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the Spirit of Play

a memoir for Stan Ulam by David Hawkins

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ometime in early 1944 I passed the open door of a small office near my own: S. ULAM. He had arrived at Los Alamos only a few days before and seemed unoccupied. We introduced ourselves-he a young mathematician, I an even younger philosopher, one with mathematical leanings. My field of work was the philosophy of mathematics and science. I had listened in on the shop talk of the theoretical physicists at Berkeley and knew their style. They thought of me for managerial chores in the newly created Los Alamos laboratory. So I came, as an administrative assistant to Robert Oppenheimer. Only later was I given the job of writing a wartime history. I was in fact the sole representative of my trade at Los Alamos, and the label "philosopher" usually caught curious attention. But Stan ignored it. He had come as a new member of the Theoretical Division, although no one (he slyly suggested) knew quite why. I later guessed that he had indeed been invited for no particular reason other than the urging of John von Neumann. Stan's version was characteristic: "Physicists don't know what to do with mathematicians."

It was the beginning of a long personal and family friendship. But here I shall restrict my recollections to associations of the thinner, more mathematical kind. We soon discovered one strong common interest, in the foundations and uses of probability theory. Some of Stan's work (Lomnicki and Ulam 1934) had preceded that of Kolmogorov on the measure-theoretic formulation of probability. Mine had been on the conceptual foundations, battled over since the time of Bernoulli and Leibnitz and closer to the philosopy of physics.

ne day Stan threw a problem at me, as if to bring our academic discussions back

Oto the concerns of a wartime laboratory. In the chain reaction that was to power

the atomic bomb, some fraction of the neutrons liberated by a fission induce other fissions, which in turn liberate more neutrons that induce more fissions, and so on. Suppose the number of induced fissions per fission is a random variable that can take on the values i = 0, 1, 2,... with probability p;. (That is, po is the probability that the neutrons from a single fission induce no further fissions, p1 is the probability that they induce one further fission, and so on.) What then is the probability distribution of the number of fissions occurring in the nth "generation" of such a process started by a single fission? Although we didn't know it at the time, the same problem-stated differently-had been solved long before. One earlier version had been posed in terms of the proliferation of a family name through male descendants. Assume that each male Jones produces i male offspring with probability p;, i = 0, 1, 2,... (and that this probability does not vary from one generation to another). What then is the probability that a given Jones has k males in the nth generation of his descendants?

I spent several evenings on the problem. By persistence rather than insight I found the very simple solution (Hawkins and Ulam 1944). A lot of algebraic solvent evaporated and left behind an unexpected little crystal of a formula, the sort of outcome that makes you ask why it hadn't been obvious all along.

Let f(x) be the Laplace generating function of the sequence of probabilities {Po P1, P2....}. (That is, let f(x) be the function to which the infinite series po + P1x + P2x2 + converges.) Then the probability that Jones has k grandsons (or k second-generation male descendants) is the coefficient of x* when f2(x) = ƒ (f(x)) is expanded in powers of x. And in general the probability that Jones has k nth-generation male descendants is the coefficient of x* when f(x) = f (fn-1(x)) is similarly expanded. Thus, to the biological process, that of reproduction, there corresponds an algebraic one, that of iteration, in which the argument of a function is replaced by the function itself. I'll mention other related results and further applications later, but this was the essence of our first venture into what was to develop into the theory of branching (we said "multiplicative") processes.*

...

Stan was delighted with my solution, and I, the rank amateur, was flattered. He already knew quite a lot about the deceptively simple operation of iteratively substituting a function for its own argument, and I got a lesson or two. In the course of these discussions, we got on to such topics as space-filling curves, turbulence, and what have recently come to be called catastrophes, in which deterministic laws lead rigorously to results we can only describe as chaotic. A good many years later when we were reminiscing about all of this, I complained that we had almost been pioneers in such matters. Why hadn't we pursued them? Stan's reply: "It's because there are so many of them guys and so few of us!"

*I should also mention a prior Los Alamos paper by S. Frankel, which may lie buried in the 1943 series of Los Alamos reports. Frankel had thought in terms of a continuous time parameter instead of discrete generations. That approach leads to a one- parameter family of generating functions embedding our f (x). The problem actually has an even earlier origin. It was discussed by Darwin's cousin Francis Galton in 1889 and then by A. Lotka in 1939. Later, in 1945, Erwin Schrödinger addressed the problem, and I recall seeing the title of a relevant Russian paper (obviously declassified!) of about the same date. A section of Feller's classic text on probability theory (Feller 1968) is devoted to branching processes; a full development is that of T. E. Harris (Harris 1963).

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know very little in detail of the wide range of Stan's work and his repertoire. In

memoir I shall confine myself to matters we corresponded about or worked on

jointly. I do this partly because some of these may not be otherwise known and partly because they affected my own mathematical avocation in a way that throws some light on the character of Stan Ulam, teacher. I never sat in on any of his courses, to be sure, though I sometimes heard him lecture. The teaching I shall speak of is that I occasionally received, over many years, one-to-one. In talking about all this I shall refer to some work of mine that shows the nature of the Ulam influence; it is minor work but still a mirror of our associations. And I enjoy bringing these pieces together for the first time.

