La première partie de cet article a un caractère introductif; on reprend l'étude des mesures vectorielles sur un espace localement compact, présenté dans (8) pour les espaces compacts.

Dans la seconde partie on donne un théorème de représentation intégrale d'une mesure vectorielle par rapport à sa variation (théorème 2), duquel on déduit une généralisation du théorème de Lebesgue-Nikodym (théorème 3) et une généralisation du théorème de Lebesgue sur la décomposition d'une mesure (théorème 5). Ces théorèmes ont été déjà démontrés par les auteurs sous des conditions un peu plus restrictives, (10) et (13), mais ici les démonstrations sont nouvelles, et l'ordre dont on les déduit l'un de l'autre, est inversée.

In 1949 I published in the Acta Mathematica (vol. 81) a rather long paper: “L'intégrale de Riemann-Liouville et le problème de Cauchy.” This work will be quoted in the sequel as Acta paper. Only minor local references to this paper will be made here, and knowledge of it is not required for the reading of the present article. The notations used here are slightly different from those used in my former paper.

In the Acta paper I introduce multiple integrals and of the Riemann- Liouville type depending on a parameter α and converging for sufficiently large values of α. I give the solution of the Cauchy problem for the wave equation in a unique formula, the same for space-time of odd or even dimensions, implying an analytic continuation with respect to the parameter α.

Let ∧ be a lattice in Euclidean n-space, that is, ∧ is a set of points ε1a1 + … + εnan where a1… , an are linearly independent vectors and the ε run over all integers. Let μ denote the Lebesgue measure. A closed convex set F is called a fundamental region for ∧ if the sets F + x (x ∈∧) cover the whole space without overlapping; that is, if F0 is the interior of F, and 0 ≠ x ∈ ∧, then F0 ∩ (F0 + x) = ϕ.

A number of combinatorial problems may be regarded as particular instances of the following rather general situation. Given a set X composed of n elements x1, x2, ..., xn, and m subsets X1 X2, … , Xm of X, find a minimal system of representatives for X1, X2, … , Xm. That is, single out a subset X* of X such that Xi ∩ X* is non-empty for i = 1,2, … ,m, and no subset of X containing fewer elements than X* has this property. To illustrate, each of the following can be thought of in these terms.

In the classification of almost alternative algebras relative quasiequivalence by Albert two new classes of algebras of type (γ, δ) were introduced, namely those of type (1, 1) and ( —1, 0) (1, equations (34), (35), and Theorem 6). Since rings of type (1, 1) and ( — 1, 0) are anti-isomorphic it suffices to consider those of type (1, 1). They may be defined as rings satisfying

1

and

2

for all elements x, y, and z of the ring, where the associator (a, b, c) is given by (a, b, c) = (ab)c — a (bc).

Actually the identity

2'

together with (1) implies (2) whenever the characteristic of the ring is different from 2. This may readily be verified by linearizing (2’). Consequently we may use (1) and (2’) as the defining relations for a ring of type (1, 1).

Les propriétés anallagmatiques de groupes de sphères ont été étudiées dans des contextes divers ces dernières années (voir (4), (5)). Dans la note qui suit nous étudions les propriétés anallagmatiques de certains groupes de 4 sphères, nous plaçant à un point de vue élémentaire.

A fin de ne pas alourdir la rédaction nous omettrons de spécifier chaque fois que les points, droites ou sphères considérés sont toujours supposés être dans la position relative la plus générale possible compatible avec les définitions données.

L'index i pourra toujours prendre les valeurs i = 1, 2, 3, 4.

Throughout this discussion R will be an integral domain with quotient field Q and K = Q/R ≠ 0. If A is an R-module, then A is said to be torsion-free (resp. divisible), if for every r ≠ 0 ∈ R the endomorphism of A defined by x → rx, x ∈ A, is a monomorphism (resp. epimorphism). If A is torsion-free, the rank of A is defined to be the dimension over Q of the vector space A ⊗R Q; (we note that a torsion-free R-module of rank one is the same thing as a non-zero R-submodule of Q). A will be said to be indecomposable, if A has no proper, non-zero, direct summands. We shall say that A has D.C.C., if A satisfies the descending chain condition for submodules. By dim R we shall mean the maximal length of a chain of prime ideals in R.

In this paper we shall be concerned with two problems: (i) the asymptotic behavior of solutions of parabolic inequalities and (ii) the uniqueness of the Cauchy problem for such inequalities when the data are prescribed on a portion of a time-like surface. The unifying feature of these rather separate problems is the employment of integral estimates of the same type in both cases.

We consider parabolic operators in self-adjoint form

(1)

as well as the non-self-adjoint operator

(2)

where the coefficients aij(x, t) = aij (x1, x2... , xn, t) are C1 functions of x and t and the bij= bij(x, t) are C2 functions of x and t.

Various results exist which permit real Banach algebras satisfying some sort of “reality condition” to be identified with the algebra of all continuous real-valued functions on a suitable compact space (or with the algebra of all continuous real-valued functions that * Vanish at infinity“ on a suitable non-compact, locally compact space in case the algebra has no unit). (Terminology follows (4), so that compact and locally compact spaces must be Hausdorff spaces, in addition to satisfying the usual requirements.) Kadison established such a result in (6, Theorem 6.6) for any Banach algebra with unit, if the algebra satisfies an appropriate reality condition and has a norm N such that N(x)2 — N(x2) for all x.

The basic problem of linear programming is to minimize (or maximize) a linear function of a finite number of variables constrained by a finite number of linear inequalities. From a mathematical point of view the subject may be regarded as being divided into two areas. One is primarily analytical and deals with certain questions of duality and consistency. The other is algorithmic and is concerned with computational questions and methods.

