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**Analogies between analogies. The mathematical reports of S. M. Ulam and his Los Alamos collaborators. Edited and with a foreword by A. R. Bednarek and Françoise Ulam. With a bibliography of Ulam by Barbara Hendry.**
*(English)*
Zbl 0917.00003

Los Alamos Series in Basic and Applied Sciences. 10. Berkeley, CA: University of California Press. xviii, 565 p. (1990).

This is a collection of reports, known up to now only to very few people, on the work done by Ulam and collaborators at the Los Alamos National Laboratory in the years 1944 to 1982. Some of these reports remain classified to the present day, others have been declassified since their appearance and are now accessible to the general public. In a sense they complement other selections of the work of S. M. Ulam like the ones in his book ‘Sets, numbers, and universes: selected works’ [edited by W. A. Beyer et al., MIT Press (1974; Zbl 0558.00017)], mostly in pure mathematics, and of more general thoughts and reflections of this remarkable scientist as in ‘From cardinals to chaos’ [edited by N. G. Cooper, Cambridge Univ. Press (1989; Zbl 0691.01029)] or in another book by S. M. Ulam [Science, computers and people, Birkhäuser (1986; Zbl 0595.00029)].

The papers in the present collection mainly treat problems arising from the applied sciences such as physics, biology, etc. The main themes of the present papers recurring again and again are: the iteration of functions and their limit behaviour, the need for computers to understand their complex structure, and the development of probabilistic methods in the treatment of systems of physics or biology which are somehow perturbed by external noise that is hard to describe in detail. Surprisingly enough, these are exactly the themes one can find in present-day journals of physics or other fields engaged in the discussion of nonlinear phenomena. This shows how far-reaching Ulam’s ideas and feelings for scientifically important developments really have been.

Starting with three papers on the foundations of discrete multiplicative random processes in one and several dimensions originating in the problem, in nuclear physics, of understanding the proliferation of neutrons in nuclear reactions from one to the next generation, the iteration of functions makes its first appearance. There it turns out that the generating function for the distribution of the number of neutrons in the \(k\)th generation is just the \(k\)th iterate of the generating function in the first generation. The natural problem then is the behaviour of these iterations for large \(k\). In modern terms, this is just the asymptotic behaviour of a finite-dimensional dynamical system with discrete time steps. Since there are almost no general methods available to treat this problem analytically, it was clear to Ulam that only computers could provide us with the necessary hints to ask the right questions. And indeed several reports deal just with the numerical treatment of rather simple polynomial, low-dimensional transformations. What Ulam and his coworkers found in this connection can be considered the origin of what today is called chaos theory: rather complex structures which by no means can be predicted from the simple form of the transformations.

In Report 12, another object of great interest of our days is discussed; namely, cellular automata and the wonderful space-time structures they can generate from simple deterministic rules. Other papers in the volume discuss such diverse problems as the possibility of accelerating rockets by nuclear explosions or of extracting energy from the gravitational field for space vehicles, an idea which indeed has been used recently for satellites travelling to outer space, and the problem of defining the notion of complexity for numbers or other algorithms.

The remaining reports are mainly on problems in the field of biology, such as the evolution of populations, and possible mechanisms for recognition and discrimination.

In Report 20, “On the notion of analogy and complexity in some constructive mathematical schemata”, Ulam quotes another great Polish mathematician, S. Banach, who often remarked that “good mathematicians see analogies between theorems and theories, while the very best ones see analogies between analogies”.

The remarkable scientist Ulam certainly belongs to this latter category, as the papers in the volume under review prove in such a convincing manner. The book should not be missing on the desk of anyone interested in the application of mathematics to the sciences.

The papers in the present collection mainly treat problems arising from the applied sciences such as physics, biology, etc. The main themes of the present papers recurring again and again are: the iteration of functions and their limit behaviour, the need for computers to understand their complex structure, and the development of probabilistic methods in the treatment of systems of physics or biology which are somehow perturbed by external noise that is hard to describe in detail. Surprisingly enough, these are exactly the themes one can find in present-day journals of physics or other fields engaged in the discussion of nonlinear phenomena. This shows how far-reaching Ulam’s ideas and feelings for scientifically important developments really have been.

Starting with three papers on the foundations of discrete multiplicative random processes in one and several dimensions originating in the problem, in nuclear physics, of understanding the proliferation of neutrons in nuclear reactions from one to the next generation, the iteration of functions makes its first appearance. There it turns out that the generating function for the distribution of the number of neutrons in the \(k\)th generation is just the \(k\)th iterate of the generating function in the first generation. The natural problem then is the behaviour of these iterations for large \(k\). In modern terms, this is just the asymptotic behaviour of a finite-dimensional dynamical system with discrete time steps. Since there are almost no general methods available to treat this problem analytically, it was clear to Ulam that only computers could provide us with the necessary hints to ask the right questions. And indeed several reports deal just with the numerical treatment of rather simple polynomial, low-dimensional transformations. What Ulam and his coworkers found in this connection can be considered the origin of what today is called chaos theory: rather complex structures which by no means can be predicted from the simple form of the transformations.

In Report 12, another object of great interest of our days is discussed; namely, cellular automata and the wonderful space-time structures they can generate from simple deterministic rules. Other papers in the volume discuss such diverse problems as the possibility of accelerating rockets by nuclear explosions or of extracting energy from the gravitational field for space vehicles, an idea which indeed has been used recently for satellites travelling to outer space, and the problem of defining the notion of complexity for numbers or other algorithms.

The remaining reports are mainly on problems in the field of biology, such as the evolution of populations, and possible mechanisms for recognition and discrimination.

In Report 20, “On the notion of analogy and complexity in some constructive mathematical schemata”, Ulam quotes another great Polish mathematician, S. Banach, who often remarked that “good mathematicians see analogies between theorems and theories, while the very best ones see analogies between analogies”.

The remarkable scientist Ulam certainly belongs to this latter category, as the papers in the volume under review prove in such a convincing manner. The book should not be missing on the desk of anyone interested in the application of mathematics to the sciences.

Reviewer: D.H.Mayer (MR 92i:01033)

### MSC:

00A69 | General applied mathematics |

01A75 | Collected or selected works; reprintings or translations of classics |

37B99 | Topological dynamics |

60J80 | Branching processes (Galton-Watson, birth-and-death, etc.) |

60K35 | Interacting random processes; statistical mechanics type models; percolation theory |

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\textit{S. M. Ulam} et al., Analogies between analogies. The mathematical reports of S. M. Ulam and his Los Alamos collaborators. Edited and with a foreword by A. R. Bednarek and Françoise Ulam. With a bibliography of Ulam by Barbara Hendry. Berkeley, CA: University of California Press (1990; Zbl 0917.00003)