On the 100th anniversary of birthday of academician Nikolai Nikolaevich Bogolyubov (1909–2009)

ISSN 0965�5425, Computational Mathematics and Mathematical Physics, 2011, Vol. 51, No. 6, pp. 901–914. © Pleiades Publishing, Ltd., 2011.Original Russian Text © M.K. Kerimov, 2011, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2011, Vol. 51, No. 6, pp. 963–978.

901

Recently the scientific community of this country marked the 100th anniversary of the birthday ofNikolai Nikolaevich Bogolyubov, a prominent 20th�century mathematician, dynamist, and theoreticalphysicist and an academician of the Russian and Ukrainian Academies of Sciences.

Bogolyubov is rightfully regarded not only as a mathematician but also as a dynamist and theoreticalphysicist. However, in the first place, he was an outstanding mathematician who made fundamental con�tributions to mathematics and its applications, starting from his young age. Numerous articles and evenbooks are available about Bogolyubov’s life and works. However, they primarily cover his discoveries innonlinear mechanics and theoretical physics, while only a brief account is given of his research on the cal�culus of variations, computational mathematics, and applications. Bogolyubov’s major contributions tothese areas are scarcely described even in large books concerning his accomplishments, such asA.N. Bogolyubov, N.N. Bogolyubov: Life and Works (Joint Institute for Nuclear Research, Dubna, 1996);

On the 100th Anniversary of Birthday of Academician Nikolai Nikolaevich Bogolyubov (1909–2009)

M. K. Kerimov

Dorodnicyn Computing Center, Russian Academy of Sciences, ul. Vavilova 40, Moscow, 119333 Russia

e�mail: [email protected]

Received December 27, 2010

DOI: 10.1134/S096554251106011X

902

COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS Vol. 51 No. 6 2011

KERIMOV

Nikolai Nikolaevich Bogolyubov: Mathematician, Engineer, and Physicist, Ed. by A. N. Sisakyan andD.V. Shirkov (Joint Institute for Nuclear Research, Dubna, 2009); and V. S. Vladimirov, “N.N. Bogoly�ubov and Mathematics,” Usp. Mat. Nauk 56 (3), 185–190 (2001). Nevertheless, it is for the research onthe calculus of variations and applied mathematics that Bogolyubov was awarded both his scientificdegrees (candidate’s and doctoral) when he was fairly young.

I acquainted myself with Bogolyubov’s works when I was young and pursued my candidate’s degree atthe Steklov Institute of Mathematics. Since I majored in the calculus of variations, my supervisor, Corre�sponding Member of the USSR Academy of Sciences L.A. Lusternik advised me to read papers on directvariational methods written by the notable mathematicians M.A. Lavrent’ev and Bogolyubov. Their pro�found works prompted me to gain a deeper insight into this area, which was then developed by renownedItalian mathematicians, such as L. Tonelli and his students. As a rule, their works were published in Ital�ian, so I had to study Italian and even passed an Italian examination as part of my graduate�level curricu�lum. I am so grateful to my destiny and my unforgettable teacher of Italian. This language has helped mein my scientific studies thus far.

In view of what was said above, I decided to focus in this article on Bogolyubov’s contribution to thecalculus of variations and incidentally outline his results in computational and applied mathematics.However, first I describe his life as based on published sources.

Bogolyubov was born on August 8 (21) 1909 in the city of Nizhni Novgorod into an intellectual family.His father, Nikolai Mikhailovich Bogolyubov, was professor of philosophy and psychology at the NizhniyNovgorod Orthodox seminary, and his mother, Olga Nikolaevna, was a teacher of music. Details concern�ing Bogolyubov’s family can be found in his brother’s book mentioned above. In 1921 the family movedto Kiev, where the 13�year�old Bogolyubov began to attend seminars led by Academician D.A. Grave andlater seminars of Academician N.M. Krylov. The latter played a huge role in Bogolyubov’s scientific careerand life. In 1925 the Minor Presidium of what would be later known as the Ukrainian Academy of Sci�ences issued a resolution of great importance to Bogolyubov, in which, due to his phenomenal mathemat�ical abilities, he was enrolled as a graduate student in the Research Department of Mathematics in Kiev.This was the starting point of Bogolyubov’s scientific career and his longtime collaboration with Krylov(see the photo of Krylov and Bogolyubov).

Bogolyubov’s first scientific work (coauthored with Krylov) was concerned with the approximate sub�stantiation of the Rayleigh principle and was published in a highly respected French journal in 1926 whenhe was 17 years old. In 1926 he did his graduate research on new methods in the calculus of variations,which was later defended as his candidate’s dissertation and was published in a renowned Italian mathe�matical journal in 1930. Later, this work will be described in more detail. In 1927 Bogolyubov joined Kry�lov’s Department of Mathematics as a researcher.

In 1927 the Bologna Academy of Sciences announced the establishment of the Adolfo Merlani Prizefor studies of the extremal properties of the line integral

(1)

in some class of curves C by applying direct methods of the calculus of variations.

( , , , , , )

C

f x y y x x y dt∫ ' ' '' ''

N.N. Bogolyubov and academician N.M.Krylov, in 1930s.


ON THE 100th ANNIVERSARY OF BIRTHDAY 903

In 1928 Bogolyubov submitted to the prize committee his paper “Sur l’application des methodsdirectes â quelques problémes du calcul variations,” where he addressed the extremal properties of integral (1)by using methods applicable to the case of semiregular and nonemiregular integrals of form (1). In thoseyears, this area was vigorously developed by Italian mathematicians (see, e.g., Tonelli’s papers), but theyconsidered only semicontinuous functionals. Despite the large number of contestants, the prize of theBologna Academy of Sciences was awarded to Bogolyubov. His results for irregular functionals included,as special cases, the Italian mathematicians' results for regular functionals. In 1931 this work by Bogoly�ubov was published in an Italian journal. Later, we will discuss it in more detail.

In 1930, for a series of his works, 21�year�old Bogolyubov earned his doctoral degree in mathematicsfrom the Presidium of the USSR Academy of Sciences without defending a dissertation.

In 1939, he was elected a corresponding member and in 1948 a full member of the Ukrainian Academyof Sciences. In 1947, he was elected a corresponding member and in 1953 a full member of the USSRAcademy of Sciences.

Bogolyubov’s research activities were extremely intensive and covered various areas of mathematics,mechanics, and physics, such as the calculus of variations, function theory, differential equations, vibra�tion theory, stability theory, statistical physics, and quantum field theory.

As was mentioned above, I give a detailed description of Bogolyubov’s studies concerning the calculusof variations and briefly outline his research on computational and applied mathematics.

While reading Bogolyubov’s works earlier in my life, I admired his great talent and could not imaginethat he was so young. However, despite his age, he not only embraced an area of mathematics developedby many prominent researchers of that time but also obtained fundamental results in that area. It was thetime when another notable mathematician, Academician Lavrent’ev worked on the calculus of variations.In this context, we mention Lavrent’ev phenomenon, which has been addressed by various authors andhas appeared in relatively recent publications.

Before describing Bogolyubov’s major contributions to the calculus of variations, we recall some con�cepts of the theory of direct variational methods.

Consider the problem of finding the absolute minimum of the Lebesgue integral

(2)

in the space of all functions that have the first derivative integrable in the sense of

almost everywhere on and satisfy the boundary conditions

, (3)where a, b, a1, and b1 are constants. The function is continuous together with its partial derivatives up tothe second order inclusive.

