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MARK KAC August ~ 6, ~ 914~ctober 25, ~ 984 BY H. P. MCKEAN POLAND. Mark Kac was born "to the sound of the guns of August on the 16th clay of that month, 1914," in the town of Krzemieniec then in Russia, later in PolancI, now in the Soviet Ukraine (1985,1, p. 61. In this connection Kac liked to quote Hugo Steinhaus, who, when asked if he had crossed the border repliecl, "No, but the border crossed me." In the early days of the century Krzemieniec was a pre- dominantly Jewish town surrounded by a Polish society gen- erally hostile to Jews. Kac's mother's family had been mer- chants in the town for three centuries or more. His father was a highly educates! person of Galician background, a teacher by profession, holding degrees in philosophy from I,eipzig, and in history and philosophy from Moscow. As a boy Kac was educated at home and at the Lycee of Krzemieniec, a well-known Polish school of the day. At home he studied geometry with his father ant! discovered a new derivation of Cardano's formula for the solution of the cu- bic a first bite of the mathematical bug that cost Kac pere five Polish zlotys in prize money. At school, he obtainer] a splendid general education in science, literature, and history. He was grateful to his early teachers to the end of his life. In 1931 when he was seventeen, he entered the John 215
2~6 BIOGRAPHICAL MEMOIRS Casimir University of Ew6w, where he obtained the degrees M. Phil. in 1935 and Ph.D. in 1937. This was a period of awakening in Polish science. Marian Smoluchowski had spurrec! a new interest in physics, and mathematics was developing rapidly: in Warsaw, under Wac- law Sierpinski, ant! in Ewow, uncler Hugo Steinhaus. In his autobiography (1985,1, p. 29), Kac called this renaissance "wonclerful." Most wonderful for him was the chance to study with Steinhaus, a mathematician of perfect taste, wide cul- ture, and wit; his adores! teacher who became his true friend and introduced him to the then undigested subject of prob- ability. Kac would devote most of his scientific life to this field and to its cousin, statistical mechanics, beginning with a series of papers prepared jointly with Steinhaus on statistical in- depenclence ~1936, I-4; and 1937, I-21. Kac's student days saw Hitler's rise and consolidation of power, and he began to think of quitting Poland. In 1938 the opportunity presented itself in the form of a Polish fellow- ship to Johns Hopkins in Baltimore. Kac was twenty-four. He left behind] his whole family, most of whom perished in Krzemieniec in the mass executions of 1942-43. Years later he returned, not to Krzemien~ec but to nearby Kiev. ~ re- member him rapt, sniffing about him ant} saying he had not smelled such autumn air since he was a boy. On this trip he met with a surviving female cousin who asked him, at parting, "Wouic! you like to know how it was in Krzemieniec?" then a(l(lecl, "No. It is better if you don't know" (1985,1, p. 106~. These cruel memories and their attendant regrets surely stood behind Kac's devotion to the plight of Soviet refusniks and others in like distress. His own life adds poignancy to his selection of the following quote from his father's hero, Sol- omon Maimon: "In search of truth ~ left my people, my coun- try and my family. It is not therefore to be assumed that shall forsake the truth for any lesser motives" (1985,1, p. 9~.
