0. So (0, i) = (0, k)(a, a). Thus i = ka and so I = aI.
(2) ⇒ (1) Let K be a regular ideal of A I and let (a, a + i) ∈ K a regular element ofA I. By Lemma 2.20, a is a regular element of A and so ⟨(a, a + i)⟩ = (a, a + i)A I =aA (iA + aI + iI)). Since I = aI, ⟨(a, a + i)⟩ = aA I. Hence 0 × I ⊆ ⟨(a, a + i)⟩ ⊆ Kand therefore K is a homogeneous ideal of A I. Remark 2.22. Let A be a ring and I be a proper regular ideal of A. Then there existsa regular ideal of A I which is not homogeneous. Indeed, if we suppose that all regularideals of A I are homogeneous, then the ideal generated by (c, c), where c is a regularelement in I, is homogeneous. By the same argument in the proof of Lemma 2.21 we showthat I = cI and so c = cλ for some λ ∈ I (since c ∈ I). So, λ = 1 since c is regular.Therefore, I = A which is absurd since I is a proper ideal of A.
Lemma 2.23. Let I and J be two ideals of a ring A. If I and J are finitely generated,then J I is a finitely generated ideal of A I.
Proof. Assume J is generated by a family of elements a1, a2 · · · , an and I is generatedby a family of elements k1, k2 · · · , kn. Let (a, a + h) ∈ J I. Then, a =
∑ni=1 αiai
where αi ∈ A for i = 1, · · · , n, and since h ∈ I, then h =∑n
i=1 βiki where βi ∈ A fori = 1, · · · , n. Hence,
(a, a + h) = (a, a) + (0, h) =∑n
i=1(αi, αi)(ai, ai) +∑n
i=1(βi, βi)(0, ki).
Hence J I is a finitely generated ideal of A I generated by
(ai, ai)i=1,··· ,n ∪ (0, ki)i=1,··· ,n.
Proof of Theorem 2.19. (1) Assume that I is a flat ideal of A. Then A I is afaithfully flat A-module. Let J be a regular locally principal ideal of A. Then J⊗(A I) =J(A I) since A I is a faithfully flat A-module. So J ⊗ (A I) = J(A I) = J JI
On LPI rings 791
by [11, Proposition 3.1(2)]. Since J is regular, then J JI is a regular ideal of A I. ByProposition 2.1, J is a faithfully flat ideal of A and so J ⊗ (A I) = J JI is a faithfullyflat ideal of (A I). Thus it is locally principal by Proposition 2.1. Since (A I) is anLPI ring, then J ⊗ (A I) = J JI is invertible which is equivalent to J ⊗ (A I)is finitely generated. Since A I is a faithfully flat A-module, J is a finitely generatedideal of A and so invertible. Hence A is an LPI ring.
(2) Let K be a regular locally principal ideal of A I. Then K is a homogeneous idealby Lemma 2.21. So K = J I for some ideal J of A. Let M be a maximal ideal of A.Two cases are possible:Case 1: I ⊆ M . Then M I is a maximal ideal of A I and so JM IM = (J I)MI = KMI ([12, Theorem 3.8]) is a principal ideal of (A I)MI . Hence JM is aprincipal ideal of AM .Case 2: I * M . Then JM = (J I)MI ([12, Theorem 3.5]) is a principal ideal of(A I)MI . Hence JM is a principal ideal of AM .In the both cases J is locally principal. So J is regular locally principal. Hence J isinvertible and so it is finitely generated. Thus J I is a finitely generated ideal of A Iby Lemma 2.23, and therefore, J I is invertible. Hence A I is an LPI ring. Iffurthermore I is flat, then the converse follows from (1). Example 2.24. Let A = K nE where K is a field and E is an infinite dimensional vectorspace over K and let I = 0 n F where F is a finite dimensional subspace of E . Then :
(1) A I is an LPI ring (by Theorem 2.19(2)).(2) A I is not Noetherian (since A is not Noetherian by Example 2.18 and by
[12, Corollary 2.9]).
References[1] M.M. Ali, Idealization and Theorems of D.D. Anderson, Comm. Algebra 34, 4479-
4501, 2006.[2] M.M. Ali, Idealization and Theorems of D.D. Anderson II, Comm. Algebra 35, 2767-
2792, 2007.[3] D.D. Anderson and A. Mimouni, LPI domains and Pullbacks, Comm. Algebra 42,
2759-2768, 2014.[4] D.D. Anderson and M. Roitman, A characterization of cancellation ideals, Proc.
Amer. Math. Soc. 125, 2853-2854, 1997.[5] D.D. Anderson and M. Winders, Idealization of a module, J. Commut. Algebra 1 (1),
3-56, 2009.[6] D.D. Anderson and M. Zafrullah, Integral domains in which nonzero locally principal
ideals are invertible, Comm. Algebra 39, 933-941, 2011.[7] C. Bakkari, S. Kabbaj and N. Mahdou, Trivial extension definided by Prûfer condi-
tions, J. Pure App. Algebra 214, 53-60, 2014.[8] S. Bazzoni, Class semigroups of Prüfer domains, J. Algebra 184, 613-631, 1996.[9] M. D’Anna, C.A. Finocchiaro and M. Fontana, Amalgamated algebra along an ideal,
Commmutative Algebra and Applications, Walter De Gruyter, 155-172, 2009.[10] M. D’Anna, C.A. Finocchiaro and M. Fontana, Properties of chains of prime ideals
in amalgamated algebras along an ideal, J. Pure Appl. Algebra 214, 1633-1641, 2010.[11] M. D’Anna, C.A. Finocchiaro and M. Fontana, New algebraic properties of an amal-
gamated algebra along an ideal, Comm. Algebra 44, 1836-1851, 2016.[12] M. D’Anna and M. Fontana, An amalgamated duplication of a ring along an ideal:
the basic properties, J. Algebra Appl. 6, 443-459, 2007.[13] S. Glaz, Commutative coherent rings, Springer-Verlag, Lecture Notes in Mathematics,
13-71, 1989.[14] S. Glaz and W. Vasconcelos, Flat ideals II, Manuscripta Math. 22 (4), 325-341, 1977.
792 R. El Khafaoui, N. Mahdou, A. Mimouni
[15] F. Halter-Koch, Clifford semigroups of ideals in monoids and domains, Forum Math.21, 1001-1020, 2009.
[16] W.C. Holland, J. Martinez, W.Wm. McGovern and M. Tesemma, Bazzoni’s Conjec-ture, J. Algebra 320, 1764-1768, 2008.
[17] K. Hu, F.G. Wang and H. Chen, A note on LPI domains, Bull. Korean Math. Soc.50 (3), 719-725, 2013.
[18] J.A. Huckaba, Commutative rings with zero divisors, Marcel Dekker, New York-Basel,1988.
[19] K. Louartiti and N. Mahdou, Transfer of multiplication-like conditions in amalga-mated algebra along an ideal, Afr. Diaspora J. Math. 14 (1), 119-125, 2012.
[20] G. Picozza and F. Tartarone, Flat ideals and stability in integral domains, J. Algebra324, 1790-1802, 2010.
[21] J. Sally and W. Vasconcelos, Flat ideals I, Comm. Algebra 3 531-543, 1975.[22] S. Xing and F.G. Wang, Two questions on domains in which locally principal ideals
are invertible. J. Algebra Appl. 16 (6), 1750112, 8 pp, 2017.
Hacettepe Journal ofMathematics & Statistics
Hacet. J. Math. Stat.Volume 49 (2) (2020), 793 – 807
DOI : 10.15672/hujms.477534
Research Article
A minimal family of sub-basesYiliang Li1, Jinjin Li∗1,2, Yidong Lin3, Jun-e Feng1, Hongkun Wang4
1School of Mathematics and Statistics, Minnan Normal University, Zhangzhou, Fujian 363000, P.R.China
2Lab of Granular Computing, Minnan Normal University, Zhangzhou, Fujian 363000, P.R. China3School of Mathematical Sciences, Xiamen University, Xiamen, Fujian 361005, P.R. China
4Department of Biostatistics, Bioinformatics, and Biomathematics, Georgetown University, Washington,DC 20057, USA
AbstractThis paper investigates a minimal family of sub-bases. First, the concept of a minimalfamily of sub-bases is presented and its properties are studied. Then the relationshipbetween reducts in covering information systems and minimal families of sub-bases isdiscussed. Based on Boolean matrices, an approach is provided to derive a minimal familyof sub-bases. Finally, experiments are conducted to illustrate the effectiveness of theproposed approach.
Mathematics Subject Classification (2010). 54A05, 54B15, 54C05, 54C10
Keywords. a minimal family of sub-bases, Boolean matrix, covering informationsystems, reduct
1. IntroductionThe object of general topology is to study topological properties, which are invariants
of homeomorphism [1]. Based on the properties of the topological rough membershipfunction, Li and Zhang [6] presented the definition of sub-base reduct in a family of sub-bases. But there is no further research on sub-base reducts in a family of sub-bases fromthe point of view of general topology. In fact, sub-base reducts in a family of sub-bases aretopological properties. In order to illustrate this point, this paper proposes the concept ofa minimal family of sub-bases, which is equal to sub-base reducts in a family of sub-bases.In this paper, we provide a criterion of the minimal family of sub-bases. And we give anapproach based on Boolean matrices to obtain the minimal family of sub-bases.
Rough set theory, introduced by Pawlak [7], provides an approach for uncertainty man-agement. As a generalization of the classical rough set, covering rough set [21] is a usefulmathematical tool to study covering information systems. A covering information systemis a pair (X, ∆), where X is a non-empty and finite set, and ∆ = Si|i = 1, 2, . . . , n is afamily of coverings on X. According to the definition of sub-base of topological spaces, itis easy to see that a covering on X is a sub-base for a topology of finite set X. So a family∆ of coverings is a family of sub-bases. Reducts are important problems in rough set∗Corresponding Author.Email addresses: [email protected] (Y. Li), [email protected] (J. Li),
[email protected] (Y. Lin), [email protected] (J. Feng), [email protected] (H. Wang)Received: 01.11.2018; Accepted: 01.04.2019
794 Y. Li et al.
theory. In covering information systems, finding a reduct is a process to delete redundantcoverings under some conditions. It is similar to deriving a minimal family of sub-bases.Then a natural question is: is there a relationship between reducts and minimal familiesof sub-bases? This paper shows that reducts of covering information systems and minimalfamilies of sub-bases are equivalent.
In addition, reducts from the given information system become more and more diffi-cult when the amount of data increases. Hence, a new research direction investigatinghomomorphisms or mappings between two information systems gains more attention inrecent years. The motivation of study homomorphisms or mappings between informationsystems is to find a relatively small information system which has the same reduct asthe original database [16]. Grzymala-Busse et al. [2] initially introduced the concept ofhomomorphism, which is used as a tool to study the relationship between informationsystems based on rough set. Li and Ma [5] studied some features of redundancy andreduct of complete information systems under some homomorphisms. Later, many au-thors [2, 3, 5, 9, 12–20,23–28] discussed homomorphisms or mappings between informationsystems based on rough set. A consistent function related to coverings was proposed byWang et al. [16]. By analyzing the consistent function related to coverings, we find thatthe work to structure a consistent function is a process to seek a representation elementunder some conditions. This process is similar to structuring a quotient space under anequivalence relation in a topological space. Hence, this paper explains consistent functionsrelated to coverings and homomorphisms from the perspective of topology.
The remainder of this paper is organized as follows. In Section 2, the definition of aminimal family of sub-bases is presented, and its properties are investigated. Section 3discusses the relationship between reducts and minimal families of sub-bases. Based onBoolean matrices, Section 4 proposes an approach to derive a minimal family of sub-bases.In Section 5, several numerical experiments are conducted on UCI data sets to evaluatethe proposed method. Section 6 has some concluding remarks.
2. A minimal family of sub-basesSuppose Si is a sub-base for finite topological space (X, τi) for i = 1, 2, . . . , n, ∆ =
S1, S2, . . . , Sn, and S∆ =∧n
i=1 Si = ∩n
i=1 Si|Si ∈ Si, i = 1, 2, . . . , n. Then S∆ is asub-base for a topology τ∆ of finite set X. For each subfamily ∆′ of ∆, a question is: arethe topologies generated by both S∆ and S∆′ as sub-bases the same? Now we presentthe definition of a minimal family of sub-bases, which keeps the topology unchanged.
Definition 2.1. Let Si be a sub-base for finite topological space (X, τi) for i = 1, 2, . . . , nand ∆ = S1, S2, . . . , Sn. If the following statement holds, then the family ∆ of sub-bases is called a minimal family of sub-bases with respect to X. The statement is: for anysubfamily ∆′ of ∆, if S∆′ is a sub-base for finite topological space (X, τ∆), then ∆ = ∆′.
In fact, two conclusions can be obtained from Definition 2.1: (1) for any family ∆ ofsub-bases, if there exists a subfamily ∆′ of ∆ and ∆′ = ∆ such that S∆′ is a sub-basefor finite topological space (X, τ∆), then ∆ is not a minimal family of sub-bases; (2) foreach subfamily ∆′′ of ∆′ and ∆′′ = ∆′, if S∆′′ is not a sub-base for finite topologicalspace (X, τ∆), then ∆′ is a minimal family of sub-bases. An example in [10] is employedto illustrate our idea.
Example 2.2. Suppose Si is a sub-base for finite topological space (X, τi) for i = 1, 2, 3, 4with X = x1, x2, . . . , x9. Let ∆ = S1, S2, S3, S4, where
S1 = x1, x2, x4, x5, x7, x8, x2, x3, x5, x6, x8, x9,S2 = x1, x2, x3, x4, x5, x6, x4, x5, x6, x7, x8, x9,S3 = x1, x2, x3, x4, x5, x6, x7, x8, x9,S4 = x1, x2, x4, x5, x2, x3, x5, x6, x4, x5, x7, x8, x5, x6, x8, x9.
A Minimal Family of Sub-bases 795
Sub-base S∆ is derived for finite topological space (X, τ∆): S∆ = x1, x2, x2, x2,x3, x4, x5, x4, x5, x7, x8, x5, x5, x6, x5, x6, x8, x9, x5, x8. And it is easy tocheck that B∆ = S∆, where B∆ is a base generated by S∆ as a sub-base.
Suppose ∆1 = S3, S4. Then S∆1 = x1, x2, x2, x3, x4, x5, x5, x6, x4, x5, x7,x8, x5, x6, x8, x9. Assume B∆1 is a base generated by S∆1 as a sub-base. One canexamine that B∆1 = B∆, which implies that S∆1 is a sub-base for finite topologicalspace (X, τ∆). So ∆ is not a minimal family of sub-bases. Moreover, S3 and S4 are notsub-bases for finite topological space (X, τ∆). Hence, ∆1 is a minimal family of sub-bases.
Example 2.2 shows the existence of minimal families of sub-bases, then is it unique?Reconsider Example 2.2. It is easy to find that both S1, S2, S3 and S3, S4 areminimal families of sub-bases. So the minimal families of sub-bases are not unique.
From [8], each point in Alexandroff spaces has a unique minimal open neighborhood.In addition, Alexandroff spaces are special topological spaces, including finite topologicalspaces as simple cases. So each point in finite topological spaces has a unique minimalopen neighborhood. Therefore, a unique minimal open neighborhood is used to study aminimal family of sub-bases.
Suppose P is a family of subsets of X. A minimal set containing x with respect to Pis denoted by NP(x) =
∩U |x ∈ U ∈ P.
Remark 2.3. Let X be a finite topological space with a topology τ . Suppose B is a basefor the topological space X and S is a sub-base for the topological space X. Accordingto the definitions of base and sub-base, Nτ (x) = NB(x) = NS (x) for each point x ∈ X.And Nτ (x) is a unique minimal open neighborhood of x.Theorem 2.4. Let Si be a sub-base for finite topological space (X, τi) for i = 1, 2, . . . , nand ∆ = S1, S2, . . . , Sn. The family ∆ of sub-bases is minimal if and only if for anysubfamily ∆′ of ∆, if NS∆′ (x) = NS∆(x) for each point x ∈ X, then ∆ = ∆′.
Proof. First, we prove that for any subfamily ∆′ of ∆, NS∆′ (x) = NS∆(x) for each pointx ∈ X if and only if S∆′ is a sub-base for finite topological space (X, τ∆). Suppose S∆′
is a sub-base for finite topological space (X, τ∆). Then NS∆′ (x) = Nτ∆(x) = NS∆(x) foreach point x ∈ X by Remark 2.3. Assume S∆′ is a sub-base for a topology τ∆′ of finiteset X. Let B∆ = NS∆(x)|x ∈ X and B∆′ = NS∆′ (x)|x ∈ X. One can see easilythat B∆ and B∆′ are bases for finite topological spaces (X, τ∆) and (X, τ∆′), respectively.Because NS∆(x) = NS∆′ (x) for each point x ∈ X, we have B∆ = B∆′ , which impliesτ∆ = τ∆′ . Hence, S∆′ is a sub-base for finite topological space (X, τ∆).
Suppose the family ∆ of sub-bases is a minimal family of sub-bases. For any subfamily∆′ of ∆, if NS∆′ (x) = NS∆(x) for each point x ∈ X, then S∆′ is a sub-base for finitetopological space (X, τ∆). Thus, ∆ = ∆′ according to the definition of a minimal familyof sub-bases. For any subfamily ∆′ of ∆, suppose S∆′ is a sub-base for finite topologicalspace (X, τ∆). From the analysis above, NS∆′ (x) = NS∆(x) for each point x ∈ X. Then∆ = ∆′. Therefore, the family ∆ of sub-bases is the minimal one.
Theorem 2.4 shows that minimal families of sub-bases are equal to sub-base reducts ina family of sub-bases which provided by Li and Zhang [6].
Now some properties about a minimal family of sub-bases are investigated.Theorem 2.5. Let Si be a sub-base for finite topological space (Y, τi) for i = 1, 2, . . . , nand ∆ = S1, S2, . . . , Sn. Suppose (X, τ) is a finite topological space. And f : (X, τ) →(Y, τ∆) is continuous and onto. If ∆ is a minimal family of sub-bases with respect to Y ,then f−1(∆) = f−1(S1), f−1(S2), . . . , f−1(Sn) is a minimal family of sub-bases withrespect to X.Proof. Sf−1(∆) is a sub-base for a topology τf−1(∆) of finite set X because f : (X, τ) →(Y, τ∆) is continuous and onto. Obviously,
796 Y. Li et al.
f−1(S∆) = f−1(∧n
i=1 Si)= f−1(
∩ni=1 Si|Si ∈ Si, i = 1, 2, . . . , n)
= f−1(∩n
i=1 Si)|Si ∈ Si, i = 1, 2, . . . , n=
∩ni=1 f−1(Si)|f−1(Si) ∈ f−1(Si), i = 1, 2, . . . , n
=∧n
i=1 f−1(Si) = Sf−1(∆).For any subfamily f−1(∆′) of f−1(∆), suppose Sf−1(∆′) = f−1(S∆′) is a sub-base forfinite topological space (X, τf−1(∆)). Then S∆′ is a sub-base for finite topological space(Y, τ∆). If not, then there exists a point y ∈ Y such that NS∆′ (y) = NS∆(y). So
NSf−1(∆′)(f−1(y)) = Nf−1(S∆′ )(f−1(y)) = f−1(NS∆′ (y))
= f−1(NS∆(y)) = Nf−1(S∆)(f−1(y)) = NSf−1(∆)(f−1(y)).
Thus Sf−1(∆′) is not a sub-base for finite topological space (X, τf−1(∆)). It is a con-tradiction. Hence, S∆′ is a sub-base for finite topological space (Y, τ∆). Since ∆ isa minimal family of sub-bases with respect to Y , by Definition 2.1, ∆ = ∆′. That isf−1(∆) = f−1(∆′). Therefore, f−1(∆) is a minimal family of sub-bases with respect toX.
By Theorem 2.5, a minimal family of sub-bases remains invariant under f−1 : Y → Xif and only if f is continuous and onto. When a minimal family of sub-bases remainsinvariant under a mapping f , what conditions does the mapping f satisfy? In order toanswer this question, we prove a proposition first.Definition 2.6 ([4]). Let f : X → Y, g : Y → Z be two mappings. We call subset(x, z) : there exists y ∈ Y such that f(x) = y, g(y) = z a composition of f and g. Thecomposition of f and g is denoted by f g : X → Z. For each x ∈ X, (f g)(x) = g(f(x)).Proposition 2.7. Let Si be a sub-base for finite topological space (X, τi) for i = 1, 2, . . . , n,∆ = S1, S2, . . . , Sn, and Y be a finite topological space. For two points x1, x2 ∈ X,define x1Rx2 if NS∆(x1) = NS∆(x2). Define p : (X, τ∆) → (X/R, τ ′) a natural quotientmapping, where (X/R, τ ′) is a natural quotient space. Suppose a mapping g : X/R → Yis a bijection. If a mapping f : X → Y satisfies f = p g, then the following conclusionshold:
(1) For each subset S ∈ Si ∈ ∆, x1 ∈ S implies x2 ∈ S for any points x1, x2 ∈ Ssatisfying f(x1) = f(x2),
(2) For any two subsets Si ∈ Si and Sj ∈ Sj for i, j = 1, 2, . . . , n, f(Si ∩ Sj) =f(Si) ∩ f(Sj),
(3) f(∩n
i=1 Si) =∩n
i=1 f(Si),(4) S = f−1(f(S)) for each subset S ∈ S ,(5) S = f−1(f(S )) for each element S ∈ ∆.
Proof. (1) First, we prove that f(x1) = f(x2) implies NS∆(x1) = NS∆(x2) for two pointsx1, x2 ∈ X. Since f satisfies f = p g, f(x1) = f(x2) implies
g(p(x1)) = p g(x1) = p g(x2) = g(p(x2))for any two points x1, x2 ∈ X. Thus, p(x1) = p(x2) because g is a bijection, i.e., [x1] = [x2].So x1Rx2, which means NS∆(x1) = NS∆(x2). Next, for each subset S ∈ Si, x1 ∈ S meansx1 ∈ NS∆(x1) ⊂ S. NS∆(x1) = NS∆(x2) because f(x1) = f(x2). It means NS∆(x2) ⊂ S,that is x2 ∈ S.
(2) First, we prove that f(Si) ∩ f(Sj) = ∅ if Si ∩ Sj = ∅. Assume f(Si) ∩ f(Sj) = ∅.Then there exists a point y ∈ Y such that y ∈ f(Si) ∩ f(Sj), i.e., y ∈ f(Si) and y ∈ f(Sj).So there exist two points x1 ∈ Si and x2 ∈ Sj such that f(x1) = f(x2) = y. By (1),x2 ∈ Si. Then x2 ∈ Si ∩ Sj . This contradicts that f(Si) ∩ f(Sj) = ∅. Next, we prove thatif Si ∩ Sj = ∅, then f(Si ∩ Sj) = f(Si) ∩ f(Sj). Obviously, f(Si ∩ Sj) ⊂ f(Si) ∩ f(Sj)holds. For each point y ∈ f(Si) ∩ f(Sj), there exist two points x1 ∈ Si and x2 ∈ Sj
such that f(x1) = f(x2) = y. By (1), x2 ∈ Si. Then x2 ∈ Si ∩ Sj , which impliesy = f(x2) ∈ f(Si ∩ Sj). So f(Si) ∩ f(Sj) ⊂ f(Si ∩ Sj). That is f(Si ∩ Sj) = f(Si) ∩ f(Sj).
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(3) The proof is similar to (2).(4) It is obvious that S ⊂ f−1(f(S)) holds. For each point x ∈ f−1(f(S)), f(x) ∈ f(S).
Thus, there exists a point x′ ∈ S such that f(x) = f(x′). By (1), x ∈ S. So f−1(f(S)) ⊂ S.That is f−1(f(S)) = S.
(5) Obviously, S ⊂ f−1(f(S )). For each subset S ∈ f−1(f(S )), f(S) ∈ f(S ). Thenthere exists a subset S′ ∈ S such that f(S) = f(S′). From (4), S = S′ ∈ S must hold.Hence, f−1(f(S )) ⊂ S , that is S = f−1(f(S )).
Remark 2.8. By the proof of Proposition 2.7, if both Si and Sj are elements of the samesub-base, then Proposition 2.7 (2) holds. Furthermore, for each subfamily ∆′ of ∆, thefollowing statements hold:
(1) f(NS∆′ (x)) = f(∩
S|x ∈ S ∈ S∆′) =∩
f(S)|f(x) ∈ f(S) ∈ f(S∆′) =Nf(S∆′ )(f(x)) for each point x ∈ X,
(2) NS∆′ (x) = f−1(f(NS∆′ (x))) for each point x ∈ X.
For two points x1, x2 ∈ X, define x1Rx2 if NS∆(x1) = NS∆(x2). Define p : (X, τ∆) →(X/R, τ ′) a natural quotient mapping, where (X/R, τ ′) is a natural quotient space. Basedon the natural quotient mapping, Theorem 2.9 is proved.
Theorem 2.9. Let Si be a sub-base for finite topological space (X, τi) for i = 1, 2, . . . , n,∆ = S1, S2, . . . , Sn, and (Y, τ) be a finite topological space. Suppose a mappingg : X/R → Y is a bijection. If a mapping f : (X, τ∆) → (Y, τ) is an open mappingsatisfying f = p g and ∆ is a minimal family of sub-bases with respect to X, thenf(∆) = f(S1), f(S2), . . . , f(Sn) is a minimal family of sub-bases with respect to Y .
Proof. Obviously, Sf(∆) is a sub-base for a topology τf(∆) of finite set Y . By Proposition2.7 (3),
f(S∆) = f(∧n
i=1 Si)= f(
∩ni=1 Si|Si ∈ Si, i = 1, 2, . . . , n)
= f(∩n
i=1 Si)|Si ∈ Si, i = 1, 2, . . . , n=
∩ni=1 f(Si)|f(Si) ∈ f(Si), i = 1, 2, . . . , n
=∧n
i=1 f(Si) = Sf(∆).For any subfamily f(∆′) of f(∆), suppose Sf(∆′) = f(S∆′) is a sub-base for finite topo-logical space (Y, τf(∆)). Then S∆′ is a sub-base for finite topological space (X, τ∆). Ifnot, then there exists a point x ∈ X such that NS∆′ (x) = NS∆(x). From Remark 2.8,Nf(S∆′ )(f(x)) = f(NS∆′ (x)) for each point x ∈ X. Then
NSf(∆′)(f(x)) = Nf(S∆′ )(f(x)) = f(NS∆′ (x))
= f(NS∆(x)) = Nf(S∆)(f(x)) = NSf(∆)(f(x)).Thus, Sf(∆′) is not a sub-base for finite topological space (Y, τf(∆)). This contradictsthat S∆′ is a sub-base for finite topological space (X, τ∆). Since ∆ is a minimal familyof sub-bases with respect to X, by Definition 2.1, ∆ = ∆′. That means f(∆) = f(∆′).Hence, f(∆) is a minimal family of sub-bases with respect to Y .
With no doubt, one-to-one mappings satisfy results of Proposition 2.7. So the followingcorollary is provided.
Corollary 2.10. Let Si be a sub-base for finite topological space (X, τi) for i = 1, 2, . . . , nand ∆ = S1, S2, . . . , Sn. Suppose (Y, τ) is a finite topological space. And f : (X, τ∆) →(Y, τ) is open and one-to-one. If ∆ is a minimal family of sub-bases with respect to X,then f(∆) = f(S1), f(S2), . . . , f(Sn) is a minimal family of sub-bases with respect tof(X).
Proof. The proof is similar to Theorem 2.9.
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3. Relationship between minimal families of sub-bases and reducts ofcovering information systems
In this section, we restrict that universes are finite. A covering information system isa pair (X, ∆), where X is a non-empty and finite set, and ∆ = Si|i = 1, 2, . . . , n is afamily of coverings on X. According to the definition of sub-base of topological spaces, itis easy to see that a covering on X is a sub-base for a topology of finite set X. So a family∆ of coverings is a family of sub-bases. Moreover, the definition of reduct is introducedin covering information systems. And finding a reduct is a process to delete redundantcoverings under some conditions. It is similar to deriving a minimal family of sub-bases.Then is there a relationship between reducts and minimal families of sub-bases? Beforeanswering this question, we present the definition of reduct.
Definition 3.1 ([17]). Let (X, ∆) be a covering information system. ∆′ ⊂ ∆ is referredto as a reduct of ∆ if ∆′ satisfies
∩∆′ =
∩∆ and
∩∆′ =
∩(∆′ \ S ) for any covering
S ∈ ∆′, where∩
∆′ = ∩
S ∈∆′ NS (x)|x ∈ X for each subfamily ∆′ of ∆.
Theorem 3.2. Let X be a non-empty and finite set and ∆ be a family of coverings.(1) If (X, ∆) is a covering information system and a subfamily ∆′ of ∆ is a reduct of
∆, then ∆′ can be viewed as a minimal family of sub-bases.(2) If each covering Si ∈ ∆ is a sub-base for topological space (X, τi) for i = 1, 2, . . . , n
and a subfamily ∆′ of ∆ is a minimal family of sub-bases, then ∆′ can be viewed as areduct of ∆.
Proof. For any subfamily ∆′ of ∆, according to the definition of NS∆′ (x),∩
∆′ =∩
∆ ifand only if NS∆′ (x) = NS∆(x) for each point x ∈ X.
(1) NS∆′ (x) = NS∆(x) for each point x ∈ X because ∆′ is a reduct of ∆. It implies thatS∆′ is a sub-base for finite topological space (X, τ∆). Suppose for any covering S ∈ ∆′,S∆′\S is a sub-base for finite topological space (X, τ∆). Then NS∆′\S
(x) = NS∆(x)for each point x ∈ X, which implies NS∆′\S
(x) = NS∆′ (x) for each point x ∈ X. Since∆′ is reduct of ∆,
∩∆′ =
∩(∆′ \ S ) for any covering S ∈ ∆′. In other words, there
exists a point x ∈ X such that NS∆′\S (x) = NS∆′ (x). It is a contradiction. Hence,
S∆′\S is not a sub-base for finite topological space (X, τ∆). By Definition 2.1, ∆′ is aminimal family of sub-bases.
(2) The proof is similar to (1). In Section 2, a minimal family of sub-bases remains invariant under some certain map-
pings. And Theorem 3.2 states that a reduct in covering information systems and aminimal family of sub-bases are equivalent. Therefore, the reduct also remains invariantunder these certain mappings when the reduct in covering information systems is viewedas a minimal family of sub-bases. Wang et al. [16] proposed the definitions of consis-tent mapping and homomorphism, which keep a reduct unchanged. Since there are twodifferent mappings which keep a reduct unchanged, are these mappings the same? Thefollowing definition is about consistent mappings.
Definition 3.3 ([16]). Let X and Y be two universes, f : X → Y a mapping, S =S1, S2, . . . , Sn a covering on X and Cov(S ) = NS (x)|x ∈ X. The mapping f iscalled a consistent function with respect to S if for any x, y ∈ X, NS (x) = NS (y)whenever f(x) = f(y).
By Definition 3.3, the construction of consistent function with respect to S is to findrepresentation elements under some conditions. This process is similar to structure anatural quotient mapping in topological spaces. For two points x1, x2 ∈ X, define x1Rx2if NS (x1) = NS (x2). Define p : X → X/R a natural quotient mapping. Based on thenatural quotient mapping, Theorem 3.4 is proved.
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Theorem 3.4. Let X and Y be two finite topological spaces, S a covering on X, f :X → Y a mapping. If f is a consistent function with respect to S , then there exists abijection g : X/R → f(X) such that f satisfies f = p g.
Proof. Obviously, there exists a mapping g such that f = pg. Suppose g([x1]) = g([x2])for any two points x1, x2 ∈ X. p(x) = [x] for each point x ∈ X, because p is a naturalquotient mapping. Then
f(x1) = p g(x1) = g(p(x1)) = g([x1]) = g([x2]) = g(p(x2)) = p g(x2) = f(x2).Since f is a consistent function with respect to S , by Definition 3.3, NS (x1) = NS (x2).It implies x1Rx2. Then [x1] = [x2], that is g is one-to-one. Because f : X → f(X) is ontoand p is a natural quotient mapping, g is onto. So g is a bijection.
According to the definition of consistent function, homomorphisms are defined in [16].
Definition 3.5 ([16]). Let (X, ∆) be a covering information system. If f is a consistentfunction with respect to each Si ∈ ∆, then f is referred to as a homomorphism on (X, ∆).
Wang et al. [16] proves that f is a consistent function with respect to S1 and S2 if andonly if f is a consistent function with respect to S1
∩S2 = NS1(x) ∩ NS2(x)|x ∈ X.
Based on mathematical induction, if f is a consistent function with respect to each coveringS ∈ ∆, then f is a consistent function with respect to S∆. In other words, if f is ahomomorphism on (X.∆), then f is a consistent function with respect to S∆. So thefollowing corollary holds according to Theorem 3.4.
Corollary 3.6. Let X and Y be two finite topological spaces, and f : X → Y a mapping.Suppose ∆ = S1, S2, . . . , Sn is a family of coverings on X. If f is a homomorphismon (X, ∆), then there exists a bijection g : X/R → f(X) such that f satisfies f = p g.
Consider f : X → f(X). By Corollary 3.6, if f is a homomorphism on (X, ∆), thenf is the composition p g, where p is a natural quotient mapping and g is a bijection.According to Theorem 3.2 (1), a reduct ∆′ of covering information system (X, ∆) can beviewed as a minimal family of sub-bases with respect to X. By Theorem 2.9, f(∆′) is aminimal family of sub-bases with respect to f(X). Then f(∆′) can be viewed as a reductof f(∆) according to Theorem 3.2 (2). The above analysis illustrates that the necessityof the following lemma, which is proposed by Wang et al in [16], is equal to Theorem 2.9presented in this paper. On the other hand, f : X → f(X) is onto. If f(∆′) is a reduct off(∆), then f(∆′) can be viewed as a minimal family of sub-bases with respect to f(X).By Theorem 2.5, f−1(f(∆′)) is a minimal family of sub-bases with respect to X. FromProposition 2.7 (5), one can see that f−1(f(S )) = S for each sub-base S ∈ ∆′. Thenf−1(f(∆′)) = ∆′, which implies ∆′ is a minimal family of sub-bases with respect to X.Thus, ∆′ is a reduct of ∆. Therefore, we conclude that the sufficiency of the followinglemma and Theorem 2.5 introduced in this paper are equivalent.
Lemma 3.7 ([16]). Let (X, ∆) be a covering information system, ∆′ ⊂ ∆. If f is ahomomorphism on (X, ∆), then ∆′ is a reduct of ∆ if and only if f(∆′) is a reduct off(∆).
Wang et al. also defined another homomorphism in [16].
Definition 3.8 ([16]). Let (Y, ∆) be a covering information system and f : X → Y bea mapping. If f is surjective, then f−1 is referred to as a homomorphism from (Y, ∆) to(X, f−1(∆)).
Based on homomorphisms from (Y, ∆) to (X, f−1(∆)), Wang et al. obtained the fol-lowing lemma in [16].
Lemma 3.9 ([16]). Suppose (Y, ∆) is a covering information system, (X, f−1(∆)) is aninduced covering information system of (Y, ∆), and f−1 is a homomorphism from (Y, ∆)
800 Y. Li et al.
to (X, f−1(∆)) and ∆′ ⊂ ∆. Then ∆′ is a reduct of ∆ if and only if f−1(∆′) is a reductof f−1(∆).
We will show Lemma 3.9 is equal to Theorem 2.5 introduced in this paper. First, if∆′ is a reduct of ∆, by Theorem 3.2 (1), then ∆′ can be viewed as a minimal family ofsub-bases with respect to Y . Next f is onto according to the definition of homomorphismfrom (Y, ∆) to (X, f−1(∆)). By Theorem 2.5, f−1(∆′) is a minimal family of sub-baseswith respect to X. So f−1(∆′) is a reduct of f−1(∆) from Theorem 3.2 (2). In anotherword, the necessity of Lemma 3.9 and Theorem 2.5 are equivalent. Besides, f−1 is alsoonto. If f−1(∆′) is a reduct of f−1(∆), then f−1(∆′) is a minimal family of sub-baseswith respect to X. By Theorem 2.5, ∆′ = f(f−1(∆′)) is a minimal family of sub-baseswith respect to Y . So ∆′ is a reduct of ∆. Therefore, the sufficiency of Lemma 3.9 andTheorem 2.5 are equivalent.
4. An approach to obtain a minimal family of sub-basesIn Section 2, a minimal family of sub-bases is presented. And a criterion of a minimal
family of sub-bases is provided. In fact, it is hard to obtain a minimal family of sub-basesvia this criterion. So an approach based on Boolean matrices will be given to derive aminimal family of sub-bases. First, the following definition is proposed.
Definition 4.1 ([10]). Let X = x1, x2, . . . , xm and A ⊂ X. The characteristic functionis defined as f(A) = (f1, f2, . . . , fm)′ (′ denotes the transpose throughout this paper),where
fi =
1, xi ∈ A;0, xi ∈ A.
From Definition 4.1, characteristic function f(A) shows the relationship between eachpoint of X and a subset A of X. For example, if X = x1, x2, x3, x4, x5, x6 and A =x2, x3, x6, then f(A) = (0, 1, 1, 0, 0, 1)′.
Definition 4.2. Let P be a family of subsets of X with X = x1, x2, . . . , xm and P =P1, P2, . . . , Pk. The characteristic matrix of P is defined as MP = (f(P1), f(P2), . . . ,f(Pk)).
Obviously, MP is a Boolean matrix with size m × k. Let S be a sub-base for a finitetopological space X. A neighborhood Boolean matrix of sub-base S is defined.
Definition 4.3. Let Si be a sub-base for finite topological space (X, τi) for i = 1, 2, . . . , nwith X = x1, x2, . . . , xm, and ∆ = S1, S2, . . . , Sn. Define a neighborhood Booleanmatrix of each sub-base S ∈ ∆ as Nm(S ) = (f(NS (x1)), f(NS (x2)), . . . , f(NS (xm))),and a neighborhood Boolean matrix of ∆ as Nm(S∆) = (f(NS∆(x1)), f(NS∆(x2)), . . . ,f(NS∆(xm))).
Denote the i-th row and j-th column element of neighborhood Boolean matrix Nm(S )as Nm(S )(i, j). In fact, Nm(S )(i, j) = 1 means xi ∈ NS (xj) and Nm(S )(i, j) = 0means xi ∈ NS (xj). If xi ∈ NS (xj) for any points xi, xj ∈ X, then we call that xi andxj are discernible by sub-base S . Hence, Nm(S )(i, j) = 0 implies that xi and xj can bediscernible by sub-base S .
Definition 4.4 ([11]). Let M = (mij)n×m be a matrix. Define a matrix operator ∼ as∼ M = (∼ mij)n×m, where
∼ mij =
1, mij = 0,0, mij = 0.
Definition 4.5 ([22]). Let A = (aij)n×m and B = (bij)n×m be two matrices. TheHadamard product of A and B is defined as A B = (aijbij)n×m.
A Minimal Family of Sub-bases 801
Two operators introduced above will be used to calculate neighborhood Boolean matri-ces.
Theorem 4.6. Let Si be a sub-base for finite topological space (X, τi) for i = 1, 2, . . . , nand ∆ = S1, S2, . . . , Sn. Then the following statements hold:
(1) Nm(S ) =∼ ((∼ MS )M ′S ) for each sub-base S ∈ ∆,
(2) Nm(S∆) = Nm(S1) Nm(S2) · · · Nm(Sn).
Proof. For each sub-base S ∈ ∆, suppose S = S1, S2, . . . , Sk. And the i-th elementof column vector f(S) is denoted by f(S)(i). For each matrix M , denote its j-th columnvector as M(·, j), and its i-th row and j-th column element as M(i, j).
(1) Suppose xi ∈ NS (xj). It means Nm(S )(i, j) = 1. Next we prove that (∼ ((∼MS )M ′
S ))(i, j) = 1. It is obvious that((∼ MS )M ′
S )(·, j) =∑
1≤l≤k M ′S (l, j)(∼ f(Sl)).
According to Definition 4.2, M ′S (l, j) = 1 means xj ∈ Sl, and M ′
S (l, j) = 0 impliesxj ∈ Sl. Thus,
((∼ MS )M ′S )(·, j) =
∑1≤l≤k M ′
S (l, j)(∼ f(Sl)) =∑
xj∈Sl(∼ f(Sl)).
For each point xi ∈ X, xi ∈ NS (xj) means xi ∈ Sl for each subset Sl satisfying xj ∈ Sl.This implies f(Sl)(i) = 1 and (∼ f(Sl))(i) = 0 for each subset Sl satisfying xj ∈ Sl.Hence, for each point xi ∈ NS (xj),
(∑
xj∈Sl(∼ f(Sl)))(i) =
∑xj∈Sl
((∼ f(Sl))(i)) = 0.That is ∼ (
∑xj∈Sl
(∼ f(Sl)))(i) = 1. Therefore, (∼ ((∼ MS )M ′S ))(i, j) = 1. Assume
xi ∈ NS (xj). Then we have Nm(S )(i, j) = 0. It is similar to proving that (∼ ((∼MS )M ′
S ))(i, j) = 0. So it is easy to conclude thatNm(S ) =∼ ((∼ MS )M ′
S ).(2) If Nm(S∆)(i, j) = 1, then xi ∈ NS∆(xj). According to the definitions of S∆ and
minimal open neighborhood of each point x ∈ X, we get xi ∈ NS (xj) for each elementS ∈ ∆, which implies Nm(S )(i, j) = 1 for each element S ∈ ∆. So
(Nm(S1) Nm(S2) · · · Nm(Sn))(i, j) = 1.If Nm(S∆)(i, j) = 0, then it is similar to proving that
(Nm(S1) Nm(S2) · · · Nm(Sn))(i, j) = 0.Consequently, Nm(S∆) = Nm(S1) Nm(S2) · · · Nm(Sn).
Theorem 4.6 shows that the discernible power of S∆ is larger than S∆′ for each sub-family ∆′ of ∆. Or the discernible power of ∆ is larger than each subfamily ∆′ of ∆. Thefollowing example in [17] is given to calculate neighborhood Boolean matrices by Theorem4.6.
Example 4.7. Suppose Si is a sub-base for finite topological space (X, τi) for i = 1, 2, 3, 4with X = x1, x2, . . . , x15.Let ∆ = S1, S2, S3, S4, where
S1 = x1, x2, x3, x4, x5, x8, x10, x15, x3, x5, x7, x11, x12, x4, x6, x8, x9, x10, x13, x14,
S2 = x1, x2, x3, x4, x5, x7, x8, x10, x11, x12, x15, x3, x4, x5, x8, x10, x3, x5, x6, x9,x13, x14,
S3 = x1, x2, x3, x4, x5, x8, x10, x15, x6, x9, x13, x14, x4, x7, x8, x10, x11, x12,
S4 = x1, x2, x3, x5, x15, x6, x7, x9, x11, x12, x13, x14, x3, x4, x5, x8, x10.
The characteristic matrices of Si for i = 1, 2, 3, 4 are obtained:
802 Y. Li et al.
MS1 =
1 0 01 0 01 1 01 0 11 1 00 0 10 1 01 0 10 0 11 0 10 1 00 1 00 0 10 0 11 0 0
, MS2 =
1 0 01 0 01 1 11 1 01 1 10 0 11 0 01 1 00 0 11 1 01 0 01 0 00 0 10 0 11 0 0
, MS3 =
1 0 01 0 01 0 01 0 11 0 00 1 00 0 11 0 10 1 01 0 10 0 10 0 10 1 00 1 01 0 0
, MS4 =
1 0 01 0 01 0 10 0 11 0 10 1 00 1 00 0 10 1 00 0 10 1 00 1 00 1 00 1 01 0 0
.
According to Theorem 4.6, neighborhood Boolean matrices are derived.
Nm(S1) =
1 1 0 0 0 0 0 0 0 0 0 0 0 0 11 1 0 0 0 0 0 0 0 0 0 0 0 0 11 1 1 0 1 0 1 0 0 0 1 1 0 0 11 1 0 1 0 1 0 1 1 1 0 0 1 1 11 1 1 0 1 0 1 0 0 0 1 1 0 0 10 0 0 0 0 1 0 0 1 0 0 0 1 1 00 0 0 0 0 0 1 0 0 0 1 1 0 0 01 1 0 1 0 1 0 1 1 1 0 0 1 1 10 0 0 0 0 1 0 0 1 0 0 0 1 1 01 1 0 1 0 1 0 1 1 1 0 0 1 1 10 0 0 0 0 0 1 0 0 0 1 1 0 0 00 0 0 0 0 0 1 0 0 0 1 1 0 0 00 0 0 0 0 1 0 0 1 0 0 0 1 1 00 0 0 0 0 1 0 0 1 0 0 0 1 1 01 1 0 0 0 0 0 0 0 0 0 0 0 0 1
,
Nm(S2) =
1 1 0 0 0 0 1 0 0 0 1 1 0 0 11 1 0 0 0 0 1 0 0 0 1 1 0 0 11 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 0 1 0 0 1 1 0 1 1 1 0 0 11 1 1 1 1 1 1 1 1 1 1 1 1 1 10 0 0 0 0 1 0 0 1 0 0 0 1 1 01 1 0 0 0 0 1 0 0 0 1 1 0 0 11 1 0 1 0 0 1 1 0 1 1 1 0 0 10 0 0 0 0 1 0 0 1 0 0 0 1 1 01 1 0 1 0 0 1 1 0 1 1 1 0 0 11 1 0 0 0 0 1 0 0 0 1 1 0 0 11 1 0 0 0 0 1 0 0 0 1 1 0 0 10 0 0 0 0 1 0 0 1 0 0 0 1 1 00 0 0 0 0 1 0 0 1 0 0 0 1 1 01 1 0 0 0 0 1 0 0 0 1 1 0 0 1
,
Nm(S3) =
1 1 1 0 1 0 0 0 0 0 0 0 0 0 11 1 1 0 1 0 0 0 0 0 0 0 0 0 11 1 1 0 1 0 0 0 0 0 0 0 0 0 11 1 1 1 1 0 1 1 0 1 1 1 0 0 11 1 1 0 1 0 0 0 0 0 0 0 0 0 10 0 0 0 0 1 0 0 1 0 0 0 1 1 00 0 0 0 0 0 1 0 0 0 1 1 0 0 01 1 1 1 1 0 1 1 0 1 1 1 0 0 10 0 0 0 0 1 0 0 1 0 0 0 1 1 01 1 1 1 1 0 1 1 0 1 1 1 0 0 10 0 0 0 0 0 1 0 0 0 1 1 0 0 00 0 0 0 0 0 1 0 0 0 1 1 0 0 00 0 0 0 0 1 0 0 1 0 0 0 1 1 00 0 0 0 0 1 0 0 1 0 0 0 1 1 01 1 1 0 1 0 0 0 0 0 0 0 0 0 1
,
Nm(S4) =
1 1 0 0 0 0 0 0 0 0 0 0 0 0 11 1 0 0 0 0 0 0 0 0 0 0 0 0 11 1 1 1 1 0 0 1 0 1 0 0 0 0 10 0 0 1 0 0 0 1 0 1 0 0 0 0 01 1 1 1 1 0 0 1 0 1 0 0 0 0 10 0 0 0 0 1 1 0 1 0 1 1 1 1 00 0 0 0 0 1 1 0 1 0 1 1 1 1 00 0 0 1 0 0 0 1 0 1 0 0 0 0 00 0 0 0 0 1 1 0 1 0 1 1 1 1 00 0 0 1 0 0 0 1 0 1 0 0 0 0 00 0 0 0 0 1 1 0 1 0 1 1 1 1 00 0 0 0 0 1 1 0 1 0 1 1 1 1 00 0 0 0 0 1 1 0 1 0 1 1 1 1 00 0 0 0 0 1 1 0 1 0 1 1 1 1 01 1 0 0 0 0 0 0 0 0 0 0 0 0 1
.
