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Vladimir Aleksandrovich Il'in (on his 80th birthday)

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Russian Math . Surveys 63:6 1163–1172 c⃝ 2008 RAS(DoM) and LMS

Uspekhi Mat . Nauk  63:6 173–182 DOI 10.1070/RM2008v063n06ABEH004588

MATHEMATICAL LIFE

Vladimir Aleksandrovich Il’in(on his 80th birthday)

On 2 May 2008 Vladimir AleksandrovichIl’in, ull member o the Russian Academy o Sciences, observed his 80th birthday.

Il’in is known or his outstanding achieve-ments in the theory o boundary-value andmixed problems or equations o mathemati-cal physics considered in domains with ‘bad’(non-smooth) boundaries and having discontin-uous coecients, in the question o the connec-tion between classical and generalized solutionso problems in mathematical physics, in mathe-matical modelling o problems involving difrac-

tion o electromagnetic waves on non-smoothsuraces, in the spectral theory o seladjointelliptic operators, in the theory o multipleFourier series and integrals, in the spectral the-ory o non-seladjoint diferential operators, inthe spectral theory o Schrodinger operatorswith strongly singular potential, and in the theory o optimization o boundarycontrols o oscillatory processes.

He was born on 2 May 1928 in the ancient Russian town o Kozelsk, where helived in his grandather’s house with a huge garden on a street on the outskirts o town known as Maloe Zarech’e (Little Riverside). The grandather in whose househe lived was a painter o icons, and the other grandather, and grandmother, camerom very successul merchant amilies. In 1931 Il’in’s ather Aleksandr Sergeevich,a mathematics teacher, and mother Elizaveta Ivanovna, a student at the Pedagog-ical Institute (known then as ‘the second Moscow State University’) moved withtheir three-year-old son to Moscow. From this time on the boy was always with hisavourite grandmother Anna Konstantinovna, who undoubtedly played an impor-tant role in the development o his personality. She taught the our-year-old Volodyato read and write, and to love poetry and try to write his own poems. At ve years

o age he knew the multiplication table and the squares o all the natural numbersup to 20. At the same time he was engrossed in geography, knew all the largecities o the world, and could draw accurate maps o countries and continents. He

AMS  2000 Mathematics Subject Classifcation . Primary 01A70.

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1164 Vladimir Aleksandrovich Il’in

liked listening to classical music, which he easily remembered, on the newly startedMoscow radio broadcasts.

School education began at 8 years o age then, but ater only two weeks in therst orm the school suggested that Volodya be transerred to the second orm.He received excellent marks in every orm, seemingly with hardly any efort — he

learned very quickly and had an outstanding memory.

As or all those o his generation, the crucial event or the teenage Volodya wasthe war with Germany. His rst shock was learning that Paris had been declaredan open city. Then there came the year 1941, his ather’s departure to the ront,and evacuation together with the school where his mother taught physics, rst toRyazan’ Province, then on to Perm’ Province (Molotov Province at the time). Liewas hard or everyone, but evacuated people were treated with kindness, and thequality o teaching in rural schools was good.

Ater returning rom the evacuation, Volodya Il’in in his high school years wasan active participant o the extracurricular mathematics group o the Faculty o Mechanics and Mathematics at Moscow State University (MSU), supervised by thethen young A. S. Kronrod and S. B. Stechkin, and he won in mathematics olympiads(a letter o commendation to the 10th-grade winner V. A. Il’in has survived, signedby the Chairman o the Olympiad Organizing Committee I. M. Gel’and, the Pres-ident o the Moscow Mathematical Society P. S. Aleksandrov, and the Dean o theFaculty o Mechanics and Mathematics o MSU V. V. Golubev).

In 1945 Il’in graduated with a gold medal rom secondary school no. 273. Onthe advice o his ather, who knew the prominent mathematician A. N. Tikhonovteaching in the Faculty o Physics at MSU, he enrolled in that aculty and, at a timewhen students were attached to departments, became a student o Tikhonov.

