Dans un précédent article [6, §4] j'ai essayé de traiter le problème suivant, qui fait l'objet du présent travail: on donne 2n abscisses, marquées 1 , 2 , … , 2n, de gauche à droite, sur un axe horizontal. On les joint deux à deux par n arcs convexes, tracés au-dessus de l'axe, de manière que chaque abscisse soit l'origine ou l'extrémité d'un seul arc, l'origine étant à gauche et l'extrémité à droite. On obtient ainsi p2n = 1.3.5. … (2n — 1) configurations et on demande le nombre de celles qui ont p points doubles.

In an earlier paper [5], the author defined a zeta function on the real sphere , whereas in the present paper it is proposed to define one on the unitary sphere where xi's are complex numbers and their complex conjugates. Following E. Cartan, harmonics on the unitary sphere are defined and then a zeta function formed just as in the case of a real sphere. The unitary sphere is seen to behave like an even-dimensional closed manifold, since results similar to the ones proved by the author and A. Pleijel [6] for closed manifolds (of even dimensions) are observed here also.

The problem of determining the conditions under which a finite set of matrices A1A2, … , Ak has the property that their characteristic roots λ1j, λ2j, … , λki (j = 1, 2, …, n) may be so ordered that every polynomial f(A1A2 … , Ak) in these matrices has characteristic roots f(λ1j, λ2j …,λki) (j = 1, 2, … , n) was first considered by Frobenius [4]. He showed that a sufficient condition for the (Ai〉 to have this property is that they be commutative. It may be shown by an example that this condition is not necessary.

J. Williamson [9] considered this problem for two matrices under the restriction that one of them be non-derogatory. He then showed that a necessary and sufficient condition that these two matrices have the above property is that they satisfy a certain finite set of matric equations.

The well-known results of Krull concerning the minimal prime divisors and the radical of an ideal in a commutative ring have been extended to the noncommutative case in a recent paper [5] by N. H. McCoy. In that paper systematic use was made of the concept of an m-system, a set M of elements of the ring such that if a ∈ M and b ∈ M then axb ∈ M for some element x of the ring.

A real number, expressed as a decimal, is said to be normal (in the scale of 10) if every combination of digits occurs in the decimal with the proper frequency. If a1a2 … ak is any combination of k digits, and N(t) is the number of times this combination occurs among the first t digits, the condition is that

1

Let = {A1, … , As}, = {B1 … , Bt} be sets of elements of a group of finite order g. We define

By (), (), … we shall denote the number of elements in , , … respectively and by , , … the sets of elements of not in ,, ….

In [3], H. Rademacher obtained a convergent series for the Fourier coefficients of the modular invariant J(τ). He found that in the expansion

Until recently the literature contained little on the axiomatic foundations of elliptic geometry that was non-analytical and independent of projective geometry. During the past decade this subject has come in for further study, notably by Busemann [2] and Blumenthal [1], who supplied such foundations. This paper presents another and, it is believed, simpler effort in the same general direction, proceeding by the familiar synthetic methods of elementary geometry and using only elementary topological notions and ideas concerning metric spaces. Specifically, elliptic 2-space is obtained on the basis of six axioms, most notable of which is one assuming the existence of translations. The writer wishes to express his deep appreciation to Herbert Busemann for his invaluable help.

The geometry of slides and turns of oriented lineal elements in the plane was first studied by Kasner [10]. Slides and turns generate whirls, which constitute a three-parameter group W3. The product of W3 and M3, the three-parameter group of Euclidean displacements in the plane, yields a sixparameter group of whirl-motionsG6. The geometry of turbines, and also of general series of lineal elements, under G6 was investigated by Kasner in [10] and, in subsequent papers, by Kasner and DeCicco, particularly in [3], [4], [11], [12].

In an earlier paper [3] I studied the property С and related properties С′ and С; but the principal problem, viz, to prove, with the axiom of choice only (without any other hypothesis), the existence of a nondenumerable set of property С, remains open.

A family F of functions is said to have a denumerable base if there exists a sequence of functions {fn(x)} (not necessarily ∈ F) such that any function f ∈ F is the limit of a subsequence of {fn (x)}. The domain X of a function f(x) is the set of x´s for which f(x) is defined ; we say f(x) is a function on X. A dyadic function is a function taking only the values 0 and 1.

Let f be a ternary quadratic form whose matrix F has integral elements with g.c.d. 1, that is, an improperly or properly primitive form according as all diagonal elements are even or not. Let d be the determinant of f (denoted by ) Ω, the g.c.d. of the 2-rowed minors of F. Then d = Ω2 Δ determines an integer Δ. Two forms f in the same genus have the same invariants Ω, Δ, d. The form whose matrix is adj F/Ω, is called the reciprocal form of f.

The only known simple alternative rings which are not associative are the Cayley algebras. Every such algebra has a scalar extension which is isomorphic over its center F to the algebra where . The elements e11 and e00 are orthogonal idempotents an , for every xij of . The multiplication table of C is then completed by the relations

Let F(z, u) denote

1

where F1(z),… , Fn(z) are rational functions of z with complex coefficients. We shall speak of F (z, u) = 0 as the fundamental algebraic equation and shall adopt z as the independent variable and u as the dependent, except in § 4, where we use x and y instead of them, and where it is understood that x and y are connected birationally with z and u.

In his work on Laplace and Stieltjes transforms Widder [6, ch. 8] has investigated relationships of the type

1

2

3

In this paper we deal with a group-theoretic configuration of the following type: G is an additive group (not necessarily abelian) for which an operator system M and a complete lattice ø of M admissible subgroups are defined; we call G and M-ø group. In §1 we make various definitions and note that analogues of some of the classical theorems of group theory hold.

Dans un espace finslérien à n dimensions rapporté à un système quelconque de coordonnées x1, x2 …, xn la distance de deux points infiniment voisins est donnée par la formule

1

In this paper we give a simple, necessary, and sufficient condition for a sequence of orthogonal functions to be closed in L2. In theory the question of closure is reduced to the evaluation of certain integrals and the summation of an infinite series whose terms depend only upon the index n. Our principal result is

Let p(t) be a function whose zeros and discontinuities have Jordan content zero, such that for each x ∊ (a, b), p(t) ∊ L2 on min (c, x) < t < max (c, x), where a ≤ c ≤ b. (a, b, and c may be infinite.)

On a surface F of genus pg = pa = p and linear genus p(1) = n + 1 whose canonical system is irreducible, and which we shall ordinarily think of as simple and free from exceptional curves, the characteristic series of the canonical system is a semicanonical since the adjoint system of the canonical system is its double, so that the canonical series on a curve of the canonical system is its characteristic series doubled.

A set a1 …, ak of different residues mod v is called a difference set (v, k, λ) (v>k > λ) if the congruence ai — aj ≡ d (mod v) has exactly λ solutions for d ≢ 0 (mod v). Singer [4] has demonstrated the existence of a difference set (v, k, 1) if k — 1 is a prime power, and difference sets for λ > 1 have been constructed by various authors; but necessary and sufficient conditions for the existence of a (v, k, λ) are not known. It has not been possible so far to find a difference set with λ = 1 if k — 1 is not a prime power and it has therefore been conjectured that no such difference set exists.