Assume that the condition

, (3')

where N is a positive constant, holds for all y and z and all x on the interval . Then we have

,

which implies that has an infimum on that satisfies i.

Therefore, by the definition of the infimum, there exists a minimizing sequence , inthat satisfies the limiting relation

as . (4)

Moreover, we need to prove that there is an infinite converging subsequence of , that sat�isfies the assumptions of the Arzela theorem, according to which every uniformly continuous and uni�formly bounded sequence of functions , on an interval has a subsequence suchthat uniformly on , where is a continuous bounded function given on (a, b). Toapply the Arzela theorem in the case of the minimizing sequence under study, we have to show that it isuniformly continuous and uniformly bounded. This can be done if a more restrictive condition than (3')is imposed on :

, (5)

( ) , ,

b

a

dxI y f x y dxdy

⎛ ⎞= ⎜ ⎟

⎝ ⎠∫

� ( )y x ( )x

ay y a y dx= + ∫ '

( , )a b

1 1( ) , ( )y a a y b b= =

( , , )f x y z N≥ −

( , )a b

( ) ( )I y N b a≥ − −

( )I y ( )i N b a≥ − − �

( )ny x 1,2,...,n =

�

( ( ))nI y x i→ n → ∞

( )ny x 1,2,...n =

( )nl x 1,2,...n = ( , )a b ( )n xΘ

( ) ( )n x xΘ →Θ ( , )a b ( )xΘ

( , , )f x y z

1( , , )f x y z z N+β

≥ α −| |

904


KERIMOV

where α, β, and N are positive constants such that α and β are nonzero.Thus, the following assertions can be proved:

(i) There exists a minimizing sequence ( ) for the integral in , i.e., a sequence suchthat as , where i is the infimum of in .

(ii) The sequence has a subsequence ь ( ) such that

,

uniformly on (a, b); here, is a function from .

Since the derivative exists, the functional takes a particular value, finite or infinite.Next, we prove the limiting relation

, (6)

which is used to obtain , i.e., to prove that the limiting function of the minimizing sequencegives an absolute minimum.

In the calculus of variations, , though can be regarded as a function of , is not continuouswith respect to . More exactly, if and are two functions from that satisfy on (a, b) the ine�quality

,

where ε is an arbitrary small number, then and can generally differ from one another by anarbitrarily large number.

Therefore, to prove (6), in addition to the relation

,

we need to prove that

. (7)

In the general case, relation (7) cannot be proved directly. However, such a proof is possible under addi�tional conditions imposed on the integrand . For all y, z and all x on the interval (a, b), these con�ditions have the form

(8)

where γ, P, Q, and R are positive constants independent of x, y, or z and .

Under conditions (8), the integral is called regular.

Under conditions (8), there exists a continuous function in that provides an absolute minimumof .

The proof can be extended to the case where conditions (8) are replaced by for all and arbitrary y and z.

Various methods are available for the construction of minimizing sequences, for example, the Rietzmethod, finite differences, and others. From a practical point of view, the finite�difference method is gen�erally superior to the Rietz method, since finite�difference equations can usually be constructed withoutcalculating very complicated quadratures, which are inevitable in the Rietz method.

There are methods for analyzing the convergence of minimizing sequences.Despite the importance of regular functionals in the calculus of variations, many problems in this area

do not satisfy the regularity conditions. Therefore, we have to consider irregular functionals, for which therelation cannot be proved using the usual method. For this reason, Tonelli intro�

duced the concept of a semicontinuous functional by analogy with the semicontinuity of functionsintroduced by Baire in calculus.

( )ny x 1,2,...n = ( )I y �

( ( ))nI y x i→ n → ∞ ( )I y �

( )ny x ( )nu x 1,2,...n =

( ) ( )nu x u x→

( )u x �

'( )u x ( )I y

lim ( ( )) ( ( ))nn

I u x I u x→∞

=

( ( ))I u x i=

( ( ))I y x ( )y x( )y x ( )w x ( )z x �

( ) ( )w x z x− ≤ ε| |

( ( ))I w x ( ( ))I z x

( ) ( ), ,nu x u x a x b n→ ≤ ≤ →∞

( ) ( ), ,'nu x u x a x b n→ ≤ ≤ → ∞'

( , , )f x y z

''( , , ) ,

'' ''( , , ) , ( , , ) ,

zz

yz yz

R f x y z

f x y z Q f x y z P

> > γ

≤ ≤| | | |

0γ ≠

( )I y

( )u x �

( )I y

''( , , )zzf x y z ≥ γ

( , )x a b∈

lim ( ( )) ( ( ))nn

i I u x I u x→∞

= =

( )I y



A functional is called lower semicontinuous near a function from some function space ifone of the following conditions holds:

(i) has an infinite value: for any positive number ε, there exists a positive number such that theinequality holds for each function from that satisfies the relation on (a,b).

(ii) has an infinite value (i.e., ): for any arbitrarily large positive number , the inequality holds for an arbitrary function from that satisfies the relation on .

If is semicontinuous near all functions from , then it is called semicontinuous everywhere in .

In the calculus of variations, it is proved that, if is semicontinuous near the boundary function ofa minimizing sequence for , then

. (9)

If is an absolute minimizer of in , then is semicontinuous near . Thus, a necessary andsufficient condition for the fulfillment of equality (9), which ensures that the functional has an absoluteminimum at the limiting function of the minimizing sequences, is that is lower semicontinuousnear . A number of necessary and sufficient conditions for the semicontinuity of near some givenfunction from were established by Tonelli. For the limiting function of a minimizing sequence, no spe�cial equation was derived from which the properties of the limiting function could be judged. For this rea�son, to ensure the semicontinuity of near only this function, Tonelli had to assume the semicontinuityof everywhere in . Explaining this requirement, he established a series of necessary and sufficientconditions for to be semicontinuous everywhere in . However, these conditions are more restrictivethan those for semicontinuity near only one given function. For example, if condition (5) holds, then anecessary and sufficient condition for semicontinuity everywhere in is the inequality

, (10)

where x, y, and z are arbitrary numbers lying on the intervals (a, b), , and , respectively.Tonelli called (10) the quasi�regularity condition for the functional. It is only somewhat more general thanthe simple regularity condition. The quasiregular and regular cases do not cover the most interesting situ�

ation, when can change its sign. The difficulty of this case lies in that the Euler differential equa�tion corresponding to then has a singularity and becomes

,

where is a regular function of its arguments.Due to these difficulties arising in the use of direct methods in the quasiregular case, researchers did

not consider it (except for a very special case that Tonelli called incompletamente regulare). It is this dif�ficult case that Bogolyubov addressed in his works on direct methods of the calculus of variations.

Now we focus on his main results concerning the calculus of variations. To begin with, we consider thework that became his candidate’s dissertation: “Sur quelques methods nouvelles dans le calcul des varia�tions,” Ann. Mat. Pura Appl. 4 (7), 249–271 (1930).

