Le théoréme qui suit résoud une question qui m'avait été posée indépendamment par M. H. Mirkil et Professeur E. Farah, dans le cas particulier de p = 2 :

Théorème 1. Soit M un espace localement compact muni d'une mesure bornée μ, et soit 1 ≤ p < + ∞. Soit H un sous-espace vectoriel de L∞(μ), fermé dans LP(μ). Alors H est de dimension finie.

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

.

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.

α1α2, …, αp are n-dimensional vectors,

,;

they are arranged to form a closed polygon

.

Denote by R(α1, α2, …, αp) the radius of the smallest circumscribed hypersphere with centre at 0 ; by R(α1, α2, …, αp) the minimum of

(α1, α2, …, αp)

for all possible reorderings

of α2, …, αp−1 and by cn the least possible constant such that

for all possible choices of p and α1α2, … , αp.

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.

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.

In the theory of the asymptotic solution or stability of ordinary differential equations most attention has been given to linear or nearly-linear cases. Investigations in this field, starting primarily with those of Kneser (7) on the equation y″ + f(x)y = 0, have by now mostly been summed up in results on the vector-matrix system dy/dx = Ay + f (y, x), where y and f denote n-vectors of functions, and A an n-by-n matrix, frequently assumed constant.

1. Introduction. Under the non-parametric assumption that a set of observations is a sample from an absolutely continuous distribution, the order statistics are known to form a complete sufficient statistic. It is proved in this note that it suffices to have the class of uniform distributions over finite numbers of intervals or the class of uniform distributions over sets of a ring which is a basis for the σ-algebra of Borel sets. This result is derived as a particular case of that of several samples from more general distributions.

We consider the problem of finding a unique canonical form for complex matrices under unitary transformation, the analogue of the Jordan form (1, p. 305, §3), and of determining the transforming unitary matrix (1, p. 298, 1. 2). The term “canonical form” appears in the literature with different meanings. It might mean merely a general pattern as a triangular form (the Jacobi canonical form (8, p. 64)). Again it might mean a certain matrix which can be obtained from a given matrix only by following a specific set of instructions (1). More generally, and this is the sense in which we take it, it might mean a form that can actually be described, which is independent of the method used to obtain it, and with the property that any two matrices in this form which are unitarily equivalent are identical.

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.

In earlier papers (4; 5; 6), certain space curves, invariant under cyclic involutions of periods three, five, and seven, have been mapped. Lucien Godeaux (2; 3) in 1916 mapped plane cubic curves, invariant under an involution of period three, onto a cubic surface in ordinary three-space. Mlle. J. Dessart (1) in 1931 mapped plane quintic curves, invariant under an involution of order five, onto a fifth order surface in a space of four dimensions.

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).

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.

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.

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.

Baer [2] and Neumann [5] have discussed groups in which there is a limitation on the number of conjugates which an element may have. For a given group G, let H1 be the set of all elements of G which have only a finite number of conjugates in G, let H2 be the set of those elements of G, the conjugates of each of which lie in only a finite number of cosets of H1 in G; and in this fashion define H3, H4, …. We shall show that the Hi are strictly characteristic subgroups of G.

Kothe spaces, in the terminology of Diendonné [2], are certain spaces X of real valued integrable functions. In this paper we consider the problem of representation of continuous linear functional on vector valued Kothe spaces. The elements of a Kôthe space X(B) are functions with values in a Banach space B (see §2).

In a paper recently published in this journal [1], R. E. Graves proved a closure criterion for orthonormal sets of functions. A refined form of it may be stated as follows :

A method of proving the impossibility of certain Affine Resolvable Balanced Incomplete Block Designs (A.R.B.I.B.D.) has been given by the author elsewhere [9]. More complete results in the same direction are obtained here using the ideas of a paper by Connor [4].

The independence of postulates for well-known systems is a question of general interest. A closely related question is whether or not, by altering one or more of the postulates in an independent set, the set remains independent. From this standpoint the best sets of postulates are those which involve, first, the fewest postulates and, next, the least number of variables. As a rule progress is made in this direction at the sacrifice of simplicity of postulates. In this paper, in counting postulates, we ignore properties such as closure under the operations and count only identities or those stating that one equation implies another.