A Minimal Family of Sub-bases 803
Thus,
Nm(S∆) =
1 1 0 0 0 0 0 0 0 0 0 0 0 0 11 1 0 0 0 0 0 0 0 0 0 0 0 0 11 1 1 0 1 0 0 0 0 0 0 0 0 0 10 0 0 1 0 0 0 1 0 1 0 0 0 0 01 1 1 0 1 0 0 0 0 0 0 0 0 0 10 0 0 0 0 1 0 0 1 0 0 0 1 1 00 0 0 0 0 0 1 0 0 0 1 1 0 0 00 0 0 1 0 0 0 1 0 1 0 0 0 0 00 0 0 0 0 1 0 0 1 0 0 0 1 1 00 0 0 1 0 0 0 1 0 1 0 0 0 0 00 0 0 0 0 0 1 0 0 0 1 1 0 0 00 0 0 0 0 0 1 0 0 0 1 1 0 0 00 0 0 0 0 1 0 0 1 0 0 0 1 1 00 0 0 0 0 1 0 0 1 0 0 0 1 1 01 1 0 0 0 0 0 0 0 0 0 0 0 0 1
.
From Example 2.2, we find that S3 is the common element of minimal families of sub-bases S1, S2, S3 and S3, S4. Then S3 is more important than others for a minimalfamily of sub-bases. A natural question is: is there an element S ∈ ∆ such that Sbelongs to all the minimal families of sub-bases? If there is, then what properties does theelement have? The concept of core is proposed in the following definition.
Definition 4.8. Let Si be a sub-base for finite topological space (X, τi) for i = 1, 2, . . . , nand ∆ = S1, S2, . . . , Sn. The element S is called a core if S is an element of all theminimal families of sub-bases.
The characteristics of a minimal family of sub-bases and core can be described vianeighborhood Boolean matrices.
Theorem 4.9. Let Si be a sub-base for finite topological space (X, τi) for i = 1, 2, . . . , n,and ∆ = S1, S2, . . . , Sn. Then the following results hold:
(1) For each subfamily ∆′ of ∆, S∆′ is a sub-base for finite topological space (X, τ∆) ifand only if Nm(S∆) = Nm(S∆′);
(2) S is a core if and only if Nm(S∆\S ) = Nm(S∆) for each element S ∈ ∆.
Proof. (1) From the proof of Theorem 2.4, for each subfamily ∆′ of ∆, S∆′ is a sub-basefor finite topological space (X, τ∆) if and only if NS∆(x) = NS∆′ (x) for each point x ∈ X.By the definition of neighborhood Boolean matrix, it is equal to Nm(S∆) = Nm(S∆′).
(2) Assume Nm(S∆\S ) = Nm(S∆). From (1), S∆\S is a sub-base for finitetopological space (X, τ∆). Obviously, there exists a subfamily ∆′ of ∆ \ S such that ∆′
is a minimal family of sub-bases, and S ∈ ∆′. Because S is a core, S is an element of allthe minimal families of sub-bases. It is a contradiction. Hence, Nm(S∆\S ) = Nm(S∆).
For each matrix M , denote its i-th row and j-th column element as M(i, j). Sup-pose a subfamily ∆′ of ∆ \ S is a minimal family of sub-bases. From (1), Nm(S∆) =Nm(S∆′). Because Nm(S∆) = Nm(S∆\S ), there exist Nm(S∆)(i, j) and Nm(S∆\S )(i, j) such that Nm(S∆)(i, j) = 0 but Nm(S∆\S )(i, j) = 1. That is Nm(S∆′)(i, j) = 0but Nm(S∆\S )(i, j) = 1. By Theorem 4.6, Nm(S∆\S )(i, j) = 1 means Nm(S ′)(i, j)= 1 for each element S ′ ∈ ∆ \ S . It is contradictory to Nm(S∆′)(i, j) = 0. So ∆′ isnot a minimal family of sub-bases. Therefore, S is a core. Corollary 4.10. Let Si be a sub-base for finite topological space (X, τi) for i = 1, 2, . . . , nand ∆ = S1, S2, . . . , Sn. Then a subfamily ∆′ of ∆ is a minimal family of sub-bases ifand only if ∆′ is a minimal subfamily of ∆ satisfying Nm(S∆) = Nm(S∆′).
Based on the results above, a heuristic algorithm is presented to find a minimal familyof sub-bases.
Steps 3-5 are to compute all the cores and their time complexity is not more thanO(
∑S ∈∆ |X|2|S |). A sub-base with the maximal discernible power is added into a min-
imal family ∆′ of sub-bases in step 9, whose time complexity is O(∑|∆|−1
i=1 |X|2(|∆| − i)).So the time complexity of Algorithm 1 is O(
∑S ∈∆ |X|2|S | +
∑|∆|−1i=0 |X|2(|∆| − i)).
804 Y. Li et al.
Algorithm 1 A matrix-based algorithm for finding a minimal family of sub-basesRequire: A family ∆ of sub-bases.Ensure: A minimal family ∆′ of sub-bases.
1: Let ∆′ = ∅;2: for each S ∈ ∆ do3: Compute Nm(S∆\S ) according to Theorem 4.6;4: if Nm(S∆\S ) = Nm(S∆); then5: Let ∆′ = ∆′ ∪ S .//find all the cores;6: end if7: end for8: while Nm(S∆′) = Nm(S∆) do9: Let ∆′ = ∆′ ∪ S0,
where S0 satisfies |Nm(S∆′∪S0)| = max|Nm(S∆′∪S )| | S ∈ ∆ \ ∆′ and | · |is the total number of 0 in a matrix;
10: end while11: Return ∆′.
The following example uses Algorithm 1 to compute a minimal family of sub-bases.
Example 4.11. Re-discuss Example 4.7. According to Algorithm 1, we can obtain that:Nm(S∆\S1) = Nm(S∆), Nm(S∆\S2) = Nm(S∆), Nm(S∆\S3) = Nm(S∆),Nm(S∆\S4) = Nm(S∆). So S4 is a core. In addition, |Nm(SS1,S4)| = max|Nm(SS4∪S )| | S ∈ ∆ \ S4 and Nm(S∆1) = Nm(S∆). Hence, ∆1 = S1, S4is a minimal family of sub-bases on X.
5. Numerical experimentsTo further illustrate the effectiveness of the proposed algorithm, we choose 18 data sets
from UCI to conduct several numerical experiments. These data sets are described in Table1. Decision attributes in data sets are removed, and the rest of these data are normalized.In order to obtain an original family of sub-bases from data sets, a parameter ε is used tocontrol the size of the neighborhood of each object in each data set. For each attributein each data set, each object has a neighborhood. And the family of neighborhoods ofall objects is regarded as a covering. Since a covering can be viewed as a sub-base fora topological space, the family of neighborhoods of all objects will be a sub-base for atopological space. Hence, given an ε, a family of sub-bases is generated by all attributesin each data set. In our numerical experiments, we take ε = 0.4 to generate a family ofsub-bases with respect to each data set.
The experiments are performed on a personal computer with Windows 10 and an Inter(R) Core (TM) i7-6700 CPU @ 3.40 GHz 3.41 GHz with 8 GB of memory. The algorithmsare implemented using Matlab R2017a.
Table 1. Data set description
Data sets Sample Attributes Data sets Sample Attributes Data sets Sample Attributes
breast 84 9216 crx 690 15 derm 366 34diabe 768 8 gearboxA 1603 72 gearboxB 1603 72
gearboxC 1603 72 gearboxD 1603 72 gene1 84 9216heart 270 13 hepatitis 155 19 horse 368 22sonar 208 60 wdbc 569 30 wine 178 13wpbc 198 33 yale 165 1024 zoo 101 16
A Minimal Family of Sub-bases 805
As seen in Table 2, sub-bases’ number in a minimal family of sub-bases is dramati-cally reduced. It shows that Algorithm 1 is effective in deleting redundant elements andproviding a minimal family of sub-bases.
Table 2. Sub-bases’ numbers in a minimal family of sub-bases
Date sets Sub-bases’ numbers in a original family of sub-bases Sub-bases’ numbers in a minimal family of sub-bases
breast 9216 2crx 15 12
derm 34 2diabe 8 8
gearboxA 72 5gearboxB 72 7gearboxC 72 7gearboxD 72 6
gene1 9216 2heart 13 1
hepatitis 19 1horse 22 13sonar 60 9wdbc 30 18wine 13 7wpbc 33 7yale 1024 5zoo 16 1
Time consumption on finding a minimal family of sub-bases is shown in Table 3. It canbe observed that Algorithm 1 does not cost much time on all the data sets. It further turnsout that Algorithm 1 is feasible in finding a minimal family of sub-bases. In addition, theeffect of objects’ number on time consumption is larger than that of sub-bases’ numbers.For example, for data set breast with 84 objects and 9216 sub-bases, the time for finding aminimal family of sub-bases is 113.76s. But for data set gearboxD with 1603 objects and72 sub-bases, the time for finding a minimal family of sub-bases is 365.01s. Between thetwo data sets, the gap of sub-bases’ number is far greater than that of objects’ number.However, time consumption of data set breast with more sub-bases is less than data setgearboxD with more objects. The reason for it is as follows, that sub-bases’ numbers donot affect objects’ number, but objects’ number affects subsets’ number in each sub-base,which affects the computing speed of neighborhood Boolean matrix.
Table 3. Time consumptions on finding a minimal family of sub-bases
Data sets Times(s) Data sets Times(s) Data sets Times(s)breast 113.76 crx 2.48 derm 0.94diabe 2.66 gearboxA 330.69 gearboxB 354.69
gearboxC 335.53 gearboxD 365.01 gene1 122.23heart 0.50 hepatitis 0.31 horse 2.35sonar 3.91 wdbc 12.68 wine 0.60wpbc 2.03 yale 48.29 zoo 0.07
6. ConclusionIn this paper, the definition of a minimal family of sub-bases has been presented. More-
over, a criterion of a minimal family of sub-bases has been provided. According to thiscriterion, an approach based on Boolean matrices has been proposed to obtain a minimalfamily of sub-bases. In order to illustrate the effectiveness of the obtained approach, wehave conducted several numerical experiments on UCI data sets. Although the presentedapproach is feasible in finding a minimal family of sub-bases, there is no way to showthat the derived result is optimal. Moreover, the relationship between reducts in coveringinformation systems and minimal families of sub-bases has been discussed.
806 Y. Li et al.
Acknowledgment. This work is supported by National Natural Science Foundation ofChina (Nos. 11871259, 61379021, 11701258, 61773371) and Natural Science Foundationof Fujian Province (Nos. 2019J01748). The authors thank the anonymous reviewers fortheir constructive comments.
References[1] R. Engelking, General Topology, Heldermann Verlag, Berlin, 1989.[2] J.M. Graymala-Busse, Algebraic properties of knowledge representation systems, in:
Proceedings of the ACM SIGART international symposium on Methodologies forintelligent systems, ACM, 432-440, 1986.
[3] G.M. Lang, Q.G. Li and L.K. Guo, Homomorphisms-based attribute reduction ofdynamic fuzzy covering information systems, Int. J. Gen. Syst. 44 (7-8), 791-811,2015.
[4] J.J. Li, K.D. Li and S. Lin, Introduction of Basic Topology, Science Press, Beijing,2009 (in Chinese).
[5] D.Y. Li and Y.C. Ma, Invariant characters of information systems under some homo-morphisms, Inf. Sci. 129 (1-4), 211-220, 2000.
[6] J.J. Li and Y.L. Zhang, Reduction of subbases and its applications, Utilitas Math.82 (3), 179-192, 2010.
[7] Z. Pawlak, Rough sets, Int. J. Comput. Inf. Sci. 11 (5), 341-356, 1982.[8] T. Speer, A short study of Alexandroff spaces, at http://arxiv.org/abs/0708.2136,
2007.[9] A.H. Tan, J.J. Li and G.P. Lin, Extended results on the relationship between infor-
mation systems, Inf. Sci. 290, 156-173, 2015.[10] A.H. Tan, J.J. Li and G.P. Lin and Y.J. Lin, Fast approach to knowledge acquistion in
covering information systems using matrix operations, Knowl.-Based Syst. 79, 90-98,2015.
[11] A.H. Tan, J.J. Li, Y.J. Lin and G.P. Lin, Matrix-based set approximations and re-ductions in covering decision information systems, Int. J. Approx. Reason. 59, 68-80,2015.
[12] E.C.C. Tsang, C.Z. Wang, D.G. Chen, C.X. Wu and Q.H. Hu, Communication be-tween information systems using fuzzy rough sets, IEEE Trans. Fuzzy syst. 21 (3),527-540, 2013.
[13] C.Z. Wang, W.Y. Bao, X.X. Wang and Q. He, A mapping between fuzzy coveringinformation systems, Proc. 2013 IEEE International Conference on Granular Com-puting, 315-319.
[14] C.Z. Wang, D.G. Chen and Q.H. Hu, Some invariant properties of ordered informationsystems under homomorphism, Sci. China: Inf. Sci. 53 (9), 1816-1825, 2010.
[15] C.Z. Wang, D.G. Chen and Q.H. Hu, Fuzzy information systems and their homomor-phisms, Fuzzy Sets Syst. 249, 128-138, 2014.
[16] C.Z. Wang, D.G. Chen, B.Q. Sun and Q.H. Hu, Communication between informationsystems with covering based rough sets, Inf. Sci. 216, 17-33, 2012.
[17] C.Z. Wang, D.G. Chen, C. Wu and Q.H. Hu, Data compression with homomorphismin covering information systems, Int. J. Approx. Reason. 52 (4), 519-525, 2011.
[18] C.Z. Wang, D.G. Chen and L.K. Zhu, Homomorphisms between fuzzy informationsystems, Appl. Math. Lett. 22 (7), 1045-1050, 2009.
[19] C.Z. Wang, C.X. Wu, D.G. Chen and W.J. Du, Some properties of relation informa-tion systems under homomorphisms, Appl. Math. Lett. 21 (9), 940-945, 2008.
[20] C.Z. Wang, C.X. Wu, D.G. Chen, Q.H. Hu and C. Wu, Communicating betweeninformation systems, Inf. Sci. 178 (16), 3228-3239, 2008.
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[21] W. Żakowski, Approximations in the space (U, Π), Demonstr. Math. 16 (3), 761-769,1983.
[22] X.D. Zhang, Matrix analysis and applications, Tsinghua University Press, Beijing,2004 (in Chinese).
[23] P. Zhu, Covering rough sets based on neighborhoods: An approach without usingneighborhoods, Int. J. Approx. Reason. 52 (3), 461-472, 2011.
[24] P. Zhu and Q.Y. Wen, Some improved results on communication between informationsystems, Inf. Sci. 180 (18), 3521-3531, 2010.
[25] P. Zhu and Q.Y. Wen, Homomorphisms between fuzzy information systems revisited,Appl. Math. Lett. 24 (9), 1548-1553, 2011.
[26] P. Zhu and Q.Y. Wen, A note on communicating between information systems basedon including degrees, Int. J. Gen. Syst. 40 (8), 837-840, 2011.
[27] P. Zhu, H.Y. Xie and Q.Y. Wen, A unified definition of consistent functions, Fund.Inf. 135 (3), 331-340, 2014.
[28] P. Zhu, H.Y. Xie and Q.Y. Wen, A unified view of consistent functions, Soft Comput.21 (9), 2189-2199, 2017.
Hacettepe Journal ofMathematics & Statistics
Hacet. J. Math. Stat.Volume 49 (2) (2020), 808 – 821
DOI : 10.15672/hujms.624000
Research Article
On C -coherent rings, strongly C -coherent ringsand C -semihereditary rings
Zhu Zhanmin
Department of Mathematics, Jiaxing University, Jiaxing, Zhejiang Province, 314001, P.R.China
AbstractLet R be a ring and C be a class of some finitely presented left R-modules. A left R-module M is called C -injective if Ext1
R(C,M) = 0 for every C ∈ C ; a left R-moduleM is called C -projective if Ext1
R(M,E) = 0 for any C -injective module E. R is calledleft C -coherent if every C ∈ C is 2-presented; R is called left strongly C -coherent, ifwhenever 0 → K → P → C → 0 is exact, where C ∈ C and P is finitely generatedprojective, then K is C -projective; a ring R is called left C -semihereditary, if whenever0 → K → P → C → 0 is exact, where C ∈ C , P is finitely generated projective, thenK is projective. In this paper, we give some new characterizations and properties of leftC -coherent rings, left strongly C -coherent rings and left C -semihereditary rings.
Mathematics Subject Classification (2010). 16D40, 16D50, 16E60, 16P70.
Keywords. C -coherent ring, strongly C -coherent ring, C -semihereditary ring.
1. IntroductionRecall that a ring R is said to be left coherent [1, 19] if every finitely generated left
ideal of R is finitely presented, a ring R is said to be left semihereditary if every finitelygenerated left ideal of R is projective. Coherent rings, semihereditary rings and theirgeneralizations have been studied extensively by many authors (see, for example, [1, 2,4, 6, 11, 13–15, 19, 24, 26]). In [27], we introduced the concepts of left C -coherent ringsand left C -semihereditary rings, and in [28], we introduced the concept of left stronglyC -coherent rings. Let C be a class of some finitely presented left R-modules. Following[27], a ring R is called left C -coherent if every C ∈ C is 2-presented; a ring R is calledleft C -semihereditary, if whenever 0 → K → P → C → 0 is exact , where C ∈ C , P isfinitely generated projective, then K is projective. To characterize left C -coherent ringsand left C -semihereditary rings , in [27], we also introduced the concepts of C -injectivemodules and C -flat modules. According to [27], a left R-module M is called C -injective ifExt1
R(C,M) = 0 for every C ∈ C , a right R-module M is called C -flat if TorR1 (M,C) = 0
for every C ∈ C . In [28], we introduced the concepts of C -projective modules and leftstrongly C -coherent rings. Following [28], a left R-module M is called C -projective ifExt1
R(M,E) = 0 for any C -injective module E; a ring R is called left strongly C -coherent,if whenever 0 → K → P → C → 0 is exact, where C ∈ C and P is finitely generated
Email address: [email protected]: 30.05.2018; Accepted: 05.04.2019
On C -coherent rings 809
projective, then K is C -projective. We shall denote the class of C -flat (resp., C -injective,C -projective) modules by CF (resp., C I , CP).
In this article, we continues to study left C -coherent rings, left strongly C -coherentrings and left C -semihereditary rings. Series characterizations and properties of theserings will be given respectively.
Next, we recall some known notions and facts needed in the sequel.Given a class L of R-modules, we shall denote by L ⊥ = M : Ext1
R(L,M) = 0, L ∈ L the right orthogonal class of L , and by ⊥L = M : Ext1
R(M,L) = 0, L ∈ L the leftorthogonal class of L .
Let F be a class of R-modules and M an R-module. Following [9], we say that ahomomorphism φ : M → F where F ∈ F is an F-preenvelope of M if for any morphismf : M → F ′ with F ′ ∈ F, there is a g : F → F ′ such that gφ = f . An F-preenvelopeφ : M → F is said to be an F-envelope if every endomorphism g : F → F such thatgφ = φ is an isomorphism. Dually, we have the definitions of F-precovers and F-covers.F-envelopes (F-covers) may not exist in general, but if they exist, they are unique upto isomorphism. It is easy to see that every C -injective preenvelope is monic, and everyC -projective precover is epic.
Following [9], a pair (A ,B) of classes of R-modules is called a cotorsion pair if A ⊥ = Band ⊥B = A . A cotorsion pair (A ,B) is called hereditary [10, Definition 1.1] if whenever0 → A′ → A → A′′ → 0 is exact with A,A′′ ∈ A then A′ is also in A . By [10, Proposition1.2], a cotorsion pair (A ,B) is hereditary if and only if whenever 0 → B′ → B → B′′ → 0is exact with B′, B ∈ B then B′′ is also in B. A cotorsion pair (A ,B) is called perfect[10] if every R-module has an A -cover and a B-envelope. A cotorsion pair (A ,B) iscalled complete (see [9, Definition 7.16] and [20, Lemma 1.13]) if for any R-module M ,there are exact sequences 0 → M → B → A → 0 with A ∈ A and B ∈ B, and0 → B′ → A′ → M → 0 with A′ ∈ A and B′ ∈ B.
Throughout this paper, R is an associative ring with identity and all modules consideredare unitary, C is a class of some finitely presented left R-modules. For any R-module M ,E(M) will denote the injective envelope of M , M+ = Hom(M,Q/Z) will be the charactermodule of M and M∗ = Hom(M,R) will be the dual module of M .
2. C -coherent ringsTheorem 2.1. The following statements are equivalent for a ring R:
(1) R is a left C -coherent ring.(2) For any projective left R-module P, P ∗ is C -flat.(3) For any free left R-module F, F ∗ is C -flat.
Proof. (1)⇒(2). For any projective left R-module P , there is an index set I and anR-module Q such that P ⊕ Q ∼= R(I). So we have P ∗ ⊕ Q∗ ∼= (R(I))∗ ∼= RI , and thus P ∗
is C -flat by [27, Theorem 3.3(4) and Proposition 2.6].(2)⇒(3). It is clear.(3)⇒(1). Let I be any index set. Then by (3), RI ∼= (R(I))∗ is C -flat, and so R is
C -coherent by [27, Theorem 3.3(4)].
Recall that a left R-module M is said to be FP-injective [19] if Ext1R(A,M) = 0 for
every finitely presented left R-module A; a left R-module M is said to be P-injective[16] if every homomorphism from a principal left ideal of R to M can be extended to ahomomorphism of R to M , it is easy to see that a left R-module M is P-injective if andonly if Ext1
R(R/Ra,M) = 0 for any a ∈ R. We recall also that a left R-module M is saidto be FI-injective [13] (resp., D-injective [14], copure injective [8] ) if Ext1
R(G,M) = 0for every FP-injective (resp., P-injective, injective) left R-module G; a right R-module Nis said to be FI-flat [13] (resp., D-flat [14], copure flat [8]) if TorR
1 (N,G) = 0 for every
810 Z. Zhanmin
FP-injective (resp., P-injective, injective) left R-module G. Inspired by these concepts,we have the following concepts.
Definition 2.2. A left R-module M is said to be C I-injective if Ext1R(G,M) = 0 for every
C -injective left R-module G; a right R-module F is said to be C I-flat if TorR1 (F,G) = 0
for every C -injective left R-module G.
Proposition 2.3. The following statements are equivalent for a left R-module M:(1) M is C I-injective.(2) For every exact sequence 0 → M → E → L → 0 with E C -injective, E → L is a
C -injective precover of L.(3) M is the kernel of a C -injective precover f : E → L with E injective.(4) M is injective with respect to every exact sequence 0 → A → B → C → 0 with C
C -injective.
Proof. (1)⇒(2) and (1)⇒(4) are clear.(2)⇒(3). It follows from the exact sequence 0 → M → E(M) → E(M)/M → 0.(3)⇒(1). Let M be the kernel of a C -injective precover f : E → L with E injective.
Then f : E → im(f) is a C -injective precover, so, for any C -injective module N , the mapHom(N,E) → Hom(N, im(f)) is epic and hence the map Hom(N,E) → Hom(N,E/M)is epic. Thus, by the exactness of the sequence 0 → Hom(N,E) → Hom(N,E/M) →Ext1
R(N,M) → 0, we have Ext1R(N,M) = 0.
(4) ⇒ (1). For any C -injective module N , there exists an exact sequence 0 → K →P → N → 0, where P is projective. Hence we get an exact sequence Hom(P,M) →Hom(K,M) → Ext1
R(N,M) → Ext1R(P,M) = 0, and thus Ext1
R(N,M) = 0 by (4).Therefore, M is C I-injective. Remark 2.4. Since the class of all C -injective modules is closed under extensions, byWakamutsu’s Lemma (see [23, Lemma 2.1.1]), any kernel of a C -injective cover is C I-injective .
Recall that a left R-module M is called reduced [9] if M has no nonzero injectivesubmodules.
Proposition 2.5. Let R be a left C -coherent ring. Then the following statements areequivalent for a left R-module M:
(1) M is a reduced C I-injective module.(2) M is the kernel of a C -injective cover f : E → L with E injective.
Proof. (1)⇒(2). Since M is C I-injective, by proposition 2.3, the natural mapping π :E(M) → E(M)/M is a C -injective precover. Since R is left C -coherent, by [27, Corollary3.7], E(M)/M has a C -injective cover. Note that there is no nonzero summand K ofE(M) contained in M as M is reduced, by [23, Corollary 1.2.8], π : E(M) → E(M)/M isa C -injective cover.
(2)⇒(1). Let M be the kernel of a C -injective cover f : E → L with E injective. Thenby proposition 2.3(3), M is a C I-injective module. Now let K be an injective submoduleof M . Suppose E = K ⊕ N, p : E → N is the projective and i : N → E is the inclusionfor some submodule N of M . It is easy to see that f(ip) = f since f(K) = 0. So ip is anisomorphism since f is a cover. Thus i is epic and hence E = N,K = 0. Therefore M isreduced.
Recall that a submodule A of left R-module B is said to be a pure submodule if forall right R-module M , the induced map M ⊗R A → M ⊗R B is monic, or equivalently,every finitely presented left R-module is projective with respect to the exact sequence0 → A → B → B/A → 0. In this case, the exact sequence 0 → A → B → B/A → 0
On C -coherent rings 811
is called pure exact. An exact sequence 0 → A → B → L → 0 is called RD-exact [14]if, for any a ∈ R, R/Ra is projective with respect to this sequence. We call a shortexact sequence of left R-modules 0 → A → B → L → 0 C -pure exact if every C ∈ C isprojective with respect to this sequence. Let A be a submodule of B, if the short exactsequence of left R-modules 0 → A → B → B/A → 0 is C -pure exact, then we call A aC -pure submodule of B and B/A a C -pure quotient module of B.
Next, we give some characterizations of C -injective modules.
Theorem 2.6. Let M be a left R-module, then the following statements are equivalent:(1) M is C -injective.(2) M is injective with respect to every exact sequence 0 → A → B → C → 0 of left
R-modules with C ∈ C .(3) M is injective with respect to every exact sequence 0 → K → P → C → 0 of left
R-modules with C ∈ C and P finitely generated projective.(4) Every exact sequence 0 → M → M ′ → M ′′ → 0 is C -pure.(5) There exists a C -pure exact sequence 0 → M → M ′ → M ′′ → 0 of left R-modules
with M ′ injective.(6) There exists a C -pure exact sequence 0 → M → M ′ → M ′′ → 0 of left R-modules
with M ′ FP-injective.(7) There exists a C -pure exact sequence 0 → M → M ′ → M ′′ → 0 of left R-modules
with M ′ C -injective.
Proof. (1) ⇒ (2). It follows from the exact sequence
Hom(B,M) → Hom(A,M) → Ext1R(C,M) = 0.
(2) ⇒ (3). It is obvious.(3) ⇒ (1). It follows from the exact sequence
Hom(P,M) → Hom(K,M) → Ext1R(C,M) → Ext1
R(P,M) = 0.
(1) ⇒ (4).Assume (1). Then we have an exact sequence Hom(C,M ′) → Hom(C,M ′′) →Ext1
R(C,M) = 0 for every C ∈ C , and so (4) follows.(4) ⇒ (5) ⇒ (6) ⇒ (7) is obvious.(7) ⇒ (1). By (7), we have a C -pure exact sequence 0 → M → M ′ f→ M ′′ → 0 of left
R-modules where M ′ is C -injective, and so, for each C ∈ C , we have an exact sequenceHom(C,M ′) f∗→ Hom(C,M ′′) → Ext1
R(C,M) → Ext1R(C,M ′) = 0 with f∗ epic. Which
implies that Ext1R(C,M) = 0, and (1) follows.
Recall that a left R-module M is called pure injective [9, Definition 5.3.6] if it is injectivewith respect to every pure exact sequence of left R-modules; a left R-module M is calledRD-injective [14] if it is injective with respect to every RD-exact sequence of left R-modules. We call a left R-module M C -pure injective if it is injective with respect toevery C -pure exact sequence of left R-modules.
Proposition 2.7. Let R be a left C -coherent ring. Then every C -pure injective moduleM has a C -injective cover f : N → M with N injective. Moreover, Ker(f) is a reducedC I-injective left R-module.
Proof. By [27, Corollary 3.7], M has a C -injective cover f : N → M . Since N is C -injective, by Theorem 2.6(4), the exact sequence 0 → N
i→ E(N) → E(N)/N → 0 isC -pure exact, and so there exists g : E(N) → M such that gi = f . Note that f is acover, there exists h : E(N) → N such that fh = g. Thus fhi = f and hence hi is anisormorphism. It follows that N is isomorphic to a direct summand of E(N) and so N isinjective. By Proposition 2.5, Ker(f) is a reduced C I-injective left R-module.
812 Z. Zhanmin
Theorem 2.8. Let R be a left C -coherent ring. Then a left R-module M is C I-injectiveif and only if M is a direct sum of an injective left R-module and a reduced C I-injectiveleft R-module.Proof. “ ⇐ ”. It is clear.
“ ⇒ ”. Let M be a C I-injective left R-module . Then by Proposition 2.3, E(M) →E(M)/M is a C -injective precover. Since R is left C -coherent, E(M)/M has a C -injectivecover L g→ E(M)/M by [27, Corollary 3.7], so we have the following commutative diagramwith exact rows:
0 −−−−→ Kf−−−−→ L
g−−−−→ E(M)/M −−−−→ 0yϕ
yγ∥∥∥
0 −−−−→ Mα−−−−→ E(M) π−−−−→ E(M)/M −−−−→ 0yσ
yβ
∥∥∥0 −−−−→ K
f−−−−→ Lg−−−−→ E(M)/M −−−−→ 0
where K is a reduced C I-injective left R-module by Proposition 2.5. Note that g = g(βγ),we have that βγ is an isomorphism, so E(M) = Ker(β) ⊕ im(γ), and thus Ker(β) isinjective. Since σϕ is an isomorphism by the Five Lemma, we have that M = Ker(σ) ⊕im(ϕ) and im(ϕ) ∼= K. Moreover, by the Snake Lemma [17, Theorem 6.5], we have thatKer(σ) ∼= Ker(β) is injective. This completes the proof. Proposition 2.9. Let M be a right R-module. Then M is C I-flat if and only if M+ isC I-injective.Proof. It follows from the isomorphism TorR
1 (M,G)+ ∼= Ext1R(G,M+).
Corollary 2.10. A pure submodule of a C I-flat module is C I-flat.Proof. Let M be a C I-flat module and M1 a pure submodule of M , then the pure exactsequence 0 → M1 → M → M/M1 → 0 induces a split exact sequence 0 → (M/M1)+ →M+ → M+
1 → 0. By Proposition 2.9, M+ is C I-injective, so M+1 is C I-injective, and
hence M1 is C I-flat by Proposition 2.9 again. Proposition 2.11. Let R be a ring and C be a class of some finitely presented left R-modules.
(1) If M is a finitely presented C I-flat module, then it is a cokernet of a C -flat preen-velope.
(2) If R is left C -coherent and L is the cokernet of a C I-flat preenvelope f : M → F ,then L is C I-flat.
Proof. (1). Let M be a finitely presented C I-flat module. Then there exists an exactsequence of right R-modules 0 → K → P → M → 0 with P finitely generated projectiveand K finitely generated. We claim that K → P is a C -flat preenvelope. In fact, for anyC -flat module F , we have F+ is C -injective by [27, Theorem 2.7], and so TorR
1 (M,F+) = 0since M is C I-flat. Hence, we have the following commutative diagram with α monic:
K ⊗ F+ α−−−−→ P ⊗ F+
τ1
y yτ2
Hom(K,F )+ β−−−−→ Hom(P, F )+
Since K is finitely generated and P is finitely presented, by [3, Lemma 2], τ1 is epic andτ2 is an isomorphism, this follows that β is monic, and hence Hom(P, F ) → Hom(K,F ) isepic, as required.
On C -coherent rings 813
(2). There is an exact sequence 0 → im(f) i→ F → L → 0. We claim that i : im(f) →F is a C -flat preenvelope. In fact, for any C -flat module F1 and any homomorphismφ : im(f) → F1, φf is a homomorphism from M to F1. Since f : M → F is a C -flatpreenvelope, there exists a ψ : F → F1 such that φf = ψf . Now, for any y ∈ im(f),write y = f(x). Then φf(x) = ψif(x), i.e., φ(y) = ψi(y). It shows that φ = ψi, and soi : im(f) → F is a C -flat preenvelope. Let N be any C -injective module. Since R is leftC -coherent, N+ is C -flat by [27, Theorem 3.3(8)], and so, the mapping Hom(F,N+) →Hom(im(f), N+) is epic. Then, from the following commutative diagram :
Hom(F,N+) α−−−−→ Hom(im(f), N+)
σ1
y yσ2
(F ⊗N)+ β−−−−→ (im(f) ⊗N)+
where σ1 and σ2 are isomorphisms, we have that the mapping (F⊗N)+ → (im(f)⊗N)+ isepic. Thus, the mapping im(f)⊗N → F⊗N is monic. But the C I-flatness of F implies theexactness of 0 → TorR
1 (L,N) → im(f) ⊗N → F ⊗N , and therefore TorR1 (L,N) = 0.
3. Strongly C -coherent ringsTheorem 3.1. The following statements are equivalent for a ring R:
(1) R is a left strongly C -coherent ring.(2) If 0 → K → E → L → 0 is an exact sequence of left R-modules with K C -injective
and E FP-injective, then L is C -injective.(3) If 0 → K → E → L → 0 is an exact sequence of left R-modules with K C -injective
and E injective, then L is C -injective.(4) R is left C -coherent, and if 0 → N → M → Q → 0 is an exact sequence of right
R-modules with M and Q C -flat, then N is C -flat.(5) R is left C -coherent, and if 0 → N → M → Q → 0 is an exact sequence of right
R-modules with M flat and Q C -flat, then N is C -flat.(6) R is left C -coherent, and if 0 → N → P → Q → 0 is an exact sequence of right
R-modules with P projective and Q C -flat, then N is C -flat.
Proof. (1)⇒(2). It follows from [28, Theorem 1(7)].(2)⇒(3); and (4)⇒ (5) ⇒ (6) are trivial.(3)⇒(1). Let M be a C -injective left R-module. Then by (2), E(M)/M is C -injective.
And so R is left strongly C -coherent by [28, Theorem 1(8)].(1)⇒(4). It follows from [28, Theorem 1(9)] and [27, Proposition 3.11(2)].(6)⇒(1). For any C -flat right R-module N , there exists an exact sequence 0 →
K → P → N → 0 with P projective. So K is C -flat by (6), and thus TorR2 (N,C) ∼=
TorR1 (K,C) = 0 for any C ∈ C . Therefore R is left strongly C -coherent by [28, Theorem
1(11)]. Proposition 3.2. Let R be a left strongly C -coherent ring. Then the following statementsare equivalent for a left R-module M:
(1) M is injective.(2) M is both C -injective and C I-injective.(3) There exists a C -injective cover f : M → N with N C I-injective.
Proof. (1)⇒(2). It is trivial.(2)⇒(3). It is clear because M → M is a C -injective cover of M.(3)⇒(1). Consider the exact sequence 0 → M
i→ E(M) → E(M)/M → 0. SinceR is a left strongly C -coherent ring, by [28, Theorem 1(7)], E(M)/M is C -injective, soExt1
R(E(M)/M,N) = 0. Thus there exists a homomorphism g : E(M) → N such that
814 Z. Zhanmin
f = gi. Since f is a cover, there exists a homomorphism h : E(M) → M such that g = fh.Hence f(hi) = f , and so hi is an isomorphism, this follows that i is left split, and thereforeM = E(M) is injective.
Theorem 3.3. The following statements are equivalent for a ring R:(1) R is a left strongly C -coherent ring.(2) R is left C -coherent, and every C -injective C I-injective left R-module is injective.(3) Each left R-module has a C -injective cover, and every C -injective C I-injective left
R-module is injective.(4) R is left C -coherent, and for every C I-injective left R-module L, there there exists
a C -injective cover E → L with E injective.(5) Each left R-module has a C -injective cover, and for every C I-injective left R-
module L, there there exists a C -injective cover E → L with E injective.(6) Every C -pure quotient of a C -injective left R-module has a C -injective cover, and
for every C I-injective left R-module L, there exists a C -injective cover E → L withE injective.
(7) Every C -pure quotient of a C -injective left R-module has a C -injective cover, andevery C -injective C I-injective left R-module is injective.
Proof. (1)⇒(2). Since R is left strongly C -coherent, by [28, Theorem 1(10)], it is leftC -coherent. Moreover, by Proposition 3.2, every C -injective C I-injective left R-module isinjective.
(2)⇒(3). It follows from [27, Corollary 3.7].(1)⇒(4). It is clear that R is left C -coherent. Let L be any C I-injective left R-module.
Then by [27, Corollary 3.7], L has a C -injective cover f : E → L, and by Proposition 3.2,E is injective.
(4)⇒(5). It follows from [27, Corollary 3.7].(3)⇒(7), and (5)⇒(6) are trivial.(6)⇒(7). Let M be a C -injective C I-injective left R-module. Then by (6), there exists
a C -injective cover f : E → M with E injective. Note that 1M : M → M is also aC -injective cover of M , we have that M ∼= E, and hence M is injective.
(7)⇒(1). Let 0 → Ni→ E
f→ L → 0 be an exact sequence of left R-modules with NC -injective and E injective. Then by Theorem 2.6(4), this exact sequence is C -pure, andso L has a C -injective cover φ : E′ → L. Thus there exists a homomorphism g : E → E′
such that f = φg. Since f is epic, φ is also epic. Now, forming a pullback we obtain thefollowing commutative diagram with exact rows and columns (see [21, 10.3(1)]).
0 0y yK Kyα
y0 −−−−→ N −−−−→ P −−−−→
h2E′ −−−−→ 0∥∥∥ yh1
yφ
0 −−−−→ N −−−−→ E −−−−→f
L −−−−→ 0y y0 0
On C -coherent rings 815
where P = (x, y) ∈ E′ ⊕ E | φ(x) = f(y), K = Ker(φ), α : K → P, k 7→ (k, 0), h1(x, y) = x, h2(x, y) = y. Let β : P → E′, (x, y) 7→ x − g(y). Then φβ(x, y) =φ(x) − φg(y) = φ(x) − f(y) = 0, so β(x, y) ∈ K , and hence β is a homomorphism fromP to K. Note that βα(k) = β(k, 0) = k − g(0) = k, we have that βα = 1K . Since Nand E′ are both C -injective, P is also C -injective, and so K is C -injective. Note that Kis C I-injective by [9, Corollary 7.2.3], we have that K is injective by conditions, so L isC -injective, and hence R is a left strongly C -coherent ring by Theorem 3.1(3).
Let F be a class of R-modules. According to [5], an F-cover ϕ : F → M is said to havethe unique mapping property if for any homomorphism f : F ′ → M with F ′ ∈ F, there isa unique homomorphism g : F ′ → F such that f = ϕg.
Theorem 3.4. The following statements are equivalent for a ring R:(1) Every left R-module is C -projective.(2) Every nonzero left R-module has a nonzero C -projective submodule.(3) R is left strongly C -coherent, and every (C -injective) left R-module has a C -
projective cover with the unique mapping property.
Proof. (1)⇒(2) and (1)⇒ (3) are obvious.(2)⇒(1). Assume (2). To prove (1), we need only to prove that every C -injective
module E is injective by [28, Theorem 6(3)].Let I be a left ideal of R , i : I → R be the inclusion map and f : I → E be any
homomorphism. It suffices to show that there is g : R → E that extends f . Let A consistof all pair (I ′, g′), where I ⊆ I ′ ⊆ R and g′ : I ′ → E extends f . Since (I, f) ∈ A , A 6= ϕ.A is a partially set by saying (I ′, g′) ≤ (I ′′, g′′) if I ′ ⊆ I ′′ and g′′ extends g′. By Zorn’sLemma, there is a maximal element (I0, g0) in A . If I0 6= R, then R/I0 6= 0. By (2), thereis a nonzero C -projective submodule K/I0 of R/I0. Note that Ext1
R(K/I0, E) = 0, wehave that g0 can be extended to K, this contradicts to the maximality of (I0, g0). Thus,I0 = R and E is injective, as required.
(3)⇒(1). Assume (3). To prove (1), we need only to prove that every C -injective moduleE is C -projective by [28, Theorem 6(4)]. By (3), E has a C -projective cover ϕ : P → Ewith the unique mapping property. Let K = Ker(ϕ), i : K → P be the inclusion map andφ : P ′ → K be a C -projective cover of K. Then ϕiφ = 0 = ϕ0, and so iφ = 0 by theunique mapping property. Since every C -projective cover is epic, φ and ϕ are epic, so ϕis an isomorphism, and thus E is C -projective. This completes the proof.
According to [28], the C -injective dimension of a module RM is defined byC I-dim(RM) = infn : Extn+1
R (C,M) = 0 for every C ∈ C ;the C -injective global dimension of a ring R is defined by
C I-GLD(R)=supC I-dim(M): M is a left R-module;the C -flat dimension of a module MR is defined by
CF-dim(MR) = infn : TorRn+1(M,C) = 0 for every C ∈ C ;
the C -weak global dimension of a ring R is defined byC -WD(R)=supCF-dim(M): M is a right R-module.
Theorem 3.5. Let R be a left strongly C -coherent ring, M a left R-module and n anonnegative integer. Then the following statements are equivalent:
(1) C I-dim(RM) ≤ n.(2) Extn+k
R (P,M) = 0 for all C -projective module P and all positive integers k.(3) Extn+1
R (P,M) = 0 for all C -projective module P.
816 Z. Zhanmin
Proof. (1)⇒(2). Assume (1). Then since R is left strongly C -coherent, by [28, The-orem 2], there exists an exact sequence of left R-modules 0 → M
ε→ E0d0→ · · · →
En−1dn−1→ En → 0 such that E0, · · · , En−1, En are C -injective. Thus, by [28, The-
orem 1(12)], we have Extn+1R (P,M) ∼= Extn
R(P, im(d0)) ∼= Extn−1R (P, im(d1)) ∼= · · · ∼=
Ext1R(P, im(dn−1)) = Ext1
R(P,En) = 0 for any C -projective module P , and Extn+kR (P,M) ∼=
Ext1R(P, 0) = 0 for any k > 1. So (2) follows.
(2)⇒ (3) ⇒(1). It is trivial. Corollary 3.6. Let R be a left strongly C -coherent ring and 0 → A → B → C → 0 anexact sequence of left R-modules. If two of C I-dim(A),C I-dim(B),C I-dim(C) are finite,then so is the third. Moreover:
(1) C I-dim(B)≤ sup C I-dim(A), C I-dim(C).(2) C I-dim(A)≤ sup C I-dim(B), C I-dim(C) + 1.(3) C I-dim(C)≤ sup C I-dim(B), C I-dim(A) − 1.
In particular, C I-dim(A⊕ C)= sup C I-dim(A), C I-dim(C).
Let n be a positive integer. then according to [4], a left R-module M is said to ben-presented in case there is an exact sequence of left R-modules Fn → Fn−1 → · · · →F1 → F0 → M → 0 in which every Fi is finitely generated free. It is easy to see thata left R-module M is n-presented if and only if there exists an exact sequence of leftR-modules 0 → Kn → Fn−1 → · · · → F1 → F0 → M → 0 such that F0, · · · , Fn−1 arefinitely generated free and Kn is finitely generated.Lemma 3.7. Let R be a left strongly C -coherent ring. Then every C ∈ C is n-presentedfor any positive integer n.Proof. Use induction on n. If n = 1, then it is clear that the result holds. Assume thatevery C ∈ C is n-presented. Then for any C ∈ C and any FP-injective module N , wehave Extn+1
R (C,N) = 0 by [28, Theorem 1(5)] because R is left strongly C -coherent. Let0 → Kn → Fn−1 → · · · → F1 → F0 → C → 0 be an exact sequence of left R-moduleswith F0, · · · , Fn−1 finitely generated free left R–modules and Kn finitely generated. ThenExt1
R(Kn, N) ∼= Extn+1R (C,N) = 0 , so Kn is finitely presented by [7], and hence C is
(n+ 1)-presented. Theorem 3.8. Let R be a left strongly C -coherent ring and M a left R-module. ThenC I-dim(M)=CF-dim(M+).Proof. Let n be a positive integer, C ∈ C . Since R is left strongly C -coherent, byLemma 3.7, C is (n+ 2)-presented. So, by [2, Lemma 2.7(2)], we have TorR
n+1(M+, C) ∼=Extn+1
R (C,M)+. Consequently, C I-dim(M) =CF-dim(M+) by [28, Theorem 2, Theorem3]. Theorem 3.9. Let R be left strongly C -coherent and RR be C -injective. If RM is C -projective with finite projective dimension, then RM is projective.Proof. Suppose that RM is C -projective with pd(M) = n < ∞. Then by [28, Theorem5], there exists an exact sequence of left R-modules
0 → Pndn→ Pn−1
dn−1→ · · · → P1d1→ P0
d0→ M → 0such that P0, · · · , Pn−1, Pn are projective. Since RR is C -injective and direct sums anddirect summands of C -injective modules are C -injective by [28, Proposition 2.5], each Pi
is C -injective for i = 0, 1, · · · , n. Clearly, im(dn) ∼= Pn is C -injective. Note that R isleft strongly C -coherent , by [28, Theorem 1(7)], im(dn−1) is C -injective. Continues inthis way, one can get that im(d1) is C -injective, so Ext1
R(M, im(d1)) = 0, and thus theexact sequence 0 → im(d1) → P0 → M → 0 is split, this follows that RM is projective, asrequired.
On C -coherent rings 817
Recall that, by [28, Example 1], a left C -coherent ring need not be left strongly C -coherent. As the end of this section, we give another example which shows that even if Ris a left artinian ring, it need not be left strongly C -coherent.
Example 3.10. Let K be a field and L be a proper subfield of K such that ρ : K → Lis an isomorphism. Let K[x; ρ] be the ring of twisted right polynomials over K wherekx = xρ(k) for all k ∈ K. Set R = K[x; ρ]/(x2), and C = R/Ra : a ∈ R. If b1, b2 isa basis for K as a vector space over L, then R is left artinian and hence left C -coherent,but it is not left strongly C -coherent.
Proof. Since K has finite vector space dimension over L, by [18, Example 1], R is leftartinian . Since the only proper right ideal of R is rR(x) = xR = xK, it is readily verifiedthat rRlR(a) = aR for any a ∈ R, so RR is P-injective by [16, Lemma 1.1]. Now, we definef : Rxb1 + Rxb2 → R by f(r1xb1 + r2xb2) = r1x + r2x, then it is easy to see that f isa left R-homomorphism. We claim that this homomorphism can not be extended to anendomorphism of R. Otherwise, there exists a c = k0 +xk′
0 ∈ R such that f = ·c. Clearly,k0 6= 0. Thus, f(xb1 −xb2) = (xb1 −xb2)(k0 +xk′
0), and so 0 = x−x = (xb1 −xb2)k0, thisfollows that b1 = b2, a contradiction. Observing that lR(x) = xK = xR = Rxb1 + Rxb2,we have Ext1
R(Rx,R) ∼= Ext1R(R/(Rxb1 +Rxb2), R) 6= 0, and hence R is not left strongly
C -coherent.