It dealt with the determination of an absolute minimum of the functional

, (11)

which is not almost regular in the sense of Tonelli. Bogolyubov considerably extended Tonelli’s results toirregular functionals. Following Tonelli’s constructions, along with (11), he considered the almost regularintegral

(12)

( )I y 0( )y x Γ

0( )I yε

ρ

0( ) ( )I y I y≥ − ε ( )y x Γ 0y yε

− ≤ ρ| |

0( )I y +∞ kρ

( )I y k≥ ( )y x Γ 0 ky y− ≤ ρ| | ( , )a b

( )I y Γ Γ

( )I y( )I y

lim ( ( )) ( ( ))nn

I u x I u x→∞

=

( )u x ( )I y � ( )I y ( )u x

( )u x ( )I y( )u x ( )I y

�

( )I y( )I y �

( )I y �

�

''( , , ) 0zzf x y z ≥

( , )−∞ +∞ ( , )−∞ +∞

''( , , )zzf x y z( )I y

( , , )

''( , , )zz

A x y yy

f x y y=

'''

'

( , , )A x y y'

( , , )

b

C

a

I f x y y dx= ∫ '

( , , )

b

a

J x y y dx= ∫� � '

906


KERIMOV

and proved that, in the class of admissible functions for (11) whose ordinates take fixed values at x = a and, the absolute minimum problem for the integral in has solutions along which

, (13)

in addition to the possible existence of pseudoarcs of “singular” curves along which

. (14)

Thus, the absolute minimum problem for the integral in the class has solutions if and only if,among the curves delivering an absolute minimum of in , there is one that does not containpseudoarcs of singular curves with a nonzero measure.

The following results were proved in the study of an absolute minimum of in some special class offunctions:

(i) Through any two points, one can always draw at least one extremaloid (according to Tonelli’s ter�minology) that consists of a finite number of extreme curves.

(ii) For any positive ε there always exists an extremaloid of in such that the integral in questiontakes a value different from its infimum in by a value not exceeding ε.

Recall the definition of an extremaloid for integral (12) according to Tonelli. A curve , i.e., the cor�responding function is an extremaloid if the condition

. (15)

holds almost everywhere on .Assertion (ii) implies that, if there is only a finite set of extremaloids in the class , then the absolute

minimum problem for the integral in has solutions.The work for which Bogolyubov was awarded the prize from the Bologna Academy of Sciences was

“Sur l’application des methods directes à quelques problémes du calcul des variations,” Ann. Mat. PuraAppl. 4 (9), 195–241 (1931). Let us briefly outline its contents.

In this work, Bogolyubov considered the problem of an absolute minimum of the line integral

, (16)

where x, y, θ, and s are the coordinates, the direction angle, and the arc for the current point of the curve С; isthe length of С; and is a continuous function that is periodic in θ with period 2π. A minimumis sought in the class М of curves С (satisfying certain conditions) such that their direction angles havea bounded variation. Bogolyubov showed that the minimum problem for integral (16) is equivalent tofinding the minimum of the line integral

, (17)

i.e., the integral for which the prize was announced. In (17), the function is continuous

for .Integrals (16) and (17) are equivalent and there is a one�to�one correspondence between them, since

,

.

For this reason, Bogolyubov studied the minimal properties of integral (16).He proved the following main results:(i) Generally speaking, the absolute minimum problem for in the field of curves М has no solution.(ii) If the function satisfies the inequality

, (18)

x b= CI �

( , , ) ( , , )f x y y x y y= �' '

( , , ) ( , , )f x y y x y y> �' '

CI �

CI �

CI

CI ��

0C

0( )y x

0 0 0 0

0

' ' ' '[ , ( ), ( )] [ , ( ), ( )]

x

y yG x y x y x G x y x y x dx− =∫ const

( , )a b

�

CI �

( )0

( ) , , ,CL

dI C f x y dsdsθ

= θ∫

CL( , , , )f x y zθ

( )sθ

( ) ( , , , , , )

C

I C F x y x y x y dt= ∫ ' ' '' ''

( , , , , , )F x y x y x y' ' '' ''2 2' ' 0x y+ ≠

( , , , ) ( , , cos , sin , sin , cos )f x y z F x y z zθ = θ θ − θ θ

2 2 1/2

2 2 3/2

' ' '( , , , , , ) , , , ( )' '( )

y y x x yF x y x y x y f x y x y

x x y

⎛ ⎞−= +⎜ ⎟⎜ ⎟

+⎝ ⎠

'' ' '' '' ' '' '' arctg

'

CI( , , , )f x y zθ

( , , , )f x y z A z Bδθ ≤ +| |



where and are bounded for bounded x and y and a, δ is a fixed number such that δ < 1, thenthe absolute minimum problem for in the field of curves with absolutely continuous direction anglesthat satisfy the same boundary conditions as curves from М generally has no solution.

(iii) Let be a twice differentiable function satisfying the inequality

, (19)

where k and δ are fixed positive numbers and and are bounded for bounded x and y, thenthe limit as ε 0 of the infimum of integral (16) in the field of curves С from the class М that have,together with С, a neighborhood (ε) of order 1 is equal to the line integral

, (20)

where

. (21)

Here, К and δ are fixed positive constants.(iv) If satisfies certain conditions (listed in the work), then a curve on which the integral

has an absolute minimum in exists if and only if, among the curves (always existing) on which the inte�gral has an absolute minimum in , there is one that does not have a common arc with the “singularcurves” of the minimum problem under study.

(v) If satisfies the conditions mentioned in (iv), then, in the field of curves , the Eulerequation has at least one solution corresponding to the functional at which it reaches a value different fromthe infimum of the integral by at most ε.

In the same years, Bogolyubov published in Ukrainian a monograph concerning direct methods in thecalculus of variations (New Methods in the Calculus of Variations (Tekhnteorizat, Kiev, 1932)). This rarebook develops and supplements Tonelli’s well�known treatise Fondamenti di calcolo dell variazioni (Zan�ichelli, Bologna 1922), Vols. 1, 2. The subjects covered in Bogolyubov’s book are clear from its contents:

1. Introduction.2. Idea of Direct Methods.3. Regular Cases in the Calculus of Variations.4. Methods of Real Formation of Minimizing Sequences.5. Concept of Semicontinuous Functionals. Tonelli’s Studies.6. The General (Nonquasiregular) Case of the Calculus of Variations.7. Some Properties of Solutions of the Euler Equation.Specifically, concerning the solutions of the Euler equation for the functional , it was proved that,

under certain conditions on , everywhere in the Euler equation

, (22)

has a solution that satisfies the boundary conditions

and consists of a finite number of extremals (extremaloids).For any positive ε, the Euler equation (22) always has a solution (in ) that consists of a finite number

of extremals such that the integral has a value different from the infimum i by at most ε. It follows that,if the Euler equation has a finite number of solutions, then the absolute minimum problem for the func�tional in can be solved.

The first scientific work by Bogolyubov was performed in 1924 and was entitled “On the Behavior ofSolutions of Linear Differential Equations at Infinity.” In 1926 (jointly with Krylov) he published the fol�lowing paper in the French Academy of Sciences journal: “Sur la justification du principe de Rayleigh purl’orde de l’erreur commis à la n�ieme approximation,” Comp. Rend. Acad. Sci. Paris 183, 476–479(1926).

In 1929 Bogolyubov published (jointly with Krylov) the article “La solution approchée du probleme deDirichlet,” Dokl. Akad. Nauk SSSR, No. 12, 284–288 (1929).