MARK KAC AMERICA 217 Kac came to Baltimore in 1938 and wrote of his reaction to his new-found land: "l find it difficult . . . to convey the feeling of decompression, of freedom, of being caught in the sweep of unimagined and unimaginable grandeur. It was life on a different scale with more of everything more air to breathe, more things to see, more people to know. The friendliness and warmth from all sides, the ease and naturalness of social contacts. The contrast to Poland . . . defied description." (1985,1, p. 85) After spending 1938-39 in Baltimore, Kac moved to Ith- aca, where he would remain until 1961. Cornell was at that time a fine place for probability: Kai-Lai Chung, Feller, Hunt, and occasionally the peripatetic Paul Erdos formed, with Kac, a talented and productive group. His mathematics bloomed there. He also courter! and married Katherine Mayberry, shortly finding himself the father of a family. So began, as he saicl, the healing of the past. From 1943 to 1947 Kac was associated off end on with the Racliation Lab at MIT, where he met and began to collaborate with George UhIenbeck. This was an important event for him. It reawakened his interest in statistical mechanics and was a decisive factor in his moving to be with UhIenbeck at The Rockefeller University in 1962. There DetIev Bronk, with his inimitable enthusiasm, was trying to build up a small, top-flight school. While this ideal was not fully realized either then or afterwards, it afforded Kac the opportunity to im- merse himself in the statistical mechanics of phase transitions in the company of Tect Berlin and UhIenbeck, among others. Retiring in 1981, Kac moved to the University of Southern California, where he stayer! until his cleath on October 25, 1984, at the age of seventy. He is survives} by his wife Kitty,
218 BIOGRAPHICAL MEMOIRS his son Michael, his daughter Deborah, and his grand- chilciren. MATHEMATICAL WORK Independence and the Normal Law In the beginning was the notion of statistical indepen- clence to which Steinhaus introduced Kac. The basic idea is that the probability of the joint occurrence of indepenclent events should be the product of their incliviclual probabilities, as in I/2 x I/2 = I/4 for a run of two heads in the tossing of an honest coin. The most famous consequence of this type of inclependence is the fact that, if #(n) is the number of heads in n tosses of such an honest coin, then the normal law of errors hoIcis: Pea (~21~ c be ~ ~ (27r)-~2e-X22d.x in which P signifies the probability of the event indicated between the brackets, the subtracted n/2 is the mean of #(n), and the approximation to the right-handecl integral im- proves inclefinitely as n gets large. The fact goes back to A. cle Moivre (1667-1754) and was extended to a vague but much more inclusive statement by Gauss and Laplace. It was put on a better technical footing by P. L. Cebysev (~821-~890) and A. A. Markov (~856-1922), but as Poincare complained, "Tout le monde y croft (la loi des erreurs) parce que les mathe'mati- ciens s'imaganent que c'est un fait d'observation et les observateurs que c'est un the'oreme de mathe'matiques." The missing ingredient, suppliecl by Steinhaus, was an unambiguous concept of in- depenclence. But that was only the start. All his life Kac de- lighted in extending the sway of the normal law over new and unforseen clomains. ~ mention two instances:
MARK KAC 219 Let W~, ,Mn be n ~ 2) independent frequencies meaning that no integral combination of them vanishes. Then T-l measure [O s t C T a c sin ()It I/ N~ sin (I)nt C bl ~ Jb (fir)- 1/2 e-x2/2 dx Ja for large T and n in which measure signifies the sum of the lengths of the several subintervals on which the indicated inequality takes place. In short sinusoids of independent fre- quencies behave as if they were statistically independent though strictly speaking they are not (1937 2; 1943 2). On another occasion Kac looker! to a vastly different do- main: Let d(n) be the number of ctistinct prime divisors of the whole number n = ~ 2 3 .... Then for large N Non N an (~)~g2 cub i;b(21r)-~2e-x22`lx in which # denotes the number of integers having the prop- erty indicated in the brackets and [g2n is the iterated loga- rithm [g(Ign). In short there is some kind of statistical inde- pendence in number theory too. Kac maple this beautiful discovery jointly with P. Erdos (1940 4). These and other examples of statistical independence are explained in Kac s delightful Carus Monograph Independence in Probability, Analysis and Number Theory (I 95 ~ I). Brownian Motion and Integration in Function Space The Brownian motion typified by the incessant move- ment of dust motes in a beam of sunlight was first discussed from a physical standpoint by M. Smoluchowski and A. Einstein (1905). N. Wiener later put the discussion on a solic!
220 BIOGRAPHICAL MEMOIRS mathematical footing. Kac was introduced to both develop- ments during his association with MIT from 1943 to 1947. Now the statistical law of the Brownian motion is normal: if Aft) is the displacement of the Brownian traveler in some fixed direction, then PLa c x~t) ' b1 fib = J (2Ut)-l/2e-X2/2tdX a The fact is that Brownian motion is nothing but an ap- · . . · . proxlmatlon to honest coln-tosslng: x~t) ~ ~ x t+ (the number of heacis in T tosses) (the number of taits)], in which T is the whole number nearest to tN and N is large. The normal law for coin-tossing cited before is the simplest version of this approximation. Kac, with the help of UhIen- beck, perceived the general principal at work, of which the following is a pretty instance: Let ptn) be the number of times that heads outnumber tails in n tosses of an honest coin. Then the arcsine law holds: PEn-~p~n) c c] IrC dx ~_ ~ = ~ Jo W] - x) arcsine By, the right-hand side being precisely the probability that the Brownian path x~t): O c t c I, starting at x(O) = 0, spends a total time, T ' c, to the right of the origin (1947,21. Kac's next application of Brownian motion was suggested In a quantum-mechanical form by R. Feynman. It has to do with the so-called elementary solution eft,x,y) of the Schrodinger equation: .
