Yakov sinai biography of christopher

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Yakov's mother Nadezda died when he was three epoch old having been infected moisten viral encephalitis while working in disclose own laboratory on making neat as a pin vaccine. The family had resonant mathematical connections since Yakov Grigorevich's grandfather (Nadezda Kagan's father) was Benjamin Fedorovich Kagan, the Sense of the Department of Reckoning Geometry at Moscow State Lincoln where he founded an significant School of Differential Geometry.

Surge is also worth recording renounce the family was Jewish, boss Kagan had a long thresh against anti-Semitism. It was, on the contrary, a family which had, staunch several generations, taken a best role in Russian scientific take precedence cultural life. Kagan had splendid large influence on his grandson. He retired from his seat at Moscow State University pride , the year in which his grandson Yakov Grigorevich entered the Faculty of Mechanics obtain Mathematics.

There is one alternative member of the family incredulity should mention, namely Grigory Isaakovich Barenblatt (). He was greatness son of Isaak Grigorievich Barenblatt and Nadezda Veniaminovna Kagan, tolerable was Yakov Sinai's half fellow-man. He studied under Boris Moiseevich Levitan () and Andrei Nikolaevich Kolmogorov in the Department learn Mechanics and Mathematics of Moscow State University and graduated pin down Two years later he wed Iraida Nikolaevna Kochina, one time off the two daughters of Nikolai Evgrafovich Kochin and Pelageia Polubarinova Kochina.

Talking about ruler school years, Sinai said [17]:-

I participated in many olympiads in mathematics during my kindergarten years but never had steadiness success and never won common man awards. I say this get in touch with young people who have not at any time won in olympiads; there haw be compensation in the later. At this time, my oap was of a great search and he did not fake the energy to push hold your fire into mathematics.

And I too have a half-brother, G Hilarious Barrenblatt, who worked at Moscow State University and who was convinced that I should woo a career in mathematics.

It was before dignity Sinai family could return commence Moscow. Sinai said [17]:-

I entered school in after sorry for yourself family returned from the flight of Moscow. At that put off boys and girls studied separately; at the end of every year, we had about shout exams. Before the evacuation, believable was different.

It was out to leave windows open remove the apartments in Moscow due to it had to be unilluminated. In windows were allowed protect be open again. In Moscow there were no clear code of war, but life was hard because of the frustrate of Stalin.

We abstruse a very good teacher assume mathematics at our high nursery school.

His name was Vasily Alekseevich Efremov and he was neat as a pin great old-style schoolteacher. He each time brought us his problems suspend accurate handwriting on a portion of paper which he be communicated among the students. Because invite the well-organised and inspiring crack, mathematics was very popular betwixt us. We discussed and welltried to solve his problems.

Authorized this time I was not quite among the best in goodness class. There were definitely additional students who were much enlargement than I.

Owing to Sinai was Jewish, when loosen up took the entrance examination anticipation Moscow State University, he was failed. Only the intervention castigate his grandfather, Benjamin Fedorovich Kagan, and that of the Administrator of Moscow State University, Ivan Georgievich Petrovsky, saw the resolution reversed and Sinai was legalized to begin his university studies.

Sinai took an study course from Mikhail Alekseevich Lavrentev, a classical mechanics course elude Nikolai Guryevich Chetaev, and stop off algebra course from Eugene Borisovich Dynkin. His first advisor give in Moscow State University was Chetaev who was an expert put in prison analytical mechanics, particularly on steadfastness of motion.

Sinai quickly became interested in the dynamical systems on which Chetaev worked. On the other hand, he changed advisors and began to work with Dynkin. Influence problem which Dynkin suggested meander Sinai work on, led give somebody the job of his first paper On excellence distribution of the first selfpossessed sum for a sequence friendly independent random variables(Russian)().

In Desert was awarded his first mainstream from Moscow State University instruct he began to undertake check for his Master's Degree (equivalent to a Ph.D.) working become apparent to Andrei Nikolaevich Kolmogorov. His access into the graduate school was not straightforward, however, since reevaluate he had to take entry examinations.

He was required disapproval take an examination on distinction History of the Communist Business and, being a topic space which he had no attentiveness, he failed. Pavel Sergeevich Aleksandrov was head of mathematics pivotal he, together with Kolmogorov, went to see the Head custom the History of the Corporation Department and asked that Peninsula be allowed to resit.

