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Booklet of abstracts

SPAS 2019

2nd International Conference onStochastic Processes and Algebraic Structures

From Theory Towards Applications

Vasteras, Sweden, September 30 – October 2, 2019

Conference is organized by:Division of Applied Mathematics

Research environment MAMMalardalen University, Vasteras, Sweden

Conference scope and goals

The goal of this conference is to showcase the frontiers of research in severalimportant topics of mathematics, mathematical statistics and its applications.The conference is organised along the following three tracks:

• AS track: Algebraic Structures and applications

• SP&MS track: Stochastic Processes/Modern Statistical methods in theoryand practice

• EM track: Engineering Mathematics

This conference brings together a selected group of mathematicians, re-searchers from related subjects and researchers and practitioners from industrywho actively contribute to theory and applications of stochastic and algebraicstructures, methods and models.

A special volume with selected contributions devoted to the topics of theconference will be published with an international publisher.

The conference will also provide earlier stage researchers with opportunitiesto learn from the subject experts, to present their research as well as to estab-lish valuable research contacts and collaboration in Sweden and internationallyimportant for their future.

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Organising and Programme Committee

Sergei Silvestrov, Division of Aplied Mathematics,Malardalen University, Vasteras, Sweden

Anatoliy Malyarenko, Division of Applied Mathematics,Malardalen University, Vasteras, Sweden

Ying Ni, Division of Applied MathematicsMalardalen University, Vasteras, Sweden

Olha Bodnar, Orebro University School of BusinessOrebro University, Sweden

Milica Rancic, Division of Applied MathematicsMalardalen University, Vasteras, Sweden

Karl Lundengard, Division of Applied MathematicsMalardalen University, Vasteras, Sweden

Christopher Engstrom, Division of Applied MathematicsMalardalen University, Vasteras, Sweden

Venue & Contact

Venue Contact

House U, Rooms: Beta, Gamma, Kappa Milica RancicMalardalen UniversityHogskoleplan 1 E-mail: [email protected] 23 Vasteras, Sweden

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Contents

SEPTEMBER 30, 2019

AS track

KEYNOTE LECTURE AS - 1Multiresolutions and Multivariable Operator Theory and Noncommu-

tative Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

COMMUNICATION SESSION

Algebraic Structures and Applications (AS - 1)Continuous Integral Operators on Subspaces of the Space of Bounded

Vector Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 13On the Existence of Nonnegative Integral Solutions of Some Classes of

Linear Equations Over Integers . . . . . . . . . . . . . . . . . . . 14Non-commuting Operators on Hilbert and Banach Spaces Satisfying

Commutation Relations Associated to Iterations of Several Maps 15

KEYNOTE LECTURE AS - 2Group Gradings on Simple Lie and Related Algebras . . . . . . . . . . 16


Algebraic Structures and Applications (AS - 2)Graded Quasi Hom-Lie Algebras . . . . . . . . . . . . . . . . . . . . . 18Hom-Lie Algebras and Deformations . . . . . . . . . . . . . . . . . . . 20Index of Hom-Lie Algebas . . . . . . . . . . . . . . . . . . . . . . . . . 21

KEYNOTE LECTURE AS - 3Drinfeld Elements and Hom-Bialgebras . . . . . . . . . . . . . . . . . . 22


Algebraic Structures and Applications (AS - 3)Strong Hom-associativity . . . . . . . . . . . . . . . . . . . . . . . . . 24Hom-Lie Structures on 3-dimensional Skew Symmetric Algebras . . . 25Hom-Gerstenhaber Algebras and Hom-Lie Algebroids . . . . . . . . . 26On Solvability and Nilpotency of (n + 1)-Hom-Lie Algebras Induced

by n-Hom-Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . 28

3

SP&MS track

KEYNOTE LECTURE SP&MS - 1Asymptotic Algorithms of Phase Space Reduction and Ergodic Theo-

rems for Perturbed Semi-Markov Type Processes . . . . . . . . . 29


Stochastic Processes / Modern Statistical methods in theory andpractice (SP&MS - 1)Information Networks and Perturbed Markov Chains with Damping

Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31PageRank and Perturbed Markov Chains . . . . . . . . . . . . . . . . 32A Variant of Updating PageRank in Evolving Tree Graphs . . . . . . 33

KEYNOTE LECTURE SP&MS - 2The Generalized Calculation of Pure Premium for Non-Life Insurance

by Generalized Non-Homogeneous Markov Reward Processes . . 34


Stochastic Processes / Modern Statistical methods in theory andpractice (SP&MS - 2)Entropy Analytics of Mortality Data Series by Mixed Binomial Distri-

bution: Sweden, France, Italy . . . . . . . . . . . . . . . . . . . . 37On Ruin Probability of Bivariate Correlated Risk Processes . . . . . . 38Risk-based Optimal Portfolio of an Insurer with Regime Switching and

Noisy Memory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Stochastic Processes / Modern Statistical methods in theory andpractice (SP&MS - 3)Extreme Points for Joint Probability Density of Wishart Matrix and

Vandermonde Determinants in Financial Mathematics . . . . . . 41Calibration of Two-Factor Stochastic Volatility Model . . . . . . . . . 42Valuation and Optimal Strategies for American Option under a Marko-

vian Regime Switching Model . . . . . . . . . . . . . . . . . . . . 43Pricing Overnight Index Swap in a Large Market Model Using a Cu-

bature Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44Analytical Studies of Implied Volatility in the Gatheral Model . . . . 45Using the Investment Problem for Option Pricing . . . . . . . . . . . . 46

EM track

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COMMUNICATION SESSIONS

Engineering Mathematics (EM - 1)Stochastic Matrix Method in Physics of Noncrystalline Condensed

Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49Accuracy of Cell Problems for Computational Homogenization of Mul-

tiscale Partial Differential Equations in Locally Periodic Media . 50Stochastic Simulations of Classical PDEs with Non-trivial Boundary

Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

Engineering Mathematics (EM - 2)Mathematical Basis of the Stochastic Digital Measurement Method . . 55Stochastic Smart Grid Energy Meter - From an Obscure Idea to the

Practical Prototype . . . . . . . . . . . . . . . . . . . . . . . . . . 57Goodness of Fit Test for Impulse Response Function . . . . . . . . . . 58

Engineering Mathematics (EM - 3)Comparison Between Finite Element Analysis and Rheological Models

for Chip Formation . . . . . . . . . . . . . . . . . . . . . . . . . . 61Magnetic Force Calculation between Truncated Cone Shaped Perma-

nent Magnet and Soft Magnetic Cylinder Using Hybrid BoundaryElement Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

Modelling Overtones in Electric Car Charging using Power-ExponentialFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

OCTOBER 1, 2019

AS track

KEYNOTE LECTURE AS - 4The Generalized Weyl Poisson Algebras and their Poisson Simplicity

Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64


Algebraic Structures and Applications (AS - 4)Hom-associative Deformations of Non-commutative Polynomial Rings 66Constructions and Modules of Hom-Poisson Color Algebras . . . . . . 67On the Ternary (Hom-)Nambu-Poisson Algebras . . . . . . . . . . . . 68

KEYNOTE LECTURE AS - 5

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From Graded Extension of the Lorentz Algebra to Colour Dirac Equa-tion and Quark Dynamics . . . . . . . . . . . . . . . . . . . . . . 69


Algebraic Structures and Applications (AS - 5)Centralizers in Crossed Product Algebras for Piece-wise Constant Func-

tion Algebras and Multiresolution Analysis . . . . . . . . . . . . 71Cellular Structure of Seam Algebras and Avenues for Deformations . . 72Two-sided Noncommutative Grobner Basis on Quiver Algebras . . . . 73

KEYNOTE LECTURE AS - 6Double Constructions of (Bi)Hom-Frobenius Algebras . . . . . . . . . 74


Algebraic Structures and Applications (AS - 6)Complex of Higher Categories and Derivations of Group Algebras . . . 76Derivations in Group Algebras and Stollings Theorem . . . . . . . . . 77Tensor Hierarchy Algebras and Extended Geometry . . . . . . . . . . 79Algebras and Operations in Statistical Systems . . . . . . . . . . . . . 80

SP&MS track

KEYNOTE LECTURE SP&MS - 3An Extended Inverse Gaussian to Model the First Exit Time Process . 81


Stochastic Processes / Modern Statistical methods in theory andpractice (SP&MS - 4)Heat Kernel Estimates for Time Fractional Equations With Nonlocal

Markov Generators . . . . . . . . . . . . . . . . . . . . . . . . . . 84Determining Influential Factors in Spatio-Temporal Models . . . . . . 85Arbitrage Theory with State-Price Deflators . . . . . . . . . . . . . . . 86

Stochastic Processes / Modern Statistical methods in theory andpractice (SP&MS - 5)On the Rate of Convergence in Central Limit Theorem for Nearly

Critical Branching Processes with Immigration . . . . . . . . . . 88On the Inequalities for Moments of Branching Processes with Immi-

gration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89On Asymptotics of Branching Processes with Immigration . . . . . . . 91

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Stochastic Processes / Modern Statistical methods in theory andpractice (SP&MS - 6)Lie Symmetry Analysis on Pricing Weather Derivatives by Partial Dif-

ferential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 93Suggestion of a Hidden Markov Model for First Order Markov Depen-

dent Observations . . . . . . . . . . . . . . . . . . . . . . . . . . 94Investigating some Attributes of Periodicity in DNA Sequences via

Semi-Markov Modeling . . . . . . . . . . . . . . . . . . . . . . . . 95Inferring Social Clustering from Register Data . . . . . . . . . . . . . 97The Relaxed Stochastic Maximum Principle for G-SDE . . . . . . . . 99

EM track


Engineering Mathematics (EM - 4)The Localized Orthogonal Decomposition Method for Flow and Wave

Propagation Problems in Fractured Porous Media . . . . . . . . . 101Homotopy Analysis Method (HAM) for Differential Equations Per-

taining to the Mixed Convection Boundary-Layer Flow over aVertical Surface Embedded in a Porous Medium . . . . . . . . . 103

Thin Film Flow of Carreau Nanofluid Past a Heated Nonlinear Stretch-ing Sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

B-spline Approach for Approximate Solution of MRLW Equation viaBFRK Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

Engineering Mathematics (EM - 5)On the Approximation of Unstructured Population Model with Stage-

structured Population Model in a Grazing System . . . . . . . . 107A Mathematical Model for Harvesting in a Stage-Structured Canni-

balistic System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108Study of a Nonlinear Problem from Geology . . . . . . . . . . . . . . . 109

OCTOBER 2, 2019

AS track

KEYNOTE LECTURE AS - 7

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Homomorphisms of Pseudo-Riemannian Calculi and NoncommutativeMinimial Submanifolds . . . . . . . . . . . . . . . . . . . . . . . . 110


Algebraic Structures and Applications (AS - 7)Applications of Matroid and Polymatroid Theory to Algebraic Coding

and Information Theory . . . . . . . . . . . . . . . . . . . . . . . 112On a Relation Between Rook Polynomials and Graph Theory . . . . . 113Algorithms for Recalculating Centrality Measures Based on Powers of

the Adjacency Matrix . . . . . . . . . . . . . . . . . . . . . . . . 114Vertex Connectivity of Line Graph of Γ(Zn) . . . . . . . . . . . . . . . 115

KEYNOTE LECTURE AS - 8Orthogonal Systems in the Complex Plane and Applications in Nu-

merical Integration . . . . . . . . . . . . . . . . . . . . . . . . . . 116


Algebraic Structures and Applications (AS - 8)On the Exponential and Trigonometric q, ω-special Functions . . . . . 119Heap Cohomology and Ternary Self-distributive Cohomology . . . . . 120Common Fixed Point Results for Perov Type Weakly Commuting

Maps in Cone Metric Spaces . . . . . . . . . . . . . . . . . . . . . 121Wallis Type Formula in Quantum Calculus . . . . . . . . . . . . . . . 122

KEYNOTE LECTURE AS - 9Extension of Nambu theory to Superspace . . . . . . . . . . . . . . . 123


Algebraic Structures and Applications (AS - 9)About Compact Monothetic Semirings . . . . . . . . . . . . . . . . . . 125Semilattice Decomposition of Semigroups: from Theory to Applications127A Note on Cyclic Codes over the Ring Zpk . . . . . . . . . . . . . . . 128Public Key Cryptography and Twisted Dihedral Group Rings . . . . . 129

SP&MS track

KEYNOTE LECTURE SP&MS - 4On the ROCOF of Higher Order for Semi-Markov and Generalized

Semi-Markov Systems with Applications in Finance . . . . . . . 130


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Stochastic Processes / Modern Statistical methods in theory andpractice (SP&MS - 7)Stochastic Optimal Control with a Multiscale Volatility Model . . . . 133Functional Data Analysis for Stochastic Volatility . . . . . . . . . . . . 134Structured Maintenance Policy for Deteriorating Systems under Cu-

mulative Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135Stochastic Approach for Nonlinear Stiff Behaviour . . . . . . . . . . . 136

Stochastic Processes / Modern Statistical methods in theory andpractice (SP&MS - 8)Are these Moments known? . . . . . . . . . . . . . . . . . . . . . . . . 138Fourth Cumulant for the Random Sum of Random Vectors . . . . . . 140Closed-form Estimator for the Matrix-Variate Gamma Distribution . . 141Noninformative Bayesian Approach in Extended Random Effects Meta-

Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

Stochastic Processes / Modern Statistical methods in theory andpractice (SP&MS - 9)Improved Uncertainty Modelling of Process Variations in an Industry

4.0 Manufacturing Facility by use of Probability Box UncertaintyModels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

The Algorithm Solution of the Problem of Optimal Control in a Dy-namic One-Sector Economic Model with a Discrete Time Basedon Dynamic Programming Method . . . . . . . . . . . . . . . . . 146

On Steady-state Analysis of Retrial Queues with State-Dependent In-put Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

On the Solution of the Optimal Control Problem of Inventory of aDiscrete Product in Stochastic Model of Regeneration . . . . . . 149

Analysis of a Recurrent Multi-Server Queue with Different Service Rules150Modelling and Estimation Study of Complex Reliability Systems . . . 151

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Keynote lecture

Date: September 30, 2019Time: 10.30 - 11.00Venue: Beta, House U

Malardalen University, Vasteras, Sweden

Multiresolutions and Multivariable OperatorTheory and Noncommutative Analysis

Palle Jorgensen

Department of Mathematics, The University of Iowa, Iowa City, Iowa, USA,[email protected]

Abstract - We offer an operator theoretic approach to multiresolutions in non-commutative analysis and applications. Multiresolutions are part of multivari-able operator theory, and noncommutative mathematics. In applications, theyare important as they offer fast algorithms, and have a host of other applica-tions. One such is to wavelets; a proven and successful alternative to classicalFourier methods, Fourier series and integrals. In general, with multiresolutions,one obtains recursive and computational spectral resolutions for multivariableoperator systems. They are localized, so better adapted to discontinuities. Andthey offer better numerical schemes. Multiresolutions are further useful in thestudy of self-similarity, in the analysis of fractals, and of non-linear dynamicalsystems. A special case of this is illustrated by the renormalization property forscaling functions from wavelet theory; and renormalization more generally.

Recursive multiresolutions basis constructions in Hilbert spaces are key toolsin analysis of fractals and of iterated function systems in dynamics. Indeed, withmultiresolutions, notions of selfsimilarity and locality yield much better point-wise approximations than is possible for traditional Fourier bases. The talk willfurther cover frequency sub-bands in signal or image-processing, and associatedmulti-band filters: One builds recursive subdivisions of signals into frequencybands. This suggests a representation theoretic framework; realizations of cer-tain representations on Hilbert spaces H having associated families of closedsubspaces in such a way that ”non-overlapping frequency bands” correspondto orthogonal subspaces in H; or equivalently to systems of orthogonal projec-tions. Since the different frequency bands must exhaust the range for signals inthe entire system, one looks for orthogonal projections which add to the iden-tity operator in H. Representations of Cuntz algebras is a case in point. From

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representations we obtain classification of families of multi-band filters; andrepresentations allow us to deal with non-commutativity as it appears in bothtime/frequency analysis, and in scale-similarity. The representations further of-fer canonical selections of special families of commuting orthogonal projections.

Keywords - Multiresolutions, Fractals, Iterated function systems, Noncomm-mutative operator systems, Representations of Cuntz algebras.

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Communication Session:Algebraic Structures and Applications (AS - 1)

Date: September 30, 2019Time: 11.15 - 12.15Venue: Room Beta, House U


Continuous Integral Operators on Subspaces ofthe Space of Bounded Vector Functions 1

Manuel Alves1, Joao Munembe2 and Yury Nepomnyashchikh3

1 Eduardo Mondlane University/Department of Mathematics and Informatics, PO.Box 257, Maputo, Mozambique, [email protected]



Abstract - We prove the action and continuity criterion for a linear integraloperator

(Kx)(t) =

∫S

k(t, s)x(s) ds

for one class of sub-spaces of the space of bounded functions on S with valuesin a Banach space X.

Given an expression for the norm of the operator K in terms of the kernel kand some estimate of the norm in the case of finite-dimensionality of the spaceX using the 1-absolutely summing constant.

The criterion covers the cases of functional spaces that are interesting forapplications, such as the space of continuous functions bounded on the real axisor semi-axis, its sub-spaces of functions vanishing at infinity, almost periodic,and others.

Also obtained a similar result on the action and uniform continuity of thenon-linear integral Urysohn operator

(Ux)(t) =

∫S

k(t, s, x(s)) ds

with the expression of its modulus of continuity through k(t, s, v).

Keywords - Absolutely summing constant, Integral operator, Modulus of con-tinuity, Uniform space, Urysohn operator.

1The work is supported by SIDA under the subprogramme Capacity Building in Mathe-matics, Statistics and Its Applications

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On the Existence of Nonnegative IntegralSolutions of Some Classes of Linear Equations

Over Integers

Masood Aryapoor1

1 Division of Applied Mathematics, Malardalen University, Vasteras, Sweden,[email protected]

Abstract - Linear programming deals with the problem of optimizing a linearfunctional subject to some linear inequalities. Integer linear programming refersto techniques used to solve linear programming problems in which the variablesare restricted to integers. As a central result of linear programming, Farkas’lemma provides a set of necessary and sufficient conditions for a system of linearequations over real (resp., rational) numbers to have a nonnegative solution inreal (resp., rational) numbers. These conditions, although necessary, are notsufficient for a system of linear equations over integers to have a nonnegativeintegral solution. In fact, no satisfactory version of Farkas’ lemma over integershas been found. Here, we show that if the necessary conditions provided byFarkas’ lemma over real numbers hold for a system of linear equations overintegers then the system has a nonnegative rational solution of a particular form.We also investigate certain classes of systems of linear equations over integersfor which some “Farkas-type” conditions are both necessary and sufficient forthe existence of a nonnegative integral solution.

Keywords - Farkas’ lemma, Integer linear programming.

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Non-commuting Operators on Hilbert andBanach Spaces Satisfying Commutation

Relations Associated to Iterations of SeveralMaps

Domingos Celso Djinja1, Sergei Silvestrov2

1 Department of Mathematics and Computer Sciences, Faculty of Sciences, EduardoMondlane University, Box 257, Maputo, Mozambique.

2 Division of Applied Mathematics, UKK, Malardalen University, Box 883, SE-72123 Vasteras, Sweden.

(e-mails: [email protected]; [email protected];

[email protected])

Abstract - In this talk We will present some constructions of families of non-commuting operators on Hilbert and Banach spaces satisfying commutationrelations associated to iterations of one or several maps. Many of these operatorsare infinite-dimensional. Some of them are bounded and some of them areunbounded with spectral properties and structure which depends on orbits ofiterations of the maps defining the commutation relations.

Keywords - Banach space, Commutation relations, Hilbert space, Iterations ofmaps, Non-commuting operators, Spectrum.

