Mathematical Circles Research Articles

The joy of formula manipulation

In mathematical circles, there is the overall opinion that formulae, in their capacity of syntactic units, are dead things, and that formula manipulation tears the heart out of mathematics. In these circles, formulae largely live by virtue of what they stand for, of what they mean, of how they feel and appeal to our intuition—our(?) intuition? And their meanings then tell which formulae to consider next. Poor Leibniz, poor Lagrange, poor Boole, poor Hilbert, and all others who shifted their attention towards uninterpreted formulae manipulation: they were all wrong, weren't they? Oh, and poor we, Edsger W. Dijkstra and all those programmers who converted themselves to formulae manipulators, because their profession demanded it. (This cultural gap in doing mathematics was once expressed quite aptly by Dijkstra when he remarked, “Ik hou van wiskunde, maar spaar me de mathematen”. [“I love mathematics, but it's the mathematicians I cannot stand”.]) Well, our profession of programming demanded a conscious and active engagement in formula manipulation and therefore we entered that field of endeavour; and we learned how mighty and powerful, how prosperous and effective, and how indicative of designing this change in attitude turned out to be, not only for the benefit of programming, but for vast parts of mathematics as well. And moreover, we learned to enjoy the activity. In this note we try to convey the effectiveness and joy of formula manipulation through a small number of simple examples from both mathematics and programming.

Hypnotism and Medicine in 1888 Paris: Contemporary Observations by Sofia Kovalevskaya

The name of the acclaimed Russian mathematician Sofia Vasilyevna Kovalevskaya (1850-1891) is virtually unknown in the West outside of mathematical circles. In Russia, by contrast, she is a textbook icon. Most high school students can easily identify her--next to the woman cosmonaut and the woman symphony conductor. But in Russia, too, awareness of her work and achievements rarely extends beyond name recognition. This relative obscurity seems undeserved, given what an engaging figure she is, and given the exceptional scientific stature she achieved in her short and turbulent life.

The Flowering of Applied Mathematics in America

Mathematicians are notoriously bad historians; they describe development of an idea as it should logically have unfolded rather than as it actually did, by fits and starts, often false starts, and buffeted by forces outside of mathematics. In this sketchy account I will describe twists and turns as well as thrusts of in America. Applied is alive and well in America today. Looking at 18 lectures [presented at Centennial Meeting of American Mathematical Society] chosen to describe frontiers of research, we see that topics include physiological modeling, fluid flow and combustion, computer science, and formation of atoms within framework of statistical mechanics. We also see that subject of one lecture and starting point of several others are physical theories; conclusions reached are of interest to physicists and mathematicians alike. This was not always so. For a few decades, late 1930s through early 1950s, predominant view in American mathematical circles was same as Bourbaki's: is an autonomous abstract subject, with no need of any input from real world, with its own criteria of depth and beauty, and with an internal compass for guiding further growth. Applications come later by accident; mathematical ideas filter down to sciences and engineering. Most of creators of modern mathematics-certainly Gauss, Riemann, Poincare, Hilbert, Hadamard, Birkhoff, Weyl, Wiener, and von Neumann-would have regarded this view as utterly wrongheaded. Today we can safely say that tide of purity has turned; most mathematicians are keenly aware that does not trickle down to applications, but that and sciences, mainly but by no means only physics, are equal partners, feeding ideas, concepts, problems, and solutions to each other. Whereas in not so distant past a mathematician asserting applied is bad mathematics or the best is pure mathematics could count on a measure of assent and applause, today a person making such statements would be regarded as ignorant. How did this change come about? Several plausible reasons can be discerned. But first a bit of selective history. World War II, a watershed for our social institutions, concepts and thinking, permanently changed status of in America. That is not to say that there was no worthwhile in America before 1945; after all, already in nineteenth century, Gibbs' contributions to statistical mechanics as well as to vector analysis and Fourier series, and Hill's studies of Hill's equation, had put America on map. The leading American analysts in 1920s and 1930s were Birkhoff, renowned worldwide for his work in dynamics, and Wiener, a pioneer in study of physical processes driven by chance influences such

The Use of Tape as a ‘Lead-Medium’

Mathematical circles in the schools of the Federal Republic of Germany are engaged in the examination and discussion of two proposed teaching reforms, the first concerned with the content of teaching and the second with its methods. The proposals stem from the 1968 Conference of Cultural Ministers, a subject of which was the establishment of ‘A Framework of Guiding Principles of Reform’. These principles provide for reforms in mathematics teaching in schools, to start in the school year 1972/73, and preparatory work has been going on in a number of places. The research includes a mathematical project on the theme: ‘Value of Digit Systems’, work on which has been carried out by the Institut fur Film und Bild in association with the publishing firm of Ernest Klett.