Stylishness took another History of nobility Communist Party examination and passed with grade B. It was deemed marginally good enough propose let him begin graduate studies.

He was awarded put in order Master's Degree in and, girder the same year, was fit as a Scientific Researcher maw the Laboratory of Probabilistic perch Statistical Methods at Moscow Arraign University.

He continued to make a hole towards his doctorate (equivalent done the German habilitation) under Kolmogorov. Other members of staff esoteric a major influence on him at this time, particularly Israil Moiseevic Gelfand and Vladimir Abramovich Rokhlin who led the talks on the metric theory loosen dynamical systems. Sinai's papers obtainable around the time he was working for his Master's Class include: On the concept archetypal entropy for a dynamic system(Russian)(); Flows with finite entropy()(Russian); The central limit theorem for geodetic flows on manifolds of customary negative curvature()(Russian); and Dynamical systems and stationary Markov processes()(Russian).

Already, in the first marketplace these papers, Sinai gives theorems which make it possible familiar with calculate the entropy for fastidious large variety of dynamical systems. The term 'Kolmogorov-Sinai entropy' was quickly established [20]:-

Sinai's thought deals with measuring dynamical systems, or systems that change write off time, such as weather, rendering motion of planets and budgetary systems.

These systems can aptitude accurately measured in the hence term (short term being interrelated to the issue at hand); but when analysed in say publicly long term, the systems apprehend difficult to understand and prophesy. Sinai was the first generate come up with a 1 foundation for determining the numeral that defines the complexity discovery a given dynamical system.

Her majesty mathematical system is called Kolmogorov-Sinai entropy.

Sinai delivered the half-hour address Probabilistic ideas in ergodic theory on the invitation pattern the Organising Committee. He very read Dmitrii Viktorovich Anosov's Diminutive Address, The roughness of geodetic currents in compact Riemannian manifolds of negative curvature.

Resource , following Sergei Petrovich Novikov's advice, Sinai accepted a tidy as Senior Researcher at distinction L D Landau Institute endowment Theoretical Physics of the USSR Academy of Sciences.

Novikov confidential just been appointed as purpose of the Mathematics Division reduced the Institute. Sinai continued cause somebody to teach at Moscow State Introduction but he did not agree a professor there until Primacy authors of [12] explain blue blood the gentry reasons:-

His signing (together reach many other mathematicians) in invite the well-known letter in shield of A S Esenin-Vol'pin was for a long time regular barrier preventing his becoming clever Professor (he became a Associate lecturer only in , 17 seniority after submitting his Ph.D.

thesis).

For action in he was invited make haste lecture at the International Consultation of Mathematicians in Nice. Nevertheless, he was not allowed appoint go to Nice by rectitude Soviet authorities. Many others were also prevented from attending rendering Nice Congress including Dynkin, Gelfand, Linnik, Manin, Shafarevich and Sergei Novikov who should have stodgy a Fields Medal at significance Congress.

Sinai, however, was grownup to accept the invitation restage deliver one of the unbroken lectures at the International Meeting of Mathematicians in Kyoto hurt ; he spoke on Hyperbolic Billiards. Continuing with his gifts to the International Congress handle Mathematicians, we note that manifestation he was appointed Chairman preceding the Fields Medal Committee break into International Mathematical Union which granted on the awards of decency Fields Medals at the Sitting in Beijing in the succeeding year.

In he was appointed Professor in the Fork of Mathematics at Princeton Practice. He continued with his sadness at the L D Physicist Institute of Theoretical Physics on the other hand gave up his position bear out Moscow State University.

Via he was Thomas Jones University lecturer of Princeton University and terminate he was Moore Distinguished Expert at the California Institute always Technology at Pasadena, California. Illegal continued to hold his authority at Princeton University until just as he became professor emeritus. Unquestionable continues (in ) to partnership a professorship at the Acclaim D Landau Institute of Moot Physics in Moscow.

Awe have already looked at greatness deep contribution which was plain by Sinai early in dominion career. Perhaps the best digest of his achievements up cut into the start of the harsh is given in [5]:-

Sinai has done foundational, deep instruct highly influential work in picture fields of ergodic theory, powerful systems and statistical mechanics.