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Keynote lecture



Group Gradings on Simple Lie and RelatedAlgebras

Mikhail Kochetov

Department of Mathematics and Statistics, Memorial University of Newfoundland,St. John’s, Canada, [email protected]

Abstract - If A is an algebra and G is a group, then a G-grading on A is a vectorspace decomposition A =

⊕g∈GAg satisfying AgAh ⊆ Agh for all g, h ∈ G; the

support of this grading is the set g ∈ G | Ag 6= 0. For example, the root spacedecomposition of a semisimple Lie algebra with respect to a Cartan subalgebrais a grading by a free abelian group (the root lattice); the support consists ofthe root system and zero.

In the 1960’s, V. Kac classified all gradings by cyclic groups on finite-dimensional simple Lie algebras over the field of complex numbers. Since then,there has been considerable interest in gradings on Lie algebras by arbitraryabelian groups. Such gradings arise in the theory of algebraic groups, compactLie groups, symmetric spaces, infinite-dimensional Lie algebras, and color Liesuperalgebras.

In the case of a simple Lie algebra, it turns out that the elements of the sup-port always commute, so we may suppose that the grading group G is abelian.Through the efforts of many authors, group gradings have recently been clas-sified for all finite-dimensional simple Lie algebras over an algebraically closedfield of characteristic 0.

This talk will review some of the classification results for simple Lie andrelated algebras as well as methods and open problems in this area.

Keywords - Graded, Lie algebra, Semisimple, Simple.

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Graded Quasi Hom-Lie Algebras

Sergei Silvestrov

Division of Applied Mathematics, Malardalen University, Vasteras, Sweden,[email protected]

Abstract - In this talk, an overview will be presented about hom-algebra struc-tures, with focus on foundations and recent advances on graded (color) quasiLie algebras, quasi-hom Lie algebras, hom-Lie algebras and related hom-algebrastructures. These interesting algebraic structures appear for example when dis-cretizing the differential calculus as well as in constructions of differential cal-culus on non-commutative spaces. Quasi Lie algebras encompass in a naturalway the Lie algebras, Lie superalgebras, color Lie algebras, hom-Lie algebras,q-Lie algebras and various algebras of discrete and twisted vector fields arisingfor example in connection to algebras of twisted discretized derivations, Oreextension algebras, q-deformed vertex operators structures and q-deferentialcalculus, multi-parameter deformations of associative and non-associative alge-bras, one-parameter and multi-parameter deformations of infinite-dimensionalLie algebras of Witt and Virasoro type, multi-parameter families of quadraticand almost quadratic algebras that include for special choices of parametersalgebras appearing in non-commutative algebraic geometry, universal envelop-ing algebras of Lie algebras, Lie superalgebras and color Lie algebras and theirdeformations. Common unifying feature for all these algebras is appearance ofsome twisted generalizations of Jacoby identities providing new structures ofinterest for investigation from the side of associative algebras, non-associativealgebras, generalizations of Hopf algebras, non-commutative differential calculibeyond usual differential calculus and generalized quasi-Lie algebra central ex-tensions and Hom-algebra formal deformations and co-homology. Hom-algebrageneralizations of Nambu algebras, associative algebras and Lie algebras to n-ary structures are also actively studied and some constructions and results onn-ary hom-Lie algebras will be presented in this talk.

Keywords - Hom-Lie algebras, n-ary Hom-Lie algebras, Quasi-Hom Lie alge-bras.

References

[1] J. T. Hartwig, D. Larsson and S. D. Silvestrov, Deformations of Lie algebrasusing sigma-derivations, J. Algebra 295 (2006) 314–361. (Preprints in Math-ematical Sciences 2003:32, LUTFMA-5036-2003, Centre for MathematicalSciences, Lund University, 2003; arxiv.org: math.QA/0408064

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[2] D. Larsson, S. Silvestrov, Quasi-hom-Lie, Algebras, Central Extensions and2-cocycle-like Identities, J. Algebra, 288, 2, (2005), 321–344.

[3] D. Larsson, S. Silvestrov, Quasi-Lie algebras, in Noncommutative Geom-etry and Representation Theory in Mathematical Physics, ContemporaryMathematics, Vol. 391, 2005, 7 pp.

[4] D. Larsson, S. D. Silvestrov, Quasi-deformations of sl2(F ) using twistedderivations, Comm. Algebra 35 (2007), 4303–4318. (Preprints in Mathemat-ical Sciences 2004:26, LUTFMA-5047-2004, Centre for Mathematical Sci-ences, Lund University 2004, 19pp; arxiv.org: math.RA/0506172

[5] A. Makhlouf, S. D. Silvestrov, Hom-algebra structures, J. Gen. Lie The-ory. Appl. 2, 2, (2008), 51–64. (Preprints in Mathematical Sciences, Centrefor Mathematical Sciences, Lund University (2006:10) LUTFMA-5074-2006;arxiv.org/math/0609501v3)

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Hom-Lie Algebras and Deformations

German Garcia Butenegro1 and Sergei Silvestrov2

1 Division of Applied Mathematics, Mlardalen University, Vsteras, Sweden,[email protected]

2 Division of Applied Mathematics, Mlardalen University, Vsteras, Sweden,[email protected]

Abstract - The article examines twisted versions of Leibnizs product rulefor derivation operators. Different twistings of the rule provide once-twistedsigmaσ-derivations or twice-twisted (σ, τ)-derivations.

The (σ, τ)-derivations are often conditioned to σ and τ being endomorphismsin the underlying coefficient algebra. This property will proved to be excessiveand a new, more flexible condition relating σ and τ will be established.

Properties of quasi-Lie algebras will be used to find new examples of twistedquasi- Hom-Lie deformations of the Witt algebra.

Latest results on extensions of quasi-Lie algebras suggest the existence ofa Virasoro-type 1-dimensional central extension for a certain quasi-Hom-Liedeformation of the Witt algebra. Relations between generators will be provided,and the canonical q-deformed Witt algebra is obtained as a case.

The final section is dedicated to finding a proper linear section that allowsto characterize said extension. It will be found that the method generalizesthe one established by Silvestrov, Hartwig and Larsson (2003) and an equationsystem whose solutions provide the desired relations is provided.

Keywords - Central extension, Hom-Lie algebras,Twisted derivations, Quasi-Lie algebras.

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Index of Hom-Lie Algebas

Hadjer Adimi

Departement of mathematics, Bordj Bou-Arreridj University, Algeria,[email protected]

Abstract - The index is an important concept in the representation theory andinvariant theory. A complete theory of the index exist for Lie algebra howeverwe do not have this theory in the case of Hom Lie algebras. The first mainpurpose of this paper is to introduce the index of the Hom-Lie algebras, as wellas to provide some key constructions and discuss index of semi-direct productsof Hom-Lie algebras.

Keywords - Adjoint representation, Coadjoint representation, Hom-Lie alge-bras, Index, Lie algebras.

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Keynote lecture



Drinfeld Elements and Hom-Bialgebras

Blas Torrecillas

Department of Mathematics, University of Almerıa, 04071 Almerıa, [email protected]

Abstract - We review the concept of a twisting element based on a Hom-bialgebra, introduced in a joint work with A.Makhlouf, and use it to providetwists or deformations of Hom-associative algebras. Moreover, we will studythe module theory in Hom-setting and show that a twisting element based ona bialgebra gives rise to a twisting element based on a Hom-bialgebra. Wewill present examples and connections with related structures and topics whichcould be interesting to explore from the point of view of establishing furtherlinks to Hom-algebras structures.

Keywords - Categories of modules, Deformations, Drinfeld elements, Hom-bialgebras.

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Strong Hom-associativity

Lars Hellstrom


Abstract - Hom-associativity is a generalisation of the ordinary associativ-ity axiom of an algebra, wherein an additional unary operation—the ‘hom’—balances the binary multiplication operation. Whereas the consequences of or-dinary associativity are quite regular, a fact that may be seen in the structureof the associativity operad, the consequences of the hom-associative axiom seemnot to be; there is for example to date no known combinatorial construction ofthe general free hom-associative algebra, and somewhat surprisingly that freealgebra turns out to have zero divisors.

Strong hom-associativity is a strengthening (though rather mild) of the ba-sic hom-associativity axiom, which renders its consequences more regular; inparticular there is a finitely presented combinatorial model of the free stronglyhom-associative algebra. Methods that have been employed to study stronghom-associativity include rewriting in the free operad, to map out the differ-ence between strong and weak hom-associativity.

All Yau twist algebras are strongly hom-associative, but the converse is farfrom true. In this talk, the hom is not assumed to be multiplicative, nor is itassumed to be invertible.

Keywords - Hom-associative algebra, Operad, Diamond lemma, Rewriting,Canyon axioms

24

Hom-Lie Structures on 3-dimensional SkewSymmetric Algebras

Ongong’a Elvice1, Richter Johan2 and Silvestrov Sergei3

1 Division of Applied Mathematics, Malardalen University, Vasteras, Sweden andSchool of Mathematics, University of Nairobi, Nairobi, Kenya,

[email protected] Division of Applied Mathematics, Malardalen University, Vasteras, Sweden,

[email protected] Division of Applied Mathematics, Malardalen University, Vasteras, Sweden,

[email protected]

Abstract - We describe the dimension of the space of possible linear endo-morphisms that turn skew-symmetric three-dimensional algebras into Hom-Liealgebra. We show that this dimension can be fully described by the matrixof structure constants of the bilinear product of the skew-symmetric algebra.This follows from results showing correspondence between this dimension andthe rank of the matrix of such structure constants. Examples from classicalcomplex Lie algebras are given to demonstrate this correspondence.

Keywords - Hom-Lie algebras, Hom-Lie structures.

25

Hom-Gerstenhaber Algebras and Hom-LieAlgebroids

Satyendra Kumar Mishra1 and Sergei Silvestrov2

1 Stat-Math. Unit, Indian Statistical Institute, Bangalore, India,[email protected]

2 Division of Applied Mathematics, Mlardalen University, Vsters, Sweden,[email protected]

Abstract - In recent years, there is a growing interest in new non-associativealgebra structures, known as Hom-algebraic structures due to their close rela-tionship with deformed vector fields and differential calculus. The first appear-ance of such an algebra was the notion of Hom-Lie algebra in the context ofq-deformations of Witt and Virasoro algebras. Later on, several Hom-algebraicstructures were studied in the subsequent works. In this talk, we are interestedin a geometric generalisation of Hom-Lie algebras, the notion of Hom-Lie alge-broid, which was defined through a formulation of Hom-Gerstenhaber algebras.Hom-Gerstenhaber algebras are graded Hom-Lie algebras with a compatiblegraded commutative algebra structure. It is known that there is a bijectivecorrespondence between Hom-Lie algebroid structures on a vector bundle andHom-Gerstenhaber algebra structures on the associated exterior algebra of mul-tisections of the vector bundle.

Some of the fundamental results for Lie algebroids are the existence of adual description in terms of a differential graded commutative algebra and adifferential calculus. In the Hom-context, this feat was achieved recently forregular Hom-Lie algebroids with a modification in the original definition. Afterthis modification, the new definition became more concise and calculations wereeasier. However, there is a disadvantage to this modification, which lies in thefact that the bijective correspondence mentioned above, one major motivationto define Hom-Lie algebroids, is not available anymore.

The purpose of this talk is two-fold. First, we obtain a differential calculusand a dual description for (regular) Hom-Lie algebroids without any modifica-tions in the original definition. The second aim of this talk is to analyse therelationship between different geometric structures on a vector bundle and Hom-Gerstenhaber algebra structures on the exterior algebra of its multisections. Toachieve these goals, we take an algebraic approach. Most of the techniques weuse are homological. To summarise, we discuss the following features.

• Special classes of Hom-Gerstenhaber algebras, namely Hom-Batalin-Vilkovi-sky algebras and differential Hom-Gerstenhaber algebras.

• Representations of Hom-Lie algebroids and a cohomology for regular Hom-Lie algebroids with coefficients in a representation.

26

• Relationship between different geometric structures on a vector bundle andspecial classes of Hom-Gerstenhaber algebras on the associated exterioralgebra of multisections.

Keywords - Hom-Lie algebra, Hom-Lie algebroid, Hom-Lie-Rinehart algebra,Hom-Gerstenhaber algebra.

27

On Solvability and Nilpotency of (n+ 1)-Hom-LieAlgebras Induced by n-Hom-Lie Algebras

Abdennour Kitouni1, Abdenacer Makhlouf2 and Sergei Silvestrov3

1 Division of Applied Mathematics, School of Education, Culture andCommunication, Malardalen University, Box 883, 72123 Vasteras, Sweden,

[email protected] IRIMAS, Universite de Haute-Alsace, 12 rue des Freres Lumiere, 68093 Mulhouse,

France, [email protected] Division of Applied Mathematics, School of Education, Culture and

Communication, Malardalen University, Box 883, 72123 Vasteras, Sweden,[email protected]

Abstract - Ternary Lie algebras, and more generally n-Lie algebras, first ap-peared in a generalization of Hamiltonian mechanics by Nambu. They wereexplicitly mentioned by Takhtajan in his algebraic formulation of Nambu me-chanics. They were also introduced and their structure studied independentlyby Filippov and Kasymov. Besides Nambu mechanics, n-Lie algebras revealedto have many applications in physics. Hom generalizations of n-ary algebraincluding n-Hom-Lie algebras and other n-ary Hom algebras of Lie type andassociative type have been introduced by Ataguema, Makhlouf and Silvestrov.

The idea of (n+1)-Hom-Lie algebras induced by n-Lie algebras started withthe work of Awata et al. on the quantization of the Nambu brackets, expressedusing the matrices commutator and trace. It was generalized using the bracketof any Hom-Lie algebra or n-Hom-Lie and linear form having similar propertiesby Arnlind, Makhlouf and Silvestrov.

In n-Lie algebras, the n-ary bracket allows to define different derived seriesand central descending series that all reduce to the well known definitions in Liealgebras when n = 2. They depend on a number 2 ≤ k ≤ n and they are usedto define k-solvability and k-nilpotency.

The purpose of this work is to generalize the concepts of k-solvability andk-nilpotency to n-Hom-Lie algebras and to study their properties. We definek-derived series, k-central descending series and study their properties, we showthat k-solvability is a radical property and we apply all of the above to the case of(n+ 1)-Hom-Lie algebras induced by n-Hom-Lie algebras, namely we comparek-derived series and k-central descending series of an ideal in the original n-Hom-Lie algebra and the k-derived series and k-central descending series of thesame ideal in the induced (n+ 1)-Hom-Lie algebra.

Keywords - Hom-algebra, k-Nilpotent, k-Radical, k-Solvable, n-Hom-Lie alge-bra.

28

Keynote lecture



Asymptotic Algorithms of Phase SpaceReduction and Ergodic Theorems for Perturbed

Semi-Markov Type Processes

Dmitrii Silvestrov

Department of Mathematics, Stockholm University, Stockholm, Sweden,[email protected]

Abstract - New asymptotic recurrent algorithms of phase space reduction andtheir applications to ergodic theorems for regularly and singularly perturbedsemi-Markov type processes are presented. The classification of short, long andsuper-long time ergodic theorems for regularly and singularly perturbed modelsis given as well as new limit theorems for hitting times and related functionals.

Keywords - Ergodic theorem, Hitting time, Perturbation, Phase space reduc-tion, Semi-Markov type process.

29

Communication Session:Stochastic Processes / Modern Statistical

methods in theory and practice (SP&MS - 1)

Date: September 30, 2019Time: 11.15 - 12.15Venue: Room Gamma, House U


Information Networks and Perturbed MarkovChains with Damping Components

Dmitrii Silvestrov1, Sergei Silvestrov2, Benard Abola2, Pitos SelekaBiganda2, Christopher Engstrom2, John Magero Mango3 and

Godwin Kakuba3

1 Stockholm University, Stockholm, Sweden, [email protected] Malardalen University, Vasteras, Sweden3 Makerere University, Kampala, Uganda

Abstract - Markov chains with damping components are popular models usedfor analysis of information networks. New results on asymptotic expansionsfor stationary distributions and coupling estimates for the rate of convergencein ergodic theorems for regularly and singularly perturbed Markov chains withdamping components are presented as well as results of related numerical ex-periments.

Keywords - Asymptotic expansion, Information network, Markov chain withdamping component, Rate of convergence, Stationary distribution.

31

PageRank and Perturbed Markov Chains

Pitos Seleka Biganda1, Benard Abola1, Christopher Engstrom1,John Magero Mango2, Godwin Kakuba2 and Sergei Silvestrov1

1 Divison of Applied Mathematics, Malardalen University, Vasteras, Sweden, e-mails:[email protected], [email protected], [email protected],

[email protected] Department of Mathematics, Makerere University, Kampala, Uganda, e-mails:

[email protected], [email protected]

Abstract - PageRank is a widely used hyperlink-based algorithm to estimatethe relative importance of nodes in networks. Since many real-world networksare large sparse networks, this makes efficient calculation of PageRank compli-cated. Moreover, one needs to escape from dangling effects in some cases aswell as slow convergence of the transition matrix. Primitivity adjustment witha damping (perturbation) parameter ε ∈ (0, ε0] (for fixed ε0 ≈ 0.15) is one ofthe essential procedures known to ensure convergence of the transition matrix.If is large, the transition matrix loses information due to shift of informationto teleportation matrix. In this talk, we aim to formulate PageRank problem asthe first and second order Markov chain perturbation problem. Using numer-ical experiments, we will compare convergence rates for the two problems fordifferent values of ε on different graph structures and investigate the differencein ranks for the two problems.

Keywords - Markov chains, PageRank, Perturbation problem.

32

A Variant of Updating PageRank in EvolvingTree Graphs

Benard Abola1, Pitos Seleka Biganda1, Christopher Engstrom1,John Magero Mango2, Godwin Kakuba2 and Sergei Silvestrov1

1 Divison of Applied Mathematics, Malardalen University, Vasteras, Sweden, e-mails:[email protected], [email protected], [email protected],

[email protected] Department of Mathematics, Makerere University, Kampala, Uganda, e-mails:

[email protected], [email protected]

Abstract- PageRank update refers to the process of computing new PageRankvalues after change(s) like addition or removal of links / vertices occurred inreal life networks. The purpose of updating is to avoid re-calculating the valuesfrom scratch. It is well known that handling nodes importance is problematic,particularly when links and nodes change. In this talk, we are concerned withthe problem of updating PageRank in changing tree graph. We will present somenumerical experiments on the proposed algorithm that maintain level structuresand update PageRank of evolving graph. Further, we will describe how to handlePageRanks update when cyclic components.

Keywords - Markov chains, PageRank, Tree graph.

33

Keynote lecture

Date: September 30, 2019Time: 13.15 - 13.45Venue: Gamma, House U


The Generalized Calculation of Pure Premiumfor Non-Life Insurance by Generalized

Non-Homogeneous Markov Reward Processes

Raimondo Manca

University of Roma ”La Sapienza/MEMOTEF, University of Roma, Roma, Italy,e-mail - [email protected]

Abstract - This paper describes the calculation, in a simple and precise way, ofthe aggregate claim amount and the claim number using generalized Markov re-ward models in a non-homogeneous time environment. The evolution equationsof the generalized non-homogeneous Markov reward processes in a discountedenvironment is the tool that will be used for the calculation of the aggregateclaim amount and in a non-discounted case for the calculation of the mean claimnumber. -This paper can be considered as a generalization of the paper DAmicoat al. (2017) that proposed the introduction of the age as main time variable,The main new results presented in this paper are: - the simultaneous introduc-tion inside the model of the calendar time as second time and taking into accountthe age as the most relevant time variable. The introduction of two time vari-ables gives the opportunity of: - a more precise usage of the non-homogeneoussetting, - the possibility of following, by means of the Markov setting in a betterway: - the evolution equation of a non-life insurance contract, - the evolutionof the technology, - the evolution of inflation and, consequently of costs andearnings of each cohort of an Insurance Company. In a cohort there are all theinsureds with the same age, sex and driving experiences. In the last section, willbe presented a non-life Insurance application. The database that will be usedfor the calculation of the aggregate claim amount and the mean claim numberwill be done using a database with more than 2000000 records in which eachrecord report the history of driving behaviour of the insureds. The proposedmodel will be presented using the real data obtained by an Insurance Company.The results will show the importance of the age of the insured people and thepossibility to consider simultaneously the calendar time in the calculation of theactuarial costs and revenues.