In 1950 Il’in graduated with distinction rom the Faculty o Physics (havingnever received a mark below ‘excellent’) and was recommended by Tikhonov orgraduate study in mathematical physics. In 1953 Il’in deended his Ph.D. thesis“Difraction o electromagnetic waves on certain inhomogeneities” and began work-ing as an assistant in the Department o Mathematics headed by Tikhonov in theFaculty o Physics. In February 1958 Il’in deended his D.Sc. thesis “On conver-gence o expansions in eigenunctions o the Laplace operator” and a year later wasappointed to a proessorship at MSU.

In 1970 he transerred, at Tikhonov’s suggestion, to the newly ormed MSUFaculty o Computational Mathematics and Cybernetics. Since July 1974 he hasheaded the Department o General Mathematics that he ormed within that aculty.Thus, Il’in’s 55 years o research and teaching are inseparably linked to MSU.

In parallel with this, Il’in has worked productively since 1973 at the SteklovMathematical Institute, rst as a senior researcher in the Department o PartialDiferential Equations and presently as chie researcher in the Department o Func-

tion Theory. In his 35 years o work at the Steklov Institute his results have beendeclared the best work within the Division o Mathematical Sciences eight times,and twice the best work within the whole Academy o Sciences.

In 1987 Il’in was elected a corresponding member and in 1990 a ull member o the USSR Academy o Sciences.

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Vladimir Aleksandrovich Il’in 1165

His rst original results, obtained already during his student years on a topicsuggested by Tikhonov, were established in the problem o convergence o bilinearseries in a cylindrical domain G, where the terms o the series had numerators con-taining products o eigenunctions o the Laplace operator in G evaluated at twogenerally distinct points P  and Q and denominators containing the corresponding

eigenvalues raised to an arbitrary positive power. In this area he obtained a num-ber o exhaustive results published as two papers in Doklady Akademii Nauk  andpresented at a session o the Moscow Mathematical Society. Among other results,he reuted a statement in the well-known book Methods o Mathematical Physics

by R. Courant and D. Hilbert which asserted that the bilinear series representingthe Green’s unction or the Laplace operator in a rectangular domain G is abso-lutely convergent everywhere or P  = Q. He proved that this bilinear series doesnot converge absolutely or any pair o interior points P  and Q o the rectangulardomain G.

The research he carried out in his graduate-student years dealt with mathe-matical models or solving the problem o difraction o electromagnetic waveson well-conducting cylindrical suraces with sharp edges, the problem o coast-line reraction o radio waves, and the problem o excitation o non-ideal radiowaveguides with sharp edges. These topics were chosen or his Ph.D. thesis.

Three and a hal years ater deending his Ph.D. thesis, Il’in submitted to theDissertation Council o the Faculty o Mechanics and Mathematics his D.Sc. thesison a purely mathematical topic. In it he included his recently obtained results onexact conditions or absolute and uniorm convergence o eigenunction expansionsor the rst three boundary-value problems (essentially rening the conditionsobtained our years earlier by O. A. Ladyzhenskaya), his results generalizing theconcept o a sourcewise representable unction and the Hilbert–Schmidt theorem,his results on ractional powers o the integral operator whose kernel is the Green’sunction o the boundary-value problem under consideration, and his exhaustiveresults on denitive conditions in the Sobolev classes W l p with integer order l oruniorm convergence o eigenunction expansions or the rst three boundary-valueproblems in the case o summation in the order o increasing eigenvalues.

He published the most important o these results in an article or Uspekhi Matem-

aticheskikh Nauk  o almost a hundred pages and in a paper “Kernels o ractionalorder”. As commented by M. A. Krasnosel’skii, these results were, among otherthings, an excellent tool or obtaining many embedding theorems.

To begin a review o subsequent scientic achievements o Il’in, we should notethe appraisal o his results by his teacher Tikhonov: “The characteristic eatureo the entire creative contribution o V. A. Il’in is the prooundness and the clarity inthe ormulations o the problems and the exhaustive nature o the results obtained.”

Furthermore, the above list o areas in which Il’in conducted investigations

demonstrates the exceptional breadth and diversity o his scientic interests. Forinstance, several years ago his attention was attracted by problems completely newto him. He studied them with great enthusiasm and achieved very signicant andimpressive results. We shall describe this topic at greater length and then morebriey treat the whole spectrum o his interests.