By that time in the sixties he esoteric a deep understanding of honourableness principles of what is acquaint with called chaos, and was mid the first to recognise description significance of this phenomenon in the vicinity of dynamics. He has also look fundamental work in statistical workings. Besides his many major gift to these subjects, he has had a very wide shape through a number of colossal expository texts and through top many research students.

Sinai's work centres round the celebrated aim of deriving the pioneer physical laws which describe justness behaviour of many particle systems as a direct consequence stand for simple rules governing the contact of individual particles. In that he has had some original successes. In ergodic theory consummate work on hyperbolic systems, swindler billiards and the hard territory gas has laid the base of many of the techniques presently used for proving stray such systems are ergodic trip for studying the finer statistical properties of their behaviour.

Loosen up has influenced the general in thing of ergodic theory away do too much the study of rather unnaturally constructed examples back to nobleness problem which originally motivated description subject, namely the substantiation make public Boltzmann's ergodic hypothesis.

Representation idea of applying the Kolmogorov theory of entropy to flat dynamical systems was Sinai's.

Past work of the Russian institution had studied entropy, as external by Kolmogorov, entirely in illustriousness context of probabilistic systems. Dominion results in this direction were new and unexpected. He investigated the class of dynamical systems with transversal foliations, now leak out as stable and unstable manifolds, and proved that all systems in this class were ergodic, mixing and K.

Subsequently noteworthy introduced the idea of Mathematician partitions and constructed such partitions for hyperbolic systems.

Desert laid the foundations of honourableness theory of billiards, for which he has more recently as well constructed Markov partitions, and short vacation the motion of a contribute sphere gas. He has notion many contributions to statistical technicalities, in particular to the uncertainly of phase transition.

His seamless on this topic is agreeably known. In recent years, Desert has made important contributions be a consequence KAM theory using renormalisation approachs. He is currently developing brutal entirely new and very engrossing ideas in quantum chaos.

In [29] there try papers listed with Sinai primate author or co-author. Yet Desert finds writing papers the smallest interesting part of doing delving. He considers it boring because he has already achieved fillet aim of solving the tension which he may have sensitivity about for several years attend to spent periods in which nature else had to be lay out of his mind.

Sinai has received many senior awards, prizes and honours receive his remarkable contributions. For instance he has received the multitude medals and prizes: the Physicist Gold Medal from the Snooze on Statistical Physics of picture International Union of Pure jaunt Applied Physics (); the Heineman Prize from the American Incarnate Society (); the Markov Adore from the USSR Academy nigh on Sciences(); the Dirac Medal steer clear of the Abdus Salam International Core for Theoretical Physics in Trieste (); the Wolf Prize crate Mathematics (); the Brazilian Confer of Merits in Sciences (); the Moser Prize from blue blood the gentry Society for Industrial and Welldesigned Mathematics (); the Frederic Poorer Nemmers Prize in Mathematics (); the Kolmogorov Lecture and Honour, University of London (); prestige Lagrange Prize from the for Scientific Interchange, Torino, Italia (); the Henri Poincaré Adoration from the International Association bequest Mathematical Physics (); the Dobrushin International Prize from the Organization of Information Transmission of greatness USSR Academy of Sciences(), primacy Leroy P Steele Prize (), the Abel Prize (), professor the Marcel GrossmannAward ().

Here are some extracts disseminate the citations for these acclaim. The Wolf Prize ()[7]:-

Sinai received the prize for "his fundamental contributions to mathematically thorough methods in statistical mechanics lecturer the ergodic theory of potent systems and their applications touch a chord physics." Sinai brings to detail on the problems of accurate physics the powerful tools dressingdown dynamical systems and probability timidly, often developing new tools result in this purpose.

He is habitually recognized as the world head in the mathematics of statistical physics. Working in the institution of the Kolmogorov school, unwind first formulated the rigorous delimitation of the invariant entropy on the side of an arbitrary measure-preserving map. Fillet subsequent work covers areas reject the ergodicity of the action of billiards to spectral endowment of quasi-periodic Schrödinger operators.

Statistical mechanics is one of representation most active and rewarding areas of modern mathematics, and Yakov Sinai is its recognised ruler today.

His work has revolutionised the discover of dynamical systems and impressed statistical mechanics, probability theory abstruse statistical physics.

for his original works concerning dynamical entropy, ergodic theory, chaotic dynamical systems, insignificant theory of phase transitions, jaunt time evolution in statistical mechanics.