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Keywords - Age and calendar times, Aggregate claim amount process, Claimnumber process, Markov chains, Non-homogeneity.

35





Entropy Analytics of Mortality Data Series byMixed Binomial Distribution: Sweden, France,

Italy

Yiannis Dimotikalis1 and Christos H Skiadas2

1 Dept of Management Science & Technology, Hellenic Mediterranean University(Crete), Greece, e-mail: [email protected]

2 ManLab, Technical University of Crete, Chania, Greece, e-mail: [email protected]

Abstract - In this paper the max entropy principle is applied to human mor-tality life table data. The entropy framework is presented and adjusted for fiveyears age groups of life tables. By entropy constrained optimization the expo-nential, discrete Truncated Geometric distribution, is the max entropy distri-bution subject to expected value constraint. The Binomial distribution appearsas max entropy distribution subject to expected value and variance constraintasymptotically. For the time evolution of mortality data series, the paths ap-pear to follow inside the Geometric upper limit and Binomial lower limit. Amixed Binomial Distribution is suggested and introduced to explain the ob-served paths of mortality data. The mortality data of Sweden (1755-2015),France (1820-2015) and Italy (1872-2014) are used to exhibit and explore thepractical results of this approach. The implications of those findings to humanmortality are discussed, and some future tasks are suggested. It seems that asimple combination of two Binomials is capable to represent the complicatedphenomena observed historically to mortality data in last few centuries.

Keywords - Maximum Entropy Principle, Geometric and Binomial Distribu-tion, Mixed Binomial Distribution, Mortality Data, Nonlinear Constrained Op-timization.

37

On Ruin Probability of Bivariate CorrelatedRisk Processes

Mohammad Jamsher Ali1, Kalev Parna2

1 Institute of Mathematics and Statistics, University of Tartu, Tartu, Estonia,[email protected]

2 Institute of Mathematics and Statistics, University of Tartu, Tartu, Estonia,[email protected]

Abstract - In actuarial science, measuring ruin probability is one of centralissues. Ruin probability has been studied by number of researchers using dif-ferent models. In their pioneering works, F. Lundberg (1903) and H. Cramer(1930) introduced classical compound Poisson risk model, and obtained sev-eral important results including asymptotic formula for ruin probability for ageneral class of claim distributions. Later on, researchers have studied variousgeneralizations of the classical risk model.

Recently, many authors have also shown an interest in multivariate riskprocesses where several risk processes are running simultaneously. For example,a single traffic accident may create two types of claims, motor and health. Bothindependent and dependent claim arrival processes and the claim sizes havebeen considered.

In this paper the ruin probability of the sum of two risk processes is stud-ied under the assumption that the two Poisson processes of claim arrivals arecorrelated. Let the sizes of claims on the companies 1 and 2 be i.i.d. ran-dom variables Xi and Yi, respectively. The merged risk process with constantpremium earning p over time interval (0, t] is defined as

R(t) = pt−N

(1)t∑i=1

Xi −N

(2)t∑i=1

Yi

The correlation between the claim number processes N(1)t and N

(1)t is modeled in

two ways: by common shock (common component) model, and by randomizedintensities. The idea is to represent the merged risk process in the form ofa single risk process where initial claim size distributions are replaced by aproperly chosen phase-type distribution. This allows to construct exact formulafor the ruin probability of the merged risk process.

Keywords - Bivariate risk processes, Common shock model, Phase-type distri-bution, Ruin probability.

38

Risk-based Optimal Portfolio of an Insurer withRegime Switching and Noisy Memory

Calisto Guambe1, Rodwell Kufakunesu2 and Lesedi Mabitsela2

1 Department of Mathematics and Computer Sciences, Eduardo Mondlane,University, 257, Maputo, Mozambique, [email protected]

2 Department of Mathematics and Applied Mathematics, University of Pretoria,0002, Pretoria, South Africa, [email protected]

2 Department of Mathematics and Applied Mathematics, University of Pretoria,0002, Pretoria, South Africa, [email protected]

Abstract - In this talk, we consider a risk-based optimal investment problemof an insurer in a regime-switching jump diffusion model with noisy memory.Using the model uncertainty modelling, we formulate the investment problemas a zero-sum, stochastic differential delay game between the insurer and themarket, with a convex risk measure of the terminal surplus and the Browniandelay surplus over a period [T − %, T ]. Then, by the BSDE approach, the gameproblem is solved. Finally, we derive analytical solutions of the game problem,for a particular case of a quadratic penalty function and a numerical exampleis considered.

Keywords - BSDE, Convex risk measures, Jump-diffusion, Noisy memory, Op-timal investment, Regime-switching.

39





Extreme Points for Joint Probability Density ofWishart Matrix and Vandermonde

Determinants in Financial Mathematics

Asaph Keikara Muhumuza1,2, Karl Lundengard2, AnatoliyMalyalenko1, Sergei Silvestrov1, John Magero Mango3 and Godwin

Kakuba3

1 Division of Applied Mathematics, Malardalen University, Box 883, SE 721 23Vasteras, Sweden.

[email protected], [email protected],[email protected], [email protected]

2 Department of Mathematics, Busitema University, Box 238 Tororo, Kampala,Uganda, East Africa.

[email protected] Department of Mathematics, College of Natural Sciences, Makerere University, Box

7062, Kampala, Uganda, East [email protected], [email protected]

Abstract - In this study, we apply the extreme points of the joint probabil-ity density distribution of the random Wishart matrix using properties of theVandermonde determinant. We also illustrate with examples the applications ofthese extreme points to option pricing and optimal portfolio selection in financialmathematics.Keywords - Joint probability distribution, Option pricing, Optimal portfolioselection, Vandermonde determinant, Wihart matrix.

41

Calibration of Two-Factor Stochastic VolatilityModel

Betuel Canhanga1, Ying Ni2 and Calisto Guambe1

1 Universidade Eduardo Faculty of Sciences, Department of Mathematics andComputer Sciences, Eduardo Mondlane University, Box 257, Maputo, Mozambique,

[email protected], [email protected] Division of Applied Mathematics, School of Education, Culture and

Communication, Malardalen University, Box 883, SE-721 23 Vasteras, Sweden,[email protected]

Abstract - The two factor double mean-reverting stochastic volatility modelproposed in [1] has the advantage of being more flexible in fitting the mar-ket implied volatility surfaces. Under the assumption of fast mean-revertingand slow mean-reverting respectively for the two stochastic volatility factors,Canhanga et al. in [2] has presented a calibration procedure for this modelvia a first-order asymptotic expansion approach. The calibration was done forreal-market option data for one day.

In the present study, we derive a second-order asymptotic expansion for-mula for calibration procedure, which is implemented for daily option data dur-ing one-year. We compare the calibration qualities between the first-order andsecond-order asymptotic expansion approaches. We compare also the asymp-totic expansion approaches to more traditional calibration methods. Extensivenumerical studies are performed to investigate the properties of the calibrationprocedures and for choosing an optimal calibration approach.

Keywords - Asymptotic expansion, Calibration, Multifactor stochastic volatil-ity model, Stochastic volatility model.

References

[1] Christoffersen, C., Heston, S., and Jacobs, K. The shape and term structureof the index option smirk: Why multifactor stochastic volatility models workso well. Management Science, 55(12):1914-1932, 2009.

[2] Canhanga, B., Malyarenko, A. Murara, J.P., Ni, Y., Silvestrov, S. NumericalStudies on Asymptotics of European Option Under Multiscale StochasticVolatility. Methodology and Computing in Applied Probability. vol 19 (4),December 2017

42

Valuation and Optimal Strategies for AmericanOption under a Markovian Regime Switching

Model

Lu Jin1 and Ying Ni2

1 University of Electro-Communications, Japan, [email protected] Malardalen University, Sweden, [email protected]

Abstract - In this research, we consider the evaluation of American optionswhen the price dynamics of the underlying risky asset are governed by a Marko-vian regime switching process. We assume the price dynamics depend on theeconomy, the state of which transits based on a discrete-time Markov chain.The real state of the economy cannot be known directly but is partially ob-servable by receiving a signal stochastically related to the real economy. Theevaluation procedure is formulated using a partially observable Markov decisionprocess, and the optimal strategy for investors under call-type options is inves-tigated. Some properties of the optimal activity regions for call type option arediscussed.

Keywords - Decision policy, Hidden Markov chain, Partially observable Markovdecision process, Total positivity of order 2.

43

Pricing Overnight Index Swap in a LargeMarket Model Using a Cubature Method

Hossein Nohrouzian1, Anatoliy Malyarenko1, Ying Ni1 andChristopher Engstrm1

1 Division of Applied Mathematics, Malardalen University, Vasteras, Sweden,[email protected], [email protected], [email protected],

[email protected]

Abstract - Cubature is an effective way to calculate integrals in a finite dimen-sional space. Extending the idea of cubature to the infinite-dimensional Wienerspace would have practical usages in pricing financial instruments. In this pa-per, we calculate and use cubature formulas of degree 5 and 7 on Wiener spaceto price European options in the classical Black–Scholes model. This problemhas a closed form solution and thus we will compare the obtained numericalresults with the above solution. In the procedure, we study some characteristicsof the obtained cubature formulas and discuss some immediate applications ofthem.

Keywords - Cubature on Wiener Space, Forward Spread Curves, Infinite-Dimensional SDEs, Large Market Model, Overnight Index Swap (OIS).

44

Analytical Studies of Implied Volatility in theGatheral Model

Mohammed Albuhayri1, Anatoliy Malyarenko2, Sergei Silvestrov3,Ying Ni4 and Cristopher Engstrom5

1 Division of Applied Mathematics, Malardalen University, PhD Student, Vasteras,Sweden, [email protected]

2 Division of Applied Mathematics, Malardalen University, Professor, Vasteras,Sweden, [email protected]

3 Division of Applied Mathematics, Malardalen University, Professor, Vasteras,Sweden, [email protected]

4 Division of Applied Mathematics, Malardalen University, Senior Lecturer,Vasteras, Sweden, [email protected]

5 Division of Applied Mathematics, Malardalen University, Senior Lecturer,Vasteras, Sweden, [email protected]

Abstract — An implied volatility of a given market model is the value of volatil-ity in the classical Black–Scholes model that equalises the price of the Europeancall option in both models if their initial stock prices, spot interest rates, andthe option’s strikes and maturities are equal. We consider the Gatheral modelwith two stochastic volatilities and calculate the coefficients of the asymptoticexpansion of the implied volatility in a parabolic region near maturity and withnearly equal stock and strike prices. The error of our expansion is little-o ofthe third power of the difference in prices and of the 1.5th power of time tomaturity.

Keywords — Asymptotic expansion, European call, Gatheral model, Impliedvolatility.

45

Using the Investment Problem for OptionPricing

Nataliya Shchestyuk1 and Olga Gavryliuk2

1 National University ”Kyiv-Mohyla Academy”/Department of Mathematics,associating prof., Kyiv, Ukraine, [email protected]

2 National University ”Kyiv-Mohyla Academy”/Department of Mathematics, masterstudent, Kyiv, Ukraine, [email protected]

Abstract - Option pricing models are part of the theory of the stochastic pro-cesses and powerful tools for finance professionals involved in options trading.In this paper we consider the approach to the opting pricing which is based onthe problem of investor.

Suppose that an investor has to make decision about the amount X of investif the actual available income is given to him by random variable Y . The presentvalue of each unit of investment is 1. If the actual income is less than thecommitted sum X, then shortfall occurs. For the shortfall, a price of u(Y −X)+

has to be paid, where u > 1. On the other hand, if the income is larger than Xthe remaining amount has only a value of l(Y −X)− with l < 1. The total sum(profit) function is

H(X,Y ) = X − u(Y −X)− + l(Y −X)+.

The optimal decision X should maximize the expected profit E(H). Pflug,G. and Romisch, W. in their book “Modeling, Measuring and Managing Risk”showed that the optimal decision is

X∗ = G−1k ,

where

k =1− lu− l

.

And expected profit takes the maximal value

(1− l)V@Rk(Y ) + lE(Y ).

A similar investment problem appears in option trading. Then for example forcall option with strike price K the income Y of investor is payoff (S − K)+,where distribution of underlying risky assets is given to him. The Black-Scholesmodel assumes the price of heavily traded assets S follows a geometric Brownianmotion and log normal distribution.

In the paper we discuss Student-like model with dependence (FAT models)and determine optimal amount of investX∗ for this case. Meanwhile we measureand manage risk for the FAT model.

46

Keywords - FAT model, Measuring and Managmet of the risk, Option pricing,Problem of investor.

47

Communication Session:Engineering Mathematics (EM - 1)

Date: September 30, 2019Time: 11.15 - 12.15Venue: Room Kappa, House U


Stochastic Matrix Method in Physics ofNoncrystalline Condensed Matter

Richard Kerner

Sorbonne-Universite, Laboratorie de Physique Theorique de la Matiere Condensee,CNRS-URA 7600, Boıte courrier 121, 4 Place Jussieu, 75005 Paris, France,

[email protected]

Abstract - Several models of nucleation and agglomeration of various types ofnon-crystalline solids, especially glasses, are presented. They are based on theinvestigation of differential systems using stochastic matrices which representstatistical evolution of finite sets of local configurations during the process ofslow cooling or epitaxial growth. A lot of useful physical information can beobtained from such models, including the dependence of glass transition tem-perature on chemical composition and rapid cooling, rigidity, stability, electricconductivity, etc. For more detailed information, see the book”Models of Agglomeration and Glass Transition”, Imperial College Press 2007:https://www.worldscientific.com/worldscibooks/10.1142/p486.

Keywords - Agglomeration, Glass transition, Non-crystalline solids, Nucle-ation.

49

Accuracy of Cell Problems for ComputationalHomogenization of Multiscale Partial

Differential Equations in Locally Periodic Media

Doghonay Arjmand1,2

1 ANMC, Ecole polytechnique Federale de Lausanne, Lausanne, Switzerland,[email protected]

2 UKK, Malardalen University, Vasteras, Sweden, [email protected]

Abstract - The present work concerns the numerical homogenization of themultiscale elliptic PDE −∇ · (aε(x)∇uε(x)) = f(x) in a bounded domain Ω,which models e.g. the thermal properties of composite materials. Here aε isa positive, uniformly elliptic, and rapidly varying matrix function, where ε |Ω| represents the characteristic wavelength of the variations in aε. A directnumerical approximation of such a problem is prohibitively expensive as thesmall scales need to be resolved over a much larger computational domain Ω.An alternative approach is to look for a corresponding homogenized system,which describes the local average of uε. Finding such homogenized quantities, byanalytical means available from the classical homogenization theory, is possibleonly in a few academic cases of interests, such as the purely periodic media,and more general multiscale methods are needed to treat more complicatedcoefficients.

Several multiscale approaches, such as the heterogeneous multiscale methods(HMM), have been proposed in the past. These approaches often rely on up-scaling the homogenized quantities via local simulations over small microscopicdomains. Once the homogenized quantities, such as the homogenized coeffi-cients, are computed, the resulting macroscopic system can be approximatedby a standard numerical scheme. The accuracy of the upscaling procedures hasbeen studied mostly in purely periodic media, [1, 2], although physical problemsare often non-periodic and may contain fast and slow variations at the sametime. The aim of this work is to extend the analysis of existing microscopicmodels to locally periodic media, which will result in a better understanding ofthe cost and accuracy of the methods in more practical cases of interests.

Keywords - Homogenization, Multiscale methods, Multiscale PDEs.

References

[1] A. Abdulle, D. Arjmand, E. Paganoni. Exponential decay of the resonanceerror in numerical homogenization via parabolic and elliptic cell problems.

50

C. R. Acad. Sci. Paris, Ser. 1 (2019),https://doi.org/10.1016/j.crma.2019.05.011

[2] D. Arjmand, C. Stohrer. A finite element heterogeneous multiscale methodwith improved control over the modeling error. Communiations in Mathe-matical Sciences 14 (2)(2016), pp. 463–487.

51

Stochastic Simulations of Classical PDEs withNon-trivial Boundary Conditions

Magnus Ogren

School of Science and Technology, Orebro University, 701 82 Orebro, Sweden,[email protected]

Abstract - In Engineering Mathematics large and complex geometries can bemanifested as e.g. digital 3D images from CT-scanning. Solving PDEs to modelvarious physical and chemical processes on such domains can be practicallyimpossible with traditional numerical methods that depends on meshes, due tomemory restrictions as well as difficulties in meshing imperfect geometries. Herestochastic methods, such as discrete randow walks on the digital domains, canbe utilised to sample solutions to many PDEs, for example those specified inthe Feynman-Kac formula. We will report on numerical methods and results fortime dependent diffusion equations for two different examples with non-trivialboundary conditions.

The first example is motivated from modeling dynamics of nuclear magneticresonance (NMR) signals in the mineral chalk [1, 2], as is richly present underseabeds of the North Sea.

The second example is motivated by Stefan problems, i.e. several phasesseparated by moving boundaries. Historically the moving boundary was aninterface between liquid water and frozen water (ice). Nowadays moving phaseproblems may arise in many applications, for example in the modeling of lithiumion batteries [3]. We have implemented the classical Stefan problem with a timedependent boundary condition and benchmarked a random walk method againstanalytic formulae and numerical finite differences results [4].

Keywords - Diffusion equation, Feynman-Kac formula, Moving phase bound-aries, Robin boundary condition, Stefan problem.

References

[1] Local boundary conditions for NMR-relaxation in digitized porous mediaM. Ogren Eur. Phys. J. B (2014) 87:255, (arXiv:1312.6581).

[2] Numerical simulations of NMR relaxation in chalk using local Robin bound-ary conditions. M. Ogren, D. Jha, S. Dobberschtz, D. Muter, M. Carlsson,M. Gulliksson, S. L. S. Stipp and H. O. Sørensen, Journal of Magnetic Res-onance (2019), https://doi.org/10.1016/j.jmr.2019.106597

52

[3] Numerical simulation of phase separation in cathode materials of lithium ionbatteries. T. Hoffman et al., International Journal of Solids and Structures100-101, 456 (2016).

[4] A random walk method for the heat equation with moving boundary. AndreasLockby, Bachelor thesis 2016:http://oru.diva-portal.org/smash/record.jsf?pid=diva2%3A935914&dswid=3345

Solution of the Stefan problem with general time-dependent boundaryconditions using a random walk method. Daniel Stoor, Bachelor thesis 2019:http://uu.diva-portal.org/smash/record.jsf?pid=diva2%3A1325632&dswid=-6545

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Mathematical Basis of the Stochastic DigitalMeasurement Method

Vladimir Vujicic1, Jelena Djordjevic Kozarov2, Platon Sovilj1 andBojan Vujicic1

1 University of Novi Sad, Faculty of Technical Sciences, Trg Dositeja Obradovica 6,Novi Sad, Serbia, [email protected], [email protected], [email protected]

2 University of Nis, Faculty of Electronic Engineering, Aleksandra Medvedeva 14,Nis, Serbia, [email protected]

Abstract - Stochastic digital measurement method (SDMM) is a significantlydifferent paradigm of digital measurements compared to the standard samplingmethod (SSM). Namely, unlike SSM, this method does not assume that thequantization error is negligibly small. Furthermore, the quantization error canbe 100 percent of full-scale range, but the final (measurement) result will beprecise as well as accurate.

