Professor Eliopoulous studied almost tangent structures on manifolds M2n in [1; 2]. An almost tangent structure F is a field of class C∞ of linear operations on M2n such that at each point x in M2n, Fx maps the complexified tangent space into itself and that Fx is of rank n everywhere and satisfies that F2 = 0. In this note, we consider a (1,1) tensor field . on a Riemannian M2n which satisfies everywhere and is such that the rank of F is n everywhere. Such gives an almost tangent structure F on M2n.

An nth order homogeneous linear differential equation is said to be disconjugate on the interval I of the real line in case no non-trivial solution of the equation has more than n - 1 zeros (counting multiplicity) on I. It is the purpose of this paper to establish several necessary and sufficient conditions for disconjugacy of the third order linear differential equation

(1.1)

where pi(t) is continuous on the compact interval [a, b], i = 0, 1, 2.

Let an open, quasi-compact subscheme of an affine scheme be called subaffine. This note will centre on an elementary characterization of such schemes in terms of their topology and global sections. Thence one can obtain simplifications and generalizations of some well-known theorems, such as Serre's Criterion [2, Thm. 1].

In this paper we consider a special class of linear operators defined on a cone K in a Banach space X. This class of operators is the natural generalization of a class of operators which has applications in the theory of interpolation spaces. In particular, using the criteria developed in Theorem 1, it is possible to characterize those sequence spaces X such that every linear operator A of weak types (p, p) and (q, q) is a continuous mapping of X into itself. For details of this we refer the reader to [3].

Let A and B be spaces having the homotopy type of countable CW-complexes. Then we prove the following theorems.

Theorem 1. If conil(A × B) ≤ 1, then for each integer n ≥ 1, the inclusion j: ∑n A ∨ ∑n B → ∑nA × ∑n B is a homotopy equivalence.

This result is obtained as a corollary of Theorem 2.

All modules under consideration are finitely generated left modules. A G-lattice is a G-module which is R-torsion-free.

The generating series for the elementary symmetric function Er, the complete symmetric function Hr, are defined by

respectively.

In [6], H. Sharp gives a matrix characterization of each topology on a finite set X = {x1, x2,…, xn}. The study of quasi-uniform spaces provides a more natural and obviously equivalent characterization of finite topological spaces. With this alternate characterization, results of quasi-uniform theory can be used to obtain simple proofs of some of the major theorems of [1], [3] and [6]. Moreover, the class of finite topological spaces has a quasi-uniform property which is of interest in its own right. All facts concerning quasi-uniform spaces which are used in this paper can be found in [4].

A celebrated theorem of Birkhoff ([1], [6]) states that the set of n × n doubly stochastic matrices is identical with the convex hull of the set of n × n permutation matrices. Birkhoff [2, p. 266] proposed the problem of extending his theorem to the set of infinite doubly stochastic matrices. This problem, which is often known as Birkhoffs Problem III, was solved by Isbell ([3], [4]), Rattray and Peck [7], Kendall [5] and Révész [8].

T. J. Willmore has shown that if a differentiable manifold's rank (the maximum number of everywhere linearly independent commuting vector fields definable on it) equals the manifold's dimension, then the manifold is a torus of the appropriate dimension [1]. This theorem is proved more simply and without any differentiability hypothesis in the present note.

We consider the second order differential equation

(1)

with the assumptions that

(2) f(x) is continuous (- ∞ < x < ∞) and p(t) is continuous and bounded: |p(t)| ≤ E, - ∞ < t < ∞.

Also, throughout this paper, F(x) denotes an antiderivative of f(x).

Let E be a finite set containing n elements, n ≡ 1, 3 (mod 6), S = S(E) a Steiner triple system on E, i.e. each unordered pair of elements of E is a subset of exactly one triple in S. Let T be a subset of E such that none of the triples of elements of T is a member of S. Erdös has asked (in a recent letter to the authors) for the maximal size of such a set T. Denote max |T| for fixed n and S by f(n, S). We prove in this note the following result:

1. (i)

1. (ii) for every n ≡ 1, 3 (mod 6) there exists a Steiner triple system S0 such that equality holds in i.

In 1957, Mordell [3] proved

Theorem. If p is an odd prime there exist non-negative integers x, y ≤ A p3/4 log p, where A is a positive absolute constant, such that

(1.1)

provided (abc, p) = 1.

Recently Smith [5] has obtained a sharp asymptotic formula for the sum where r(n) denotes the number of representations of n as the sum of two squares.

Topology of positively curved compact Kähler manifolds had been studied by several authors (cf. [6; 2]); these manifolds are simply connected and their second Betti number is one [1]. We will restrict ourselves to the study of some compact homogeneous Kähler manifolds. The aim of this paper is to supplement some results in [9]. We prove, among other results, that a compact, simply connected homogeneous complex manifold whose Euler number is a prime p ≥ 2 is isomorphic to the complex projective space Pp-1 (C); in the p-1 case of surfaces, we prove that a compact, simply connected, homogeneous almost complex surface with Euler-Poincaré characteristic positive, is hermitian symmetric.

Consider a system of differential equations . Solutions of this system are said to be convergent if, given any pair of solutions x(t), y(t), x(t) - y(t) → 0 as t → ∞. In this case the system is also said to be extremely stable.

In [6] a technique was developed which yielded the convergence of solutions of the forced Lienard equation. Here a similar technique i s applied to forced third order equations. A critical step in [6] was to show that a certain matrix was negative definite. This could be done directly in [6] since the matrix was only 2 × 2. With third and higher order equations, direct use of necessary and sufficient conditions is not feasible since the computations become unwieldy.

R. L. Graham has posed the following question:

Given n positive integers a1, < a2 <… < an, does there exists a pair of indices i, j such that ai/(ai, aj) ⩾ n? ((ai, aj) = g.c.d. of ai and aj).

Call an m × n array an m × n; k array if its mn entries come from a set of k elements. An m × n; 1 array has mn like entries. We write

(1)

if every m × n; k array contains a p × q; 1 sub-array. The negation of (1) is written

and means that there is an m × n; k array containing no p × q; 1 sub-array. Relations (1) are called "polarized partition relations among cardinal numbers" by P. Erdös and R. Rado [2]. In this note we prove the following theorems.

We wish to study the problem of finding conditions under which a family of maps from one space into another, with a suitable topology, is compact. Some of the results obtained in this direction are in [1; 2; 3]. We propose to give conditions, to be called uniformly regular and regular (the terminology is motivated by [4]), under which "Ascoli" theorems can be proved. These notions turn out to be equivalent to even continuity of Kelley [1, page 235] under such conditions that all the theorems in the section on even continuity in it still hold when in their statements even continuity is replaced by either uniform regularity or regularity (see Theorem A below).

I is an interval of R, the set of real numbers, n is a positive integer and F ⊂ Cj (I) for j ≥ 0 large enough so that the following definitions are possible:

(i) Let λ(n) = (λ1, λ2,…,λk) where k, λ1, λ2,…, λk, are positive integers and λ1 + λ2 +… +λk = n. Then λ(n) is an ordered partition of n. The set of all such partitions of n is denoted by P(n).

In this note we apply a well-known theorem due to Weierstrass to show that under certain conditions convergence in distribution of a sequence of distribution functions implies the convergence of moments.

This note may be understood by an undergraduate student who has an introductory course of complex variables and a second course of statistics.