Introduction. This paper deals with questions of the following type. Problem (A) : Let F(x) be the nth integral of a positive non-decreasing function for all large positive x, the problem is to find a function f(x), being the nth integral of a non-decreasing function for all x (— ∞ < x < ∞), with the property

A.

1. Introduction. Let X be a Banach space (B-space). A sequence {s(i)} in X is unconditionally summable if and only if every rearrangement of the series Σis(i) is convergent. The set of unconditionally summable sequences in X will be written as U(X). In this paper several classes of summable sequences in X will be compared with one another. Each class to be considered is identical with U(X) when X has finite dimension.

1. Introduction. The purpose of this note is to establish some limit theorems for the non-linear recurrence relations

1.1, i= 1,2, …, N; n ≥ 0

under certain assumptions concerning the initial values ci = xi(0), and the coefficient matrices A (q) = (aij(q)).

1. Introduction. The quasi-linear elliptic partial differential equation to be studied here has the form

(1.1) Δu = − F(P,u).

Here Δ is the Laplacian while F(P,u) is a continuous function of a point P and the dependent variable u. We shall study the Dirichlet problem for (1.1) and will find that the usual formulation must be modified by the inclusion of a parameter in the data or the differential equation, together with a further numerical condition on the solution.

1. Introduction. Let us consider all power series of the form

It was shown first by Borel (1) that to each such series there corresponds a non-empty class of functions such that each function in the class has the given series as its asymptotic expansion about z = 0, the expansion being valid in a sector of the right half z-plane with vertex at the origin.

1. Introduction. Let a1 a2, …, am be a set of real non-negative numbers and let

1.1 P(x) = a1x + a2x2 + … + amxm (am ≠ 0).

Many combinatorial problems can be reduced to the study of numbers Bn generated by

1.2.

Some problems of this type were treated by Touchard (7), Jacobsthal (3), Chowla, Herstein, Moore and Scott (1; 2), and the present authors (4).

1. Introduction. This paper is concerned with problems connected with the permutations of the integers 1, 2, … , n subject to certain special restrictions. One such class of problems, the so-called “card matching” problems, deals with conditions of the type, “the number i is in the jth position,” “the number k is in the mth position,” etc.

1. Introduction. Let Q be a matrix of order v, all of whose entries are 0's and l's. Let the total number of l's in Q be t, and let the absolute value of the determinant of Q be denoted by |det Q|. In this paper we study the problem of determining the maximum of |det Q| for fixed t and v. It turns out that this problem is closely related to the v, k, λ problem, which has been extensively studied of late.

Introduction. In the study of almost alternative algebras (2) relative to quasiequivalence an important class called algebras of (γ, δ) type arises. An algebra of (γ, δ) type is a finite dimensional algebra over a field satisfying the identities

1,

and

2

where γ and δ are elements of satisfying γ2 − δ2 + δ = 1.

1. Introduction. We consider the group of proper orthogonal transformations (rotations) in three-dimensional Euclidean space, represented by real orthogonal matrices (aik) (i, k = 1,2,3) with determinant + 1 . It is known that this rotation group contains free (non-abelian) subgroups; in fact Hausdorff (5) showed how to find two rotations P and Q generating a group with only two non-trivial relations

P2 = Q3 = I.

1.1 Baer has shown (1) that if the fact that the exponent of a group is m (that is, m is the least common multiple of the periods of the elements) implies a limitation on the class of the group, then m must be a prime. Graham Higman has extended this result by proving (3) that for any given integer M there are at most a finite number of prime powers q other than primes, such that the fact that a group has exponent q implies a limitation on the class of the Mth derived subgroup.

Various problems concerning the general theory of centralizers of modules which are not assumed to be completely reducible have been discussed by Fitting (3), Brauer (2), and Nakayama. In this paper we present a new approach to some of these questions, which has its origin in Weyl's discussion (15) of the centralizer of a finite group of collineations.

Bambah (1) has recently determined the most economical covering of three dimensional space by equal spheres whose centres form a lattice, the density of this covering being

1.1.

As is well known, this problem may be interpreted in terms of the inhomogeneous minimum of a positive definite quadratic form.

Introduction. Les nombres entiers positifs a0, a1, … , an, … définis par la fonction génératrice

et que l'on appelle nombres exponentiels jouent, pour la sommation de certaines séries, un rôle qui rappelle le rôle sommatoire des nombres de Bernoulli. Nous avons rassemblé ici les principales propriétés des nombres an dont plusieurs sont, croyons nous, nouvelles.

In the preceding paper Touchard considers a set of polynomials Qn(x) defined by

1.

Touchard uses (1) to compute Qn(x) for 0 ≤ n ≤ 9 and also finds Qn(-½).

He remarks however “l'expression générale des polynômes Qn(x) nous echappe.”

The object of this note is to derive an explicit expression for Qn(x).

Boolean rings (B, ✗, + ) and Boolean logics ( = Boolean algebras) (B, ∩, *) are equationally interdefinable in a familiar way (6). Foster's theory of ring-logics (1; 2; 3) raises this interdefinability and indeed the entire Boolean theory to a more general level. In this theory a ring (or an algebra) R is studied modulo K, where K is an arbitrary transformation group (or “Coordinate transformations”) in R.

We use the following notation throughout:

.

1. Introduction. A ring is defined to be standard (1) in case the following two identities hold :

(1) (wx, y, z) + (xz, y, w) + (wz, y, x) = 0,

(2) (x, y, z) + (z, x, y) − (x, z, y) = 0,

where the associator (x, y, z) is defined by (x, y, z) = (xy)z − x(yz). Albert has determined the structure of finite-dimensional, standard algebras (1).

The simple ones turn out to be either Jordan algebras or associative ones.

1. Introduction. In a well-known paper (7) Jacobson has shown how his structure theory for arbitrary rings can be applied to give more precise information about the so-called “algebraic” algebras. This specialization of his general theory is, however, perhaps not completely satisfying in that it deals only with algebras, i.e. rings admitting a field of operators, whereas neither the general structure theory nor the definition of the property of being “algebraic” seems to depend in any essential way on the precise nature of the operators.

Introduction. Let R be a ring and let R[x] be the ring of all polynomials in a commutative indeterminate x over R. Let J(R) denote the Jacobson radical (5) of the ring R and let L(R) be the lower radical (4) of R. The main object of the present note is to determine the radicals J(R[x]) and L(R[x]). The Jacobson radical J(R[x]) is shown to be a polynomial ring N[x] over a nil ideal N of R and the lower radical L(R[x]) is the polynomial ring L(R)[x].