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The waves considered here are solutions of a first-order strictly hyperbolic system of differential equations, written in the form $${u_t} + a(u){u_x} = 0$$ (1) , where u = u(x, t) is a vector with n components u 1,· ·· , u n depending on two scalar independent variables x, t, and a = a(u) is an n-th order square matrix. The question to be discussed is the formation of singularities of a solution u of (1) corresponding to initial data (2) u(x, 0) = f(x). It will be shown that if the system (1) is “genuinely nonlinear” in a sense to be defined below, and if the initial data are “sufficiently small” (but not identically 0), the first derivatives of u will become infinite for certain (x, t) with t > 0. The result is well known for n = 1 and n = 2 (see the papers by Lax and by Glimm and Lax [1], [2], [10]). In many cases such a singular behavior can be identified physically with the formation of a shock, and considerable interest attaches to the study of the subsequent behavior of the solution. In the present paper we shall be content just to reach the onset of singular behavior, without attempting to define and to follow a solution for all times. For that we would need a physical interpretation of our system to guide us in formulating proper shock conditions.

Let L be a linear elliptic differential operator in n-space of order m. Let M denote the adjoint operator to L. An integrable function u is called a weak solution of the equation L[u] = f in a domain D, if for every w of class C ∞, which vanishes outside a compact subset of D, the relation $$ \int_D {(uM[w]) - fw)d{x_1}...d{x_n}} $$ holds; u will be called a strict solution, if u is of class C m and satisfies L[u] =f in the ordinary sense. One of the remarkable facts concerning elliptic equations is that under suitable regularity assumptions on f and the coefficients of L a weak solution can be differentiated any number of times and is a strict solution. In a recent paper F. E. Browder1 gives the theorem: a weak solution which is square integrable on every compact subset of D is almost everywhere equal to a strict solution, provided f is in C 1 and the coefficients of the j-th derivatives in the operator L are in C m+i . Browder’s proof of this generalization of “Weyl’s lemma” makes use of the fundamental solution of elliptic equations with analytic coefficients.2 Results contained in a paper by K. O. Friedrichs,3 which appears in the same issue, imply the theorem that a weak solution is a strict solution, provided f is in C (n+1)/2 and the coefficients of L are in C m/2.

In this paper we show that the so-called scattering frequencies of the Laplace operator over an exterior domain, subject to Robin or Dirichlet boundary condition, cannot lie in certain portions of the upper half-plane. The excluded sets depend only on the type of boundary condition and the radius of the smallest sphere containing the scattering obstacle.

In this paper the exact distribution function of Votaw's likelihood ratio criterion L has been obtained in the form of an infinite series by applying Mellin's transformation on the moments and then by using the asymptotic expansion for log Γ(x + h). The author has actually calculated the values of the constants for some particular cases and has shown that the first four or five terms give a high degree of accuracy. An expression for the Integral probability of the distribution has also been obtained.

We consider here the problem of determining when two elements in a free group will have the same character under all possible representations of the given group in the special linear group of 2 x 2 matrices with determinant 1. Problems involving the representation of free groups in terms of 2 x 2 matrices with real entries and determinant 1 have been investigated by R. Fricke in connection with certain problems in the theory of Riemann surfaces [2]. Fricke’s main concern is to find those representations which define discontinuous groups of the conformal self-mappings of the upper half-plane. His method is based on arguments in non-Euclidean geometry, and has only recently been made fully rigorous and transparent by Linda Keen [4]. Apart from the analytic problems inherent in Fricke’s method there are several algebraic problems which have been answered by Fricke either incompletely or not at all. It is the algebraic aspects of Fricke’s theory which we shall investigate here. Independently of Fricke’s problems however, the subsequent development may be considered as a straightforward contribution to the representation theory of free groups. We shall consider the following general situation. K will denote a fixed commutative ring with identity of characteristic zero, and SL(2, K) the group of all 2 x 2 matrices with determinant 1 and entries from K. It will be shown that the character of any element in a free group under an arbitrary representation of the group in SL(2, K) can be constructively represented as a polynomial expression with integer coefficients in the characters of certain simple products of the group generators. Two polynomials will represent the same character if and only if they are congruent modulo a member of a certain class of ideals. This will reduce the problem of determining whether two elements have the same character to the problem of determining the structure of these ideals. The structure of the ideals will then be determined for the free groups of rank 1, 2, and for the free group of rank 3 under restricted conditions on K. We shall show that if two elements u, v in a free group F have the same character under all possible * The work for this paper was done while the author was at the Courant Institute of Mathematical Sciences and was supported by the National Science Foundation Grant GP-28538. Reproduction in whole or in part is permitted for any purpose of the United States Government. 635