A Survey of the Development of Operational Calculus

Summary Although the name of Oliver Heaviside is usually associated with the foundations of operational calculus, he was in fact preceded in introducing operational methods by a number of mathematicians. Heaviside's purely operational approach was often intuitive rather than rigorous and fell into disfavour in mathematical circles, being replaced by methods involving the concept of an integral transformation. In particular, the Laplace transformation came to be adopted by applied mathematicians and engineers despite the need for the Schwartz theory of distributions to account satisfactorily for the Dirac and other impulse functions. Jan Mikusinski,∗ ∗ It is intended that an expository article on Jan Mikusinski's operational calculus by H. G. Flegg will appear in a later issue ‐ Editor. however, has returned to the earlier operational approach and, by placing operational calculus upon a rigorous algebraic basis, has provided a calculus which is more general than those based upon integral transformations, and one which includes generalized functions without recourse to any specialized external theory.

The Disordered Lattice Problem: A Review

Introduction. In the eighteenth century a mathematician would probably have been familiar with the latest advances and problems in physics. In the twentieth century, due to the increasing tendency to specialize in both physics and mathematics the problem of communication is a real one, notwithstanding the obvious mutual benefits which result from exchanges between the two sciences. It is the purpose of this paper to describe a mathematical problem of some interest in current investigations of solid state physics which does not seem to have received any previous publicity in mathematical circles. This problem, that of determining the vibrational properties of a disordered lattice of masses and springs, has not been solved in a satisfactory fashion. Since the main difficulties are mathematical it is our hope that this paper will interest mathematicians who would not ordinarily be acquainted with solid state physics. We begin by giving some of the preliminaries necessary to a discussion of the problem. The heat capacity at constant volume, Cv, is defined as the amount of energy needed to raise the temperature of a solid by one degree centigrade. If E is the energy, C, equals ( 3E/T)X , where T is the absolute temperature and the subscript v refers to the fact that the volume of the solid is held constant during this process. A modern theory of the heat capacity of solids was originated by Einstein [1] in 1907 through the application of the Planck quantum theory. Einstein approximated a crystal by an assembly of 3N independent harmonic oscillators each vibrating with the same frequency P. The energy of such an ensemble is given by

Electronic Computation in Economic Statistics

(1953). Electronic Computation in Economic Statistics. Journal of the American Statistical Association: Vol. 48, No. 263, pp. 414-428.

On the Theory of Prediction of Nonstationary Stochastic Processes

We consider the following problem of prediction: During a finite time interval T the real valued function S(t)+N(t) is observed, in which S(t) is a signal and N(t) is a linearly superimposed noise disturbance. The problem is to predict the value of a given linear functional of S(t), the predictor formula having certain preassigned ``optimum properties'' among a certain class of predictors. In the case in which the mean value of S(t) is known, the random components of S(t) and N(t) are strictly stationary, and the time interval T is infinite, a complete solution to this problem has been given by N. Wiener. (In the case of discrete time series, the solution was given by A. Kolmogoroff.) This theory has been extended by L. Zadeh and J. Ragazzini [J. Appl. Phys. 21, 645 (1950] to the case in which T is a finite time interval and the mean value of S(t) is unknown but is restricted to be a polynomial in time. We extend the above theories to the case in which the random components of both S(t) and N(t) are nonstationary in time and merely possess finite continuous covariance and cross-covariance functions. The analytical tools of probability theory which we use are those developed independently by M. Loève and K. Karhunen in their studies of a class of stochastic processes usually termed processes of second order. Apparently these techniques are practically unknown outside of certain mathematical circles. In the opinion of the author these techniques are extremely powerful in the analysis of transient random phenomena in linear systems. Finally we give an exposition of a method of prediction by the theory of conditional probabilities. This method is applicable when the form of the joint probability distribution of signal and noise is known and hence is applicable to many practical problems, since this distribution is often Gaussian. In this particular case the predictor formula given by the theory of conditional probabilities is identical with the usual linear predictor formula given by the theory of least squares. In case the signal is a Markoff process, the method of conditional probabilities yields a much simpler prediction formula than the usual method of least squares.

Living Mathematics

Why Is It that speakers in the field of general education so frequently draw their illustrations of uninspired teaching, of the use of dead subject matter, or of almost any type of undesirable procedure, from the mathematics classroom? Why do many people consider mathematics a necessary evil which must be endured for supposedly practical though rather indefinite ends? Is there any reason why mathematics in itself and of itself may not be as vitally interesting as any other field of study? Is the pupil's past experience so completely devoid of mathematical content as to give no true teaching basis? We hear and read much, in mathematical circles, of methods by which we may vitalize our teaching. But do we succeed in leading our pupils to see the life in mathematics and the mathematics in life?

The Nomenclature and Notation of Aeroplane Rigid Dynamics

In applying the methods of rigid dynamics of three dimensions to the study of aeroplane motions, one of the first questions which had to be settled by myself and Mr. Harper in the preparation of “Stability in Aviation” was the choice of co–ordinate axes and the notation to be adopted for the various forces, couples and rates of variation of components of air resistance. Where, as in this case, the necessary symbols more than exhaust the ordinary alphabet, the problem is a difficult one, and it was therefore intended to insert a separate section explaining the advantages which rendered the choice adopted the only logically possible one.Unfortunately, at the time when “ Stability in Aviation “ was published, very great doubts existed as to the practical utility of any such investigation except as a subject of academic study in orthodox mathematical circles, and the work of preparing the book in the face of obstacles, partly arising from opposition, had proved such a strain that any further discussions on the subject would have been incompatible with my academic duties.