Rufus Isaacs

On applied mathematics

This paper covers some aspects, problems, and episodes of applied mathematics intended to be enjoyable, instructive, and advisory to the young.

Airfoil Theory for Flows of Variable Velocity

In many cases of aircraft practice—notably that of the rotating wing—the lift surfaces are in air streams of variable velocity. There are two resulting effects tha t are excluded from a direct application of the usual airfoil formulas. One arises from the continuous wake of vortexes which are shed from the trailing edge with each change of circulation; the other, from the impulsive pressure because the air masses are accelerating. The purpose of this paper is to study this aerodynamic problem, taking into account these effects with a certain but sufficient degree of approximation. The first section deals with the general problem of an airfoil at constant angle of attack but with a variable stream velocity. The main result is an integro-differential relation between the velocity and circulation which is t rue for general flows. I t is given in two forms: the realistic, where the airfoil is assumed at some time to have started from rest; and the periodic, where the flow is periodic. The second section is a complete solution to the important practical case: velocity = constant + sinusoidal variation. In the third section the lift is calculated. In particular, an explicit formula is given for the case of the second section in the form of a Fourier series. The moment is not evaluated since the technique is similar. Finally, the solution to a numerical example is given.

ITERATES OF FRACTIONAL ORDER

Abstract : This paper is concerned with the following question: Let E be any space whatever. g(x) is a function mapping E into E. When does there exist a function f(x), of the same type, such that f(f(x)) = g(x).

Differential games: Their scope, nature, and future

There is a profound distinction between classical mathematical analysis and game theory which comes into especial prominence with the advent of differential games. There is a hierarchy of theories of applied mathematics in which the classical theory is the bottom row. Thus, it is important in the inevitable pending developments of higher forms of game theory to be prepared for ideas and concepts which break with tradition. These general thoughts, not at present widely understood, are expounded here with some simple examples which already illustrate the novelties of future research.

The finite differences of polygenic functions

By a polygenic function ƒ(z) we shall mean a function analytic in x and y separately, but whose real and imaginary parts are not required to satisfy the Cauchy-Riemann equations. At any point z the derivative of such a function will depend on 0, the angle at which the incremented point (used in defining the derivative) approaches z. The set of these numbers, for a fixed z} but for different 0, form a circle. The equation for the derivative was given by Riemann in his classic dissertation (1851), but Kasner was the first to point out that it was a circle and make a detailed study of its geometry. Hedrick called it the Kasner circle. In this paper we shall be concerned with the finite difference quotients of polygenic functions. We shall show how a surface can be constructed for each point z representing the difference quotient, and the derivative circle is a cross section of this surface.

The Past and Some Bits of the Future

Since my book Differential Games [1] appeared in 1965, the subject has burgeoned vastly. I have not followed all its developments — my research efforts have lately turned elsewhere — and so there is some doubt as to my now qualifying as an expert. A second printing of [1] is due shortly and therein I describe some advances, particularly those that resolve quandaries appearing in the first. Here the material will be limited — narrowly or broadly, as you deem — with much of it essentially also in the introduction to the second printing.