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fter I had “noticed” these simple relations (with Jan Mycielski's forbearance), I

went to some of the literature of mathematical demography and population genetics and learned, of course, that it dealt with much more recondite problems, which I was loth to become involved in. Not equipped to make judgements, I nevertheless wondered why it seemed to skip over these simple zeroth approximations. And then I realized why I wished to talk about all this in a personal memoir about Stan Ulam. He, the "village idiot,” the one who had the necessary “don't-know-how," did not skip over them. It was his style to value the art of successive approximation, of evading the big complexities until he was ready for them, the art sometimes called common sense. Many of his computer simulations were rough sketches of this kind yet could lead into deep water, such as his work on iterated nonlinear transformations. (See "Iteration of Maps, Strange Attractors, and Number Theory-An Ulamian Potpourri.")

With a few further modifications this mathematical genealogy begins to resemble a real biological story, possibly our own, and with all kinds of further questions in tow. I bring a closure to this writing by mentioning two such modifications, neither of which is so complex as to obscure the essential simplicity. The first recapitulates our early work on branching processes (Hawkins and Ulam 1944). Such processes appear within the scheme of sexual reproduction as soon as we shift from pure genealogy to genetics and to an interest in evolution. I shall describe briefly the simplest example. The second modification is necessary to give context for the first. It generalizes the original scheme, moving it away from the unrealistic assumption of random pairing toward a pattern of “assortative" mating. This move is curiously parallel to the later work of Everett and Ulam on branching processes in several dimensions (Everett and Ulam 1948).

Genetically considered, sexual reproduction is not only quadratic but also biquadratic: Each partner contributes to an offspring half of a diploid genome. But once inherited, the genetic makeup of the offspring remains constant, apart from mutations. Consider then the fate, within our model, of any individual genetic token, taken to be the only one of its kind. It will or will not be transmitted to an offspring with probability Ż. So the probability of its transmission to 0, 1, or 2 offspring is the coefficient of the corresponding power of x in the generating function g(x) = ({ +x)?. Its appearance in subsequent generations is described by a simple chain reaction with 8n(x) = 8 (8n–1(x)), one just at the level of transition from a subcritical to a supercritical condition. In any later generation the expected number of descendants with the token is a constant, namely 1. The probability that the token eventually disappears is 1, but its expected lifetime is infinite. The model itself forbids any evolutionary consequences. All of the model's essential properties are preserved, however, by allowing a variation of family size, insisting only on a mean value of 2. (Indeed even a slow exponential rate of population growth leaves essentials unchanged.) Then inheritance of any given “bad” gene will be decreased, and that token will have a finite expected lifetime. For a “good” gene the chain goes supercritical; with probability greater than { the number of descendants with the “good” gene will grow exponentially with time and eventually dominate the population.

In such a way we can mimic stochastic adaptation. That is a necessary condition for evolution, but not sufficient. Diveregent adaptation is also necessary. If different environmental conditions face two subpopulations, "good" genetic changes in one might be "bad" in the other. If the two are long separated, genealogical distances become very great, and the original gene pool may finally fission into those of separate species.

For such reasons we may consider a pattern of assortative mating that involves random pairing within subpopulations and rates of migration between them that decrease with some measure of distance. Successive generations in one subpopulation will gradually acquire more ancestors in the others. In the long run complete mixing will occur, but genealogical distances can now spread over a wide range. If the rate of mutation is assumed to be low but constant, genetic distances will increase with genealogical.

All this seemed at first quite difficult to mathematize, but surprisingly it is not. Shared ancestries and genealogical distances can be expressed in closed algebraic forms that depend only on the rates of diffusion between the subpopulations and their sizes. I leave the subject at this point. Stan's work in biomathematics went further in other areas, but this extension of early work I think would have pleased him.

