The purpose of this note is to establish Theorem A below for the two-point homogeneous vector boundary problem

where the Pi(x) are given real m × m symmetric matrix functions of x with P0(x) positive definite and Pi(x) of class C2−i on an infinite interval [a, ∞), and where by a solution of (1.1) — (1.2) for a ≤ x1 < x2 < ∞ we understand a real m-dimensional column vector u = u(x) of class C2 on [a, ∞) which is such that Pi(x)u(2−i) is of class C2−i on [a, ∞) and which satisfies (1.1) — (1.2) with the former a vector identity on [a, ∞).

Suppose that p(t) > 0, that both p(t) and f(t) are continuous functions on the half-line 0 ≤ t < ∞, and that λ denotes a real parameter. Only real-valued functions will be considered in this paper. Let the differential equation

,

be of the limit-point type (3, p. 238), so that (1) and a linear homogeneous boundary condition

, 0 ≤ α < π,

determine a boundary value problem on 0 ≤ t < ∞ for every fixed α.

The boundary value problems of generalized potential theory on finite Riemannian manifolds may be regarded as extensions of the Dirichlet and Neumann problems for harmonic functions. In the tensor theory there is, in fact, a greater variety of such problems; that is to say, these generalizations from classical potential theory can be made in various ways. We here introduce yet another pair of boundary value problems for the tensor equation of Laplace.

The results presented here were motivated by a desire to give a simple treatment of the non-linear elliptic partial differential equation governing the steady irrotational subsonic flow of an ideal compressible fluid. For two independent space variables x and y, the stream function Ψ of such a flow satisfies an equation

,

where is an analytic, increasing, convex function of whose explicit form depends on the equation of state of the fluid in question.

It has been shown by Dickson (1) that if (i, j)8 is the number of solutions of

(mod p),

then 64(i,j)8 is expressible for each i,j, as a linear combination with integer coefficients of p, x, y, a, and b where

,

and

a ≡ b ≡ 1 (mod 4),

while the sign of y and b depends on the choice of the primitive root g. There are actually four sets of such formulas depending on whether p is of the form 16n + 1 or 16n + 9 and whether 2 is a quartic residue or not.

Let

(W): W0, W1,…, Wn,…

be a linear integral recurring sequence of order r ≥ 2; that is, a particular solution of the recurrence

(1.1) ,

where P1, P2,… , Pr ≠ 0 are integers, and the initial values W0, W1, … , Wr−1 are integers.

Consider the classes of positive, primitive binary quadratic forms ax2 + bxy + cy2 of discriminant — Δ = d = b2 − 4ac < 0. Dickson (2, p. 89) lists 101 values of A such that — Δ is a discriminant having a single class in each genus. The largest value given is 7392, and Swift (7) has shown that there are no more up to 107.

Let J be an integral domain (i.e., a commutative ring without divisors of zero) with unit element, F its quotient field and J[x] the integral domain of polynomials with coefficients from J . The domain J is called integrally closed if every root of a monic polynomial over J which is in F also is in J.

Let Ω be a cyclic extension of degree l over the field Σ. Correcting an error which for some time had been haunting the literature, Hasse (1, p. 272) noted that for l = 2, the field Ω may contain a unit ξ such that

.

In this paper certain relations will be proved between ∊-normality of integers, (k, ∊)-normality of integers, and normality of real numbers. Also a new type of normality of numbers will be introduced, namely, quasi-normality, as defined below.

A simply normal number is a real number, expressed in some scale B, in which each digit of the scale B occurs with the asymptotic frequency 1/B.

The study of the modular representation theory of the symmetric group has been greatly facilitated lately by the introduction of the graph (9, III ), the q-graph (5) and the hook-graph (4) of a Young diagram [λ]. In the present paper we seek to coordinate these ideas and relate them to the r-inducing and restricting processes (9, II ).

The method of A. Young for finding the set of primitive idempotents of the group algebra of the symmetric group is classical; it was first given by Frobenius (4) using results of Young (10 and 11). A concise account can be found in (9) and a very detailed treatment in (6).

Considerable advance has been made recently towards a systematic method of evaluating the “product” {μ} ꕕ {λ}, most notably in the methods of Robinson (3), Littlewood (2), and Todd (5) and the differential operator technique of H. O. Foulkes.

Let p be a fixed prime number. We denote by k(n) the number of partitions of n. As is well known, the number of ordinary irreducible characters of the symmetric group Sn is k(n). We set k(0)=1.

The purpose of this paper is to give a short proof of the Cartwright-Littlewood fixed point theorem (2, p. 3, Theorem A).

Theorem A. If T is a (1-1) continuous and orientation preserving transformation of the Euclidean plane E onto itself which leaves a bounded continuum M invariant and if M does not separate E, then some point of M is left fixed by T.

Let P be a set partially ordered by a (reflexive, antisymmetric, and transitive) binary relation ≺. Let be the family of all subsets K of P having the property that x ∈ P and y ∈ K and y ≺ x imply x ∈ K.

In §1, we define a differential-integral operator, which for positive real indices is commonly known as the Liouville-Riemann generalized integral. For positive integer indices, we obtain an iterated integral. For negative real indices we obtain the Riemann-Holmgren (5; 9) generalized derivative, which for negative integer indices gives the ordinary derivative of order corresponding to the negative of such an integer. Following M. Riesz (10) we extend these ideas to include complex indices.

In this paper we consider the Cauchy problem for the equation

(1) h(x, y) K(y) vxx − vyy + a(x, y) vx + b(x, y) vy + c(x, y) v + f(x, y) = 0

with initial values prescribed on a segment of the x-axis. The coefficients in (1) are assumed to possess two continuous derivatives with respect to x and one continuous derivative with respect to y in the closure of the domain under consideration.

Titchmarsh (4) has shown how the classical method of complex variables can be used to obtain expansion theorems for the singular cases of the second order equation

(1) .

The purpose of this paper is to indicate how these results can be generalized to the singular cases of the pair of first order equations

,