A general overview of the SDMM is given in [1]. Its most important mathe-matical characteristic is summarized in the following quotation from [1]: ”Themathematics that governs the operation all types of the above presented instru-ments is relatively simple but very interesting. Rather than discussing the detailsof mathematical modelling, here we want to point out an element of philosophy.The average value on an interval is actually the integral of the measured func-tion. As the digital integration is a process of adding, the commutative principleapplies. Therefore, the elements of the sum can be added in a deterministicfashion (as they are sampled in time), in an arbitrary chosen order, or in anabsolutely random sequence - it is completely unimportant. Consequently, fromthe average value point of view, the time within the interval can be treated asa stochastic variable with a uniform distribution. In this way, the problem ofmeasurement over the interval can be classified in the Probability theory and thearea of Statistic theory of samples”.

The SDMM is accepted among engineers and it is constantly evolving. Thiscan be seen from [2]. However, so far, none of the mathematicians has writtenany comments about the SDMM. Therefore it is unknown whether or not thismethod is accepted by the mathematicians.The authors believe that this con-ference is a competent forum for the discussion of this issue. This paper givesan overview of the mathematical (theoretical) basis of the SDMM with an em-phasis on a two-bit SDMM and measurements in the transformation (Fourier)domain.

”There is nothing more practical than a good theory” - J.C. Maxwell. Theauthors (they all are engineers) are wondering continuously: is the theory ofthe SDMM good enough? In the last two decades, they have been continuouslyworking on its theoretical/practical expansion and refinement.

55

Keywords - Measurement uncertainty, Probability, Sampling method, Stochas-tic measurement.

56

Stochastic Smart Grid Energy Meter - From anObscure Idea to the Practical Prototype

Marjan Urekar1, Jelena Djordjevic Kozarov2

1 University of Novi Sad, Faculty of Technical Sciences, Trg Dositeja Obradovica 6,Novi Sad, Serbia, [email protected]

2 University of Nis, Faculty of Electronic Engineering, Aleksandra Medvedeva 14,Nis, Serbia, [email protected]

Abstract - Smart Grids are complex systems of electrical power distributiongrids with high level of integration with modern communication and process-ing technology. There are demands for digital high precision electrical energymeters but with low price, as millions of units are needed just for householdsaround the EU. Most modern solutions are based on standard sampling analogto digital convertors (ADC), where input signal is measured in a number of dis-crete points over the sampling period. A different approach, Stochastic DigitalMeasurement Method (SDMM) uses cheap, high speed flash ADCs with lowresolution with added statistical process - dither, a well-defined and controlledrandom noise signal with the uniform probability density function. Dither is astatistical process that, based on the Central limit theorem, in effect decouplesthe quantization error of low resolution ADCs and produce the average valueof the input signal. High precision and high speed/ bandwidth are two op-posing demands for regular sampling ADCs, resulting in complicated hardwareand sources of systematic error. By measuring signals over continuous but fi-nite periods of time with this method, high precision results could be obtained.It is shown that by prolonging the measurement time, measurement precisionincreases (error decreases). Low resolution Stochastic Flash ADC based onSDMM is simple, reliable and enables high sampling speeds with high accu-racy, and less sources of error. Measurement of slowly changing quantities overprolonged periods of time, like electrical energy consumption in a power grid,enables realization of a high precision instrument. Based on detailed mathe-matical model and simulations, a hardware practical prototype of a StochasticSmart Grid Energy Meter was produced at Faculty of Technical Sciences in NoviSad, Serbia. The working prototype of the stochastic instrument with only 4bits of resolution has measurement uncertainty of 0.0080 %, that is up to 25times better than commercially available models with standard ADCs with 16and 18 bits of resolution.

Keywords - Electrical energy, Measurement, Precision, Smart Grid, Stochasticprocess.

57

Goodness of Fit Test for Impulse ResponseFunction

Yuriy Kozachenko1 and Iryna Rozora2

1 Department of Probability Theory, Statistics and Actuarial Mathematics, TarasShevchenko National University of Kyiv, Kyiv, Ukraine, [email protected]

2 Department of Applied Statistics, Taras Shevchenko National University of Kyiv,Kyiv, Ukraine, [email protected]

Abstract - The problem of estimation of a stochastic linear system has beena matter of active research for the last years. One of the simplest models is a’black box’ with some input and giving a certain output. The input may besingle or multiple and there is the same choice for the output. This generatesa great amount of models that can be considered. The sphere of applicationsof these models is very extensive, ranging from signal processing and automaticcontrol to econometrics (errors-in-variables models). An impulse response isthe reaction of any dynamic system in response to some external change. Theimpulse response describes the reaction of the system as a function of time.Any system in a large class known as linear, time-invariant (LTI) is completelycharacterized by its impulse response. That is, for any input, the output can becalculated in terms of the input and the impulse response.

Consider a LTI continuous system with a real-valued square integrable im-pulse response function H(τ) which is defined on a domain τ ∈ [0,Λ]. Thismeans that the response of the system to an input signal X(t) has the followingform

Y (t) =

∫ Λ

0

H(τ)X(t− τ)dτ,

where H ∈ L2([0,Λ]). One of the problems arising in the theory of linearsystems is to estimate the function H from observations of responses of thesystem to certain input signals. A sample input-output cross-correlogram istaken as an estimator of the response function

H(τ) =1

T

∫ T

0

Y (t)X(t− τ)dt,

where T > 0 is a parameter for averaging. The input processes are sup-posed to be zero-mean stationary Gaussian process and can be represented intruncated series of Fourier expansion.

A goodness of fit test on the shape of the impulse response function is giventhat are based on the rates of convergence of the estimator H to H in Banachspace L2([0,Λ]).

58

Keywords - Fourier expansion, Gaussian Process, Goodness of fit test, Impulseresponse function, Linear time-invariant system.

59




Comparison Between Finite Element Analysisand Rheological Models for Chip Formation

Olga Liivapuu1, Juri Olt2 and Tanel Targla3

1 Institute of Technology, Estonian University of Life Sciences, Tartu, Estonia,[email protected]



Abstract - In the process of cutting, often the selection of cutting parametersis done considering empirical methods. This approach is more expensive anddoes not usually lead to the best solutions. Numerical methods for simulatingthe chip formation have been under development over the last thirty years. Theaim of the present research is to compare models based on rheological propertiesof metals with 2D finite element models of chip formation process.

Keywords - Chip formation, Elastic and plastic deformation, Finite elementmethod, Rheological model.

61

Magnetic Force Calculation between TruncatedCone Shaped Permanent Magnet and SoftMagnetic Cylinder Using Hybrid Boundary

Element Method

Ana Vuckovic1, Dusan Vuckovic1, Mirjana Peric1 and NebojsaRaicevic1

1 Faculty of Electronic Engineering, University of Nis, Nis, Serbia, e-mail:[email protected]




Abstract - The field and force modeling of permanent magnets in literaturehave mainly been focused on magnet assemblies with cuboidal, cylindrical orring permanent magnets. However, in many electromechanical devices the useof other permanent magnet shapes may result a performance improvement. Be-cause of that there is a constant need for permanent magnet shape optimizationand size reduction. Besides, there is a special group of problems that considerspermanent magnet structures in the vicinity of different bodies with magneticmaterial composition. In the survey of existing literature, in most of the casesonly field or force between permanent magnet and infinite magnetic plain cal-culation was presented.This paper presents modeling of truncated cone shaped permanent magnet inthe vicinity of a body of finite dimensions made of soft magnetic material. Theforce between permanent magnet and soft magnetic cylinder is performed us-ing the hybrid boundary element method along with semi-analytical approachbased on fictitious magnetization charges and discretization technique. Thehybrid boundary element method (HBEM) was developed at the Departmentof Theoretical Electrical Engineering, Faculty of Electronic Engineering of Nis,and was successfully used for the analysis of multilayered electromagnetic prob-lems. Results of presented approach are compared with results of Finite elementmethod (FEMM software). Magnetic force versus configuration parameters willbe presented in more graphical form.

Keywords - Hybrid Boundary Element Method (HBEM), Magnetic charges,Magnetic force, Permanent magnet.

62

Modelling Overtones in Electric Car Chargingusing Power-Exponential Functions

Karl Lundengard


Abstract - In power delivery systems based on alternating current, harmonicdistortion is an important phenomenon to consider. Therefore, being able toaccurately simulate typical harmonic overtones for typical loads is a useful toolwhen designing and managing power delivery systems. It is believed that charg-ing electric vehicles will become an increasingly important part of the load profileof power delivery systems and the electrical vehicles on the market today areknown to introduce noticeable harmonic distortion. In this presentation we willshow how power-exponential functions can be used to create phenomenologicalmodels of harmonics based on measurements from electrical vehicles.

Keywords - Electrical vehicles, Harmonic distortion, Harmonic overtones, Power-exponential function.

63

Keynote lecture

Date: October 1, 2019Time: 10.00 - 10.30Venue: Beta, House U


The Generalized Weyl Poisson Algebras andtheir Poisson Simplicity Criterion

Vladimir Bavula

Department of Pure Mathematics, University of Sheffield, Sheffield, UK.E-mail: [email protected]

Abstract - A new large class of Poisson algebras, the class of generalized WeylPoisson algebras, is introduced. It can be seen as Poisson algebra analogue ofgeneralized Weyl algebras. A Poisson simplicity criterion is given for generalizedWeyl Poisson algebras and explicit descriptions of the Poisson centre and theabsolute Poisson centre are obtained. Many examples are considered.

Keywords - Poisson algebras, Generalized Weyl Poisson algebras, Poisson cen-tre, Absolute Poisson centre

64


Date: October 1, 2019Time: 10.35 - 12.15Venue: Room Beta, House U


Hom-associative Deformations ofNon-commutative Polynomial Rings

Per Back1, Johan Richter2 and Sergei Silvestrov1

1 Division of Applied Mathematics, The School for Education, Culture andCommunication, Malardalen University, Vasteras, Sweden, [email protected],

[email protected] Department of Mathematics and Natural Sciences, Blekinge Institute of

Technology, Karlskrona, Sweden, [email protected]

Abstract - In this talk, I will give a short introduction to the non-associativeand non-commutative polynomial rings known as hom-associative Ore exten-sions. I will then show how one within this framework can deform otherwiserigid algebras such as the Weyl algebras, and how a hom-associative analogue ofthe Dixmier conjecture can be proven to hold true. If time permits, I will demon-strate how the classical Hilbert’s basis theorem for non-commutative polynomialrings can be generalized to a non-associative setting, and an application thereof.

Keywords - Dixmier conjecture, Hom-associative Ore extensions, Hom-associativeWeyl algebras, Hom-associative deformations, Formal hom-Lie deformations.

66

Constructions and Modules of Hom-PoissonColor Algebras

Ibrahima Bakayoko

University of N’Zerekore/Department of Mathematiques,N’Zerekore, Guinea,

[email protected]

Abstract - We introduce Hom-Poisson color algebras and give the followingvarious constructions:

1. From a non-commutative Hom-associative color algebra by using the com-mutator bracket.

2. From a Hom-associative color algebra which twisting map is an averagingoperator.

3. From a given Hom-Poisson color algebra and an averaging operator.

4. From a given Hom-Poisson color algebra and an endomomorphism.

5. From a Hom-post-Poisson color algebra.

As consequences, we get that:

1. a sequence of Hom-Poisson color algebras may gives rise from either a non-commutative associative color algebra or a Hom-Poisson color algebra andan endomorphism.

2. a Hom-Poisson color algebra may come from a Hom-pre-Poisson coloralgebra.

3. a morphism of Poisson color algebras turns to morphism of Hom-Poissoncolor algebras by twisting the color Poisson structure.

Next we show that associative multplication and lie braket of any Hom-Poissoncolor algebra can be twisted to obtain a module over the given Hom-poissoncolor algebra. Then, we prove that module over a Hom-associative color alge-bra A extends to module over the Hom-Lie color algebra L(A), where L(A) isthe Hom-Lie color algebra associated to the Hom-associative color algebra A.Moreover, by twisting a color Hom-Poisson module structure map by a Hom-Poisson color algebra endomorphism, we get another one. As corollary, we getthat module over a Hom-associative color algebra obtained by twisting an asso-ciative color algebra and the corresponding Hom-Poisson color algebra has themodule structure map.

Keywords - Averaging operator, Hom-Poisson color algebra, Hom-post-Poissoncolor algebras, Homomorphism, Hom-modules.

67

On the Ternary (Hom-)Nambu-Poisson Algebras

Amri Hanene

LaPS, Annaba University, Algeria, [email protected]

Abstract - The main purpose of this talk is to study ternary Nambu-Poissonalgebras and their Hom-type version. We explore twisting principles of ternaryNambu-Poisson algebras along with an algebra morphism that lead to con-struct ternary Hom-Nambu-Poisson algebras, or to deform a classical structureinto a corresponding Hom-structure with an algebra morphism. Furthermore,a 3-dimensional classification of ternary non-commutative Hom-Nambu-Poissonalgebra is giving in the classical case and used twisting principal a 3-dimensionalclassification is giving too. Finally, we provide a proceeding to construct ternary(Hom-)Nambu-Poisson algebras from (Hom-)Poisson algebras equipped with atrace function satisfying some conditions.

Keywords - Algebra morphism, Classification, Hom-structure, Poisson algebra,Trace function, Twisting principle.

68

Keynote lecture



From Graded Extension of the Lorentz Algebrato Colour Dirac Equation and Quark Dynamics

Richard Kerner

Sorbonne-Universite, Laboratorie de Physique Theorique de la Matiere Condensee,CNRS-URA 7600, Boıte courrier 121, 4 Place Jussieu, 75005 Paris, France,

e-mail: [email protected]

Abstract - We propose a modification of standard QCD description of thecolour triplet of quarks describing quark fields endowed with colour degree offreedom We propose to interpret color variables as a generalization of spin, withZ3 cyclic group replacing he usual Z2 symmetry. This leads naturally to “colourDirac equations”, in which the SU(3) colour symmetry is entangled with theZ3-graded generalization of Lorentz symmetry, containing three 6-parametersectors related by Z3 maps. The generalized Lorentz covariance requires simul-taneous presence of 12 colour Dirac multiplets, which lead to the description ofall internal symmetries of quarks: besides SU(3) × SU(2) × U(1), the flavoursymmetries and three quark families. A Z3-graded generalization of Pauli’sexclusion principle is also briefly discussed.

Keywords - Color Dynamics, Dirac equation, Extended Lorentz Symmetry,Z3-graded algebras.

69




Centralizers in Crossed Product Algebras forPiece-wise Constant Function Algebras and

Multiresolution Analysis

Alex Behakanira Tumwesigye1 and Sergei Silvestrov2

1 Department of Mathematics, Makerere University, Kampala, Uganda,[email protected]

2 Division of Applied Mathematics, UKK, Malardalen University, Vasteras, Sweden,[email protected]

Abstract - In this talk we consider the commutant (or centralizer) of the algebraof piecewise constant functions generated by the Haar scaling function in thecrossed product of the same algebra with the integers. Such algebras forman increasing sequence of subalgebras. We derive conditions under which therespective crossed products form an increasing family of algebras and describethe commutant of the coefficient algebra in the crossed product algebra. Wefinish by describing the relative commutant of the crossed product algebra inthe containing crossed product algebra.

Keywords - Centralizer, Crossed product algebras, Multiresolution analysis.

71

Cellular Structure of Seam Algebras andAvenues for Deformations

Alexis Langlois-Remillard1 and Yvan Saint-Aubin2

1 Ghent University, Department of Applied Mathematics, Computer Sciences andStatistic, Krijslaan 281-S9, 9000 Ghent, Belgium, [email protected]

2 Universite de Montreal, Departement de mathematiques et de statistique, H3C 3J7Montreal, Quebec, Canada, [email protected]

Abstract – The seam algebras were introduced by Morin-Duchesne, Rasmussenand Ridout to give algebraic formulation of boundary conditions for loop modelsin statistical physics. They are examples of a family of deformations of theTemperley-Lieb algebras by left- and right-multiplication by an idempotent.

In this talk, we will present a method of study of the representation theoryof seam algebras by constructing the irreducible, the cellular and the princi-pal modules, and expliciting their structure via short exact non-split sequences.The underlying goal will be to identify the key steps in the method to ease itsapplication to other algebras of the form eAe with A a cellular algebra and ean idempotent compatible with the cellular datum. The first part follows:

Langlois-Remillard, A. and Saint-Aubin, Y. The representation theory ofseam algebras (in preparation).

In the last part of this talk, I will present possible avenues of applications ofthis method. The key part of the study was the cellular datum, so it is logical toconsider first the impact deforming the settings has on cellularity. Using a ring-theoretical formulation of cellularity due to Konig and Xi, I claim it is possibleto formulate a Hom-associative version of cellular algebras and will present mypreliminary thoughts on the matter.

Keywords – Cellular algebras, Representation theory of algebras, Seam alge-bras, Temperley-Lieb algebras

72

Two-sided Noncommutative Grobner Basis onQuiver Algebras

Daniel Kariuki1, Damian Maingi2

1 Najing University of Aeronautics/Departmen of Communication and InformationEngineering, Nanjing, China, [email protected]

2 Sultan Qaboos University/Department of Mathematics, Muscat, Oman,[email protected]

Abstract - For a quiver Q, we define a path algebra KQ as a span of all thepaths of positive length. We studied the left and right ideals and their GrobnerBases. We show that we can find Grobner Basis for a two-sided ideal I of KQgiven an admissible ordering.

Keywords - Algebra and Grobner Basis, Quiver.

73

Keynote lecture



Double Constructions of (Bi)Hom-FrobeniusAlgebras

Mahouton Norbert Hounkonnou1 and Damien Houndedji2

1 University of Abomey-Calavi/International Chair in Mathematical Physics andApplications, ICMPA- UNESCO Chair, 072 BP 50, Cotonou, Benin Republic,

[email protected], with copy to [email protected] University of Abomey-Calavi/International Chair in Mathematical Physics and

Applications, ICMPA- UNESCO Chair, 072 BP 50, Cotonou, Benin Republic,[email protected]

Abstract - This paper addresses a hom-associative algebra built as a direct sumof a given hom- associative algebra (A, ·, α) and its dual (A∗, , α∗), endowedwith a non-degenerate symmetric bilinear form B, where · and are the prod-ucts defined on A and A∗, respectively, α and α∗ stand for the correspondingalgebra homomorphisms. Such a double construction, also called hom-Frobeniusalgebra, is interpreted in terms of an infinitesimal hom-bialgebra. The same pro-cedure is applied to characterize the double construction of bihom-associativealgebras, also called bihom-Frobenius algebra. Finally, a double constructionof hom-dendriform algebras, also called double construction of Connes cocycleor symplectic hom-associative algebra, is performed. Besides, the concept ofbihom-dendriform algebras is introduced and discussed. Their bimodules andmatched pairs are also constructed, and related relevant properties are given.

Keywords - (Bi)hom-Frobenius algebra, (Bi)hom-dendriform algebra and in-finitesimal (bi)hom-bialgebra.

74




Complex of Higher Categories and Derivationsof Group Algebras

Aleksandr V. Alekseev1, Andronick Arutyunov1

1 Moscow Institute of Physics and Technologies, Moscow, Russia

Abstract - There is a well-known problem called a derivation problem of B. E.Johnson. In [4], a method was proposed to describe derivations by consideringthe character space of a certain groupoid (where the character χ is defined as amorphism from the groupoid map set to the complex numbers which is for everypair of maps ϕ,ψ the following condition is satisfied χ(ψ ϕ) = χ(ψ) + χ(ϕ))associated with a given group algebra.

We developed this result, proposed a more geometric and general approachfor description of algebras of derivations in our terms of 2-groupoid and exactsequence and got appropriate formula.

It turns out that a given n−category can be associated with a set of vectorspaces, the so-called n−characters, i.e. mappings from the set of n−morphismsof the n−category to complex numbers preserving the composition. In theparticular case when this category is a 2−groupoid associated with a groupalgebra, the complex gives a description of the derivation algebra in the groupalgebra.