4. C -semihereditary ringsWe begin with the following definition.
Definition 4.1. A ring R is called weakly C -semihereditary, if whenever 0 → K → P →C → 0 is exact , where C ∈ C , P is finitely generated projective , then K is flat.
Recall that a ring R is called left weakly n-semihereditary [25] if every n-generated leftideal is flat; a ring R is called a left p.f ring [11] if every principal left ideal of R is flat.By [11, Theorem 2.2], a ring R is left p.f if and only if it is right p.f; a ring R is called aleft FS-ring [12, 22] if Soc(RR) is flat.
Example 4.2. (1). Let C = R/I : I is an n-generated left ideal of R. Then the ring Ris weakly C -semihereditary if and only if R is left weakly n-semihereditary.
(2). Let C = R/Ra : a ∈ R. Then the ring R is weakly C -semihereditary if and onlyif R is left p.f.
(3). Let C = R/Ra : Ra is a minimal left ideal of R. Then the ring R is weaklyC -semihereditary if and only if every minimal left ideal of R is flat, if and only if R is aleft FS-ring .
Theorem 4.3. The following statements are equivalent for a ring R:(1) R is a left weakly C -semihereditary ring.(2) Every submodule of a C -flat right R-module is C -flat.(3) Every submodule of a flat right R-module is C -flat.(4) Every submodule of a projective right R-module is C -flat.(5) Every submodule of a free right R-module is C -flat.(6) Every finitely generated right ideal of R is C -flat.
Proof. (2)⇒(3)⇒ (4)⇒ (5)⇒ (6) is trivial.(1)⇒(2). Assume (1). Let A be a submodule of a C -flat right R-module B and let
C ∈ C . Then there exists an exact sequence of left R-modules 0 → K → P → C →0, where P is finitely generated projective. By (1), K is flat. Then the exactness of0 = TorR
2 (B/A,P ) → TorR2 (B/A,C) → TorR
1 (B/A,K) = 0 implies that TorR2 (B/A,C) =
0. And thus from the exactness of the sequence 0 = TorR2 (B/A,C) → TorR
1 (A,C) →TorR
1 (B,C) = 0 we have TorR1 (A,C) = 0. It shows that A is C -flat.
818 Z. Zhanmin
(6)⇒(1). Let C ∈ C . There exists an exact sequence of left R-modules 0 → K →P → C → 0, where P is finitely generated projective. For any finitely generated rightideal I of R, we have an exact sequence 0 → TorR
2 (R/I,C) → TorR1 (I, C) = 0 since I is
C -flat. So TorR2 (R/I,C) = 0, and hence we obtain an exact sequence 0 = TorR
2 (R/I,C) →TorR
1 (R/I,K) → 0. Thus, TorR1 (R/I,K) = 0. And so K is flat.
Proposition 4.4. If R is a left weakly C -semihereditary ring, then C -WD(R)≤ 1.
Proof. Let M be any right R-module and let C ∈ C . Then there exists an exact sequenceof left R-modules 0 → K → P → C → 0, where P is finitely generated projective. SinceR is left weakly C -semihereditary, K is flat. So TorR
2 (M,C) ∼= TorR1 (M,K) = 0. It shows
that C -WD(R)≤ 1.
Lemma 4.5. Let F be a class of some right R-modules. If N f1→ N1 and Nf2→ N2 are
F-preenvelopes , then N1 ⊕N2/f2(N) ∼= N2 ⊕N1/f1(N).
Proof. Let εi : Ni → N1 ⊕ N2 be the injections, i = 1, 2. We obtain a morphismq∗ = ε1f1 + ε2f2 : N → N1 ⊕ N2. Let ε1 : N1 → Coker(q∗);n1 7→ (n1, 0) + im(q∗), ε2 : N2 → Coker(q∗);n2 7→ (0, n2) + im(q∗) and Q = Coker(q∗). Then we get thefollowing pushout diagram:
Nf2−−−−→ N2
f1
y ε2
yN1
ε1−−−−→ Q
And so, by the proof of [21, 10.6(1)(i)], we have the following commutative diagram withexact rows, where g : Q → N2/f2(N); (n1, n2) + im(q∗) 7→ n2 + f2(N):
Nf2−−−−→ N2 −−−−→ N2/f2(N) −−−−→ 0
f1
y ε2
y 1y
N1ε1−−−−→ Q
g−−−−→ N2/f2(N) −−−−→ 0
Since N f2→ N2 is an F-preenvelope and N1 ∈ F, there exists a homomorphism α : N2 → N1such that f1 = αf2. If ε1(n1) = 0, then (n1, 0) = q∗(n) = (f1(n), f2(n)) for some n ∈ N ,so f2(n) = 0, f1(n) = n1, and hence n1 = f1(n) = αf2(n) = 0. It shows that ε1 is monic.Now, we define h : Q → N1 by (n1, n2) + im(q∗) 7→ n1 − α(n2). Then h is well-defined,and hε1(n1) = h((n1, 0) + im(q∗)) = n1 − α(0) = n1 for each n1 ∈ N1, so hε1 = 1N1 , andthen ε1 is left split. Thus, we have Q ∼= N1 ⊕ N2/f2(N). Similarly, we have also thatQ ∼= N2 ⊕N1/f1(N) and so N1 ⊕N2/f2(N) ∼= N2 ⊕N1/f1(N).
Next, we give some new characterizations of left C -semihereditary rings.
Theorem 4.6. The following statements are equivalent for a ring R:(1) R is left C -semihereditary.(2) R is left C -coherent and left weakly C -semihereditary.(3) R is left strongly C -coherent and every C -projective left R-module has a monic
C -injective cover.(4) Every C -projective left R-module has projective dimension at most 1.(5) R is left C -coherent and every C I-injective module is injective.(6) Every left R-module has a C -injective cover and every C I-injective module is injec-
tive.(7) Every C -pure quotient of a C -injective left R-module has a C -injective cover and
every C I-injective module is injective.(8) R is left strongly C -coherent and every C I-injective module is C -injective.
On C -coherent rings 819
(9) R is left strongly C -coherent and the kernel of any C -injective precover of a leftR-module is C -injective.
(10) R is left strongly C -coherent and the kernel of any C -injective cover of a leftR-module is C -injective.
(11) R is left strongly C -coherent and the cokernel of any C -injective preenvelope of aleft R-module is C -injective.
(12) R is left strongly C -coherent and the kernel of any C -flat precover of a right R-module is C -flat.
(13) R is left strongly C -coherent and the kernel of any C -flat cover of a right R-moduleis C -flat.
(14) R is left strongly C -coherent and the cokernel of any C -flat preenvelope of a rightR-module is C -flat.
Proof. (1)⇔ (2). It follows from [27, Theorem 4.3(2)] and Theorem 4.3(2).(1)⇒(3). Suppose that R is left C -semihereditary. Then it is left strongly C -coherent
by [28, Theorem 4]. Moreover, by [27, Theorem 4.3(7)], every C -projective left R-modulehas a monic C -injective cover.
(3)⇒(1). Let E be any injective left R-module and K any submodule of E. By [27,Theorem 4.3(6)], we need only to prove that E/K is C -injective. In fact, since (CP,C I)is a complete cotorsion pair by [27, Theorem 2.10(1)], there exists an exact sequences0 → K → E1
f→ P → 0 with P C -projective and E1 C -injective. By (3), P has a monicC -injective cover φ : E2 → P . So, there exists a homomorphism g : E1 → E2 such thatf = φg. Thus φ is epic, and hence φ is an isomorphism. This implies that P is C -injective.For any C ∈ C , we have the exact sequence
0 = Ext1R(C,P ) → Ext2
R(C,K) → Ext2R(C,E1).
But R is left strongly C -coherent, by [28, Theorem 1(6)], Ext2R(C,E1) = 0, and so
Ext2R(C,K) = 0. On the other hand, the short exact sequence 0 → K → E → E/K → 0
induces the exact sequence0 = Ext1
R(C,E) → Ext1R(C,E/K) → Ext2
R(C,K) = 0.so, we have Ext1
R(C,E/K) = 0, and hence E/K is C -injective. Consequently, R is leftC -semihereditary by [27, Theorem 4.3(6)].
(1)⇒(4). Let M be a C -projective module and N be any left R-module. Since R is leftC -semihereditary, by [27, Theorem 4.3(6)], E(N)/N is C -injective. So, by the exactnessof the sequence
0 = Ext1R(M,E(N)/N) → Ext2
R(M,N) → Ext2R(M,E(N)) = 0.
We have Ext2R(M,N) = 0, and hence M has projective dimension at most 1.
(4)⇒(1). Let C ∈ C and 0 → K → P → C → 0 be exact, where P is finitely generatedprojective. Note that C is C -projective, by (4), pd(C) ≤ 1, and so K is projective bySchanuel’s Lemma.
(1)⇒(5). Since R is left C -semihereditary, by [27, Theorem 4.3], R is left C -coherentand every quotient module of an injective left R-module is C -injective . Let M be aC I-injective left R-module. Then E(M)/M is C -injective, so M is injective with respectto the exact sequence 0 → M → E(M) → E(M)/M → 0 by Proposition 2.3, and henceM = E(M) is injective.
(5)⇒(6). It follows from [27, Corollary 3.7].(6) ⇒ (1). Let M be a quotient of an injective left R-module. By (6), M has a C -
injective cover. Suppose f : F → M is a C -injective cover of M . Then f is epic. ByRemark 2.4, Ker(f) is C I-injective, and so it is injective by (6). Thus, M is isomorphicto a direct summand of F and hence it is C -injective. Hence, by [27, Theorem 4.3(6)], Ris left C -semihereditary.
820 Z. Zhanmin
(6) ⇒ (7). It is obvious.(7)⇒(8). It follows from Theorem 3.3(7).(8) ⇒ (5). Assume (8). Then by [28, Theorem 1(10)], R is left C -coherent. Let M be a
C I-injective module. Then by (8), M is C -injective. But R is left strongly C -coherent, by[28, Theorem 1(7)], E(M)/M is C -injective. Thus, by Proposition 2.3(4), M is injective .
(1) ⇒ (9). Clearly, R is left strongly C -coherent . Let f : F → M be a C -injectiveprecover and K = Ker(f) . Since R is left C -semihereditary, by [27, Theorem 4.3(7)],there exists a monic C -injective cover φ : G → M . Thus, by [9, Lemma 8.6.3], we haveK ⊕G ∼= F , and so K is C -injective.
(9)⇒(10). It is obvious.(10) ⇒ (1). Let M be a quotient of a C -injective left R-module. Since R is left C -
coherent, by [27, Corollary 3.7], M has a C -injective cover f : F → M . Clearly, f is epic.So, by (10), we have that Ker(f) is C -injective , this implies that M is also C -injective by[28, Theorem 1(7)] as R is left strongly C -coherent. Therefore, by [27, Theorem 4.3(5)],R is left C -semihereditary.
(1) ⇒ (11). Clearly, R is left strongly C -coherent. And by [27, Theorem 4.3(5)], everyquotient module of a C -injective module is C -injective, so the cokernel of any C -injectivepreenvelope of a left R-module is C -injective.
(11) ⇒ (1). LetM be any left R-module. Since the class of all C -injective left R-modulesis closed under pure submodules , isomorphisms and direct product, by [29, Theorem 2.6],M has a C -injective preenvelope f : M → E. By (11), E/im(f) is C -injective . It iseasy to see that f is monic. Since R is left strongly C -coherent, by [28, Theorem 2(5)],C I-dim(RM) ≤ 1. And so , C I-GLD(R)≤ 1. Therefore, by [28, Theorem 4(2)], R is leftC -semihereditary.
(1) ⇒ (12). Clearly, R is left strongly C -coherent. And by [27, Theorem 4.3(2)], thekernel of any C -flat precover of a right R-module is C -flat.
(12)⇒(13). It is obvious.(13) ⇒ (1). Let N be any right R-module. Then by [27, Theorem 2.10(2)], N has a C -
flat cover f : F → N . Clearly, f is epic. By (13), we have that Ker(f) is C -flat. But R isleft strongly C -coherent, by [28, Theorem 3(5)], CF-dim(NR) ≤ 1. Thus, C -WD(R)≤ 1.Consequently, by [28, Theorem 4(3)], we have that R is left C -semihereditary.
(1) ⇒ (14). Clearly, R is left strongly C -coherent. Let φ : N → F be a C -flat preen-velope of a right R-module N and L = coker(φ). Since R is left C -semihereditary, by[27, Theorem 4.3(8)], N has an epic C -flat envelope ϕ : N → G. Hence, by Lemma 4.5,we have F ∼= G⊕ L, and so L is C -flat.
(14) ⇒ (1). Let N be a submodule of a C -flat module. Since R is left C -coherent, by[27, Theorem 3.3(12)], N has a C -flat preenvelope f : N → F . It is easy to see that f ismonic. By (14), F/im(f) is C -flat. Note that R is left strongly C -coherent, by Theorem3.1(4), N is C -flat. Therefore, by [27, Theorem 4.3(2)], R is left C -semihereditary.
Acknowledgment. The authors would like to thank the referee for the useful sugges-tions and comments. This research was supported by the Natural Science Foundation ofZhejiang Province, China(LY18A010018).
References[1] S. U. Chase, Direct products of modules, Trans. Amer. Math. Soc. 97 , 457-473, 1960.[2] J. L. Chen and N. Q. Ding, On n-coherent rings, Comm. Algebra 24 (3), 3211-3216,
1996.[3] R. R. Colby, Rings which have flat injective modules, J. Algebra 35, 239-252, 1975.[4] D. L. Costa, Parameterizing families of non-noetherian rings, Comm. Algebra 22
(10), 3997-4011, 1994.
On C -coherent rings 821
[5] N. Q. Ding and J. L. Chen, On envelopes with the unique mapping property, Comm.Algebra 24 (4), 1459-1470, 1996.
[6] D. E. Dobbs, S. Kabbaj, and N. Mahdou, n-coherent rings and modules, LectureNotes in Pure and Appl. Math. 185, 269-281, 1997
[7] E. Enochs, A note on absolutely pure modules, Canad. Math. Bull. 19 (3), 361-362,1976.
[8] E. E. Enochs and O. M. G. Jenda, Copure injective resolutions, flat resolutions anddimensions, Comment. Math. Univ. Carolin 34 (2), 203-211, 1993.
[9] E. E. Enochs and O. M. G. Jenda, Relative Homological Algebra, Walter de Gruyter.Berlin-New York, 2000.
[10] E. E. Enochs, O. M. G. Jenda and J. A. Lopez-Ramos, The existence of Gorensteinflat covers, Math. Scand 94 (1), 46-62, 2004.
[11] S. Jøndrup, p.p.rings and finitely generated flat ideals, Proc. Amer. Math. Soc. 28(2), 431-435, 1971.
[12] Z. K. Liu, Rings with flat left socles, Comm. Algebra 23 (5), 1645-1656, 1995.[13] L. X. Mao and N. Q. Ding, FI-injective and FI-flat modules, J. Algebra 309 (1),
367-385, 2007.[14] L. X. Mao and N. Q. Ding, On divisible and torsionfree modules, Comm. Algebra 36
(2), 708-731, 2008.[15] L. X. Mao, On mininjective and min-flat modules, Publ. Math. Debrecen 72 (3-4),
347-358, 2008.[16] W. K. Nicholson and M. F. Yousif, Principally injective rings, J. Algebra 174 (1),
77-93, 1995.[17] J. J. Rotman, An Introduction to Homological Algebra, Academic press, New Yock,
1979.[18] E. A. Rutter, Rings with the principle extension property, Comm. Algebra 3 (3),
203-212, 1975.[19] B. Stenström, Coherent rings and FP-injective modules, J. London Math. Soc. 2,
323-329, 1970.[20] J. Trlifaj, Cover, envelopes, and cotorsion theories, in:Homological Methods in Module
Theory. Lecture notes for the workshop, Cortona, 10-16, 2000.[21] R. Wisbauer, Foundations of Module and Ring Theory, London-Tokyo: Gordon and
Breach 1991.[22] Y. F. Xiao, Rings with flat socles, Proc. Amer. Math. Soc. 123 (8), 2391-2395, 1995.[23] J. Z. Xu, Flat Covers of Modules, Lecture Note in Math. Springer-Verlag, Berlin-
Heidelberg-New York, 1634, 1996.[24] X. X. Zhang , J. L. Chen and J. Zhang, On (m,n)-injective modules and (m,n)-
coherent rings, Algebra Colloq. 12 (1) , 149-160, 2005.[25] X. X. Zhang and J. L. Chen, On n-semihereditary and n-coherent rings, Int. Electron.
J. Algebra 1, 1-10, 2007.[26] Z. M. Zhu, On n-coherent rings, n-hereditary rings and n-regular rings, Bull. Iranian
Math. Soc. 37 (4), 251-267, 2011.[27] Z. M. Zhu, C -coherent rings, C -semihereditary rings and C -regular rings, Studia Sci.
Math. Hungar 50 (4), 491-508, 2013.[28] Z. M. Zhu, Strongly C -coherent rings, Math. Rep. 19 (4), 367-380, 2017.[29] Z. M. Zhu, On Π-coherence of rings, Hacet. J. Math. Stat. 46 (5), 875-886, 2017.
Hacettepe Journal ofMathematics & Statistics
Hacet. J. Math. Stat.Volume 49 (2) (2020), 822 – 834
DOI : 10.15672/hujms.458085
Research Article
Slant submersions in paracontact geometry
Yılmaz Gündüzalp
Department of Mathematics, Dicle University, 21280, Diyarbakır, Turkey
AbstractIn this paper, we investigate some geometric properties of three types of slant submersionswhose total space is an almost paracontact metric manifold.
Mathematics Subject Classification (2010). 53C15,53C40
Keywords. almost paracontact metric manifold, semi-Riemannian submersion, properslant submersion
1. IntroductionGiven a C∞−submersion ψ from a (semi)-Riemannian manifold (N, gN ) onto a (semi)-
Riemannian manifold (B, gB), according to the circumstances on the map ψ : (N, gN ) →(B, gB), we get the following: a (semi)-Riemannian submersion ([3, 8, 14, 20]), an almostHermitian submersion ([27]), a paracontact submersion ([9]), a paracontact paracom-plex submersion ([10]), a (para) quaternionic submersion ([6, 17]), a slant submersion([12, 19, 22, 23]), an anti-invariant submersion ([11, 24]), a conformal semi-slant submer-sion ([1, 13]), a conformal anti-invariant submersion ([2]), a hemi-slant submersion ([25]),etc. As we know, Riemannian submersions were severally introduced by B. O’Neill ([20])and A. Gray ([14]) in 1960s. In particular, by using the concept of almost Hermitiansubmersions, B. Watson ([27]) gave some differential geometric properties among fibers,base manifolds, and total manifolds. After that, there are lots of results on this issue. Itis well-known that Riemannian submersions are associated with physics and have theirapplications in the Yang-Mills theory ([5]), Kaluza-Klein theory ([4,15]), supergravity andsuperstring theories ([16]), etc.
The paper is organized as follows. In Section 2, we remind some concepts, which areneeded in the following part. In Section 3, we study some geometric properties of threetypes of proper slant submersions from an almost paracontact metric manifold onto asemi-Riemannian manifold. We present examples, investigate the geometry of leaves ofdistributions. We obtain a necessary and sufficient circumstance for such submersions tobe totally geodesic map, as well.
Email address: [email protected]: 07.09.2018; Accepted: 10.04.2019
Slant submersions in paracontact geometry 823
2. Preliminaries2.1. Semi-Riemannian submersions
A C∞− submersion ψ : N → B between two pseudo-Riemannian manifolds (N, gN )and (B, gB) is called a semi-Riemannian submersion if it satisfies circumstances:(i) the fibers ψ−1(b), b ∈ B, are r− dimensional pseudo-Riemannian submanifolds of N,where r = dim(N) − dim(B).(ii) ψ∗ preserves scalar products of vectors normal to fibres.The tangent bundle TN of the total space N has an orthogonal decomposition
TN = kerψ∗ ⊕ (kerψ∗)⊥,
where kerψ∗ is the vertical distribution while (kerψ∗)⊥ designates the horizontal one.In ([20]), O’Neill has defined two configuration tensors T and A, of the total space of asemi-Riemannian submersion by setting
TX1X2 = h∇vX1vX2 + v∇vX1hX2 (2.1)and
AX1X2 = v∇hX1hX2 + h∇hX1vX2 (2.2)for any X1, X2 ∈ χ(N), here v and h are the vertical and horizontal projections respec-tively.Using (2.1) and (2.2), we get
∇X1X2 = TX1X2 + ∇X1X2; (2.3)
∇X1X3 = TX1X3 + h(∇X1X3); (2.4)∇X3X1 = AX3X1 + v(∇X3X1), (2.5)∇X3X4 = AX3X4 + h(∇X3X4), (2.6)
for any X3, X4 ∈ Γ((kerψ∗)⊥), X1, X2 ∈ Γ(kerψ∗). In addition, if X3 is basic thenh(∇X1X3) = h(∇X3X1) = AX3X1.
The fundamental tensor fields T,A satisfy:
TX1X2 = TX2X1, X1, X2 ∈ Γ(kerψ∗); (2.7)
AX3X4 = −AX4X3 = 12v[X3, X4], X3, X4 ∈ Γ((kerψ∗)⊥). (2.8)
Lemma 2.1. If ψ : (N, gN ) → (B, gB) is a (semi-)Riemannian submersion and X3, X4fundamental vector fields on N, ψ−related to X∗3 and X∗4 vector fields on base manifoldB, at that time we obtain the following features
(1) h[X3, X4] is a fundamental vector field and ψ∗h[X3, X4] = [X3∗, X∗4] ψ;(2) h(∇X3X4) is a fundamental vector field ψ−related to (∇∗
X∗3X∗4), here ∇ and ∇∗
are the Riemannian connection on N and B;(3) [E,X1] ∈ Γ(kerψ∗), for any X1 ∈ Γ(kerψ∗) and for any fundamental vector field
E([8, 21]).
Let (N, gN ) and (B, gB) be (semi-)Riemannian manifolds and ψ : (N, gN ) → (B, gB) isa differentiable map. At that time, the second fundamental form of ψ is given by
(∇ψ∗)(X1, X2) = ∇ψX1ψ∗X2 − ψ∗(∇X1X2) (2.9)
for X1, X2 ∈ Γ(N), here we show conveniently by ∇ the Riemannian connections of themetrics gN and gB. Recall that ψ is said to be harmonic if trace(∇ψ∗) = 0 and ψ is calleda totally geodesic map if (∇ψ∗)(X1, X2) = 0 for X1, X2 ∈ Γ(TN), [18].
824 Y. Gündüzalp
2.2. Almost paracontact metric manifoldsLet N be a differentiable manifold of dimensional (2n + 1). An almost paracontact
structure on N is a triple (φ, ξ, η), where:(1) ξ is a Reeb vector field,(2) η is a one-form such that η(ξ) = 1, and(3) φ is a tensor field of type (1, 1) satisfying
φ2 = Id− η ⊗ ξ, η φ = 0, φ(ξ) = 0. (2.10)
If N is equipped with a pseudo-Riemannian metric gN such that
gN (φX1, φX2) = −gN (X1, X2) + η(X1)η(X2), X1, X2 ∈ χ(N), (2.11)
then (φ, ξ, η, gN ) is an almost paracontact metric structure. So, the quintuple(N2n+1, φ, ξ, η, gN ) is an almost paracontact metric manifold ([26,28]).Observe that, since (2.11) holds, any compatible with metric gN has got sign (n + 1, n)and by (2.10) and (2.11) we have η(X1) = gN (ξ,X1). Furthermore, we can determine ananti-symmetric two-form Φ by Φ(X1, X2) = gN (X1, φX2), which is called the fundamental2-form corresponding to the structure.
An almost paracontact metric structure (φ, ξ, η, gN ) is said to be paracosymplectic, if∇η = 0 and ∇Φ = 0 are closed ([7]), and the structure equation of a paracosymplecticmanifold is given by
(∇X1φ)X2 = 0, X1, X2 ∈ χ(N), (2.12)where ∇ denotes the Riemannian connection of the metric gN on N . Moreover, for aparacosymplectic manifold, we know that
∇X1ξ = 0. (2.13)
3. Proper slant submersionsLet ψ be a semi-Riemannian submersion from an almost paracontact metric manifold
N with the structure (φ, ξ, η, gN ) onto a semi-Riemannian manifold (B, gB). Then forX1 ∈ Γ(kerψ∗), we write
φX1 = αX1 + βX1, (3.1)where αX1 and βX1 are vertical and horizontal parts of φX1.
In addition to for X2 ∈ Γ((kerψ∗)⊥), we get
φX2 = tX2 + rX2, (3.2)
where tX2 and rX2 are vertical and horizontal components of φX2.
If for any spacelike or timelike vertical vector field X1 ∈ kerψ∗ − ξ, the quotientgN (αX1,αX1)gN (φX1,φX1) is constant, i.e. it is independent of the choice of the point p ∈ N and choiceof the spacelike or timelike vertical vector field X1 in kerψ∗ − ξ, at that time we callthat ψ is a slant submersion. In this case, the angle ω is called the slant angle of the slantsubmersion.We note that Reeb vector field ξ is a spacelike vertical vector field.
Let E1, E2, ξ be a local orthonormal frame of vertical vector fields with gN (E1, E1) =1, i.e., such that E1 is spacelike (if both E1 and E2 are timelike, the situation would besimilar). From (2.11) and (3.1), we have
−1 = gN (φE1, φE1) = gN (αE1, αE1) + gN (βE1, βE1).
Slant submersions in paracontact geometry 825
On the other hand, αE1 = ρE2. Let us suppose ρ = 0,±1; these conditions would cor-respond to invariant ([9]) and anti-invariant submersions . Clearly, αE1 and E2 have thesame causal character. Depending on it and the value of ρ, we can separate the followingthree conditions:(1) If αE1 is a timelike and ∥ρ∥ > 1, at that time gN (βE1, βE1) = −1 + ρ2 and so βE1 isspacelike.(2) If αE1 is a timelike and ∥ρ∥ < 1, at that time gN (βE1, βE1) = −1 + ρ2 and so βE1 istimelike.(3) If αE1 is a spacelike, gN (βE1, βE1) = −1 − ρ2, and βE1 is a timelike vector field.
These three conditions will correspond to three different types of proper slant submer-sions.
Definition 3.1. Let ψ be a proper slant submersion from an almost paracontact manifoldN with the structure (φ, ξ, η, gN ) onto a semi-Riemannian manifold (B, gB). We say thatit is oftype 1 if for any spacelike (timelike) vertical vector field X1 ∈ Γ(kerψ∗), αX1 is timelike(spacelike), and ∥αX1∥
∥φX1∥ > 1,type 2 if for any spacelike (timelike) vertical vector field X1 ∈ Γ(kerψ∗), αX1 is timelike(spacelike), and ∥αX1∥
∥φX1∥ < 1,type 3 if for any spacelike (timelike) vertical vector field X1 ∈ Γ(kerψ∗), αX1 is timelike(spacelike).
It is known that the distribution (kerψ∗) is integrable for a semi-Riemannian submer-sion between semi-Riemannian manifolds. In fact, its leaves are ψ−1(b), b ∈ B, i.e., fibres.Thus it follows from above definition that the fibers of a slant submersion are slant sub-manifolds of N .
Theorem 3.2. Let ψ be a proper slant submersion from an almost paracontact manifoldN with the structure (φ, ξ, η, gN ) onto a semi-Riemannian manifold (B, gB). Then,(i) ψ is slant submersion of type 1 if and only if for any spacelike (timelike) vector fieldX1 ∈ Γ(kerψ∗), αX1 is timelike (spacelike), and there exists a constant µ ∈ (1,∞) suchthat
α2X1 = µ(X1 − η(X1)ξ). (3.3)If ψ is a proper slant submersion of type 1, then µ = cosh2 ω, with ω > 0.(ii) ψ is a proper slant submersion of type 2 if and only if for any spacelike (timelike) vectorfield X1 ∈ Γ(kerψ∗), αX1 is timelike (spacelike), and there exists a constant µ ∈ (0, 1)such that
α2X1 = µ(X1 − η(X1)ξ). (3.4)If ψ is a proper slant submersion of type 2, then µ = cos2 ω, with 0 < ω < 2π.(iii) ψ is slant submersion of type 3 if and only if for any spacelike (timelike) vector fieldX1 ∈ Γ(kerψ∗), αX1 is timelike (spacelike), and there exists a constant µ ∈ (−∞, 0) suchthat
α2X1 = µ(X1 − η(X1)ξ). (3.5)If ψ is a proper slant submersion of type 3, then µ = − sinh2 ω, with ω > 0.In every case, the angle ω is called the slant angle of the slant submersion.
Proof. (i) If ψ is slant submersion of type 1, for any spacelike vertical vector field X1 ∈Γ(kerψ∗), αX1 is timelike, and, by virtue of (2.11), φX1 is timelike. Furthermore, they
826 Y. Gündüzalp
satisfy ∥αX1∥∥φX1∥ > 1. So, there exists ω > 0 such that
coshω = ∥αX1∥∥φX1∥
=√
−gN (αX1, αX1)√−gN (φX1, φX1)
. (3.6)
By using (2.10), (2.11, (3.1)) and (3.6) we obtaingN (α2X1, X1) = −gN (αX1, αX1)
= − cosh2 ωgN (φX1, φX1)= cosh2 ωgN (φ2X1, X1)= cosh2 ωgN (X1 − η(X1)ξ,X1) (3.7)
for all X1 ∈ Γ(kerψ∗). Since gN is a semi-Riemannian metric, from (3.7) we getα2X1 = cosh2 ω(X1 − η(X1)ξ), X1 ∈ Γ(kerψ∗). (3.8)
Let µ = cosh2 ω. Then it is obvious that µ ∈ (1,∞) and α2 = µ(I − η ⊗ ξ).Everything works in a similar way for any timelike vector field X2 ∈ Γ(kerψ∗), but now,αX2 and φX2 are spacelike and hence, instead of (3.6) we can write:
coshω = ∥αX2∥∥φX2∥
=√gN (αX2, αX2)√gN (φX1, φX1)
.
Since α2X1 = µ(X1 − η(X1)ξ), for any spacelike or timelike X1 we have that α2 =µ(I − η ⊗ ξ). The converse is just a easy computation.(ii) is obtained in a similar way.
(iii) If ψ is proper slant submersion of type 3, for any spacelike vector field X1 ∈Γ(kerψ∗), αX1 is spacelike,as well and hence, there exists ω > 0 such that
sinhω = ∥αX1∥∥φX1∥
=√gN (αX1, αX1)√
−gN (φX1, φX1).
Once more, we can demonstrate that gN (α2X1, X1) = − sinh2 ωgN (X1 − η(X1)ξ,X1). Letµ = − sinh2 ω. At that time it is clear that µ ∈ (−∞, 0) and α2 = µ(I − η ⊗ ξ).The converse is just a easy computation.
For slant submersion of type 2, the slant angle coincides with the Wirtinger angle, i.e.,the slant angle between φX1 and αX1.
Theorem 3.3. Let ψ be a proper slant submersion from an almost paracontact manifoldN with the structure (φ, ξ, η, gN ) onto a semi-Riemannian manifold (B, gB). Then,(i) ψ is slant submersion of type 1 if and only if α2X1 = cosh2 ω(X1 − η(X1)ξ) for everyspacelike vector field X1 ∈ Γ(kerψ∗).(ii) ψ is slant submersion of type 2 if and only if α2X1 = cos2 ω(X1 − η(X1)ξ) for everyspacelike vector field X1 ∈ Γ(kerψ∗).
Proof. (i) For every timelike vector field X2 ∈ Γ(kerψ∗), there exists a spacelike vectorfield X1 ∈ Γ(kerψ∗) such as αX1 = X2. Then:
α2X2 = α2αX1 = αα2X1 = cosh2 ω(αX1 − η(αX1)ξ) = cosh2 ω(X2 − η(X2)ξ).The same proof is valid for (ii), but α2X1 = cos2 ω(X1 − η(X1)ξ). Theorem 3.4. Let ψ be a proper slant submersion from an almost paracontact manifoldN with the structure (φ, ξ, η, gN ) onto a semi-Riemannian manifold (B, gB). Then ψ isslant submersion oftype 1 if and only if tβX1 = − sinh2 ω(X1 − η(X1)ξ) for every spacelike (timelike) vertical
Slant submersions in paracontact geometry 827
vector field X1 ∈ Γ(kerψ∗).type 2 if and only if tβX1 = sin2 ω(X1 − η(X1)ξ) for every spacelike (timelike) verticalvector field X1 ∈ Γ(kerψ∗).type 3 if and only if tβX1 = cosh2 ω(X1 − η(X1)ξ) for every spacelike (timelike) verticalvector field X1 ∈ Γ(kerψ∗).Proof. For any vertical vector field X1 ∈ Γ(kerψ∗), it holds
X1 − η(X1)ξ = φ2X1 = α2X1 + βαX1 + tβX1 + rβX1.
Equalizing the vertical and the horizontal parts of the above equation, we obtain:α2X1 + tβX1 = X1 − η(X1)ξ, βαX1 + rβX1 = 0. (3.9)
Hence, for a slant submersion of type 1,tβX1 = X1 − η(X1)ξ − α2X1 = (1 − cosh2 ω)(X1 − η(X1)ξ) = − sinh2 ω(X1 − η(X1)ξ),
while for a slant submersion of type 2,tβX1 = X1 − η(X1)ξ − α2X1 = (1 − cos2 ω)(X1 − η(X1)ξ) = sin2 ω(X1 − η(X1)ξ),
and, for a slant submersion of type 3,tβX1 = X1 − η(X1)ξ − α2X1 = (1 + sinh2 ω)(X1 − η(X1)ξ) = cosh2 ω(X1 − η(X1)ξ).
The converse results are deduced from the same equations. Theorem 3.5. Let ψ be a semi-Riemannian submersion from an almost paracontact met-ric manifold (N4n+1
2n , φ, η, ξ, gN ) onto a semi-Riemannian manifold (B2nn , gB). Then ψ is
a slant submersion oftype 1 if and only if r2X2 = cosh2 ωX2 for every spacelike (timelike) horizontal vector fieldX2 ∈ Γ((kerψ∗)⊥).type 2 if and only if r2X2 = cos2 ωX2 for every spacelike (timelike) horizontal vector fieldX2 ∈ Γ((kerψ∗)⊥).Proof. In the case of a slant submersion oftype 1, for every horizontal timelike (spacelike) vector field X2 ∈ Γ((kerψ∗)⊥), there existsa spacelike (timelike) vertical vector field X1 ∈ Γ(kerψ∗) such as βX1 = X2. From (3.9),we obtain
r2X2 = r2βX1 = −rβαX1 = βα2X1 = β(cosh2 ω(X1 − η(X1)ξ)). (3.10)From (3.10), we get r2X2 = cosh2 ω(βX1 −η(βX1)ξ). Since βX1⊥ξ, we obtain η(βX1) = 0and thus r2X2 == cosh2 ωX2.
In the case of a slant submersion oftype 2, in a similar way, we get
r2X2 = cos2 ωX2.
The converse results follow from the fact that t((kerψ∗)⊥) = (kerψ∗)⊕ < ξ > . Theorem 3.6. Let ψ be a semi-Riemannian submersion from an almost paracontact met-ric manifold (N4n+1
2n , φ, η, ξ, gN ) onto a semi-Riemannian manifold (B2n2j , gB)(0 < j < n).
At that time, ψ is a slant submersion of type 3 if and only if r2X2 = − sinh2 ωX2 for everyhorizontal vector field X2 ∈ Γ((kerψ∗)⊥).Proof. If X1 is a spacelike (timelike) vertical vector field, αX1 is also spacelike (timelike)and βX1 is timelike (spacelike). Therefore, given that the dimension of B is half thedimension of N, if X2 is a timelike (spacelike) horizontal vector field, then there existsa vertical vector field X1 ∈ Γ(kerψ∗) such that βX1 = X2. Then, from (3.10) we haver2X2 = β(− sinh2 ω(X1 − η(X1)ξ)) = − sinh2 ωX2. The converse results follow from thefact that t((kerψ∗)⊥) = (kerψ∗)⊕ < ξ > .
828 Y. Gündüzalp
From Theorem 3.2, (2.11) and (3.1) we obtain the following result.
Lemma 3.7. Let ψ be a semi-Riemannian submersion from an almost paracontact metricmanifold (N,φ, η, ξ, gN ) onto a semi-Riemannian manifold (B, gB).If ψ is a proper slant submersion of type 1, then, for any spacelike (timelike) vector fieldsX1, X2 ∈ Γ(kerψ∗), we have
gN (αX1, αX2) = cosh2 ω(−gN (X1, X2) + η(X1)η(X2)) (3.11)gN (βX1, βX2) = − sinh2 ω(−gN (X1, X2) + η(X1)η(X2)) (3.12)
If ψ is a proper slant submersion of type 2, then, for any spacelike (timelike) vector fieldsX1, X2 ∈ Γ(kerψ∗), we have
gN (αX1, αX2) = cos2 ω(−gN (X1, X2) + η(X1)η(X2)) (3.13)gN (βX1, βX2) = sin2 ω(−gN (X1, X2) + η(X1)η(X2)). (3.14)
If ψ is a proper slant submersion of type 3, then, for any spacelike (timelike) vector fieldsX1, X2 ∈ Γ(kerψ∗), we have
gN (αX1, αX2) = − sinh2 ω(−gN (X1, X2) + η(X1)η(X2)) (3.15)gN (βX1, βX2) = cosh2 ω(−gN (X1, X2) + η(X1)η(X2)). (3.16)
Note that given a semi-Euclidean space R2n+1n with coordinates (x1, ..., x2n, z) on R2n+1
n ,we can naturally choose an almost paracontact structure (φ, ξ, η) on R2n+1
n as follows:
η = dz, ξ = ∂
∂z, φ( ∂
∂x2i) = ∂
∂x2i−1, φ( ∂
∂x2i−1) = ∂
∂x2i, φ(ξ) = 0
where i = 1, ..., n. Let R2n+1n be a semi-Euclidean space of signature (+,-,+,-,...,+) with
respect to the canonical basis ( ∂∂x1
, ..., ∂∂x2n
, ∂∂z ).Now, we can present four examples of proper slant submersions.
Example 3.8. Determine a map ψ : R52 → R2
1 by
ψ(x1, x2, x3, x4, z) = (x1 − x3√2
, x2).
At that time, by direct calculations we obtain
kerψ∗ = spanU1 = ∂
∂x1+ ∂
∂x3, U2 = ∂
∂x4, U3 = ξ = ∂
∂z
and(kerψ∗)⊥ = spanX1 = ∂
∂x1− ∂
∂x3, X2 = ∂
∂x2.
Thus, the map ψ is a slant submersion of type 2 with the slant angle ω with cos−1( 1√2).
Example 3.9. Define a map ψ : R52 → R2
1 by
ψ(x1, x2, x3, x4, z) = (x2 sinh x+ x3 cosh x, x1 sinh y + x4 cosh y),
any for x, y ∈ R. Then, by direct calculations we get
kerψ∗ = spanU1 = cosh x ∂
∂x2− sinh x ∂
∂x3, U2 = cosh y ∂
∂x1− sinh y ∂
∂x4, U3 = ξ = ∂
∂z
and
(kerψ∗)⊥ = spanX1 = − sinh x ∂
∂x2+ cosh x ∂
∂x3, X2 = sinh y ∂
∂x1− cosh y ∂
∂x4.
Thus,the map ψ is a slant submersion of type 1 with the slant angle coshω = cosh(x− y).
Slant submersions in paracontact geometry 829
Example 3.10. Define a map ψ : R52 → R2
1 byψ(x1, x2, x3, x4, z) = (x1 sin x+ x3 cosx, x2 sin y + x4 cos y),
for any x, y ∈ R. The map ψ is a slant submersion of type 2 with the slant angle cosω =cos(x− y).
Example 3.11. Define a map ψ : R52 → R2
1 byψ(x1, x2, x3, x4, z) = (x2 cosh x+ x3 sinh x, x4),
for any x ∈ R+. The map ψ is a slant submersion of type 3 with the slant angle α2 =− sinh2 x.
Let ψ be proper slant submersions of type 1,2 and 3 from a paracosymplectic manifold Nwith the structure (gN , φ, η, ξ) onto a semi-Riemannian manifold (B, gB). From (2.11),(3.1)and (3.2), one can simply see that
gN (X1, αX2) = −gN (αX1, X2) (3.17)and
gN (βX1, X3) = −gN (X1, tX3), (3.18)for spacelike (timelike) vector fields X1, X2 ∈ Γ(kerψ∗), X3 ∈ Γ((kerψ∗)⊥).
Using (2.3),(2.5) and (2.13) we haveTX1ξ = 0, AX3ξ = 0 (3.19)
for spacelike (timelike) vector fields X1 ∈ Γ(kerψ∗), X3 ∈ Γ(kerψ∗)⊥.
We determine the covariant derivatives of α and β as follows(∇X1α)X2 = ∇X1αX2 − α∇X1X2 (3.20)
and(∇X1β)Y = h∇X1βX2 − β∇X1X2 (3.21)
for spacelike (timelike) vector fields X1, X2 ∈ Γ(kerF∗), where ∇X1X2 = v∇X1X2. Thenwe easily have
Lemma 3.12. Let (N, gN , φ, η, ξ) be a paracosymplectic manifold and (B, gB) a semi-Riemannian manifold. Let ψ : N → B be proper slant submersions of type 1, 2 and 3.Then, we have
∇X1αX2 + TX1βX2 = α∇X1X2 + tTX1X2
TX1αX2 + h∇X1βX2 = β∇X1X2 + rTX1X2
for any spacelike(timelike)vector fields X1, X2 ∈ Γ(kerψ∗).
We say that β is parallel with respect to the Riemannian connection ∇ on (kerψ∗) ifits covariant derivative with respect to ∇ vanishes, i.e., we get
(∇X1β)X2 = h∇X1βX2 − β∇X1X2 = 0 (3.22)for any spacelike (timelike) vertical vector fields X1, X2 ∈ Γ(kerψ∗).
Theorem 3.13. Let ψ be a proper slant submersions of type 1, 2 and 3 from a paracosym-plectic manifold (N, gN , φ, η, ξ) onto a semi-Riemannian manifold (B, gB). At that time,the fibres are not proper totally umbilical.
Proof. See [19]. We now indicate the orthogonal complementary distribution to β(kerψ∗) in (kerψ∗)⊥
by τ. At that time, we obtain the following.
830 Y. Gündüzalp
Theorem 3.14. Let ψ : (N, gN , φ, η, ξ) → (B, gB) be a proper slant submersions of typeof 1, 2 and 3. If N is a paracosymplectic manifold, then τ is an invariant distribution of(kerψ∗)⊥,with respect to φ.
Proof. For X2 ∈ Γ(τ), using (2.11) and (3.1), we getgN (φX2, βX1) = −gN (X2, X1) − gN (φX2, αX1) + η(X1)η(X2)
= −gN (φX2, αX1)= gN (X2, φαX1) = 0
for any spacelike (timelike) vector field X1 ∈ Γ(kerψ∗).In a similar way, we have gN (φX2, X3) = −gN (X2, φX3) = 0 due to φX3 ∈ Γ((kerψ∗) ⊕β(kerψ∗)) for any spacelike (timelike) vector field X2 ∈ Γ(τ) and X3 ∈ Γ(kerψ∗). Hence,proof is complete. Corollary 3.15. Let ψ : (N4n+1
2n , gN , φ, η, ξ) → (B2n2j , gB)(0 < j < n) be a proper slant
submersion of type 3. If N is a paracosymplectic manifold and E1, ..., E2n, ξis a local or-thonormal basis of (kerψ∗), at that time 1
coshωβE1, ...,1
coshωβE2n is a local orthonormalbasis of β(kerψ∗).
Proof. It will be enough to demonstrate that gN ( 1coshωβEi,
1coshωβEj) = ϵiδij , for any
i, j ∈ 1, ..., n, where ϵi = sgn(gN (E1, E1)) = ±1. By using (3.16), we get
gN ( 1coshω
βEi,1
coshωβEj) = ( 1
coshω)2 cosh2 ωgN (Ei, Ej) = ϵiδij ,
which proves the assertion. In a similar way, we get the following.
Lemma 3.16. Let ψ : (N4n+12n , gN , φ, η, ξ) → (B2n
2j , gB)(0 < j < n) be a proper slantsubmersion of type 3. If N is a paracosymplectic manifold and E1, ..., En, ξ are orthogonalunit vector fields in (kerψ∗), then
E1,1
sinhωαE1, E2,
1sinhω
αE2, ..., En,1
sinhωαEn, ξ
is a local orthonormal basis of (kerψ∗).
Let ψ be a proper slant submersion of type 3 from a paracosymplectic manifold(N4n+1, gN , φ, η, ξ) onto a semi-Riemannian manifold (B2n, gB). We call such an orthonor-mal frame
E1,1
sinhωαE1, E2,
1sinhω
αE2, ..., E2n,1
sinhωαEn,
1coshω
βE1, ...,1
coshωβE2n
an adapted slant frame for proper slant submersion of type 3.
Proposition 3.17. Let ψ : (N, gN , φ, η, ξ) → (B, gB) be a proper slant submersion of type1. If N is a paracosymplectic manifold and β is parallel with respect to ∇ on (kerψ∗), thenwe have
TαX1αX1 = cosh2 ωTX1X1 (3.23)for any spacelike (timelike) vector field X1 ∈ Γ(kerψ∗).
Proof. If β is parallel, at that time from Lemma 3.12 we get rTX1X2 = TX1αX2 for anyspacelike (timelike) vector fields X1, X2 ∈ Γ(kerψ∗). Interchanging the role of X1 and X2,we get rTX2X1 = TX2αX1. Thus we have
rTX1X2 − rTX2X1 = TX1αX2 − TX2αX1
Since T is symmetric, we derive TX1αX2 = TX2αX1. Then substituting X2 by αX1 we getTX1α
2X1 = TαX1αX1. Using (3.3) and (3.19) we obtain (3.23). In a similar way, we obtain the following.
Slant submersions in paracontact geometry 831
Corollary 3.18. Let ψ : (N, gN , φ, η, ξ) → (B, gB) be a proper slant submersion of type2. If N is a paracosymplectic manifold and β is parallel with respect to ∇ on (kerψ∗), thenwe have
TαX1αX1 = cos2 ωTX1X1 (3.24)for any spacelike (timelike) vector field X1 ∈ Γ(kerψ∗).
Corollary 3.19. Let ψ : (N, gN , φ, η, ξ) → (B, gB) be a proper slant submersion of type3. If N is a paracosymplectic manifold and β is parallel with respect to ∇ on (kerψ∗), thenwe have
TαX1αX1 = − sinh2 ωTX1X1 (3.25)for any spacelike (timelike) vector field X1 ∈ Γ(kerψ∗).
Theorem 3.20. Let ψ : (N4n+12n , gN , φ, η, ξ) → (B2n
2j , gB)(0 < j < n) be a proper slantsubmersion of type 3. If N is a paracosymplectic manifold and β is parallel with respect to∇ on (kerψ∗), at that time ψ is a harmonic map.
