1. Introduction. In a study of the commuting algebra of tensor space representations of the orthogonal group W. P. Brown encountered a class of algebras for which the existence of a unity element was equivalent to semisimplicity, but which were of interest whether or not semisimple. He gave these algebras the name generalized-total matrix algebras and proved (2) that each such algebra was characterized by three integers l, r, m and was isomorphic to the algebra of all square matrices of degree r + I + m which have zeros in the first l rows and in the last r columns.

1. Introduction. It has been observed (2) that the number of p-regular classes of Sn, i.e. the number of classes of order prime to p, is equal to the number of partitions (λ) of n in which no summand is repeated p or more times. For this relation to hold it is essential that p be prime. It seems natural to call the Young diagram [λ] associated with (λ) p-regular if no p of its rows are of equal length, otherwise p-singular.

Introduction. An element x is said to be right-quasi-regular (r.q.r.) if there exists an element y such that x + y + xy = 0. This concept had its inception in the fact that (for rings with unity) if 1 + x has an inverse, written as 1 + y, then (1 + x)(l + y) = 1, x + y + xy = 0. Thus in rings without unity elements it seemed (1; 3; 12) profitable to consider this latter equation.

In (1) and (2) we proved that under certain conditions a given ring R must be commutative. The conditions used there were “global” in the sense that they were imposed at once on the relation of a given element to all the other elements of the ring R.

1. Let be a finite projective plane (8, §17), i.e. a projective space of dimension 2 over a Galois field γ. We suppose that γ has characteristic p ≠ 2, hence order q = pn, where p is an odd prime and h is a positive integer. It is well known that every straight line and every non-singular conic of then contains q + 1 points exactly.

Consider a projective plane with n + 1 points on each line, An oval is a set of n + 1 points, no three of which are collinear.

DEFINITION. (1) A line which contains two points of will be called a secant of .

(2) A line which contains exactly one point of will be called a tangent of or an absolute line.

A classical theorem of n-dimensional projective geometry asserts: if Desargues' theorem holds (in particular, if n ≥ 3) then the points of the geometry can be assigned homogeneous coordinates (α1, … , αn+l) with αi not all zero in a suitable division ring (2, p. 104; 3, Theorems 10, 11, ex. 19, p. 204).

In this paper, results obtained by the author for Riemannian Spaces Vn imbedded in Euclidean Spaces EN (3; 4) are extended to Vn imbedded in VN.

The first section is introductory. In §2 the general result is obtained. This is the establishment of a certain dependency among the three basic sets of equations of the Vn with respect to the VN, namely the equations of Gauss, Codazzi and Kuehne.

Let be the class of bounded non-decreasing functions defined on the real line which are normalized by the conditions ϕ(− ∞) = 0 , ϕ(t + 0) = ϕ(t).

Let be the class of Fourier-Stieltjes transforms of elements of i.e. the elements of and are connected by the relation

where ϕ ∊ and Φ ∊ .It is well known, and easy to verify that this mapping from to is one to one (1, p. 67, Satz 18).

1. Introduction. Let , n = 0, 1, … , be the normalized Legendre polynomials. If and if

then we write

.

1. Introduction. In this paper, infinite fields K will be discussed which satisfy the following condition :

CONDITION C: for any polynomial p(t, x) in x, coefficients in K(t), t transcendental with respect to K, which has no zeros in K (t), there is a t* in K for which p(t*, x) has no zeros in K.

We shall consider here two generalizations to rings of the concept of induced representation as it occurs in the representation theory of finite groups (6).

If A is a ring, S a subring of A, we shall associate with each S-module M an induced pair (I(M), K) consisting of an A -modulo I(M) and an S-homomorphism K: M → I(M).

Introduction. The present note began from the observation that the arguments produced by J-M. Maranda in developing his very interesting theory of representations of groups by automorphisms of modules over Dedekind rings (4, 5) were applicable without essential change to arbitrary orders, instead of just group rings, provided that a suitable generalization of Theorem 1 of (4) could be supplied.

Let t be a Dedekind ring whose quotient field we denote by K. If is a prime ideal of t, let o denote the ring of all -regular elements of K. If is a torsion free i-module, let K denote the smallest K-module into which can be embedded and let o denote the o-submodule of K generated by .

1. Introduction. In one of his papers on limits for the characteristic roots of a Matrix, Brauer (1) has stated a theorem, which connects the roots of a given square matrix A, with those of a matrix A* derived from A by a certain process. The proof of this theorem involves a continuity argument and in a recent paper on the construction of stochastic matrices Hazel Perfect (5) has given a proof which avoids considerations of continuity.

1. Introduction. We consider first in §2 the asymptotic behavior as t → ∞ of the solutions of the vector-matrix differential equation

(1.1) ,

where A is a constant n-square complex matrix, B{t) a continuous complex valued n-square matrix defined on [0, ∞ ), and x a complex n-vector.

In classical analysis ordinary differential equations and partial differential equations are distinct concepts, and the transition from one derivation to several partial derivations changes some of their properties distinctly. On the other hand, the algebraic theories of modified ordinary and partial differential equations (5; 6), based on the differentiations in the sense of Hasse (2) and Schmidt (3) and the multidifferentiations in the sense of Jaeger (4), turn out to be strikingly similar in the case of fields of prime number characteristic.

1. Introduction. A linearly ordered space S in which neighborhoods are segments is called a Souslin space if

(i) S is not separable, but

(ii) every collection of disjoint segments of S is countable.

Whether a Souslin space exists is not known; this is the problem referred to in the title and was proposed by Souslin in (2).

An open interval of a simply ordered set S is a subset I of S such that either

(1) for some ,

(2) for some , or

(3) for some .

A simply ordered set with its interval topology (i.e., the topology in which “neighborhood of x” means “open interval containing x”) will be called an ordered space.

Soit C un espace de Banach (nous ne parlerons que d'espaces sur le corps des nombres réels, pour simplifier). On dit que C possède la propriété de Nachbin ou est un espace (N), si pour toute application linéaire continue u dans C d'un sous-espace vectoriel F d'un espace de Banach E, il existe une application linéaire de même norme de E dans C qui prolonge u.