( , )A x y ( , )B x y

CI �

( , , , )f x y zθ

1 1( , ) ( , ) ( , , , )A x y z B x y f x y z k z+δ +δ

+ ≥ θ ≥| | | |

( , )A x y ( , )B x y

ε( )С�

0

( ) ( , , , )CL

J C x y ds= Φ θ θ∫ '

1( , , , )x y K z +δ

Φ θ θ ≥' | |

( , , , )f x y zθ ( )I C�

( )J C �

( , , , )f x y zθ �

( )I y( , , )f x y z �

( )''( , , ) ' , , 0zy

df x y dy dx dyf x y

dx dx− =

/

1 1( ) , ( )y a a y b b= =

�

( )I y

( )I y �

908


KERIMOV

A series of his works concerning approximate methods for the solution of differential equations werepublished in the journal Izvestiya AN SSSR, Otd. OMEN. During the first years of Bogolyubov’s scientificcareer, many of his scientific works addressed approximate and numerical methods for the solution of dif�ferential equations. This can be seen from the list of his publications presented at the end of this article.

While a young man, Bogolyubov also obtained his fundamental results in the theory of almost�periodicfunctions (where he, in fact, created a new direction), in approximate methods for the solution of differ�ential equations, in the theory of dynamical systems, etc.

His next major accomplishment was the creation of the theory of nonlinear oscillations. In this newarea, his unique results were highly acclaimed not only by mathematicians but also by dynamists and engi�neers. His fundamental contributions to the theory of nonlinear oscillations can be found in any textbookin this area and are studied by students all over the world. These results of Bogolyubov laid the foundationsof the stability theory of motion, control and stabilization theory, space flight mechanics, etc.

Bogolyubov also obtained fundamental results in statistical physics, quantum field theory, and particlephysics.

Let us briefly describe his works concerning the approximate solution of boundary value problems fordifferential equations and mathematical physics. A series of his early works were devoted to error estima�tion for approximately determined eigenvalues and eigenfunctions of a boundary value problem. Theapproximate solution of boundary problems was based on the Rayleigh principle in statistical mechanics,which deals with the transition from a difference equation to a differential one when we pass from themechanics of a discrete system to those of a continuous system. To substantiate the Rayleigh principle, itis necessary to show that the solutions of the difference equation tend to those of the corresponding dif�ferential equation. For the problem of the extremum of functional (2), Bogolyubov showed that, undercertain constraints imposed on the derivatives of the integrands, the Euler equation for this functional withgiven boundary conditions is limiting for the corresponding finite�difference equation with the sameboundary conditions. This approximation method is applicable to the approximate computation of eigen�values and eigenfunctions for both ordinary and partial differential equations.

Particularly worth noting are Bogolyubov’s works in the theory of nonlinear oscillations, where,together with Krylov, he created the new discipline of nonlinear mechanics. In this area, they developedmethods for the asymptotic integration of nonlinear equations governing oscillatory processes and pro�posed their mathematical substantiation based on the general theory of dynamical systems. These meth�ods are used for asymptotically solving differential equations with a small or large parameter and for deriv�ing approximate formulas in applications. Before publishing these works, asymptotic methods wereapplied only to conservative systems. This considerably limited the application of perturbation methodsto the study of oscillatory systems, which are, as a rule, conservative. Overcoming fundamental difficul�ties, Bogolyubov and Krylov extended perturbation methods to general nonconservative systems and con�structed new asymptotic methods for nonlinear mechanics. Rigorously mathematically substantiated,these methods are able to produce solutions in the first and higher approximations and are applicable tothe study of both periodic and quasi�periodic oscillatory processes. The methods are simple and visual asapplied to the computation of particular problems. They produce the nth approximation to the desiredsolution. If the formula for the nth approximation is treated as a change of variables, then an exact equa�tion can be reduced to a form convenient for theoretical studies, thus providing the opportunity to esti�mate the approximation error and establish a number of properties of the exact solution. The methodsapply to various systems with a small or large parameter, including those with an infinite number ofdegrees of freedom. Particularly worth noting are the effective equivalent linearization principle and sym�bolic methods. The asymptotic methods can be directly applied to important real�life problems. In thequalitative theory of nonlinear mechanics, Bogolyubov made contributions to the theory of dynamicalsystems. He proved the existence of an invariant measure, introduced the important concept of an ergodicset, and proved theorems concerning the partition of an invariant measure into “indecomposable” invari�ant measures “localized” in ergodic sets.

In his first works on physics, Bogolyubov further developed his asymptotic methods and applied themto the many�body problem in classical statistical mechanics. Even the first applications of asymptoticmethods to statistical mechanics produced fundamental results of great interest. They were summarizedin his book On Statistical Methods in Mathematical Physics, which came out in 1945. Bogolyubov devel�oped methods of distribution functions and generating functionals as applied to the fundamental problemof statistical physics, namely, the computation of thermodynamic functions in terms of the molecularcharacteristics of matter. He used the method of kinetic distribution functions to study the problem ofderiving fluid dynamics equations as based on the classical mechanics of a collection of interacting mol�ecules. His seminal ideas and methods in quantum statistical physics gave rise to the microscopic theory



of the fundamental phenomena of superfluidity and superconductivity. These studies also laid the foun�dations of the modern theory of imperfect quantum microscopic systems and atomic nucleus theory.Bogolyubov created the axiomatic quantum field theory, for which he was the first to formulate the fun�damental causality condition. Based on this theory, he used a unified approach to investigate many phe�nomena in particle physics and established deep intrinsic interrelations between them. These works byBogolyubov gave rise to new mathematical disciplines, such as multidimensional complex analysis(Bogolyubov’s edge�of�the�wedge theorem) and the Tauberian theory of functions of many variables. Forthe series of studies concerning the microscopic theory of superconductivity and quantum field theory,Bogolyubov was awarded the Lenin Prize in 1958.

In the 1960s–1970s, Bogolyubov and his students introduced a new physical characteristic for quarksthat is now known as color charge. This concept made it possible to solve the well�known problem of quarkstatistics and formed the foundation for the development of quantum chromodynamics, i.e., the moderngauge theory of strong interactions.

Bogolyubov’s major contribution to statistical mechanics was the derivation of equations for equilib�rium and nonequilibrium multiparticle distribution functions. Based on these results, he derived kineticequations playing a fundamental role in plasma theory and neutron physics.

In the early 1950s, Bogolyubov took an active part in the creation of thermonuclear weapons in theUSSR.

Simultaneously with his scientific activities, Bogolyubov put much effort into the training of young andparticipated in science management activities. From 1936 he worked as a department head at Kiev StateUniversity and, later, at Moscow State University. For four years, Bogolyubov was the dean of the Facultyof Mechanics and Mathematics of Kiev State University, headed several departments at the Academy ofSciences of the Ukrainian SSR (Department of Nonlinear Mechanics at the Institute of StructuralMechanics and the Department of Mathematical Physics at the Institute of Mathematics). He was thelong�time head of the Department of Theoretical Physics at the Steklov Mathematical Institute and theLaboratory of Theoretical Physics at the Joint Institute for Nuclear Research in Dubna, which was namedafter him. From 1969 to 1992, he was the editor�in�chief of this journal.

Several times Bogolyubov was elected a deputy of the USSR Supreme Council.For many years, he headed the Joint Institute for Nuclear Research in Dubna, was director of the Stek�

lov Institute of Mathematics of the USSR Academy of Sciences, and worked as an academician�secretaryof the Branch of Mathematics of the USSR Academy of Sciences.

Bogolyubov nurtured several generations of mathematicians and theoretical physicists. Many promi�nent scientists are proud to refer to him as their teacher.

Bogolyubov created fruitful scientific schools of linear mechanics in Kiev and theoretical physics inMoscow and Dubna, which include dozens of doctors and candidates of sciences, some of whom occupyhigh�rank academic positions and have their own scientific schools.