MARK KAC ~ 64f/13t = 32/2 - V(X)4f. Y21 The formula states that, with the left-hand imaginary unit removed: eft,X,y) = foxy I X (47rt)-~/2e_(X_y)2'4t in which the final factor is the free elementary solution (for V = 0) and Exy is the Brownian mean taken over the class of paths starting at x(O) = x ant! ending at x(~) = y. This is not really as explicit as it looks, as the mean is not readily expressible in closed form for any but the simplest cases, but it floes exhibit just how e depends upon V in a transparent way. It can be used very electively, as Kac illustrated by a beautiful derivation of the WKB approximation of classical quantum mechanics (1946,31. ~ will describe one more application of Brownian motion contained in Kac's Chauvenet Prize paper, Can One Hear the Shape of a Drum? (1976,1~. The story goes back to H. Wey1's proof of a conjecture of H. A. Lorentz. Let D be a plane region bounded by one or more nice curves, holes being per- mitted, anc! let oh, W2, etc., be its fundamental tones, i.e., let _~2~, ~22, etc., be the eigenvalues of Laplace's operator ~ = d2/8x2~ + 82/3x2 acting upon smooth functions that vanish at the boundary of D. Then Lorentz conjectured and Wey} proved: #in: O)n C (do ~ U-~2 X the area of D for large m. Kac found a remarkably simple proof of this fact based upon the self-evident principle that the Brownian trav- eller, starting inside D, does not fee] the boundary of D until it gets there. He also speculated as to whether you could de- duce the shape of D (up to rigid motions) if you could "hear" all of its fundamental tones and showed that, indeed, you can
222 BIOGRAPHICAL MEMOIRS hear the length of the boundary, and the number of holes if any. The full question is still open. Statistical Mechanics As noted before, Kac's interest in this subject had been reawakened by UhIenbeck at MIT. A famous conundrum of the fielcl was the superficial incompatibility of the (obvious) irreversibility of natural processes and the reversibility of the underlying molecular mechanics. Boltzmann struggled con- tinually with the problem, best epitomized by UhIenbeck's teachers, P. and T. Ehrenfest, in what they called the "dog- flea" model. Kac's debut in statistical mechanics was to pro- vide its complete solution, put forth in his second Chauvenet Prize paper (1947,4~. Next, Kac took up Boltzmann's equation describing the development, in time, of the distribution of velocities in a dilute gas of like molecules subject to streaming and to col- lisions (in pairs). ~ think this work was not wholly successful, but it did prompt Kac to produce a stimulating stucly of Boltzmann's idea of"molecular chaos" (Stosszahiansatz) and a typically elegant, Kac-type "caricature" of the Boltzmann equation itself. ~ pass on to the eminently successful papers on phase transitions. The basic question which Maxwell and Gibbs an- swerec! in principle is this: How does steam know it should be water if the pressure is high or the temperature is low, and how does that come out of the molecular model? There are as many variants of the question as there are substances. A famous one is the Ising mode! of a ferromagnet, brilliantly solved by L. Onsager in the two-climensional case. Kac and I. C. Warc! found a different ant! much simpler derivation (1952,21. The related "spherical model" invented by Kac was solved by T. Berlin ~ ~ 952, ~ ). But to my mind, Kac's most inspiring work in this line is
