Showing papers in "The Mathematical Gazette in 1962"

An Introduction to Magneto-Fluid Mechanics. By V. C. A. Fekrabo and C. Plumpton. Pp. 181. 25s. 1961. (Oxford University Press)

Abstract: ing journals. At the head of each abstract is the name of the author and the title of the paper, although not its date or origin, and also a pair of numbers. The reader is able to find by trial that one of these numbers is the serial number of the same paper as entered in the bibliographical list, the other being a new serial number for the paper in the sequence of abstracts. At the end of the volume are three lists. The first, a short one, is of numbers entitled ‘List of the entries included in bibliography no. 1’; bibliography no. 1 is apparently what this bibliography used to be when it was issued informally by AGARD in February 1960. The second is an index to authors, which presumably serves to identify the number of a paper written by a known author who is one of several joint authors and not the first named, the latter alone being used in the bibliographical list. The last is a subject-index, in which there are seven main divisions, like ‘magnetohydrodynamics ’ and ‘magneto-gas-dynamics ),with afew sub-divisions for each. The value of the subject index will be limited by the fact that the subheading ‘ General’ swallows up about half the entries in four of the main divisions. It is obviously useful to prepare and disseminate an up-to-date and ordered list of papers on this important subject, and the initiative and work contributed by AGARD have been widely appreciated. But why print a second time, publish, and sell at a substantial price, a bibliography which pays little attention to the ordinary principles of fact-gathering and indexing? As well as being rather wasteful, the book does not maintain accepted standards in scientific publications. G. K. BATCHELOR

Time Series Analysis. By E. J. Hannan. Pp. viii, 152. 21s. 1960. - Stochastic Processes, Problems and Solutions. By L. Takács. Pp. xi, 137. 18s. 1960. - Stochastic Population Models in Ecology and Epidemiology. By M. S. Bartlett. Pp. x, 90. 12s. 6d. 1960. (Methuen and Co., Ltd., London)

Abstract: This paper begins with a discussion of the properties of the noise occurring in the structures considered. This noise is taken to be generated by a stationary, ergodic, purely nondeterministic process. In case the observed vector sequence is generated by an autoregressive-moving average process then (a little more than) the additional requirement that the best predictor be the best linear predictor suffices for the development of an asymptotic inference theory. Signal measurement problems are considered, first where the signal is directly observed except for some unknown parameters and second where the signal is not directly observable and some characteristics, such as the velocity of propagation, have to be measured. Finally nonstationary models, nonlinear models for prediction, transient signals, and irregularly spaced samples are briefly discussed. Throughout, the methods are based on the use of the fast Fourier transforms of the data and their relation to the use of quasimaximum likelihoods in terms of those transforms is discussed.

The Theory of Linear Economie Models. By David Gale. Pp. 330. 74s. 1960. (McGraw-Hill Book Company)

Abstract: In the past few decades, methods of linear algebra have become central to economic analysis, replacing older tools such as the calculus. David Gale has provided the first complete and lucid treatment of important topics in mathematical economics which can be analyzed by linear models. This self-contained work requires few mathematical prerequisites and provides all necessary groundwork in the first few chapters. After introducing basic geometric concepts of vectors and vector spaces, Gale proceeds to give the main theorems on linear inequalities—theorems underpinning the theory of games, linear programming, and the Neumann model of growth. He then explores such subjects as linear programming; the theory of two-person games; static and dynamic theories of linear exchange models, including problems of equilibrium prices and dynamic stability; and methods of play, optimal strategies, and solutions of matrix games. This book should prove an invaluable reference source and text for mathematicians, engineers, economists, and those in many related areas.

Playing with Infinity. By Rózsa Péter. Pp. 268. 25s. 1961. (G. Bell and Sons Ltd., London)

Abstract: counting to infinity as a realizable computation define a region beyond which indeterminateness reigns. There are numbers, and processes derived from numbers, that are too large to be operated upon in our physical universe and are thus beyond the scope of certain (apodictic, Kant would say) knowledge. Using finite—but often rather long, complex, and clausally convoluted—sentences, Brian Rotman develops a most ingenious refutation of the naturalness of the natural numbers in his latest work: Ad Infinitum: The Ghost in Turing’s Machine: Taking God Out of Mathematics and Putting the Body Back in: An Essay in Corporeal Semiotics. This essay is well worth the parsing effort. Brian Rotman is one of the most creative contemporary philosophers of mathematics thinking today. Briefly put, he argues that the uniform infinite extent of the natural numbers is not natural at all. Resonating with two of the most important themes in current scientific thinking, nonlinear dynamics and computational complexity, Rotman develops a new strongly finitistic—and somewhat startling—view of the nature of the mathematical infinite. He insists that the physical processes of entropy and the corresponding physical limits of computation must be taken into account in explaining how mathematicians prove and communicate theorems. He fashions his sophisticated semiotic approach to mathematical discourse in order to bypass key deficiencies in each of the three dominant philosophical orthodoxies in interpreting mathematics: Platonism, formalism and intuitionism. A sophisticated triadic model of mathematical discourse lets him reinterpret each of these schools as having too simple-minded an acceptance of the infinite. Rotman is sympathetic to the basic thrust of Wittgenstein’s way of philosophizing about mathematics—as a specialized language game for making assertions about mathematical inscriptions. He focusses his analysis on one particular move in the game: the use of the ellipsis (...) to represent the unending extension of a the sequence of integers off to infinity, the “ad infinitum.” Though sympathetic to Wittgenstein, Rotman’s approach is far richer and more substantial in its detail. He reinterprets mathematical proof as a type of conscious waking dream, a controlled thought experiment. The approach via semiotics—the study of how symbols embody meaning—is most directly descended from the seminal work Book Review

On Geometric Problems of Erdös and Oppenheim

Abstract: The American Mathematical Monthly in Vol. 68 (1961) page 380 gave a solution by Leon Bankoff of the following problem proposed by A. Oppenheim of the University of Malaya in Vol. 67 (1960) page 802.