Bogolyubov was repeatedly invited to give his lectures and talks at universities, research institutions,international conferences, and congresses all over the world. Some of his books were translated into for�eign languages.

Bogolyubov’s scientific and public activities were highly appreciated by the Government. He was twiceawarded the Stalin Prize, once the Lenin Prize, and received many orders and medals, including six LeninOrders and two Orders of the Red Banner of Labor. He was twice Hero of Socialist Labor and the recipientof the Lomonosov Gold Medal.

Bogolyubov was elected to several foreign academies of sciences and was awarded honorary degrees ofdoctor of sciences from renowned universities around the world. He was the recipient of many interna�tional prizes and medals.

Bogolyubov’s interests extended beyond his professional activities. He was a man of broad mind whospoke several languages and was interested in history, philosophy, and classical literature. He was a respon�sive and kind person. Bogolyubov’s life and work are a distinguished example of unselfish service to sci�ence, homeland, and mankind. Due to his invaluable works, the name of Bogolyubov will be a role modelfor many generations of researchers all over the world.

Bogolyubov had more than 400 scientific publications. Some of them were compiled in the 12�volumeSelected Works, which were issued by the Publishing Company “Nauka” in 2005–2009.

LIST OF N.N. BOGOLYUBOV’S SELECTED PUBLICATIONS

“Sur la justification du principe de Rayleigh par l’ordre de l’erreur commise la n�iéme approximation,” Compt.Rend. Acad. Sci 183, 476–479 (1926) (with N. M. Krylov).

910


KERIMOV

“On Rayleigh’s Principle in the Theory of Differential Equations of Mathematical Physics and Euler’s Methodin Calculus of Variations,” Tr. Fiz.�Mat. Vidd. Vseukr. Akad. Nauk 3 (3), 39–57 (1926).

“Approximate Solution of Differential Equations,” Tr. Fiz.�Mat. Vidd. Vseukr. Akad. Nauk 5 (5) (1927).“Sur quelques criteres, concernant l’existance des dérivees d’une function d’une variable reelle,” Collected

Papers of the Section of Mathematics, Medicine and Natural Sciences of Shevchenko Scientific Society (Lviv, 1928),pp. 215–221 (with N. M. Krylov).

“Sur les methods des differences finies pour la résolution approchee des problemes fondamentaux de la physiquemathémetique,” Compt. Rend. Acad. Sci. (Paris) 186, 422–425 (1928) (with N. M. Krylov).

“Sur la theorie mathématique des oscillographes,” Compt. Rend. Acad. Sci. (Paris) 187, 938–940 (1928) (withN. M. Krylov).

“On Rayleigh’s Principle in the Theory of Differential Equations of Mathematical Physics and on Euler’sMethod in Calculus of Variations,” Ann. Math. Ser. II 29, 255–276 (1928) (with N. M. Krylov).

“Sopra il methodo dei coefficienti constanti (methodo dei tronconi) per l’integrazione approsimata delle equazi�oni differenziali della fisica mathematica,” Boll. Un. Math. Ital. 7 (1), 72–77 (1928) (with N. M. Krylov).

“La solution approchee du probleme de Dirichlet,” Dokl. Akad. Nauk SSSR A, No. 12, 284–288 (1929) (withN. M. Krylov).

“Sur le calcul des raciness de la transcendante de Fredholm les plus voisines d’un nombre donne par des methodsdes moindres carres et de l’algorithme variationnel,” Izv. Akad. Nauk SSSR Otd. Fiz. Mat. Nauk, No. 5, 471–488(1929) (with N. M. Krylov).

“Sur l’approximation des functions par des sommes trigonometriques,” Dokl. Akad. Nauk SSSR A, No. 6, 147–152 (1930).

“Application de la methode de l’algorithme variationel a la solution approchee des equations differentiielles auxderives partielles du type elliptique. Estimation des erreurs qu’on commet en s’arretant a la n�eme approximationdans le calcul des valeurs et des functions singulieres. Cas general de l’equation non homogene,” I: Izv. Akad. NaukSSSR Otd. Fiz. Mat. Nauk, No. 1, 43–71; II: No. 2, 105–114 (1930) (with N. M. Krylov).

“On the Synchronization Theory,” Zap. Fiz.�Mat. Vidd. Vseukr. Akad. Nauk 4, 299–302 (1930) (withN. M. Krylov).

“La solution approchee du probleme de Dirichlet (Resume),” Vortrage aus dem Gebiete der Aerodynamik und ver�wandter Gebiete. Aachen (1929) (Springer�Verlag, Berlin 1930), pp. 53–55 (with N. M. Krylov).

“Application de la méthode des réduites la solution approchee des équations differentielles aux derives partiellesdu type elliptique,” Vortrage aus dem Gebite der Aerodynamik und verwandter Gebriete. Aachen, 1929 (Springer�Ver�lag, Berlin, 1930), pp. 55–57 (with N. M. Krylov).

“Sur quelques méthodes nouvelles dans le calcul dés variations,” Ann Math. Pura Appl. Ser. IV 7, 249–271(1930).

“Sur l’approximation trigonometrique des functions dans l’intervalle infiini,” Izv. Akad. Nauk SSSR OMEN,No. 1, 23–54; No. 2, 149–160 (1931) (with N. M. Krylov).

“On Some Theorems Concerning the Existence of Integrals of Hyperbolic Partial Differential Equations,” Izv.Akad. Nauk SSSR OMEN, No. 3, 323–344 (1931) (with N. M. Krylov).

“Determination of Maximum Values of Some Quantities (Displacements, Moments, etc.) Using Special Meth�ods Reducing Their Majorants,” Izv. Akad. Nauk SSSR OMEN, No. 6, 771–785 (1931) (with N. M. Krylov).

“Sur un probleme de l'électrostatique, Tr. Khar’kov. Elrktrotekh. Inst., No. 1, 7–19 (1931) (with N. M. Krylov).“Sur l’application des methods directes quelques problemes du calcul des variations,” Ann. Math. Pura Appl.

Ser. IV 9, 195–241 (1931).Investigation of the Longitudinal Stability of Aircraft (Aviaavtoizdat, Moscow, 1932) (with N. M. Krylov) [in Rus�

sian].Basic Problems in Nonlinear Mechanics (Gostekhizdat, Moscow, 1932) [in Russian].New Methods in the Calculus of Variations (Tekhnteorizat, Kiev, 1932) [in Ukrainian].Oscillations of Synchronous Machines. 2: On the Stability of Parallel Operation of Synchronous Machines (Ener�

govidav, Kharkov, 1932) (with N. M. Krylov).Méthodes nouvelles pour la solution quelques problemes mathématiques se rencontrant dans la science des construc�

tions (Kiev, 1932) (with N. M. Krylov) [in French].“Recherches sur la stabilite statique et la stabilite dynamique des machines synchrones,” Compt. Rend. Travaux

de la Troisieme section (Gauthier�Villars, Paris, 1932), pp. 179–205 (Congres internat. d’electricite 4, 3 Sect. T. 1.Rap. No. 14) (with N. M. Krylov).

“Fundamental Problems of the Nonlinear Mechanics,” Congres Internat. des Math. (Zurich, 1932), Vol. 2,pp. 270–272 (with N. M. Krylov).

“Quelques exemples d’oscillation non linéares,” Compt. Rend. Acad. Sci. (Paris) 194, 957–960 (1932) (withN. M. Krylov).