MARK KAC 223 contained in the three papers written jointly with P. C. Hemmer and UhIenbeck (1963,1-3), in which they relater! the phase transition of a one-dimensional mode} of a gas to the splitting of the lowest eigenvalue of an allied integral equation and derived, for that model, the (previously ad hoc) van der Waal's equation of state, Maxwell's rule of equal areas included a real tour de force. PERSONAL APPRECIATION ~ am sure ~ speak for all of Kac's friends when ~ remember him for his wit, his personal kinkiness, and his scientific style. One summer when ~ was quite young and at loose ends, ~ went to MIT to stucly mathematics, not really knowing what that was. ~ hac! the luck to have as my instructor one M. Kac ant! was enchanted not only by the content of the lectures but by the person of the lecturer. ~ had never seen mathe- matics like that nor anybody who could impart such (to me) difficult material with so much charm. As ~ understood! more fully later, his attitude toward the subject was in itself special. Kac was fond of Poincare's dis- tinction between GocI-given and man-made problems. He was particularly skillful at pruning away superfluous details from problems he considered to be of the first kind, leaving the question in its simplest interesting form. He mistrusted as insufficiently digestecl anything that required fancy tech- nical machinery to the extent that he would sometimes in- sist on clumsy but elementary methods. ~ user! to kic! him that he tract made a career of noting with mock surprise that ex = ~ + x + x2/2 + etc. when the whole thing could have been done without expanding anything. But he clid wonders with these sometimes awkward tools. Indeed, he loved com- putation (Desperazionsmatematik includecI) and was a prodi- gious, if secret, calculator all his life. ~ cannot close this section without a Kac story to illustrate
224 BIOGRAPHICAL MEMOIRS his wit and kindliness. Such stories are innumerable, but ~ reproduce here a favorite Kac himself recorded in his auto- biography: "The candidate [at an oral examination] was not terribly god in math- ematics at least. After he had failed a couple of questions, I asked him a really simple one . . . to describe the behavior of the function 1/z in the complex plane. 'The function is analytic, sir, except at z = 0, where it has a singularity,' he answered, and it was perfectly correct. 'What is the sin- gularity called?' I continued. The student stopped in his tracks. 'Look at me,' I said. 'What am I?' His face lit up. 'A simple Pole, sir,' which was the correct answer." ( 1985,1, p. 126)
MARK KAC HONORS, PRIZES, AND SERVICE 1950 Chauvenet Prize, Mathematical Association of America 1959 American Academy of Sciences 1963 Lorentz Visiting Professor, Leiden 1965 National Academy of Sciences 1965-1966 Vice President, American Mathematical Society 1966-1967 Chairman, Division of Mathematical Sciences, National Research Council Chauvenet Prize, Mathematical Association of America Nordita Visitor, Trondheim Visiting Fellow, Brasenose College, Oxford American Philosophical Society Royal Norwegian Academy of Sciences Solvay Lecturer, Brussels Alfred Jurzykowski Award G. D. Birkhoff Prize, American Mathematical Society Kramers Professor, Utrecht Fermi Lecturer, Scuola Normale, Pisa 968 968 969 969 969 971 976 978 980 980 225
226 BIOGRAPHICAL MEMOIRS SELECTED BIBLIOGRAPHY 1934 A trigonometrical series. ~. London Math. Soc., 9:116-18. 1935 Une remarque sur les series trigonometriques. Stud. Math., 5:99- 102. 1936 Une remarque sur les equations fonctionnelles. Comment. Math Helv., 9:170-71. Sur les fonctions independantes I. Stud. Math., 6:46-58. Quelques remarques sur les fonctions independantes. Comptes Rendus Acad. Sci. (Paris), 202:1963-65. With H. Steinhaus. Sur les fonctions independantes II. Stud. Math., 6:89-97. With H. Steinhaus. Sur les fonctions independantes III. Stud. Math., 89-97. 1937 With H. Steinhaus. Sur les fonctions independantes IV. Stud. Math., 7:1-15. Sur les fonctions independantes V. Stud. Math., 7:96-100. Une remarque sur les polynomes de M. S. Bernstein. Stud. Math., 7:49-51. On the stochastical independence offunctions (Doctoral dissertation, in Polish). Wiadomosci Matematyczne, 44:83-112. 1938 Quelques remarques sur les zeros des integrales de Fourier. J. Lon- don Math. Soc., 13: 128-30. Sur les fonctions 2nt t2nt] - 1/2. J. London Math. Soc., 13: 131-34. 1939 Note on power series with big gaps. Am. J. Math., 61:473-76. On a characterization of the normal distribution. Am. J. Math., 61 :726-28.