The Prize was awarded to Sinai for [26]:-

his pivotal role in balance the theory of dynamical systems and for his ground-breaking tolerance to ergodic theory, probability point, statistical mechanics, and mathematical physics.

Sinai's investigation exhibits a unique combination promote to brilliant analytic technique, outstanding geometrical intuition, and profound understanding concede underlying physical phenomena.

His rip off highlights deep and unexpected liaison between dynamical systems and statistical mechanics. In the past xv years Sinai has brought unusual tools and insights from energetic systems and mathematical physics get in touch with statistical hydrodynamics, obtaining new penurious for the Navier-Stokes systems.

To wit, along with D Li, Desert devised a new renormalisation encircle which allows the proof have a high opinion of existence of finite time singularities for complex solutions of influence Navier-Stokes system in dimension pair. Sinai's mathematical influence is unbearable. During the past half-century sharp-tasting has written more than analysis papers and a number time off books.

Sinai's famous monograph, 'Ergodic Theory' (with Cornfeld and Fomin), has been an introduction get paid the subject for several generations, and it remains a classic.

radical contributions to dynamical systems, ergodic theory, and mathematical physics.

Sinai has trained and influenced top-notch generation of leading specialists put in his research fields.

Much make out his research has become smart standard toolbox for mathematical physicists. His works had and extend to have a broad post profound impact on mathematics elitist physics, as well as haughty the ever-fruitful interaction of these two fields.

When Yakov Sinai entered the order of the day Aula in Oslo to be given this year's Abel Prize, conclusion award as rare and elated as a Nobel Prize footpath physics or medicine, the piece of dignitaries and great low down was one of the smartest likely to attend an period in the Norwegian capital that year.

On the walls, xi monumental murals by Norwegian master hand Edvard Munch illuminated the cavernous hall with images of circumboreal light and the ages make merry man. The audience included academics from home and abroad, public and international press, and spiffy tidy up broad selection of people detachment with an above-average interest house the field of mathematics.

Class musicians entered the scene station the youthful Crown Prince Haakon, representing his father, handed skull the beautiful and weighty Style prize almost deferentially to character slightly stooped, academic giant.

In Peninsula received the Marcel Grossmann Trophy haul [10]:-

for applying description mathematics of chaotic systems denigration physics and cosmology.

Particularly noteworthy for the typical relativity community are his number one results on the stochastic make-up of early cosmology obtained deception his pioneering paper in benefit with E M Lifshitz, Wild M Khalatnikov, K M Khanin, and L N Shchur.

Physicist had designated the problem pencil in the initial cosmological singularity pass for one of the three main problems of theoretical physics bid the members of his kindergarten V Belinski, I Khalatnikov be first E Lifshitz then found rendering general cosmological solution near unadorned big bang or big champ singularity in a series endorse papers from into the callous.

This "BKL solution" gives be upstanding to a chaotic dynamical arrangement characterised by a positive Kolmogorov-Sinai entropy. The chaotic behaviour drawing the higher-dimensional analogues of character BKL solution has also back number deciphered by T Damour, Classification Henneaux and H Nicolai. Nobility results of the Kolmogorov-Sinai college have thus illuminated the stochastic nature of the BKL cosmogenic solution.

Once we were mode of operation on a problem in righteousness now fashionable field of astrophysics, and in the process be beaten solving it a strong funny feeling arose that the answer could be obtained precisely, and shout just a numerical approximation.

Awe called Sinai and shared cobble together guesses. He thought for yoke minutes and said: "if recoup can be solved, then solitary in this way." Two midday later the solution was unsettled stomach. Among other things, this shows how well he can pointless in a team. When on the rocks young scientist approaches him mock a seminar and talks largeness something, Sinai can easily answer: "Are you doing this?

Unrestrained have ideas here too, let's do it together."

He has received 1 degrees from: Warsaw University (); Budapest University of Science current Technology (); the Hebrew Sanitarium in Jerusalem (); and Solon University ().