“Sur le théoreme fundamental de l’algebra,” Boll. Un. Math. Ital. 5, 65–66 (1932).New Methods for Solving Some Mathematical Problems Encountered in Engineering (Budvidav, Kharkov, 1933)

(with N. M. Krylov) [in Russian].“Fundamental Problems in Nonlinear Mechanics,” Izv. Akad. Nauk SSSR OMEN, No. 4, 475–498 (1933)

(with N. M. Krylov).“Sur quelques propriétes generales des resonances dans la mécanique non linéaire,” Compt. Rend. Acad. Sci.

(Paris) 197, 908–910 (1933) (with N. M. Krylov).



New Methods in Nonlinear Mechanics and Their Application to the Study of Operation of Electronic Generators(ONTI, Moscow, 1934), Part I (with N. M. Krylov) [in Russian].

Application of Nonlinear Mechanics Methods to the Theory of Stationary Oscillations (Vseukr. Akad. Nauk, Kiev,1934) (with N. M. Krylov) [in Russian].

“Symbolic Methods in Nonlinear Mechanics and Their Application to the Study of Resonances in ElectronicGenerators,” Izv. Akad. Nauk SSSR OMEN, No. 1, 7–34 (1934) (with N. M. Krylov).

“Sur les solutions quasi�périodiques des équations de la mecanique non linéare,” Compt. Rend. Acad. Sci.(Paris) 199, 1592–1593 (1934) (with N. M. Krylov).

“Uber einige Methoden der nicht linearen Mechanik in ihren Anwendungen zur Theorie der nicht linearen Res�onanz,” Schweiz. Bauzig 103 (22), 255–257 (1934); (23), 267–270 (1934) (with N. M. Krylov).

“Study of Resonance in Transverse Vibrations of Rods Experiencing Periodic Normal Forces Applied to One ofthe Rod Ends,” in Studies of Structural Vibrations (Nauchn.�Tekh. Izd., Kharkov, 1935) (with N. M. Krylov).

“New Methods in Nonlinear Mechanics and Their Application to the Study of Longitudinal Stability of Air�craft,” Proceedings of All�Union Conference on Aerodynamics, December 23–27, 1993 (Moscow, Tsentr. Aerogidro�din. Inst., 1993), Part I, pp. 101–109 (with N. M. Krylov).

“Sur l’étude du cas de résonance dans les problémes de la mécanique non linéaire,” Compt. Rend. Acad. Sci.(Paris) 200, 113–115 (1935) (with N. M. Krylov).

“Sur quelques théorems de la théorie génerale de la mesure,” Compt. Rend. Acad. Sci. (Paris) 201, 1002–1003(1935) (with N. M. Krylov).

“Les measures invariants et la transivite,” Compt. Rend. Acad. Sci. (Paris) 201, 1454–1456 (1935) (with N. M. Krylov).“Méthodes de la méchanique non linéaire appliqués la theorie des oscillations stationnaires,” Cas. Pest. Math.

64, 107–115 (1935) (with N. M. Krylov).“Les measures invariantes et transitives dans la mécanique non linéaire,” Mat. Sb. I, No. 5, 707–710 (1936)

(with N. M. Krylov).“Les mouvements stationnaires généraux dans les systémes dynamiques de la mécanique non linéaire,” Compt.

Rend. Acad. Sci. (Paris) 202, 200–201 (1936) (with N. M. Krylov).“Sur les propriétes ergodiques de l’équation de Smoluchovsky, Soc. Math. 64, 49–56 (1936) (with N. M. Kry�

lov).“Upon Some New Results in the Domain of Nonlinear Mechanics,” Proc. Ind. Acad. Sci. Ser. A 3, 523–526

(1936) (with N. M. Krylov).“Application de la mécanique linéaire quelques problémes de la radiotechnique moderne,” Onde Electr. 15,

508–531 (1936) (with N. M. Krylov).Introduction in Nonlinear Mechanics (Approximate and Asymptotic Methods in Nonlinear Mechanics) (Akad. Nauk

Ukr. SSR, Kiev, 1937) (with N. M. Krylov) [in Ukrainian].“On Repeated Iteration with Variable Parameters, in Collected Papers on Nonlinear Mechanics (Akad. Nauk Ukr.

SSR, Kiev, 1937), pp. 191–200 (with N. M. Krylov) [in Ukrainian].“Remarks on Rietz Theorem,” Nauk. Zap. Kiev. Derzh. Univ. 3 (1) (1937); Fiz.�Mat. Zb., No. 3, 9–23 (1937).“Sur les probabilitiés en chaine,” Compt. Rend. Acad. Sci. (Paris) 204, 1386–1388 (1937) (with N. M. Krylov).“Les proprietés ergodiques des suites de probabilitiés en chaine,” Compt. Rend. Acad. Sci. (Paris). 204, 1454–

1456 (1937) (with N. M. Krylov).“La théorie generale de la mesure dans son application l’étude des systemes dynamiques de la mecanique non

lineaire,” Ann. Math. Ser. II 38 (1), 65–113 (1937) (with N. M. Krylov).“Fokker–Planck Equations Derived in Perturbation Theory by a Method Based on the Spectral Properties of a

Perturbed Hamiltonian,” Zap. Kaf. Fiz. Akad. Nauk Ukr. SSR 4, 5–80 (1939) (with N. M. Krylov).“Some Problems in the Ergodic Theory of Stochastic Systems,” Zap. Kaf. Fiz. Akad. Nauk Ukr. SSR 4, 243–

287 (1939) (with N. M. Krylov).“Certain Ergodic Properties of Continuous Transformation Groups,” Nauk. Zap. Kiev. Derzh. Univ. 4 (5); Fiz.�

Mat. Zb., No. 4, 45–52 (1939).“The Steepest Descent Method as Applied to the Proof of Some Asymptotic Inequalities,” Nauk. Zap. Kiev.

Derzh. Univ. 4 (5); Fiz.�Mat. Zb., No. 4, 221–250 (1939) (with N. M. Krylov).“On Linearization of Transitive Compact Transformation Groups,” Nauk. Zap. Mekh.�Mat. Kiev. Derzh. Univ.

5, 17–24 (1941).“Asymptotic Inequalities Applied to Some Issues of Statistical Dynamics of Systems with a Large Number of

Degrees of Freedom,” Nauk. Zap. Mekh.�Mat. Fak. Kiev. Derzh. Univ. 5, 49–68 (1941) (with N. M. Krylov).Introduction to Nonlinear Mechanics by N. Kryloff and N. Bogoliuboff: A Free Translation by Solomon Lefschets

from Two Russia Monographs (Princeton Univ. Press, Princeton, 1943).On Statistical Methods in Mathematical Physics (Akad. Nauk Ukr. SSR, Kiev, 1945) [in Russian].“On Certain Limiting Distributions for Sums Depending on Arbitrary Phases,” Uch. Zap. Mosk. Gos. Univ. Fiz.

77 (3), 43–50 (1945).“On the Influence of Random Force on a Harmonic Oscillator,” Uch. Zap. Mosk. Gos. Univ. Fiz. 77 (3), 51–

73 (1945).“Statistical Perturbation Theory,” Uch. Zap. Mosk. Gos. Univ. Fiz. 77 (3), 74–100 (1945).Problems of Dynamic Theory in Statistical Physics (Gostekhizdat, Moscow, 1946) [in Russian].“Expansions in Powers of a Small Parameter in Statistical Equilibrium Theory,” Zh. Eksp. Teor. Fiz. 16, 681–

690 (1946).“Kinetic Equations,” Zh. Eksp. Teor. Fiz. 16, 691–702 (1946).

