We consider the product of two power series with positive coefficients: .

What conditions will ensure that the coefficients wn shall be either (i) monotonic, or (ii) logarithmically convex? By the latter, we mean that wn2 ≤ Wn-1Wn+1 for n = 1, 2, … In investigating this question, which was suggested by a special example, we have found it convenient to express the conditions in terms of the ratios of un, vn to certain binomial coefficients, rather than in terms of un, vn themselves.

Nous nous occupons dans ce travail d'une classe de surfaces dont l'équation différentielle est du cinquième ordre, et qui jouissent de la propriété que, sur elles, les lignes d'égale courbure moyenne forment, avec leurs trajectoires orthogonales, un système isotherme. Dans cette classe rentrent les surfaces admettant un groupe de déplacements à un paramètre (cylindres, surfaces de révolution et hélicoïsian Bonnet).

Modern physical theory, both classical and quantal, faces serious difficulties which arise from the divergence of certain integrals. Perhaps the best known of these “infinities” is the self-energy of the point electron. Most of the simpler devices used to eliminate the infinities, such as the introduction of a finite electron radius, are non-relativistic and must therefore be rejected. Relativistic theories1 which do avoid some or all of the infinities are very complicated and often suffer from difficulty in physical interpretation.

The following note which deals with the effect of a certain determinantal operator when it acts upon a product of determinants was suggested by the original proof which Dr. Alfred Young gave of the property

subsisting between the positive P and the negative N substitutional operators, θ being a positive integer. This result which establishes the idempotency of the expression θ−1NP within an appropriate algebra is fundamental in the Quantitative Substitutional Analysis that Young developed.

A function η(i1, …, if) of ƒ quantities i, varying over the finite range i = 1,2, …, n, is usually called an n-dimensional tensor of rank ƒ. Any permutation p: 1 → 1, … , ƒ → ƒ… changes this tensor into a tensor pη according to the equation pη(i1, ...,if) = η(i1', ...,if')Thus the permutation p appears as a linear operator p in the n-dimensional space Σ = Σn,ƒ of all n-dimensional tensors of rank ƒ, η is symmetric if pη = η for all permutations p, it is antisymmetric if pη δp.η where δp = + 1 for the even and — 1 for the odd permutations.

Every proper orthogonal matrix A can be written

where Q is a skew matrix [6], and conversely every such matrix A is orthogonal. It is also known that every proper orthogonal transformation in real Euclidean four-space may be characterized in term of quaternions [1, 3] by the equation

determines with the origin a vector having the coordinates (XQ, XI, x2, x3). The relationship between these two representations was clearly shown by Murnaghan [5].

Since the classical work of Minkowski and Jensen it is well known that many of the inequalities used in analysis may be considered as consequences of the convexity of certain functions. In several of these inequalities pairs of “conjugate” functions occur, for instance pairs of powers with exponents a and a related by 1/a + 1/a = 1. A more general example is the pair of positively homogeneous convex functions denned by Minkowski and known as the distance (or gauge) function and the function of support of a convex body. The purpose of the present paper is to explain the general (by the way rather elementary) idea underlying this correspondence.

In this note, I shall establish necessary and sufficient conditions for the existence of critical lattices of an arbitrary point set, and I shall construct a non-trivial example of a point set without any critical lattice. In a previous paper, I proved that every star body of the finite type possesses at least one critical lattice.

A projective plane geometry π is a mathematical system composed of undefined elements called points and undefined sets of points (at least two in number) called lines, subject to the following three postulates:

(P1) Two distinct points are contained in a unique line.

(P2) Two distinct lines contain a unique common point.

(P3) Each line contains at least three points.

During the last fifty years, the concept of the Euclidean space (an ndimensional coordinate space with a Pythagorean distance) has undergone various profound generalizations.

Hilbert introduced the infinitely-dimensional Euclidean space whose points are infinite sequences of coordinates having from the origin, and thus from each other, finite Pythagorean distances.

Let G be a compact group. According to the celebrated theorem of Peter-Weyl there exists a complete set of finite-dimensional irreducible unitary representations of G, the completeness meaning that for any group element other than the identity there is a representation sending it into a matrix other than the unit matrix. If G is commutative, the representations are necessarily one-dimensional. It is an immediate consequence of the Peter-Weyl theorem that the converse also holds: if every representation is one-dimensional, G is commutative. The main theorem in the present paper is a generalization of this result to the case where the representations have bounded degree.