Within the framework of this paper, we will also propose a natural modifica-tion of the construction of internal and external derivations, which significantlyencapsulates the study of the space of derivations, taking into account the geo-metric constructions that we introduce.

References

[1] Charles Ehresmann, Catgories et structures, Dunod, Paris 1965

[2] Jean Bnabou, Introduction to bicategories, Reports of the Midwest Cate-gory Seminar, Springer, Berlin, 1967, pp. 1–77

[3] Saunders Mac Lane, Categories for the Working Mathematician, 2004. 352. ISBN 0-387-90036-5

[4] Arutyunov A. A., Mishchenko A. S., Shtern A. I., Derivations of GroupAlgebras, arXiv:1708.05005

76

Derivations in Group Algebras and StollingsTheorem

Andronick Arutyunov1

1 Moscow Institute of Physics and Technologies, Moscow, Russia,[email protected]

Abstract - The main theme of the report is a combinatorial description ofderivations on a group algebra. The linear mapping d : C[G]→ C[G] is said tobe the derivation if the Leibniz rule hold

d(uv) = d(u)v + ud(v),∀u, v ∈ C[G].

We can consider a gruppoid Γ in the following way.The set of objects is a set of elements Obj(Γ) = g ∈ G. The set of arrows

is a set of pairs Hom(Γ) := (u, v)|u, v ∈ G. An arrow φ = (u, v) has a sorces(φ) = v−1u and a target t(φ) = uv−1. For arrows φ = (u1, v1) and ψ = (u2, v2)such that t(φ) = s(ψ) we will define a composition

ψ φ := (v2u1, v2v1).

We will call a charachter any mapping χ : Hom(Γ) → C such that χ(ψ φ) = χ(ψ) + χ(φ) when t(φ) = s(ψ). We will call a charachter a locally finitecharachter if ∀v ∈ Gχ((u, v)) = 0, for all except of finite number of elementsu ∈ G.Theorem For any derivation d exist a character χd such that ∀a ∈ G

d(a) = a

∑t∈[u]

χd(at, a)t

.

We get a new look at the study of differentiations from the position ofcategories and their characters. It becomes reasonable to define the conceptof quasiinner derivations QInn, as a derivations which characters are trivialon loops. It turns out that such differentiations form an ideal. This givesan interesting connection with one old topological invariant, namely with theends of topological spaces. And let Γ be the conjugacy diagram for a givenpresentation of a group. Understanding this graph as a metric space we denoteby e(Γ) the number of ends of this space.Theorem The number e(Γ) doesn’t depend on a corepresentation of the groupG. And the following formula holds

dim(QInn/Inn) = e(Γ)− 1.

77

This approach, among other things, allows us to calculate quasiouter deriva-tions QOut := Der/QInn. It turns out that such differentiations can also beunderstood as characters on a 2-groupoid.Keywords - Derivations, Noncommutative algebras, Noncommutative geome-try, Stollings theorem.

78

Tensor Hierarchy Algebras and ExtendedGeometry

Jakob Palmkvist1

1 Division for Algebra and Geometry, Department of Mathematical Sciences,Chalmers University of Technology and University of Gothenburg, Goteborg,

Sweden, [email protected]

Abstract - Tensor hierarchy algebras constitute a new class of non-contragredientLie superalgebras, whose finite-dimensional members are the simple Lie superal-gebras of Cartan type in Kac’s classification. I will review their construction bygenerators and relations and how they can be used to describe gauge structuresin physical models, especially in extended geometry, where ordinary diffeomor-phisms are unified with internal gauge transformations. The corresponding gen-eralised Lie derivatives do not form a Lie algebra but a generalisation thereof,an L∞ algebra, which can be derived from the tensor hierarchy algebra.

Keywords - Extended geometry, Double and exceptional field theory, L∞ al-gebras, Lie (super)algebras, Tensor hierarchies.

79

Algebras and Operations in Statistical Systems

Ralph Kaufmann and Birgit Wehefritz-Kaufmann

Purdue University, West Lafayette, IN, USA, [email protected]

Abstract - There is a well-known connection between quantum spin chains,integrable models and their underlying algebras in statistical physics. The mostprominent example is the connection between the Temperley-Lieb algebra andthe XXZ quantum spin chain. Applications of this can be found in reaction-diffusion systems, where the spectrum of the Hamiltonian governing the Masterequation can often be found analytically due to the connection to the quantumgroup. In the present work, we connect well known algebras, like the Temperley-Lieb algebra or other algebras present in topological settings, such as loop spacesand knots. This adds a new differential graded (dg) aspect to the statistical side.We also expect relation to Jones’ planar algebras.

80

Keynote lecture

Date: October 1, 2019Time: 10.00 - 10.30Venue: Gamma, House U


An Extended Inverse Gaussian to Model theFirst Exit Time Process

Christos H Skiadas1 and Charilaos Skiadas2

1 ManLab, Technical University of Crete, Chania, Greece, e-mail: [email protected] Department of Mathematics and Computer Science, Hanover College, IN, USA,

e-mail: [email protected]

Abstract - We have proposed a geometric like methodology for the derivationof an extended form of the Inverse Gaussian distribution g(t). First, we startwith the following form of the Inverse Gaussian, that is:

g(t) = |L|σ√

2πt3exp

[− [L−bt]2

2σ2t

], where L is the intercept parameter, σ2 is the

variance and L− bt = H(t) is the state function of the system at time t.Our concern is to find the form of the distribution for an extended form of

the state function H(t) of the system H(t). As we are interested in decayingsystems we have selected a smooth function of the form: H(t) = L−(bt)c, wherec is a shape parameter.

For c = 1 the Inverse Gaussian appear. In this case the state function issimply a line. For other values of this parameter an Extended Inverse Gaussianis expected for the probability density function. The line for the shape functionis replaced by a smooth curve. However, we can, at time t, to assume a verysmall part of the curve H(t) to appear as linear if it is quite small, smooth andcontinuous. In this case the only needed is to consider the following form for

the first exit time density g(t) = |L(t)|t p(t),

Where p(t) is derived in Janssen and Skiadas (1995) in the ASMDA Journal,

that is: p(t) = 1σ√

2πtexp

[− [H(t)]2

2σ2t

].

By using a geometric approach presented with the use of the appropriategraph, we estimate the time varying function L(t) to be: L(t) = H(t)− tH(t)′.

Note that this form can also be derived by expanding H(t) in a Taylor

series of the form: H(t) = H(0) + tH(t)′ + t2

2 H(t)′′ + .... Considering onlythe linear term we arrive to H(0) = H(t) − tH(t)′ where H(0) is the so-called

81

intercept parameter, that is the L parameter in the Inverse Gaussian. Note thatL(t) = H(0) = H(t) − tH(t)′. Then, the Extended Inverse Gaussian is of the

form: g(t) = |H−tH′|σ√

2πt3exp

[− [H(t)]2

2σ2t

].

Clearly the derivation based on the Taylor series has omitted the secondorder derivative and higher order derivatives as well. The next approach isto consider a correction based on the second derivative. This was done in arecent publication (Skiadas et al. in a Springer book, 2018 on Fractional Dy-namics,...) by considering a fractional approach where only a fraction of thesecond derivative is included. This methodology enables to take into accountthe curvature of the state function when estimating the intercept parameter andto find a convenient density form for the death probabilities of a population.

g(t) = 1σ√

2π

[|H−tH′|√

t3+ k

√t3H′′

2|H−tH′|

]exp

[− [H(t)]2

2σ2t

].

Keywords - Extended Inverse Gaussian, First exit time.

82



Date: October 1, 2019Time: 10.35 - 12.15Venue: Room Gamma, House U


Heat Kernel Estimates for Time FractionalEquations With Nonlocal Markov Generators

Jose Luıs da Silva1, Anatoly Kochubei2 and Yuri Kondratiev3

1Faculty of Exact Sciences and Engineering, CIMA – University of Madeira, Funchal,Portugal, [email protected]

2Institute of Mathematics, National Academy of Sciences of Ukraine, Kyiv, Ukraine3Department of Mathematics, University of Bielefeld, Bielefeld, Germany

Abstract - In this talk we investigate the long time behavior of the random timechange of the fundamental solution of a class of evolution equations given byconvolution type operators with regular kernels. We assume that these operatorsare generator of a Markov jump processes, and that their convolution kernelssatisfies certain conditions. We use the Karamata-Tauberian theorem in orderto obtain the Cesaro limit time behavior of the density associated to the timechange process.

References

[1] A. Kochubei, Y. Kondratiev and J. L. da Silva. Heat Kernel Estimates forTime Fractional Equations With Nonlocal Markov Generators. In prepara-tion, 2019.

84

Determining Influential Factors inSpatio-Temporal Models

Rebecca Nalule Muhumuza1,2,3, Olha Bodnar1, Sergei Silvestrov1,Joseph Nzabanita4 and Rebecca Nsubuga5

1 Division of Applied Mathematics, Malardalen University, Box 883, SE 721 23Vasteras, Sweden.

[email protected], [email protected], [email protected] Department of Mathematics, Busitema University, Box 238 Tororo, Kampala,

Uganda, East [email protected]

3 Department of Mathematics, College of Natural Sciences, Makerere University, Box7062, Kampala, Uganda, East Africa.

[email protected] Department of mathematics, CST-University of Rwanda, Kigali, Rwanda, East

[email protected]

5 Uganda Virus Research Institute, Box 49 Entebbe, Uganda, East [email protected]

Abstract - In various areas of modern statistical applications such as in En-vironmetrics, Image Processing, Epidemiology, Biology, Astronomy, IndustrialMathematics, and many others, we encounter challenges of analyzing massivedata sets which are spatially observable, often presented as maps, and tempo-rally correlated. The analysis of such data is usually performed with the goalto obtain both the spatial interpolation and the temporal prediction. In bothcases, the data-generating process has to be fitted by an appropriate stochasticmodel which should have two main properties: (i) it should provide a good fitto the true underlying model; (ii) its structure could not be too complicatedavoiding considerable estimation error appeared by fitting the model to realdata. Consequently, achieving the reasonable trade-off between the model un-certainty and the parameter uncertainty is one of the most difficult questions ofmodern statistical theory.

We deal with this problem in the case of general spatio-temporal models.New approaches are developed to determine the most influential factors to beincluded into the model. We also discuss the computational aspects in the caseof large-dimensional data and apply the theoretical findings to real data.

Keywords - Hierarchical model, Latent process, Statistical inference, Variableselection.

85

Arbitrage Theory with State-Price Deflators

Salvador Cruz Rambaud1

1 Universidad de Almerıa / Departamento de Economıa y Empresa, Spain,[email protected]

Abstract - The aim of this work is to derive some mathematical results for theexistence of asset price bubbles, which is consistent with the martingale analy-sis of financial markets. More specifically, this manuscript analyzes the relationbetween the divergence of the sum of dividend-price ratios and the absence ofbubbles. The framework describes a contemporary model of a securities marketbuilt in an environment with both uncertainty and a countable number of trad-ing dates. The market allows trading of a finite number of corporate securities.In this context, no arbitrage condition is characterized by the existence of astate-price deflator across an infinite time horizon.

Keywords - Arbitrage, Bubble, Convergence, Martingale, State-price deflator.

86





On the Rate of Convergence in Central LimitTheorem for Nearly Critical Branching

Processes with Immigration

Sadillo Sharipov

Institute of Mathematics named after V.I.Romanovskiy, Uzbek Academy of Sciences,Tashkent, Uzbekistan, [email protected]

Abstract - Let for all n ≥ 1,ξ

(n)k,j , k, j ≥ 1

and

ε

(n)k , k ≥ 1

be two indepen-

dent, non-negative, integer valued random variables such thatξ

(n)k,j , k, j ≥ 1

and

ε

(n)k , k ≥ 1

for each n ≥ 1 are identically distributed. Define a sequence

of branching processes with immigration X(n)k , k ≥ 0, n ≥ 1, recursively as

X(n)k =

X(n)k−1∑j=1

ξ(n)k,j + ε

(n)k , k, n ≥ 1, X

(n)0 = 0. (1)

We assume that for all n ≥ 1, mn = Eξ(n)1,1 , σ2

n = V arξ(n)1,1 , λn = Eε

(n)1 ,

b2n = V arε(n)1 are exist and finite. Set β2

n = V arX(n)n and denote by Φσ2 (x)

distribution function of normal law with zero mean and variance σ2 > 0 and put

Fn (x) = P((nb2n)−1/2

(X

(n)n − EX

(n)n

)< x

). The random process (1)is called

nearly critical if mn → 1 as n → ∞. M.Ispany [2008, Theorem 2.9] provedfunctional limit theorems for fluctuation of process (1). Our main result is thefollowing inequality which asserts the rate of convergence for one-dimensionalanalog of Theorem 2.9.

For all n ≥ 2 the following inequality holds:

ρ (Fn,Φβ) = supx|Fn (x)− Φβ (x)| ≤ C√

nb2n+

∣∣∣β2n

b2n− β2

∣∣∣min

(β2n

b2n, β2) ,

where C may dependence from α, σ2, λ, b2.

Keywords - Branching process, Immigration.

88

On the Inequalities for Moments of BranchingProcesses with Immigration

Ya.M. Khusanbaev1 , Kh.E. Kudratov2

1 Institute of Mathematics, Uzbekistan Academy of Sciences, [email protected] National University of Uzbekistan, qudratovh−[email protected]

Abstract - One of the main problems of probability theory is to estimate thecentral moments of random variables. It was obtained several estimates for mo-ments sums of independent random variables (see [1]). In this paper we presentthe estimate for central moments of the branching processes with immigration.

Let ξk,i, k, i ≥ 1 and εk, k ≥ 1 be the independent sequences of indepen-dent, nonnegative, integer valued and identically distributed random variables.

We consider the branching processes with immigration Xk, k ≥ 0 given byrecursion

X0 = η, Xk =

Xk−1∑j=1

ξk,j + εk, k ≥ 1

where η is an nonnegative, integer valued random variable and independent ofξk,i, εk, k, i ≥ 1(see [2]).

We introduce

m = Eξ1,1, γp = Eξp1,1, θp = E | ξ1,1 −m |p, ν = Eη

λ = Eε1, τp = Eεp1, δp = E | ε1 − λ |p, νp = Eηp

Theorem. Suppose m 6= 1, θp < ∞ and δp < ∞ for any p > 0. Then thefollowing statements holds:

(i) If 0 < p ≤ 1, then

E | Xn − EXn |p≤ θp[mp(n−1)(mn(1−p) − 1)

m(1−p) − 1(ν +

λ

m− 1)−

− λ(mnp − 1)

(mp − 1)(m− 1)] + δp

mnp − 1

mp − 1

(ii) If 1 < p ≤ 2, then

E | Xn − EXn |p≤ 2p+1θp[mp(n−1)(mn(1−p) − 1)

m(1−p) − 1(ν +

λ

m− 1)−

− λ(mnp − 1)

(mp − 1)(m− 1)] + 2pδp

mnp − 1

mp − 1

89

(iii) If p > 2, and 2p2−1γ p

26= 1, then

E | Xn − EXn |p≤ 2p−1C(p)np2−1[νpθp

(2p2−1γ p

2)n −mnp

2p2−1γ p

2−mp

+

+2

p2−1τ p

2θp

2p2−1γ p

2− 1

((2

p2−1γ p

2)n −mnp

2p2−1γ p

2−mp

−

−mnp − 1

mp − 1) + δp

mnp − 1

mp − 1]

where C(p) is a constant, which depends only on p.

References

[1] V.V. Petrov, Limit theorems for sum independent random variables, 1987,Moskov (in Russian)

[2] Athreya K.B., Ney P.E., Branching processes, Springer - Verlag, Berlin.1972.287p.

90

On Asymptotics of Branching Processes withImmigration

Gairat Rakhimov1, Lola Sharipova2

1 V.I. Romanovskiy Institute of Mathematics/Department of Probability Theory andMathematical Statistics, senior researcher, Tashkent, Uzbekistan,

[email protected] Tashkent institute of railway transport engineers/Department of Higher

Mathematics, assistant professor, Tashkent, Uzbekistan, [email protected]

Abstract - Let ξk,j , k, j ∈ N and εk, k ∈ N be two independent sets ofrandom variables taking non-negative integer values, moreover ξk,j , k, j ∈ Nbe independent and equally distributed. Let the sequence of random variablesXk, k ≥ 0 be defined by the following recurrence relations

X0 = 0, Xk =

Xk−1∑j=1

ξk,j + εk, k = 1, 2, . . .

Thus defined random processes often arise in population theory and are calledbranching processes with immigration.

For large n, finding the distribution of the variable Xn is an importantproblem. Many works have been devoted to finding the asymptotics of thedistribution of Xn, in the case when the random variables εk, k ∈ N are in-dependent and equally distributed. In the present work, we consider the casewhen εk, k ∈ N are variously distributed, means λk = Eεk and variationsb2k = varεk of these variables increase with the growth in the index k andchange correctly at infinity. We have obtained functional limit theorems forstep processes constructed by the values of the branching process with immi-gration Xk, k ≥ 0. Moreover, no conditions are imposed on the dependence ofrandom variables ξk,j , k, j ∈ N and εk, k ∈ N .

Keywords - Asymptotics, Branching process, Functional limit theorem.

91





Lie Symmetry Analysis on Pricing WeatherDerivatives by Partial Differential Equations

Clarinda Nhangumbe1,2, Ebrahim Fredericks1 and BetuelCanhanga2

1 Department of Mathematic and Applied Mathematics, University of Cape Town,Cape Town, South Africa, [email protected]

2 Department of Mathematic and Informatic, Eduardo Mondlane University,Maputo, Mozambique, [email protected], [email protected]

Abstract - Weather derivatives are priced considering weather variables such asrainfall, temperature, humidity and wind as the underlying asset. The OrnsteinUhlenbeck Process was proposed in [1] to model the amount of rainfall. Byusing the Feynman-Kac theorem and the rainfall index we derive the partialdifferential equation(PDE) that governs the price of an option. We apply theLie analysis theory to solve this PDE, we provide the group classification andwe use it to find the invariant analytical solutions.

Keywords - Lie symmetry analysis, Ornstein-Uhlenbeck process, Partial dif-ferential equation, Rainfall index, Weather derivatives.

References

[1] Unami, K., Abagale, F.K., Yangyuoru, M., Alam, A.H.M.B., Kranjac-Berisavljevic, G.: A stochastic differential equation model for assessingdrought and flood risks. Stochastic environmental research and risk as-sessment. 24,725–733 (2010)

93

Suggestion of a Hidden Markov Model for FirstOrder Markov Dependent Observations

Ozgur Danisman1 and Umay Uzunoglu Kocer2

1 Department of Statistics, Dokuz Eylul University, Izmir, Turkey,[email protected]

2 Department of Statistics, Dokuz Eylul University,Izmir, Turkey,[email protected]

Abstract - A hidden Markov model (HMM) is a two dimensional stochasticprocess which can be applied in many areas such as earthquake analysis, signalprocessing, speech processing, communications or text recognition. The processmakes transitions between unobservable (hidden) states in accordance with theMarkovian property while emitting another observable stochastic random vari-able. The main assumption of HMMs is that the observations emitted after eachtransition are independent and identically distributed. However, observationsmay be dependent not only the current hidden state but also the previous ob-servation in many practical problems. In this study, a modification of the modelis studied when a first order Markov-dependency exists between two successiveobservations. Assumptions and complete parameter set of the proposed modelis defined. For the estimation of parameters, modified Baum-Welch algorithm ispresented. An experimental study is conducted to examine the performance ofthe proposed model. Finally, a comparison between the classical and modifiedHMMs is given.

Keywords - Baum-Welch algorithm, Dependent observations, Markov models,Parameter estimation, Simulation.