912


KERIMOV

“On the Theory of Superfluidity,” Izv. Akad. Nauk SSSR Ser. Fiz. 11 (1), 77–90 (1947).“Kinetic Equations in Quantum Mechanics,” Zh. Eksp. Teor. Fiz. 17, 614–628 (1947) (with K. P. Gurov).“Energy Levels of an Imperfect Bose–Einstein Gas,” Vestn. Mosk. Gos. Univ., No. 7, 43–53 (1947).“On the Theory of Superfluidity,” J. Phys. 11 (1), 23–32 (1947).“Single�Frequency Free Oscillations in Nonlinear Systems with Many Degrees of Freedom,” Tr. Inst. Stroit.

Mekh. Akad. Nauk Ukr. SSR 10, 9–21 (1948).“Application of Methods of Nonlinear Mechanics to Kinetic Problems,” Zb. Prats. Inst. Bud. Mekh., No. 8, 3–

17 (1948).“On the Theory of Superfluidity,” Zb. Prats. Inst. Bud. Mekh., No. 9, 89–103 (1948).“On Positive Completely Continuous Operators,” Zb. Prats. Inst. Bud. Mekh., No. 9, 130–139 (1948) (with

S. G. Krein).“Fluid Dynamic Equations in Statistical Mechanics,” Zb. Prats. Inst. Bud. Mekh., No. 10, 41–59 (1948).“An Application of the Theory of Positive Definite Functions,” Tr. Inst. Mat. Akad. Nauk SSSR, No. 11, 113–

120 (1948).“Kinetic Equations in the Theory of Superfluidity,” Zh. Eksp. Teor. Fiz. 18, 622–630 (1948).Lectures on Quantum Statistics (Rad. Shk., Kiev, 1949; Gordon and Breach, London, 1967/1970).“An Application of Perturbation Theory to the Polar Model of Metals,” Zh. Eksp. Teor. Fiz. 19, 251–255 (1949)

(with S. V. Tyablikov).“Approximate Method for Finding the Lowest Energy Levels of Electrons in Metals,” Zh. Eksp. Teor. Fiz. 19,

256–258 (1949) (with S. V. Tyablikov).“Perturbation Theory Method for a Degenerate Level in the Polar Model of Metals,” Vestn. Mosk. Gos. Univ.,

No. 3, 35–48 (1949) (with S. V. Tyablikov).“On the Elimination of the Self�energy Divergence in Nonrelativistic Field Theory,” Dokl. Akad. Nauk Ukr.

SSR, No. 5, 10–16 (1949) (with S. V. Tyablikov).“On the Invariant Formulation of Quantum Field Theory,” Dokl. Akad. Nauk SSSR 74, 681–684 (1950) (with

V. L. Bonch�Bruevich and B. V. Medvedev).“Perturbation Theory in Nonlinear Mechanics,” Tr. Inst. Stroit. Mekh. Akad. Nauk Ukr. SSR, No. 14, 9–34

(1950).“Oscillations,” in Mechanics in the USSR over Thirty Years: 1917–1947 (Gostekhizdat, Moscow, 1950), pp. 99–

114 [in Russian].“On a New Form of Adiabatic Perturbation Theory in the Problem of the Interaction of a Particle with a Quan�

tized Field,” Ukr. Mat. Zh. 2 (2), 3–24 (1950).“On the Fundamental Equations of Relativistic Quantum Field Theory,” Dokl. Akad. Nauk SSSR 81, 757–760

(1951).“On a Class of Fundamental Equations of Relativistic Quantum Field Theory,” Dokl. Akad. Nauk SSSR 81,

1051–1018 (1951).“Variational Equations of Quantum Field Theory,” Dokl. Akad. Nauk SSSR 82, 217–220 (1952).“On the Representation of the Green–Schwinger Functions by Functional Integrals,” Dokl. Akad. Nauk SSSR

99, 225–226 (1954).Asymptotic Methods in the Theory of Nonlinear Oscillations (Gostekhizdat, Moscow, 1955; Gordon and Breach,

New York, 1962) (with Yu. A. Mitropol’skii).“On the Theory of Multiplication of Causal Singular Functions,” Dokl. Akad. Nauk SSSR 100, 25–28 (1955)

(with O. S. Parasyuk).“On the Subtraction Formalism in the Multiplication of Causal Singular Functions,” Dokl. Akad. Nauk SSSR

100, 429–432 (1955) (with O. S. Parasyuk).“On the Renormalization Group in Quantum Electrodynamics,” Dokl. Akad. Nauk SSSR 103, 203–206 (1955)

(with D. V. Shirkov).“Application of the Renormalization Group to Improving Perturbation Theory Formulas,” Dokl. Akad. Nauk

SSSR 103, 391–394 (1955) (with D. V. Shirkov).“Lie�Type Model in Quantum Electrodynamics,” Dokl. Akad. Nauk SSSR 105, 685–688 (1955) (with

D. V. Shirkov).“On the Condition of Causality in Quantum Field Theory,” Izv. Akad. Nauk SSSR Ser. Fiz. 19 (2), 237–246

(1955).“Equations with Variational Derivatives in Statistical Physics and Quantum Field Theory,” Vestn. Mosk. Gos.

Univ., No. 4–5, 115–124 (1955).“Issues of Quantum Field Theory I,” Usp. Fiz. Nauk 55 (2), 149–214 (1955) (with D. V. Shirkov).“Issues of Quantum Field Theory II: Elimination of Divergence from the Scattering Matrix,” Usp. Fiz. Nauk 57

(1), 3–92 (1955) (with D. V. Shirkov).“Method of Asymptotic Approximation for Systems with a Rotating Phase and Its Application to the Motion of

Charged Particles in a Magnetic Field,” Ukr. Mat. Zh. 7 (1), 5–17 (1955) (with D. N. Zubarev).“On Analytic Continuation of Generalized Functions,” Dokl. Akad. Nauk SSSR 109, 717–719 (1956) (with

O. S. Parasyuk).“On the Subtraction Formalism in the Multiplication of Causal Functions,” Izv. Akad. Nauk SSSR Ser. Mat. 20,

585–610 (1956) (with O. S. Parasyuk).“Multiplicative Renormalization Group in Quantum Field Theory,” Zh. Eksp. Teor. Fiz. 30 (1), 77–86 (1956)

(with D. V. Shirkov).



Introduction to the Theory of Quantized Fields (Gostekhizdat, Moscow, 1957; Wiley, New York, 1980) (withD. V. Shirkov).

“Approximate Secondary Quantization Methods in the Quantum Theory of Magnetism,” Izv. Akad. Nauk SSSRSer. Fiz. 21, 849–853 (1957) (with S. V. Tyablikov).

“Dispersion Relations for Compton Scattering by Nucleons,” Dokl. Akad. Nauk SSSR 113, 529–532 (1957)(with D. V. Shirkov).

“Dispersion Relations for Weak Interaction,” Dokl. Akad. Nauk SSSR 115, 891–893 (1957) (with S. M. Belen’kii andA. A. Logunov).

“A Theorem on Analytic Continuation of Generalized Functions,” Nauchn. Dokl. Vyssh. Shkoly Fiz.�Mat.Nauki 3, 26–35 (1958) (with V. S. Vladimirov).