Proof. Using (2.9), we obtain(∇ψ∗)(X3, X4) = 0
for any spacelike (timelike) vector fields X3, X4 ∈ (kerψ∗)⊥. A proper slant submersion oftype 3 ψ is harmonic if and only if
∑2ni=1(∇ψ∗)(E∗
i , E∗i ) =
∑2ni=1(∇ψ∗)(TE∗
iE∗i ) = 0, here
E∗i 2ni=1 is an orthonormal basis of (kerψ∗). Hence, using Lemma 3.16 we should write
κ = −n∑i=1
ψ∗(TEiEi + T 1sinh ω
αEi
1sinhω
αEi) − Tξξ.
From (3.19), we have
κ = −n∑i=1
ψ∗(TEiEi + 1sinh2 φ
TαEiαEi).
Then using (3.25) we have
κ = −n∑i=1
ψ∗(TEiEi − TEiEi) = 0
which shows that ψ is harmonic. Putting θ = α2, we define ∇θ by
(∇X1θ)X2 = v∇X1θX2 − θ∇X1X2 (3.26)for any spacelike(timelike) vector fields X1, X2 ∈ Γ(kerψ∗).
Theorem 3.21. Let ψ : (N, gN , φ, η, ξ) → (B, gB) be a proper slant submersion of type 1.If N is a paracosymplectic manifold, then ∇θ = 0.
Proof. Using (3.3),we have
θ∇X1X2 = cosh2 ω(∇X1X2 − η(∇X1X2)ξ) (3.27)for all spacelike(timelike) vector fields X1, X2 ∈ Γ(kerψ∗). On the other hand, from (3.3)and (2.13) we obtain
v(∇X1θX2) = cosh2 ω(∇X1X2 − (∇X1η(X2))ξ)= cosh2 ω(∇X1X2 − η(∇X1X2) − gN (X2,∇X1ξ))= cosh2 ω(∇X1X2 − η(∇X1X2). (3.28)
Using (3.27) and (3.28), we get (∇X1θ)X2 = 0. Now, we examine the geometry of the leaves of the distributions (kerψ∗) and (kerψ∗)⊥.
832 Y. Gündüzalp
Theorem 3.22. Let ψ : (N, gN , φ, η, ξ) → (B, gB) be a proper slant submersion of type 1.If N is a paracosymplectic manifold, then the distribution (kerψ∗) defines a totally geodesicfoliation on N if and only if
gN (h∇X1βαX2, X3) = gN (h∇X1βX2, rX3) + gN (TX1βX2, tX3) (3.29)for spacelike (timelike) vector fields X1, X2 ∈ Γ(kerψ∗) and X3 ∈ Γ((kerψ∗)⊥).
Proof. For spacelike (timelike) vector fields X1, X2 ∈ Γ(kerψ∗) and X3 ∈ Γ((kerψ∗)⊥),since (2.10) and (2.12) we obtain
gN (∇X1X2, X3) = gN (φ∇X1φX2, X3) + gN (∇X1X2, ξ)η(X3)Using (3.1) and (3.2) we get
gN (∇X1X2, X3) = gN (∇X1α2X2, X3) + gN (∇X1βαX2, X3)
− gN (∇X1βX2, tX3) − gN (∇X1βX2, rX3).Then from (2.4),(2.13) and (3.3) we obtain
gN (∇X1X2, X3) = cosh2 ωgN (∇X1X2, X3) + gN (h∇X1βαX2, X3)− gN (TX1βX2, tX3) − gN (h∇X1βX2, rX3).
Hence we have− sinh2 ωgN (∇X1X2, X3) = gN (h∇X1βαX2, X3)
− gN (TX1βX2, tX3) − gN (h∇X1βX2, rX3)which proves assertion. Theorem 3.23. Let ψ : (N, gN , φ, η, ξ) → (B, gB) be a proper slant submersion of type1. If N is a paracosymplectic manifold, then the distribution (kerψ∗)⊥ defines a totallygeodesic foliation on N if and only if
gN (h∇X1X2, βαX3) = gN (AX1tX2 + h∇X1rX2, βX3) (3.30)for any spacelike (timelike)vector fields X3 ∈ Γ(kerψ∗) and X1, X2 ∈ Γ((kerψ∗)⊥).
Proof. For X3 ∈ Γ(kerψ∗) and X1, X2 ∈ Γ((kerψ∗)⊥), from (2.12) and (3.1) we obtaingN (∇X1X2, X3) = −gN (∇X1φX2, φX3) + gN (∇X1X2, ξ)η(X3)
= −gN (φ∇X1X2, αX3) − gN (∇X1φX2, βX3)+ gN (∇X1X2, ξ)η(X3). (3.31)
Using (3.1) in (3.31), we getgN (∇X1X2, X3) = gN (∇X1X2, α
2X3) + gN (∇X1X2, βαX3)− gN (∇X1φX2, βX3) + gN (∇X1X2, ξ)η(X3). (3.32)
Using (3.2) and (3.3) we getgN (∇X1X2, X3) = cosh2 ωgN (∇X1X2, X3) − cosh2 ωη(∇X1X2)η(X3)
+ gN (∇X1X2, βαX3) − gN (∇X1tX2, βX3)− gN (∇X1rX2, βX3) + gN (∇X1X2, ξ)η(X3). (3.33)
Using (2.5),(2.6) and (2.13) in (3.33), we get− sinh2 ωgN (∇X1X2, X3) = gN (h∇X1X2, βαX3) − gN (AX1tX2 + h∇X1rX2, βX3).
Thus, we have (3.30). Now, we show necessary and sufficient conditions for a proper slant submersion of
type 1 to be totally geodesic. Recall that a smooth map ψ between (semi-) Riemannianmanifolds (N, gN ) and (B, gB) is called a totally geodesic map if (∇ψ∗)(X1, X2) = 0 forall X1, X2 ∈ Γ(TN).
Slant submersions in paracontact geometry 833
Theorem 3.24. Let ψ : (N, gN , φ, η, ξ) → (B, gB) be a proper slant submersion of type 1.If N is a paracosymplectic manifold, at that time ψ is totally geodesic if and only if
gN (h∇X1βαX2, X3) = gN (h∇X1βX2, rX3) + gN (TX1βX2, tX3) (3.34)and
gN (h∇X4βαX1, X5) = −gN (AX4tX5 + h∇X4rX5, βX1) (3.35)for any spacelike (timelike) vector fields X3, X4, X5 ∈ Γ((kerψ∗)⊥) and X1, X2 ∈ Γ(kerψ∗).
Proof. First of all, since ψ is a semi-Riemannian submersion we get(∇F∗)(X4, X5) = 0
for sapacelike (timelike) vector fields X4, X4 ∈ Γ((kerψ∗)⊥).For sapacelike (timelike) vector fields X1, X2 ∈ Γ(kerψ∗) and X3, X4, X5 ∈ Γ((kerψ∗)⊥),from (2.10) and (2.12) we have
∇X1X2 = φ∇X1φX2 + η(∇X1X2)ξ. (3.36)Using (2.9),(3.1) and (3.36) we get
gB((∇ψ∗)(X1, X2), ψ∗X3) = −gN (∇X1φαX2, X3) + gN (∇X1βX2, φX3).Using (3.1) and (3.2) we get
gB((∇ψ∗)(X1, X2), ψ∗X3) = −gN (∇X1α2X2, X3) − gN (∇X1βαX2, X3)
+ gN (∇X1βX2, tX3) + gN (∇X1βX2, rX3).Using (2.3), (2.4) and (3.3) we have
gB((∇ψ∗)(X1, X2), ψ∗X3) = − cosh2 ωgN (∇X1X2, X3) − gN (h∇X1βαX2, X3)+ gN (TX1βX2, tX3) + gN (h∇X1βX2, rX3).
Hence we obtain− sinh2 ωgB((∇ψ∗)(X1, X2), ψ∗X3) = −gN (h∇X1βαX2, X3) + gN (TX1
βX2, tX3)+ gN (h∇X1βX2, rX3). (3.37)
Similarly, we get− sinh2 ωgB((∇ψ∗)(X1, X4), ψ∗X5) = −gN (AX4tX5 + h∇X4rX5, βX1)
− gN (h∇X4βαX1, X5). (3.38)Thus from (3.37) and (3.38), we get (3.34) and (3.35). Acknowledgment. The author is grateful to the referees for their valuable commentsand suggestions.
References[1] M.A. Akyol, Conformal semi-slant submersions, Int. J. Geom. Methods Mod. Phys.
14 (7), 2017.[2] M.A. Akyol, Conformal anti-invariant submersions from cosymplectic manifolds,
Hacet. J. Math. Stat. 46 (2), 177-192, 2017.[3] G. Baditoiu and S. Ianus, Semi-Riemannian submersions from real and complex
pseudo-hyperbolic spaces, Diff. Geom. and appl. 16, 79-84, 2002.[4] J.P. Bourguignon and H.B. Lawson, A mathematician’s visit to Kaluza- Klein theory,
Rend. Sem. Mat. Univ. Politec. Torino, Special Issue, 143-163, 1989.[5] J.P. Bourguignon and H.B. Lawson, Stability and isolation phenomena for Yang-Mills
fields, Comm. Math. Phys. 79, 189-230, 1981.[6] A.V. Caldarella, On para-quaternionic submersions between para-quaternionic Kähler
manifolds, Acta Applicandae Mathematicae 112, 1-14, 2010.
834 Y. Gündüzalp
[7] P. Dacko, On almost para-cosymplectic manifolds, Tsukuba J. Math. 28 (1), 193-213,2004.
[8] M. Falcitelli, S. Ianus and A. M. Pastore, Riemannian Submersions and Related Top-ics, World Scientific, 2004.
[9] Y. Gündüzalp and B. S. ahin, Paracontact semi-Riemannian submersions, Turk.J.Math. 37 (1), 114-128, 2013.
[10] Y. Gündüzalp, B. S. ahin, Para-contact para-complex semi-Riemannian submersions,Bull. Malays. Math. Sci. Soc. 37 (1), 139-152, 2014.
[11] Y. Gündüzalp, Anti-invariant semi-Riemannian submersions from almost para-Hermitian manifolds, J. Funct. Spaces 2013, ID 720623, 2013.
[12] Y. Gündüzalp, Slant submersions from almost product Riemannian manifolds,Turk. J. Math. 37, 863-873, 2013.
[13] Y. Gündüzalp and M.A. Akyol, Conformal slant submersions from cosymplectic man-ifolds, Turk. J. Math. 42 (5), 2672-2689, 2018.
[14] A. Gray, Pseudo-Riemannian almost product manifolds and submersions, J. Math.Mech. 16, 715-737, 1967.
[15] S. Ianus and M. Visinescu, Kaluza-Klein theory with scalar fields and generalised Hopfmanifolds, Classical Quantum Gravity 4, 1317-1352, 1987.
[16] S. Ianus, and M. Visinescu, Space-time compactification and Riemannian submer-sions, The mathematical heritage of C.F. Gauss, World Sci. Publ., River Edge, NJ,358-371, 1991.
[17] S. Ianus, R. Mazzocco and G. E. Vilcu, Riemannian submersions from quaternionicmanifolds, Acta Appl. Math. 104, 83-89, 2008.
[18] S. Ianus, G.E. Vilcu and R.C. Voicu, Harmonic maps and Riemannian submersionsbetween manifolds endowed with special structures, Banach Center Publications 93,277-288, 2011.
[19] I.K. Erken and C. Murathan, On slant Riemannian submersions for cosymplecticmanifolds, Bull. Korean Math. Soc. 51 (6), 1749-1771, 2014.
[20] B. O‘Neill, The fundamental equations of a submersion, Michigan Math. J. 13, 459-469, 1966.
[21] B. O‘Neill, Semi-Riemannian Geometry with Application to Relativity, AcademicPress, New York, 1983.
[22] K.S.Park, H-slant submersions, Bull. Korean Math. Soc. 49, 329-338, 2012.[23] B. S. ahin, Slant submersions from almost Hermitian manifolds, Bull. Math. Soc.Sci.
Math. Roumanie Tome. 54, 93-105, 2011.[24] B. S. ahin, Anti-invariant Riemannian submersions from almost Hermitian manifolds,
Central European J.Math 8 (3), 437-447, 2010.[25] H.M. Tas.tan , B. S. ahin and S. . Yanan , Hemi-slant submersions, Mediterr. J. Math.
13, 2171-2184, 2016.[26] J. Welyczko, On Legendre curves in 3-dimensional normal almost paracontact metric
manifolds, Results Math. 54, 377-387, 2009.[27] B. Watson, Almost Hermitian submersions, J. Differential Geom. 11, 147-165, 1976.[28] S. Zamkovoy, Canonical connections on paracontact manifolds, Ann. Global Anal.
Geometry 36, 37-60, 2009.
Hacettepe Journal ofMathematics & Statistics
Hacet. J. Math. Stat.Volume 49 (2) (2020), 835 – 842
DOI : 10.15672/hujms.522814
Research Article
Lorentz-Schatten classes of direct sum ofoperators
Pembe Ipek Al
Karadeniz Technical University, Department of Mathematics, 61080, Trabzon, Turkey
AbstractIn this paper, the relations between Lorentz-Schatten property of the direct sum of op-erators and Lorentz-Schatten property of its coordinate operators are studied. Then, theresults are supported by applications.
Mathematics Subject Classification (2010). 47A05, 47A10
Keywords. direct sum of Hilbert spaces and operators, compact operators,Lorentz-Schatten operator classes
1. IntroductionThe general theory of singular numbers and operator ideals was given by Pietsch [13,14]
and the case of linear compact operators was investigated by Gohberg and Krein [5].However, the first result in this area can be found in the works of Schmidt [16] andSchatten, von Neumann [15]. They used these concepts in the theory of non-selfadjointintegral equations.
Later on, the main aim of mini-workshop held in Oberwolfach (Germany) was to presentand discuss some modern applications of the functional-analytic concepts of s−numbersand operator ideals in areas like numerical analysis, theory of function spaces, signalprocessing, approximation theory, probability of Banach spaces and statistical learningtheory (see [3]).
Let H be a Hilbert space, S∞(H) be a class of linear compact operators in H and sn(T )be the n−th singular numbers of the operator T ∈ S∞(H). The Lorentz-Schatten operatorideals are defined as
Sp,q(H) =
T ∈ S∞(H) :∞∑
n=1n
qp
−1sq
n(T ) < ∞
, 0 < p ≤ ∞, 0 < q < ∞
and
Sp,∞(H) =
T ∈ S∞(H) : supn≥1
n1p sn(T ) < ∞
, 0 < p ≤ ∞
in [1, 13,14,17].Let α be a positive real number. If sn(T ) ∼ cn−α, c > 0, n → ∞ for any linear compact
operator T in a Hilbert space H, then for each p ∈(
1α , ∞
]and q ∈ (0, ∞), T ∈ Sp,q(H). In
Email address: [email protected]: 05.02.2019; Accepted: 10.04.2019
836 P. Ipek Al
this case, the necessary and sufficient condition for the series∞∑
n=1n
qp
−1−αq to be convergent
is p >1α
. Moreover, the necessary and sufficient condition for T ∈ Sp,∞(H) is p ∈[
1α , ∞
].
The infinite direct sum of Hilbert spaces and the infinite direct sum of operators havebeen studied in [4]. Namely, the infinite direct sum of Hilbert spaces Hn, n ≥ 1 and theinfinite direct sum of operators An in Hn, n ≥ 1 are defined as
H =∞⊕
n=1Hn =
u = (un) : un ∈ Hn, n ≥ 1,
∞∑n=1
∥un∥2Hn
< +∞
,
A =∞⊕
n=1An,
D(A) = u = (un) ∈ H : un ∈ D(An), n ≥ 1, Au = (Anun) ∈ H .
Recall that H is a Hilbert space with the norm induced by the inner product
(u, v)H =∞∑
n=1(un, vn)Hn , u, v ∈ H.
Our aim in this paper is to study the relations between Lorentz-Schatten property ofthe direct sum of operators and Lorentz-Schatten property of its coordinate operators.
It should be noted that the analogous problems in special cases have been investigatedin [8].
The problem of belonging to the Schatten-von Neuman classes of the resolvent oper-ators of the normal extensions of the minimal operator generated by the direct sum ofdifferential-operator expression for first order with suitable operator coefficients in the di-rect sum of Hilbert spaces in finite interval has been studied in [7].
In [6,9], the same problem for normal and hyponormal extensions of the minimal oper-ators generated by corresponding differential-operator expressions under some conditionsto operator coefficients in a finite interval has been investigated.
Later on, some more general Schatten-von Neumann classes of compact operators inHilbert spaces have been defined and characterized in [10] in terms of Berezin symbols.In [2], the question raised by Nordgren and Rosenthal about the Schatten-von Neumannclass membership of operators in standard reproducing kernel Hilbert spaces in terms oftheir Berezin symbols has been answered.
2. Lorentz-Schatten property of block diagonal operator matricesLet Hn be a Hilbert space, An ∈ L(Hn) for n ≥ 1 and
H =∞⊕
n=1Hn, A =
∞⊕n=1
An.
Recall that, in order to A ∈ L(H) the necessary and sufficient condition is supn≥1
∥An∥ < ∞.
Moreover, ∥A∥ = supn≥1
∥An∥ (see [11]).
It is known that if An ∈ S∞(Hn) for n ≥ 1, then the necessary and sufficient conditionfor A ∈ S∞(H) is lim
n→∞∥An∥ = 0 (see [12]).
The following result on singular numbers of the operator A ∈ S∞(H)
sm(A) : m ≥ 1 =∞∪
n=1sm(An) : m ≥ 1
can be found in [8].Throughout this paper, for the simplicity we assume that:
Lorentz-Schatten classes of direct sum of operators 837
(1) for any n, k ≥ 1 with n = k, sm(An) : m ≥ 1 ∩ sm(Ak) : m ≥ 1 = ∅ or 0;(2) for any n ≥ 1 in the sequence (sm(An)) , if for some k > 1, sk(An) > 0, then sk(An) <sk−1(An).
Proposition 2.1. For n ≥ 1 there is a strongly increasing sequence k(n)m : N → N such
that sk
(n)m
(A) = sm(An) holds for m ≥ 1 and∞∪
n=1
∞∪m=1
k
(n)m
= N. Moreover, it is clear
that k(n)m ≥ m for n, m ≥ 1.
Indeed, in the Hilbert space H =∞⊕
n=1Hn = l2(R), where Hn = (R, | . |), consider the
following infinite matrices with reel entries in forms
A =
a1a2
a3 0. . .
0 an
. . .
: H → H
and
B =
b1b2
b3 0. . .
0 bn
. . .
: H → H,
where for any n, m ≥ 1, n = m, an = am, an > 0 and bn = an + an+12
with propertylim
n→∞an = 0.
In this case, A, B ∈ S∞(H) and the singular numbers of the operators A, B are givenin the following forms
sm(An) : m ≥ 1 = an : n ≥ 1 ,
sm(Bn) : m ≥ 1 = bn : n ≥ 1 ,
respectively. Then, by [12] it implies that T = A⊕B ∈ S∞(H⊕H) and sm(T ) : m ≥ 1 =an, bn : n ≥ 1. In this case, it is easy to see that
k(1)m = 2m − 1, m ≥ 1,
k(2)m = 2m, m ≥ 1.
Theorem 2.2. Let 0 < p, q < ∞. A ∈ Sp,q(H) if and only if the series∞∑
n=1
∞∑m=1
(k(n)
m
) qp
−1sq
m(An)
is convergent.
Proof. If A ∈ Sp,q(H), it is clear that the series∞∑
m=1m
qp
−1sq
m(A)
838 P. Ipek Al
is convergent. From the structure of the set of the singular numbers of the operator Aand the important theorem on the convergent of the rearrangement series it is obtainedthat the series ∞∑
n=1
∞∑m=1
(k(n)
m
) qp
−1sq
m(An)
is convergent.
Conversely, if the series in the theorem is convergent, then∞∑
m=1m
qp
−1sq
m(A), which is
the rearrangement of the above series, is convergent. So, A ∈ Sp,q(H).
Now, in Theorem 2.3-2.5, we will investigate the problem of belonging to Lorentz-Schatten classes of its coordinate operators, if the direct sum of operators belongs toLorentz-Schatten classes.Theorem 2.3. Let A ∈ S∞(H) and 0 < p ≤ q < ∞. If A ∈ Sp,q(H), then An ∈ Sp,q(Hn)for n ≥ 1.
Proof. In the special case 0 < p = q < ∞, the result has been proved in [8].
In the case of p < q, we havem ≤ k(n)
m and sk
(n)m
(A) = sm(An)
for n, m ≥ 1. Consequently, for n ≥ 1 we get∞∑
m=1m
qp
−1sq
m(An) ≤∞∑
m=1
(k(n)
m
) qp
−1sq
m(An)
≤∞∑
n=1
∞∑m=1
(k(n)
m
) qp
−1sq
m(An)
=∞∑
m=1m
qp
−1sq
m(A) < ∞.
Hence, An ∈ Sp,q(Hn) for n ≥ 1.
Theorem 2.4. Let 0 < q < p < ∞ and for n ≥ 1, supm≥1
(k
(n)m
m
)≤ γ < ∞. If A ∈ Sp,q(H),
then An ∈ Sp,q(Hn) for n ≥ 1.
Proof. Under the assumptions in the theorem, we have∞∑
m=1m
qp
−1sq
m(An) =∞∑
m=1
(m
k(n)m
) qp
−1 (k(n)
m
) qp
−1sq
m(An)
≤ supm≥1
(k
(n)m
m
)1− qp ∞∑
m=1
(k(n)
m
) qp
−1sq
m(An)
≤ γ1− q
p
∞∑n=1
∞∑m=1
(k(n)
m
) qp
−1sq
m(An)
= γ1− q
p
∞∑j=1
jqp
−1sq
j(A) < ∞.
Therefore, An ∈ Sp,q(Hn) for n ≥ 1. Now, we will investigate the case of q = ∞.
Lorentz-Schatten classes of direct sum of operators 839
Theorem 2.5. Let 0 < p ≤ ∞. If A ∈ Sp,∞(H), then An ∈ Sp,∞(Hn) for n ≥ 1.
Proof. Since A ∈ Sp,∞(H), we have supm≥1
m1p sm(A) < ∞. Hence, sup
m≥1
(k
(n)m
) 1p sm(An) <
∞. On the other hand, we get
supm≥1
m1p sm(An) = sup
m≥1
(k(n)
m
) 1p sm(An)
(m
k(n)m
) 1p
≤ supm≥1
(k(n)
m
) 1p sm(An) < ∞.
Then, An ∈ Sp,∞(Hn) for n ≥ 1.
Now, in Theorem 2.6-2.8, we will investigate the problem of belonging to Lorentz-Schatten classes of the direct sum of operators, if its coordinate operators belong toLorentz-Schatten classes.
Theorem 2.6. Let 0 < q ≤ p < ∞. If An ∈ Sp,q(Hn) for n ≥ 1 and the series∞∑
n=1
∞∑m=1
mqp
−1sq
m(An) is convergent, then A ∈ Sp,q(H).
Proof. For 0 < q ≤ p < ∞, we have∞∑
m=1m
qp
−1sq
m(A) =∞∑
n=1
∞∑m=1
(k(n)
m
) qp
−1sq
k(n)m
(A)
=∞∑
n=1
∞∑m=1
(k
(n)m
m
) qp
−1
mqp
−1sq
m(An)
≤∞∑
n=1
∞∑m=1
mqp
−1sq
m(An) < ∞.
This completes the proof.
Theorem 2.7. Let 0 < p < q < ∞, for n ≥ 1∞∑
m=1m
qp
−1sq
m(An) ≤ βn < ∞, supm≥1
(k
(n)m
m
)≤
γn < ∞ and∞∑
n=1γ
qp
−1n βn < ∞. If An ∈ Sp,q(Hn) for n ≥ 1, then A ∈ Sp,q(H).
Proof. The validity of this claim is clear from the following inequality∞∑
m=1m
qp
−1sq
m(A) =∞∑
n=1
∞∑m=1
(k(n)
m
) qp
−1sq
k(n)m
(A)
=∞∑
n=1
∞∑m=1
(k
(n)m
m
) qp
−1
mqp
−1sq
m(An)
≤∞∑
n=1
(supm≥1
(k
(n)m
m
)) qp
−1 ∞∑m=1
mqp
−1sq
m(An)
≤∞∑
n=1γ
qp
−1n βn.
Now, we will investigate in the case of q = ∞.
840 P. Ipek Al
Theorem 2.8. Let 0 < p < ∞, for n ≥ 1 αn = supm≥1
(k
(n)m
m
) 1p
< ∞, γn = supm≥1
m1p sm(An)
and supn≥1
αnγn < ∞. If An ∈ Sp,∞(Hn) for n ≥ 1, then A ∈ Sp,∞(H).
Proof. This result is clear from the following relation
supm≥1
m1p sm(A) = sup
n,m≥1
(k(n)
m
) 1p s
k(n)m
(A)
= supn,m≥1
(k(n)
m
) 1p sm(An)
≤ supn≥1
supm≥1
(k
(n)m
m
) 1p
supm≥1
m1p sm(An)
= sup
n≥1αnγn < ∞.
Theorem 2.9. Let 0 < pn, qn < ∞, An ∈ Spn,qn(Hn) for n ≥ 1 and p = sup
n≥1pn < ∞, q =
supn≥1
qn < ∞. Then, A ∈ Sp,q(H) if and only if the series∞∑
n=1
∞∑m=1
(k
(n)m
) qp
−1sq
m(An) is
convergent.
Proof. From the result in [1], we have An ∈ Sp,q(Hn) for n ≥ 1. Therefore, the validityof this claim is implied by Theorem 2.2. Remark 2.10. Using this method, the analogous researches for the following operators
B =
0 B10 B2
0 B3 0. . . . . .
0 0 Bn
. . . . . .
: H =
∞⊕n=1
Hn → H
and
C =
0C1 0
C2 0 0. . . . . .
0 Cn 0. . . . . .
: H =
∞⊕n=1
Hn → H
can be studied.
Lorentz-Schatten classes of direct sum of operators 841
3. ExamplesIn this section, we provide some examples as applications of our theorems.
Example 3.1. In the Hilbert space H =∞⊕
n=1Hn = l2(C), where Hn := (C, | . |), n ≥ 1,
consider the following diagonal infinite matrix with complex entries
A =
a1a2
a3 0. . .
0 an
. . .
: H → H
under the condition |an| < r < 1, n ≥ 1. Then, limn→∞
an = 0. In this case, A ∈ S∞(H). Ifwe define An := an for n ≥ 1, then sm(An) = |λ(An)| = |an|, 0 , m ≥ 1.Hence, the singular numbers of the operator A are given as
sm(A) : m ≥ 1 = |an| : n ≥ 1 .
On the other hand, for n ≥ 1 and 0 < q ≤ p < ∞ we get∞∑
m=1m
qp
−1sq
m(An) = |an|q.
Then, An ∈ Sp,q(Hn), n ≥ 1, 0 < q ≤ p < ∞. Therefore, we have∞∑
n=1
∞∑m=1
mqp
−1sq
m(An) =∞∑
n=1|an|q < ∞.
Hence, by Theorem 2.6, A ∈ Sp,q(H).
Example 3.2. Let Hn := (C2, | . |2), H :=∞⊕
n=1Hn = l2(C2), An =
(0 α2n−1
α2n 0
)for
n ≥ 1, 0 < |α| < 1 and A =∞⊕
n=1An : H → H. Then A ∈ S∞(H) (see [12]).
In this case, for n ≥ 1 we get∥An∥ = |α|2n−1,
sm(An) : m ≥ 1 = |α|2n−1, |α|2nand
sm(A) : m ≥ 1 = |α|n : n ≥ 1.
On the other hand, for n ≥ 1 and 0 < q ≤ p < ∞ we obtain∞∑
m=1m
qp
−1sq
m(An) = |α|(2n−1)q + 2qp
−1|α|2nq < ∞.
Hence, An ∈ Sp,q(Hn), n ≥ 1, 0 < q ≤ p < ∞. Therefore, we have∞∑
n=1
∞∑m=1
mqp
−1sq
m(An) =∞∑
n=1
(|α|(2n−1)q + 2
qp
−1|α|2nq)
= |α|q
1 − |α|2q
(1 + 2
qp
−1|α|q)
< ∞.
Hence, by Theorem 2.6, A ∈ Sp,q(H).
842 P. Ipek Al
Acknowledgment. The author would like to thank Professor Z. I. Ismailov (KaradenizTechnical University, Department of Mathematics, Turkey) for his various comments andsuggestions.
References[1] M.Sh. Birman and M.Z. Solomyak, Estimates of singular numbers of integral opera-
tors, Russian Math. Survey 32 (1), 15-89, 1977 (Translated from Uspekhi Mat. Nauk32 (1), 17-84, 1977).
[2] I. Chalendar, E. Fricain, M. Gürdal and M. T. Karaev, Compactness and Berezinsymbols, Acta Sci. Math. (Szeged) 78 (1), 315-329, 2012.
[3] F. Cobos, D.D. Haroske, T. Kühn and T. Ullrich, Mini-workshop: modern applicationsof s-numbers and operator ideals, Mathematisches Forschungs Institute Oberwolfach,Germany, 369-397, 8-14 February 2015.
[4] N. Dunford and J.T. Schwartz, Linear Operators I, Interscience Publishers, 1958.[5] I.C. Gohberg and M.G. Krein, Introduction to the Theory of Linear Non-Selfadjoint
Operators in Hilbert Space, American Mathematical Society, 1969.[6] Z.I. Ismailov, Compact inverses of first-order normal differential operators, J. Math.
Anal. Appl. 320, 266-278, 2006.[7] Z.I. Ismailov, Multipoint normal differential operators for first order, Opuscula Math.
29, 399-414, 2009.[8] Z.I. Ismailov, E. Otkun Çevik and E. Unluyol, Compact inverses of multipoint normal
differential operators for first order, Electron. J. Differential Equations 89, 1-11, 2011.[9] Z.I. Ismailov and E. Unluyol, Hyponormal differential operators with discrete spec-
trum, Opuscula Math. 30, 79-94, 2010.[10] M.T. Karaev, M. Gürdal and U. Yamancı, Special operator classes and their proper-
ties, Banach J. Math. Anal. 7 (2), 74-88, 2013.[11] M.A. Naimark and S.V. Fomin, Continuous direct sums of Hilbert spaces and some
of their applications, Uspehi Mat. Nauk 10, 111-142, 1955, (in Russian).[12] E. Otkun Çevik and Z.I. Ismailov, Spectrum of the direct sum of operators, Electron.
J. Differential Equations 210, 1-8, 2012.[13] A. Pietsch, Operators Ideals, North-Holland Publishing Company, 1980.[14] A. Pietsch, Eigenvalues and s−Numbers, Cambridge University Press, 1987.[15] R. Schatten and J. von Neumann, The cross-space of linear transformations, Ann. of
Math. 47, 608-630, 1946.[16] E. Schmidt, Zur theorie der linearen und nichtlinearen integralgleichungen, Math.
Ann. 64, 433-476, 1907.[17] H. Triebel, Über die verteilung der approximationszahlen kompakter operatoren in
Sobolev-Besov-Raumen, Invent. Math. 4, 275-293, 1967.
Hacettepe Journal ofMathematics & Statistics
Hacet. J. Math. Stat.Volume 49 (2) (2020), 843 – 853
DOI : 10.15672/hujms.479184
Research Article
On submanifolds of Kenmotsu manifold withTorqued vector field
Halil İbrahim Yoldaş∗, Şemsi Eken Meriç, Erol YaşarDepartment of Mathematics, Faculty of Science and Arts, Mersin University, 33343 Mersin, Turkey
AbstractIn this paper, we consider the submanifold M of a Kenmotsu manifold M endowed withtorqued vector field T. Also, we study the submanifold M admitting a Ricci soliton ofboth Kenmotsu manifold M and Kenmotsu space form M(c). Indeed, we provide somenecessary conditions for which such a submanifold M is an η−Einstein. We have presentedsome related results and classified. Finally, we obtain an important characterization whichclassifies the submanifold M admitting a Ricci soliton of Kenmotsu space form M(c).
Mathematics Subject Classification (2010). 53C25, 53C40.
Keywords. Kenmotsu manifold, Ricci soliton, Torqued vector field.
1. Introduction
Hamilton introduced the concept of Ricci soliton, which is a natural generalizationof Einstein manifold, in 1982 [11]. This notion actually corresponds to the self-similar
solution of Hamilton’s Ricci flow: ∂
∂tg = −2S, viewed as a dynamical system on the space
of Riemannian metrics modulo diffeomorphims and scaling, (for details, see [12]).In the framework of the contact geometry, Sharma started the studying of the problem
of the Ricci solitons in K-contact manifolds in [18]. After this work, Ricci solitons havebeen investiaged in some different classes of contact geometry. For instance, it is provedby Ghosh that the constant curvature of a Kenmotsu 3−manifold as Ricci soliton is −1 in[10]. Then, Perktaş and Keleş proved that if a 3−dimensional normal almost paracontactmetric manifold admits a Ricci soliton then it is shrinking in [17]. For more details, see([1, 2, 8, 9, 16,19,21]).
Consider the following equation on a Riemannian manifold (M, g)(£V g)(X, Y ) + 2S(X, Y ) + 2λg(X, Y ) = 0, (1.1)
where £V g is the Lie-derivative of the metric tensor g in the direction vector field V , S isthe Ricci tensor of M and λ is a constant. (M, g) is called a Ricci soliton if the equation(1.1) holds for vector fields X, Y on M . The vector field V is called the potential field ofRicci soliton (M, g). If £V g = ρg, then potential field V is said to be conformal Killing,∗Corresponding Author.Email addresses: [email protected](H. İ. Yoldaş), [email protected](Ş. E. Meriç),
[email protected](E. Yaşar)Received: 06.11.2018; Accepted: 17.04.2019
844 H. İ. Yoldaş, Ş. E. Meriç, E. Yaşar
where ρ is a function. If ρ vanishes identically, then V is said to be Killing. Also, if V iszero or Killing in (1.1), then the Ricci soliton is called trivial and in this case, the metricis an Einstein. In addition, a Ricci soliton is called a gradient if the potential field V isthe gradient of a potential function −f (i.e., V = −∇f) and is called shrinking, steady orexpanding depending on λ < 0, λ = 0 or λ > 0, respectively.
On the other hand, Riemannian manifolds which admit torqued vector fields (as acombination of concurrent and recurrent vector fields) were first defined by Chen in [6].According to this definition, a nowhere zero vector field T on a Riemannian manifold(M, g) is called torqued vector field, if it satisfies the following two conditions
∇XT = fX + α(X)T and α(T) = 0, (1.2)
where ∇ is the Levi-Civita connection on M , for any X ∈ Γ(TM). The function f iscalled the torqued function and 1−form α is called the torqued form of T. Here, Chencharacterized rectifying submanifolds for a Riemannian manifold endowed with torquedvector field in [6]. Then, Chen proved that every Ricci soliton with torqued potential fieldis an almost quasi-Einstein under some conditions (see [7]).
The paper is organized as follows:
In Section 2, we recall some basic notions which are going to be needed.
In Section 3, we consider the submanifold M of Kenmotsu manifold M endowed with atorqued vector field T and find that the characteristic vector field ξ of M is never torquedon the ambient space M . Also, we give a necessary and sufficient condition for which thetangential part T⊤ of T is torse-forming on M .
In Section 4, we deal with Kenmotsu space form M(c) endowed with a torqued vectorfield T and give some characterizations on a submanifold admitting a Ricci soliton of M(c).
The last section is devoted to conclusion. Here, we present our results which are obtainedin this paper.
2. Preliminaries
In this section, we shall review some basic definitions and formulas of almost contactmetric manifolds from [3,4, 15,20] and [22].
Let M be an (2n + 1)−dimensional almost contact metric manifold with an almostcontact metric structure (φ, ξ, η, g) such that φ is a tensor field of type (1, 1), ξ is a vectorfield (called the characteristic vector field) of type (0, 1), 1− form η is a tensor field oftype (1, 0) on M and the Riemannian metric g satisfies the following relations:
φ2X = −X + η(X)ξ, η(ξ) = 1, φξ = 0, η φ = 0, η(X) = g(X, ξ) (2.1)
and
g(φX, φY ) = g(X, Y ) − η(X)η(Y ), g(φX, Y ) = −g(X, φY ) (2.2)
for any X, Y ∈ Γ(TM).
If the following condition is satisfied for an almost contact metric manifold (M, φ, ξ, η, g),then it is called a Kenmotsu manifold
(∇Xφ)Y = g(φX, Y )ξ − η(Y )φX, (2.3)
where ∇ is the Levi-Civita connection on M , for any X, Y ∈ Γ(TM). From (2.3), for aKenmotsu manifold we also have
∇Xξ = X − η(X)ξ. (2.4)
On submanifolds of Kenmotsu manifold with Torqued vector field 845
On the other hand, a Kenmotsu manifold M with constant φ−sectional curvature c issaid to be a Kenmotsu space form and it is denoted by M(c). The curvature tensor R ofa Kenmotsu space form is given by
R(X, Y )Z = c − 34
g(Y, Z)X − g(X, Z)Y
+c + 1
4
[η(X)Y − η(Y )X
]η(Z)
+[g(X, Z)η(Y ) − g(Y, Z)η(X)
]ξ
+g(φY, Z)φX − g(φX, Z)φY − 2g(φX, Y )φZ
(2.5)
for any X, Y, Z ∈ Γ(TM).Let M be isometrically immersed submanifold of Kenmotsu manifold M . For any
X, Y ∈ Γ(TM), we have∇XY = ∇XY + h(X, Y ), (2.6)
where ∇ and ∇ stand for the Levi-Civita connections on M and M , respectively. Then,the equality (2.6) is called the Gauss formula and h is called the second fundamentalform of M . Also, if the second fundamental form h vanishes identically in (2.6), then thesubmanifold M is called totally geodesic. Similarly, one has
∇U V = −AV U + ∇⊥U V, (2.7)
where AV and ∇⊥ denote the shape operator and the normal connection of M in theambient space M , respectively, for any U ∈ Γ(TM) and V ∈ Γ(TM⊥). Using (2.4) and(2.6), it follows that
∇Xξ = X − η(X)ξ, (2.8)h(X, ξ) = 0, (2.9)
where ∇ is the Levi-Civita connection of M .Also, it is well known that the relation between second fundamental form h and the
shape operator AV are related byg(AV X, Y ) = g(h(X, Y ), V ) (2.10)
for any X, Y ∈ Γ(TM). Here, we denote by the same symbol g the Riemannian metricinduced by g on M .
The equation of Gauss is given byg(R(X, Y )Z, W ) = g(R(X, Y )Z, W ) + g(h(X, W ), h(Y, Z))
−g(h(X, Z), h(Y, W )) (2.11)for any X, Y, Z, W ∈ Γ(TM).
We denote by H the mean curvature vector, that is,
H(p) = 1n
n∑i=1
h(ei, ei),
where e1, e2, ..., en = ξ is an orthonormal basis of the tangent space TpM , p ∈ M . As itis known, M is called minimal if H vanishes identically.
The submanifold M is ω−umbilical with respect to a normal vector field ω if its shapeoperator satisfies Aω = µI, where µ is a function on M and I is the identity map.
Furthermore, the submanifold M is said to be totally umbilical if and only if one hash(X, Y ) = g(X, Y )H (2.12)
846 H. İ. Yoldaş, Ş. E. Meriç, E. Yaşar
for any X, Y ∈ Γ(TM), where h and H denote the second fundamental form and the meancurvature vector, respectively.
The scalar curvature r of (M, g) is defined by
r =n∑
i=1S(ei, ei),
where e1, e2, ..., en = ξ is an orthonormal frame of TM and S is the Ricci tensor of M .
Now, we recall some definitions from ([7, 14,22]), as follows:
A Riemannian manifold (M, g) is called η−Einstein if there exists two real constants aand b such that the Ricci curvature tensor field S satisfies
S(X, Y ) = ag(X, Y ) + bη(X)η(Y )
for any X, Y ∈ Γ(TM). If the constant b is equal to zero, then M becomes an Einstein.
The Ricci tensor S of a Kenmotsu manifold (M, g) is called η−parallel if it satisfies
(∇X S)(φY, φZ) = 0
such that
(∇X S)(φY, φZ) = ∇X S(φY, φZ) − S(∇XφY, φZ) − S(φY, ∇XφZ)
for any X, Y, Z ∈ Γ(TM).
A vector field v on a Riemannian manifold (M, g) is called torse-forming if it satisfies
∇Xv = fX + α(X)v, (2.13)
where f is a function, α is a 1−form and ∇ is the Levi-Civita connection on M , for anyX ∈ Γ(TM). The 1−form α is called the generating form and the function f is called theconformal scalar of v.
If the 1−form α in (2.13) vanishes identically, then the vector field v is called concircular[5]. If f = 1 and α = 0, then the vector field v is called concurrent [23]. The vector fieldv is called recurrent if it satisfies (2.13) with f = 0. Also, if f = α = 0, the vector field vin (2.13) is called parallel.
Let M be a Kenmotsu manifold endowed with a torqued vector field T and ϕ : M → Mbe an isometric immersion. Then, we get
T = T⊤ + T⊥, (2.14)
where T⊤ and T⊥ the tangential and normal components of T on M , respectively.
3. The submanifolds admitting Ricci soliton of Kenmotsu manifoldsIn this section, we deal with the submanifold M of Kenmotsu manifold M endowed
with torqued vector field T.
From now on, we make the following:
Assumption. Throughout the paper, we suppose that the characteristic vector field ξ istangent to M .
Theorem 3.1. Let M be a Kenmotsu manifold endowed with a torqued vector field T.Then, the characteristic vector field ξ is never torqued vector field on M .
On submanifolds of Kenmotsu manifold with Torqued vector field 847
Proof. Since T is a torqued vector field on M , then we have∇XT = fX + α(X)T and α(T) = 0, (3.1)
where ∇ stands for the Levi-Civita connection on M , for any X ∈ Γ(TM).Suppose that ξ is a torqued vector field on M . Using ξ instead of T in equation (3.1),
one has∇Xξ = fX + α(X)ξ and α(ξ) = 0. (3.2)
Also, taking the inner product of (3.2) with ξ, we haveα(X) = −fη(X).
Therefore, the equation (3.2) reduces to∇Xξ = f(X − η(X)ξ). (3.3)
It follows from (2.8) and (3.3),f = 1 and α(X) = −η(X) (3.4)
are found.On the other hand, if we take the characteristic vector field X = ξ in (3.4), then we
findα(ξ) = −1 (3.5)
which is a contradiction. Hence, ξ is never torqued vector field on Kenmotsu manifoldM .
The next example supports Theorem 3.1, as follows:
Example 3.2. ([13]). We consider the three-dimensional Riemannian manifoldM = (x, y, z) ∈ R3, (x, y, z) = (0, 0, 0),
and the linearly independent vector fields
e1 = z∂
∂x, e2 = z
∂
∂y, e3 = −z
∂
∂z,
where (x, y, z) are the Cartesian coordinates in R3. Let g be the Riemannian metricdefined by
g(ei, ei) = 1g(ei, ej) = 0 for i = j.
and is given by
g = 1z2
dx ⊗ dx + dy ⊗ dy + dz ⊗ dz
.
Also, let η, φ be the 1− form and the (1, 1)−tensor field, respectively defined byη(Z, e3) = 1, φ(e1) = −e2, φ(e2) = e1, φ(e3) = 0
for any Z ∈ Γ(TM). Hence, (M, φ, ξ, η, g) becomes an almost contact metric manifoldwith the characteristic vector field e3 = ξ.
By direct calculations, we have[e1, e2] = 0, [e1, e3] = e1 and [e2, e3] = e2.
On the other hand, using Koszul’s formula for the Riemannian metric g, we have:∇e1e3 = e1, ∇e2e3 = e2, ∇e3e3 = 0 (3.6)
848 H. İ. Yoldaş, Ş. E. Meriç, E. Yaşar
and others∇e1e2 = ∇e2e1 = ∇e3e1 = ∇e3e2 = 0, ∇e1e1 = ∇e2e2 = −e3. (3.7)
Therefore, the manifold M is a 3−dimensional Kenmotsu manifold. Now, we suppose thate3 = ξ is a torqued vector field on M . Then,
∇e1ξ = fe1 + α(e1)ξ and α(ξ) = 0 (3.8)∇e2ξ = fe2 + α(e2)ξ and α(ξ) = 0 (3.9)∇e3ξ = fe3 + α(e3)ξ and α(ξ) = 0. (3.10)
are satisfied. From (3.6), (3.8), (3.9) and (3.10), we getf = 1 and α(e3) = α(ξ) = −1 = 0, (3.11)
which is a contradiction. Therefore, e3 = ξ is never torqued vector field on Kenmotsumanifold M .
Considering Theorem 3.1, we get the following:
Remark 3.3. Let M be a submanifold endowed with a torqued vector field T of a Ken-motsu manifold M . Then, the characteristic vector field ξ is never torqued on M .
Next, we have the following theorem.
Theorem 3.4. Let M be a submanifold of a Kenmotsu manifold M endowed with atorqued vector field T. The submanifold M is totally geodesic if and only if the tangentialcomponent T⊤ of T is a torse-forming vector field on M whose conformal scalar is therestriction of the torqued function and whose generating form is the restriction of thetorqued function of T on M .
Proof. Since T is a torqued vector field on the ambient space M , it follows from (1.2),(2.14) and the formulas of Gauss and Weingarten, one has
∇XT⊤ + h(X,T⊤) − AT⊥X + ∇⊥XT⊥ = fX + α(X)T⊤ + α(X)T⊥, (3.12)
where ∇ stands for the Levi-Civita connection on M , for any X ∈ Γ(TM). By comparingthe tangential and normal components of (3.12), we get
h(X,T⊤) + ∇⊥XT⊥ = α(X)T⊥,
∇XT⊤ − AT⊥X = fX + α(X)T⊤. (3.13)If M is a totally geodesic submanifold of M , then the equation (3.13) becomes
∇XT⊤ = fX + α(X)T⊤, (3.14)which implies that T⊤ is a torse-forming on M . The proof of the converse part is straight-forward.
Considering the equality (3.13), we have the following cases:From now on, we suppose that the submanifold M admits a Ricci soliton in Theorem 3.4.
Case I: If we take T⊤ ∈ Γ(D), then from (2.4), (2.9), (2.10) and (3.13) we get
g(∇XT⊤, ξ) = g(fX, ξ), (3.15)where TM = D ⊕ Spanξ, for any X ∈ Γ(TM). Since the Riemannian metric g isnon-degenere, we have
∇XT⊤ = fX, (3.16)which shows that the vector field T⊤ is a concircular on M .
On submanifolds of Kenmotsu manifold with Torqued vector field 849
On the other hand, from the definition of Lie-derivative and (3.16) one has(£T⊤g)(X, Y ) = g(∇XT⊤, Y ) + g(∇Y T
⊤, X)= 2fg(X, Y ) (3.17)
for any X, Y ∈ Γ(TM), which means that the vector field T⊤ is a conformal Killing. Also,from (1.1) and (3.17), we obtain
S(X, Y ) = −(λ + f)g(X, Y ),where S is the Ricci tensor of M . Hence, M is an Einstein.
Case II: If we take T⊤ ∈ Γ(D), then it follows from (3.15), we haveg(∇XT⊤, ξ) = 0 (3.18)
for any X ∈ Γ(D). As a consequence of the equation (3.18), T⊤ is a parallel vector fieldon distribution D and thus, T⊤ is a D−Killing vector field.