Irel

mentioned above that Stan was a bit standoffish about my involvement in work Lrelating to the education of children. I was playing with them instead of him, my mathematical mentor! But I heartily forgive him. Some of what I had learned from him, that very spirit of play, I could take to the struggles for better science and mathematics teaching in the schools. Children don't have to be taught how to engage in serious play, usually, but teachers and other “educators” frequently do. They too often have lost the art, overwhelmed by mistaken notions of some puritan or utilitarian origin. Stan never lost it. I

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combined interest in the humanities and science that

continues to this day.) In 1943, after short teaching stints at Stanford and Berkeley, he joined the newly created Los Alamos laboratory, serving first as administrative aide to J. Robert Oppenheimer and later as historian. A year at George Washington University was followed in 1947 by a move, which proved permanent, to the University of Colorado, Boulder. He is now a Distinguished Professor Emeritus at that institution. Hawkins has devoted much of his professional life to projects concerning the teaching of mathematics and science. In 1970 he helped create the University of Colorado's Mountain View Center for Environmental Education, an advisory center for preschool and elementary teachers, and is still a participant in its activities. He has enjoyed leaves of absence at several colleges and universities in the United States and abroad and has been honored with a fellowship at the Institute for Advanced Study, a MacArthur Fellowship, membership in the Council of the Smithsonian Institution, and chairmanship of the Colorado Humanities Program. In addition to numerous journal articles, he has written four books: Science and the Creative Spirit: Essays on Humanistic Aspects of Science (Harcourt Brown, editor, 1958), The Language of Nature: An Essay in the Philosophy of Science (1964), The Informed Vision: Essays on Learning and Human Nature (1974), and The Science and Ethics of Equality (1977).

David Hawkins earned his academic degrees in philosophy: an A.B. and M.A. from Stanford University and a Ph.D. from the University of California, Berkeley. (The title of his doctoral dissertation, "A Causal Interpretation of Probability,” reflects a

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PROBABILITYand NONLINEAR SYSTEMS S

by R. Daniel Mauldin

Part 1
An Introduction

tan Ulam, at sixty-five, was vig-
orous, handsome, full of ideas.
It was the spring of 1974, and he

had come to lecture at the Uni-
versity of Florida, where I was a young
assistant professor. I had known Stan by
reputation for several years. In fact, the
very
first
paper

that I read as a part of my German language requirement in graduate school was his landmark 1930 paper on measurable cardinals, “Zur Masstheorie in der allgemeinen Mengenlehre." But listening to him in person was quite an inspiration. He did not lecture in the usual sense but presented snapshots of mathematical ideas, a style reminiscent of Steinhaus, one of Stan's teachers in Poland. Afterwards, several of us talked with him for a remarkably long time. I was immediately impressed with his ability to take up a mathematical topic and

breathe new life into the subject.

The following year Stan took a position at Florida. His weekly seminar was similar to his book A Collection of Mathematical Problems. A topic would be brought up for discussion, and if it appeared to intrigue someone, we would return to it at a slightly deeper level. Stan soon became a stimulating source of encouragement to the younger mathematicians, and to me he became a mentor. As always, he was very generous in sharing his ideas. Throughout his life Stan nourished mathematics

sations became a witty (on his part) and very productive exchange. Like a master of reflecting boundaries, he would bounce ideas back to us from an endless variety of angles, especially humorous ones. The amplification of an idea could occur in a time span varying from a coffee conversation to a number of years. Although we would repeatedly go over the same topics, it wasn't exactly like working the beads on a rosary. Every so often an idea would undergo some adjustment or transformation, and something new, perhaps unexpected, would emerge. I don't know whether it was always his way to have short, quick discussions of some central idea, but that is certainly the impression one gets from perusing his comments and problems in The Scottish Book. (This famous notebook of problems was jotted down at the Scottish Café in Lwów during

in that manner.

At first he would listen to us for a very short time—and then expound his own ideas. Eventually, however, our conver

framework. Instead of considering just one possible outcome of a process, one can consider an infinite number of possible outcomes at once by randomizing the process. Then one can apply the powerful tools of probability, such as the laws of large numbers, to determine the likelihood of a given outcome. The famous Monte Carlo method is a perfect example of that approach. In fact, one of the favorite sayings of Erdös and Ulam, both of whom worked in combinatorics (in which the number of outcomes is finite) and probability, was

the 1930s and first published in this country in 1957. See "Excerpts from The Scottish Book.")