Asymptotic Methods in the Theory of Nonlinear Oscillations (Fizmatgiz, Moscow, 1958; Gordon and Breach, NewYork, 1962) (with Yu. A. Mitropol’skii).

Problems in the Theory of Dispersion Relations (Fizmatgiz, Moscow, 1958) (with B. V. Medvedev and M. K. Poli�vanov) [in Russian].

A New Method in the Theory of Superconductivity (Akad. Nauk SSSR, Moscow, 1958; Consultants Bureau, NewYork, 1959) (with V. V. Tolmachev and D. V. Shirkov).

“On Analytic Continuation of Generalized Functions,” Izv. Akad. Nauk SSSR Ser. Mat. 22 (1), 15–48 (1958)(with V. S. Vladimirov).

“On the Condition for Superfluidity in the Theory of Nuclear Matter,” Dokl. Akad. Nauk SSSR 119, 52–55(1958).

“On a Variational Principle in the Many Body Problem,” Dokl. Akad. Nauk SSSR 119, 244–246 (1958).“A New Method in the Theory of Superconductivity I, III,” Zh. Eksp. Teor. Fiz. 34 (1), 58–65, 73–79 (1958).“On Indefinite Metric in Quantum Field Theory,” Nauchn. Dokl. Vyssh. Shkoly Fiz.�Mat. Nauki 2, 137–142

(1958) (with B. V. Medvedev, M. K. Polivanov).“Some Problems in Quantum Field Theory,” Proceedings of 3rd All�Union Congress of Mathematicians (Akad.

Nauk SSSR, Moscow, 1958), Vol. 3, pp. 514–521 (with D. V. Shirkov).“On a Variant of the Theory with Indefinite Metric,” Proceedings of 8th International Conference on High Energy

Physics (Geneva, 1958), pp. 129–130.“Probleme der Theorie der Dispersionsbeziehungen,” Fortschr. Phys. 6, 169–245 (1958) (with B. V. Medvedev

and M. K. Polivanov).“A New Method in the Theory of Superconductivity,” Fortschr. Phys. 6, 605–682 (1958) (with V. V. Tolmachev

and D. V. Shirkov).“The Compensation Principle and the Self�Consistent Field Method,” Usp. Fiz. Nauk 67, 549–580 (1959).“The Method of Dispersion Relations and Perturbation Theory,” Zh. Eksp. Teor. Fiz. 37, 805–815 (1959) (with

A.A. Logunov and D. V. Shirkov).“On Some Problems of the Theory of Superconductivity,” Physica 26, Suppl. 1, 1–16 (1960).“On Analytic Continuation of Generalized Functions,” in Research on Modern Problems in the Theory of Func�

tions of Complex Variable (Moscow, 1960), pp. 535–536 (with V. S. Vladimirov) [in Russian].“Methodes analytiques de la theorie des oscillations nonlineaires,” Proceedings of 10th International Congress on

Applied Mechanics (Amsterdam, 1960), pp. 9–25 (with Yu. A. Mitropol’skii).“On Some Mathematical Problems of Quantum Field Theory,” Proceedings of International Congress of Mathe�

maticians, Edinburgh, 1958 (Cambridge, 1960), pp. 19–32 (with V. S. Vladimirov).“Application of N.I. Muskhelishvili’s Methods to the Solution of Singular Integral Equations in Quantum Field

Theory,” Problems in Continuum Mechanics (Akad. Nauk SSSR, Moscow, 1961), pp. 45–59 (with B. V. Medvedevand A. N. Tavkhelidze).

“Analytical Methods in the Theory of Nonlinear Oscillations,” Proceedings of 1st All�Union Congress on Theoret�ical and Applied Mechanics (Moscow, 1962), pp. 25–35 (with Yu. A. Mitropol’skii).

“Certain Mathematical Problems in Quantum Field Theory,” Proceedings of International Congress of Mathema�ticians, Edinburgh, 1958 (Moscow, 1957, 1962), pp. 27–47 (with V. S. Vladimirov).

Asymptotic Methods in the Theory of Nonlinear Oscillations (Gordon and Breach, New York, 1962; Fizmatgiz,Moscow, 1963) (with Yu. A. Mitropol’skii).

“Method of Integral Manifolds in Nonlinear Mechanics,” Proceedings of International Symposium on NonlinearOscillations (Kiev, 1963), pp. 93–154 (with Yu. A. Mitropol’skii).

“Fields and Quanta: Quantum Field Theory—Science of Elementary Particles and Their Interactions,” inThrough the Eyes of a Scientist (Moscow, Akad. Nauk SSSR, 1963), pp. 159–173 (with M. K. Polivanov) [in Rus�sian].

“Method of Integral Manifolds in the Theory of Differential Equations,” Proceedings of 4th All�Union Congressof Mathematicians, Leningrad, 1961 (Nauka, Leningrad, 1964), Vol. 2, pp. 432–437 (with Yu. A. Mitropol’skii).

“Mathematical Problems in Quantum Field Theory,” Usp. Mat. Nauk 20 (3), 31–40 (1964).“Study of Quasi�Periodic Regimes in Nonlinear Oscillating Systems,” in Les vibrations forcees dans les systemes

non�lineaires (Paris, 1964), pp. 181–192 (with Yu. A. Mitropol’skii).The Soviet School of Mathematics (Moscow, 1967), pp. 1–63 (with S. N. Mergelyan) [in Russian].“Modern Mathematics,” October Revolution and Scientific Progress (APN, Moscow, 1967), Vol. 1, pp. 21–69

(with S. N. Mergelyan) [in Russian].“Symmetry Theory of Elementary Particles,” in High Energy Physics and Theory of Elementary Particles (Nauk�

ova Dumka, Kiev, 1967), pp. 5–112 [in Russian].

914


KERIMOV

“Adler�Weisenberger Relation and Dispersion Sum Rules in the Theory of Strong Interactions,” Nuovo Cimento48 (1), 132–139 (1967) (with V. A. Matveev and A. N. Tavkhelidze).

“Dynamic Moments of Local Types of Composite Particles in Quantum Field Theory,” in Aspects of the Theoryof Elementary Particles: Proceedings of International Symposium, Varna (Dubna, 1968), pp. 269–279 (withV. A. Matveev and A. N. Tavkhelidze).

Introduction to Axiomatic Quantum Field Theory (Nauka, Moscow, 1969; Benjamin, New York, 1975) (withA. A. Logunov and I. T. Todorov).

Selected Works, 3 vols. (Naukova Dumka, Kiev, 1969) [in Russian].“Representation of n�Point Functions,” Tr. Mat. Inst. im. V.A. Steklova 112, 5–21 (1971) (with

V. S. Vladimirov).“Self�Similar Asymptotics in Quantum Field Theory,” Zh. Teor. Mat. Fiz. 12, 305–330 (1972) (with

V. S. Vladimirov and A. N. Tavkhelidze).Selected Works, Part I: Dynamical Theory (Gordon and Breach, New York, 1990).Selected Works, Part II: Quantum and Classical Statistical Mechanics (Gordon and Breach, New York, 1991).Selected Works, Part III: Nonlinear Mechanics and Pure Mathematics (Gordon and Breach, New York, 1995).Selected Works, Part IV: Quantum Field Theory (Gordon and Breach, New York, 1995).Collection of Scientific Works, 12 vols. (Nauka, Moscow, 2005–2009) [in Russian].

Documents

On the 100th anniversary of birthday of academician Nikolai Nikolaevich Bogolyubov (1909–2009)