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1166 Vladimir Aleksandrovich Il’in

In 1999 he began to study boundary control problems or processes described byhyperbolic equations, rst and oremost the wave equation

utt(x, t) − uxx(x, t) = 0,

considered on an interval 0 x l and over a time interval 0 t T .In his rst papers on the subject he introduced on the rectangle QT  =

[0 x l] × [0 t T ] a class o unctions that was to play a decisive role

in the subsequent investigation, namely, the class  W 12 (QT ) o unctions u(x, t)o two variables that are continuous in the closed rectangle QT  and have in itboth generalized partial derivatives ux(x, t) and ut(x, t) belonging not only to theclass L2(QT ) but also to the class L2[0 x l] or all t ∈ [0, T ] and to the classL2[0 t T ] or all x ∈ [0, l].

All boundary control problems are treated in terms o generalized solutions in  W 

12 (QT ) o the mixed problem or the wave equation with given initial conditions

u(x, 0) = φ(x), ut(x, 0) = ψ(x)

and with boundary conditions at x = 0 and x = l ensuring that specied terminalconditions

u(x, T ) =  φ(x), ut(x, T ) =  ψ(x)

are satised at the terminal moment o time t = T .The condition that the solutions u(x, t) belong to the class  W 12 (QT ) allows one

to describe exactly the smoothness requirements imposed on the unctions in theinitial, terminal, and boundary conditions.

For T  such that T  l, Il’in rst ound necessary and sucient conditions on theinitial unctions φ(x) and ψ(x) and the terminal unctions  φ(x) and  ψ(x) guarantee-ing the existence o boundary controls o the endpoint displacements u(0, t) = µ(t)and u(l, t) = ν (t) which transorm the oscillation process rom the prescribedinitial state {u(x, 0) = φ(x), ut(x, 0) = ψ(x)} to the prescribed terminal state

{u(x, T ) =  φ(x), ut(x, T ) =  ψ(x)}, and under those conditions he produced thedesired boundary controls in explicit analytic orm. He obtained a similar result

or the boundary control o the displacement at one endpoint with the second end-point xed or T  such that T  2l.

Boundary control problems have innitely many solutions when T > l in thecase o controls at both endpoints and when T > 2l in the case o a control atone endpoint (a non-trivial characterization o this set o solutions can be ound inIl’in’s papers o 2000).

The results obtained by Il’in were included in the list o the most importantachievements o the Russian Academy o Sciences in 2001.

In 2003 he ound an explicit analytic orm or boundary controls o spherical-

ly symmetric oscillations o a three-dimensional ball occurring over a time intervalequal to two radii o the ball (the speed o wave propagation is assumed equal to 1).

In 2002 Il’in was joined in this search or explicit orms o boundary controlsby E. I. Moiseev, and in their joint papers o 2002 and 2004 they presented inexplicit analytic orm the boundary controls or a process described by the telegraph

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Vladimir Aleksandrovich Il’in 1167

equation over the time interval T  = 2l or a control at one endpoint, and over thetime interval T  = l or controls at both endpoints (these problems simulate controlo oil or gas pressure in a pipeline).

Starting rom the end o 2004, they directed their eforts towards nding, orvarious orms o boundary controls and or arbitrary suciently large time inter-

vals T , an explicit analytic orm or optimal boundary controls minimizing thecorresponding boundary energy integral in the presence o constraints ollowingrom the validity o the given initial conditions, the given terminal conditions, andthe matching condition or the initial and the terminal displacements. Optimizationwas carried out or the ollowing six boundary control problems:

• boundary control o the displacement or the elastic orce at one endpointwith the other endpoint xed or ree;

• boundary control o the displacements at both endpoints;• boundary control o the elastic orces at both endpoints.

In papers published between the end o 2004 and the end o 2005 Il’in and Moiseevrst presented an explicit analytic orm or optimal boundary controls in each o the above six problems under the assumption that the time interval T  is a multipleo 2l or boundary conditions o the same kind at both endpoints and a multiple o 4l or boundary conditions o diferent kinds at both endpoints.