On the other side, using (1.1) and (3.18) the Ricci tensor SD of the distribution D
SD(X, Y ) = −λg(X, Y )is found. Therefore, the distribution D is an Einstein.
Case III: If we use ξ instead of T⊤ in (3.14), we have∇Xξ = fX + α(X)ξ (3.19)
for any X ∈ Γ(TM). Taking the inner product of (3.19) with ξ, we getg(∇Xξ, ξ) = fη(X) + α(X)
which yieldsα(X) = −fη(X).
It is easy to see that α(ξ) = 0. So, ξ is a torse-forming on M .Using the equality (3.13), we have the following:
Corollary 3.5. Let M be a submanifold of a Kenmotsu manifold M endowed with atorqued vector field T. If M is T⊥−umbilical, then T⊤ is a torse-forming on M .
The next theorem gives a characterization as follows:
Theorem 3.6. Let M be a Kenmotsu manifold endowed with a torqued vector field T andM be a submanifold admitting a Ricci soliton of M . Then, (M, g, ξ, λ) is an η−Einstein.
Proof. If we take ξ instead of T⊤ in (3.13), we have∇Xξ − AT⊥X = fX + α(X)ξ. (3.20)
From the equalities (2.4), (2.6) and (3.20), we getAT⊥X = (1 − f)X − (η(X) + α(X))ξ. (3.21)
Also, if we use the relations (2.1), (2.10) and (3.21), one hasg(h(X, Y ),T⊥) = (1 − f)g(X, Y ) − (η(X) + α(X))η(Y ). (3.22)
Interchanging the roles of X and Y in (3.22) givesg(h(Y, X),T⊥) = (1 − f)g(Y, X) − (η(Y ) + α(Y ))η(X). (3.23)
Since h and g are symmetric, from (3.22) and (3.23) we have2g(h(X, Y ),T⊥) = 2(1 − f)g(X, Y ) − 2η(X)η(Y )
−α(X)η(Y ) − α(Y )η(X) (3.24)for any X, Y ∈ Γ(TM).
850 H. İ. Yoldaş, Ş. E. Meriç, E. Yaşar
On the other hand, from the definition of Lie-derivative and (2.1), (2.10), (3.20) and(3.24), we obtain
(£ξg)(X, Y ) = g(∇Xξ, Y ) + g(∇Y ξ, X)= g(fX + α(X)ξ + AT⊥X, Y )
+ g(fY + α(Y )ξ + AT⊥Y, X)= 2g(X, Y ) − 2η(X)η(Y ). (3.25)
Since M is a submanifold admitting a Ricci soliton and from the equalities (1.1) and(3.25), the Ricci tensor S of M
S(X, Y ) = −(λ + 1)g(X, Y ) + η(X)η(Y ) (3.26)
is satisfied. This means M is an η-Einstein.
As a consequence of Theorem 3.6, we can state the followings:
Corollary 3.7. Let M be a Kenmotsu manifold endowed with a torqued vector field T andM be a submanifold admitting a Ricci soliton as its potential field ξ of M . Then, M hasη−parallel Ricci tensor.
Corollary 3.8. Let M be a Kenmotsu manifold endowed with a torqued vector field T andM be a n−dimensional submanifold admitting a Ricci soliton as its potential field ξ of M .Then, M has constant scalar curvature r given by
r = 1 − n(λ + 1).
4. Ricci solitons in Kenmotsu space form with torqued vector fieldIn this section, we investigate the submanifolds admitting a Ricci soliton of Kenmotsu
space form M(c) endowed with torqued vector field T.
Now, we are ready to give the next theorem as follows:
Theorem 4.1. Let M(c) be a Kenmotsu space form and M be a n−dimensional subman-ifold of M(c). If M is totally umbilical and the mean curvature ∥H∥ is constant, then Mis η−Einstein.
Proof. Let e1, e2, ..., en−1, ξ be an orthonormal basis of TpM , p ∈ M . From the defini-tion of the Ricci tensor, we have
S(Y, Z) =n−1∑i=1
g(R(ei, Y )Z, ei) + g(R(ξ, Y )Z, ξ), (4.1)
where R is the Riemann curvature tensor of the submanifold M .
On submanifolds of Kenmotsu manifold with Torqued vector field 851
If we put X = W = ei in (2.11) and use the equalities (2.1), (2.2), (2.5), (2.9) and(2.12), then one has
n−1∑i=1
g(R(ei, Y )Z, , ei) =n−1∑i=1
g(R(ei, Y )Z, ei) − g(h(ei, ei), h(Y, Z))
+g(h(ei, Z), h(Y, ei))
=n−1∑i=1
c − 34
g(Y, Z)g(ei, ei) − g(ei, Z)g(Y, ei)
+c + 1
4
3g(ei, φY )g(φZ, ei) − η(Y )η(Z)g(ei, ei)
+
n−1∑i=1
(g(ei, Z)g(Y, ei) − (g(ei, ei)g(Y, Z)
)∥H∥2
= c − 34
(n − 2)g(Y, Z) + η(Y )η(Z)
+c + 1
4
3g(Y, Z) − (n + 2)η(Y )η(Z)
+
((n − 2)g(Y, Z)) + η(Y )η(Z)
)∥H∥2 . (4.2)
Similarly, taking X = W = ξ in (2.11), we getg(R(ξ, Y )Z, ξ) = g(R(ξ, Y )Z, ξ) = η(Y )η(Z) − g(Y, Z) (4.3)
for any Y, Z ∈ Γ(TM). Then, using (4.2) and (4.3) in (4.1), the Ricci tensor S of M
S(Y, Z) =(c(n + 1) − 3n + 5
4+ (n − 2)∥H∥2
)g(Y, Z)
−(c(n + 1) + n + 1
4− ∥H∥2
)η(Y )η(Z) (4.4)
is obtained which means that M is an η−Einstein. This completes the proof. Theorem 4.2. Let M(c) be a Kenmotsu space form endowed with a torqued vector fieldT and M be an n−dimensional (n > 1) totally umbilical submanifold admitting a Riccisoliton of M . Then, M has a constant mean curvature.
Proof. If we put Y = Z = ξ in (3.26) and using (2.1) and (2.2), we getS(ξ, ξ) = −λ. (4.5)
Similarly, if we take Y = Z = ξ in (4.4) and also using (2.1) and (2.2), then we haveS(ξ, ξ) = (1 − n)(1 − ∥H∥2 ). (4.6)
Since M is a Ricci soliton, from the equalities (4.5) and (4.6),
∥H∥2 = 1 − λ
n − 1(4.7)
is obtained which completes the proof of the theorem.
Using the equality (4.7), we can state the following corollary:
Corollary 4.3. Let M(c) be a Kenmotsu space form endowed with a torqued vector fieldT and M be an n−dimensional (n > 1) totally umbilical submanifold admitting a Riccisoliton of M . Then, we have the following:i) If ∥H∥ < 1, then the Ricci soliton (M, g, ξ, λ) is expanding.ii) If ∥H∥ > 1, then the Ricci soliton (M, g, ξ, λ) is shrinking.iii) The Ricci soliton (M, g, ξ, λ) is steady if and only if ∥H∥ = 1.
852 H. İ. Yoldaş, Ş. E. Meriç, E. Yaşar
5. ConclusionRicci soliton is a natural generalization of Einstein manifold. This notion corresponds
to the self-similar solution of Hamilton’s Ricci flow. Over the last decades, the geometry ofRicci solitons has been studied by many mathematicians. In 2008, Sharma applied Riccisolitons to K−contact manifolds and launched the study of Ricci solitons. Since then,Ricci solitons have been studied. In this paper, we deal with the submanifold admitting aRicci soliton of a Kenmotsu manifold endowed with torqued vector field T. We find thatthe characteristic vector field ξ is never torqued on submanifold M of Kenmotsu manifoldM . We obtain a necessary and sufficient condition for the tangential part T⊤ of T tobe a torse-forming on M . Also, we prove that if M admits a Ricci soliton, then it is anη−Einstein. Finally, we study the submanifold M admitting a Ricci soliton of a Kenmotsuspace form M(c) endowed with a torqued vector field T and obtain that if M admits aRicci soliton as its potential field ξ, then it is an expanding.
References[1] C. S. Bagewadi, and G. Ingalahalli, Ricci Solitons in Lorentzian α−Sasakian Mani-
folds, Acta Math. Acad. Paedagog. Nyházi. (N.S), 28 (1), 59-68, 2012.[2] C. L. Bejan and M. Crasmareanu, Second Order Parallel Tensors and Ricci Solitons
in 3-Dimensional Normal Paracontact Geometry, Ann. Glob. Anal. Geom., 46 , 117-127, 2014.
[3] D. E. Blair, Contact Manifolds in Riemannian Geometry, Lecture Notes in Mathe-matics, 509, Springer-Verlag, Berlin, 1976.
[4] B.-Y. Chen, Geometry of Submanifolds, Marcel Dekker, New York, 1973.[5] B.-Y. Chen, Some Results on Concircular Vector Fields and Their Applications to
Ricci Solitons, Bull. Korean Math. Soc., 52 (5), 1535-1547, 2015.[6] B.-Y. Chen, Rectifying Submanifolds of Riemannian Manifolds and Torqued Vector
Fields, Kragujevac J. Math., 41 (1), 93-103, 2017.[7] B.-Y. Chen, Classification of Torqued Vector Fields and Its Applications to Ricci
Solitons, Kragujevac J. Math., 41 (2), 239-250, 2017.[8] J. T. Cho and J. Park, Gradient Ricci Solitons with Semi-Symmetry, Bull. Korean
Math. Soc., 51 (1), 213-219, 2014.[9] A. Ghosh, Certain Contact Metrics as Ricci Almost Solitons, Results Maths., 65,
81-94, 2014.[10] A. Ghosh, Kenmotsu 3-Metric as a Ricci Soliton, Chaos, Solitons & Fractals, 44 (8),
647-650, 2011.[11] R. S. Hamilton, Three-Manifolds with Positive Ricci Curvature, J. Diff. Geom., 17
(2), 255-306, 1982.[12] R. S. Hamilton, The Ricci Flow on Surfaces, Mathematics and General Relativity
(Santa Cruz, CA, 1986), Contemp. Math., A.M.S, 71, 237-262, 1988.[13] S. K. Hui, S. K. Yadav and A. Patra, Almost Conformal Ricci Solitons on
f−Kenmotsu Manifolds, Khayyam J. Math., 5 (1), 89-104, 2019.[14] J.-B. Jun, U. C. De and G. Pathak, On Kenmotsu Manifolds, J. Korean Math. Soc.,
42 (3), 435-445, 2005.[15] K. Kenmotsu, A Class of Almost Contact Riemannian Manifolds, Tohoku Math. J.,
24, 93-103, 1972.[16] H. G. Nagaraja and C. R. Premalatha, Ricci Solitons in Kenmotsu Manifolds, J.
Math. Anal., 3 (2), 18-24, 2012.[17] S. Y. Perktaş and S. Keleş, Ricci Solitons in 3-Dimensional Normal Almost Paracon-
tact Metric Manifolds, Int. Electron. J. Geom., 8 (2), 34-45, 2015.[18] R. Sharma, Certain Results on K-Contact and (k, µ)−Contact Manifolds, J. Geom.,
89, (1-2), 138-147, 2008.
On submanifolds of Kenmotsu manifold with Torqued vector field 853
[19] R. Sharma and A. Ghosh, Sasakian 3-Manifolds as a Ricci Soliton Represents theHeisenberg Group, Int. J. Geom. Methods Mod. Phys, 8 (1), 149-154., 2011.
[20] S. Sular and C. Özgür, On Some Submanifolds of Kenmotsu Manifolds, Chaos, Soli-tons & Fractals, 4 (2), 1990-1995, 2009.
[21] M. M. Tripathi, Ricci Solitons in Contact Metric Manifolds, arXiv:0801.4222v1,[math DG], 2008.
[22] K. Yano and M. Kon, Structures on Manifolds, Series in Mathematics, World Scien-tific Publishing, Springer, 1984.
[23] H. İ. Yoldaş, Ş. E. Meriç, E. Yaşar, On Generic Submanifold of Sasakian Manifoldwith Concurrent Vector Field, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat.,68 (2), 1983-1994, 2019.
Hacettepe Journal ofMathematics & Statistics
Hacet. J. Math. Stat.Volume 49 (2) (2020), 854 – 868
DOI : 10.15672/hujms.624015
Research Article
Rings of frame maps from P(R) to frames whichvanish at infinity
Ali Akbar Estaji∗, Ahmad Mahmoudi DarghadamFaculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran.
AbstractLet FP(L) be the set of all frame maps from P(R) to L, which is an f -ring. In this paper,we introduce the subrings FP∞(L) of all frame maps from P(R) to L which vanish atinfinity and FPK
(L) of all frame maps from P(R) to L with compact support. We proveFP∞(L) is a subring of FP(L) that may not be an ideal of FP(L) in general and we obtainnecessary and sufficient conditions for FP∞(L) to be an ideal of FP(L). Also, we showthat FPK
(L) is an ideal of FP(L) and it is a regular ring. For f ∈ FP(L), we obtain asufficient condition for f to be an element of FP∞(L) (FPK
(L)). Next, we give necessaryand sufficient conditions for a frame to be compact. We introduce FP-pseudocompact andnext we establish equivalent condition for an FP-pseudocompact frame to be a compactframe. Finally, we study when for some frame L with FP∞(L) 6= (0), there is a locallycompact frame M such that FP∞(L) ∼= FP∞(M) and FPK
(L) ∼= FPK(M).
Mathematics Subject Classification (2010). 06D22, 54C05, 54C30
Keywords. Frame, compact frame, locally compact frame, zero-dimensional frame,vanish at infinity
1. IntroductionLet C(X) denote the ring of all real-valued continuous functions on a topological space
X; and C∞(X) is the subring of all functions C(X) which vanish at infinity. Aliabadet al. in [1] have shown that for every completely regular Hausdorff space X, wheneverC∞(X) 6= (0), then there exists a locally compact space Y such that C∞(X) ∼= C∞(Y ).
Let L be a completely regular frame and RL be the ring of real-valued continuousfunctions on L and R∗L be the ring of bounded real-valued continuous functions on L (see[2, 4]). R∞L, the family of all functions f ∈ RL for which ↑f(−1
n,
1n
) is compact for eachn ∈ N and RKL, the family of all functions f ∈ RL for which ↑ coz(f)∗ is compact, wereintroduced by Dube in [5]. Estaji and Mahmoudi Darghadam in [8] studied when for aframe L with R∞L 6= (0), there is a locally compact frame M such that R∞L ∼= R∞Mand RKL ∼= RKM (also, see [9]).
The f -ring FP(L) := Frm(P(R), L) was introduced by Karimi Feizabadi et al. in [11].Estaji et al. in [7] showed that for every frame L, there is a zero-dimensional frame M suchthat FP(L) ∼= FP(M). Hence, for study FP(L), we assume that L is a zero-dimensional∗Corresponding Author.Email addresses: [email protected] (A.A. Estaji), [email protected] (A. Mahmoudi Darghadam)Received: 08.06.2018; Accepted: 19.04.2019
Rings of frame maps from P(R) to frames which vanish at infinity 855
frame. Let C(X,Rd) denote the set of continuous functions from a space X into thediscrete space of real-numbers Rd. It is known that C(X,Rd) ≤ C(X). If X is discrete,then
C(X,Rd) = C(X) = RX ∼= FP(P(X)).In this manner, FP(L) is the generalization of the f -ring C(X,Rd).
In [3] an element α ∈ RL is called locally constant if there exists a partition P of L,meaning P is a cover of L and its elements are pairwise disjoint, such that α|a is constantfor each a ∈ P , where α|a : L(R) → ↓a given by α|a(v) = α(v) ∧ a for every v ∈ L(R).The set of all locally constant elements of RL is denoted by SL. In [3], Banaschewskishowed that FPL ∼= SL as f -ring.
In this paper, we introduce the subring FP∞(L) of all frame maps from P(R) to L forwhich vanish at infinity and FPK
(L) of all frame maps from P(R) to L with compactsupport (see Definition 3.1 and Definition 3.2). We show that FP∞(L) is a subring ofFP(L) and is an ideal of F∗P(L) (see Proposition 3.6 and Proposition 3.8). We provethat FP∞(L) may not be regular and an ideal of FP(L) in general (see Example 7.7).Also, we give necessary and sufficient conditions for FP∞(L) to be an ideal of FP(L) (seeProposition 4.14). We prove that FPK
(L) is an ideal of both FP(L) and F∗P(L) and alsoit is a regular ring (see Lemma 3.5). We introduce an FP-pseudocompact frame and nextwe establish equivalent condition for an FP-pseudocompact frame to be a compact frame(see Definition 4.1 and Lemma 4.7). For every frame L with FP∞(L) 6= (0), there is alocally compact frame M such that FP∞(L) and FPK
(L) are isomorphic with an f -subringof FP∞(M) and an f -subring FPK
(M) respectively, see Lemma 7.3, and if c :=∨
a ∈ L :↑a∗ is a compact frame is complemented then ↓c is a locally compact frame such thatFP∞(L) ∼= FP∞(↓c) and FPK
(L) ∼= FPK(↓c) (see Propositions 5.6, 7.5 and 7.8).
2. PreliminariesIn this section, we represent several concepts and definitions that are necessary in this
paper. Throughout this paper L denotes a zero-dimensional frame, that is, L generatedby their complemented elements. An element a of L is called compact if, for any subsetS of L, a =
∨S implies a =
∨T for some finite T ⊆ S. A frame L is called compact
whenever its the top element > of L is compact. For every a, b ∈ L, we recall from [5]that if ↑a and ↑b are compact frames then ↑(a ∧ b) is a compact frame and also, if ↑a is acompact frame and a ≤ b, then ↑b is a compact frame. For general background regardingframes we refer to [12].
For each set X, we can form the set P(X) of all subsets of X (called the power set ofX). Also, (P(X), ⊆) is a complete Boolean algebra. Let FP(L) be the set of all framemaps from P(R) to L. Details regarding FP(L) can be found in [7, 10, 11]. In [11] theauthors showed that, the set FPL by operation : R × R → R is a sub-f -ring of RL inwhich for all f, g ∈ FPL, f g : P (R) → L by
(f g)(X) =∨
f(Y ) ∧ g(Z) : Y Z ⊂ X =∨
f(y) ∧ g(z) : y y ∈ X,
where ∈ +, −, ∧, ∨ and Y Z := y z : y ∈ Y, z ∈ Z. Also, for every r ∈ R, thecorresponding constant function r : P (R) → L such that r(X) = > if r ∈ X and r(X) = ⊥otherwise. According to [11], for every f ∈ FPL, f(0) (f(R \ 0)) is denoted by z(f)(coz(f)) and is called a zero-element (cozero-element). We put Z(A) := z(f) : f ∈ Aand coz(A) := coz(f) : f ∈ A, for every A ⊆ FP(L). Also, for every f ∈ FPL, z(f) = ⊥if and only if f is a unit element of FPL (see [10]). The bounded part, in the f -ring sense,of FPL is denoted by F∗P(L) and is characterized by:
f ∈ F∗P(L) ⇔ f(p, q) = 1 for some p, q ∈ R,
where (p, q) = r ∈ R : a < r < b.
856 A.A. Estaji and A. Mahmoudi Darghadam
We recall from [7] that for any set S, an S-trail on L is a function t : S −→ L such that∨x∈R t(x) = > and t(x) ∧ t(y) = ⊥ for any x, y ∈ S with x 6= y and an R-trail is called
real-trail. Also, for any S-trail t on a frame L,φt : P (S) −→ L
X 7−→∨
x∈X t(x)is a frame map. Throughout this paper, this notation will be used. Also, if f ∈ FPL, thentf : R −→ L by tf (r) = f(r) is a real-trail on a frame L. The correspondences betweenreal-trails on a frame L and the f -ring FPL are powerful tools in the study of FPL. If ais a complemented element of L, then ta : R −→ L by
ta(x) =
a if x = 1a′ if x = 0⊥ if x 6∈ 0, 1
is a real-trail on L, coz(φta) = a, φ2ta
= φta and
fφta(X) =
a ∧ f(X) if 0 6∈ X
a′ ∨ f(X) if 0 ∈ X
for every f ∈ FPL and every X ⊆ R, throughout this notation will be used (see [10]). Itis clear that for S-trail t : S → L on L, φt is a monomorphism frame map if and only ift(s) 6= ⊥ for any s ∈ S. Let B(L) denote the sublattice of complemented elements of aframe L. Hence,
z(FPL) = B(L) = coz(FPL)and also, for every x ∈ L, there exists a subset A of B(L) such that x =
∨a∈A coz(φta).
3. The f-subrings FP∞(L) and FPK(L) of FP(L)
In this section, we introduce FP∞(L) and FPK(L) and prove that FP∞(L) is the f -
subrings of FP(L) that may not be both regular ring and an ideal of FP(L) in general butis an ideal of F∗P(L). We prove that FPK
(L) is an ideal of both FP(L) and F∗P(L) and isa regular f -subring of FP(L). Also, we establish several equivalent conditions for the setFP∞(L) to be an ideal of FP(L).
We begin with the following basic definitions.
Definition 3.1. We say f ∈ FP(L) vanishes at infinity if ↑f(− 1n , 1
n) is a compact framefor any n ∈ N. We denote the family of all f ∈ FP(L) vanishing at infinity with FP∞(L).
Definition 3.2. We say f ∈ FP(L) has compact support if ↑z(f) is a compact frame, orequivalently, coz(f) is a compact element of L. We denote the family of all f ∈ FP(L)with compact support by FPK
(L).
It is obvious that FPK(L) ⊆ FP∞(L).
Example 3.3. We recall a frame M is called connected, if B(M) = ⊥, >. Let M bea connected frame. Consider 0 6= f ∈ FP(M). Then coz(f) = > and z(f) = ⊥, whichimplies that there exists an 0 6= r ∈ R such that f(r) 6= ⊥ and so we clearly see thatf = r. Therefore, FP(M) = r : r ∈ R ∼= R ∼= FP(2). Since for every 0 6= r ∈ R, thereis an element n in N such that |r| > 1
n , we conclude that r ∈ FP∞(M) if and only if Mis a compact frame if and only if r ∈ FPK
(M). Hence for every connected frame M , thefollowing statements are equivalent.
(1) M is a compact frame(2) FP∞(M) = FP(M).(3) FPK
(M) = FP(M).
Rings of frame maps from P(R) to frames which vanish at infinity 857
Estaji et al. in [7] showed that FP(L) is a regular ring. In the following we prove thatFPK
(L) is a regular ring, too.
Lemma 3.4. For every f ∈ FPK(L), x ∈ R : f(x) 6= ⊥ is a finite subset of R and
f ∈ F∗P(L).
Proof. Consider f ∈ FPK(L). Since
∨x∈R f(0, x) = >, there are x1, x2, ..., xn ∈ R
such that f(0, x1, . . . , xn) = >, and so f(R \ 0, x1, . . . , xn) = ⊥, which implies thatx ∈ R : f(x) 6= ⊥ is a finite subset of R and f ∈ F∗P(L). Proposition 3.5. The following statements hold.
(1) The set FPK(L) is an ideal of FP(L).
(2) The set FPK(L) is an ideal of F∗P(L).
(3) The set FPK(L) is a regular ring.
Proof. (1). Let f, g ∈ FPK(L) and h ∈ FP(L). Since ↑(z(f) ∧ z(g)) is a compact frame
and z(f +g) ≥ z(f)∧z(g), we conclude that ↑(z(f +g)) is a compact frame, which impliesthat f +g ∈ FPK
(L). Also, from ↑z(f) is a compact frame and z(fh) = z(f)∨z(h) ≥ z(f),we infer that ↑z(fh) is a compact frame, which implies that fh ∈ FPK
(L).(2). Since, by Lemma 3.4, FPK
(L) ⊆ F∗P(L), the proof is similar to the first statement.(3). Consider f ∈ FPK
(L). We define the real-trail t : R → L on the frame L by
t(x) =
f( 1x) if x 6= 0
f(0) if x = 0.
Then
fφt(x) =
z(f) if x = 0coz(f) if x = 1⊥ if x ∈ R \ 0, 1,
which implies that f2φt = f. Since ↑z(φt) = ↑z(f) is a compact frame, we conclude thatφt ∈ FPK
(L), which implies that FPK(L) is a regular ring.
Proposition 3.6. The set FP∞(L) is a subring of FP(L).
Proof. Consider f, g ∈ FP∞(L) and n ∈ N. Since
(f + g)((− 1
n,
1n
))
≥ f(− 12n
,1
2n) ∧ g(− 1
2n,
12n
)
and ↑(f(− 1
2n , 12n) ∧ g(− 1
2n , 12n)
)is a compact frame, we conclude that ↑ (f + g)(− 1
n , 1n) is
a compact frame, which implies that f + g ∈ FP∞(L).Consider m ∈ N with m > [
√n]. From ↑
(f(− 1
m , 1m) ∧ g(− 1
m , 1m)
)is a compact frame
and(fg)
((− 1
n,
1n
))
≥ f(− 1m
,1m
) ∧ g(− 1m
,1m
),
we infer that ↑ (fg)(− 1n , 1
n) is a compact frame, which implies that fg ∈ FP∞(L). Lemma 3.7. For every f ∈ FP∞(L), the following statements hold.
(1) The set x ∈ R \ (− 1n , 1
n) : f(x) 6= ⊥ is finite for every n ∈ N.(2) f ∈ F∗P(L).(3) The set x ∈ R : f(x) 6= ⊥ is an at most countable set.
Proof. (1). Consider n ∈ N. Since∨
x∈R\(− 1n
, 1n
) f((− 1n , 1
n) ∪ x) = >, there arex1, x2, ..., xm ∈ R \ (− 1
n , 1n) such that f((− 1
n , 1n) ∪ x1, . . . xm) = >, which implies that
f(R \ (x1, x2, ..., xm ∪ (− 1n , 1
n))) = ⊥. Hence x ∈ R \ (− 1n , 1
n) : f(x) 6= ⊥ is a finitesubset of R.
(2) and (3), by the first statement, are obvious.
858 A.A. Estaji and A. Mahmoudi Darghadam
If L is not compact, then 1 6∈ FP∞(L), because 1(− 1n , 1
n) = ⊥ and ↑⊥ is not compact.
Proposition 3.8. The set FP∞(L) is an ideal of F∗P(L).
Proof. By Proposition 3.6 and Lemma 3.7, FP∞(L) is a subring of F∗P(L). Now we assumef ∈ F∗P(L) and g ∈ FP∞(L). Then f(−m, m) = > for some m ∈ N. Hence ↑fg(− 1
n , 1n) is
a compact frame, because
fg(− 1n
,1n
) ≥ f(−m, m) ∧ g(− 1mn
,1
mn) = g(− 1
mn,
1mn
),
which follows that fg ∈ FP∞(L).
The following example shows that FP∞(L) may not be an ideal of FP(L) in general andalso FP∞(L) my not be a regular ring in general.
Example 3.9. Consider L = P(N). We define the real-trail t : R → L on the frame L by
t(x) =
1x if 1
x ∈ N⊥ otherwise.
Then z(φt) = ⊥ and so φt is a unit element of FP(L). Since 1 6∈ FP∞(L) and φt ∈ FP∞(L),we conclude that FP∞(L) is not an ideal of FP(L). Also, if there is an element f in FP∞(L)such that φ2
t f = φt then φtf = 1 ∈ FP∞(L), which is contradiction. Therefore, FP∞(L) isnot a regular ring.
Definition 3.10. Let I be any ideal in FP(L). If∨
f∈I coz(f) is the non-top element ofL, we call I a fixed ideal; if
∨f∈I coz(f) = >, then I is a free ideal.
Lemma 3.11. If c is a compact element of L, then c ∈ coz(I) for every free ideal I ofFP(L) and every c ∈ B(L).
Proof. From c is a compact element of L and there exists a subset A of B(L) suchthat c =
∨a∈A coz(φta), we conclude that there a finite subset B of A such that c =
coz(Σa∈Bφ2ta
) ∈ B(L). Let I be a free ideal of FP(L) and c = coz(f) for some f ∈ FP(L).
c = c ∧ > = coz(f) ∧∨g∈I
coz(g) =∨g∈I
coz(fg),
and so, there are g1, g2, . . . gn ∈ I such that c = coz(Σni=1(fgi)2) ∈ coz(I).
Corollary 3.12. The set of all compact elements of L is a subset of∩coz(I) : I is a free ideal of FP(L)
.
Proof. By Lemma 3.11, it is clear.
Definition 3.13. An element a of a frame M is called σ-compact if there exists a familyan : n ∈ N of compact elements of M such that a =
∨n∈N an. A frame M is called
σ-compact whenever its the top element > of M is σ-compact.
By Lemma 3.11, if a ∈ L is a σ-compact element of L, then there exists an ascendingsequence ann∈N of B(L) such that a =
∨n∈N an and ↑a′n is compact, for every n ∈ N.
Proposition 3.14. The following statements hold.(1) Every element of coz(FP∞(L)) is a σ-compact element of L.(2) If B(L) is a sub-σ-frame of L and a ∈ L is a σ-compact element of L then a ∈
coz(FP∞(L)).
Rings of frame maps from P(R) to frames which vanish at infinity 859
Proof. (1). Consider f ∈ FP∞(L) and a = coz(f). We put
an := f((−∞, − 1n
] ∪ [ 1n
, +∞)),
for every n ∈ N. Then ↑a′n = ↑f(− 1n , 1
n) is a compact frame and a =∨
n∈N an, whichimplies that a is a σ-compact element of L.
(2). Let ann∈N be an ascending sequence of B(L) such that a =∨
n∈N an and ↑a′n iscompact for every n ∈ N. We put b1 := a1 and bn := an ∧ a′n−1 for every 2 ≤ n ∈ N. Thenfor every n ∈ N,
∨ni=1 bi = an, which implies that a =
∨∞i=1 bi and also bi ∧ bj = ⊥ for
every i 6= j. We define the real-trail t : R → L on the frame L by
t(x) =
bn if there exists an element n of N such that 1
x = n
a′ if x = 0⊥ otherwise.
Since
↑φt(−1n
, − 1n
) =↑ (a′ ∨∞∨
i=n+1bi) = ↑a′n
is a compact frame, we conclude that φt ∈ FP∞(L) and coz(φt) = a.
4. Compact and FP-pseudocompact framesIn this section, we introduce FP-pseudocompact frame and give several equivalent con-
ditions for it.For any element a of a frame M , we have the frame map M → ↓a taking x to x∧a, and
the associated θ : FP(M) → FP(↓a) will be denoted f 7→ f |a, where f |a(A) = f(A) ∧ a forevery A ⊆ R. Evidently, this is the counterpart of restricting functions of RX on a subsetof X. Throughout this paper, this notation will be used.
We begin with the following basic definition.
Definition 4.1. An element a of a frame M is called FP-pseudocompact if f |a is bounded,for every f ∈ FP(M). If > is FP-pseudocompact we say L is an FP-pseudocompact frame,in fact FP(M) = F∗P(M).
Proposition 4.2. L is a compact frame if and only if FP∞(L) = FP(L).
Proof. Necessity. Consider f ∈ FP(L) and n ∈ N. From ↑⊥ = L is compact and⊥ ≤ f(− 1
n , 1n), we infer that ↑f(− 1
n , 1n) is a compact frame, which implies that f ∈ FP∞(L).
Also, we have, by Lemma 3.7, FP∞(L) ⊆ F∗P(L) = FP(L) and this completes the proof.Sufficiency. It is clear that L = ↑⊥ = ↑1(−1, 1) is compact, since 1 ∈ FP∞(L).
Lemma 4.3. Let L be a compact frame. If f ∈ FP(L), then there exists a finite subset Xof R such that f(R \ X) = ⊥.
Proof. Since∨
x∈R f(x) = >, we conclude that there are x1, x2, ..., xn ∈ R such that∨ni=1 f(xi) = >, which implies that f(R \ x1, x2, ..., xn) = ⊥.
It is well known that t : R(βM) → R∗M given by t(f) = jM f is the ring isomorphism forevery completely regular frame M , where jM : βM → M given by I 7→
∨I (see [6]). We
define tP : FP(βM) → F∗P(M) by tP(f) = jM f for every f ∈ FP(βM). Now, it is naturalto ask whether tP is a ring isomorphism. It is clear that tP is a ring monomorphism.
The following example shows that tP is a ring monomorphism, my not be a ring iso-morphism.
860 A.A. Estaji and A. Mahmoudi Darghadam
Example 4.4. Consider L = P(N). We define the real-trail t : R → L on the frame L by
t(x) =
x if 1x ∈ N
⊥ if 1x 6∈ N.
Since x ∈ R : φt(x) 6= ⊥ is an infinite subset of R, we conclude from Lemma 4.3 thatφt 6∈ Im(tP), which implies that tP is not an isomorphism.
Now, we ask this question: When is tP a ring isomorphism?
Proposition 4.5. For tP : FP(βL) → F∗P(L) given by f 7→ jLf , the following statementshold.
(1) If tP is a ring isomorphism then L is a compact frame.(2) If L is a compact frame and B(L) is a sub-σ-frame of L then tP is a ring isomor-
phism.
Proof. (1). Consider f ∈ F∗P(L). Then there are x1, x2, ..., xn ∈ R, such that∨n
i=1 f(xi) =>. We define the real-trail t : R → βL on the frame βL by t(x) = ↓f(x). ThentP(φt) = f , which implies that tP is a ring isomorphism.
(2). Let L be not compact and S ⊆ L such that∨
S = > and∨
F 6= > for everyfinite subset F of S. For every s ∈ S, there is a subset Cs of B(L) such that s =
∨Cs.
Consider C =∪
s∈S Cs. Therefore∨
F 6= > for every finite subset F of C. Thereforewithout losing generality we may assume that
∨(C \ c) 6= > for every c ∈ C. Let
B := cn+1 ∈ C : n ∈ N be an infinite countable subset of C. Since B(L) is a σ-frame,we conclude that a =
∨B ∈ B(L) has a complement in L. We put bn =
∨ni=2 ci, for every
n ∈ N \ 1 and define the real-trail t : R → L on L by
t(x) =
a′ if x = 1b2 if x = 1
2bn ∧ b′n−1 if there is an element n of N \ 1, 2 such that x = 1
n
⊥ otherwise.It is clear that φt ∈ F∗P(L), and by Lemma 4.3, φt 6∈ Im(tP). Therefore tP is not anisomorphism. Proposition 4.6. The following statements are equivalent.
(1) L is compact.(2) Every proper ideal of FP(L) (F∗p(L)) is fixed.(3) Every maximal ideal of FP(L) (F∗p(L)) is fixed.
Proof. (1) ⇒ (2). Let I be a free proper ideal of FP(L). Since, by Lemma 3.11, > ∈ coz(I),we conclude that I = FP(L), which is a contradiction.
(2) ⇒ (3). It is clear.(3) ⇒ (1). Let aλλ∈Λ ⊆ L such that > =
∨λ∈Λ aλ. It is clear that
I = φ ∈ FP(L) : coz(φ) ≤∨
λ∈Λ′ aλ, for a finite subset Λ′ of Λis an ideal of FP(L). If I 6= FP(L), then there exists a maximal ideal M such that I ⊆ Mand so
> =∨
λ∈Λaλ =
∨coz(I) ≤
∨coz(M),
which is a contradiction. Therefore I = FP(L) and there exists a finite subset Λ′ of Λ suchthat > = coz(1) =
∨λ∈Λ′ aλ. This completes the proof.
Proposition 4.7. The following statements hold.(1) If L is compact then FP(L) = F∗P(L).(2) If B(L) is a sub-σ-frame of L and FP(L) = F∗P(L) then L is compact.
Rings of frame maps from P(R) to frames which vanish at infinity 861
Proof. (1). By Proposition 4.2, it is obvious.(2). Let L be not compact and S ⊆ L such that
∨S = > and
∨F 6= > for every finite
subset F of S. For every a ∈ S, there is a subset Ca of B(L) such that a =∨
Ca. ConsiderC =
∪a∈A Ca. Then
∨F 6= > for every finite subset F of C. Therefore without losing
generality we may assume that∨
(C \ c) 6= > for every c ∈ C. Let B := cn+1 ∈ C :n ∈ N be an infinite countable subset of C. Since B(L) is a σ-frame, we conclude that∨
B ∈ B(L) has a complement in L, say c1. We put bn =∨n
i=1 ci for every n ∈ N, anddefine the real-trail t : R → L on L by
t(x) =
b1 if x = 1bx ∧ b′x−1 if x ∈ N \ 1⊥ otherwise.
It is clear that φt ∈ FP(L) \ F∗P(L), which is a contradiction. Definition 4.8. A onto frame map h : L → M is called FP-quotient if for every f ∈FP(M), there is an element f in FP(L) such that hf = f , i.e., the following diagramcommutes.
Lh // // M
P(R)f
aa
f
<<yyyyyyyy
Also, an onto frame map h : L → M is called cozFP-onto if for every c ∈ coz(FP(M)),
there is an element c in coz(FP(L)) such that h(c) = c.
Corollary 4.9. A frame map h : L → M is cozFP-onto if and only if it is FP-quotient.
Proof. It is obvious. Any frame map h : M → N between frames gives rise to an f -ring homomorphism
FPh : FP(M) → FP(N)f 7→ h f,
and this results in a variant functor FP from the category Frm of frames and frame mapsto AfR from archimedean f -rings, and morphisms which are f -ring homomorphisms, forif ∈ +, ., ∨, ∧ and f, g ∈ FP(M), then
FPh(f g)(a) = h((f g)(a))
= h(∨
fx ∧ g(y) : x y = a)
=∨
h(fx) ∧ h(g(y)) : x y = a
, since h is the frame map
= FPh(f) FPh(g)(a),for every a ∈ R, which implies that FPh(f g) = FPh(f) FPh(g). Hence we have
Proposition 4.10. If FP-quotient map h : M → N is codense then the f -ring homomor-phism FPh : FP(M) → FP(N) given by f 7→ h f is an f -ring isomorphism. Also, h iscozFP
-onto.
Proof. If f ∈ ker(FPh), then FPh(f) = 0, which implies that FPh(f)(0) = h(f(0)) => and so z(f) = >, i.e., f = 0. It is clear that FPh is onto.
862 A.A. Estaji and A. Mahmoudi Darghadam
Lemma 4.11. Let L be σ-compact and not compact. Then L ∼= P(N) and there is anf -ring isomorphism η : FP(P (N)) → FP(L) such that
(1) f ∈ F∗P(P(N)) if and only if η(f) ∈ F∗P(L).(2) f ∈ FP∞(P(N)) if and only if η(f) ∈ FP∞(L).(3) f ∈ FPK
(P(N)) if and only if η(f) ∈ FPK(L).
Proof. Similar to the proof of Proposition 4.7, there exists an infinite countable subsetcn : n ∈ N of B(L) such that c′1 =
∨n∈Nn6=1
cn and∨
F 6= > for every finite subset F of
cn : n ∈ N\1. We put bn =∨n
i=1 ci for every n ∈ N, and define the real-trail t : R → Lon L by
t(x) =
b1 if x = 1bx ∧ b′x−1 if x ∈ N \ 1⊥ otherwise.
It is clear that φt ∈ FP(L) \ F∗P(L). We define the N-trail t : N → L on L by
t(x) =
φt((−∞, 1]) if x = 1φt((x − 1, x]) if x ∈ N \ 1
Hence φt : P(N) → L given by φt(X) =∨
x∈X t(x) is an isomorphism FP-quotient map.By Proposition 4.10, η = FPφt : FP(P(N)) → FP(L) given by f 7→ φt f is an f -ringisomorphism. Proposition 4.12. For every c ∈ B(L), there exists an f -ring monomorphism θ :FP(↓c) → FP(L) such that
(1) f ∈ FP∞(↓c) if and only if θ(f) ∈ FP∞(L).(2) f ∈ FPK
(↓c) if and only if θ(f) ∈ FPK(L).
Proof. We define θ : FP(↓c) → FP(L) by θ(f) = f , where f : P(R) → L give by
f(X) =
f(X) if 0 6∈ X
f(X) ∨ c′ if 0 ∈ X
is a frame map. Consider f, g ∈ FP(↓c) and ∈ +, ., ∨, ∧. Then we have
θ(f) θ(g)(0) =∨
f(x) ∧ g(y) : x y = 0, x 6= 0 or y 6= 0 ∨ c′
= (f g)(0) ∨ c′
= θ(f g)(0).Consider 0 6= x ∈ R. Since for every r ∈ R,
f(r) ∧(g(0) ∨ c′
)= f(r) ∧ g(0)
and (f(0) ∨ c′
)∧ g(r) = f(0) ∧ g(r),
we conclude thatθ(f) θ(g)(x) = θ(f g)(x).
Therefore, θ is an f -ring homomorphism. Let f be an element of ker(θ). From f(0)∨c′ =θ(f)(0) = 0(0) = > and f(0)∧c′ ≤ c∧c′ = ⊥, we infer that f(0) = c and since forevery 0 6= x ∈ R, f(x) = θ(f)(x) = 0(x) = ⊥, we conclude that f = 0. Therefore,θ is an f -ring monomorphism.
We recall from [7] that a proper ideal I in FPL is called a zFP-ideal if z(f) = z(g) and
f ∈ I implies that g ∈ I. We will also need the following results which appear in [7], forthe proof of the following proposition.
Rings of frame maps from P(R) to frames which vanish at infinity 863
Proposition 4.13. Every proper ideal in FPL is a zFP-ideal.
Proposition 4.14. Let B(L) be a sub-σ-frame of L. The following statements are equiv-alent.
(1) FP∞(L) is an ideal of FP(L).(2) Every σ-compact element a of L is FP-pseudocompact.(3) Every σ-compact element of L is compact.(4) If ann∈N is a family of compact elements of L such that
a1 ≤ a2 ≤ · · · ≤ an ≤ an+1 ≤ · · · ,
then there exists an element k of N such that ak = ak+i for all i ∈ N.(5) FP∞(L) = FPK
(L).(6) FP∞(L) is a regular ring.
Proof. (1) ⇒ (2) and (1) ⇒ (3). Let a be a σ-compact element of L. Then, by Proposition3.14, there is an element f of FP∞(L) such that coz(f) = a. Since coz(φta) = a = coz(f),we conclude from Proposition 4.13 that φta ∈ FP∞(L), which implies that ↑φta(− 1
n , 1n) =
↑a′ is compact for any n ∈ N and so a is compact. Therefore, by Lemma 4.7, ↓a is anFP-pseudocompact frame.
(2) ⇔ (3). By Lemma 4.7, it is clear.(3) ⇒ (4). It is clear.(4) ⇒ (5). Consider f ∈ FP∞(L). Since for every n ∈ N, f((−∞, − 1
n ] ∨ [ 1n , +∞)) is
compact, we conclude from the fourth statement that there exists an element m of N suchthat
coz(f) =∨
n∈Nf((−∞, − 1
n] ∨ [ 1
n, +∞)) = f((−∞, − 1
m] ∨ [ 1
m, +∞)),
which implies that coz(f) is compact and so f ∈ FPK(L). Therefore, FP∞(L) = FPK
(L).(5) ⇒ (1) and (5) ⇒ (6). By Proposition 3.5, it is clear.(6) ⇒ (2). Let a be a σ-compact element of L and not a compact element of L. Let t
and φt be the same in Proposition 3.14. Because φt ∈ FP∞(L) and FP∞(L) is the regularring, there exists an element f of FP∞(L) such that φt = φ2
t f . Since for every x ∈ R\ 0,
φt(x) = φt(x) ∧ φ2t f(x)
= φt(x) ∧∨
φty ∧ fφt(y′) : yy′ = x
=∨
φt(x) ∧ φt(y) ∧ fφt(y′) : yy′ = x= φt(x) ∧ fφt(1),
we infer that coz(φt) ≤ fφt(1) ≤ coz(fφt) ≤ coz(φt) and hence coz(f) ≥ coz(φt). Sincecoz(φt|a) = a = >↓a, we conclude that φt|a is a unit element of FP(↓a) and φt|af |a = 1,which implies that f |a(n) = φt|a( 1
n) = bn 6= ⊥ for every n ∈ N. Therefore f |a 6∈F∗P(↓a), which is a contradiction.
It is clear that if I is an ideal of the f -ring FP(L), then coz(I) is an ideal of B(L).
Corollary 4.15. For every f, g ∈ FP(L), if coz(f) ≤ coz(g) then there exists an elementh of FP(L) such that f = gh.
Proof. Consider f, g ∈ FP(L) and I is the ideal generated by g. Since coz(I) is an ideal ofB(L) and coz(f) ≤ coz(g) ∈ coz(I), we conclude from Proposition 4.13 that f ∈ I, whichimplies that there exists an element h of FP(L) such that f = gh.
If A is an ideal of frame L then coz←(A) := f ∈ FP(L) : coz(f) ∈ A is an ideal ofFP(L).
864 A.A. Estaji and A. Mahmoudi Darghadam
Proposition 4.16. If I is a free proper ideal in FP(L) then f(− 1n , 1
n) 6∈ coz(I) for everyf ∈ FP∞(L) and every n ∈ N.
Proof. Consider f ∈ FP∞(L) and n ∈ N. From
> =∨
coz(I) =∨
coz(g) ∨ f(− 1n
,1n
) : g ∈ I
and ↑f(− 1n , 1
n) is compact, we conclude that there exists an element g of I such that> = coz(g) ∨ f(− 1
n , 1n). If f(− 1
n , 1n) ∈ coz(I), then > ∈ coz(I), i.e., I = L, which is a
contradiction. Hence f(− 1n , 1
n) 6∈ coz(I).
5. Locally compact framesIn this section, we consider C := a ∈ L : ↑a∗ is a compact frame and c :=
∨C. We
show that if FP∞(L) 6= (0), then ↓c is a locally compact frame and∨φ∈FP∞ (L)
coz(φ) = c =∨
φ∈FPK(L)
coz(φ).
Next, we prove that FP∞(L) ∼= FP∞(↓c) if c is complemented.
Proposition 5.1. The following statements hold.(1) c =
∨coz(FP∞(L)).
(2) If FP∞(L) 6= (0) then FPK(L) 6= (0) and c =
∨coz(FPK
(L)).
Proof. (1). Consider f ∈ FP∞(L). For every n ∈ N, we put vn = f(−∞,−1n
]∨φ[ 1n
, +∞).
From f(−1n
,1n
) = v′n and ↑f(−1n
,1n
) is a compact frame, we conclude that vn ∈ C for everyn ∈ N. Then coz(f) =
∨n∈N vn ≤ c, it implies that
∨f∈FP∞ (L) coz(f) ≤ c. Now, assume
that a ∈ C and fλλ∈Λ ⊆ FP(L) with a =∨
λ∈Λ coz(fλ). From a∗ ≤ coz(fλ)∗ and↑a∗ is a compact frame, we conclude that ↑z(fλ) is a compact frame for every λ ∈ Λ.Hence, fλλ∈Λ ⊆ FPK
(L) ⊆ FP∞(L) and a ≤∨
coz(FP∞(L)), which implies that c ≤∨f∈FP∞ (L) coz(f), and hence c =
∨f∈FP∞ (L) coz(f).
(2). The proof is similar to the part (1). From the Proposition 5.1, we conclude the following corollary.
Corollary 5.2. FP∞(L) 6= (0) if and only if C 6= ⊥ if and only if FPK(L) 6= (0).