Ulam's incredible feel for mathematics was due to a rare combination of intuitions, a common feature of almost all great mathematicians.

He had a very good sense of combinatorics and orders of magnitude, which included the ability to make quick, crude, but in-the-ballpark estimates. Those talents, combined with the more ordinary abilities to analyze a problem by means of logic, geometry, or probability theory, already made him very unusual. Besides, he had a good intuition for physical phenomena, which motivated many of his ideas.

Ulam's intuition, as exhibited in numerous problems formulated over a span of more than fifty years, covered an enormous range of subjects. The problems on computing, physical systems, evolution and biology were stimulated by new developments in those fields. Many others seemed to spring from his head. He usually had some prime examples in mind that motivated his choice of mathematical model or method. In this regard one of his favorite quotes, from Shakespeare's Henry VIII, was

The infinite we do right away; the finite takes a little longer.

the earliest work that Stan and he did on branching processes.)

Stan's background in probability made him a leader among the outstanding group of intellects who, during the late 1940s and early 1950s, recognized the potential value of the computer for doing experimental mathematics. They realized that the computer was an ideal tool for analyzing stochastic, or random, processes. While formal theorems gave rules on how to determine a probability measure on a space of objects, the computer opened up the possibility of generating those objects at random. Simply stated constructions that yield complicated objects could be implemented on the computer, and if one was lucky, demonstrable guesses could be made about their asymptotic, or longterm, behavior. That was the approach Stan took in studying deterministic as well as random recursions. In addition he invented cellular automata (lattices of cells and rules for evolution at each cell) and used them to simulate growth patterns on the computer.

The experimental approach to mathematics has since become very popular and has tremendously enhanced our vision of complex physical, chemical, and biological systems. Without the fortuitous conjunction of the computer and probability theory, it is very unlikely that we would have reached today's understanding of those nonlinear systems. Such systems present a challenge analogous to that Newton would have faced if the earth were part of a close binary or tertiary star system. (One can speculate whether Newton could have ever unraveled the law of gravitation from the complicated motions of such a system.) At present researchers are trying to formulate limiting laws governing the long-term dynamics of nonlinear systems that are analogous to the major limiting theorems in classical probability theory. The attempt to construct appropriate probability measures for such systems is one of the topics I will discuss in more depth.

Things done without example

in their issue Are to be feared.

Stan's interest in probability dates back to the early 1930s, when he and Lomnicki proved several theorems concerning its foundations. In particular, they showed how to construct consistent probability measures for systems involving infinite (as opposed to finite) sequences of independent random variables and, more generally, for Markov processes. (In Markov processes probabilities governing the future depend only on the present and are independent of the past.) At about the same time Kolmogorov, independently, proved his consistency theorem, which includes the Ulam and Lomnicki results as well as many more.

Those results guarantee the existence of a probability measure on classes of objects generated by various random processes. The objects might be infinite sequences of numbers or more general geometrical or topological objects, such as the homeomorphisms (one-to-one, onto maps) discussed in detail later in this article. Stan's interest in probability continued after World War II, when he and Everett wrote fundamental papers on “multiplicative” processes (better known as branching processes). Those papers were stimulated by the need to calculate neutron multiplication in fission and fusion devices. (David Hawkins, in "The Spirit of Play," discusses some of

In approaching a complicated problem Stan first searched for simplicity. He had no patience for complicated theories about simple objects, much less complex objects. That philosophical dictum happened to match his personality. He could not hold still for the time it would take to learn, let's say, modern abstract algebraic geometry, nor could he put up with the generalities of category theory. Also, he was familiar with, and early in his career obtained fundamental results in, measure and probability theories. That background led him to approach many problems by placing them in a probabilistic

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