For positive integral n let Cn denote the n-dimensional unit cube with vertices (δ1, δ2,…, δn) where δi = 0 or 1 for i=1, 2,…, n. Call two vertices of Cn adjacent if the distance between them is 1.

A (round-robin) tournament Tn consists of n nodes p1, p2, …, Pn such that each pair of distinct nodes pi and pj is joined by one of the oriented arcs or . If the arc is in Tn, then we say that pi. dominates pj. The set of all dominance-preserving permutations a of the nodes T form a group, the automorphism group G(T ) of Tn. It is known (see [1]) that there exist tournaments Tn whose group n G(Tn) is abstractly isomorphic to a given group H if and only if the order g(H) of H is odd.

There is considerable literature concerning the century old result that for arbitrary positive integers a and m,

1.1

where μ(m) is the usual Mobius function. For earlier work on this we refer to L.E. Dickson [4, pp. 84–86] and L. Carlitz [1,2]. Another reference not noted by the above authors is R. Vaidyanathaswamy [6], who noted that the left member of (1.1) represents the number of special fixed points of the m th power of a rational transformation of the n th degree.

Let p denote a prime and n a positive integer. Write q = pn and let kq denote the Galois field with q elements. The unique factorization domain of polynomials in m(≤ 2) indeterminâtes x1,…, xq with coefficients in k is denoted by kq [x,…, xm. It is the purpose of this note to prove the foliowing generalization of Eisenstein's irreducibility criteria and to point out some of its consequences.

A fundamental construction of a category ζ((2), Appendice) is a triple (S, p, k), where S is a functor from ζ to itself and 2 where p:S2→S and k:1ζ→S are natural transformations such that

It is assumed that the reader is acquainted with the first two parts of the present paper, [1] and [2], in which there was developed informally the theory of a certain class of hypothetical computing devices, the Q-machines. In the present part of the paper we develop a way of describing Q-programs and Q-computations; then, following the theorem in [2], we obtain some further connections between the arithmetical nature of a number x and the computational schémas for generating sequences of rational numbers converging to x; and finally, by means of the usual device of Goedel numbering we begin to examine some properties of programs which operate on programs.

Un demi-groupe d'applications d'un ensemble est dit séparateur si, pour tout triple d'éléments x, y, z de l'ensemble, avec x ≠ y, il existe une application α du demi-groupe telle que x.α ≠ z et y α ≠ z. Un tel demi-groupe d'applications est évidemment transitif. Tout demi-groupe isomorphe à un demigroupe séparateur d'applications est dit un demi-groupe séparateur.

Let Mn be an n-dimensional manifold of differentiability class C∞ with an almost product structure . Let have eigenvalue +1 of multiplicity p and eigenvalue -1 of multiplicity q where p+q = n and p≧1, q≧1. Let T(Mn) be the tangent bundle of M. T(Mn) is a 2n dimensional manifold of class C∞. Let xi be the local coordinates of a point P of Mn. The local coordinates of T(Mn) can be expressed by 2n variables (xi, yi) where xi are coordinates of the point P and yi are components of a tangent vector at P with respect to the natural frame constituted by the vectior ∂/∂xi at P.

In [1], p. 132, it is shown that there exist two disjoint subsets of a square each connecting a pair of opposite vertices. Our interest in obtaining covers by infinite collections of such sets was prompted by a question posed by R.B. Reed in [3], We are grateful also to Dr. M. Edelstein for his guidance and encouragement of our work on the subject.

It is an obvious property of convergence that implies that sn+k exists and equals s for k = -1 (left translativity ) and for k = 1 (right translativity). Not so for sumrnability.

The triangular matrix (A) = (X ), where n = 0, 1, 2,…; k = 0, 1, 2, …; and λn, k = 0 for k > n is regular (in the sense of defining a regular sequence to sequence transform) if

• for every fixed k ;

• independently of n;

• 

In 1958, Egerváry and Turán [3] proposed and solved the problem of finding a stable interpolation process of minimal degree on a finite interval. Later [4] they investigated the same problem for an infinite interval with a suitable modification of the definition of stability. For the interval (-∞, ∞) their definition naturally differs from the one for the semi-infinite interval.

Let T be a tetrahedron and let V(T) and L(T) denote its volume and the sum of its edge-lengths. In this note we prove

Theorem 1.

with equality if and only if the tetrahedron T is regular.

The theorem that the homotopy sequence is exact splits into six statements. Scherk ([4]) obviates the use of homotopy extension in the proof of one of these statements. The purpose of this note is to show that the method can be adapted to give a direct proof of the corresponding statement in the theorem that the Eckmann-Hilton homotopy sequence ([l]) is exact. The note is based on Eckmann' s exposition ([2]). We are concerned with the proof of b2, pp. 34–35.

We call k integers x1 < x2 … < xk chosen from 1, 2, …, n} a k-choice (combination) from n. With 1, 2, …, n arranged in a circle, so that 1 and n are consecutive, we have a circular k-choice from n. A part of a k-choice from n is a sequence of consecutive integers not contained in a longer one. Let denote the number of circular k-choices from n with exactly r parts all ≤ w.

The primordial problems of linear algebra are the solution of a system of linear equations

and the solution of the eigenvalue problem

for the eigenvalues λk, and corresponding eigenvectors of a given matrix A.