They published the undamental complete results between November 2006 andJanuary 2008: they obtained explicit analytic orms or optimal boundary controlson arbitrary suciently large time intervals T  or each o the six problems. Theseresults were included in the list o the most important achievements o the Russian

Academy o Sciences in 2007.It should be noted that even though a number o well-known authors (J.-L. Lions,

F. P. Vasil’ev, A. G. Butkovskii, A. I. Egorov, L. D. Akulenko, and others) hadtreated boundary control problems and their optimization beore the work o Il’inand Moiseev, none o them had ound an explicit analytic orm or optimal bound-ary controls.

Very recently Il’in proved that the optimal boundary controls are independent o the choice o a point at which the initial and terminal displacements are matched.

We now describe selected results rom the large number o papers by Il’in dealing

with problems in mathematical physics and the theory o unctions and unctionalanalysis.

In two years ater receiving his D.Sc. degree, Il’in obtained undamental results onthe classical solubility o the mixed problem or a second-order hyperbolic equation.He proved the solubility o this mixed problem in an arbitrary normal cylinder, thatis, a cylinder whose cross-section is a domain in which the Dirichlet problem orthe Laplace operator is soluble or any continuous boundary unction. Beore Il’in’spapers, Ladyzhenskaya and Kh. L. Smolitskii had proved the solubility o this mixedproblem only or a cylindrical domain with boundary satisying conditions o very

high smoothness which grows unboundedly with the dimension.These results o Il’in, in combination with earlier results or parabolic and elliptic

equations due to Tikhonov, O. A. Oleinik, and H. Tautz, showed that in the sense o the constraints imposed on the boundary, the solubility o boundary-value problemsand mixed problems or equations o all three types reduces to the solubility o the

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Vladimir Aleksandrovich Il’in 1169

generalized sense (that is, in the sense o convergence almost everywhere in a neigh-bourhood o a given point). His idea about the generalized orm o the localizationprinciple has been developed in the D.Sc. thesis o his student I. L. Bloshanskii.

In 1971 Il’in published a negative solution o the Gel’and problem on the validityo the theorem on equiconvergence o the spectral expansion and the Fourier integral

expansion in the case when the spectral expansion itsel is not uniormly convergent.He also obtained a similar result or the problem o Riesz equisummability. Theseresults showed that the subtle theorems he established earlier on conditions notensuring uniorm convergence o spectral expansions or their Riesz means cannotbe obtained by using theorems on equiconvergence or equisummability.

In 1972 he published a negative solution o the problem posed by Sobolev onconvergence in the W l p-metric, p = 2, o the spectral expansion o a compactly

supported unction in the class W l p.

A new method developed by Il’in or estimating the remainder term o the spec-

tral unction o an elliptic operator in both the L∞-metric and the L2-metric wasan outstanding result.

Beore this work o Il’in there were attempts to estimate the remainder o thespectral unction in the L∞-metric by many well-known authors (T. Carleman,V. G. Avakumovic, B. M. Levitan, L. Garding, L. Hormander, and others) usingvarious modications o Carleman’s original method that were based on investi-gation o the kernel o a certain unction o the operator and then applicationo a Tauberian theorem. Il’in’s method, based on a sophisticated analysis o theFourier transorm o the leading term o the spectral unction, has nothing in com-

mon with the methods o his predecessors.He developed another original method or estimating the remainder term o 

the spectral unction o an elliptic operator, based on the idea that i  ρ denotes thegeodesic distance induced by the Riemannian metric determined by the leadingcoecients o an elliptic operator, and Θ0(ρ,τ,λ) denotes the spectral unction o the singular ordinary diferential operator obtained rom this elliptic operator byconsidering unctions dependent only on ρ, then the leading term o the spectralunction o the elliptic operator can be taken in the orm Θ0(|x − y|, 0, λ).

Il’in’s main results on the spectral theory o seladjoint elliptic operators were

presented in his 1991 monograph, published in English by Plenum in 1995.His papers on the spectral theory o the Schrodinger operator belong to the same

cycle o publications. For a seladjoint extension on the whole innite line R o theSchrodinger operator with a singular potential satisying only the so-called Katocondition, Il’in proved in 1995 the theorem on the uniorm equiconvergence on R

o the spectral expansion o an arbitrary unction f (x) in L p(R) (1 p 2) andthe Fourier integral expansion o the same unction.