94

Investigating some Attributes of Periodicity inDNA Sequences via Semi-Markov Modeling

Pavlos Kolias1 and Aleka Papadopoulou1

1 Department of Mathematics, Aristotle University of Thessaloniki, Thessaloniki,Greece, [email protected], [email protected]

Abstract - Periodicity is a structural property of DNA sequences. It is ex-pressed as either nucleotides or words of nucleotides that appear with specificfixed distances in-between. Mainly, there have been observed two types of peri-odic behaviours in DNA. The first one has been observed in chromatin, whichis a basic element of the cell nucleus. The researchers observed that certaindi-nucleotides in the DNA of chromatin tends to appear at approximately every10 to 11 bases. Subsequent studies suggested that the period of chromatin se-quences converges to 10.4 bases. Also, more recent studies, which investigatedthe genome of three organisms, A. thaliana, C.elegans and H.sapiens, suggestedthat the di-nucleotide AA has almost perfect 10.5-base periodic behaviour inthose organisms. One explanation about this type of periodicity is that thedistance of 10.5 bases is exactly the ”step” of the double strand, which curvesthe DNA chain and allows these long sequences to suppress into the small areaof the nucleus. The second type of periodicity has been observed in areas of thegenome that are transcribed and later translated into proteins, also called cod-ing regions. Previous studies have used methods from mathematical analysis,such as the spectral density, and they have shown that in coding regions, thereis a tendency of certain nucleotides to reappear every 3-bases. Also, this typeof periodicity has only been observed in coding regions, while for non-codingregions similar periodic behaviour has not been observed. As each of the aminoacids is encoded with a triplet of nucleotides (codons) and some specific aminoacids are more abundant than others, authors concluded that the periodic be-haviour, in fact exists, due to this higher frequency of certain amino acids andthe period of 3-bases is sue to the triplet nature of the DNA. As the wholegenome of each organism is frequently of several billions bases, the informationabout the periodic behaviour of the coding regions of the DNA would be reallyhelpful into detecting those regions and distinguish between protein encodingregions and non-coding regions. Some algorithmic techniques have already beenimplemented using this information and they have used mathematical methods,such as the Fourier transformation. Also, some other well-known algorithms usehidden-Markov models, in order to classify between different regions of DNA.In this paper, a semi-Markov model is applied to 3-base periodic sequences,which characterize the protein-coding regions of the gene. Analytic forms of therelated probabilities and the corresponding indexes are provided, which yield adescription of the underlying periodic pattern. Last, the previous theoretical

95

results are illustrated with synthesized and real data from different organisms.

Keywords - DNA sequences, Periodicity, Semi-Markov chain.

96

Inferring Social Clustering from Register Data

Fredrik Jansson1, Gunn E. Birkelund2 and Mats Lillehagen2


2 Department of Sociology and Human Geography, University of Oslo, Norway

There is an increasing availability of large-scale data, documenting people’schoices and behaviour. The wealth of data raises new possibilities and challengesfor data analysis. What this data typically does not contain, is information onthe mechanism behind these choices, such as the social networks in which peopleinteract and influence each other. We will here present a new statistical methodfor inferring patterns of social clustering based on similarity of choices. Themethod will be applied to the case of social influence in school and drawinginferences from register data on educational choices, by comparing within- andbetween-cohort designs, to separate endogenous from exogenous effects.

We develop a three-step analysis to distinguish between endogenous (throughsocial influence in structured populations) and exogenous (related to the inde-pendent variable) preferences. First, we develop a measure of exogenous pref-erences. We then use the same measure for estimating preferences that includeboth endogenous and exogenous factors. Finally, we compare the two to find ameasure of social clustering controlled for exogenous preferences.

Inferring social clustering from choice data poses methodological challenges.For example, most linear and discrete choice models study choices in socialisolation. There are relatively recent attempts at incorporating the impact ofsocial networks, but generally these assume the underlying graph is known, andinclude a measure of overall, instead of dyadic, peer effects. Our method allowsfor studying choices that are made simultaneously and the extent to which theyare the same. The unit of analysis is thus the dyad, which is, in turn, a subgraph.

Dyadic units impose several statistical challenges. First, there are high de-pendencies, e.g. due to transitivity. If A and B make the same choice, and sodo B and C, then A and C must also have made the same choice. Further, cor-relation measures between cohorts are not directly comparable, meaning that itis not straightforward to compute aggregated measures.

We know of no alternative method that could directly address simultaneousdyadic choices, controlling for structural dependencies (and exogenous effects).E.g., in a McFadden discrete choice model, the peer effects terms would endup being infinitely recursive. Therefore, we approach these challenges by usinga combination of statistical methods, including modifications of the quadraticassignment procedure (QAP), to find graph correlations between the existenceof edges derived from the independent and dependent variables, controlling forstructure, and meta-analytic statistical models, to find an aggregate measure.

We illustrate the method on Scandinavian register data and find that im-migrant students tend to cluster in educational choices based on country oforigin (but not based on cultural similarity measures on weighted graphs), evencontrolled for exogenous effects, with higher propensities for social ties.

Keywords - Applications, Data analysis, Graphs, Simulation, Statistical meth-ods.

98

The Relaxed Stochastic Maximum Principle forG-SDE

Amel Redjil1

1 Department of Mathematics and LaPS laboratory, UBM University, Annaba,Algeria, [email protected]

Abstract - We are interested by systems subject to model uncertainty or am-biguity due to incomplete or inaccurate information, or vague concepts andprinciples.

Climate or weather and financial markets are typical fields where informa-tion is subject to uncertainty.

We study stochastic optimal control problem in the G-frame work, we estab-lish existence of an optimal stochastic relaxed control for stochastic differentialequations driven by a G-Brownian motion and we prove the stability results.

We study the maximum principle in order to derive the necessary conditionsof optimality.

Keywords - G-chattering lemma, G-Brownian motion, Relaxed optimal con-trol.

References

[1] A. Redjil and S. E. Choutri, On Relaxed Stochastic Optimal Control forStochastic Differential Equations Driven by G−Brownian Motion, ALEA,Lat. Am. J. Probab. Math. Stat. 15(2018), 201− 212.

[2] S. Peng, Nonlinear expectations and stochastic calculus under uncertainty,arXiv preprint arXiv:1002.4546, 2010.

[3] S. Bahlali, B. Mezerdi and B. Djehiche, Approximation and optimality neces-sary conditions in relaxed stochastic control problems, International Journalof Stochastic Analysis , (2006), 1− 23

[4] N. El Karoui, D. H. Nguyen, and M.Jeanblanc-Picqu, Compactificationmethods in the control of degenerate diffusions: existence of an optimalcontrol, Stochastics, 20, no.3, (1987), 169− 219.

99


Date: October 1, 2019Time: 10.35 - 12.15Venue: Room Kappa, House U


The Localized Orthogonal DecompositionMethod for Flow and Wave Propagation

Problems in Fractured Porous Media

Fredrik Hellman1, Axel Malqvist1 and Siyang Wang2

1 Department of Mathematical Sciences, Chalmers University of Technology andUniversity of Gothenburg, Gothenburg, Sweden.

2 School of Education, Culture and Communication, Malardalen University, Vsters,Sweden, [email protected]

Abstract - To solve partial differential equations with rapidly varying coef-ficients, a major challenge is the high resolution needed to resolve the datavariation. In this work, we consider a Darcy flow problem in fractured porousmedia. Besides the data variation, an additional difficulty here is to efficientlyhandle thin and highly conductive fractures. They typically have a width sev-eral orders of magnitude smaller than the length scale, and resolving their thinstructures is computationally infeasible.

Instead, we consider an asymptotic model [1], where thin structures arereduced to internal interfaces in the governing equations. We develop a varia-tional multiscale finite element method in the localized orthogonal decomposi-tion (LOD) framework [2]. A special type of Scott-Zhang interpolation operatoris constructed to ensure that, in the presence of interfaces, the corrected basisfunctions decay exponentially from the supports. This leads to a sparse systemin the upscaled representation, and can be solved efficiently. We derive a priorierror estimate to prove that the method converges to the optimal first orderin energy norm, even when the mesh does not resolve the fine scale features[3]. We present numerical experiments to demonstrate the effectiveness of theproposed method for both flow and wave propagation problems.

Keywords - Darcy flow, Fractures, Localized orthogonal decomposition, Mul-tiscale, Porous media, Wave Propagation.

References

[1] V. Martin, J. Jaffre and J.E. Roberts, Modeling fractures and barriers asinterfaces for flow in porous media, SIAM J. Sci. Comput., 26(2005), pp.1667–1691.

[2] A. Malqvist and D. Peterseim, Localization of elliptic multiscale problems,Math. Comp., 83(2014), pp. 2583–2603.

101

[3] F. Hellman, A. Malqvist and S. Wang, Numerical upscaling for het-erogeneous materials in fractured domains, Submitted, available onlinearXiv:1908.03822.

102

Homotopy Analysis Method (HAM) forDifferential Equations Pertaining to the Mixed

Convection Boundary-Layer Flow over a VerticalSurface Embedded in a Porous Medium

Imran Chandarki1 and Brijbhan Singh3

1 Department of Mathematics(GSE),N. B. Navale Sinhgad College of Engineering, Solapur-413255 (M.S.) India

2 Department of Mathematics,Dr. Babasaheb Ambedkar Technological University, Lonere,402103 Tal-Mangaon,

Dit. Raigad(M.S.) India

Abstract - The objective of present paper is to revisit the problem pertainingto a vertically flowing fluid passed a model of a thin vertical fin in a saturatedporous medium. The governing equations have been simplified using the simi-larity transformation to yield ordinary differential equations. These equationshave been solved by homotopy analysis method (HAM). It is shown that thesolution has two branches in a certain range of the mixed convection and surfacetemperature parameters. The effects of these parameters on the velocity dis-tribution have been presented graphically. The results obtained by HAM havebeen found in good agreement with the corresponding results obtained by otherauthors.

Keywords - Homotopy analysis method (HAM), Mixed convection, Similarityboundary layer equations.

103

Thin Film Flow of Carreau Nanofluid Past aHeated Nonlinear Stretching Sheet

Prashant G Metri1 and Mahesha Narayana2

1 Division of Applied Mathematics, Malardalen University,UKK, Vasteras, Sweden,[email protected]

2 aDepartment of Mathematics, The University of the West Indies, Kingston 7,Jamaica, [email protected]

Abstract - The impact of magnetic field and nanoparticle on two diemesionalthin film flow of generalized Newtonian Carreau fluid flow over nonlinear heatedstretching sheet has been analyzed. An evolution equation for the film thick-ness is derived using long-wave approximation theory of thin liquid film. Thegoverning nonlinear PDEs with boundary condition are transformed into a sys-tem system of nonlinear ODEs with appropriate boundary condition by usingdiemensionaless terms. It is solved numerically for some representative valuesof non-diemensional parameters. The impacts of non-diemensional controllingparameters on velocity and temperature profiles are scrutinized with the aid ofgraphs. The effect of Maragoni and Prandtl numbers are explored in presenceof magnetic field. The physical explanations are provided to understand thediffetrent effects of film thinning.

Keywords - Carreau nanofluid, Heat transfer, Thin liquid film.

104

B-spline Approach for Approximate Solution ofMRLW Equation via BFRK Scheme

Saumya Rajan Jena1, Guesh Simretab Gebremedhin1

1 KIIT DT University, India, [email protected]

Abstract - In this note, an approximate solution of a modified regularized longwave (MRLW) equation has been determined directly neither transformationnor linearization of the nonlinear terms of the MRLW equation using septicB spline approach with the help of Butchers fifth order Runge Kutta (BFRK)scheme which is best fit to the solution of system of ODEs without using finitedifference scheme. The validity of the present approach has been performed formass, momentum and energy of the motion to test the conservative propertiesand also the results have been compared with other researchers.

Keywords - B-spline approach, Butcher’s fifth order Runge Kutta scheme,Invariants sep partial differential equation, Solitary wave.

105


Date: October 1, 2019Time: 13.45 - 14.45Venue: Room Kappa, House U


On the Approximation of UnstructuredPopulation Model with Stage-structuredPopulation Model in a Grazing System

Sam Canpwonyi1, Linus Carlsson1,2 Betty K. Nannyonga3

1 Makerere University, Department of Mathematics, Makerere University, Kampala,Uganda, [email protected]

2 Malardalen University, Department of Applied Mathematics, MalardalenUniversity,Vestaras, Sweden, [email protected]


Abstract - A great deal of ecological theory is based on simple Lotka-Volterratype of unstructured population models in the study of complex population dy-namics and communities in order to obtain important information for predictingtheir future evolution. In these models, they assumed that all individuals be-have equally with respect to the population dynamics and have the same birthand death rates, and contribute to competition for shared resources in the sameway with many oversimplifying assumptions which are not biologically realistic.

Applying this paradigm on the grazing system consisting of coupled ODEs de-scribing the dynamics of forage resource and livestock population in an ecosys-tem, we study the two models and see how the unstructured population modelcan be approximated by a stage-structured population model by looking at thediscrete dynamics of the individuals at different life stages (juveniles and adults).Graphical approach will be used to perform these analyses.

Keywords - Ecosystem, Forage, Grazing, Life history stages, Unstructured-population model.

107

A Mathematical Model for Harvesting in aStage-Structured Cannibalistic System

Loy Nankinga1, Linus Carlsson2


2 Malardalen University, Division of Applied Mathematics, Malardalen University ,Vasteras, Sweden, [email protected]

Abstract - To increase the production of proteins in East Africa, aquaculturegained increased attention recently. We study the interactions of a consumer-resource system with harvesting, in which African Catfish consume a food re-source. The cannibalistic behaviour of African Catfish is captured by using afour stage-structured system.

The dynamics of food resource and African Catfish result in a system ofordinary differential equations called a stage-structured fish population model.Existence and stability steady of states are analysed qualitatively and quanti-tatively.

Results from the simulations will help us to determine the most appropriateharvesting strategy that gives maximum sustainability yield as well as financialyield. The main goal of our findings will be to inform policy in order to improvethe fishing strategies and hence increase fish production and avoid extinction.

Keywords - Harvesting, Cannibalism and Stage-structured.

108

Study of a Nonlinear Problem from Geology

Ngonn Seam1

1 Department of Mathematics, Royal University of Phnom Penh, RussianConfederation Boulevard, Phnom Penh, Cambodia, [email protected]

Abstract - This work deals with the study to a nonlinear degenerated pseudoparabolic problem involving a nonlinear term f (∂tu). By the way of an implicittime discretisation, we would prove the existence of a solution to the problem

f (∂tu)−∆u− τ∆∂tu = 0

where f is a non-decreasing Lipschitz-continuous function, null at the origin.As an application, if H denotes the maximal monotone graph of the Heavysidefunction, we should prove the existence of a solution to the differential inclusion

0 ∈ H (∂tu)−∆u− τ∆∂tu.

We finally simulate the numerical solution by finite element method to the aboveproblem.

Keywords - Differential inclusion, Existence results, Pseudo parabolic prob-lems, Time discretization.

References

[1] S.N. Antontsev, G. Gagneux, A. Mokrani, G. Vallet, Stratigraphic modellingby the way of a pseudoparabolic problem with constraint, Adv. Math. Sci.Appl. 19 (2009) 195209.

[2] H. Brzis, Oprateurs maximaux monotones et semi-groupes de contractionsdans les espaces de Hilbert, in: Notas de Matematica (50), in: North- Hol-land Mathematics Studies, vol. 5, North-Holland Publishing Comp., Ams-terdam, London, 1973, p. 183. American Elsevier Publishing Comp. Inc.,

[3] G. Gagneux, G. Vallet, Sur des problmes dasservissements stratigraphiques,ESAIM Control Optim. Calc. Var. 8 (2002) 715739

[4] G. Gagneux, G. Vallet, A result of existence for an original convectiondiffu-sion equation, RACSAM Rev. R. Acad. Cienc. Exactas Fs. Nat. Ser. A Mat.99 (1) (2005) 125131.

[5] G. Vallet, Sur une loi de conservation issue de la gologie, C. R. Math. Acad.Sci. Paris 337 (8) (2003) 559564.

109

Keynote lecture



Homomorphisms of Pseudo-Riemannian Calculiand Noncommutative Minimial Submanifolds

Joakim Arnlind1, Axel Tiger Norkvist2

1 Dept. of Math., Linkoping University, Linkoping, Sweden, [email protected] Dept. of Math., Linkoping University, Linkoping, Sweden, [email protected]

Abstract - In this talk I will review the concept of pseudo-Riemannian calculias a framework for noncommutative Riemannian geometry. In this setting, onecan consider metric and torsion-free connections and prove that there exists atmost one such (“Levi-Civita”) connection. By introducing homomorphisms ofsuch calculi, one can discuss noncommutative embeddings and we prove thatthere exists a theory of submanifolds much in analogy with the classical case,including second fundamental forms, the Weingarten map and Gauss’ equations.Via the trace of the second fundamental form, one readily introduces the meancurvature, and the definition of a noncommutative minimal submanifold is givenas an embedding with zero mean curvature. As a motivating example, we con-sider the noncommutative torus as a minimal surface in the noncommutative3-sphere.

Keywords - Noncommutative Riemannian geometry, Noncommutative sub-manifolds, Noncommutative minimal surfaces.

110




Applications of Matroid and PolymatroidTheory to Algebraic Coding and Information

Theory

Thomas Westerback

Division of Applied Mathematics, Malardalens University, Vasteras, Sweden,[email protected]

Abstract - Matroid theory, a branch of algebraic combinatorics, has been suc-cessfully used to solve problems in many areas of mathematics and computerscience. The theory has links to areas such as algebra, combinatorics, geom-etry and topology. For example, graphs and matrices over fields give rise tomatroids. Polymatroids, a generalization of matroids, can be considered bothas a set-combinatorial object and as a special class of polytopes. The theoryof polymatroids has especially been important for combinatorial optimizationwhen submodular functions are considered. Polymatroids can also be used inconnection with many other concepts, for example Shannon entropy and hyper-graphs.

Matroid theory and polymatroid theory are closely related to coding andinformation theory via matrices over fields and Shannon entropy. Despite thisfacts, matorid and polymatroid theory has only rather recently played an impor-tant role in the development of coding, communication and information theory.For example in areas such as distributed storage for large network data storage,network coding, index coding and machine learning. In this talk, we will presenthow certain properties of different algebraic structures can be captured by ma-troids and polymatroids and and some applications of matroid and polymatroidtheory to algebraic coding and information theory.

Keywords - Coding, Information, Matroid, Polymatroid.

112

On a Relation Between Rook Polynomials andGraph Theory

Aleksandra Rosic1, Branislav Randjelovic2 and DraganaStanojevic3

1 Information Technology School, Belgrade, Serbia, [email protected] University of Nis, Faculty of Electronic Engineering, Nis, Serbia,

[email protected] Institute for Education Quality and Evaluation, Belgrade, Serbia,

[email protected]

Abstract - In this paper we give an overview on two-dimensional boards, appro-priate rook polynomials and their correspondence with graphs. This paper aimsto illustrate that the concept of rook polynomials can be extended to graph the-ory. We also discuss possibilities of transformation of square boards into boardsthat can be tiled with regular polygons, generated by the movement of the rookin more than two directions, all through corresponding rook polynomials andgraphs.

Keywords - Graph theory, Permutations, Rook polynomials.

113

Algorithms for Recalculating CentralityMeasures Based on Powers of the Adjacency

Matrix

Collins Anguzu1, Christopher Engstrom2, John Magero Mango1,and Sergei Silvestrov2

1 Department of Mathematics, Makerere University, Kampala, Uganda,[email protected], [email protected]

2 Divison of Applied Mathematics, Malardalen University, Vasteras, Sweden,[email protected], [email protected]

Abstract - In graph theory, centrality measures are very crucial in rankingnodes of the graph in order of their importance. Numerous types of centralitymeasures do exist with each of them being applied in quite different metrics.Our focus is on those types based on the powers of the adjacency matrix orrandom walks. In this work we will look at recalculating alpha and eigenvectorcentralities for graphs we first partition into components. Particularly we shallconsider in-component changes and weight change between components.

