Let Ⓗ be an abstract space and for every positive integer n let Fn, θ(x, y), θ ∈ Ⓗ, be a family of distribution n, 0 function s of random variables (Xn, Yn)θ, θ ∈ Ⓗ. For every θ ∈ Ⓗ, Eθg (Xn, Yn) will denote the expected value of the function g of (Xn, Yn )θ. The following proposition is proved.

If K is a finite geometric (i.e. admitting a rectilinear triangulation) n-complex and σn is an n - simplex of K which is n -1 not a face of any n + 1 simplex of K, and if σn-1 is an n-1 face of σn which is not a face of any other n - simplex in K, then the complex K - σn - σn-1 (the complex whose simplexes are those of K except for σn and σn-1) is called an elementary contraction of K of order n. The correspondence K → K - σn - σn-1 will also be called an elementary contraction, there being no possibility of confusion.

K. Morita has shown that, given two rings R and S, there is an isomorphism between the category of left R-modules and the category of left S-modules if and only if there exists an R-S bimodule U such that

(1) U is a progenerator in the category of left R-modules, and

(2) S ≅ (EndR U)opp as rings.

In another paper [l] measures determined from base functions, that have finite right and left limits everywhere and are of generalized bounded variation in the restricted sense, are studied and used to define non absolutely convergent integrals of Denjoy type. In this paper base functions of bounded variation and the corresponding measures are studied as a background for that paper. The results supplement parts of [2].

The space co consists of the sequences tending to zero with addition and scalar multiplication defined coordinate-wise and with the sup norm. The space ℓ1 consists of the sequences b = (bi) under coordinate»wise arithmetic for which

Let A and X be spaces. Then as is wellknown, [∑A, X] is a group where ∑ denotes the suspension. We wish to find conditions on A which will imply that this group is abelian for all spaces X, that is, ∑A is homotopy-commutative. This is equivalent to saying that conii A≤ 1 (see [2] for definition). Our results contain relations between conil A and the generalised Whitehead product of [1]. We work in the category of complexes with base points.

The purpose of this paper is to extend results of Ball [1] concerning maximal subgroups of the group S(X) of all permutations of the infinite set X. The basic idea is to consider S(X) as a group of operators on objects more complicated than X. The objects we consider here are subspaces of the Stone-Čech compactification of the discrete space X and the Boolean algebra of “big setoids” of X.

Let κ be a class of real valued functions defined on an interval which need not be bounded. The class κ is said to be characterized in terms of associated sets if there exists a family of sets of real numbers P such that f ∈ κ if and only if for every real number α the sets a remembers of P. Many classes of functions have been characterized in terms of associated sets. The chart below summarizes a few such characterizations.

The different results proved in this paper do not have very much in common. Since they all deal with the location of the zeros of a polynomial, we have decided to put them in one place. Improving upon a classical result of Cauchy we obtain in § 2 a circle containing all the zeros of a polynomial. In § 3 we obtain an extension of the well known theorem of Enestrőm and Kakeya concerning the zeros of a polynomial whose coefficients are non-negative and monotonie.

Let μ∗ be an outer measure on (X, S) with σ- algebra S and let μ* be the inner measure induced by μ∗. A set M is a measurable cover of a set A ⊆ X if A ⊆ M, M is measurable, and μ∗ (M-A) = 0. We assume that every subset of X has a measurable cover; this holds, for example, if μ∗ is the outer measure induced by a measure which is σ- finite on X [2, theorem C, p. 50].

In [1[ Trypanis has proved the following extension of Fermat's theorem: If a is any integer, paprime, , then

1

This result is to be understood in the sense that a(p-1)/pn-1=p1/pnα where α is an algebraic integer.

Let (x1, x2, …, xn) denote the coordinates of a point of Euclidean n-space En. Let be a set of n+1 points of En with the property that

form a linearly independent set and define a lattice Λ of points

by allowing u1, …, un to take all integer values.

Throughout this paper the arithmetic functions L(n) and w(n) denote respectively the number and product of the distinct prime divisors of the integer n > 1, with L(1) = 0 and w(1) = 1. Also let

We recall that an arithmetic function f(n) is said to be multiplicative if f(1) = 1 and f(mn) = f(m)f(n) whenever (m, n) = 1, where (m, n) denotes as usual the greatest common divisor of m and n.

A vector measure (countable additive set function with values in a Banach space) on a field may be extended to a vector measure on the generated σ- field, under certain hypotheses. For example, the extension is established for the bounded variation case [2, 5, 8], and there are more general conditions under which the extension exists [ 1 ]. The above results have as hypotheses fairly strong boundedness conditions on the n o rm of the measure to be extended. In this paper we prove an extension theorem of the same type with a restriction on the range, supposing further that the measure is merely bounded. In fact a vector measure on a σ- field is bounded (III. 4. 5 of [3]) but it is conceivable that a vector measure on a field could be unbounded.

Suppose X is a simply-connected CW- complex whose homotopy groups are finite. For each n, let be the class of torsion groups whose p - primary components are zero for all primes p which do not figure in the homotopy groups πi(X) for i ≤ n.

If x, x1,…, xn are independent stochastic variables and if the conditional distribution of x given x1 +…+ xn is known, what can be said about the marginal distributions of x, x1,…, xn? In this paper we will show that if the conditional distribution of x given a subset of x1, x2,…, xn-1, x+ x1 +…+ xn has a certain structural form then x, x1,…, xn are distributed as members of the linear exponential family of distributions and further x1,…, xn are identically distributed.

In 1955, Suranyi and P. Turán [8] considered the problem of existence and uniqueness of interpolatory polynomials of degree ≤ 2n-1 when their values and second derivatives are prescribed on n given nodes. Around this kind of interpolation - aptly termed (0, 2) interpolation - considerable literature has grown up since then. For more complete bibliography on this subject we refer to J. Balazs [3], Later we considered [10] the problem of modified (0, 2) interpolation when 2 the abscissas are the zeros of (1-x2) Tn(x), where Tn(x) is the Tchebycheff polynomial of the first kind (Tn(x) = cos n θ, x = cos θ).

In this paper we are dealing with differential equations of the forms:

Ei

where the functions pi are positive.

By a solution of an equation of the above forms, we mean a function x(t) ∈ C2 [c, + ∞) where c is a non-negative constant, which satisfies the corresponding equation on the whole interval [c, +∞). By an oscillatory solution of (Ei), we mean a solution with arbitrarily large zeros.

In [1] J. O. C. Ezeilo asks whether the equation

1

has periodic solutions for a ≠ 0. Since (1) has a two-dimensional space of solutions of period 2π if sin x is approximated by x, it is plausible to conclude, by analogy with x″ + sin x = 0, that (1) does have periodic solutions. However, when one applies the standard theory of perturbation of periodic solutions (treating a as small, see [2]), one finds that the only real periodic solutions obtainable in this manner are the trivial ones x(t, a) = nπ for some integer n.

The densest lattice packings of spheres in Euclidean spaces En of n dimensions are known for n ≤ 8 (for full n — references see [6]). However, it i s not known for any n ≥ 3 whether there can be any non-lattice sphere packing with density exceeding that of the corresponding densest lattice packing.