MARK KAC 227 With E. R. van Kampen. Circular equidistribution and statistical independence. Am. }. Math., 61:677-82. With E. R. van Kampen and A. Wintner. On Buffon's needle prob- lem and its generalizations. Am. I. Math., 61:672-76. With E. R. van Kampen and A. Wintner. On the distribution of the values of real almost periodic functions. Am. I. Math., 61:985- 91. 1940 On a problem concerning probability and its connection with the theory of diffusion. Bull. Am. Math. Soc., 46:534-37. With P. Erdos, E. R. van Kampen, and A. Wintner. Ramanujan sums and almost periodic functions. Stud. Math., 9:43-53. Also in: Am. I. Math., 62:107-14. Almost periodicity and the representation of integers as sums of squares. Am. }. Math., 62: 122-26. With P. Erdos. The Gaussian law of errors in the theory of additive number-theoretic functions. Am. I. Math., 62:738-42. 1941 With R. P. Agnew. Translated functions and statistical indepen- dence. Bull. Am. Math. Soc., 47: 148-54. Convergence and divergence of non-harmonic gap series. Duke Math. I., 8:541-45. Note on the distribution of values of the arithmetic function dime. Bull. Am. Math. Soc., 47:815-17. Two number theoretic remarks. Rev. Ciencias, 43: 177-82. 1942 Note on the partial sums of the exponential series. Rev. Univ. Nac. Tucuman Ser. A., 3:151-53. 1943 On the average number of real roots of a random algebraic equa- tion. Bull. Am. Math. Soc., 49:314-20. On the distribution of values of trigonometric sums with linearly independent frequencies. Am. J. Math., 65:609-15. Convergence of certain gap series. Ann. Math., 44~2~:411-15.
228 BIOGRAPHICAL MEMOIRS 1944 With Henry Hurwitz,3r. Statistical analysis of certain types of ran- dom functions. Ann. Math. Stat., 15: 173 -81. 1945 Random walk in the presence of absorbing barriers. Ann. Math. Stat., 16:62-67. With R. P. Boas, Jr. Inequalities for Fourier transforms of positive functions. Duke Math. I., 12: 189-206. A remark on independence of linear and quadratic forms in- volving independent Gaussian variables. Ann. Math. Stat., 16:400-1. 1946 On the distribution of values of sums of the type It. Ann. Math., 47(2):33-49. With P. Erdos. On certain limit theorems of the theory of proba- bility. Bull. Am. Math. Soc., 52:292-302. On the average of a certain Weiner functional and a related limit theorem in calculus of probability. Trans. Am. Math. Soc., 59:401-14. 1947 With A. J. F. Siegert. On the theory of random noise in radio re- ceivers with square law detectors. I. Appl. Phys., 18:383-97. With P. Erdos. On the number of positive sums of independent random variables. Bull. Am. Math. Soc., 53:1011-20. On the notion of recurrence in discrete stochastic processes. Bull. Am. Math. Soc., 53: 1002-10. Random walk and the theory of Brownian motion. Am. Math. Month., 54:369 - 91. With A. I. F. Siegert. An explicit representation of a stationary Gaussian process. Ann. Math. Stat. 18:438-42. 1948 With R. Salem and A. Zygmund. A gap theorem. Trans. Am. Math. Soc., 63:235-48. On the characteristic functions of the distributions of estimates of
MARK KAC 229 various deviations in samples from a normal population. Ann. Math. Stat., 19:257-61. 1949 On distributions of certain Wiener functionals. Trans. Am. Math. Soc., 65:1-13. On deviations between theoretical and empirical distributions. Proc. Natl. Acad. Sci., 35:252-57. On the average number of real roots of a random algebraic equa- tion (II). Proc. London Math. Soc., 50:390-408. Probability methods in some problems of analysis and number theory. Bull. Am. Math. Soc., 55:641-65. 1950 Distribution problems in the theory of random noise. Proc. Symp. Appl. Math., 2:87-88. With H. Pollard. The distribution of the maximum of partial sums of independent random variables. Canad. I. Math., 2:375-84. 1951 Independence in Probability, Analysis, and Number Theory. New York: John Wiley and Sons. With K. L. Chung. Remarks on fluctuations of sums of indepen- dent random variables. Mem. Am. Math. Soc., no. 6. (See also: corrections in Proc. Am. Math. Soc., 4:560-63.) Cry 1 On some connections between probability theory and differential and integral equations. Proc. 2d Berkeley Symp. Math. Stat. Prob., ed. J. Neyman, Berkeley: University of California Press, pp. 189-215. On a theorem of Zygmund. Proc. Camb. Philos. Soc., 47:475-76. With M. D. Donsker. A sampling method for determining the low- est eigenvalue and the principal eigenfunction of Schrodinger's equation. J. Res. Nat. Burl Standards, 44:551-57. 1952 With T. H. Berlin. The spherical model of a ferromagnet. Phys. Rev., 86~2~:821-35. With J. C. Ward. A combinatorial solution of the two-dimensional Ising model. Phys. Rev., 88: 1332-37.