Keywords - Centrality, Eigenvector, PageRank, Random walks.

114

Vertex Connectivity of Line Graph of Γ(Zn)

Vijay Kumar Bhat and Pradeep Singh

School of Mathematics, SMVD University, Katra, India,[email protected]

[email protected]

Abstract - For a ring R, the Line graph L(Γ(R)) of a zero divisor graph Γ(R)is defined as a graph with vertex set V (L(Γ(R))) = E(Γ(R)) i.e., a vertex of aline graph L(Γ(R)) represents an edge of Γ(R), and distinct vertices in L(Γ(R))are connected if and only if their corresponding edges in Γ(R) share a commonvertex.

V (L(Γ(R))) = (a, b) | a.b = 0, a and b are non-zero zero divisors of RE(L(Γ(R))) = e1, e2 | e1 and e2 are incident to common vertex in Γ(R)

Line graph of zero divisor graph of Zn has been discussed in [1].

A graph is connected if there is a path joining every pair of vertices. Ver-tex connectivity of a graph is the minimum number of vertices whose removalresults in disconnection of the graph.

Vertex connectivity of L(Γ(Zn)) for different values of n has been discussedby several authors. In this article, we derive a relation to find the vertex con-nectivity of L(Γ(Zn)) for n = pm, where p is a prime number. We prove thefollowing:

The vertex connectivity of L(Γ(Zpm)) is:

1. 2(pm2 − 3), when m is even

2. pm+1

2 + pm−1

2 − 5, when m is odd.

Keywords - Line graph, Vertex connectivity, Zero divisor graph.

References

[1] Aponte, C. I., Johnson, P. S. and Mims, N. A; Line Graphs of Zero DivisorGraphs, Summer Undergraduate Mathematical Sciences Research InstituteJournal, Miami University, 2005.

115

Keynote lecture



Orthogonal Systems in the Complex Plane andApplications in Numerical Integration

Gradimir V. Milovanovic

Serbian Academy of Sciences and Arts, Mathematical Institute, Belgrade, Serbia,[email protected]

Abstract - A few classes of polynomials and rational functions, orthogonalin the complex plane with respect to the Hermitian and non-Hermitian innerproducts are considered, as well as some applications in numerical analysis.

The first results on this subject were obtained for orthogonal polynomialson the semicircle Γ = z = eiθ | 0 ≤ θ ≤ π with respect to the non-Hermitianinner product defined by

(f, g) =

∫Γ

f(z)g(z)w(z)(iz)−1dz,

where z 7→ w(z) is a complex weight function holomorphic in the half diskD+ = z ∈ C | |z| < 1, = z > 0 (see [1], [2], [3]).

Beside the construction and analysis of these new orthogonal systems, thecorresponding quadrature formulas of the highest degree of exactness are alsoconstructed. The obtained results allow a very efficient application in the inte-gration of quasi-singular integrals, as well as in numerical differentiation.

Keywords - Complex orthogonal systems, Quadrature formula, Quasi-singularintegrals, Recurrence relations, Zeros.

References

[1] W. Gautschi, G. V. Milovanovic, Polynomials orthogonal on the semicircle,J. Approx. Theory, 46 (1986), pp. 230–250.

116

[2] W. Gautschi, H. J. Landau, G. V. Milovanovic, Polynomials orthogonal onthe semicircle. II, Constr. Approx., 3 (1987), pp. 389–404.

[3] G. V. Milovanovic, Complex orthogonality on the semicircle with respect toGegenbauer weight: theory and applications, In: Topics in MathematicalAnalysis (Th. M. Rassias, ed.), pp. 695–722, Ser. Pure Math., 11, WorldSci. Publ., Teaneck, NJ, 1989.

117




On the Exponential and Trigonometricq, ω-special Functions

Thomas Ernst

Department of Mathematics, Uppsala University, Uppsala, Sweden,[email protected]

Abstract – The purpose of this talk is to continue the study of q, ω-specialfunctions in the spirit of Wolfgang Hahn from the previous papers by Annabyet.al. and Varma et.al. By introducing the new variable ω, we develop a quitesimilar calculus consisting of two dual exponential, hyperbolic and trigonometricfunctions. The concept even and odd functions is replaced by x, ω-even and odd,since a change of sign in x is always accompanied by a change of sign in ω. Inthe same way, formulas for chain rule, Leibniz theorem, q, ω-additions for thethree above functions are introduced. Graphs for these functions are shown,which closely resemble the original ones. To enable q, ω-trigonometric formulaswith half argument and de Moivre theorem, Ward numbers and q, ω-rationalnumbers are introduced.

References

[1] M. H. Annaby, A. E. Hamza, K. A. Aldwoah, Hahn difference operator andassociated Jackson–Nørlund integrals. J. Optim. Theory Appl. 154, No. 1,133–153 (2012).

[2] T. Ernst, A comprehensive treatment of q-calculus. Birkhauser 2012.

[3] S. Varma, B. Yasar, M. Ozarslan, Hahn–Appell polynomials and their d-orthogonality. Rev. R. Acad. Cienc. Exact. Fıs. Nat. (2018).

119

Heap Cohomology and Ternary Self-distributiveCohomology

Mohamed Elhamdadi1, Masahico Saito2 and Emanuele Zappala3

1 University of South Florida, Department of Mathematics, Tampa, FL, USA,[email protected]

2 University of South Florida, Department of Mathematics, Tampa, FL, USA,[email protected]

3University of South Florida, Department of Mathematics, Tampa, FL, USA,[email protected]

Abstract - Heaps are para-associative ternary operations bijectively exempli-fied by groups endowed with the operation (x, y, z) 7→ xy−1z. They are alsoternary self-distributive, and therefore have a diagrammatic interpretation byframed links. Motivated by these properties, I will introduce heap cohomologyand ternary self-distributive cohomology with abelian heap coefficients, with thepurpose of defining framed link cocycle invariants. I will also explain the relationintercurring between heap cohomology, and ternary self-distributive and groupcohomologies. I will briefly discuss, lastly, heap objects in symmetric monoidalcategories.

Keywords - Cocycle invariant, Cohomology, Framed link, Group, Heap, Self-distributive operation.

120

Common Fixed Point Results for Perov TypeWeakly Commuting Maps in Cone Metric

Spaces

Talat Nazir1 and Mujahid Abbas2

1 Iepartment of Mathematics, COMSATS University Islamabad, AbbottabadCampus, Pakistan, [email protected]

2 Department of Mathematics, Government College University, Lahore, Pakistan andDepartment of Mathematics, University of Pretoria, Pretoria, South Africa,

[email protected]

Abstract - The aim of this paper is to present coincidence and common fixedpoint results of Perov type weakly commuting mappings satisfying certain gen-eralized contractive conditions in the context of cone metric spaces. It is worthmentioning that these results are obtained without appealing to any form ofcontinuity of mappings involved herein. Some examples are presented to sup-port the results proved in this paper. We study also well-posedness of a relatedfixed point and common fixed point problems. Our results unify, generalize andextend various comparable results in the existing literature.

Keywords - Coincidence point, Common fixed point, Generalized contractionpair, Perov type mapping, Weakly commuting maps, Well-posedness.

121

Wallis Type Formula in Quantum Calculus

Predrag M. Rajkovic1, Sergei D. Silvestrov2, Sladjana D.Marinkovic3

1 Faculty of Mechanical Engineering at University of Nis, Serbia,[email protected]

2 Division of Applied Mathematics, The School of Education, Culture andCommunication (UKK), Malardalen University, Vasteras, Sweden,

[email protected] Faculty of Electronic Engineering at University of Nis, Serbia,

[email protected]

Abstract - John Wallis has discovered in 1655. the famous formula

π

2=

∞∏n=1

(2k)2

(2k − 1)(2k + 1),

by a relation between an integral and the area of the quarter of the unit circle.Later, Euler discovered another proof of Wallis’ formula based on the product

development. Also, Wallis’ formula is often connected with the evaluation ofthe definite integrals

Wallis’ formula inspired a few mathematicians to avoid integration and proveit another way. For example, A.M. Yaglom proved it by the trigonometricidentities and J. Wastlund by the special geometrical figures.

In this paper, we will use the previous idea to justify an approach to thenumber π in q–calculus. We will expose a geometrical interpretation of the q–Wallis formula. We construct plane regions which consist of rectangles whoseedges’ lengths are directly connected with factors in this formula. These regionsare bounded by quarters of inside and outside circles from which we get estimatesand conclusions about the number πq.

Keywords - Infinite products, Number π, Q–numbers, Wallis formula.

122

Keynote lecture



Extension of Nambu theory to Superspace

Viktor Abramov

Institute of Mathematics and Statistics, University of Tartu, Tartu, Estonia,[email protected]

Abstract - We propose a generalization of the NambuHamilton equation insuperspace with three real and two Grassmann coordinates. We construct theeven degree vector field in the superspace by means of the right-hand sides ofthe proposed generalization of the NambuHamilton equation and show that thisvector field is divergenceless in superspace. Then we show that our generaliza-tion of the NambuHamilton equation in superspace leads to a family of ternarybrackets of even degree functions defined with the help of a Berezinian. Thisfamily of ternary brackets is parametrized by the infinite dimensional group ofinvertible second order matrices, whose entries are differentiable functions onthe space R3. We study the structure of the ternary bracket in a more generalcase of a superspace Rn|2 with n real and two Grassmann coordinates and showthat for any invertible second order functional matrix it splits into the sum oftwo ternary brackets, where one is the usual NambuPoisson bracket, extendedin a natural way to even degree functions in a superspace Rn|2, and the secondis a new ternary bracket, which we call the Ψ-bracket, where Ψ can be identifiedwith an invertible second order functional matrix. We prove that the ternaryΨ-bracket as well as the whole ternary bracket (the sum of the Ψ-bracket withthe usual NambuPoisson bracket) is totally skew-symmetric, and satisfies theLeibniz rule and the FilippovJacobi identity ( Fundamental Identity).

Keywords - Nambu-Hamilton mechanics, Ternary Nambu-Poisson brackets,Superspace.

123




About Compact Monothetic Semirings

Boris Tanana1, Bhangy Cassy2 and Sergei Silvestrov3

1 ISCTEM, Maputo, Mozambique, [email protected] DMI - UEM, Maputo, Mozambique, [email protected]

3 Malardalen University - Division of Applied Mathematics, Vasteras, Sweden,[email protected]

Abstract - In this communication we present the structure of compact mono-thetic semirings. The classification of these semirings be based on known des-cription of discrete cyclic semirings and compact monothetic semigroups. Wegive, initially, the necessary definitions and notions. Semiring is an algebraicstructure < S,+, x> where < S,+ > and < S, x> are semigroups (bothoperations, addition ”+” and multiplication ”×” are associative) and these op-erations satisfy the laws of distributivity for both sides. A Semiring < S,+, x>is a topological semiring if S is a topological Hausdorff space and both opera-tions ”+” and ”x” are continuous. A cyclic semiring is a semiring with cyclicmultiplication in which every element is an entire non-negative power of somegenerating element a of S. This fact we will denote by S =< a >. Let Sbe a semigroup and a topological Hausdorff space where an operation that iscontinuous than S is a topological semigroup. So, a topological semigroup S issaid to be monothetic semigroup if there exists an element a of S such that thesmallest closed subsemigroup in S containing an element a coincide with S. Atopological semiring S is a monothetic semiring if there exists an element a ofS such that S =< a >∗. Where X∗ is a closure of any subset X of a topologicalspace. Now we will reduce the description of a compact monothetic semigroupsbelonging to E. Hewitt [1]. So, a compact monothetic semigroup S =< a >∗

has one of the three kinds: i) S is a compact monothetic group. ii) S containsa compact monothetic group H (with generating element b and unit element eand a finite number of distinct discreet elements a, a2, . . . , as which don’t belongto H,S = HUa, a2, . . . , as, and ae = b, ak belongs to H for k > s. iii). Scontains a compact monothetic group H (with generating element b and unit el-ement e and infinite number of distinct discreet elements a, a2, . . . , which don’tbelong to H,S = HUa, a2, . . . , and ae = b.Theorem Let S =< S,+, x>, be a compact monothetic semiring. Then berealized one of the next kinds: i) S is a discreet finite cyclic semiring. ii)< S,x>, is a no discreet compact monothetic semigroup and < S,+ > or hasleft addition (x + y = x for all x, y of S) or has right addition (x + y = y forall x, y of S) or < S,+ > is a chain at relation of order x ≤ y, if and only if,x+ y = y for all x, y of S.

Keywords - Compact, Cyclic, Monothetic, Semigroup, Semiring.

125

References

[1] Hewitt E., Compact monothetic semigroups. Duke Math. J., 23, N3, 1956,pp. 447 - 457.

126

Semilattice Decomposition of Semigroups: fromTheory to Applications

Sergei Silvestrov1, Melanija Mitrovic1,2

1 Division of Applied Mathematics, School of Education, Culture andCommunication, Malardalen University, Vasteras, Sweden, [email protected] Department of Mathematics and Informatics, Faculty of Mechanical Engineering,

University of Nis, Nis, Serbia, [email protected]

Abstract - A semigroup is an algebraic structure consisting of a set with anassociative binary operation defined on it. We can say that most of the workwithin theory is done on semigroups with a finiteness condition, i.e. a semi-groups possessing any property which is valid for all finite semigroups - like, forexample, π-regularity, completely π-regularity, periodicity, finite generation are.There are many different techniques for describing various kinds of semigroups.Among the methods with general applications is a semilattice decomposition ofsemigroups. Certain types of semigroups being decomposable into a semilatticeof archimedean semigroups occurred in semigroup-theoretic investigations of di-verse directions. Here, we are interested, in particular, in the decomposabilityof a certain type of semigroups with finiteness conditions into a semilattice ofarchimedean semigroups. Roughly speaking this talk will be mostly about the“games” which elements (special and/or “ordinary”) of the semigroup can playand influence of that “games” on semigroup structure, in first place in makingconnection between semilattice decomposition, hereditarness and periodicity.

“Semigroups aren’t a barren, sterile flower on the tree of algebra, they are anatural algebraic approach to some of the most fundamental concepts of algebra(and mathematics in general), this is why they have been in existence for morethen half a century, and this is why they are here to stay,” (B. M. Schein inSemigroup Forum, 54, 1997, 264-268). Semigroup-theoretic approach becomesquite substantial in more complex branches of mathematics, like theories ofautomata, formal languages, codes, and combinatorics in general. However, thistalk does not pretend to mention all of the existing applications of semigrouptheory. Throughout this talk we are going to list some of the applications ofpresented classes of semigroups and their semilattice decompositions in certaintypes of ring constructions. The results which will be presented are mainlybased on the ones given in authors’ book and (overview) papers (some of themalready published, some of them accepted for publication).

Keywords - GVS-semigroup, Hereditary GVS-semigroup, MBC-semigroup,Semilattice decomposition of semigroups, Semilattice of archimedean semigroups.

127

A Note on Cyclic Codes over the Ring Zpk

Om Prakash1, Habibul Islam2 and Syamantak Das3

1 Department of Mathematics, Indian Institute of Technology Patna, Patna, India-801 106, [email protected]



Abstract - For a prime p and an integer k > 1, let Zpk be the ring of integersmodulo pk. This article completely determines the structure of cyclic codes overZpk . Also, minimal generating sets for those codes are obtained. Finally, somecomputational examples are given to show the importance of our results.

Keywords - Cyclic code, Hamming distance, Generating set.

128

Public Key Cryptography and Twisted DihedralGroup Rings

M.D. Gomez Olvera1, J.A. Lopez Ramos2 and B. Torrecillas3

1 Department of Mathematics, University of Almerıa, Almerıa, Spain,[email protected]

2 Department of Mathematics, University of Almerıa, Almerıa, Spain, [email protected] Department of Mathematics, University of Almerıa, Almerıa, Spain, [email protected]

Abstract - Public Key Cryptography is a system that allows to share a se-cret through an insecure channel. This secret might be used later as a key toprovide confidentiality of communications. Current standards include schemesbased in either classical Number Theory results or Elliptic Curves over a finitefield. These were shown to be insecure in a theoretical way using an applica-tion of the the Fast Fourier Transform. However this algorithm that threatsthese cryptographic schemes could be developed only on a quantum computer.Recent reports of some international organisms recommend to start develop-ing new constructions to provide security of communications for the so calledPost-Quantum Cryptography. One of the recommendations includes consideringconjugations of elements in non-commutative structures.

In this work we consider a twisted dihedral group ring by a 2-cocycle andestablish a key exchange in this setting based on the so called DecompositionProblem, i.e., given two elements a and b, finding other two elements x, y suchthat a = xby, which is clearly an extension of the preceding recommendation.We show the existence of an associated public key cryptosystem and how we canextend our scheme to secure communications of group of users. We will showhow the non-commutaivity plays a fundamental role, not only in the securityof the scheme, but also when establishing the different protocols, since theyclearly differ from the traditional ones given on commutative settings or cyclicsubgroups (and thus commutative) of non-commutative structures.

Keywords - Decomposition Problem, Post-Quantum Cryptography, TwistedGroup Ring.

129

Keynote lecture

Date: October 2, 2019Time: 10.00 - 10.30Venue: Gamma, House U


On the ROCOF of Higher Order forSemi-Markov and Generalized Semi-Markov

Systems with Applications in Finance

Guglielmo D’Amico1 and Filippo Petroni2

1 University ”G. d’Annunzio” of Chieti-Pescara/Department of Pharmacy,Laboratory of Mathematics, Chieti, Italy,[email protected]

2 Marche Polytechnic University/Department of Management, Laboratory ofMathematics, Ancona, Italy, [email protected]

Abstract - In this paper we study the rate of occurrence of failures (ROCOF)of continuous time semi-Markov systems and generalized semi-Markov systemsof indexed type.The ROCOF is an important reliability indicator which has been originally con-sidered for a finite Markov process by Shi (1985, Acta Math. Appl. Sin.) andsuccessively extended for Markov processes (MP) with denoumerable state spaceby Lam (1997 J. Appl. Probab.). A further extension to semi-Markov processes(SMP) was advanced by Ouhbi and Limnios (2002 Stat. Probab. Lett.) wherea formula for computing the ROCOF was derived and a statistical estimatorwas studied by assessing its asymptotic properties.

In general, the ROCOF gives information whether there are a lot of failuresor only a few within a time interval. In the study of failures of a system, it is alsointeresting the study of the relative positioning of pairs of failures and more ingeneral of tuples of failures. Consequently, an extension of the ROCOF, calledROCOF of higher order, was advanced for MP in D’Amico (2015, Methodol.Comput. Appl. Probab.).

Here we consider SMP and a mixed discrete-continuous probability distri-bution for the initial law of the system taking into account the possible randomstarting from any state of the system with any duration. Furthermore, we de-termine a formula for the ROCOF of higher order for SMP and we recover asparticular cases the results contained in the above quoted papers. The results

130

are further extended to the framework of indexed semi-Markov models. Dif-ferent applications to financial data demonstrates the possibility of using thisindex as a measure of volatility in diverse financial contests.

Keywords - Semi-Markov system, Rocof, Volatility.

131





Stochastic Optimal Control with a MultiscaleVolatility Model

Jean-Paul Murara1 and Betuel Canhanga2

1 Malardalen University, Vasteras, Sweden, [email protected] Eduardo Mondlane University / Faculty of Sciences, Department of Mathematics

and Computer Sciences, Box 257, Maputo, Mozambique, [email protected]

Abstract - In this paper, using generalised reference probability space we startby giving a strong and also a week formulation of an optimal control prob-lem. We extend the general problem to the particular stochastic optimal contolproblem. After, we study the application of the above in the case of a port-folio optimization with a utility function that follows a multiscale stochasticvolatilitity model.