Remark 5.3. Consider a ∈ C and f ∈ FP(L). Since ↑a∗ is a compact frame and
> =∨
p,q∈Qf(p, q) =
∨p,q∈Q
f(p, q) ∨ a∗,
we conclude that there exist p, q ∈ Q such that f(p, q) ∨ a∗ = >, which follows thata ≺ f(p, q). Therefore, for any a ∈ C and any f ∈ FP(L) there exist p, q ∈ Q such thata ≺≺ f(p, q).
Remark 5.4. Let J be a free ideal of FP(L) and a ∈ C. Since ↑a∗ is a compact frame and
> =∨
f∈J
coz(f) =∨
f∈J
coz(f) ∨ a∗,
we conclude that there exists an element f of J such that coz(f) ∨ a∗ = >. Hence, if J isa free ideal of FP(L) or F∗P(L), then for every a ∈ C, there exists an element f of J suchthat a ≺≺ coz(f).
Rings of frame maps from P(R) to frames which vanish at infinity 865
Lemma 5.5. The following statements hold.(1) C is an ideal of L.(2) If x ≺ a then x << a for every (x, a) ∈ L × C.(3) For any a ∈ C, a =
∨x<<a x.
Proof. (1). Consider a, b ∈ L such that b ≤ a and a ∈ C. From ↑a∗ is a compact frameand a∗ ≤ b∗, we conclude that ↑b∗ is a compact frame, which implies that b ∈ C. HenceM is a down set in L. Also, for a, b ∈ C, ↑(a ∨ b)∗ = ↑a∗ ∧ b∗ is a compact frame, whichimplies that a ∨ b ∈ C, that implies C is an up directed subset of L, Therefore, C is anideal of L.
(2). Consider (x, a) ∈ L × C with x ≺ a. If aλλ∈Λ ⊆ L such that a ≤∨
aλλ∈Λ, then∨λ∈Λ
(x∗ ∨ aλ) = x∗ ∨ (∨
λ∈Λaλ) = x∗ ∨ a = >.
From the first statement we conclude x ∈ C, and hence ↑x∗ is a compact frame. Since(x∗∨ aλλ∈Λ ⊆ ↑x∗, we infer that there are λ1, λ2...λn ∈ Λ such that > = x∗∨ (
∨ki=1 aλi
),which implies that x ≤ (
∨ki=1 aλi
). Hence x << a.(3). Consider a ∈ C. Since L is a completely regular frame, we conclude from the
statement (2) that a =∨
x≺a x =∨
x<<a x and so, the proof is now complete. Proposition 5.6. If FP∞(L) 6= (0), then ↓ c is a locally compact frame.
Proof. Consider a ∈↓ c. Then a =∨
m∈C(a ∧ m). By Lemma 5.5, a ∧ m ∈ C anda ∧ m =
∨x<<a∧m x ≤ a for every m ∈ C. Hence a =
∨x<<a x. This completes the
proof. Consider S ⊆ C and a ∈ C is an upper bound of S. Since
∨S ≤ a, we conclude that∨
S ∈ C. Therefore, if S ⊆ C is bounded in C, then∨
S ∈ C.
6. The relation between the generated subframe by coz(FP∞(L)) andcoz(FPK
(L)) in L
In this section, we show that coz(FPK(L)) and coz(FP∞(L)) are the bases of ↓c.
Lemma 6.1. If FP∞(L) 6= (0) then the following statements hold.(1) For any f ∈ FP(L), if coz(f) ≤ c then there is a subset fλλ∈Λ of FPK
(L) suchthat coz(f) =
∨λ∈Λ coz(fλ).
(2) For any f ∈ FP(L), if coz(f) ≤ c then there is a subset fλλ∈Λ of FP∞(L) suchthat coz(f) =
∨λ∈Λ coz(fλ).
Proof. (1). Consider f ∈ FP(L) with coz(f) ≤ c. we havecoz(f) = coz(f) ∧ c
= coz(f) ∧( ∨
g∈FPK(L)
coz(g)), by Proposition 5.1
=∨
g∈FPK(L)
(coz(f) ∧ coz(g)
)=
∨g∈FPK
(L)coz(fg).
Since, by Lemma 3.5, FPK(L) is an ideal of FP(L), we conclude that fg ∈ FPK
(L) forevery g ∈ FPK
(L) and every f ∈ FP(L).(2). By the first statement, it is clear. A base B of a frame L is a subset of L such that every element of L is a join of elements
of B.
866 A.A. Estaji and A. Mahmoudi Darghadam
Proposition 6.2. If FP∞(L) 6= (0) then the following statements hold.(1) coz(FPK
(L)) is a base of ↓c.(2) coz(FP∞(L)) is a base of ↓c.
Proof. (1). Consider x ≤ c and fλλ∈Λ ⊆ FP(L) with x =∨
λ∈Λ coz(fλ). Sincecoz(fλ) ≤ x ≤ c. Lemma 6.1 implies that there exists a subset Bλ of FPK
(L) suchthat coz(fλ) =
∨g∈Bλ
coz(g) for every λ ∈ Λ. We put B =∪
λ∈Λ Bλ then B ⊆ FPK(L)
and x =∨
g∈B coz(g). The proof is now complete.(2). By the first statement, it is clear.
By Proposition 6.2, we have the following Corollary.
Corollary 6.3. The subframes produced by coz(FP∞(L)) and coz(FPK(L)) in L are the
same.
7. The relationship between FP∞(L) and FP∞(↓ c)In this section, we assume that FP∞(L) 6= (0) and c =
∨C.
Lemma 7.1. The map θ : FP(L) → FP(↓c) given by θ(f) = f |c is an f -ring homomor-phism.
Proof. Straightforward.
Lemma 7.2. If f ∈ FP∞(L) then f(r, s) ∨ c = > for every r, s ∈ R with r < 0 < s.
Proof. Consider f ∈ FP∞(L) and r, s ∈ R with r < 0 < s. There exists an elementn of N such that (−1
n , 1n) ≤ (r, s). Since ↑ f(−1
n , 1n) is a compact frame, we infer that
f(−∞,−1n
] ∨ f [ 1n
, +∞) ∈ C, which implies that
f(r, s) ∨ c ≥ f(−1n
,1n
) ∨ c ≥ f(−1n
,1n
) ∨ f(−∞,−1n
] ∨ f [ 1n
, +∞) = >.
The proof is now completed.
For every a, b ∈ L, we put [a, b] := x ∈ L : a ≤ x ≤ b. Consider 0 6= f ∈ FP∞(L),r, s ∈ R with r < 0 < s and S ⊆ [f(r, s) ∧ c, c] with
∨S = c. By the Lemma 7.2,
> = c ∨ f(r, s) =∨
x∈S
(x ∨ f(r, s)
).
Consider n ∈ N such that (−1n , 1
n) ≤ (r, s). From ↑f(−1n , 1
n) is a compact frame, weconclude that ↑f(r, s) is a compact frame, it implies that there exist x1, x2, ..., xk ∈ S suchthat > = f(r, s) ∨
∨ki=1 xi. Since xi ∈ S ⊆ [f(r, s) ∧ c, c], we have
c =(c ∧ f(r, s)
)∨
( k∨i=1
(c ∧ xi)
)=
k∨i=1
xi ≤∨
S = c.
Therefore [f(r, s) ∧ c, c] is a compact frame. Hence f |c ∈ FP∞(↓c), which implies that
θ∞ = θ|FP∞ (L) : FP∞(L) → FP∞(↓c)
is an f -ring homomorphism. If f ∈ ker θ∞, then f |c(− 1n , 1
n) = f(− 1n , 1
n) ∧ c = c, thereforef(− 1
n , 1n) ≥ c for any n ∈ N. By Lemma 7.2, f(− 1
n , 1n) = f(− 1
n , 1n) ∨ c = > for any n ∈ N.
We show that f = 0. C 0 6= x ∈ R, there is an element m in N, such that x 6∈ (− 1m , 1
m) ,we infer that
f(x) = f(x) ∧ > = f(x) ∧ f(− 1m
,1m
) = ⊥.
We infer that f = 0. Hence, we have
Rings of frame maps from P(R) to frames which vanish at infinity 867
Proposition 7.3. The map
θ∞ := θ|FP∞ (L) : FP∞(L) → FP∞(↓c)
is an f -ring monomorphism.
In what follows, for every f ∈ FP(↓c), we define the real-trail tf : R −→ L on L by
tf (x) =
f(x) ∨ c∗ if x = 0f(x) if x 6= 0.
Lemma 7.4. If c is complemented and f ∈ FP(↓c) then the following statements hold.(1) coz(φtf
) = coz(f) and z(φtf) = z(f) ∨ c′.
(2) φtf|c = f .
(3) f ∈ FP∞(↓c) if and only if φtf∈ FP∞(L).
Proof. (1) and (2) are clear.(3). If f ∈ FP∞(↓c), then [f(− 1
n , 1n), c] is compact, for every n ∈ N. Hence ↑ (f(− 1
n , 1n)∨
c′) = ↑φtf(− 1
n , 1n) is compact for every n ∈ N, therefore φtf
∈ FP∞(L). Conversely, ifφtf
∈ FP∞(L) then, by the second statement and Proposition 7.3, φtf|c = f ∈ FP∞(L).
Proposition 7.5. If c is complemented, then
θ∞ := θ|FP∞ (L) : FP∞(L) → FP∞(↓c)
is an f -ring isomorphism.
Proof. By Proposition 7.3 and lemma 7.4, θ∞ is an f -ring isomorphism.
Proposition 7.6. If c is complemented, then there is a locally compact frame L′ such thatFP∞(L) ∼= FP∞(L′).
Proof. We consider L′ = ↓c, by Propositions 5.6 and 7.5, it is obvious.
Lemma 7.7. If c is complemented, then f ∈ FPK(↓c) if and only if φtf
∈ FPK(L).
Proof. f ∈ FPK(↓c) if and only if [z(f), c] is compact if and only if ↑(z(f) ∨ c′) is compact
if and only if ↑z(φtf) is compact, by Lemma 7.4, if and only if φtf
∈ FPK(L).
Proposition 7.8. If c is complemented, then
θK := θ|FPK(L) : FPK
(L) → FPK(↓c)
is an f -ring isomorphism.
Proof. By Proposition 7.5 and Lemma 7.7, θK is an f -ring isomorphism.
Proposition 7.9. If c is complemented, then there is a locally compact frame L′ such thatFPK
(L) ∼= FPK(L′).
Proof. Put L′ = ↓c.
Acknowledgment. The authors would like to thank the anonymous referees for theirhelpful comments.
868 A.A. Estaji and A. Mahmoudi Darghadam
References[1] A.R. Aliabad, F. Azarpanah and M. Namdari, Rings of continuous functions vanish-
ing at infinity, Comment. Mat. Univ. Carolinae 45 (3), 519–533, 2004.[2] R.N. Ball and J. Walters-Wayland, C- and C∗-quotients in pointfree topology, Disser-
tationes Math. (Rozprawy Mat.) 412, 1–61, 2002.[3] B. Banaschewski, Remarks Concerning Certain Function Rings in Pointfree Topology,
Appl. Categor. Struct, 26 (5), 873–881, 2018.[4] B. Banaschewski, The real numbers in pointfree topology, Textos de Mathematica
(Series B) 12, 1–96, 1997.[5] T. Dube, On the ideal of functions with compact support in pointfree function rings,
Acta Math. Hungar 129 (3), 205–226, 2010.[6] T. Dube, Extending and contracting maximal ideals in the function rings of pointfree
topology, Bull. Math. Soc. Sci. Math. Roumanie 55 (103) No.4, 365–374, 2012.[7] A.A. Estaji, M. Abedi and A. Mahmoudi Darghadam, On self-injectivity of the f -ring
Frm(P(R), L), Math. Slovaca Accepted.[8] A.A. Estaji and A. Mahmoudi Darghadam, Rings of continuous functions vanishing
at infinity on a frame, Quaest. Math., 2018, DOI:10.2989/16073606.2018.1509151.[9] A.A. Estaji and A. Mahmoudi Darghadam, Ring of real measurable functions vanish-
ing at infinity on a measurable space, submitted.[10] A.As. Estaji, E. Hashemi and A.A. Estaji, On property (A) and the socle of the f -ring
Frm(P(R), L), Categ. Gen. Algebr. Struct. Appl. 8 (1), 61–80, January 2018.[11] A. Karimi Feizabadi, A.A. Estaji and M. Zarghani, The ring of real-valued functions
on a frame, Categ. Gen. Algebr. Struct. Appl. 5 (1), 85–102, July 2016.[12] J. Picado and A. Pultr, Frames and locales: Topology without points, Frontiers in
Mathematics, Springer Basel, 2012.
Hacettepe Journal ofMathematics & Statistics
Hacet. J. Math. Stat.Volume 49 (2) (2020), 869 – 886
DOI : 10.15672/hujms.630402
Research Article
Visual research on the trustability of classicalvariable selection methods in Cox regression
Nihal Ata Tutkun∗, Yasemin Kayhan Atilgan
Department of Statistics, Hacettepe University, Ankara, Turkey
Abstract
Multivariate models such as the Cox regression model, if developed carefully, are power-ful tools for making prognostic prediction which are frequently used in studies of clinicaloutcomes. Many applications require a large number of variables to be modelled by usinga relatively small patient sample. Determination of the important variables in a model iscritical to understand the behaviour of phenomena as the independent variables contributethe most to the outcome. From a practical perspective, a small subset of independent vari-ables are usually selected from a large data set without the loss of any predictive efficiency.Automatic variable selection algorithms in scientific studies are commonly used for ob-taining interpretable and practically applicable models. However, the careless use of thesemethods may lead to statistical problems. The performance of the generated models maybe poor due to the violation of assumption, omission of the important variables, problemsof overfitting, and the problem of multicollinearity and outliers. In order to enhance theaccuracy of a model, it is essential to explore the data and its main characteristics beforemaking any statistical inference. This study suggests an approach for acquiring a trustwor-thy model selection procedure for survival data by performing classical variables selectionmethods, accompanied by a graphical visualization method, namely robust coplot. Thus,it enables us to investigate the discrimination of observations, clusters of the variables andclusters of the observations that are highly characterized by a particular variable in a onegraph. We present an application of combined method, as an integral part of statisticalmodelling, on survival data on multiple myeloma to show how coplot results are used inautomatic variable selection algorithm in Cox regression model-building.
Mathematics Subject Classification (2010). 62-07,62-09,62N01,62P10
Keywords. Cox regression model, graphical visualization,multidimensional scaling,robust coplot, variable selection
∗Corresponding Author.Email addresses: [email protected] (N. A. Tutkun), [email protected] (Y. K. Atilgan)Received: 07.10.2019; Accepted: 11.03.2020
870 N.A. Tutkun, Y.K. Atilgan
1. IntroductionMany events of life, whether scientific, environmental or social have multiple and specific
reasons and these reasons are usually connected to one another. A multivariate model,which is a statistical analysis tool, allows us to determine the relative contributions ofdifferent independent variables to an outcome. The strength of multivariate model isits ability to determine how multiple independent variables, which are connected to oneanother, influence an outcome. Clinical studies, in particular, are in need of a multivariatemodel because most researches have been done on a prognosis which is usually determinedby a large number of variables [13, 15]. It is generally an unknown fact that variables aresignificantly connected to the outcome and thus, they should be included in the generatedmodel. Researchers may often collect data from a large scale of candidate demographicand clinical variables for the purpose of an accurate determination of a subset of variablesthat explains the predicted variable best. Identification of the best subset among themany variables which will be included in a model so-called variable selection procedure isa critical part of building a model. If the sample size is not sufficiently large, the numberof independent variables should be decreased in the analysis [15]. Even though an increasein the number of observations is more desirable, it may not be possible in the process ofanalysis. Besides, if two variables are highly correlated with each other, the model maynot be reliable to assess the independent impact of each variable on the outcome. Thisproblem is named as multicollinearity and it requires only one entry of the very highlyrelated variables to the model [9]. In order to determine the correlation structure betweenvariables, one may need to evaluate a correlation coefficient matrix with all the proposedindependent variables. The problem with a correlation matrix is that it only evaluates therelationship between two variables, without the adjustment of the other variables [15]. Asanother approach, multiple bivariate comparisons can be performed. However, it requirestoo many couple comparisons and bivariate associations of two independent variables andit cannot reflect the simultaneous contribution of a number of independent variable to theoutcome.
In the literature, several approaches are used for decreasing the number of independentvariables. For example, one may exclude the variables that are uncorrelated with theoutcome variable; one may combine two or more strongly/moderately correlated variablesinto a single variable; one may use variable selection algorithms to exclude the variablesthat have minimal impact on the outcome variable etc. However, these approaches mighthave some drawbacks [9, 15]. In the factor analysis, the number of independent variablescan be reduced without omitting a variable. This method turns the cluster of variablesinto a factor and the original variables which are the major interest of medically orienteddata analysis would be lost. Obtained factors may not be useful or interpretable for theclinician because one can measure the importance of a factor from the outcome but cannotmeasure the importance of any other variables from the outcome.
Automatic variable selection procedure is another approach for decreasing the numberof independent variables in the analysis. This procedure helps us decide which indepen-dent variables will be included in a multivariate model and this model determines theminimum number of independent variable which are necessary to estimate the outcomeaccurately. Most of the statistical software packages present a diverse range of variableselection techniques such as backward, forward and stepwise selection. Automatic variableselection algorithms are available in any statistical software package and they are com-monly used for obtaining interpretable models that are practically applicable in scientificstudies [14]. However, the careless use of these methods may lead to statistical problems[22]. In these selection techniques, selection or deletion of variables continues to evaluateeach variable for the way it improves the fit of the model. It is natural to think that theseselection procedures eventually produce the same subset of variables. When the selection
Visual research on the trustability of classical variable selection methods in Cox regression 871
algorithms present the same subset of variables to the researcher, it may be seen as a signof a trustworthy model; however, this is not always the case. Moreover, highly correlatedindependent variables in the model, might make it impossible to solve the equation with-out deleting one or more variables from the analysis. Apart from the above mentionedmulticollinearity problem particularly in clinical studies, continuous independent variablesmay have nonlinear relations with the outcome. Modelling a nonlinear relation with a lin-ear model would not be desirable. Determining of such variables is also crucial while usingthe automatic variable selection algorithms, because the researcher can keep the variablein the model by a simple transformation and this variable may provide valuable infor-mation. Detection of the possible observation(s) that does not follow a similar patternwith the majority of the data is also another important issue in the modelling processsince a model generated by the selection algorithms may be negatively influenced by theoutliers. Lastly, determining the possible cluster(s) of observations and a variable thatdefines this cluster(s) can be also informative in modelling process. Due to the situationsdescribed above, researchers may require a simple pre-examination of the data to preventsuch unexpected statistical problem before the application of variable selection techniquesinto the empirical multidimensional data.
The main contribution of the present study is to display the benefits that comes fromcombining the two methods that exist in both variable selection and multidimensionalgraphical representation fields. In order to build precise models that can explain theamount of predicted variable with minimum number of predictor variables without thedisadvantages of outlier observations and multicollinearity problem, robust coplot whichis a data visualization technique, is used as an auxiliary technique. Making a visualinvestigation of the multidimensional data by robust coplot method before having furtherstatistical inferences provides the researcher information about the relations among allindependent variables which are all taken from the outcome, the relation between eachindependent variable and the outcome variable, the cluster of observations, the cluster ofvariables and suspicious observation. Additionally, robust coplot presents the correlationcoefficients of the variables which enables us to measure how strong the linear relationsbetween variable, and the data are. If the correlation coefficient of an independent variableis low, this variable has little contribution to the model or has a non-linear correlation.As a result, it should be expected that the relevant variable should not be included in theCox regression model.
Many multivariate statistical methods analyze the observations and the variables sep-arately. In robust coplot method, clusters of variables, clusters of observations and thecharacterization of observations can be seen in one graph [3]. Among a wide spectrum ofgraphical techniques which are available for the management of multidimensional dataset,coplot method has attracted much more attention for various purposes from a wide rangeof areas in recent years [3]. Studies that focus on robust coplot - an approach developedfor reducing the impact of outliers - and a more flexible software, called RobCoP, whichcan produce coplot and robust coplot graphs are also available in the literature [4]. Robustcoplot method is specifically convenient for visualizing and interpreting clinical data set.In contrast to many other multivariate methods such as principal component, cluster andfactor analysis which produce the composite of variables, coplot uses original variables,and representation and interpretation of the original variables and observations and theseare more crucial and meaningful in clinical studies [6].
In life sciences, the data set may consist of many variables and the decision of whichvariables should be in the model poses as a difficult and confusing problem. Furthermore,the outliers or the multicollinearity problems negatively affect the choosing of the correctmodel. Conventional variable selection techniques based on information criterion such asAIC (Akaike [1]), BIC (Schwarz[23]), and Cp (Mallows[19]), are widely used for selectingan appropriate model. AIC and BIC are also used in survival analysis. The leading
872 N.A. Tutkun, Y.K. Atilgan
researchers studying on information criterion are Tibshirani [25], Faraggi and Simon [11],Volinsky and Raftery [26], Fan and Li [10], Liang [17]. Although these criteria work welland are efficiently implemented in well-developed statistical softwares such as R and SAS[17] for existing models, for new-developed models they should be inferred theoreticallyand added to the package programs. It is useful for researchers to initially examine thedata which is independent from the model. In this study, robust coplot analysis is usedas a preliminary examination of the data before building one of the most popular survivalmodels i.e. Cox regression model (CRM). The aim of this study is to compare the resultsacquired from using this approach with conventional variable selection methods in CRM.
The rest of the paper is organized as follows: In Section 2, robust coplot method andCRM are briefly explained. Multiple myeloma data set is described in Section 3. Inaddition to the results from robust coplot findings, the findings from conventional variableselection methods, and the comparison of the obtained results are discussed. The paperis concluded with discussions in Section 4.
2. Materials and methods2.1. Cox regression model
The most common approach for modelling the effects of variable on survival is Coxregression model as it takes into account the effect of censored observations into account[8]. In survival analysis, regression models for survival data is traditionally based on CRM.The effect of the variables on survival acts multiplicatively on some unknown baselinehazard rate which makes it difficult to model variable effects as they change over time.Although the model is based on the assumption of proportional hazards, no particularform of probability distribution is assumed for the survival times. The model is thereforereferred as a semi-parametric model [2].
The data based on a sample of size n, consists of (ti, di, xi), i = 1, ..., n where ti isthe time on study for the ith individual, di is the event indicator ( di=1 if the event hasoccurred and di=0 if the lifetime is censored) and xi is the vector of variables for the ith
individual. Hazard function for CRM is given by
hi(t) = h0(t) exp(β′xi)where h0(t) is the baseline hazard function and β is a p×1 vector of unknown parameters
[7]. The ordered death times are denoted by t1 < . . . < tk and the set of individuals whoare at risk at the time tj are denoted by R(tj), so R(tj) is the set of individuals who arealive and uncensored at a time prior to tj . Then, the likelihood for CRM is given by
L(β) =k∏
j=1
exp (β′xj)∑ℓ∈R(tj)
exp (β′xℓ)
where xj is the vector of variables for the individual who dies at the jth ordered deathtime, tj [7]. Newton-Raphson iteration is most commonly used algorithm for the estima-tion of regression coefficients.
Regression coefficients (or transforms thereof such as exp(.)) are easily interpretable andthis makes CRM popular in life sciences. The assumption of the model is linearity, that isexpected outcome value is thought to be modeled by a linear combination of independentvariables, and additivity, that is the effects of the independent variables can be added[1]. In a setting with several independent variables, the fundamental interpretation of aregression coefficient βj in a linear predictor model is the expected change in outcome(or log odds or log hazard) if Xj changes by one unit and all other variables are heldconstant. Consequently, βj measures the conditional effect of Xj . A single true model and
Visual research on the trustability of classical variable selection methods in Cox regression 873
the correct model specification can be rarely assumed. This implies that the interpretationof βj changes if the set of independent variables in the model changes and Xj is correlatedwith the other independent variables [14].
When there is few amount of independent variables in data set, building and developinga model is much more easier. However, in routine work, which variables should be includedin a model is priorly not known and we often have the candidate variables within the rangeof 10-30. This number is often too large to be considered in a statistical model [14]. Also,the number of possible models that are required to be fitted is computationally timeconsuming.
In multivariate Cox regression modelling, the selection of variables and the fit of finalmodel is very important. Since all comments are made according to the final model andthe assessment of obtained results are crucial in life sciences. There are numerous variableselection methods based on significance and/or information criteria, penalized likelihood,the change-in-estimate criterion, background knowledge, or combinations thereof [14].
In practical applications, the first and most common applicable approach is the use ofautomatic routines based on forward selection, backward elimination or stepwise proce-dures for variable selection that are used. Collett [7]recommends using a likelihood ratiotest for all variable inclusion/exclusion decisions. Iterated testing between the modelsyields forward selection (FS) or backward elimination (BE) variable selection algorithms,depending on whether one starts with an empty model or with a model that all inde-pendent variables are considered upfront. Most statisticians prefer backward elimination(BE) over forward selection (FS), especially when collinearity is present [20]. However,when models can become complex, for example in the context of highdimensional data,then FS is still possible[1]. The second easily applicable approach is the use of significancecriteria which are applied to include or exclude independent variables from a model andselect a model from a set of plausible models.
2.2. Robust coplot methodTypically, clinical studies require the analysis of multidimensional data which include
many clinical, demographic, socioeconomic, and interested outcome variables collectedfrom patients. Robust coplot graph, based on two superimposed graphs, is a simple pictureof a multidimensional data set. The first graph shows the embedding of n observationsinto two- dimensional space. This representation conserves relative distance between theobservations which means two observations that are close to each other in p dimensions areembedding closely in two dimensions. The second graph consists of p vectors that representthe variables, and reflects the relations among the variables. This method provides asimultaneous investigation of the relationship patterns between both observations andvariables in a dataset. When the data contain outliers, obtained results from robustcoplot are unaffected by these outliers. Robust coplot output is mainly generated withthree steps [3].
At the first step, for the purpose of treating the different scale variable equally, thedata matrix Xnxp is normalized into Znxp. The elements of standardized data matrixare deviations from column median, med(.) which is divided by their median absolutedeviation value, MAD(xj) = 1.4826med (|xj − med (xj)|) as follows:
zij = xij − med (xj)MAD (xj)
where zij is the ith row and jth column element of the standardized matrix Znxp, xj
is the jth column of the data matrix Xnxp. After the data matrix is standardized ina robust way, multidimensional scaling embeddings of p-dimensional n observations intotwo-dimensional space are determined at the second step.
874 N.A. Tutkun, Y.K. Atilgan
At the second step, multidimensional embedding of the data set are formed. Robustmultidimensional scaling (RMDS) is used for visualing dispersion of n observations intotwo-dimensional space [12]. RMDS uses the outlier aware cost function given in thefollowing equation;
f (O, Y ) =∑i<j
[δij − dij (Y ) − oij ]2 + λ∑i<j
|oij |
where, δij is the dissimilarity metric among ith and jth row of the Znxp, Ynx2 is thecoordinate matrix for two-dimensional space, ith row, jth column element of the outliermatrix, O, is oij = sgn (dij − dij (Y )) max (0, |dij − dij (Y )| − 1/2) which defines the out-lier variable, and λ > 0 is the parameter that controls the number of presumed outliersin the dissimilarity matrix[3]. Kruskall stress value,(σ), is used as a measure for decidinghow good the fit of the configuration of n observations obtained by RMDS is.
At the last step, p vectors are drawn on the graph which is obtained in the secondstep. Each variable is denoted by a vector acquired from the center of gravity of then observations. The direction and the magnitude of the vector are determined by themedian absolute deviation correlation coefficient (MADCC) robust against the outliers[24]. Direction of the each vector is chosen in a way that the correlation between theoriginal values of the corresponding variable and their projections on the chosen vector atmaximum value. The degree that shows how good a vectors fit is assessed by correlationvalue. A decision should be made to keep or delete the variables that do not fit thegraphical representation, in other words, variables that have low correlation values beforefurther statistical analysis. The magnitude of the vector is proportional to the evaluatedcorrelation value. Additionally, observations with high value in this vector are located inthe graph where the vector points to. MADCC is defined as follows:
ρj,MADCC = MAD2 (uj) − MAD2 (kj)MAD2 (uj) + MAD2 (kj)
where, uj and kj are the robust principal variables which are defined as follows:where, zj is the jth column of Znxp, and vj represents the projection values of all points
in the MDS graph on the jth variable vector for a specific direction.In the robust coplot representation, observations are colorized according to the selected
categorical variable in order to understand the available observations discrimination better.The outcome of the robust coplot analysis gives the following inferences about multidimen-sional data. Two highly correlated variables are represented by two vectors that are closein the same direction, and if the correlation of the variables is negative, the correspondingvectors will lie in opposite directions. Two uncorrelated variables are represented by twovectors which are perpendicular to each other. Observations which are highly character-ized by a specific variable are embedded close to each other and this mass is placed inthe same direction as the variables vector. Possible outlier observation(s) is embedded farfrom the mass of observations [24].
3. Results and discussion3.1. A descriptive look at the dataset
In this study, multiple myeloma data set of Krall et al. [16] is used for illustration.Krall, et. al. [16] analyzed the data from a study on multiple myeloma and the studyresearches 65 patients which are treated with alkylating agents. Out of 65 patients 48patients died in the study and 17 of them survived. The variable time represents thesurvival time (survtime) in months, started from the time of diagnosis. The censoringvariable consists of two values 0 and 1, that indicate whether the patient is alive or dead,respectively. Therefore, the censoring rate is 26% and the type of censoring is right.
Visual research on the trustability of classical variable selection methods in Cox regression 875
Table 1. Descriptive statistics .
X01 X02 X05 X07 X09 X10 X11 X12 X14 X15 X16Mean 1.39 10.2 60.15 3.76 1.55 6.74 30.35 3.62 8.6 5.22 10.12Std. Error ofMean
0.039 0.317 1.282 0.03 0.045 0.781 2.485 0.746 0.279 0.272 0.225
Std.Deviation
0.313 2.558 10.334 0.242 0.364 6.3 20.032 6.012 2.249 2.19 1.816
Skewness 0.872 -0.304 0.062 2.156 -0.792 0.925 -0.332 2.337 0.834 0.913 1.928Std. Error ofSkewness
0.297 0.297 0.297 0.297 0.297 0.297 0.297 0.297 0.297 0.297 0.297
Kurtosis 0.436 -0.501 -0.642 9.115 0.202 0.239 -1.218 5.114 1.778 0.524 5.486Std. Error ofKurtosis
0.586 0.586 0.586 0.586 0.586 0.586 0.586 0.586 0.586 0.586 0.586
Minimum 1 5 38 3 0.477 0 0 0 4 2 7Maximum 2 15 82 5 2 24 68 27 17 12 18Q1 1.15 8.8 51 3.63 1.35 1 10.5 0 7 3 9Q2 1.32 10.2 60 3.73 1.62 6 35 1 9 5 10Q3 1.58 12.05 67.5 3.87 1.85 10 49.5 4 10 7 10.5
The data about myeloma consists of 16 demographic and clinical variables which areas follows: X01: Log BUN at diagnosis, X02: Hemoglobin at diagnosis, X03: Platelets atdiagnosis (0: abnormal, 1: normal), X04: Infections at diagnosis (0: none, 1: present),X05: Age at diagnosis (complete years), X06: Sex (1: male, 2: female), X07: Log WBC atdiagnosis, X08: Fractures at diagnosis (0: none, 1: present), X09: Log ZBII at diagnosis(log % of plasma cells in bone marrow), X10: % Lyiaphocytes in peripieral blood atdiagnosis, X11: % Myeloid cells in peripheral blood at diagnosis, X12: Proteinuria atdiagnosis, X13: Bence Jone protein in urine at diagnosis(1: present, 2: none), X14: Totalserum protein at diagnosis, X15: Serum globin (gm%) at diagnosis, X16: Serum calcium(mgm%) at diagnosis.
We use 11 variables which are continuous, survival time and censoring variable. Robustcoplot representation of categorical variables are not meaningful. When working withcategorical variables, standardization of data matrix with the median and MAD estimatorsmay result in an undefined Z matrix. Additionally, PCC and MADCC are the correlationcoefficients that measures the relation between two continuous variables. Thus, categoricalvariables are used as color coded variables and the results are presented in Figure 3 (a) and(b). The use of categorical variables by means of color coding is quite helpful in identifyingclusters of observations and possible outliers. Since categorical variables are quite commonin clinical studies, one may want to include these variables in the Cox regression model asindependent variables. In such cases, the researcher may use the classical coplot methodinstead of robust coplot [6].
Descriptive statistics for the variables used in this application are given in Table 1.
3.2. Robust coplot findingsRobust coplot method is used for revealing the relations among a set of variables in
order to have an idea about the variables which are the backbone of CRM. Before start-ing the classical variable selection procedures, robust coplot graph is used for decidingpossible variables to be eliminated. Robust coplot software, RobCop, which is publiclyavailable (see Atilgan [24] for details about software), is used for visualing myeloma dataand exploring potential associations.
876 N.A. Tutkun, Y.K. Atilgan
Figure 1. Initial robust coplot map of myeloma data
Figure 1 is the robust coplot graph of myeloma data and it includes all variables inthe data set. Each vector in Figure 1 corresponds to a variable. Observations, since anextreme outlier, censoring value, in order to display the spread of observations on thereduced two-dimensional space. Failed observations are colored as the black cross, whilecensored observations as the red square.
The Kruskall stress value of MDS fit is found as 0.128 which is close to fair fit, andFigure 1 is available for interpretation. MADCC correlation values of each variables areseen in parenthetical in Figure 1.
X01, X02, X05, X12, and X16 are found to be low correlated variables (evaluated cor-relation coefficient values are lower than 0.50). Removal of a variable from the data setrequires a redraw of robust coplot graph because this procedure affects the previous steps.Different variable combinations produce different graphs. Therefore, one-by-one subtrac-tion is performed instead of subtracting all of the variables at once. All combinations ofeliminated variable(s) are tried. This enables us to decide the redundant variables, beforestarting the model building process for myeloma data.
Figure 1 helps us to find the variables that have high positive or negative correlationswith the other variables. For example, X10 and X11 are highly positively variables, andX02 is also moderately positively correlated with these variables whereas X09 and X11are highly negatively correlated variables. Survtime and X01, X05, and X15 are in highnegative correlation. Survtime and X12 are in high positive correlation. The anglesbetween the survtime and X10, X11 and X16 are close to 90 degree, it is implicated that
Visual research on the trustability of classical variable selection methods in Cox regression 877
Table 2. Angles between the 12 variables in myeloma data.
X01 X02 X05 X07 X09 X10 X11 X12 X14 X15 X16 SurvtimeX01 0X02 127 0X05 11 116 0X07 107 126 118 0X09 62 171 73 45 0X10 109 18 98 144 171 0X11 113 14 102 140 175 4 0X12 130 103 141 23 68 121 117 0X14 37 90 26 144 99 72 76 167 0X15 6 121 5 113 68 103 107 136 31 0X16 72 161 83 35 10 179 175 58 109 78 0Survtime 164 69 175 57 102 87 83 34 159 170 92 0
survtime and these variables to be uncorrelated. The angles of the variable vectors whichare relative to the other variable vectors are given in Table 2. It is obvious that the anglebetween the two vectors is an indicator of the correlations between their correspondingvariables.
After all variables are used for generating the robust coplot graph, we run the robustcoplot method for myeloma data set in several times. Low correlation variables are removedfrom the analysis one at a time, and we run the method on different combinations on theremaining data to see which variables are stable. It is found out that some of the lowcorrelation variables consistently have low correlations. Instead of giving all of the steps,we present the removed variables in this paper.
Four variables had correlations below 0.50, X12, X16, X07, and X05, have been removedfrom the analysis one at a time, respectively. Removing these variables had no major effecton previous robust coplot findings, namely, the association patterns among the remainingvariables which have high MADCC values, but the correlation values of remaining variableshave increased. Figure 2 presents the robust coplot graph of reduced data set.
The Kruskall stress value of this graph is 0.117, and the correlations of remainingvariables are higher than 0.60. These are usually considered as acceptable goodness-of-fitvalues [3].
Figure 2 demonstrates that the correlations between X01, X14 and X15 variables arepositive and high. This variable cluster is highly negatively correlated with survtime. Itmay be assumed that the variables that grow together are nearly duplicated to provideinformation for making inference about survtime. The variables X10 and X11 are posi-tively highly correlated variables, and these variables are nearly ortogonal to the variablesurvtime. Additionally, variable X09 is also nearly ortogonal to the variable survtime.The contents of information in the uncorrelated variables with survtime explain the vari-ation in the amount of survtime which is aspected as low. Since robust coplot analyzesobservations and variables simultaneously, clusters of observations and their nature areable to be seen. Extreme outlier(s) does not appear in myeloma data set, and survivedand non-survived patients are not clustered. The projection of a point on a vector shouldbe proportional to its distance from the corresponding variable’s average, where higherthan the average is in the direction of the vector and vice versa. For instance, observation50 have higher values in X01, X14, and X15, but small value in survtime.
Two models are inferred from myeloma data with the help of robust coplot findings.First, all variables are considered for constituting a CRM. Afterwards, reduced data set are
878 N.A. Tutkun, Y.K. Atilgan
Figure 2. Robust coplot map of reduced myeloma data
considered for building a CRM. These two processes are compared and we have achievedthe following findings; Variables which have low MADCC values are eliminated from theCRM. Variables which are orthogonal with the survtime are eliminated from the CRM.One of the variables that are highly correlated with each other is eliminated from the CRMduring the variable selection procedures. Furthermore, the final models inferred from thefull data set and the reduced data set are the same.
3.3. Cox regression model findingsFirst of all, univariate CRM is applied to the data set for the purpose of seeing the
univariate effects of the variables on survival. The obtained results are given in Table 3The results show that the Log BUN (X01) and Hemoglobin (X02) at diagnosis are
important variables as they affect the risk of death whereas the others are not significiantat a 95% confidence level. The estimated hazard of Log BUN is 5.912 that means, the riskof death increases a 5.912 unit for 1-unit increase in log BUN. There is a 0.117% decreasein risk of death for one unit increase in Hemoglobin at diagnosis (HR=0.883).
Then the two scenarios are set up according to the results of robust coplot analysis.First scenario examines CRM with variables X01, X02, X05, X07, X09, X10, X11, X12,X14, X15, and X16. Second scenario builds the full model with variables X01, X02, X09,X10, X11, X12, X14, and X15.
Visual research on the trustability of classical variable selection methods in Cox regression 879
Table 3. The results of univariate Cox regression model
Hazard Ratio Std. Error95% Confidence
Intervalz P>|z|
X01 5.912 3.583 1.803 19.391 2.93 0.003X02 0.883 0.049 0.791 0.985 -2.22 0.026X05 0.998 0.016 0.967 1.03 -0.11 0.912X07 2.415 1.756 0.581 10.04 1.21 0.225X09 1.39 0.602 0.594 3.25 0.76 0.448X10 0.983 0.023 0.939 1.029 -0.74 0.462X11 0.997 0.008 0.982 1.013 -0.34 0.736X12 1.008 0.021 0.968 1.05 0.38 0.705X14 1.093 0.074 0.957 1.248 1.32 0.188X15 1.067 0.076 0.927 1.227 0.9 0.368X16 1.11 0.112 0.912 1.352 1.04 0.298
Table 4. The summary of multivariate Cox regression model
Cox regresion model Variables in the model AIC BIC
Scen
ario
1 Full modelX01 X02 X05 X07 X09
X10 X11 X12 X14 X15 X16310.3952 334.3134
Stepwise selection X01 X02 301.3336 305.6824Forward selection X01 X02 301.3336 305.6824Backward selection X01 X02 301.3336 305.6824
Scen
ario
2 Full modelX01 X02 X09 X10 X11
X14 X15305.5589 305.5589
Stepwise selection X01 X02 301.3336 305.6824Forward selection X01 X02 301.3336 305.6824Backward selection X01 X02 301.3336 305.6824
We have run the full model and the models with variable selection procedures. Themodel selection criteria are given in Table 4. The results of multivariate CRM withoutthe variable selection are given in Table 5 and the results of multivariate CRM with thevariable selection are given in Table 6.
For Scenario 1, the model selection criteria lead us to use the variables X01 and X02for the final model, and these results are concordant with the univariate CRMs. For thisscenario, the variable selection for the data set is consistent with robust coplot findings.
For Scenario 2, X01 and X02 are statistically significant in univariate and multivariateCRM without variable selection. Additionally the variable selection procedures lead usto use the variables X01 and X02 in the final model. In summary, the results of secondscenario are same with the first scenario and robust coplot findings in terms of variableselection.
Consequently, regarding the myeloma data set, Log BUN (X01) and Hemoglobin (X02)at diagnosis are determined as important variables which affect the risk of death. Theestimated hazard of Log BUN is 5.474 that means the risk of death increases a 5.912 unit(HR=exp(1.7)=5.474) for a 1-unit increase in log BUN. There is a 0.11% decrease in riskof death for a one unit increase in Hemoglobin at diagnosis (HR=exp(-0.118)=0.89).
880 N.A. Tutkun, Y.K. Atilgan
Table 5. The results of multivariate Cox regression model without variable selection
Coef.Std.
Error95% Conf.
IntervalHazardRatio
Std.95% Conf.
Intervalz P>|z|
Scen
ario
1
X01 1.741 0.676 0.415 3.067 5.703 3.857 1.515 21.467 2.57 0.01X02 -0.181 0.068 -0.314 -0.048 0.835 0.057 0.731 0.953 -2.66 0.008X05 -0.008 0.021 -0.049 0.034 0.992 0.021 0.952 1.034 -0.37 0.715X07 0.773 0.713 -0.624 2.17 2.167 1.545 0.536 8.761 1.09 0.278X09 0.293 0.875 -1.422 2.009 1.341 1.174 0.241 7.456 0.34 0.737X10 -0.042 0.035 -0.111 0.027 0.959 0.034 0.895 1.027 -1.2 0.23X11 0.005 0.017 -0.027 0.038 1.005 0.017 0.973 1.039 0.32 0.748X12 0.032 0.029 -0.024 0.089 1.033 0.03 0.976 1.093 1.12 0.262X14 0.288 0.168 -0.041 0.616 1.333 0.223 0.96 1.852 1.72 0.086X15 -0.18 0.154 -0.481 0.121 0.835 0.128 0.618 1.129 -1.17 0.242X16 0.027 0.13 -0.228 0.281 1.027 0.133 0.796 1.325 0.2 0.838
Scen
ario
2
X01 1.819 0.658 0.529 3.109 6.166 4.058 1.697 22.397 2.76 0.006X02 -0.158 0.065 -0.285 -0.031 0.854 0.055 0.752 0.969 -2.45 0.014X09 0.116 0.818 -1.487 1.718 1.123 0.918 0.226 5.575 0.14 0.887X10 -0.025 0.031 -0.087 0.037 0.975 0.031 0.917 1.037 -0.79 0.428X11 -0.005 0.014 -0.033 0.024 0.995 0.014 0.967 1.024 -0.33 0.743X14 0.287 0.142 0.01 0.565 1.333 0.189 1.01 1.759 2.03 0.042X15 -0.209 0.143 -0.49 0.072 0.811 0.116 0.613 1.074 -1.46 0.144
Table 6. The results of multivariate Cox regression model with variable selection
Coef. Std. Error95% Conf.
Intervalz P>|z|
Scenario 1X01 1.7 0.613 0.498 2.901 2.77 0.006X02 -0.118 0.058 -0.231 -0.005 -2.05 0.041
Scenario 2X01 1.7 0.613 0.498 2.901 2.77 0.006X02 -0.118 0.058 -0.231 -0.005 -2.05 0.041
3.4. Findings from color coding variablesRobust coplot graphs of obtained model (categorical variables; Censoring variable, X03,
X04, X06, X13) are drawn according to different color coded variables. The purpose ofthis is in doing so was to reveal possible clusters of observations that share commoncharacteristics in groups.
Colorization of observations based on variable, censoring variable, sex and Bence Joneprotein in urine at diagnosis do not form cluster patterns, obtained graphs are given inSupplementary Figure 3(c-d-e).
However, it can be easily seen from Figure 3(a) that patients whose Platelets at di-agnosis are abnormal have lower Hemoglobin values. For example, hemoglobin values ofobservations 22 and 14 are 5.5 and 5.1 respectively. Log BUN values of observations 10and 4 are high and these two observations are embedded in the same direction as the X01vector. Observation 40 has the highest survtime value among the patients whose Plateletsat diagnosis are abnormal.
Visual research on the trustability of classical variable selection methods in Cox regression 881
(a) Platelets at diagnos is color codedvariable red square:0, blue plus:1
(b) Infections at diagnosis is color codedvariable red square:0, blue plus:1
Figure 3. Colorization of observations
(a) Robust Coplot Map
(b) Classical Coplot Map
Figure 4. Coplot analysis of contaminated (47) data set
A visual inspection of Figure 3(b) displays that patients who have infections at diagnosishave lower survival time. However, observation 47 does not follow this manner. A separateexamination of this patients conditions might be beneficial.
It is mentioned before that when fitting a CRM, one single outlier is enough to makethe estimator take values arbitrarily far from their true value, and lack of robustness ofCRM is widely discussed in the literature [5, 18, 21]. Identifying and removing outlyingobservations is crucial for providing more accurate relations between variables and survivaltime. An emphasis on even one single strong outlier, in classical parameter estimates,might poorly effect the classical variables selection algorithms. For instance, the LogBUN value of observation 47 is incorrectly typed as 4.3222 instead of 1.3222 and thus,obtained data set is called as contaminated data set. Graphs of robust coplot and classicalcoplot are given in Figure 4(a) and (b) respectively.
The classical coplot produces a degenerated graphical representation of the multidi-mensional data set when the underlying dataset contains one strong outlier. A single
882 N.A. Tutkun, Y.K. Atilgan
observation poorly affects the associations between the variables, and it may lead to ob-tain slightly different coefficient estimates. On the other hand, robust coplot graph pre-serves the correct relations between variables and survival time, and indicated suspiciousobservation for further examinations.
CRMs are applied for contaminated data set. The renewed results are given in Sup-plementary Table (7-8-9). The size of hazard for Log BUN (X01) severely decreases forthe both scenarios, and it also becomes insignificant. The selection methods suggest afinal model that contains only Hemoglobin (X02). There is a 0.12% decrease in the riskof death for one unit increase in Hemoglobin at diagnosis (HR=exp(-0.124)=0.88). Thevariable selection methods propose the same model that robust coplot findings suggest.
4. ConclusionIn model building studies, the researcher encounters with a large number of variables
and deals with the question of which of these variables should be included in a model.The answer cannot be known immediately because the observed variables are often highlycorrelated with the interested outcome variable. Thus, another question of whether thereis any need to put all these variables in the model building process comes up. A commonapproach for reducing the number of candidate variables is to apply firstly data reductiontechniques such as principal component analysis for the purpose of defining smaller set ofuncorrelated variables and then, automatic applying automatic selection procedures forthe purpose of reducing the risk of overfitting and multicollinearity. However, in medicalstudies, interpretation of component variables may be unreasonable. Consequently, a morepreferable approach is to rely on classical variable selection methods, expert knowledgeand clinical judgment during the process of deciding on primarily important variablesthat must be included in the model. In the case of absent or limited of expert knowledge,visual representation tools of multivariate data are very useful for better understandingthe underlying structure of the data in an unsupervised way.
