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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
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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~.
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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,
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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:
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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!
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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:
.
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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
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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
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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
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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)
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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
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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.
OCR for page 227
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.
OCR for page 228
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
OCR for page 229
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.
OCR for page 230
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.
OCR for page 231
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
OCR for page 232
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.)
OCR for page 233
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.
OCR for page 234
234
BIOGRAPHICAL MEMOIRS
1972
On applying mathematics: Reflections and examples. Q. Appl.
Math., 30:17-29.
With S. I. Putterman and G. E. Uhlenbeck. Possible origin of the
quantized vortices in He, II. Phys. Rev. Lett., 29:546-49.
William Feller, In memorium. In: Proceedings of the 6th Berkeley Sym-
posium, University of California, JunelJuly 1970, ed. J. Neyman,
Berkeley: University of California Press, vol. 2, pp. xxi-xxiii.
1973
With I. M. Luttinger. A formula for the pressure in statistical me-
chanics. J. Math. Phys., 14:583-85.
With K. M. Case. A discrete version of the inverse scattering prob-
lem. I. Math. Phys., 14:594-603.
Lectures on mathematical aspects of phase transitions: NATO
Summer School in Math. Phys., Istanbul, 1970. NATO Adv.
Stud. Inst. Ser. C, 1970, pp. 51-79.
With D. J. Gates and I. Gerst. Non-Markovian diffusion in idealized
Lorentz gases. Arch. Rational Mech. Anal., 51:106-35.
Some probabilistic aspects of the Boltzmann equation. Acta Phys.
Austriaca, suppl. 10:379-400.
With I. M. Luttinger. Bose-Einstein condensation in the presence
of impurities. l. Math. Phys., 14:1626-28.
Phase transitions. In: The Physic?st's Conception of Nature, ed.
I. Mehra, Dordrecht, Netherlands: Reidel Pub., pp. 514-26.
1974
Will computers replace humans? In: The Greatest Adventure, ed.
E. H. Kone and H. I. Jordan, New York: Rockefeller University
Press, pp. 193-206.
With I. M. Luttinger. Bose-Einstein condensation in the presence
of impurities II. I. Math. Phys., 15:183-86.
Hugo Steinhaus A reminiscence and a tribute. Am. Math.
Month., 81 :572-81.
With P. van Moerbeke. On some isospectral second order differ-
ential operators. Proc. Natl. Acad. Sci. USA, 71:2350-51.
Probabilistic methods in some problems of scattering theory. Rocky
Mountain I. Math., 4:511-37.
OCR for page 235
MARK KAC
235
A stochastic model related to the telegrapher's equation. Rocky
Mountain I. Math., 4:497-509.
The emergence of statistical thought in exact sciences. In: The Her-
itage of Copernicus, ed. J. Neyman, Cambridge: MIT Press, pp.
433-44.
Quelques problemes mathematiques en physique statistique. Col-
lection de la chaire Aisenstadt. Montreal: Les Presses de
l'Universite de Montreal.
1975
With P. van Moerbeke. On some periodic Toda lattices. Proc. Natl.
Acad. Sci. USA, 72: 1627-29.
With P. van Moerbeke. On an explicitly soluble system of nonlinear
differential equations related to certain Toda lattices. Adv.
Math., 16: 160~9.
An example of "counting without counting" (to honor C. l. Bouw-
kamp). Philips Res. Rep., 30:20-22.
With P. van Moerbeke. A complete solution of the periodic Toda
problem. Proc. Natl. Acad. Sci. USA, 72:2879-80.
The social responsibility of the academic community. Franklin
Inst., 300:225-27.
Some reflections of a mathematician on the nature and the role of
statistics. Suppl. Advances Appl. Probab., 7:5-11.
With W. W. Barrett. On some modified spherical models. Proc.
Natl. Acad. Sci. USA, 72:4723-24.
Henri Lebesgue et l'ecole mathematique Polonaise: Aperqus et sou-
venirs. Enseignement Math., 21:111-14.
With P. van Moerbeke. Some probabilistic aspects of scattering
theory. In: Functional Integration and Its Applications, eds. A. M.
Arthurs and M. R. Bhagavan, Oxford: Oxford University Press,
pp. 87-96.
With I. M. Luttinger. Scattering length and capacity. Ann. Inst.
Fourier Univ. Grenoble, 25:317-21.
1976
With I. Logan. Fluctuations and the Boltzman equation, I. Phys.
Rev. A., 13:458-70.
1985
Enigmas of Chance. (Autobiography). New York: Harper and Row.
Representative terms from entire chapter:
sur les