Keywords - Dynamic programming principle, Portfolio optimization, Stochas-tic volatility model.

133

Functional Data Analysis for StochasticVolatility

Jean-Paul Murara1and Bahaeddine Taoufik2

1 Malardalen University, Vasteras, Sweden, [email protected] Saint Joseph’s University / Departement of Mathematics, 5600 City Ave,

Philadelphia, PA 19131, Pennsylvania, USA, [email protected]

Abstract - In this paper, we present a functional volatility process for mod-eling stochastic volatility trajectories in financial markets. A description of arelated multiscale stochastic volatilities model is introduced. Simulations for im-plementing this functional stochastic volatility process and related results aregiven. The paper is organized as follows. We start by formulating a functionalvolatility process and describing a multiscale stochastic volatility model. Afterlinking the two notions, we make simulations for financial markets.

Keywords - Functional data analysis, Multiscale stochastic volatility model.

134

Structured Maintenance Policy for DeterioratingSystems under Cumulative Shocks

Shoma Matoba1 and Lu Jin1

1 Department of Informatics, University of Electro-Communications, Tokyo, Japan,[email protected], [email protected]

Abstract - An optimal maintenance policy was investigated for deterioratingsystems subject to shocks that occur in accordance with a homogeneous Pois-son process. The random variables representing the magnitudes of the shocksare assumed to be independent, identically distributed, and independent of thecounting process for shocks. Shocks that could damage the system are identifiedon the basis of whether their magnitude exceeds a determined threshold valueand recorded. System deterioration is modeled as a discrete-time Markov chain,with the state transition probabilities depending on the number of cumulativerecorded shocks. At the beginning of every time period, the decision makerselects one action from three possible actions: operate, repair, replace. Theoptimal decision-making problem is formulated as a Markov decision process.An optimal maintenance policy that minimizes the total discounted expectedcost over an infinite horizon is determined on the basis of information relatedto system deterioration and the cumulative shocks. Several structural proper-ties of an optimal maintenance policy are obtained under certain conditions ondeterioration and cost conditions. These properties are helpful for determiningan optimal maintenance policy.

Keywords - Markov decision process, Monotone property, Optimal decision-making, Poisson process, Shock model.

135

Stochastic Approach for Nonlinear StiffBehaviour

Hande Uslu1 and Murat Sari1

1 Department of Mathematics, Yildiz Technical University, Istanbul, Turkey,[email protected]

Abstract - It is important to define the behaviour of natural phenomena witha solution method. Sudden reactions taken place in the relevant real system onsmall time scales should be considered. Stiff differential equations have the abil-ity to represent unstable behaviour for very small values and are used to modelmany problems from chemical reactions to control theory. Capturing real-lifebehaviour represented by stiff problems often has many limitations, unless tradi-tional methods are rebuilt for such problems in small time steps. Despite severalresearches conducted by researchers to determine which method is the most effi-cient for years, it has been found that the methods suffer from limitations fromtime to time in terms of computational cost or issues of constructions. This factnecessitates the development of alternative approaches for stiff behaviours suchas simulation techniques. This study aims at analyzing how stiff behaviours canbe captured by stochastic approaches. Thus, a Monte Carlo based stochasticalgorithm is presented for some nonlinear stiff differential equations. It has beenseen that the considered approach gives relatively definite results. In order torealize the simulated behaviour of the processes properly, detailed discussionshave been made in the present study.

Keywords - Initial value problem, Monte Carlo method, Nonlinear process,Stiff differential equation, Stochastic approach.

136





Are these Moments known?

Lars Hellstrom1


Abstract - The Lanczos algorithm for matrices takes as input an Hermitianmatrix A and a unit vector v1 (usually chosen randomly to get a with highprobability general position). In each iteration it computes two real scalarstraditionally denoted αk−1 and βk, and another unit vector vk orthogonal toall of v1, . . . , vk−1. These vectors make up a unitary matrix V such that V ∗AVis tridiagonal, and the scalars computed are the elements of that tridiagonalmatrix. The algorithm is typically used as an initial step of computing theeigendecomposition of very large Hermitian matrices.

Being unit, the vectors vk may be interpreted as probability amplitude vec-tors (ket vectors) in matrix quantum mechanics, in which case the matrix be-comes an observable with the matrix spectrum (set of eigenvalues) as range.Under this interpretation, αk turns out to be exactly the expected value of Awith respect to the distribution determined by vk, and β2 is exactly the stan-dard deviation of A with respect to v1. Later βk are however slightly morecomplicated, and the fact that the evolution of the state vector vk is not mean-ingfully unitary (thus unphysical) makes an interpretation as a quantum processsomewhat strained.

Alternatively, one may try to express everything with respect to the distribu-tion given by the original vector v1. It then turns out that the first m numbersin the sequence α1, β2, α2, β3, . . . determine the first m moments of the observ-able A (with respect to v1). This begs the question: are these moments known,and if so what are they called?

Though these quantities may have their origin in a quantum mechanicalinterpretation of a numerical linear algebra algorithm, there is nothing thatrestricts them to that setting; they may be defined for real random variables ingeneral. Given a random variable X, one would define a sequence Yk∞k=0 ofderived random variables and the moments α1, β2, . . . recursively through

Y0 = 1, Y1 = X − α1, Yk = 1βk

(X − αk)Yk−1 − βk

βk−1Yk−2 for k > 2,

β1 = 1, βk+1 = σ[Yk], αk = E[XY 2k−1]/β2

k for k > 1.

The derived random variables Y0, Y1, . . . , though obviously dependent upon X,are by construction all pairwise uncorrelated; they constitute a sequence oforthogonal polynomials in X. The moments α1 = E[X], β2 = σ[X], α2, β3, . . .naturally all have the same unit as the underlying random variable X.

138

Keywords - Lanczos algorithm for matrix tridiagonalisation, Higher moments,Probability amplitude.

139

Fourth Cumulant for the Random Sum ofRandom Vectors

Farrukh Javed1, Nicola Loperfido2 and Stepan Mazur3

1 Unit of Statistics, Orebro University School of Business, Orebro, Sweden,[email protected]

2 Dipartimento di Economia, Societa e Politica, Universita degli Studi di Urbino”Carlo Bo”, Urbino, Italy, [email protected]

3 Unit of Statistics, Orebro University School of Business, Orebro, Sweden,[email protected]

Abstract - In this paper, we consider the random sum of random vectors thatis expressed as

s := x1 + · · ·+ xN ,

where x1, . . . ,xN are iid d-dimensional random vectors, N is the univariate ran-dom variable that is independent of x1, . . . ,xN . The fourth cumulant of s isdelivered. Two important special cases are considered. In the first one, multi-variate skewed normal distribution for xi’s is considered that is aggregated by aPoisson variable. The second case is dealing with multivariate asymmetric gen-eralized Laplace and aggregation is made by a negative binomial variable. Dueto the invariance property the latter case can be derived directly, leading to theidentity involving the cumulant of xi’s and s. There is a well established relationbetween asymmetric Laplace motion and negative binomial process that corre-sponds to the invariance principle of s for the generalized asymmetric Laplacedistribution. We explore this relation and provide multivariate continuous timeversion of the results.

Keywords - Fourth cumulant, Laplace motion, Random sum, Skew-normal.

140

Closed-form Estimator for the Matrix-VariateGamma Distribution

Gustav Alfelt1

1 Stockholm University, Department of Mathematics, Stockholm, Sweden,[email protected]

Abstract - In this paper we present a novel closed-form estimator for the param-eters of the matrix-variate gamma distribution. The new estimator relies on themoments of the diagonal elements of a transformation of the observed matrices.We compare the new estimator with the maximum likelihood estimator (MLE)in a simulation study, revealing that the suggested estimator outperforms theMLE for ill-conditioned scale matrices, in terms of mean squared error. Thestudy further reveals that, for well-conditioned scale matrices, the applicationof the proposed closed-form estimator as a starting value makes the numeri-cal optimization procedure for the MLE more efficient, compared to using anarbitrary start value.

Keywords - Ill-conditioned matrices, Parameter estimation, Random matrixdistribution, Simulation.

141

Noninformative Bayesian Approach in ExtendedRandom Effects Meta-Analysis

Olha Bodnar1,2 and Clemens Elster3

1 Unit of Statistics, School of Business, Orebro University, 70182 Orebro, Sweden2 Division of Applied Mathematics, Malardalen University, Hogskoleplan 1, 722 20

Vasteras, Sweden, [email protected] Physikalisch-Technische Bundesanstalt, Abbestrasse 2-12, 10587 Berlin, Germany,

[email protected]

Abstract - Quality management plays an important role in measurement sci-ence. It usually appears in key comparisons, where one of the main goals isthe identification of the laboratories which underestimate their uncertaintiesbudget, i.e. the laboratories whose measurement results considerably deviatefrom those obtained by other participating laboratories. The situation becomeseven more difficult when several measurement standards are circulated amongparticipants in separate petals.

We propose a new statistical method for analyzing data from a key compar-ison when two transfer standards are measured in two petals. The approachis based on a generalization of the classical random effects model, a popularprocedure in metrology. A Bayesian treatment of the model parameters as wellas of the random effects is suggested. The latter can be viewed as potentiallaboratory effects which are assessed through the proposed analysis. While theprior for the laboratory effects naturally is assigned as a Gaussian distribution,the Berger & Bernardo reference prior is taken for the remaining model param-eters. The results are presented in terms of the posterior distributions derivedfor the laboratory effects. From these distributions posterior means and credi-ble intervals are calculated. The proposed method paves the way for applyingthe established random effects model also for data arising from the measure-ment of several transfer standards in several petals, and it is illustrated formeasurements of two 500mg transfer standards carried out in key comparisonCCM.M-K7.

Keywords - Bayesian reference analysis, Credible interval, Interlaboratorycomparison, Reference prior.

142





Improved Uncertainty Modelling of Process Variations inan Industry 4.0 Manufacturing Facility by use of

Probability Box Uncertainty Models

Keivan Shariatmadar1, Frederik Debrouwere1, Ashwin Misra1,2 andMark Versteyhe1

1Dep of Mechanical Engineering-Division PMA, KU Leuven Campus Bruges,Belgium, keivan.shariatmadar,frederik.debrouwere,[email protected]

2Dep. of Mechanical Engineering, Delhi Technological University, New Delhi, India,ashwin [email protected]

Abstract - Industry 4.0 is the fourth and current working paradigm leaning ondata exchange and further automation of manufacturing technologies. This I4.0paradigm enables the possibility for optimal autonomous decision making andfull autonomous operation of the facility. Since all manufacturing processes areinherently indeterministic (machine breakdowns, stock limitations, obstructionsof resource usages, delays of jobs; continuous variations in available resources,execution times, human errors), optimal decision making will only work whenprovided with realistic uncertainty models. The authors have observed thatthe typical used uncertainty models do not suffice as they do not represent theindeterministic nature of the processes in a realistic manner.

The objective of this paper is to use the logged data to design more rele-vant uncertainty models. For this purpose, p-box models are used to representproduction variation as function of unknown indeterministic parameters. Fur-thermore, by using regression learning on the database, we show that the un-certainty can be decreased. We will show that classic stochastic models will notsuffice to capture the full indeterministic nature of the uncertainties. It makestherefore sense that autonomous decision making based on such stochastic mod-els are unreliable and suboptimal. Future research possibilities are discussed,based on the proposed uncertainty models, together with possible notions thatwill help in the development of reliable optimal decision making.

Motivated by the need for more realistic uncertainty models, this paper haspresented an alternative approach for modelling nondeterministic uncertaintiesby use of p-box models. We conclude, from observation of real data that a singleCDF does not suffice for scheduling as it varies to much as a function of nonobservable nondeterministic parameters. Instead a p-box provides a more real-istic representation of reality. We state that autonomous decision making basedon stochastic models are therefore unreliable in practice. A regression algo-rithm was proposed and observed to improve the uncertainty on the productiontime significantly: p-box models with a much narrower band are obtained. Thepresented approach is highly valuable for autonomous decision making in anIndustry 4.0 framework which is inherently nondeterministic. Furthermore, weobserved that data models could augment analytic predictions and reduce the

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indeterminism in the uncertainty.Future research will focus on the incorporation of these types of uncertainty

models for scheduling under indeterministic uncertainty.

Keywords - Data training, Imprecise uncertainty modelling, Industry 4.0,Probability box.

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The Algorithm Solution of the Problem ofOptimal Control in a Dynamic One-Sector

Economic Model with a Discrete Time Based onDynamic Programming Method

Petr Shnurkov 1, Anna Rudak2

1 Department of of Higher Mathematics, National Research University Higher Schoolof Economics, Moscow, Russian Federation, [email protected]

2 Department of Applied Mathematics, National Research University Higher Schoolof Economics, Moscow, Russian Federation, [email protected]

Abstract - In this paper we study a new formulation of the optimal controlproblem in a dynamic single-sector economic model with discrete time. In thetask, the states are the values of the specific capital, that is, the total amountof capital related to the unit of labor resources. The role of management isplayed by a parameter representing the proportion of the specific product pro-duced that is directed to investment. The target functionality is the sum oftwo components. The first one expresses the specific consumption accumulatedduring the evolution of the system. The second is expressed as a given func-tion of the value of the specific capital at the final point in time and describesthe level of technological development in the system formed at that moment.The main limitation is the dynamic ratio for the specific capital, describing itschange under the influence of management. The initial state in the system isassumed to be fixed. The study is based on the dynamic programming method.The Bellman equations for the problem are obtained. Based on the well-knowntheoretical assertions, it is established that the sequence of controls satisfyingthe Bellman equations is optimal. An algorithm has been created and describedin detail that allows one to solve the Bellman functional equations numericallyand find a sequence of optimal controls for the problem posed.

Keywords - Dynamic programming, One-sector model of an economic system.

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On Steady-state Analysis of Retrial Queues withState-Dependent Input Flow

Eugene Lebedev, Vadym Ponomarov, and Hanna Livinska1

1 Taras Shevchenko National University of Kyiv, Applied Statistics Department,Kyiv, Ukraine, [email protected], [email protected], [email protected]

Abstract - The work is devoted to studies of a multi-server retrial queue withthe rate of input flow dependent on the number of sources of retrial calls inthe system. Such systems form a class of models, that is important from apractical point of view. Unlike classic queueing systems, retrial queues assumethat each call that cannot be served immediately returns to the system andtries to get service again in a random period of time independently from othersources of retrial calls. The existence of the second flow of incoming calls allowsto model the service process more precisely, but turns the steady-state analysisof retrial queues into a complicated problem. The Markov chain describing theservice process in a retrial queue is multidimensional, and it has infinite statespace. The transition matrix of such a process usually does not have specialproperties that would streamline the explicit solution of the Kolmogorov set ofequations. Moreover, in systems with state-dependent characteristics, transientprobabilities depend on its current state, that makes the problem of findingsteady-state probabilities even harder. Therefore, only the simplest models areexplicitly investigated so far.

Let us consider a retrial queue with a Poisson input flow of calls with itsrate dependent on the number of sources of repeated calls in the system. Callsare served by c identical servers and service time is exponentially distributedwith the rate ν. If an input call finds all servers busy, it becomes a source ofretrial calls and tries to get service after a random time, that is exponentiallydistributed with the rate µ. The service process of such a retrial queue canbe described in terms of bivariate Markov process X(t) = (X1(t), X2(t)), whosestate space is a lattice semistrip S(X) = 0, 1, . . . , c×Z+. The first componentX1(t) ∈ 0, 1, ..., c indicates the number of busy servers at the instant t ≥ 0,and the second one X2(t) ∈ 0, 1, ... is the number of retrial calls.

For the described above class of retrial queues, a vector-matrix representa-tion of steady-state distribution is obtained. This representation allows to writedown the stationary probabilities via the model parameters. In partial cases ofc = 1, 2 the obtained solution turns into explicit formulas of a scalar type. Theinvestigative techniques use an approximation of the initial model by means ofthe truncated one and the direct passage to the limit.

Authors also present a multicriteria problem of the maximization of totalincome from the system. The obtained representation of steady-state proba-bilities allows to build an algorithm for solving this problem and find out an

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optimal control policy, that will grant the maximum income.

Keywords - Queueing, Repeated calls, State-dependent input flow, Stationaryregime.

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On the Solution of the Optimal ControlProblem of Inventory of a Discrete Product in

Stochastic Model of Regeneration

Nikita Vakhtanov1 and Petr Shnurkov2

1 Department of Higher Mathematics, National Research University Higher School ofEconomics, Moscow, Russian Federation, [email protected]

2 Department of Applied Mathematics, National Research University Higher Schoolof Economics, Moscow, Russian Federation, [email protected]

Abstract - The work considers a new complete model of discrete product in-ventory control in regeneration scheme with a Poisson flow of customer require-ments and random delivery delay. In the system deferred demand is allowed,the volume of which is limited by a given value N0. The control parameter r isthe level of the stock, at which achievement it is necessary to make an order forreplenishment, and this parameter is determined in accordance with a discreteprobability distribution, which plays the role of a control strategy. The momentsof change in the states of the regeneration process describing the functioningof the system are the moments of arrival of customers, if the value of deferreddemand does not exceed the value N0, and the moments of replenishment of theinventory.

As an indicator of control efficiency, we consider the average specific profitobtained during the regeneration period. In order to obtain an explicit repre-sentation for this indicator, a special version of the classical ergodic theoremwas proved for the additive cost functional, which has an additional componentassociated with regeneration moments. The optimal control problem is solvedon the basis of the statement about the extremum of a fractional-linear integralfunctional on the set of discrete probability distributions. It is established thatthe optimal control strategy is deterministic and is determined by the point ofglobal extremum of the function, which is a stationary cost efficiency indicatorand depends on the control parameter.

In the work, explicit representations are derived for the mathematical ex-pectations of the increments of the profit functional on the regeneration periodunder all possible conditions on the control parameter, in other words, the de-cision taken on the specified period. These analytical representations enable usto explicitly obtain the stationary cost indicator of control efficiency as a func-tion of the control parameter and, for given model characteristics, numericallydetermine the optimal value of the control, which contributes to solving one ofthe important applied problems of the modern economy.

Keywords - Inventory management, Controlled regenerative process, Station-ary cost indicator of control effciency.

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Analysis of a Recurrent Multi-Server Queuewith Different Service Rules

H. Okan Isguder1, Umay U. Kocer2



Abstract - In this study, a recurrent arrival GI/M/c/K queueing system withheterogeneous servers are examined under two different servivice disciplines.Interarrival times are independent random variables and have an arbitrary F (t)distribution function which has a finite mean; that is a =

∫∞0

[1− F (t)] dt.Service time in server k detoted by ηk is an exponential random variable withparameter µk (k = 1, . . . , c). Arrival and service processes are independenteachother and there is a system capacity denoted by K. When all the servers arebusy, arring customers leave the system without being served. These customersare lost. The defined queueing system is analysed under two different serviceprocedure:

• An arriving customer begins service in any available server with the sameprobability.

• An arriving customer begins service with probability 1 in the server whichhas the lowest index number.

The defined queueing system is modeled by using semi-Markov process andembedded Markov chain representing the system is obtained. Additionaly, anal-ysis for the stream of overflow is performed and the loss probabilites are com-puted. Finally some performance measures such as mean waiting time and meannumber of customers are computed for different interarrival time distributions.

Keywords - Embedded Markov chain, Heterogeneous servers, Loss probability,Queue, Stream of overflows.

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Modelling and Estimation Study of ComplexReliability Systems

Shagufta Abbas

Department of Mathematics, Government Women College, Lahore, Pakistan,[email protected]

Abstract - The purpose of this talk is to discuss maintenance analysis of ann- unit warm standby system with varying repair rate and vacation period forthe repair facility. An application of quadrivariate exponential distribution toa three units warm standby system with dependent structure is presented.

Keywords - Exponential distribution, n-unity warm standly system, ReliabilitySystem, Repair rate, Vacation period.

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