This manuscript presents the results of application of two statistical methods to analyzesurvival data and shows the usefulness of the adapted approach in the context of Coxregression model. Robust coplot outcome gives pragmatic recommendations on duplicatedvariables for the researchers, low correlated variables, and the variables which have norelations with the predicted variable. Due to the multidimensional and complex natureof the survival data, identifying outlier(s) is not an easy task. However, considering thecontaminated dataset example, it is illustrated that preliminary examination of the dataset by robust coplot leads to the identification of suspicious observations. Although themethods are standard methods; their combination is new and has potential advantagesover the classical ways of analyzing such data. However, as a future work, to explicitlyshow the usefulness of the approach a simulation experiment should be conducted fordifferent censoring types and also censoring rates.
Acknowledgment. We would like to thank the referees for their valuable reviewsand highly appreciate the comments and suggestions, which significantly contributed toimproving the quality of the publication.
References[1] H. Akaike, A new look at the statistical model identification, IEEE Transactions on
Automatic Control AC 19, 716-723, 1974.[2] N. Ata and M.T. Sozer, Cox regression models with nonproportional hazards applied
to lung cancer survival data, Hacet. J. Math. Stat. 36 (2), 157-167, 2007.[3] Y.K. Atilgan, Robust coplot analysis, Comm. Statist. Simulation Comput. 45 (5),
1763-1775, 2016.
Visual research on the trustability of classical variable selection methods in Cox regression 883
[4] Y.K. Atilgan and E.L. Atilgan, RobCoP: A Matlab Package for Robust CoPlot Anal-ysis, Open Journal of Statistics 7, 23-35, 2017.
[5] T. Bednarski, On sensitivity of Coxs estimator, Statistics and Decisions 7, 215-228,1989.
[6] D.M. Bravata, K.G. Shojania, I. Oklin and A. Raveh A tool for visualizing multivariatedata in medicine, Stat. Med. 27 (12), 2234-2247, 2007.
[7] D. Collett,Modeling Survival Data in Medical Research, 2nd Ed. New York: Chapman@ Hall/ CRS A CRC Press Company, 2003.
[8] D.R.Cox, Regression Models and Life Tables, J. R. Stat. Soc. Ser. B. Stat. Methodol.34 (2), 187-220, 1972.
[9] S. Derksen and H.J. Keselman, Backward, forward and stepwise automated subsetselection algorithms: Frequency of obtaining authentic and noise variables, Brit. J.Math. Stat. Psy. 45 (2), 265-282, 1992.
[10] J. Fan and R. Li, Variable selection for Cox’s proportional hazards model and frailtymodel, Ann. Statist. 3, 74-99, 2002.
[11] D. Faraggi and R. Simon, Bayesian variable selection method for censored survivaldata, Biometrics 54, 1475-1485, 1998.
[12] P.A. Forero and G.B. Giannakis, Robust multi-dimensional scaling via outlier spar-sity control, Robust multi-dimensional scaling via outlier sparsity control, 1183-1187,2011.
[13] Jr F. Harrell and K.L. Lee, Regression Modelling Strategies for Improved PrognosticPrediction, Stat. Med. 3, 143-152, 1984.
[14] G. Heinze, C. Wallisch and D. Dunkler, Variable selection - A review and recommen-dations for the practicing statistician, Biom J. 60 (3), 431-449, 2018.
[15] M.H. Katz,Multivariable Analysis: A Practical Guide for Clinicians and Public HealthResearchers, Third Edition, Cambridge University Press, New York, 2011.
[16] J.M. Krall, V.A. Uthoff and J.B. Harley, A step-up procedure for selecting variablesassociated with survival, Biometrics 31, 49-57, 1975.
[17] H. Liang, and G. Zou, Improved AIC selection strategy for survival analysis, Comput.Statist. Data Anal. 52 (5), 2538-2548, 2008.
[18] A. Nardi and M. Schemper, New residuals for Cox regression and their application tooutlierscreening, Biometrics 55, 523-529, 1999.
[19] C.L. Mallows Nardi and M. Schemper, Some comments on Cp, Technometrics 15,661-675, 1973.
[20] N. Mantel, Why stepdown procedures in variable selection, Technometrics 12, 621-625,1970.
[21] P.J. Rousseeuw and A.M. Leroy, Robust Regression and Outlier Detection, New York:Wiley Interscience, 1987.
[22] K.L. Sainani, Multivariate regression: The pitfalls of automated variable selection,Am. J. Phys. Med. Rehabil. 5, 791-794, 2013.
[23] G. Schwarz, Estimating the dimension of a model, Ann. Statist. 6, 461-464, 1978.[24] G. Shevlyakov and P. Smirnov, Robust estimation of the correlation coefficient: an
attempt of survey, Austrian J. Stat. 40, 147-156, 2011.[25] R. Tibshirani, The lasso method for variable selection in the Cox model, Stat. Med.
16, 385-395, 1997.[26] C.T. Volinsky and A.E. Raftery, Bayesian information criterion for censored survival
models, Biometrics 56, 256-262, 2000.
884 N.A. Tutkun, Y.K. Atilgan
SUPPLEMENTARY MATERIAL
Table 7. The summary of multivariate Cox regression model for the contaminateddata set
Cox regresion model Variables in the model AIC BIC
Scen
ario
1 Full modelX01 X02 X05 X07X09 X10 X11 X12
X14 X15 X16316.1785 340.0968
Stepwise selection X02 306.5977 308.7721Forward selection X02 306.5977 308.7721Backward selection X02 306.5977 308.7721
Scen
ario
2 Full modelX01 X02 X09 X10
X11 X14 X15312.857 328.0777
Stepwise selection X02 306.5977 308.7721Forward selection X02 306.5977 308.7721Backward selection X02 306.5977 308.7721
Table 8. The results of multivariate Cox regression model without a variableselection for the contaminated data set
Coef.Std.
Error95% Conf.
IntervalHazardRatio
Std.Error
95% Conf.Interval
z P>|z|
Scen
ario
1
X01 0.242 0.278 -0.302 0.787 1.274 0.354 0.739 2.196 0.87 0.383X02 -0.209 0.07 -0.347 -0.071 0.811 0.057 0.707 0.931 -2.97 0.003X05 0 0.021 -0.041 0.04 1 0.021 0.96 1.041 -0.02 0.987X07 1.28 0.757 -0.204 2.763 3.595 2.721 0.816 15.85 1.69 0.091X09 0.431 0.857 -1.249 2.111 1.539 1.319 0.287 8.254 0.5 0.615X10 -0.042 0.036 -0.112 0.028 0.959 0.034 0.894 1.028 -1.18 0.238X11 0.014 0.017 -0.019 0.047 1.014 0.017 0.981 1.048 0.83 0.408X12 0.037 0.029 -0.019 0.093 1.038 0.03 0.981 1.097 1.29 0.196X14 0.288 0.172 -0.049 0.625 1.334 0.229 0.952 1.869 1.67 0.094X15 -0.201 0.166 -0.527 0.125 0.818 0.136 0.591 1.133 -1.21 0.226X16 0.08 0.131 -0.177 0.336 1.083 0.142 0.838 1.4 0.61 0.542
Scen
ario
2
X01 0.138 0.273 -0.396 0.673 1.148 0.313 0.673 1.959 0.51 0.612X02 -0.178 0.067 -0.309 -0.046 0.837 0.056 0.734 0.955 -2.65 0.008X09 0.187 0.833 -1.446 1.82 1.206 1.004 0.236 6.17 0.22 0.822X10 -0.035 0.032 -0.097 0.028 0.966 0.031 0.907 1.028 -1.08 0.279X11 0.003 0.015 -0.026 0.033 1.003 0.015 0.974 1.034 0.22 0.822X14 0.307 0.158 -0.002 0.615 1.359 0.214 0.998 1.851 1.95 0.052X15 -0.227 0.166 -0.552 0.098 0.797 0.132 0.576 1.103 -1.37 0.172
Visual research on the trustability of classical variable selection methods in Cox regression 885
Figure 5. Color coded variable for censoring; red square:0, blue plus:1
Figure 6. Color coded variable for sex; red square:0, blue plus:1
886 N.A. Tutkun, Y.K. Atilgan
Table 9. TThe results of multivariate Cox regression model with a variable se-lection for the contaminated data set
Coef.Std.
Error95% Conf.
Intervalz P>|z|
Scenario 1 X02 -0.124 0.056 -2.22 0.026 -0.234 -0.015Scenario 2 X02 -0.124 0.056 -2.22 0.026 -0.234 -0.015
Figure 7. Color coded variable for Bence Jone protein in urine at diagnosis; redsquare:1, blue plus:2
Hacettepe Journal ofMathematics & Statistics
Hacet. J. Math. Stat.Volume 49 (2) (2020), 887 – 901
DOI : 10.15672/hujms.621919
Research Article
Gaussian copula of stable random vectors andapplication
Phuc Ho Dang∗1, Truc Giang Vo Thi21Institute of Mathematics - VAST 18 Hoang Quoc Viet, Ha Noi, Vietnam
2Tien Giang University 119 Ap Bac, My Tho City, Vietnam
AbstractIn this paper, we present a new method to investigate data of multivariate heavy-taileddistributions. We show that for any given number α ∈ (0; 2], each Gaussian copula isalso the copula of an α-stable random vector. Simultaneously, every random vector isα-stable if its marginals are α-stable and its copula is a Gaussian copula. The result isused to build up a formula representing density functions of α-stable random vectors withGaussian copula. Adopting a new tool, the paper points out that pairs of GPS signalsrecording latitude and longitude of a fixed point have two-dimensional stable distribution,and in the most of cases, vectors of daily returns in stock market data have multivariatestable distributions with Gaussian copulas.
Mathematics Subject Classification (2010). 60E07, 60F05, 60F10, 62E10, 62H05
Keywords. stable distributions, multivariate density function, GPS data, stock market,portfolio selection
1. IntroductionUntil the 1970’s, most of the statistical analysis methods were developed under nor-
mality assumptions, mainly for mathematical convenience. In applications, however, nor-mality is only a poor approximation of reality. In particular, normal distributions do notallow heavy tails, which are so common, especially in finance and risk management stud-ies [4, 8, 11, 14, 16, 18]. Arising as solutions to central limit problems, stable distributionsare natural heavy tailed extensions of normal distributions and have attracted a lot ofattention [1, 2, 11,12].
While the univariate stable distributions are now mostly accessible by several methods toestimate stable parameters and reliable programs to compute stable densities, cumulativedistribution functions, and quantiles for stable random variables [1,6,10,13], the use of theheavy tailed models in practice has been restricted by the lack of the tools for multivariatestable distributions.
The main challenge of dealing with multivariate data with heavy tailed distributionsis that of ambiguous dependence between coordinates of a random vector. Whilst the∗Corresponding Author.Email addresses: [email protected] (P. Ho Dang), [email protected] (T. G. Vo Thi)Received: 19.09.2019; Accepted: 02.03.2020
888 P. Ho Dang, T. G. Vo Thi
dependence can be completely determined by covariance matrix for the case of multi-normal data, the covariance matrix does not exist for heavy tailed data.
Fortunately, the problem can be solved by the tool of copula. The term copula was firstintroduced by Sklar [17], but was not of great interest until recent years. Copula functionsdescribe the dependence structure connecting random variables, giving an opportunity toseparate the dependence structure and marginal distributions.
Another way of parameterizing multivariate stable distributions is to use the well knownunivariate stable distribution results about one-dimensional projections of random vectors.However, in practice, this approach faces with challenging computational problems whichhave not been generally solved for multivariate stable distributions. The problems arecaused by the complexity of the possible distributions with an uncountable set of param-eters.
In recent years, computations are more accessible for elliptically contoured stable dis-tributions [15] which are scale mixtures of multivariate normal distributions. The tools forthe very special class of stable distributions were applied in several empirical studies [9,12].Although the method is available only for a narrow subclass of symmetric multivariatestable distributions, that approach stimulates researchers to create similar tools for othersubclasses of general stable multidimensional distributions.
The current paper attempts to develop a new method for investigating the data ofmultivariate distributions with heavy tails, trying to decrease the complexity of stablecopulas downwards to a more practicable case of Gaussian copulas.
The paper is organized as follows. Section 1 presents some auxiliary results on one-dimensional stable distributions and copulas. In Section 2 we give the main results ofmultidimensional stable distributions with Gaussian copulas, demonstrating that Gaussiancopulas are also those of some multivariate stable distributions and a random vector isstable if it has Gaussian copula and all its marginals are stable. In the last section weformulate the density function of a stable random vector with Gaussian copula, which canbe practically computed. Then the results are applied for studies of GPS data and stockmarket data.
2. Preliminaries and notationGiven a random vector X = (X1, ..., Xn)T taking values in Euclidean space Rn, its cumu-
lative distribution function (cdf hereafter) and probability density function (pdf hereafter)are denote by FX and fX, respectively. The coordinates X1, ..., Xn are called marginals,simultaneously FX1 , ..., FXn and fX1 , ..., fXn are called marginal cdf’s and marginal pdf’sof X, respectively.
A continuous random vector X is said to have normal distribution (X is called a Gauss-ian or normal random vector) and denoted by X ∼ N(µ; Σ) if its pdf is given by theformula
fX(x) = 1(2π)n/2
√|Σ|
exp[−1
2(x − µ)T Σ−1(x − µ)
],
with parameter µ ∈ Rn and positive definite matrix Σ ∈ Rn×n. For the special case ofstandard normal random variable X ∼ N(0; 1), which plays an important role in statistics,symbols φ and Φ refer to its pdf and cdf, respectively. Then
φ(x) = 1√2π
exp(−x2
2) .
By definition, a random vector X has stable distribution if for every pair (X′, X′′) of
independent random vector’s identically distributed as X, for every pair (a, b) of positivenumbers, there always exist a positive number c and a vector d ∈ Rn such that aX′ + bX′′
Gaussian copula of stable random vectors and application 889
has the same distribution as cX+d. It is shown that the constant c is uniquely determinedby the pair (a, b). Namely, there is a number α ∈ (0; 2] called stability index satisfied theequation aα + bα = cα. Then X is said to be α-stable. Moreover, it is well known thatevery stable random variable is absolutely continuous with respect to Lebesgue measure onR and its pdf has a support of the forms (−∞; +∞), [c; +∞) or (−∞; c] with some c ∈ R(see [13], Theorem 1.9 and Lemma 1.10, for instance). Besides, every 2-stable randomvariable is normally distributed.
From the definition it can be concluded that the stability of a random vector is conservedafter any linear transformation. Specifically, if a random vector X is α-stable and A :Rn → Rn is a linear transformation then the random vector AX is also α-stable.
In the one-dimensional case, for all α ∈ (0; 2], every α-stable random variable has acharacteristic function of the form
X(u) = E exp(iuX) =
exp(−γα|u|α[1 − iβ(tan πα2 )sign(u)] + iδu) α 6= 1
exp(−γ|u|[1 + iβ 2π sign(u) ln |u|] + iδu) α = 1,
with fixed β ∈ [−1; 1], γ > 0 and δ ∈ R. Then the parameters α, β, γ, and δ uniquelydetermine the distribution of X, the symbol X ∼ S(α; β; γ; δ) can be used to refer to thatsituation. Proposition 1.17 of [13] gives formulas on the parameter’s change after a lineartransformation. In particular,
Proposition 2.1. (a) If X ∼ S(α; β; γ; δ), then for any a 6= 0 and b ∈ R,
aX + b ∼
S(α; sign(a)β; |a|γ; aδ + b) α 6= 1S(1; sign(a)β; |a|γ; aδ + b − 2
π βa ln |a|) α = 1,
(b) If X1 ∼ S(α; β1; γ1; δ1) and X2 ∼ S(α; β2; γ2; δ2) are independent, then X1 + X2 ∼S(α; β; γ; δ), where
β = β1γα1 + β2γα
2γα
1 + γα2
, γα = γα1 + γα
2 , δ = δ1 + δ2 .
For a cdf G : R → [0; 1] let G←(y) = infx : G(x) ≥ y be its generalized inverse. Thencopula of a random vector X, denoted by CX, can be defined by
CX(t1, ..., tn) = FX(F←X1(t1), ..., F←Xn(tn)) , (2.1)
for 0 ≤ t1, ..., tn ≤ 1. The famous Sklar’s Theorem (see [17]) confirms the relationshipFX(x1, ..., xn) = CX(FX1(x1), ..., FXn(xn)) , (2.2)
for x1, ..., xn ∈ R = [−∞; +∞].When the random vector X is continuous, its pdf fX and marginal pdf’s fX1 , ..., fXn
exist, simultaneously F←Xk= F−1
Xkfor k = 1, ..., n. Then the copula density cX can be
defined and is given by the formula
cX(t1, ..., tn) =fX(F−1
X1(t1), ..., F−1
Xn(tn))
fX1(F−1X1
(t1))...fXn(F−1Xn
(tn)). (2.3)
Besides, the pdf fX can be calculated from cX by the identityfX(x1, ..., xn) = cX(FX1(x1), ..., FXn(xn)) × fX1(x1)...fXn(xn) . (2.4)
In general, the copula functions are invariant under strictly increasing transformations.Especially, the following proposition given by Embrechts et al. (see Proposition 5.6[3])provides an useful tool for getting the main results of this study.
Proposition 2.2. Let C be the copula of a random vector X = (X1, ..., Xn)T and supposethat all marginals X1, ..., Xn are continuous random variables. If T1, ..., Tn are strictlyincreasing functions, then C is also the copula of random vector (T1(X1), ..., Tn(Xn))T .
890 P. Ho Dang, T. G. Vo Thi
The Gaussian copula is the most popular one in applications. It is simply derived fromthe correlation matrix of a multivariate Gaussian distribution function. For instance, thecopula of any two-dimensional Gaussian random vector is completely determined by thecorrelation coefficient ρ between its marginals through the formula
C(u, v; ρ) = 12π
√1 − ρ2
Φ−1(u)∫−∞
Φ−1(v)∫−∞
exp(−(s2 − 2ρst + t2)2(1 − ρ2)
)dsdt (2.5)
for (u, v) ∈ [0; 1]2.While Gaussian copulas have evidently analytical forms as the above, it raises a question:
whether Gaussian copulas can be the copulas of some non - Gaussian stable random vectorsor not? Then, the formulas (1.2), (1.3) and (1.4) can be combined together with correlationmatrices of Gaussian random vectors to compute pdf’s and cdf’s of those stable randomvectors.
To answer that question, Theorem 2.5 in the next section points out that indeed somestable random vectors have Gaussian copulas. On the other hand, Theorem 2.6 confirmsthat a random vector X = (X1, ..., Xn)T is α-stable with some α ∈ (0; 2) if all marginalsX1, ..., Xn are α-stable random variables and the copula CX is a Gaussian copula. Thatmeans possessing Gaussian copula is a sufficient condition for a random vector with stablemarginals to be stable itself.
3. Stable random vectors with Gaussian copulaBefore main results are stated, we formulate some auxiliary lemmas.
Lemma 3.1. Let Y and Z be continuous random variables with pdf’s fY and fZ whichare positive on the images ran(Y ) and ran(Z) of Y and Z. Then there exists a strictlyincreasing function g : ran(Y ) → ran(Z) such that the random variable gY : Ω → ran(Z)has the same distribution as Z. Moreover, the function g has positive derivative g′.
Proof. By assumption, the pdf’s fY and fZ are positive, therefore the cdf’s FY and FZ arestrictly increasing on ran(Y ) and ran(Z), respectively. Then the function g : ran(Y ) →ran(Z) defined by g(u) = F−1
Z (FY (u)) is well-determined as a strictly increasing function.Besides, for each u ∈ ran(Z),
g′(u) = fY (u)fZ(g(u))
,
which is a positive function. The identity implies
fZ(g(u))g′(u)du = fY (u)du ,
that yields
FZ(g(u)) =g(u)∫−∞
fZ(g(u))g′(u)du =u∫
−∞
fY (u)du = FY (u) . (3.1)
On the other hand, for every t ∈ R,
FgY (t) = Pω : g(Y (ω)) ≤ t = Pω : Y (ω) ≤ g−1(t) = FY (g−1(t)) .
Compared the above with (2.1), putting t = g(u) implies
FZ(t) = FgY (t) .
This confirm the two random variables Z and gY have the same distribution. The lemmais proved.
Gaussian copula of stable random vectors and application 891
Although the lemma seems to be quite trivial and simple, it can be useful in practicalapplication. Namely in applied statistics, sometimes normalizing transformations thatturn a given data set to a new form with normal distribution need to be used. Oneof those normalizing transformation for continuously distributed data is proposed in thefollowing immediate consequence of the lemma.
Corollary 3.2. Let X be continuous random variable with pdf fX which is positive onran(X). Then there exists a strictly increasing function g : ran(X) → R such that therandom variable g X has normal distribution.
It is evident that all marginals of a stable random vector are stable random variables.The inverse statement is not true, a random vector with all stable marginals is not alwaysstable. However, as it is confirmed in the next lemma, the inverse statement is valid ifthose marginals are independent. Proof of that is quite simple and need not be presented.
Lemma 3.3. Let α ∈ (0; 2] and a random vector U = (U1, ..., Un)T be given. Supposedthat the marginals U1, ..., Un are independent α-stable random variables, then U is anα-stable random vector.
The above lemma shows that the independence is a strong condition that guarantees thestability of a random vector with stable marginals. However, for every 2-stable randomvector, it is always possible to rotate the space axes to get a new basis, in which therandom vector has independent marginals. Namely, the standard Cholesky decompositiontheorem immediately implies the following
Lemma 3.4. Let X be an n-dimensional normally distributed random vector with positivedefined covariance matrix Σ, X ∼ Nn(µ, Σ). Then there exists an orthogonal n×n matrixA = (aij) such that the r.v. Y = AX has independent normally distributed marginals,where A : Rn → Rn is the linear transformation defined by Ax = AxT .
We are now ready to state the main result.
Theorem 3.5. Let C be a Gaussian copula of a normally distributed random vector Xwith positive defined covariance matrix. Then for every number α ∈ (0; 2] there exists anα-stable random vector W such that C is also the copula of W.
Proof. Because both addition and multiplication by positive numbers are strictly increas-ing transformations in R, by virtue of Proposition 1.2 it can be supposed that all marginalsX1, ..., Xn of the random vector X are standard normal random variables, Xk ∼ N(0; 1)for k = 1, ..., n.
Based on the assumption, the covariance matrix Σ of X is positive defined, Lemma2.4 implies the existence of an orthogonal n × n matrix A = (aij), A−1 = AT , such thatthe normal random vector Y = AX has independent marginals Y1, ..., Yn and diagonalcovariance matrix AΣAT with diagonal elements consist of all eigenvalues of Σ.
Now α-stable random variables S1 , ... , Sn are concerned. Lemma 2.1 ensures that, foreach k = 1, ..., n, there exists a strictly increasing function gk : R → ran(Sk) such that therandom variable Uk = gk Yk has the same α-stable distribution as Sk. Simultaneously,the independence of marginals Y1, ..., Yn implies the independence of random variablesU1, ..., Un. Thus, the random vector U = (U1, ..., Un)T has independent α-stable marginals,it must be an α-stable random vector as the conclusion of Lemma 2.3.
Let define a new random vector W = (W1, ..., Wn)T = A−1U. Then it is clear that Wis also an α-stable random vector as U. We attempt to point out that W has the samecopula C as X. Firstly, the α-stability of marginals W1, ..., Wn of the random vector Wtogether with Lemma 2.1 ensures that for each k = 1, ..., n, there exist strictly increasingfunction hk such that the random variable Zk = hk Wk has standard normal distribution
892 P. Ho Dang, T. G. Vo Thi
as Xk. Consequently, due to Proposition 1.2, the random vector’s W and Z = (Z1, ..., Zn)T
have the same copula, CW = CZ. Therefore, to complete the proof, it is sufficient to showthat CZ = CX, which is equivalent to
cX = cZ . (3.2)However, for (u1, ... , un) ∈ [0; 1]n, it implies from (1.3) that
cX(u1, ... , un) =fX(F−1
X1(u1), ... , F−1
Xn(un))
fX1(F−1X1
(u1))...fXn(F−1Xn
(un))
= fX(Φ−1(u1), ... , Φ−1(un))φ(Φ−1(u1))...φ(Φ−1(un))
(3.3)
and
cZ(u1, ... , un) =fZ(F−1
Z1(u1), ... , F−1
Zn(un))
fZ1(F−1Z1
(u1))...fZn(F−1Zn
(un))
= fZ(Φ−1(u1), ... , Φ−1(un))φ(Φ−1(u1))...φ(Φ−1(un))
(3.4)
From (2.3) and (2.4), it is evident that (2.2) is equivalent tofX(Φ−1(u1), ... , Φ−1(un)) = fZ(Φ−1(u1), ... , Φ−1(un)) . (3.5)
Denoting g := (g1, ..., gn), h := (h1, ..., hn), andQ := A−1 g A ,
we see that Z = h(W) and W = Q(X). Then, due to hk = Φ−1FWkand the independence
of Y1, ... , Yn, setting x1 = F−1W1
(u1), ... , xn = F−1Wn
(un), the right hand side of (3.5) equalsto
fZ(Φ−1(u1), ... , Φ−1(un)) = fhW(Φ−1(u1), ... , Φ−1(un))= fW(h−1(Φ−1(u1), ... , Φ−1(un))) = fQX(h−1
1 (Φ−1(u1)), ... , h−1n (Φ−1(un)))
= fX(Q−1(h−11 (Φ−1(u1)), ... , h−1
n (Φ−1(un))))= fX(A−1 g−1 A(h−1
1 (Φ−1(u1)), ... , h−1n (Φ−1(un))))
= fX(A−1 g−1 A(F−1W1
(u1), ... , F−1Wn
(un)))= fA−1Y(A−1 g−1 A(x1, ... , xn)) = fY(A A−1 g−1 A(x1, ... , xn))
= fY1(g−11 (
n∑j=1
a1jxj)) · ... · fY1(g−1n (
n∑j=1
anjxj))
= fU1(n∑
j=1a1jxj) · ... · fUn(
n∑j=1
anjxj) . (3.7)
Simultaneously, setting y1 = F−1U1
(u1), ... , yn = F−1Un
(un), the left hand side of (3.5) is equalto
fA−1Y(Φ−1(u1), ... , Φ−1(un)) = fY(A(Φ−1(FU1(y1)), ... , Φ−1(FUn(yn)))= fY(A(g1(y−1
1 ), ... , fYn(g−1n (yn))))
= fY1(n∑
j=1a1jg−1
j (yj)) · ... · fYn(n∑
j=1anjg−1
j (yj))
= 1√2π
exp(−12
[n∑
j=1a1jg−1
j (yj)]2) · ... · 1√2π
exp(−12
[n∑
j=1anjg−1
j (yj)]2)
= ( 1√2π
)n exp(−12
n∑k=1
([n∑
j=1akjg−1
j (yj)]2)
Gaussian copula of stable random vectors and application 893
= ( 1√2π
)n exp(−12
n∑k=1
n∑j=1
akjg−1j (yj)
n∑i=1
akig−1i (yi))
= ( 1√2π
)n exp(−12
n∑j=1
n∑i=1
n∑k=1
akjakig−1j (yj)g−1
i (yi))
= ( 1√2π
)n exp(−12
n∑j=1
n∑i=1
δjig−1j (yj)g−1
i (xi))
= ( 1√2π
)n exp(−12
n∑j=1
(g−1j (xj))2) = φ(g−1
1 (x1))...φ(g−1n (yn))
= fY1(g−11 (y1))...fYn(g−1
n (yn)) = fU1(y1)...fUn(yn) , (3.8)where δkk = 1 for k = 1, ..., n; δki = 0 for k 6= i = 1, ..., n. Comparing (3.6) to (3.7), withy1 =
∑nj=1 a1jxj , ... , yn =
∑nj=1 anjxj , we can conclude (3.5) is true, that means (3.2)
holds, the proof completes. The next theorem presents Gaussian copula as a sufficient condition for a random vector
with stable marginals to have stable distribution.
Theorem 3.6. For given α ∈ (0; 2] let X be a Gaussian random vector with positivedefined covariance matrix such that the matrix A = (aij) defined in the Lemma 2.4 satisfiesdet(|aij |α) 6= 0. Suppose that W∗ = (W ∗
1 , ..., W ∗n)T is a random vector with α-stable
marginals W ∗1 , ... , W ∗
n such that the copulas of W∗ and of X are equal, CW∗ = CX.Then W∗ is an α-stable random vector.
Proof. Let β∗1 , ..., β∗n ∈ [−1; 1]; γ∗1 , ..., γ∗n > 0; and δ∗1 , ..., δ∗n ∈ R be the stable parametersand W ∗
1 ∼ S(α; β∗1 ; γ∗1 ; δ∗1), ..., W ∗n ∼ S(α; β∗n; γ∗n; δ∗n). Then, since α-stability of a random
vector is unchanged after invertible linear transformation(x1, ..., xn) 7→ (a1x1 + b1, ..., anxn + bn)
for any a1 > 0, ..., an > 0 and (b1, ..., bn) ∈ Rn, without loss of generality, it can besupposed that γ∗1 = ... = γ∗n = 1 and δ∗1 = ... = δ∗n = 0.
Let matrix A = (aij) and the random vector Y = (Y1, ..., Yn)T = AX be determined asin Lemma 2.4. We attempt to determine α-stable random variables S1 ∼ S(α; β1; γ1; 0),..., Sn ∼ S(α; βn; γn; 0) satisfying the equations
1 = (γ∗1)α = |a11|αγα1 + ... + |an1|αγα
n
. . .1 = (γ∗n)α = |a1n|αγα
1 + ... + |ann|αγαn
(3.9)
and β∗1 = β1|a11|αγα
1 +...+βn|an1|αγαn
|a11|αγα1 +...+|an1|αγα
n= |a11|αγα
1 β1 + ... + |an1|αγαn βn
. . .
β∗n = β1|a1n|αγα1 +...+βn|ann|αγα
n
|a1n|αγα1 +...+|ann|αγα
n= |a1n|αγα
1 β1 + ... + |ann|αγαn βn
(3.10)
with unknowns γα1 , ..., γα
n and γα1 β1, ..., γα
n βn.From the assumption, det(|aij |α) 6= 0, it is clear that the equations (3.9) and (3.10) are
solved and the α-stable random variables S1 ∼ S(α; β1; γ1; 0), ..., Sn ∼ S(α; βn; γn; 0) arecompletely defined. By virtue of Lemma 2.1, for each k = 1, ..., n, there exists a strictlyincreasing function gk : R → ran(Sk) such that the random variable Uk = gk Yk has thesame α-stable distribution as Sk. Simultaneously, the independence of marginals Y1, ..., Yn
implies the independence of α-stable random variables U1, ..., Un.With W = (W1, ..., Wn)T = A−1U, by the same argument of Theorem 2.5, it is certain
that W is an α-stable random vector that has the same copula as X, CW = CX = CW∗ .
894 P. Ho Dang, T. G. Vo Thi
On the other hand, the equations (3.9) and (3.10) together with Proposition 1.1 imply theequality in distribution of all the marginals of W to correspondent marginals of W∗. Inparticular, W1 ∼ S(α; β∗1 ; γ∗1 ; δ∗1), ..., Wn ∼ S(α; β∗n; γ∗n; δ∗n). Consequently, W∗ D= W andW∗ is an α-stable random vector, the proof is fulfilled.
The above theorem suggests a procedure to check whether a data set can be fitted toany stable random vector or not, with details as follows:
Step 1. To estimate stable parameters of data marginals and to check if the all marginalshave α-stable distributions with a common suitably chosen α.
Step 2. To estimate a Gaussian copula for the transformed data having normal dis-tributed marginals, and to check if the original data are fitted to that Gaussian copula.
If the two steps are satisfied, it can be concluded the data vector has stable distribution.
4. ApplicationThis section proposes a new method to analyze data of stable distribution with Gauss-
ian copula by using the results given in the previous section. The special structure ofGaussian copulas allows researchers to combine well - known computational tools for one-dimensional stable distributions and Gaussian copulas to compute density functions andcumulative distribution functions of data which follow stable distributions with Gaussiancopulas.
4.1. Computation with multivariate data of stable distributionNolan[12] represented an approach to determine density functions of stable random
vectors belonging to a specific class of random vectors with elliptically contoured stabledistributions. At first, the researcher dealt with an α-stable radially symmetric (around0) random vector X with density function fX(x) and amplitude R = |X|. For the case0 < α < 2, the study showed that
X D= Z1/2G1 ,
where Z ∼ S(α/2; 1; 2γ20(cos πα/4)2/α; 0) is positive stable and G1 ∼ N(0; I), Z and G1
are independent. Then R2 D= ZT , where T is a chi-squared random variable with n degreesof freedom, and independent of Z. From this equation, the density functions of R and Xare completely defined by
fR(r) = fR(r|α; γ0; n) = 2r
∞∫0
fZ(r2/t)fT (t)t
dt (4.1)
and
fX(x) =
(Γ(n/2)/(2πn/2))|x|1−nfR(|x| | α; γ0; n) x 6= 0Γ(n/α)/(α2n−1πn/2Γ(n/2)2γn
0 ) x = 0.(4.2)
A random vector Y is called elliptically contoured stable if it is an affine transformationof the α-stable radially symmetric random vector X by Y = Σ1/2X + δ, where Σ1/2 isfrom the Cholesky decomposition of a positive definite matrix Σ and δ ∈ Rn. Then
Y D= Z1/2Σ1/2G1 + δ ,
andfY(y) = | det Σ|−1/2fX(Σ−1/2y) . (4.3)
Since the tools for computation of univariate stable density functions are quite com-monly available at present, the formulas (4.1), (4.2) and (4.3) make the analyses of datawith elliptically contoured stable distributions easier.
Gaussian copula of stable random vectors and application 895
However, it is clear that all marginals of an α-stable radially symmetric random vectorare identically distributed and all marginals of an elliptically contoured stable randomvector are symmetric (around respective location parameter). Those properties of dataare quite rarely met in practice. In the following part, we propose a method of computingdensity functions of stable random vectors having Gaussian copulas. Whilst marginals ofa stable random vector with Gaussian copula are not necessarily symmetric, this model ofmultivariate stable distributions may be more acceptable in the practical data analysis.
Let G ∼ N(0; Σ) be a Gaussian random vector with covariance matrix Σ = (σij) andσii = 1 for all i = 1, ..., n. It is evident that Σ also is the correlation matrix of G and allmarginals of this random vector are standard normal random variables. Then G has thedensity function
fG(x) = (2π)−n/2|Σ|−1/2 exp(−12
xΣ−1xT )
for x = (x1, ..., xn) ∈ Rn, and from (1.1), its copula can be computed by
CG(u) = 1(2π)n/2|Σ|1/2
Φ−1(u1)∫−∞
...
Φ−1(un)∫−∞
exp(−12
xΣ−1xT ) dxn... dx1
for u = (u1, ..., un) ∈ [0; 1]n. Moreover from (1.3), its copula density is determined by
cG(t) = fG(Φ−1(t1), ..., Φ−1(tn))φ(Φ−1(t1))...φ(Φ−1(tn))
=exp(−1
2(Φ−1(t1), ..., Φ−1(tn))Σ−1(Φ−1(t1), ..., Φ−1(tn))T )(2π)n/2|Σ|1/2φ(Φ−1(t1))...φ(Φ−1(tn))
(4.4)
for t = (t1, ..., tn) ∈ [0; 1]n.For fixed α ∈ (0; 2), let S1 ∼ S(α; β1; γ1; δ1), ..., Sn ∼ S(α; βn; γn; δn) be certain α-stable
random variables with density functions fS1 , ..., fSn and cumulative distribution functionsFS1 , ..., FSn , respectively. Then formula (1.2) from Sklar’s Theorem ensures the function
FY(y1, ..., yn) = CG(FS1(y1), ..., FSn(yn)) ,
for y1, ..., yn ∈ R = [−∞; +∞], defines the cumulative distribution function of some ran-dom vector Y with copula CG and marginals S1, ..., Sn. In the case when the conditionof Theorem 2.6 fulfilled for A = Σ1/2, the random vector Y is truly an α-stable randomvector, and by virtue of formula (1.4), its density function has the following form:
fY(y1, ..., yn) = cG(FS1(y1), ..., FSn(yn)) × fS1(y1)...fSn(yn)
=exp(−1
2(Φ−1FS1(y1), ..., Φ−1FSn(yn))Σ−1(Φ−1FS1(y1), ..., Φ−1FSn(yn))T )(2π)n/2|Σ|1/2φ(Φ−1FS1(y1))...φ(Φ−1FSn(yn))
× fS1(y1)...fSn(yn). (4.5)It is evident that (3.5) can be calculate by ordinary computing without symmetry as-
sumption on the α-stable random variables S1, ..., Sn. Therefore, the approach representedhere is more flexible than the one of (4.3).
4.2. Multivariate stable distribution of GPS dataIn this subsection we examine the probability distribution of two-dimensional random
vectors of GPS data. The data used in this study are given by Bui Quang[2] and weretaken from a GPS receiver device (JMC GP-200) fixed at the window of a room on the 3rdfloor in a 5 floor’s building. The device was connected with a computer to automaticallyrecord data by a software (format-NMEA0183). With the equipment, longitude and lat-itude coordinates of the fixed location were recorded sequentially every one second. Themeasurement was conducted three times in May 2011, with the duration of 10 - 15 min-utes each time, providing three files with 812, 752, and 627 signals, respectively. Totally,
896 P. Ho Dang, T. G. Vo Thi
we got a data set with 2191 signals, each signal include a pair of longitude and latitudecoordinates recorded. Due to the changes of satellites positions on orbits, the measuresvaried moment by moment though the location was fixed. However, we can determine thedistribution of the deviations from the "true" coordinates of the location, which will bedone as follows.
Traditionally, deviations in any measurement were treated as values of a normally dis-tributed quantity. Therefore, Kolmogorov - Smirnov test (K-S test) was used to check ifthe GPS data followed the normal distribution. The results are represented in Table 1,with (X, Y ) denoted the pairs of longitude and latitude observed. In this table, the twop-values presented less than 0.05, rejecting the hypothesis of having normal distributionfor coordinates. That means the two-dimensional vectors of longitudes and latitudes arenot normally distributed.
Guessing the two-dimensional coordinate vectors have stable distribution, we use thefunctions McCullochParametersEstim and ks.test in the software R were used to estimatestable parameters for each coordinate and then do K-S test verifying the hypothesis ofstable distribution, taking α (the average of two values of α) as the common first stableparameter. The results are presented in Table 2, where the p-values are greater than 0.05,indicating every coordinate has stable distribution S(α; β; γ; δ) with the correspondingparameters. That means
X ∼ S(1.567; −0.1024; 3.15 · 10−5; 105.799922),Y ∼ S(1.567; 0.1592; 2.16 · 10−5; 21.043174).
According to Theorem 2.6, the random vector (X, Y ) will be stable if it has Gaussiancopula. By Proposition 1.1 and Corollary 2.2, the Gaussian copula of (X, Y ) is com-pletely determined by the correlation coefficient corr(X ′, Y ′), where variables X ′ and Y ′
are standard normal variables transformed from X and Y by
x′j = Φ−1(FX(xj)),
y′j = Φ−1(FY (yj)),
with j = 1, ..., N , FX and FY being the empirical cumulative distribution functions of Xand Y , respectively. Actually,
ρ = corr(X ′, Y ′) = −0.01462955 . (4.6)
We attempt now to show that the copulas defined by (2.5) with the correlation coef-ficients ρ will be truly the copula of GPS coordinates corresponding to the data vectors(X, Y ).
Genest and Rémillard[5] established the validity of the parametric bootstrap methodwhen testing the goodness of-fit of families of multivariate distributions and copulas. Basedon the reliable theoretical frame and other related ones, Kojadinovic and Yan[7] built upan R package named copula to implement the goodness-of-fit tests. Used the functionnamed gofCopula in the copula package, we tested the hypothesis
H0 : C(X,Y )(u, v) = C(u, v; ρ) ,
(u, v) ∈ [0; 1]2, where C(u, v; ρ) is the Gaussian copula defined by (1.5) with the correlationcoefficient ρ given in (4.6).
The testing result approximates p-value of 0.6489, much greater than 0.05. That meansthe hypothesis can be accepted. Then we can confirm that the GPS data vector hasGaussian copula, consequently it has two-dimensional stable distribution by virtue ofTheorem 2.6.
Gaussian copula of stable random vectors and application 897
Table 1. Kolmogorov-Smirnov test for normal distribution of GPS data
Coordinate N (Size) Mean SD p-valueX 2191 105.799922 0.000053 4.6 · 10−5
Y 2191 21.043174 0.000039 2.4 · 10−8
Table 2. Parameters and Kolmogorov-Smirnov test for stable distribution
Coordinate α α β γ δ p-valueX 1.585 1.567 -0.1024 3.15 · 10−5 105.799922 0.1756Y 1.504 1.567 0.1592 2.16 · 10−5 21.043174 0.6437
Applying (3.5) for the two-dimensional case, density functions of GPS signals are for-mulated as follows.
f(X,Y )(x, y) = exp(−[Φ−1(FX(x))]2 + 2ρΦ−1FX(x)Φ−1FY (y) − [Φ−1FY (y)]2
2(1 − ρ2))
× fX(x)fY (y)2π[1 − ρ2]1/2φ(Φ−1FX(x))φ(Φ−1FY (y))
for (x, y) ∈ R2.
4.3. Multivariate stable distribution of stock market dataThis subsection models daily return data from 6 stocks AK (Akorn), AP (Apple), FA
(Facebook), MI (Microsoft), WM (Walmart), and AM (Amazon), using a sample from 01May 2014 to 01 May 2019 to imply observations downloaded from Nasdaq Finance. Con-tinuously compounded percentage returns are considered, i.e. daily returns are measuredby log-differences of closing pricing multiplied by 100. Descriptive statistics together withKolmogorov - Smirnov for normal distribution of the univariate series are shown in Table3 and the result for the univariate stable model estimation are presented in Table 4.
Table 3 indicates that all the series have univariate distributions statistically differentfrom Gaussian distribution. In the meantime, Table 4 shows the series having univariatestable distributions with a common stability index α = 1.4885 less than 2, driving usto guess the 6-coordinates vector of daily returns have multivariate stable distribution.The preliminary exploratory analysis in Table 3 also presents all series being asymmetric,therefore it can be concluded that the vector of daily returns is not an elliptically contouredstable vector.
To check if the 6-coordinates vector of daily returns have multivariate stable distributionwith Gaussian copula, applying the same argument as that presented in the subsection3.2, we used the function named gofCopula in the copula package, testing the hypothesisof owning Gaussian copula for the vector of daily returns. However, the calculation gavea p-value of 0.0004995, rejecting the above hypothesis of Gaussian copula.
That phenomenon may be caused by the fact that the multivariate distribution of the5-years data is a "mixture" of several distributions of data collected in shorter periods, atthe same time, the relationships between daily returns of any stock market are changingyear - by - year. It is possible that vectors of daily returns in any one - year collecteddata have Gaussian copula even when their five-years "mixture" does not. In this context,we investigate one - year period’s data, dividing each of the given daily returns seriesinto samples from the one - year periods of 1/5/2014 - 1/5/2015, 1/5/2015 - 1/5/2016,1/5/2016 - 1/5/2017, 1/5/2017 - 1/5/2018, and 1/5/2018 - 1/5/2019.
898 P. Ho Dang, T. G. Vo Thi
Table 3. Normal distribution test for Nasdaq stock market daily return data
Daily return Mean SD skew kurtosis p-valueAkorn -0.18 4.65 -6.97 120.11 2.200 × 10−16
Amazon 0.15 1.89 0.48 8.19 7.293 × 10−10
Apple 0.07 1.53 -0.33 4.35 5.181 × 10−8
Facebook 0.09 1.80 -0.89 19.78 1.354 × 10−8
Microsoft 0.09 1.45 0.15 7.06 3.715 × 10−7
Walmart 0.02 1.23 -0.12 17.59 1.186 × 10−7
Table 4. K-S test for univariate stability of Nasdaq daily returns
Daily return α α β γ δ p-valueAkorn 1.323 1.4885 -0.139 1.3814575 0.0928667 0.2145Amazon 1.507 1.4885 -0.136 0.8938552 0.1625179 0.3195Apple 1.433 1.4885 -0.112 0.7530880 0.0997931 0.5816Facebook 1.564 1.4885 -0.153 0.9040878 0.1411224 0.2961Microsoft 1.495 1.4885 -0.024 0.7038934 0.0739206 0.5166Walmart 1.609 1.4885 -0.150 0.6042559 0.0620062 0.5488
Table 5. K-S tests for normal distribution of Nasdaq daily returns
Period AK AM AP FA MI WM(p-value) (p-value) (p-value) (p-value) (p-value) (p-value)
2014-2015 0.013261 0.008177 0.391102 0.744801 0.008314 0.1159032015-2016 0.018070 0.047730 0.039080 0.061702 0.049890 0.0387902016-2017 0.003373 0.120602 0.041780 0.115101 0.008741 0.0083372017-2018 2.2/1016 0.021290 0.103902 0.006812 0.011230 0.0008132018-2019 5.32/108 0.004618 0.009243 0.000610 0.029810 0.169003
The Kolmogorov - Smirnov tests checking the normal distribution hypothesis for eachdaily returns series of AK; AM; AP; FA; MI; and WM for each one - year period, providethe correspond p-values given in Table 5. Almost all p-values smaller than 5% confirm thesignificant divergence from normal distribution of the 6-dimensional vector. Simultane-ously, the greater than 5% p-values of Kolmogorov - Smirnov tests in Table 6 are crucialarguments to conclude all the daily returns series of AK; AM; AP; FA; MI; and WM haveunivariate stable distributions with common stable index α.
After the above conclusion, we guess the 6-components vectors of one - year series ofAK; AM; AP; FA; MI; and WM have multivariate stable distributions with stable index α(the average number of α’s of those returns series in each one - year period). The values ofparameter β in Table 6 show almost all the series are asymmetric, then the 6-coordinatevectors of one - year daily returns are not elliptically contoured stable vectors. Thereforethe method proposed by Nolan[12] for modeling density functions of stable random vectorscannot be applied to these data.
To solve the problem, model of multivariate stable distributions with Gaussian copulawas used. In the first step, the correlation matrices of daily returns for each year (afternormalizing by respective functions determined in Corollary 2.2) were calculated, withresults given in Table 7 (showing only upper diagonal part of each correlation matrix).