230 BIOGRAPHICAL MEMOIRS 1953 An application of probability theory to the study of Laplace's equa- tion. Ann. Soc. Math. Poll, 25:122-30. With W. L. Murdock and G. Szego. On the eigenvalues of certain Hermitian forms. I. Ration. Mech. Anal., 2:767 - 800. 1954 Signal and noise problems. Am. Math. Month., 61:23-26. Toeplitz matrices, translation kernels and a related problem in probability theory. Duke Math. I., 21:501-9. 1955 Foundations of kinetic theory. Proc. 3d Berkeley Symp. Math. Stat. Prob., ed. I. Neyman, Berkeley: University of California Press, pp. 171-97. A remark on the preceding paper by A. Renyi, Acad. Serbes des Sci., Belgrade: Extrait Publ. de ['Inst. Math., 8: 163-65. With J. Kiefer and J. Wolfowitz. On tests of normality and other tests of goodness of fit based on distance methods. Ann. Math. Stat., 26: 189-211. Distribution of eigenvalues of certain integral operators. Mich. Math. J., 3:141-48. 1956 Some remarks on the use of probability in classical statistical me- chanics. Bull. Acad. Roy. Belg. C1. Sci., 42~5~:356-61. Some Stochastic Problems in Physics and Mathematics: Collected Lectures in Pure and Applied Science, no 2. (Hectographed). Magnolia Pe- troleum Co. 1957 With D. A. Darling. On occupation times for Markoff processes. Trans. Am. Math. Soc., 84:444-58. A class of limit theorems. Trans. Am. Math. Soc., 84:459-71. Probability in classical physics. In: Proc. Symp. Appl. Math., ed. L. A. MacColl, New York: McGraw Hill Book Co., vol. 7., pp. 73-85. Uniform distribution on a sphere. Bull. Acad. Poll Sci., 5:485-86. With R. Salem. On a series of cosecants. Indag. Math., 19:265-67.
MARK KAC 231 Some remarks on stable processes. Publ. Inst. Statist. Univ. Paris, 6:303-6. 1958 With H. Kesten. On rapidly mixing transformations and an appli- cation to continued fractions. Bull. Am. Math. Soc., 64:283-87. (See also correction, 65:67.) 1959 Remark on recurrence times. Phys. Rev., 115~2~: 1. Some remarks on stable processes with independent increments. In: Probability and Statistics: The Harald Cramer Volume, ed. Ulf Grenander, Stockholm: Almqvist & Wiksell, pp. 130-38; and New York: John Wiley & Sons. On the partition function of a one-dimensional gas. Phys. Fluids, 2:8-12. With D. Slepian. Large excursions of Gaussian processes. Ann. Math. Stat., 30: 1215 -28. Probability and Related Topics in Physical Sciences. New York: Intersci- ence. Statistical Independence in Probability, Analysis, and Number Theory. New York: John Wiley & Sons. 1960 Some remarks on oscillators driven by a random force. IRE Trans. Circuit Theory. August, pp. 476-79. 1962 A note on learning signal detection. IRE Trans. Prof. Group In- form. Theory, 8:127-28. With P. E. Boudreau and I. S. Griffin. An elementary queueing problem. Am. Math. Month., 69:713-24. Probability theory: Its role and its impact. SIAM Rev., 4: 1-11. Statistical mechanics of some one-dimensional systems. In: Studies in Mathematical Analysis and Related Topics, ed. G. Szego, Stan- ford: Stanford University Press, pp. 165-69. 1963 Probability theory as a mathematical discipline and as a tool in engineering and science. In: Proceedings of the Ist Symposium on