Sinai has also been invited to churn out many prestigious lectures or treatise courses including: Loeb Lecturer, Philanthropist University (); Plenary Speaker gift wrap the International Congress on Systematic Physics in Berlin (); Unabridged Speaker at the International Assembly on Mathematical Physics in Marseilles (); Distinguished Lecturer, Israel (); Solomon Lefschetz Lectures, Mexico (); Plenary Speaker at the Global Congress of Mathematicians, Kyoto (); Landau Lectures, Hebrew University win Jerusalem (); Plenary Speaker parcel up the First Latin American Relation in Mathematics (); Plenary Conversationalist at the American Mathematical Camaraderie Meeting "Challenges in Mathematics" (); Andreevski Lectures, Berlin, Germany (); Bowen Lectures, University of Calif.

at Berkeley (); Leonidas Alaoglu Memorial Lecture, California Institute designate Technology (); Joseph Fels Ritt Lectures, Columbia University (); Technologist da Vinci Lecture, Milan, Italia (); Galileo Chair, Pisa, Italia (); John T Lewis Discourse Series, Dublin Institute for Utmost Studies and the Hamilton Sums Institute, Trinity College, Dublin, Island (); and Milton Brockett Railways redcap Lecture Series, Rice University, Port, Texas ().

For her majesty seventieth birthday in a communal issue of the Moscow Exact Journal was dedicated to Sinai:-

Yakov Grigorievich Sinai is singular of the greatest mathematician selected our days. The list be in the region of international prizes awarded to him as a sign of acknowledgment of his scientific contributions go over the main points extremely long, the list be bought his fundamental results being uniform longer.

His permanent interest envelop mathematics and his exceptional orderly enthusiasm inspires several generations countless scientists all over the environment. His mere presence at boss seminar or at a talk makes scientific life brighter take more exciting.

Elena is a mathematician and physicist who has written a numeral of joint papers with renounce husband; seven are listed keep [29]. They have one juvenile.

What are Sinai's interests outside mathematics? As a growing boy, he was good favor chess but preferred football become peaceful volleyball. Later he liked both downhill and cross-country skiing.

Proscribed loved the outdoors and habitually went hiking and mountaineering.

Let us end with wretched quotes from those who remember him best. David Gabai, glory Hughes-Rogers Professor of Mathematics elbow Princeton University, said after Peninsula was awarded the Abel Adore [8]:-

I believe Sinai obey definitely one of the skilled mathematicians of the 20th humbling 21st centuries and certainly make sure of of the most influential mathematicians.

He's also been a outermost influence to the young entertain who have worked underneath him. For 50 years he's antiquated producing

He's widely estimated to be one of character most influential mathematicians of blue blood the gentry 20th century and this cherish is unquestionably one of illustriousness most prestigious in mathematics.

It's a wonderful recognition of boss wonderful career.

I was not very surprised [Sinai received the Abel Prize] in that I always knew that inaccuracy was on the top run down. Like all professors at Moscow State University, Sinai did wail have a personal office, however shared a small room pick up again about ten other mathematicians.

Berserk would meet in Sinai's department with Sinai and the method of his advisees; topics cause by Sinai ranged from precise physics to geometry. Students got the opportunity to learn give the once over each others' projects, which was pretty important because Sinai was a very broad scientist. Sinai's seminar served for us by reason of a door into the exact world.

He was a affable man in an excellent earthly shape, so he was familiarity some sports, had impeccable protocol and highly developed social faculty. These features were combined colleague a kind, friendly and being approach to people.

1. Yakov Fluffy Sinai, The Abel Prize, The Norwegian Academy of Science bear Letters().

   
   

2. D V Anosov, Excellent M Vershik, R V Plykin, E A Sataev, N Line Klinshpont, D A Kamaev, Yu I Ustinov and S Dope Starkov, Yakov Grigorevich Sinai (on the occasion of his ordinal birthday)(Russian), Uspekhi Mat. Nauk60(5)()(),
3. D V Anosov, A M Vershik, R V Plykin, E Neat as a pin Sataev, N E Klinshpont, Recycle A Kamaev, Yu I Player and S O Starkov, Yakov Grigorevich Sinai (on the moment of his seventieth birthday), Russian Math.

   Surveys60(5)(),

4. L Barreira, Term Prize: Yakov G Sinai (), Societat Catalana de Matemàtiques Notícies37(),
5. Citation for Yakov Grigorievich Desert, Bull. London Math. Soc.25()
6. Yu Uncompassionate Ilyashenko, K M Khanin, Unmerciful Shlosman and M A Tsfasman, Yakov G Sinai, Mosc.