Gaussian copula of stable random vectors and application 899
Table 6. K-S test for univariate stability of Nasdaq daily returns
Period return Size α α β γ δ p-valueAK 253 1.585 1.749 -0.119 1.5530655 0.2911653 0.8921AM 253 1.890 1.749 0.250 1.0121948 0.0468389 0.9705
2014- AP 253 1.816 1.749 0.226 0.9143218 0.0940329 0.29432015 FA 253 1.834 1.749 -0.487 1.0605608 0.1819212 0.1691
MI 253 1.802 1.749 0.950 0.7181263 -0.0944571 0.8333WM 253 1.566 1.749 -0.164 0.5259360 0.0202864 0.6175AK 253 1.520 1.594 -0.129 1.8928297 -0.1685622 0.9705AM 253 1.551 1.594 -0.170 1.1102714 0.1187294 0.2943
2015- AP 253 1.464 1.594 0.011 0.9516475 -0.1293445 0.61752016 FA 253 1.714 1.594 -0.275 1.1203550 0.1709250 0.6175
MI 253 1.579 1.594 -0.279 0.8872402 0.0570302 0.8921WM 253 1.735 1.594 -0.342 0.7393400 -0.0387758 0.6175AK 252 1.386 1.542 -0.052 1.4137946 0.0060832 0.5412AM 252 1.571 1.542 0.040 0.6644741 0.1368053 0.5412
2016- AP 252 1.554 1.542 0.052 0.5639271 0.0694965 0.69002017 FA 252 1.700 1.542 -0.059 0.6375508 0.0656164 0.5412
MI 252 1.456 1.542 0.163 0.5062569 0.0125571 0.6900WM 252 1.587 1.542 0.013 0.4897892 0.0414158 0.9971AK 253 0.879 1.416 0.022 0.2224033 -0.0017055 0.2467AM 253 1.594 1.416 -0.242 0.8952823 0.2258286 0.7655
2017- AP 253 1.519 1.416 -0.272 0.7178885 0.0804291 0.83332018 FA 253 1.449 1.416 -0.226 0.7877504 0.1023312 0.6175
MI 253 1.383 1.416 -0.206 0.6377951 0.1471863 0.1690WM 253 1.672 1.416 -0.349 0.6182917 0.1956893 0.6924AK 252 1.475 1.486 -0.592 2.1214072 0.7908369 0.9375AM 252 1.324 1.486 -0.171 0.9626232 0.2057382 0.9375
2018- AP 252 1.420 1.486 -0.076 0.8866676 0.2004820 0.76342019 FA 252 1.636 1.486 0.163 1.0735537 0.0367324 0.2447
MI 252 1.363 1.486 -0.098 0.8032993 0.2114007 0.615WM 252 1.696 1.486 -0.420 0.6659449 0.0687611 0.4709
Then, the function named gofCopula in the copula package was used to test the hy-potheses of having Gaussian copula for each one - year 6-coordinates vector of daily returns.The calculations generated the p-values of 0.09241; 0.4001; 0.4261; 0.03247; and 0.1424for the periods of 1/5/2014-1/5/2015, 1/5/2015-1/5/2016, 1/5/2016-1/5/2017, 1/5/2017-1/5/2018, and 1/5/2018-1/5/2019, respectively. The result shows that the copula ofdaily returns vector is significantly different from Gaussian copula only for the periodof 1/5/2017-1/5/2018. Consequently, for other periods, (3.5) can be applied to determinethe density functions of 6-coordinates vector of daily returns.
In particular, the density functions are defined by the following explicit form:
fYi(y1, ..., y6) =
=exp(−1
2(Φ−1FSi1(y1), ..., Φ−1FSi6(y6))Σ−1i (Φ−1FSi1(y1), ..., Φ−1FSi6(y6))T )
(2π)3|Σi|1/2φ(Φ−1FSi1(y1))...φ(Φ−1FSi6(y6))
× fSi1(y1)...fSi6(y6).
900 P. Ho Dang, T. G. Vo Thi
Table 7. Correlation matrices of yearly Nasdaq daily returns
Period Return AK AM AP FA MI WMAK 1 0.2186643 0.2475693 0.3077665 0.2459316 0.1601064
2014- AM − 1 0.3409933 0.4551869 0.3825091 0.30969442015 AP − − 1 0.4335172 0.3217411 0.3265398(Σ1) FA − − − 1 0.3563987 0.1559727
MI − − − − 1 0.4056158WM − − − − − 1AK 1 0.3588723 0.2654328 0.2849989 0.3524828 0.1686205
2015- AM − 1 0.4593876 0.6691572 0.5978800 0.26963402016 AP − − 1 0.4907881 0.6187722 0.3225324(Σ2) FA − − − 1 0.6049646 0.2915228
MI − − − − 1 0.3800727WM − − − − − 1AK 1 0.1343061 0.1724847 0.1420178 0.1304932 0.0985428
2016- AM − 1 0.3852057 0.6213989 0.5295887 0.23049482017 AP − − 1 0.4696930 0.3770917 0.1327082(Σ3) FA − − − 1 0.4943592 0.1408454
MI − − − − 1 0.1764853WM − − − − − 1AK 1 0.1331431 0.1339723 0.0957475 0.2203172 0.1150943
2017- AM − 1 0.5391196 0.5924327 0.6516110 0.16149152018 AP − − 1 0.4973002 0.6237823 0.2566277(Σ4) FA − − − 1 0.5590985 0.1808916
MI − − − − 1 0.2446375WM − − − − − 1AK 1 0.1580187 0.0902602 0.0766932 0.0922525 0.1139641
2018- AM − 1 0.6874822 0.5985045 0.7364774 0.23538712019 AP − − 1 0.4927705 0.6563119 0.1773758(Σ5) FA − − − 1 0.5133173 0.0338976
MI − − − − 1 0.2756855WM − − − − − 1
Where i = 1, 2, 3, and 5 for the one-year periods of 1/5/2014-1/5/2015, 1/5/2015-1/5/2016,1/5/2016-1/5/2017, and 1/5/2018-1/5/2019, respectively. Simultaneously, FSi1 , ..., FSi6and fSi1 , ..., fSi6 are univariate cumulative distribution functions and density functions ofdaily returns series of AK; AM; AP; FA; MI; and WM with stable parameters (α, β, γ, δ)given in Table 6, whilst Σi is the correlation matrix presented in Table 7.
In short, the multivariate stable density functions can be directly used to compute theimplied distribution of any portfolio of 6 assets AK; AM; AP; FA; MI; and WM. As thejoint distribution of the vector of asset-returns is a multivariate stable distribution, theunivariate distribution of returns of any portfolio of these assets is also stable. This ap-proach can be used to solve many problems related to portfolio selection.
Acknowledgment. The research is funded by the Vietnam National Foundation forSciences and Technology Development (NAFOSTED) under grant number 101.03-2017.07,and is partially supported by the International Center for Research and Postgraduate
Gaussian copula of stable random vectors and application 901
Training in Mathematics (ICRTM) under grant number ICRTM01-2020.03. Thanks aredue to Dr. Bui Quang Nam for the sharing the GPS data used in the application part ofthis study.
References[1] R. J. Adler, R. E. Feldman, M. S Taqqu. A Practical Guide to Heavy Tailed Data,
Birkhäuser, Boston, 1998.[2] N. Bui Quang. On stable probability distributions and statistical application, Thesis,
Academy of military science and technology, Ha Noi, 2016.[3] P. Embrechts, F. Lindskog, A. McNeil. Modelling Dependence with Copulas and Ap-
plications to Risk Management, Handbook of Heavy Tailed Distributions in Finance,2003, ed. Rachev S., Elsevier, Chapter 8, 329–384, 2001.
[4] E. Fama. The behavior of stock prices, Journal of Business 38, 34–105, 1965.[5] C. Genest, B. Rémillard. Validity of the parametric bootstrap for goodness-of-fit test-
ing in semiparametric models, Annales de lInstitut Henri Poincaré - Probabilités etStatistiques 44 no. 6, 1096–1127, 2008.
[6] S. M. Kogon, D. B. Williams. Characteristic function based estimation of stable pa-rameters, In Adler R., Feldman R. and Taqqu M. (eds.) A Practical Guide to HeavyTailed Data, Birkhäuser, Boston, MA, 311–335, 1998.
[7] I. Kojadinovic, J. Yan. Modeling Multivariate Distributions with Continuous MarginsUsing the copula R Package, J. Stat. Softw. 34 no. 9, 1–20, 2010.
[8] R. M. Kunst. Apparently stable increments in finance data: Could ARCH effects bethe cause?, J. Statist. Comput. Simulation 45, 121–127, 1993.
[9] F. Lamantia, S. Ortobelli, S. Rachev. VaR, CVaR and Time Rules with Ellipticaland Asymmetric Stable Distributed Returns, Investment Management and FinancialInnovations 3, Issue 4, 19–39, 2006.
[10] J. H. McCulloch. Simple consistent estimators of stable distribution parameters,Comm. Statist. Simulation Comput. 15, 1109–1136, 1986.
[11] J. H. McCulloch. Financial applications of stable distributions, Handbook of Statistics14, ed. G. Maddala and C. Rao, Elsevier Science Publishers, North-Holland, 393–425,1996.
[12] J.P. Nolan. Multivariate elliptically contoured stable distributions: theory and estima-tion, Comput. Statist. 28, Issue 5, 2067–2089, 2013.
[13] J.P. Nolan. Stable Distributions - Models for Heavy Tailed Data, Birkhauser: Boston,MA, USA, 2016.
[14] K. J. Palmer, M. S. Ridout, B. J. T. Morgan. Modelling cell generation times usingthe tempered stable distribution, J. R. Stat. Soc. Ser. C. Appl. Stat. 57, 379–397, 2008.
[15] G. Samorodnitsky, M. S. Taqqu. Stable Non-Gaussian Random Processes, New York,NY, Chapman & Hall, 1994.
[16] P. Samuelson. Efficient portfolio selection for Pareto - Lévy investments, J. Financ.Quant. Anal. 2, 107–117, 1967.
[17] A. Sklar. Fonctions de rèpartition à n dimensions et leurs marges. Publications del’Institut de Statistique de l’Universitè de Paris 8, 229–231, 1959.
[18] M. S. Taqqu. The modeling of Ethernet data and of signals that are heavy-tailed withinfinite variance. Scand. J. Stat. 829, 273–295, 2002.
Hacettepe Journal ofMathematics & Statistics
Hacet. J. Math. Stat.Volume 49 (2) (2020), 902 – 913
DOI : 10.15672/hujms.510261
Research Article
An adaptation of pseudo-score confidence intervalmethod for linear mixed models
Hatice Tul Kubra Akdur∗, Deniz Ozonur, Hulya BayrakDepartment of Statistics, Faculty of Science, Gazi University, Ankara, Turkey
AbstractWe adapt a confidence interval method based on a generalized Chi-Square test for fixed ef-fect parameters of linear mixed models. Under different correlation structure of a responsevariable of a linear mixed model, the performances of the adaptation method pseudo-scoreand some of the existing confidence interval methods are investigated by carrying out aMonte Carlo simulation study. The simulation study suggests that pseudo-score methodprovides better results for small to moderate sample size cases with dependent observa-tions in terms of coverage probability rates and average expected lengths. A depressionstudy is analyzed for demonstrating the adaptation method.
Mathematics Subject Classification (2010). 62J05, 62F25, 62F05
Keywords. linear mixed models, pseudo-score, profile likelihood
1. IntroductionLinear mixed models (LMMs) including a very broad class of models are frequently
utilized to fit and analyze longitudinal, clustered, or repeated measures datasets whichare arisen from many areas of application such as agriculture, medical sciences, economicsand engineering. An important issue in the analysis of longitudinal or repeated measuresdata is that the observations usually show dependency structure within-subject. In orderto obtain valid inferences of model parameters of LMMs, the correlation structure of ob-servations within-subject should be identified accurately in the first step of an analysis.An efficient way is to use extended LMMs which allow modelling within-subject corre-lation and variation [18]. In biopharmaceutical clinical trials, LMM is quiet pervasivemethod. For instance, placebo-controlled schizophrenia study was carried out by fittingthree-level linear mixed model to the data in order to determine sample size for futurestudies in psychiatric research area [6]. In order to compare anti-depressant effects in anelderly depressed group of patients a linear mixed model including random intercept andslope was fitted to the dataset containing repeated measures of depression severity overtime [12]. By allowing both fixed effects and random effects, LMMs provide very usabletools for modeling within-subject and between subject variations. A covariance structureof repeated measures can be explained as a function of time by including random effectsto a linear model. In this study, main attention is the LMMs related to longitudinal∗Corresponding Author.
Email addresses: [email protected] (H. T. K. Akdur), [email protected] (D. Ozonur),[email protected] (H. Bayrak)Received: 08.01.2019; Accepted: 11.03.2020
An adaptation of pseudo-score confidence interval method 903
studies and repeated measures designs which include correlated measurements. Statisticalinferences in LMMs usually focus on fixed effect parameters and they are implemented byusing hypothesis tests or confidence intervals methods. In both cases, researchers need tofind a good fixed effect estimator which has small mean square error (MSE). Constructingconfidence intervals for fixed effects require estimators of fixed effect, variance estimationsof random effects and error terms in LMMs. Most common variance estimations meth-ods are ANOVA, maximum likelihood (ML), restricted (or residual) maximum likelihood(REML). ML based methods for model parameters in LMMs can be carried out by usingnumerical optimization algorithms such as Newton-Raphson, EM, Fisher-Scoring due tothe model likelihood complexity. MSE estimator for REML of fixed effect in LMMs wassuggested by expanding [13] approximation to reduce bias in small samples [14]. It hasbeen recently pointed out that variance-covariance structures of LMMs affect power ofapproximate F test, Kenward-Roger or Satterthwaite approximation of degrees of free-dom for hypothesis testing of fixed effect parameters based on simulation-based samplesize study in LMMs [3]. Most widely used methods for constructing confidence intervalsare Wald, t-naive [8], profile likelihood and Bayesian methods. Recent studies have pro-vided that parametric bootstrap approach can be better alternative for making inferenceabout fixed effect parameter of LMMs. Parametric bootstrap confidence interval meth-ods (PBCI) have been investigated and reported that bootstrap-t is the best alternativeamong bootstrap methods [21,22]. Also, some of existing confidence interval methods hasbeen recently investigated for nested error regression model and it is reported that PBCIcan be a good alternative for weak correlation under small samples [2]. Computationaltime and coverage rate accuracy of some confidence interval methods including paramet-ric bootstrap (PB) were examined in repeated measures degradation model [16]. Eventhough PB method seems a good alternative, its computational time is quiet unbearablefor a model with too many parameters. Furthermore, other drawback of PB method isthat rely on distributional assumptions which may not be held. Many contributions to thestatistics literature provide estimation theory of LMMs parameters and applications forLMMs. However, very few of these studies addresses the issue of performances of confi-dence interval (CI) methods under different correlation structures of the response variableof LMMs. Balancing the coverage accuracy and the narrowness of average length of a CImethod is desired against the relative computational cost. Since the existing CI meth-ods have their own drawbacks, our aim is to search for a new confidence interval methodin LMMs. In this study, we have adapted pseudo-score confidence interval method forfixed effect parameters of LMMs under different correlation settings for small to moderatesample sizes by using Kenward-Roger method for obtaining standard errors of fixed effectestimators. Pseudo-score confidence interval was firstly introduced for discrete model pa-rameters [1]. The generalized form of Pearson-Chi square statistic provided by [17] wasused for constructing confidence interval for fixed effect parameters of LMMs in this study.In the simulation study, well-known covariance structures for longitudinal studies and re-peated measures data-sets were exploited to model covariance structures of error termswithin-subjects in order to investigate effect of covariance structures on the performancesof the confidence interval methods.
In Section 2, general form of LMMs and the assumptions are provided briefly. InSection 3, some of existing confidence interval methods are summarized and pseudo-scoreconfidence interval method for fixed effects of LMM are introduced. Section 4 includes asimulation study for scenarios of variance-covariance matrix for the response of a randomslope model which is special case of LMMs. In Section 5, previously analyzed longitudinalstudy is used with a random slope model and the confidence intervals of fixed effects areobtained for each method.
904 H.T.K. Akdur, D. Ozonur, H.Bayrak
2. Extended Gaussian linear mixed modelsExtended Gaussian linear mixed model gives flexibility to basic LMM with indepen-
dent errors within-subject by allowing heteroscedastic correlated errors within-subject.Extended Gaussian linear mixed model was given by [18] as follow,
Yi = Xiβ + Zibi + εi i = 1, . . . , nbi ∼ N (0, G) , εi ∼ N (0, Ri) , cov (bi, εi) = 0
V (Yi) = ZiGZ′i + Ri.
(2.1)
Equation 2.1 represents the marginal formulation of linear mixed model and n representsthe number of independent experimental units (subjects or clusters). Yi denotes a (mx1)vector of observations within unit (for balanced sample size within each unit) i ; Xi is a(mxp) known covariates design matrix for unit i ; β is a (px1) unknown vector of fixedeffects parameters. Zi is a (mxq) known design matrix for unit i associated with a (qx1)unknown vector of random effects bi. bi is a (qx1) unknown random effect vector forunit i and it is distributed normally with zero mean and covariance matrix G. εi is anunobserved vector of random errors distributed normally with zero mean and covariancematrix Ri. The columns of Zi can be subset of columns of Xi. This usage allows populationparameters β vary randomly among independent experimental units. In this study, wefocus on performances of confidence interval methods described in Section 3 when thecovariance matrix of V (Yi) differs based on the covariance matrix Ri of random errors inthe LMMs setting of complete and balanced sample size within-subject.
The likelihood function for the linear mixed model in Equation 2.1 is
L (α, β) =n∏
i=1(2π)−m/2|Vi (α)|−1/2 exp
(−12
(Yi − Xiβ)′V −1
i (α) (Yi − Xiβ))
(2.2)
where α denote the vector of all variance and covariance parameters found in V (Yi) =ZiGZ
′i + Ri.
3. The confidence interval methodsThis section describes some of the existing methods of confidence interval and the pro-
posed method, namely pseudo-score confidence interval method, for fixed effect parametersof Gaussian linear mixed models.
3.1. Approximate Wald confidence interval methodIn parameter space of fixed effect parameters β, k = 1, . . . , p , an approximate Wald
test (z-test) and related confidence interval is obtained based on asymptotic normality ofmaximum likelihood estimators of fixed effect parameters. The Wald statistic is writtenfor k = 1, . . . , p
βk − βk
SE(βk)(3.1)
where βk is a maximum likelihood estimator or a restricted maximum likelihood estima-tor of (βk) and SE(βk) is a standard error of βk. The standard error of βk is obtainedby using negative expectation of Hessian matrix (Fisher information matrix) or observedinformation matrix. The Wald statistic in Equation 3.1 can be seen as a pivot in con-fidence interval framework. The pivot distribution is assumed to have standard normaldistribution for sufficient sample sizes. Hence z-distribution percentiles are used for con-structing approximate Wald confidence interval. βk ± z1− α
2SE(βk) denotes asymptotic
An adaptation of pseudo-score confidence interval method 905
100 (1 − α) % Wald confidence interval for βk where z1− α2
is the 1− α2 quantile of standard
normal distribution.
3.2. Approximate t-naive confidence interval methodUnder small samples when the variance of the fixed effect estimator is unknown, max-
imum likelihood estimator of a fixed effect may not be distributed normally. The pivotdistribution in Equation 2.2 may not have standard normal. Also the estimated standarderrors which underestimate the true variability in fixed effect estimators. Because they donot take into account the variability arisen from estimating random effects [23]. Student-t distribution cut-off is suggested [8] to use instead of z-distribution cut-off. Given thedegrees of freedom n∗m−p for the pivot statistic, asymptotic 100 (1 − α) % t-naive confi-dence interval for βk is βk±tn∗m−p;1− α
2SE
(βk
)where tn∗m−p;1− α
2is the 1− α
2 with n∗m−p
(for balanced sample size cases) degress of freedom quantile of student-t distribution [8].
3.3. Approximate profile likelihood confidence interval methodDue to skewed shape of the log-likelihood function or excessive number of nuisance
parameter in the model, Wald-type methods may fail to provide accurate interval estima-tion. Profile likelihood (PL) method is more preferable in this situation because it does notrequire asymptotic normality of maximum likelihood estimators. Assume that unknownparameter vector θ is partitioned as θ = (βk, δ) where βk is the parameter of interest, fixedeffect parameters of LMMs in this paper, and δ is the additional parameters of the model.Let L (βk, δ) represents the likelihood function of the model. For βk, profile likelihoodfunction is L1 (βk) = maxL (βk, δ). L1 (βk) is the maximum likelihood function over theremaining parameters for each fixed value of βk = βk(0). In order to obtain PL confidenceinterval for parameter of interest, the likelihood ratio test of a two-sided hypothesis shouldbe inverted. Therefore, profile likelihood method still depends on asymptotic property oflikelihood ratio test which is assumed to be distributed with chi-square distribution withone degree of freedom. Likelihood ratio test is used for comparing nested models withdifferent mean structures. For a two-sided test of null hypothesis H0 : βk = βk(0), the like-lihood ratio test statistic is the difference between log-likelihood of full model and reducedmodel:
2[logL
(βk, δ
)− logL
(βk(0), δ0
)](3.2)
where βk and δ are the MLEs for the full model and δ0 is the MLE of δ for the reducedmodel with βk = βk(0). Hence, a 100 (1 − α) % CI for βk includes the values of βk(0) wherethe test is non-significant at the α level as shown below in the inequality:
2[logL
(βk, δ
)− logL
(βk(0), δ0
)]< χ2
1,α. (3.3)
SAS Proc Mixed, Proc Glimmix procedures and R lme4 package provide a general frame-work for profile confidence interval for linear mixed model and generalized linear mixedmodel [4, 24]. However, in this paper, we used a simpler algorithm which in many situa-tions, such as random slope model which is used in the simulation study, could be relativelyeasy to implement. Consider lower bound of confidence interval and assume that profilelikelihood function is an increasing function. Firstly, ML estimates of (βk, δ) should beobtained for full model. And then implemented by the following steps:
(1) For computing a lower confidence limit, a lower bound can be obtained by β′k =
βk − 5xSE(βk).(2) From β
′k to βk define a grid of values in the range (300 points ).
(3) For each point βk(i) in the range, profile likelihood value is computed by maximizinglogL
(βk(i), δ (i)
)over δ values.
906 H.T.K. Akdur, D. Ozonur, H.Bayrak
(4) Lower bound of 95% confidence interval is taken as the smallest βk(i) value for whichits profile likelihood value, logL1(βk(i)), satisfies this inequality, logL1(βk(i)) >
logL(βk, δ
)− 1.92.
The upper confidence limit is similarly estimated using the algorithm. When necessary,the grid of values around lower bound or similarly for upper bound can be expanded fora greater accuracy. Step 4 and Step 5 are implemented using uniroot function of R whichutilizes bi-section algorithm over the interval. More points may be required when thenumber of random effects and fixed effect parameters in the model are increased.
3.4. Approximate Pseudo-score confidence interval methodAssume that Yi is maximum likelihood fitted values of Yi for a particular linear mixed
model. Let Yi0 be maximum likelihood fitted values of a reduced model or null model inwhich fixed effect parameter, βk, takes a fixed value, βk0, in a reasonable interval of itsparameter space. This study implements generalized form of Pearson statistics suggestedby [17] to construct a new confidence interval for a fixed effect for a LMM. The generalizedform of Pearson statistic is given below:
X2 =∑
i
(Yi − Yi0
)2
var(Yi0
) =(Y − Y0
)TV −1
0
(Y − Y0
)(3.4)
where var(Yi0
)is the estimated variance of Yi under the reduced/null model and V0
is the variance-covariance matrix of these values [17]. Although the Pearson statistic isused for constructing confidence interval for discrete model parameter, we adapted thisgeneralized form for continuous response model by using (adjusted) estimated covariancematrix of linear mixed model [14] to construct an asymptotic confidence interval for a fixedeffect parameter. Because Kenward and Roger stated that Wald-type test procedures andcorresponding confidence intervals that are based on a conventional estimate of asymptoticcovariance matrix ignore the variability in the estimate of V (Yi) and finally block-diagonalmatrix of LMM, V . Yi0 is obtained by fitting the reduced model under the constraint,β = βk0. χ2
1,α is the 1 − α of chi-squared distribution with one degree of freedom. Anasymptotic 100 (1 − α) % CI for any fixed effect parameter, βk, is obtained by invertingthe generalized Pearson statistic. The asymptotic CI for βk includes such values of βk0,that satisfies the following inequality,
X2(βk0) =∑
i
(Yi − Yi0
)2
var(Yi0
) < χ21,α. (3.5)
4. Simulation studyIn order to investigate performances of four confidence interval methods presented in
Section 3 in terms of coverage probability rate and average expected length, a broad simula-tion study is carried out with different simulation settings. In the context of longitudinalstudy and repeated measures design, our simulation study is formed by differentiatingwithin-subject covariance matrix, Ri. For all simulation settings, the following LMMwith random intercept and random slope that vary randomly among subjects for the ith
experimental unit at the jth measurement time point is taken into the simulation study,Yij = β0 + β1xij + β2tij + b1i + b2itij + εij i = 1, . . . , n; j = 1, . . . , m. (4.1)
In the random slope model in the equation 4.1, the fixed effect parameters are β0, β1,β2 and random effect parameters are b1i, b2i for i = 1, ..., n. The continuous covariateswere generated from the normal distribution, xij ∼ Uniform (0, 4). It is known that two
An adaptation of pseudo-score confidence interval method 907
sample size concepts are involved in LMMs. One of them represents number of experi-mental unit/ cluster, n, and the other represents number of observations/ measurementswithin each experimental unit/ subject, m. In the simulation study, β0=0, β1=3, β2=0.87are taken as values of fixed effect parameters. The variance-covariance matrix for random
effects in the model is taken as unstructured matrix, G = V ar (bi) =[
σ2b1
σb1,b2σb2,b1
σ2b2
]=[
5.55 0.81830.8183 8.0238
]for all simulation cases. We were inspired by real applications when
choosing G and Ri matrices for random slope model in the simulation cases [10]. Mul-tivariate normal distribution was used to generate random effects with zero mean andcovariance matrix, G, using mvtnorm R package [10]. Alternatively, Gaussian copula withcopula package can be also used to jointly generate b1i and b2i with a given correlation co-efficient using the normal distribution margins instead of multivariate normal distributionbecause Gaussian copula describes dependence in the same way that multivariate normaldistribution does. [7,26]. Using the same mean model and random effect covariance struc-ture, different covariance structures for random error terms, V ar (εi) = Ri , and σ2
εi= 4.41
were taken into account in the simulation study. MLE-based estimations of fixed effectparameters, Wald CIs for β1, and conventional variance-covariance matrices were obtainedby performing lme4 R package (version 1.1-15) [4]. The covariance estimations based onKenward-Roger approximation for pseudo-score method were obtained by using pbkrtestR package [11]. Two-sided 95% confidence intervals were constructed for each case. Thesimulation cases were summarized below:
Case 1. Ri is chosen as AR(1), auto-regressive correlation is assumed between errorterms within each unit. AR(1) structure provides that the correlations decline over timeas the time interval between observations of repeated measures increases. Repeated mea-surements should be measured at equal points of time in order to use this correlationstructure.
tij = (8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46); m = 20.tij = (8, 10, 12, 14, 16, 18, 20, 22, 24, 26); m = 10.tij = (8, 10, 12, 14, 16); m = 5.Case 2. Ri is chosen as compound symmetry structure, same correlation is assumed
between error terms within each unit. Same vectors with AR(1) covariance model for timecovariates tij are used for this covariance model.
Case 3. Ri is chosen as an exponential structure, it is suitable for the situation that themeasurements are not made equally spaced over time. ti1, . . . , tim represents the timepoints for the ith experimental unit under the constant variance assumption for randomerror terms within each experimental unit,
corr (eij , eik) = ρ|tij−tik|,
cov (eij , eik) = σ2ρ|tij−tik| = σ2exp (−θ |tij − tik|) , (4.2)
where θ = −log (ρ) for θ > 0. The structure allows the correlation between error termsor any pair of measurements within each experimental unit decreases exponentially as thetime intervals increase between any pair of measurements [9].
tij = (2, 3, 5, 8, 9, 10, 13, 15, 19, 20, 24, 26, 29, 30, 33, 35, 38, 40, 41, 44); m = 20.
tij = (2, 3, 5, 8, 9, 10, 13, 15, 19, 20); m = 10.
908 H.T.K. Akdur, D. Ozonur, H.Bayrak
tij = (2, 3, 5, 8, 9); m = 5.
Case 4. Ri is chosen as a banded unstructured. The banded covariance structureassumes that correlation is zero after some specified time point. A banded pattern can beapplied to any covariance structure. Same vectors with AR(1) covariance model for timecovariate tij is used for this covariance model. For the time points m = 5 and bandingparameter c = 1, unstructured banded covariance matrix example which is used in thesimulation study is given below:
Ri =
2.8 0.1 0 0 00.1 0.3 0.2 0 00 0.2 0.4 0.1 00 0 0.1 0.5 0.230 0 0 0.23 0.6
(4.3)
The simulation results of the confidence interval methods were summarized in Table 1 toTable 4 for all cases.
Table 1. Coverage probability rates and average expected lengths of the confidenceinterval methods for the Case 1 for ρ = 0.25, 0.5, 0.75 at the 95% confidence level.
Sample Sizes Estimated Coverage Probability Rates Estimated Average Lengths
ρ n m Wald Naive Pseudo-Score Profile Wald Naive Pseudo-Score Profile
0.25
5 5 0.8897 0.9083 0.9205 0.9936 3.1325 3.3145 3.5581 5.91365 10 0.9318 0.9380 0.9626 0.9944 2.1795 2.2370 2.5840 3.48515 20 0.946 0.9482 0.9538 0.9924 1.5281 1.5474 1.5932 2.255710 5 0.9237 0.9309 0.9360 0.9911 2.2827 2.3430 2.4472 3.630910 10 0.9404 0.9428 0.9474 0.9918 1.5685 1.5883 1.6418 2.256810 20 0.9454 0.9464 0.9528 0.9911 1.0883 1.0950 1.1230 1.4784
0.5
5 5 0.9032 0.9182 0.9370 0.9921 3.0426 3.2193 3.7994 5.72955 10 0.9296 0.9366 0.9398 0.9944 2.1428 2.1994 2.2912 3.43215 20 0.9386 0.9414 0.9482 0.9924 1.5100 1.5291 1.5706 2.2239410 5 0.9272 0.9328 0.9394 0.9950 2.2279 2.2867 2.3996 3.544310 10 0.9416 0.9456 0.9516 0.9926 1.5373 1.5569 1.6081 2.214110 20 0.9466 0.9488 0.9534 0.9901 1.073 1.0802 1.1070 1.4650
0.75
5 5 0.8837 0.9007 0.9371 0.9931 2.9496 3.1210 3.7253 5.53525 10 0.9313 0.9397 0.9459 0.9935 2.0723 2.1270 2.2128 3.31585 20 0.9492 0.9518 0.9554 0.9954 1.4659 1.4843 1.5240 2.182010 5 0.9256 0.9330 0.9438 0.9916 2.1742 2.2316 2.3553 3.456710 10 0.9406 0.9442 0.9504 0.9915 1.4912 1.5101 1.5602 2.155810 20 0.9496 0.9514 0.9544 0.9894 1.0461 1.0525 1.0779 1.4400
An adaptation of pseudo-score confidence interval method 909
Table 2. Coverage probability rates and average expected lengths of the confidenceinterval methods for the Case 2 for ρ = 0.25, 0.5, 0.75 at the 95% confidence level.
Sample Sizes Estimated Coverage Probability Rates Estimated Average Lengths
ρ n m Wald Naive Pseudo-Score Profile Wald Naive Pseudo-Score Profile
0.25
5 5 0.8869 0.9040 0.9632 0.9887 1.3844 1.4648 1.8342 2.34995 10 0.9240 0.9310 0.9672 0.9876 0.9352 0.9599 1.1236 1.38205 20 0.9372 0.9400 0.9488 0.9888 0.6432 0.6513 0.6665 0.904110 5 0.9212 0.9280 0.9424 0.9884 1.0221 1.0491 1.1166 1.482610 10 0.9430 0.9464 0.9534 0.9894 0.6730 0.6815 0.6997 0.907310 20 0.9456 0.9464 0.9520 0.9874 0.4570 0.4598 0.4690 0.6003
0.5
5 5 0.8896 0.9116 0.9344 0.9858 1.1531 1.2201 1.3434 1.93815 10 0.9353 0.9413 0.9705 0.9858 0.7750 0.7955 0.9238 1.14145 20 0.9428 0.9450 0.9501 0.9928 0.5275 0.5342 0.5446 0.745110 5 0.9252 0.9326 0.9470 0.9868 0.8564 0.8790 0.9255 1.232910 10 0.9430 0.9468 0.9522 0.9904 0.5545 0.5615 0.5740 0.750910 20 0.9406 0.9430 0.9478 0.9880 0.3742 0.3765 0.3836 0.4966
0.75
5 5 0.9012 0.9172 0.9644 0.9882 0.8506 0.9000 1.0810 1.40785 10 0.9390 0.9442 0.9496 0.9908 0.5524 0.5669 0.5793 0.81215 20 0.9472 0.9504 0.9540 0.9936 0.3729 0.3776 0.3837 0.529310 5 0.9282 0.9346 0.9424 0.9862 0.6221 0.6385 0.6614 0.888610 10 0.9482 0.9514 0.9554 0.9906 0.3953 0.4003 0.4075 0.535610 20 0.9482 0.9514 0.9560 0.9914 0.2648 0.2664 0.2712 0.3533
Table 3. Coverage probability rates and average expected lengths of the confidenceinterval methods for the Case 3 for ρ = 0.25, 0.5, 0.75 at the 95% confidence level.
Sample Sizes Estimated Coverage Probability Rates Estimated Average Lengths
ρ n m Wald Naive Pseudo-Score Profile Wald Naive Pseudo-Score Profile
0.25
5 5 0.8889 0.9050 0.9560 0.9860 1.4478 1.5637 1.9592 2.52335 10 0.9312 0.9364 0.9688 0.9894 1.0454 1.0730 1.2596 1.54505 20 0.9432 0.9452 0.9504 0.9878 0.7330 0.7423 0.7611 1.024010 5 0.9288 0.9364 0.9496 0.9900 1.0944 1.1233 1.1908 1.591910 10 0.9450 0.9468 0.9530 0.9902 0.7530 0.7625 0.7836 1.013010 20 0.9458 0.9469 0.9580 0.9859 0.5215 0.5247 0.5356 0.6792
0.5
5 5 0.8958 0.9126 0.9655 0.9895 1.3210 1.3978 1.7132 2.25025 10 0.9352 0.9418 0.9720 0.9908 0.9930 1.0192 1.1912 1.47075 20 0.9420 0.9452 0.9510 0.9896 0.7129 0.7218 0.7391 1.001410 5 0.9238 0.9306 0.9488 0.9880 1.0909 1.1197 1.1874 1.586310 10 0.9419 0.9452 0.9524 0.9896 0.7125 0.7214 0.7400 0.962910 20 0.9496 0.9504 0.9544 0.9874 0.5208 0.5240 0.5348 0.6789
0.75
5 5 0.9012 0.9159 0.9588 0.9866 1.0028 1.0611 1.2701 1.69215 10 0.9374 0.9442 0.9742 0.9910 0.8527 0.8752 1.0142 1.27015 20 0.9516 0.9538 0.9590 0.9942 0.6547 0.6629 0.6774 0.926510 5 0.9360 0.9436 0.9506 0.9896 0.7333 0.7526 0.7781 1.067510 10 0.9400 0.9426 0.9490 0.9880 0.6123 0.6200 0.6334 0.836210 20 0.9510 0.9520 0.9578 0.9876 0.4668 0.4697 0.4788 0.6182
As shown in Table 1, the coverage rates of Wald and t-naive methods often approxi-mately were around 0.90 for small sample size cases for n = 5 and m = 5, 10 for AR(1),compound symmetry and exponential covariance structures. Approximate pseudo-scoremethod provided higher coverage probability rates than Wald and t-naive methods. Pro-file likelihood method provided the highest coverage probability rates due to the widest
910 H.T.K. Akdur, D. Ozonur, H.Bayrak
average expected lengths. As the correlation terms of AR(1), compound symmetry and ex-ponential structures got stronger, coverage probability rates of Wald and t-naive methodsare generally slightly increased.
The performances of confidence interval methods for the compound symmetry covari-ance structure of random error terms showed similar behavior to the autoregressive caseabove. Coverage probability rates of pseudo-score method in compound symmetry struc-ture slightly were better than those in AR(1) covariance structure for especially in smallsamples.
Table 4. Coverage probability rates and average expected lengths of the confidenceinterval methods for the Case 4 at the 95% confidence level.
Sample Sizes Estimated Coverage Probability Rates Estimated Average Lengths
n m Wald Naive Pseudo-Score Profile Wald Naive Pseudo-Score Profile
5 5 0.8909 0.9084 0.9647 0.9899 1.2590 1.3331 1.6539 2.10685 10 0.9300 0.9376 0.9488 0.9886 1.1920 1.2235 1.2838 1.77075 20 0.9444 0.9466 0.9496 0.9911 1.1355 1.1499 1.1893 1.617010 5 0.9280 0.9356 0.9478 0.9854 0.9839 1.0099 1.0722 1.403710 10 0.9410 0.9440 0.9510 0.9876 0.8587 0.8695 0.8967 1.149910 20 0.9377 0.9403 0.9470 0.9887 0.8112 0.8162 0.8366 1.069
For banded covariance structure of random errors, approximate pseudo-score methodproduced the closest coverage probability rates to the nominal confidence level even forsmall sample sizes.
5. ApplicationA psychological longitudinal study is used as an example dataset and a sketch of it is
given in Table 5 to demonstrate the confidence interval methods [15]. Since we considersmall-sample cases in our simulation study, we only include a subset of the dataset whichcontains observations of ten subjects for our analysis. This longitudinal study focuses onnew treatment on the depression patients aged 18 to 75 who are not treated with anytreatment method [20]. The patients are randomly allocated to two treatment groups:computer based treatment (Beating the Blues) and usual treatment group (Tau). Theresponse variable, depression score of the patients is measured by Beck Depression Inven-tory [5] on five time points: prior to treatment (baseline), 2 months later, and at 1, 3, and6 months follow-up. The effect of duration of depression is also included in the study bycategorizing the duration time: if a patient had been ill for longer than six months codedduration=1 or for less than six months coded duration=0. In this paper, the patientswith missing values are excluded from the dataset only the patients with complete mea-surements in all time points are included to the analysis. The random slope and randomintercept model is fitted to the dataset. In the model, depression scores as a responsevariable, baseline as a continuous covariate, treatment coded 1 and 0, respectively showsBeating the Blues and Tau, as categorical covariate and duration as an another categoricalcovariate are taken . Here the following linear mixed model for the response is given onthe patient i at time tj for i = 1, . . . , 10; j = 1, . . . , 4, ,
Yij = β0 + β1Treatmenti + β2Durationi + β3Baselinei + β4tj + b1i + b2itj + εij . (5.1)
In order to choose an appropriate covariance structure, compound symmetry, exponen-tial, unstructured and AR(1) covariance models were considered and fitted to the datasetby using nlme R package [19]. Since time intervals are not equal in the application as givenin the sketch of the dataset, exponential covariance structure would be more preferable
An adaptation of pseudo-score confidence interval method 911
Table 5. A subset of a dataset of a psycological longitudinal study
Subject Duration Treatment Baseline Month Depr-Score
2 1 1 32 2 162 1 1 32 3 242 1 1 32 5 172 1 1 32 8 204 1 1 21 2 174 1 1 21 3 164 1 1 21 5 104 1 1 21 8 97 0 0 17 2 77 0 0 17 3 77 0 0 17 5 37 0 0 17 8 78 1 0 20 2 208 1 0 20 3 218 1 0 20 5 198 1 0 20 8 13
for this dataset according to Table 6. Parameter estimations, standard errors, t-valuesand confidence interval estimations of fixed effects under this covariance structure aresummarized in Table 7.
Table 6. Likelihood-based values of Several Covariance Models
Covariance Model AIC BIC log-likelihoodAR(1) 254.71 270.26 -117.35Unstructured 259.87 283.20 -114.93Compound Symmetry 256.57 272.12 -118.28Exponential 254.71 270.27 -117.35
Table 7. Parameter estimations, standard errors, t-values and confidence interval esti-mations of fixed effects at the 95% confidence level
Parameter β0 β1 β2 β3 β4Estimation(REML) -1.704 -0.800 -2.405 2.342 0.846
Standard Error 8.123 0.581 3.424 5.033 0.404K-R Stn. Error 9.230 0.581 3.954 5.812 0.466
Wald CI (-17.626; 14.218) (-1.938; 0.338) (-9.118; 4.307) (-7.522; 12.208) (0.054; 1.639)Profile CI (-15.496; 12.338) (-1.992; 0.392) (-9.757; 4.946) (-6.105; 10.790) (0.131; 1.561)t-naive CI (-18.196; 14.788) (-1.979; 0.379) (-9.358; 4.547) (-7.876; 12.561) (0.026; 1.667)
Pseudo-score CI (-16.476; 13.318) (-1.962; 0.362) (-9.314; 4.504) (-7.105; 11.790) (0.031; 1.661)
Based on the simulation study results, it was concluded that pseudo-score methodwas more preferable than the other methods. Therefore, pseudo-score method confidenceinterval results for the fixed effect parameters would be more appropriate for makinginference in this dataset.
6. DiscussionThis paper contributed the idea of using pseudo-score confidence interval method with
Kenward-Roger variance estimation method for obtaining confidence intervals for fixed
912 H.T.K. Akdur, D. Ozonur, H.Bayrak
effect parameters in linear mixed models. In this study, the covariance structure of re-sponse of a LMM was composed from two sources of variation: first variation was derivedfrom random effects and its covariance structure was called G (unstructured) and secondvariation was derived from random error terms and covariance structure was modeled byRi. Hence a hybrid covariance structure of Vi was produced for each covariance case inthe simulation study. Approximate pseudo-score method was adapted for random slopemodel when the dependent observations within-subject was existed.
The simulation study suggested that pseudo-score confidence interval method usuallyprovided close covarege rates to nominal 95% confidence level for small to moderate sam-ple sizes. Wald method provides liberal results whereas profile likelihood method providesvery conservative results. Pseudo-Score method generally produces better coverage ratesthan Wald and t-naive methods for especially small sample size cases. In cases where thecovariance structure of random error terms within subject are autoregressive, compoundsymmetry, exponential and banded it can be said that more preferable confidence inter-val method is approximate pseudo-score method in terms of coverage probability rates.Pseudo-score method not only regulates the disadvantage of wider lengths of confidenceintervals provided by profile likelihood but it also regulates poor coverage rates of Waldand t-naive methods. Therefore, pseudo-score confidence interval method usually balancesthese two disadvantages of the existing methods. There can be limitation of the simulationin this study. The proposed approach can be investigated with other covariance structuresof linear mixed models such as ARMA or MA as a future study. Recently, Wu and deLeon (2014) have proposed Gaussian copula mixed models for correlated mixed response[25]. Gaussian copula mixed models are flexible in terms of modeling multiple correlateddiscrete or continuous responses together in a joint analysis. The proposed confidenceinterval method can be extended for fixed effect model parameters of Gaussian copulamixed models as an another future study.
Acknowledgment. The authors are very grateful to Dr. Alan Agresti for his valuablesuggestions. The authors thank to Dr. Alex De Leon for contributing the paper with hisvaluable comments and suggestions. Furthermore, the part of the simulation results forthe manuscript are from the author, H.T.K. AKDUR, PhD thesis that was submitted toGazi University in July 2017. The author was supported by Scientific Council of Turkey(TUBITAK) and Gazi University by a visiting PhD student grant (2214) for studyingabroad in University of Calgary under supervision of Dr. Alex De Leon during the partof this research in 2015.
References[1] A. Agresti and E. Ryu, Pseudo-score confidence intervals for parameters in discrete
statistical models, Biometrika 97 (1), 215-222, 2010.[2] H.T.K. Akdur, D. Ozonur and H. Bayrak, A Comparison of Confidence Interval
Methods of Fixed Effect in Nested Error Regression Model, SDU J Nat Appl Sci 20(2), 2016.
[3] M.K. Bahçecitapar, Some factors affecting statistical power of approximate tests inthe linear mixed model for longitudinal data, Comm. Statist. Simulation Comput. 47(1), 2018.
[4] D. Bates, M. Machler, B. Bolker and S. Walker, Fitting linear mixed-effects modelsusing lme4, arXiv preprint 1406.5823, 2014.
[5] A.T. Beck, R.A. Steer and G.K. Brown, Beck depression inventory-II, PsychologicalCorporation 78 (2), 490-498, San Antonio, TX, 1996.
An adaptation of pseudo-score confidence interval method 913
[6] D.K. Bhaumik, A. Roy, S. Aryal, K. Hur, N. Duan, S.L.T. Normand, C.H. Brown,and R.D. Gibbons, Sample size determination for studies with repeated continuousoutcomes, Psychiatr. Ann 38 (12), 2008.
[7] A. R. De Leon and K. C. Chough, Analysis of mixed data: methods and applications,CRC Press, 2013.
[8] D. Eugene, Mixed Models: Theory and Applications, Hoboken: Wiley, 2004.[9] G.M. Fitzmaurice, N.M. Laird and J.H. Ware Applied longitudinal analysis (Vol. 998),
John Wiley and Sons, 2012.[10] A. Genz, F. Bretz, T. Miwa, ... , and M.T. Hothorn, Package mvtnorm, J. Stat. Softw.
11, 950-971, 2020.[11] U. Halekoh and S. Højsgaard, A kenward-roger approximation and parametric boot-
strap methods for tests in linear mixed modelsthe R package pbkrtest, J. Stat. Softw.59 (9), 1-30, 2014.
[12] P.R. Houck, S. Mazumdar, T. Koru-Sengul, G. Tang, B.H. Mulsant, B.G. Pollockand C.F. Reynolds III, Estimating treatment effects from longitudinal clinical trialdata with missing values: comparative analyses using different methods, PsychiatryResearch 129 (2), 209-215, 2004.
[13] R.N. Kackar and D.A. Harville, Approximations for standard errors of estimators offixed and random effects in mixed linear models, J. Am. Stat. Assoc 79 (388), 853-862,1984.
[14] M. G. Kenward and J. H. Roger, Small sample inference for fixed effects from re-stricted maximum likelihood, Biometrics 53 (3), 983-997, 1997.
[15] S. Landau, A handbook of statistical analyses using SPSS, CRC, 2004.[16] J. Liu, Statistical inference for functions of the parameters of a linear mixed model,
Unpublished doctoral dissertation, Iowa State University, 2013.[17] G. Lovison, On rao score and pearson x2 statistics in generalized linear models, Statist.
Papers 46 (4), 555-574, 2005.[18] J.C. Pinheiro and D.M. Bates, Mixed-effects models in S and S-Plus, Springer. New
York. US, 2000.[19] J.C. Pinheiro, D.M. Bates, S. DebRoy, D. Sarkar, and R. C. Team, nlme: Linear and
nonlinear mixed effects models. R package version, 3 (1), 111.[20] J. Proudfoot, D. Goldberg, A. Mann, B. Everitt, I. Marks and J. Gray, Computerized,
interactive, multimedia cognitive-behavioural program for anxiety and depression ingeneral practice, Psychol. Med. 33 (2), 217-227, 2003.
[21] V.S. Staggs, Parametric bootstrap interval approach to inference for fixed effects in themixed linear model, Unpublished doctoral dissertation, University of Kansas, 2009.
[22] V.S. Staggs, Comparison of naive, kenwardroger, and parametric bootstrap intervalapproaches to small-sample inference in linear mixed models, Comm. Statist. Simu-lation Comput. 46 (3), 1933-1943, 2017.
[23] G. Verbeke Linear mixed models for longitudinal data. In Linear mixed models inpractice, 63153, Springer, 1997.
[24] W. W. Stroup, G. A. Milliken, E. A. Claassen, R. D. Wolfinger, SAS for MixedModels: Introduction and Basic Applications., SAS Institute, 2018.
[25] B. Wu and A.R. de Leon Gaussian copula mixed models for clustered mixed outcomes,with application in developmental toxicology, J. Agric. Biol. Environ. Stat.19 (1), 39-56, 2014.
[26] J. Yan, Enjoy the joy of copulas: with a package copula, J. Stat. Softw. 21 (4), 1-21,2007.
