Our aim is to prove: If a, b, c are positive integers, if ab > 1, if a, b, c are relatively prime in pairs, and all are free of squares, if — ab is a quadratic residue of c, bc of a, ca of b, and if F(x, y, z) = ax2 + by2 — cz2, we have non-trivial solutions of

with the inequalities

Professor Dresden called to our attention the following theorem :

If S1, S2,… , Sm are m line segments parallel to the y-axis, all of equal lengths, whose projections on the x-axis are equally spaced, and if we assume that a straight line can be made to intersect every set of three among these segments, then there exists a straight line intersecting all the segments.

In a recent paper we studied systems of equations of the form

(1)

(2)

where as usual [a,b] = ab — ba and ϕ(λ) is a polynomial. Equations of this type have arisen in quantum mechanics. In our paper we gave a method of determining the matrix solutions of such equations. The starting point of our discussion was the observation that if the elements xi satisfy (1) then the elements xi , [xj ,xk ] satisfy the multiplication table of a certain basis of the Lie algebra of skew symmetric (n + 1) ⨯ (n + 1) matrices. We proved that if (2) is imposed as an added condition, then the algebra generated by the has a finite basis, and we obtained the structure of the most general associative algebra that is generated in this way.

The fact that the perimeter S(a, b) of an ellipse is not an elementary function of its semiaxes a, b has led to many suggested approximations of S in finite form.

Summary of results. The principal result of this paper is as follows: givenany set of real numbers z1, z2, & , zn and an integer t we can find an integer and a set of integers p1, p2 & , pn such that

(0.11).

Also, if n = 2, we can, given t, produce numbers z1 and z2 such that

(0.12)

This supersedes the results of Nils Pipping (Acta Aboensis, vol. 13, no. 9, 1942)that there is a q satisfying (0.11) such that , and also theclassical result of Dirichlet that there is such a q less than tn.

1. Let m be a positive square-free integer. Euclid's Algorithm is said to holdin the field if, given any non-integral element a in the field, an integera can be found so that

The definition and some of the properties of what may be called a Perron second integral (P 2-integral) were given in a previous paper [4]. This integral starts with a function f(x) defined in an interval (a, c) and goes directly to a second primitive F(x) with the property that the generalized second derivative D 2 F is equal to f(x) for almost all x in (a, c). In the present paper the definition is changed slightly and further properties are deduced.

Introduction. Dans ses mémoires “Über zweidimensionale allgemeine metrische Räume” [1] et “On Finsler and Cartan Geometries III,” [3] Berwald a définie deux invariants fondamentaux des espaces finslériens à deux dimensions. Dans le présent travail nous allons établir une formule pour le scalaire de courbure et une pour le scalaire principal, qui définissent un invariant de l'espace finslérien à n dimensions, et qui sont identiques, pour deux dimensions, aux invarients définis par L. Berwald. Notre idée fondamentale sera d'exprimer le vecteur normal (hi, hi)—qui n'est défini qu'à deux dimensions—par le vecteur d'Euler, déjà défini pour n dimensions.

Conditions to be imposed on q(x) which ensure the uniqueness of the Green's function associated with the linear second-order differential equation

H. S. M. Coxeter has shown the existence of three infinite regular polyhedra, and has proved that there are no infinite regular polyhedra other than these. In his paper he gives the definition of regularity of a polyhedron :

A polyhedron is said to be regular if it possesses two particular symmetries: one which cyclically permutes the vertices ox any face c, and one which cyclically permutes the faces that meet at a vertex C, C being a vertex of c.

In a recent paper Robinson has obtained an explicit formula for the expression of an invariant matrix of an invariant matrix as a direct sum of invariant matrices. The object of the present note is to show that this formula may be deduced from known properties of Schur functions, with the aid of a result which the author has proved elsewhere.

1. Introduction. The problem of the expression of an invariant matrix of an invariant matrix as a direct sum of invariant matrices is intimately associated with the representation theory of the full linear group on the one hand and with the representation theory of the symmetric group on the other. In a previous paper the author gave an explicit formula for this reduction in terms of characters of the symmetric group. Later J. A. Todd derived the same formula using Schur functions, i.e. characters of representations of the full linear group.

1. Introduction. The Fréchet variation of a function g defined over a 2-interval I2 was introduced by Fréchet to enable him to generalize Riesz's theorem on the representation of functionals linear over the space C [7]. Recently the authors have found this variation fundamental in the study of functionals bilinear over the Cartesian product A ⨯ B of two normed linear spaces with certain characteristic properties, and in the further use of this theory in spectral and variational analysis. The recent discovery by the authors of several new properties of the Fréchet variation has made it possible to to give new and natural tests for the convergence of multiple Fourier series generalizing the classical Jordan, de la Vallée Poussin, Dini, Young and Lebesgue tests under considerably less restrictive hypotheses than those now accepted.

1. The problem. It has been shown [1] that if Xi, + X2, … are independent random variables each of density then

(1.1)

On demande le nombre Sn de maniéres dont on peut replier une bande de n timbres-poste (TP) sur un seul timbre. Dans le graphique de Sainte Lagüe (graphique SL) [3, p. 39], les liaisons entre timbres ne doivent pas se couper (fig. 1). Nous supposerons toujours que le timbre 1 occupe la premiére place à gauche, car si Pn est, dans ce cas, le nombre de permutations bonnes, on aura Sn = nPn. Nous supposerons de plus que la liaison (1, 2) est au-dessus de l'axe. Nous appellerons arc pair ou couple pair un arc (a, a + 1) où a est pair; arc impair ou couple impair un arc (a, a + 1) où est impair. Nous appellerons aussi permutation inverse de la permutation .

1. Introduction. It is the object of this paper to investigate the function γ(m), the number of representations of m in the form

(1)

where . It is shown that γ(m) is always equal to the number of odd divisors of m, so that for example γ(2k ) = 1, this representation being the number 2k itself. From this relationship the average order of γ(m) is deduced ; this result is given in Theorem 2. By a method due to Kac [2], it is shown in §3 that the number of positive integers for which γ(m) does not exceed a rather complicated function of n and ω, a real parameter, is asymptotically nD(ω), where D(ω) is the probability integral

1. Introduction. For define C(n) to be the integer j such that , where denotes the j th iterate of the Euler ø-function. Define C(l) = C(2) = 0. This function has been studied by S. S. Pillai [1], with the notation R(n) for 1 + C(n) if , and R(l) = 0. H. Shapiro [2] has also investigated this function, proving the basic relations

(1)

The problem of finding a lattice with a given abstract group of automorphisms has been solved by Garrett Birkhoff who proved that for any group of order g there exists a distributive lattice with at most elements. That this number can be somewhat reduced by modifications of Birkhoff's original procedure has already been shown by the author; it turns out, however, that it remains rather high for finite groups of relatively low order.

Jacobi proved a curious theorem regarding the solutions of the system ofequations

for functions λ=(x1, … , xn) satisfying