Stan was indeed a superb teacher, of a kind not very common. One part of his secret was a quite extraordinary talent for turning forbidding topics into attractive problems, attractive because they seemed promising, seemed to open up some larger area. Another part is a quality I am tempted to describe as meritorious laziness. Though Stan could, on occasion, himself engage in intense and concentrated work, as a teacher he would give you the challenge and then-let you do the work. I remember feeling a bit resentful. I did all the work on that first little paper, and he could have added more! But what he really added was to my confidence. For Stan no ego was invested.

Later, when I was at the University of Colorado, Stan and I both did some further work on branching processes. He, with C. J. Everett, had generalized the whole scheme by including “particles" of different types (Everett and Ulam 1948). This generalization, in its physical applications, allowed offspring and progenitors to differ from each other, for example, in their spatial or dynamical, and hence also in their reproductive, characteristics.

My own related work was inspired partly by a conversation we had about one of the great and vital mysteries of mathematics. The Greeks got on to it, long before Euclid, in the discovery that geometrical facts could be represented arithmetically, while those of arithmetic could be seen in the mirror of geometry. In our own day the pendulum has swung far toward the arithmetical, whether analytic or digital, side. Rather typically Stan took the "wrong" side, that of geometry. "Draw a curve," he said, "of a nice simple function. Now draw another curve parallel to it. The relation is very simple to see and understand, but algebraically it can be quite messy." How is it possible that relationships that are so complicated in one domain can be mapped into another where they appear so simple, or vice versa?

The generating-function transformation I had used in that first problem of ours is an elegant elementary example; it belongs to a wider family with many applications in applied mathematics, including probability theory. We had extended its use a bit, and it was Stan's challenge to extend it further, as he did in the work with Everett. To me the challenge was to explore the relevance of this transformation to other operations of a stochastic nature. Long known of course is the fact that addition of independent random variables corresponds to multiplication of their generating functions. What could one say about other arithmetical operations-division, say, or the logarithm-when random variables take the place of simple numbers?

Consider the following example. Physics students learn that the number of alpha particles emitted per unit time by a bit of uranium is a random variable (call it F) described by the Poisson distribution, whose Laplace generating function f(x) is

e-(1-x), where X is a decay constant characteristic of uranium. The time between emission of successive alpha particles is also a random variable (call it G); it is described by the exponential distribution, whose generating function g(x) is A e-λ"x"du = A/A Inx). Now F and G are reciprocals of each other. What then is the relation between ƒ and g, the generating functions of their probability distributions? The answer is that they satisfy the remarkably simple relation

ƒ ̄1(x)g(x ̄1) = f(x ̄1)g ̄'(x) = 1,

where f-1 and g-1 denote the inverses of ƒ and g: f'(x) = (lnx + X)/A and g ̄1(x) = e^(1-1/x) ̧

One can show that this functional relation holds quite generally. Whatever the probability distribution may be for A per unit B, its generating function and that of the distribution for the reciprocal measure B per unit A will satisfy the above identity. (A mathematical nicety is that the inverses of such functions always exist.) Thus one can easily calculate means, variances, and higher moments of one distribution from those of the other.

A related topic is the "random logarithm": Find the probability distribution, for example, of the time required for a chain reaction to produce a given population size. I was in fact looking for some old notes on these matters, which Stan had asked about, when I learned of his death.

AFBI

fter the war I was absented from weaponeering-first from choice and then by the F.B.I. I became politically opposed to the arms race that supported it, but not to wartime friends. Over the following years Stan and I corresponded or talked about a good many different topics, and again it was he who got me thinking about some of these. As I write now, I realize they all concerned iterative processes, deterministic or stochastic or mixed, that seemed to lie beyond the range of "standard methods." So although it might seem a bit of a jump to go from chain reactions to prime numbers, both fitted that general category. There is an iterative definition of the prime numbers, the sieve of Eratosthenes. The process is completely deterministic, but the way the primes are scattered among the other numbers has very chancy look that has stimulated generations centuries-of investigation.

First there came from Stan some rolls of print-out: very long lists of primes, of twin primes, of successive differences between them, and so forth. All these of course were computer-generated. Others may have computed even longer lists; Stan was one of the first to do so. But soon the pattern changed. The theory of primes is a highorder specialty for number theorists, and a happy hunting ground for amateurs like me. Stan was neither, or both. I think he may have been the first to think of the sieve of Eratosthenes as merely one among many sorts of iterative processes whose products lie beyond the range of standard methods. He thought of a good many other ways of generating number sequences that were more or less prime-like in their frequency and distribution. It was a flanking maneuver: If you can't solve the original problem, think of others that resemble it, and may be easier. Some of Stan's schemes seemed to me far-fetched, and I said so. His reply: "Yes, but I am the village idiot!"

Indeed, I think that Stan often did not care whether he got to the essence of a

SIEVE OF ERATOSTHENES

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