In this paper a wide class of difference equations is described for approximating discontinuous time dependent solutions, with prescribed initial data, of hyperbolic systems of nonlinear conservation laws. Among these schemes we determine the best ones, i.e., those which have the smallest truncation error and in which the discontinuities are confined to a narrow band of 2-3 meshpoints. These schemes are tested for stability and are found to be stable under a mild strengthening of the Courant-Friedrichs-Levy criterion. Test calculations of one dimensional flows of compressible fluids with shocks, rarefaction waves and contact discontinuities show excellent agreement with exact solutions. In particular, when Lagrange coordinates are used, there is no smearing of interfaces. The additional terms introduced into the difference scheme for the purpose of keeping the shock transition narrow are similar to, although not identical with, the artificial viscosity terms, and the like of them introduced by Richtmyer and von Neumann and elaborated by other workers in this field.

In this paper we study the dependence of the set of ‘exterior’ eigenvalues {λk} of Δ on the geometry of the obstacle . In particular we show that the real eigenvalues, corresponding to purely decaying modes, depend monotonically on the obstacle , both for the Dirichlet and Neumann boundary conditions. From this we deduce, by comparison with spheres—for which the eigenvalues {λk} can be determined as roots of special functions—upper and lower bounds for the density of the real {λk}, and upper and lower bounds for λ1, the rate of decay of the fundamental real decaying mode. We also consider the wave equation with a positive potential and establish an analogous monotonicity theorem for such problems. We obtain a second proof for the above Dirichlet problem in the limit as the potential becomes infinite on . Finally we derive an integral equation for the decaying modes; this equation bears strong resemblance to one appearing in the transport theory of mono-energetic neutrons in homogeneous media, and can be used to demonstrate the existence of infinitely many modes.

The purpose of this paper is to obtain solutions which are periodic with respect to distance, describing the phenomenon called “roll-waves,” for water flow along a wide inclined channel, and to discuss the behavior of the mathematical solutions. The basic idea presented in Part I is that discontinuous periodic solutions can be constructed by joining together sections of a continuous solution through shocks (or “bores”). It is shown first that no continuous solutions can be periodic and that only one special continuous solution can be used as the basis for constructing discontinuous periodic solutions. The analysis is based upon the non-linear partial differential equations of the “shallow water theory,” augmented by the Cheay formula to allow for turbulent resistance. The Bresse profile equation is obtained in a form applicable for progressing wave flows. Shock conditions are derived for the case of an arbitrary continuous channel bed and for a flow subject to a resisting force. The special continuous solution is explicitly obtained and analyzed. Branches of it are then joined together through shocks. It is proved that roll-waves cannot occur either if the resistance is zero or if the resistance exceeds a certain critical value. As the resistance decreases, the size of the waves decreases also; and if the resistance becomes too large, the profiles reverse their direction and can no longer be joined by shocks. This critical value is reached when the (dimensionless) resistance coefficient equals one-fourth the value of the channel slope. The presence of a resistance force which varies merely with velocity is not sufficient to permit the construction of periodic solutions; the resistance must also act in such a manner that it decreases as the water depth increases. The analysis proves that the ratio of wave height to wave length of roll-waves is always independent of the speed of the waves. Explicit expressions for water height and shock height as functions of wave length are derived. The investigation studies the static discharge rate as a function of the wave speed, and asymptotic formulas for the wave speed in terms of the average discharge rate are derived. Twelve sets of curves arc presented, based on the equations obtained here, to illustrate the quantitative behavior of roll-waves; these may be used to check this theory against observed data. For prescribed values of slope, resistance, and wave speed, there is a one-parameter family of roll-wave solutions. If the wave length is also prescribed, the solution will then be unique.

The paper discusses the advantages of solving boundaryvalue problems by the use of eigen-function expansions of suitable fourth order differential equations instead of those of second order equations. Some such expansions are constructed, their convergence properties studied and their use in different types of boundary-value problems are discussed.

The solutions of problems involving compressible flows have different functional forms in different regions, separated not only by discontinuities of the fluid itself but also by shock fronts and by characteristics starting from the origins of shock fronts and from points where such characteristics meet shocks. To obtain accuracy in numerical integration, small intervals and formulas using differences of low order are convenient immediately after crossing these lines, but much larger intervals, with differences of higher order, are sufficient elsewhere. The purpose of this paper is to explain a method of exploiting the economy made possible in this way. If the equations for the one-dimensional case are transformed to characteristic variables, they form a ‘canonical hyperbolic system’. Adams' method for ordinary differential equations extends to a variable interval by using divided differences of the derivative, and it then becomes practical to use it for starting and across discontinuities. Further, it applies directly to canonical hyperbolic systems. Numerical solution is then very direct over regions determined by two characteristics and is convenient where the flow crosses already existing shocks. Finally, the origin of a new shock in any interval can be detected and arrangements can be made for it.