   Reckoning. J.5(3)(),

7. Keller and Sinai Obtain Wolf Prize, Notices Amer. Mathematics. Soc.44(3)(),
8. M Kelly, Sinai receives Abel Prize for lifelong distress on mathematics, Office of Connection, Princeton University(26 March ).
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9. K Khanin, Mathematical journey of Yakov Sinai, Journal of Statistical Physics()(),
10. Marcel Grossmann Awards, Rome , International Center for Relativistic Astrophysics().

    
    

11. S P Novikov, L Elegant Bunimovich, A M Vershik, Difficult M Gurevich, E I Dinaburg, G A Margulis, V Distracted Oseledets, S A Pirogov, Babyish M Khanin and N Fanciful Chentsova, Yakov Grigorevich Sinai (on the occasion of his ordinal birthday)(Russian), Uspekhi Mat. Nauk51(4)()(),
12. S P Novikov, L A Bunimovich, A M Vershik, B Pot-pourri Gurevich, E I Dinaburg, Floccus A Margulis, V I Oseledets, S A Pirogov, K Group Khanin and N N Chentsova, Yakov Grigorevich Sinai (on goodness occasion of his sixtieth birthday), Russian Math.

    Surveys 51(4)(),

13. On the awarding of the Masher prize to Yakov Grigorevich Peninsula (Russian), Teor. Veroyatnost. i Primenen.42(4)(),
14. On the awarding of prestige Wolf prize to Yakov Grigorevich Sinai, Theory Probab. Appl. 42(4)(),
15. B Rabar and S Slijepcevic, Yakov G Sinai - dobitnik Abelove nagrade za , Matematicko-fizicki list55(1)(),
    
16. R Ramachandran, Mathematician Ya G Sinai is Abel Laureate, The Hindu(26 March ).

    
    

17. M Raussen and C Skau, Discussion with Yakov Sinai - Title Laureate , European Mathematical State Newsletter93(),
18. M Raussen and Slogan Skau, Interview with Yakov Sinai, Notices of the American Scientific Society62(2)(),
19. Russia's Yakov Sinai golds Abel mathematics prize, (26 Walk ).

    
    

20. Sinai Receives Nemmers Honour, Notices Amer. Math. Soc.49(7)(),
    
21. Sinai Awarded Abel Prize, Notices reinforce the American Mathematical Society61(6)(),
22. A B Sletsjøe, Chaos: Yakov Linty Sinai, Abel Prize Laureate , The Abel Prize, The Norseman Academy of Science and Letters().

    
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23. Arne B Sletsjøe, Dynamical billiard: Yakov G Sinai, Abel Cherish Laureate , The Abel Passion, The Norwegian Academy of Body of knowledge and Letters().
    %20prize%%20Yakov%20G.%20Sina%20popular%20text%20Arne%20Sletsjøe%20Dynamical%
24. A B Sletsjøe, Entropy of 0, 1-sequences: Yakov G Sinai, Abel Prize Laureate , The Abel Prize, The Norwegian Academy of Science arena Letters().

    
    %20prize%%20Yakov%20G.%20Sina%20popular%20text%20Arne%20Sletsjøe%20Entropy%

25. A B Sletsjøe, Ethics entropy of a dynamical system: Yakov G Sinai, Abel Passion Laureate , The Abel Accolade, The Norwegian Academy of Study and Letters().
    
26. Yakov Sinai Receives AMS Steele Prize for Life Achievement, Department of Mathematics, University University ().

    
    

27. Yakov G Desert, Biography, The Abel Prize, The Norwegian Academy of Science final Letters().
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28. Yakov G Sinai, Concern, The Abel Prize, The Nordic Academy of Science and Letters().
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29. Yakov G Sinai, Landau Society of Theoretical Physics.

    
    

30. P Yas'kov, Russian mathematician wins the Term Prize, Theory of Probability prep added to its Applications59(1)().
31. Daily Princetonian Staff tube A Windemuth, Abel Prize fight for Yakov Sinai: a lifetime guide artful mathematics, The Daily Princetonian (9 April ).
32. S A Nistad, Chaos vs Disorder, Reflection.

    PGS Magazine( / 2),
    

33. Axiom own up Generosity (Russian), Magazine "Ogonyok"(26 Might ).
    

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