232 BIOGRAPHICAL MEMOIRS Engineering, eds. I. L. Bogdanoff and F. Kozin, New York: John Wiley & Sons, pp. 31-68. With E. Helfand. Study of several lattice systems with long-ranae forces. I. Math. Phys., 4:1078-88. v cat With G. E. Uhlenbeck and P. C. Hemmer. On the van der Waals theory of the vapor-liquid equilibrium. I. Discussion of a one- dimensional model. I. Math. Phys., 4: 216-28. With G. E. Uhlenbeck and P. C. Hemmer. On the van der Waals theory of the vapor-liquid equilibrium. II. Discussion of the distribution functions. l. Math. Phys., 4:229-47. 1964 With G. E. Uhlenbeck and P. C. Hemmer. On the van der Waals theory of the vapor-liquid equilibrium. III. Discussion of the critical region. I. Math. Phys., 5:60-74. Probability. Sci. Am., 211:92-106. The work of T. H. Berlin in statistical mechanics: A personal rem- iniscence. Phys. Today, 17:40-42. Some combinatorial aspects of the theory of Toeplitz matrices. Pro- ceedings of the IBM Scientific Computing Symposium on Com- binatorial Problems, 12: 199-208. 1965 With G. W. Ford and P. Mazur. Statistical mechanics of assemblies of coupled oscillators. I. Math. Phys., 6:504-15. A remark on Wiener's Tauberian theorem. Proc. Am. Math. Soc., 16:1155-57. 1966 Can liberal arts colleges abandon science? Proceedings of the 52nd Annual Meeting of the Association of American Colleges. Bull. Assoc. Am. Coll., 52:41 - 49. Wiener and integration in function spaces. Bull. Am. Math. Soc., 72:52-68. Can one hear the shape of a drum? Am. Math. Month., 73: 1-23. Mathematical mechanisms of phase transitions. In: 1966 Brandeis Summer Inst. Theor. Phys., ed. M. Chretien, E. P. Gross, and S. Deser, New York: Gordon and Breach. 1:243-305. With C. J. Thompson. On the mathematical mechanism of phase transition. Proc. Natl. Acad. Sci., 55:676-83. (See also correc- tion in 56: 1625.)
MARK KAC 1967 233 The physical background of Langevin's equation. In: Stochastic Dif- ferential Equations. Lecture Series in Differential Equations, session 7, vol. 2, ed. A. K. Ariz, Van Nostrand Math. Studies, 19:147- 66. New York: Van Nostrand. 1968 With S. Ulam. Mathematics and Logic: Retrospect and Prospects. New York: Frederick A. Praeger. On certain Toeplitz-like matrices and their relation to the problem of lattice vibrations. Arkiv for det Fysiske Seminar i. Tron- dheim, 11:1-22. 1969 With Z. Cielieski. Some analytic aspects of probabilistic potential theory. Zastosowania Mat., 10:75-83. Asymptotic behavior of a class of determinants. Enseignement Math., 15~2~: 177-83. With C. I. Thompson. Critical behavior of several lattice models with long-range interaction. I. Math. Phys., 10: 1373-86. Some mathematical models in science. Science, 166:695-99. With C. J. Thompson. One-dimensional gas in gravity. Norske Videnskabers Selskabs Forhandl., 42:63-73. With C. J. Thompson. Phase transition and eigenvalue degeneracy of a one-dimensional enharmonic oscillator. Stud. Appl. Math., 48:257-64. 1970 Aspects probabiltstes de la theorte du potential: Seminaire de mathematiques superzeures, Ete 1968. Montreal: Les Presses de L'Universite de Montreal. On some probabilistic aspects of classical analysis. Am. Math. Month., 77:586-97. 197 The role of models in understanding phase transitions. In: Critical Phenomena in Alloys, Magnets and Superconductors, ed. R. E. Mills, New York: McGraw-Hill, pp. 23-39. With C. J. Thompson. Spherical model and the infinite spin di- mensionality limit. Phys. Norveg., 5: 163-68.
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