We will consider the following enumeration problem. Let A and B be finite sets with α and β elements in each set respectively. Let n be some positive integer such that n ≦ αβ. A subset S of the product set A × B of exactly n distinct ordered pairs (ai, bj) is said to be admissible if given any a ∈ A and b ∈ B, there exist elements (ai, bj) and (ak, bl) (they may be the same) in S such that ai = a and bl = b. We shall find here a generating function for the number N(α, β n) of distinct admissible subsets of A × B and from this generating function, an explicit expression for N(α, β n). In obtaining this result, the idea of a cut probability is used. This approach in a problem of enumeration may be of interest.

A classical theorem states that if a square matrix B over an algebraically closed field F commutes with all matrices X over F which commute with a matrix A over F, then B must be a polynomial in A with coefficients in F (2). Recently Marcus and Khan (1) generalized this theorem to double commutators. Our purpose is to complete the generalization to commutators of any order.

Let F be an algebraically closed field and let Fn be the ring of all n by n matrices with elements in F. We define ΔYZ — = [Z, Y] = ZY — YZ for all Y, Z in Fn.

This note can be regarded as an addendum to the paper (4). On the complex Hilbert space of vectors x = (x1, x2, … ,) a matrix A is said to be bounded if there exists a constant M such that ||Ax|| ≦ M||x|| whenever ||x||2 = Σ|xk|2 < ∞ ; the least such M is denoted by ||A||. Only bounded matrices A and vectors x satisfying ||x|| < ∞ will be considered in the sequel. The spectrum of A, denoted by sp(A), is the set of values for which the resolvent R(λ) = (A — λI)-1 fails to be bounded. The notation A ≥ 0 or A > 0, where A = (aij), means that, for all i and j , aij ≥ 0 or aij 0 respectively.

In a previous paper (18), G = F/Fn was studied for F a free product of a finite number of cyclic groups, and Fn the normal subgroup generated by commutators of weight n. In that paper the following cases were completely treated:

(a) F a free product of cyclic groups of order pαi, p a prime, αi positive integers, and n = 4, 5, … , p + 1.

(b) F a free product of cyclic groups of order 2αi, and n = 4.

In this paper, the following case is completely treated:

(c) F a free product of cyclic groups of order pαi p a prime, αi positive integers, and n = p + 2.

(Note that n = 2 is well known, and n — 3 was studied by Golovin (2).) By ‘'completely treated” is meant: a unique representation of elements of the group is given, and the order of the group is indicated. In the case of n = 4, a multiplication table was given.

The problem of expanding an arbitrary function in a series of characteristic solutions of the ordinary differential equation

(1.1)

and the boundary relations

(1.2)

is well known. The various discussions are distinguished by the manner in which a parameter λ appears in the differential system and by the number of points at which the boundary conditions apply. The case in which the boundary conditions apply at intermediate as well as at the end points of a fundamental interval has been considered by Wilder (3).

This paper deals with finite sequences of integers. Typical of the problems we shall treat is the determination of the number of sequences of length n, consisting of the integers 1, 2, … , m, which have a longest increasing subsequence of length α. Throughout the first part of the paper we will deal only with sequences in which no numbers are repeated. In the second part we will extend the results to include the possibility of repetition. Our results will be stated in terms of standard Young tableaux.

The unboundedness of the sequence of Lebesgue constants (norms), at a point, of certain transforms implies, as is well known, that there exist (i) a continuous function whose transform fails to converge to the function at the point in question (the du Bois-Reymond singularity), and (ii) another such function whose transform, while converging everywhere to the function, does not do so uniformly in any neighbourhood of the stipulated point (the Lebesgue singularity). The converses also hold in our case.

Let f be a function on the product space V × W, where V and W are analytic manifolds, both either real or complex. The function f is said to be analytic (or bi-analytic) on V × W if it is analytic in the analytic structure induced on V × W by the corresponding structures on V and W. The function f is said to be separately analytic on V × W if, for each x in V, the function f(x, .) is analytic on W while, for each y in W, the function f (. ,y) is analytic on V. In the case of complex analytic manifolds, the classical theorem of Hartogs (3, chapter VII) states that the two notions of analyticity and separate analyticity are equivalent. For real analytic manifolds, it is known that such an equivalence does not hold, even if one adds the additional hypothesis that f is infinitely differentiate on V × W.

This paper deals with processes of combining convex bodies in Euclidean n-space which are, in a sense, dual to the process of Minkowski addition and some of its generalizations.

All the convex bodies considered will have a common interior point Q. Variables x and y denote vectors drawn from Q; we shall speak of their terminal points as the points x and y. Unit vectors will be denoted by u; ||x|| signifies the length of x. Convex bodies will be symbolized by K with distinguishing marks. ∂K means the boundary of K. λK will mean the image of K under a homothetic transformation in the ratio λ : 1. The centre of the homothety will always be Q.

Let C be a square matrix with complex elements. If C = C' (C' denotes the transpose of C) there exists a unitary matrix U such that

(1)

where the μ's are the non-negative square roots of the eigenvalues μ12, μ22, … , μn2 of C*C (C* is the adjoint of C) (2). If C is skew-symmetric, that is, C= — C”, there exists a unitary matrix U such that

(2)

where

(3)

and the α's are the positive square roots of the non-zero eigenvalues α12, α22, … , αk2 of C*C (1). Clearly rank C = 2k and the number of zeros appearing in (2) is n — 2k. Both (1) and (2) are classical. In a recent paper (3) Stander and Wiegman, apparently unaware of (1), give an alternative derivation of (2) with its appropriate generalization to quaternions.