The work o Il’in and Moiseev in 1996–1998 was important or physical applica-tions. For a seladjoint extension in RN  o the Schrodinger operator with a singular

potential satisying only the Kato condition, they established sharp (with respectto order) estimates or the spectral unction on the diagonal and or its increment.

Il’in’s work on the spectral theory o non-seladjoint diferential operators wasa undamental contribution to science. This work was preceded by Keldysh’swell-known papers in which the completeness o a specially chosen system (called

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1170 Vladimir Aleksandrovich Il’in

canonical by Keldysh) o eigenunctions and associated unctions was proved orwide classes o boundary-value problems. However, Keldysh’s theory did not answera question topical or applications: does the constructed system orm a basis, sothat an arbitrary unction in some class can be expanded in this basis?

To answer this question, Il’in in 1980 suggested considering, or any linear ordi-nary diferential operator L o any order n on an interval (a, b), a unction system{uk(x)} (k = 1, 2, . . . ) which is complete and minimal in the L p(a, b)-metric ( p 1xed) with each element uk(x) a regular (or a generalized) solution o the equation

Luk + λkuk = Θkuk−1

on (a, b) or some complex number λk, where Θk is 0 or 1 (in the last case λk = λk−1)and Θ1 = 0.

For all non-seladjoint boundary-value problems or the operator L, the eigen-unctions and associated unctions that have the property o being complete andminimal in some L p orm such a system {uk(x)}, and this ensures the existence o a biorthogonal conjugate system {vk(x)}.

Il’in proved that under certain conditions (quite close to necessary) on the set{λk} the existence or any compact set K 0 ⊂ (a, b) o a constant C (K 0) with

∥uk∥Lp(K0) ∥vk∥Lp/(p−1)(a,b) C (K 0) (∗)

or all k is necessary and sucient or the basis property o the system {uk(x)} in L pwhen p > 1, and is necessary and sucient or the expansion o any L p(a, b)-unctionf (x) in a biorthogonal series in the system {uk(x)} to be uniormly equiconvergenton any compact subset o (a, b) with the expansion o  f (x) in an ordinary Fouriertrigonometric series when p 1.

The theorem on a necessary and sucient condition or equiconvergence when p = 1, that is, in L1, is the most dicult and, o course, the most interest-ing here. Starting with Steklov’s amous paper, many distinguished mathemati-cians (V. A. Steklov himsel, J. D. Tamarkin, E. Titchmarsh, A. Haar, B. M. Levi-

tan, Ya. L. Geronimus, and others) established various equiconvergence theorems,but only Il’in’s theorem made it possible not only to rene many o these theo-rems but also to nd or the rst time the exact boundary beyond which equicon-vergence ails.

Considered or p = 1, Il’in’s theorem allows one to assert, in particular, that i condition (∗) is satised, then the spectral expansion o the L1-unction constructedin Kolmogorov’s amous example1 diverges almost everywhere on the interval underconsideration.

We note that the conditions or the basis property and equiconvergence ound

by Il’in are constructive, since or particular boundary-value problems the validityor ailure o condition (∗) and the conditions on the set {λk} can be seen rom the

1A. Kolmogorof, “Une serie de Fourier–Lebesgue divergente presque partout”, Fund . Math . 4(1923), 324–328.

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Vladimir Aleksandrovich Il’in 1171

rst terms o the asymptotic expansions o the eigenunctions, associated unctions,

and eigenvalues in powers o 1/λ1/nk .

A special case o the system {uk(x)} considered by Il’in is a system o exponen-tials, and although very many mathematicians had dealt with expansions in suchsystems, Il’in’s theorem made it possible to obtain new conditions or equiconver-

gence o expansions in a system o exponentials and in the trigonometric system.In 1980–1992, or a non-seladjoint ordinary diferential operator o arbitrary

order and or a non-seladjoint second-order elliptic operator, Il’in ound estimatessharp with respect to order or the L2-norm over a compact subset K  o the basedomain o an eigenunction (or an associated unction) in terms o the L2-norm overa compact set K ′ ⊂ K o an associated unction o order higher by 1. He called theseinequalities estimates o anti-apriori type (since they estimate the unction on theright-hand side o the diferential equation in terms o a solution o the equation).He showed that such estimates play a key role in the theory o non-seladjoint

operators.The theory o estimates o anti-apriori type and their applications was developed

in the papers and theses o a number o Il’in’s students (V. D. Budaev, N. B. Keri-mov, I. S. Lomov, A. S. Makin, and others).

