The theory of the Laplace integral of abstractly-valued functions of a real or complex variable has been developed, in the last few years, to an extent that it is almost approaching the degree of completeness enjoyed by the classical theory of the Laplace integral of numerically-valued functions. In certain respects, however, there are still large gaps. One of the gaps occurs in representation theory.

The unit tangent vector at a point of a curve in a hypersurface of a Riemannian space has two derived vectors along the curve, one with respect to the Riemannian space in which the hypersurface is imbedded and one with respect to the hypersurface itself. When the former vector is decomposed along the directions normal and tangent to the hypersurface, its tangential component, which is called the first curvature vector of the curve at the point in the hypersurface, is exactly the latter vector.

From time to time attention has been directed to the study of spaces in which the “distances” are selected from algebraic structures other than the real or complex numbers. For example, Menger (9) and Taussky (11) have considered spaces in which the distances are taken from a group. More recently, Ellis (4), Blumenthal (2), and Elliott (3), have investigated spaces in which the distances are elements of a lattice.

Let G be an abelian group of order [G] ≤ ∞. Let A = {a}, B = {b}, … denote non-empty finite complexes in G. Let [A] be the number of elements of A. Finally put

A + B = {a + b}.

Small italics denote integers. Let A, B, … be sets of non-negative integers. Let A (h) be the number of positive integers in A that are not greater than h. Finally let A + B denote the set of all integers of the form a + b where a ⊂ A, b ⊂ B. The following result is implicitly contained in Mann's Proposition 11 (4):

Theorem 1. Let n > 0 and

(1.1) 0⊂4, 0⊂B, n⊄C = A + B.

A linear algebra is called a Jordan algebra if it satisfies the identities

(1) ab = ba, (a2b) a = a2(ba).

It is well known that a linear algebra S over a field of characteristic different from two is a Jordan algebra if there is an isomorphism a → a of the vector-space underlying S into the vector-space of some associative algebra A such that

1,

where the dot denotes the multiplication in A. Such an algebra S is called a special Jordan algebra.

The most interesting cases of stable lattices, introduced in an earlier volume of this journal (12), were the (algebraic) modules of stable norm, or modules whose ratio of minimum absolute non-zero norm to lattice determinant (i.e., to the square root of module-discriminant) is a local maximum for small variations of the basis. We soon found that these modules were perhaps more numerous than we should have desired if we were interested only in finding an absolute maximum.

Any finite group of linear transformations on n variables leaves invariant a positive definite Hermitian form, and can therefore be expressed, after a suitable change of variables, as a group of unitary transformations (5, p. 257). Such a group may be thought of as a group of congruent transformations, keeping the origin fixed, in a unitary space Un of n dimensions, in which the points are specified by complex vectors with n components, and the distance between two points is the norm of the difference between their corresponding vectors.

Les groupes classiques de petite dimension présentent entre eux des isomorphismes que l'on peut qualifier de “génériques”, c'est-à-dire que ces isomorphismes ne dépendent que de la dimension et du type de groupe considéré, et non du corps de base de l'espace vectoriel où opère le groupe.

Each irreducible representation [λ] of the symmetric group Sn may be identified by a partition [λ] of n into non-negative integral parts λ1 ≥ λ2 ≥ … λn ≥ 0, of which the first λ'j parts are ≥j, or by a right (Young) diagram also called [λ], that contains λi nodes in its ith row and λ'j nodes in its jth column.

In this paper we are concerned with the group of formal power series of the form

,

the coefficients being elements of a commutative ring R and the group operation being substitution. Little seems to be known of the properties of groups of this type, except in special cases, although groups of formal power series in several variables with complex coefficients have been investigated from a different point of view by Bochner and Martin (1, chap. I) and Gotô (2).

A balanced incomplete block design is defined as an arrangement of v objects in b blocks, each block containing k objects all different, so that there are r blocks containing a given object and λ blocks containing any two given objects. Such designs have been studied for their combinatorial interest, as in (3), and also for their application to statistics, where the objects are usually varieties.

We define a graph as a set V of objects called vertices together with a set E of objects called edges, the two sets having no common element. With each edge there are associated just two vertices, called its ends. We say that an edge joins its ends. Two vertices may be joined by more than one edge.

The theorem that every properly primitive binary quadratic form is capable of representing infinitely many prime numbers was first proved completely by H. Weber (5). The purpose of this paper is to give an elementary proof of the case where the form is ax2 + 2bxy + cy2, with a > 0, (a, 2b, c) = 1, and D = b2 — ac not a square. The cases where the form is ax2 + bxy + cy2 with b odd, and the case where the form is ax2+ 2bxy + cy2 with D a square, can be settled very simply once the first case is taken care of, and this is done in a page and a half in the Weber paper.

The problem of giving a criterion for the eth power residue character of 2 has long interested number theorists. This paper is primarilyconcerned with the cases e = 4, 8 and 16. Gauss (8) proved that 2 is a biquadratic residue of a prime p of the form 4n + 1 if and only if p is representable as x2 + 64y2.

In discussing an isomorphism between two Banach algebras, one will ordinarily tacitly assume that the mapping is linear (i.e., preserves the complex scalars as well as the ring operations). In general this cannot be avoided; for instance if the two Banach algebras are just the field of complex numbers, then the isomorphism is unrestricted, and could be given by any one of the myriads of discontinuous automorphisms of the complex numbers.

In various numerical problems one is confronted with the task of solving a system of linear inequalities:

(1.1) (i = 1, … ,m)

assuming, of course, that the above system is consistent. Sometimes one has, in addition, to minimize a given linear form l(x). Thus, in linear programming one obtains a problem of the latter type.

Let A be a closed set of points in the n-dimensional euclidean space En. If p and p1 are points of En such that

1.1

then p1 is said to be point-wise closer than p to the set A. If p is such that there is no point p1 which is point-wise closer than p to A, then p is called a closest point to the set A.

In a number of recent papers, especially by Wilansky (4; 6), Zeller (8), and Peyerimhoff (3), the sequence-to-sequence transformation

A : (n = 0, 1, …)

has been studied under certain conditions, designated by FAK, PMI, etc. (see §3). The purpose of this note is to point out some relations among these conditions, and to show that some theorems previously obtained hold under weaker assumptions.

Recently a number of papers have been written on Bessel polynomials which arise as the solutions of the classical wave equation in spherical coordinates. Krall and Frink (5) studied in some detail the properties of these polynomials yn(x, a, b) defined as

(1) .