Showing papers in "American Journal of Mathematics in 1951"

The Existence and Limit Behavior of the One-Dimensional Shock Layer

Abstract: Introduction. We consider steady, one-dimensional flows of a viscous, heat-conducting fluid which approach finite limit values at x = +oo and x = -oo. Such flows display the character of a shock wave (for small viscosity, ,u, and heat conductivity, A), in that they differ sensibly from their end states at x + oo only in a small interval of rapid transition. In analogy with the classical boundary layer, and also to distinguish these flows from the shock waves which belong properly to the theory of ideal fluids, we follow Weyl [1] in naming such a flow a shoclk layer. The one-dimensional shock layer is in certain respects the prototype of all shock phenomena and has therefore been studied widely, with particular emphasis on the problemn of thickness of the shock front [2, 3, 5, 6, 8]. I-Towever, basic problems concerning these flows, such as those of existence, and limit behavior for small A, ,u, remain open. Their solution, which we consider here, is a step towards placing on a sound basis the relation between the theories of real and ideal fluids. The general problem of existence of the shock layer for a fluid with given A, j, and with the preassigned end states, has been studied inconclusively by Rayleigh [4] and Weyl [1]. Until now, the existence of the shock layer seems to have been definitely proved only for an exceptional set of ideal gases for which a postulated relation between A, ,u, and the specific heat at constant pressure,' permits explicit integration of the equations of motion; (Becker [2], also [5, 6]). We succeed here in obtaining an essentially complete solution of the existence problem by proving the existence andI uniqueness of the shock layer for the general class of fluids considered by Weyl, with A, ,u arbitrary functions of the state, and for arbitrary end states satisfying the shock relations (Theorem 1). This result, therefore, establishes for gene.ral fluids an exact correspondence between the steady one-dimensional shock waves and the shock layers.

On the asymptotic curves of a surface.

Abstract: hand, conditions sufficient for the truth of the classical statements will be developed in terms which take into account the specifically geometrical origin of the various differential equations defining the respective problems. Oii the other hand, examples will show that the conditions to be imposed cannot be omitted. Both an attraction and a difficulty of the theory is the circumstance that, even if the surface has a parametrization in terms of asymptotic curves, examples show that this parametrization of the surface can be less differentiable than some other parametrization of the surface (or, equivalently, less smooth than the parametrization z = z (x, y), where x, y, z are Cartesian coordinates). Surfaces of negative Gaussian curvature are dealt with in Parts I, II, III. Part IV deals with lines of curvature and applies therefore to surfaces of positive curvature also. Finally, Part V develops the corresponding theory

The Topology of the Level Curves of Harmonic Functions with Critical Points

Abstract: Introduction. In a previous paper,2 of which this is a continuation, topological properties of curve families which filled the Euclidean plane 7r, or a simply connected domain in r, were investigated. The families were assumed regular (i. e. locally homeomorphic to parallel lines) except at a possibly infinite collection of isolated singularities at each of which the family had the structure of a multiple saddle point; such families were called branched regular curve families. Further investigation of these families, in particular their relation to harmonic functions, is the aim of this paper. In what follows the definitions and theorems in [I] will be assumed, and the same notation will be used. In particular F, G will denote branched regular curve families filling the plane 7r, B will denote the set of singular points, R the domain 7r B in which F is regular, and so on. The Euclidean plane will be taken as a model for all simply connected domains. The principal result of [I] was to prove that any branched regular curve family F filling Xr can be given as the family of level curves of a function f (p) which is continuous on all of 7r and has no relative extrema. This generalizes a portion of [II] in which the same theorem is proved for a curve family without singularities in 7r. In this paper there are two main results: the first, proved in Section 1, is that F is actually homeomorphic to the level curves of a harmonic function; the second, proved in Section 2, asserts the existence of a decomposition of F into a countable collection of subfamilies of curves, each of which has the structure of the parallel lines y=constant of the upper half-plane. Such subfamilies will be called half-parallel, and this decomposition has consequences for the study of harmonic functions and analytic functions which will be mentioned below. These two results generalize

The Topology of Regular Curve Families with Multiple Saddle Points

Abstract: Introduction. It is known that the level curves of any function f (x, y) which is harmonic in a simply connected domain form a curve family which is regular (locally homeomorphic to parallel lines) in the neighborhood of every point of the domain with the exception at most of a set of isolated points at each of which the curve family has a singularity of the multiple saddle point type (Figure 1). The central task of this and a later paper 2 iS to

Homomorphisms of a semigroup onto a group.

Abstract: asked in connection with images of the form, a group with a zero element adjoined. These images exist if and only if S contain a prime ideal, according to Theorem 1. In the present paper the notions of a maximal group image and maximal group with zero image are formulated for a semigroup S and the construction of such images is discussed in terms of "normal" subsystems (generalizations of a normal subgroup in the group case) of S, using a method devised by Dubreil [2]. A set of independent defining conditions for these normal subsystems is obtained. Finally, several well known semigroups which have unique maximal images of the types under considerations are discussed. The most interesting examples are two classes of semigroups (regular sets of partial transformations of a set, and completely simple semigroups without zero) studied by Rees in [7] and [5] respectively, and one (semigroups having zeroid elements) studied by Clifford and Miller in [1]. For the first class, the group image discussed by Rees is shown to be maximal. For the second class, the maximal group image can be described in terms of a homomorphic image of the basis group for the semigroup. For the third class; the group of zeroid elements is found to be the maximal