In 1983 Il’in obtained another important result: or a second-order diferentialoperator he proved necessary and sucient conditions or the Riesz basis propertyo the system o its eigenunctions and associated unctions under minimal condi-tions on the coecients o the operator. In 1986 he generalized this result to thecase o a discontinuous operator, which is especially important or the investiga-

tion o problems with non-local boundary conditions in which the adjoint operatoris discontinuous.

The work carried out by Il’in together with Moiseev and K. V. Mal’kov in 1989is certainly very interesting. Here it was shown that Il’in’s previously obtainednecessary and sucient conditions or the basis property o the system o eigen-unctions and associated unctions o a non-seladjoint operator L are at the sametime necessary and sucient or the existence o a complete system o integrals o motion or a non-linear evolution system generated by the Lax (L-A) representation.

In 1991 Il’in considered a one-dimensional Schrodinger operator with a matrix

potential that is non-Hermitian and has complex-valued entries that are only integ-rable on the base interval. He obtained constructive necessary and sucient condi-tions or the componentwise equiconvergence o the spectral expansion o a vectorunction and its componentwise expansion in a Fourier trigonometric series. Thevalidity o the componentwise localization principle was rst proved in this work.

In 1994 he demonstrated in joint work with Moiseev the ailure o the basisproperty in L p or any p > 1 or the system o root unctions o a boundary-valueproblem posed already by Poincare or the Laplace operator with directional deriva-tive.

Il’in has authored more than 370 publications. He heads a major school o research, and he has supervised the preparation o 28 D.Sc. theses and more than100 Ph.D. theses in mathematics and physics, including Academician o the Rus-sian Academy o Sciences E. I. Moiseev, Academician o the National Academy o Sciences o Uzbekistan Sh. A. Alimov, and Corresponding Member o the Russian

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1172 Vladimir Aleksandrovich Il’in

Academy o Sciences I. A. Shishmarev. A complete list o Il’in’s publications canbe ound in the journal Diferentsial’nye Uravneniya  (34:5 (1998), 586–594; 39:5(2003), 581–583; 44:5 (2008), 587–590), and also on pp. 662–692 o vol. 2 o hisSelected works (Moscow: Maks-Press, 2008, in Russian).

Within just the last year and a hal Il’in published 7 papers in Doklady Akademii 

Nauk  (partially translated as Doklady : Mathematics ), 8 papers in Diferentsial’nyeUravneniya  (translated as Diferential Equations), and 2 papers in Avtomatika i 

Telemekhanika  (translated as Automation and Remote Control ) — an indication o the enduring intensity o his scientic activity.

At Moscow State University he is known as a brilliant lecturer who has madeenormous contributions to education there: he is the author o eight well-knowntextbooks included in the Classical University Textbook  series created in connectionwith the 250th anniversary o MSU.

His achievements have received ample recognition. He has been awarded our

State Orders, the State Prize o the USSR (1980), the Lomonosov Prize or Science(1980), the prize o the Ministry o Higher and Special Secondary Education o theUSSR or the best scientic work (1988), the Lomonosov Prize “For the creationo unique university textbooks and or excellence in teaching” (1992), the title o Proessor Emeritus o Moscow State University (1993), the title o Honoured Citizeno the town o Kozelsk (1998), the title o Best Lecturer o MSU (2000), and theprize o the President o the Russian Federation in the eld o education (2005).

V. A. Il’in has great personal charm, he is sensitive, open, and benevolent withcolleagues, especially with gited young people. He is kind, generous, and sympa-

thetic. People around him are in awe o his many-sided interests and knowledgeand o his extraordinary memory.

With undiminished energy he presses on with his research, generously sharinghis ideas with his students and implementing ever-new creative plans. We wishVladimir Aleksandrovich good health and new achievements in all his undertakings!

O. M . Belotserkovskii , I . S . Lomov , E . I . Moiseev ,

Yu . S . Osipov , V . A. Sadovnichii , and I . A. Shishmarev 

Translated by V. KISIN