G. Legrand [1] studied a generalization of the almost complex structures [2] by considering a linear operator J acting on the complexified space of a differentiable manifold Vm satisfying a relation of the form J2 = λ2 (identity) where λ is a nonzero complex constant. Such structures are called π-structures. A π-structure is defined on Vm by the knowledge of two fields, of proper supplementary subspaces T1 and T2 of the complexified tangent space at x ∈ Vm, such that dim(T1) = n1; dim(T2) = n2; n1 + n2 = m. In the remaining case, λ= 0, H. A. Eliopoulos [3] introduced almost tangent structures and discussed euclidean structures compatible with almost tangent structures [4]. In a similar way the purpose of this paper is to study singular riemannian structures compatible with π structures, briefly Rπ-structures.

The implicit function theorem has applications at all levels of mathematics from elementary calculus (implicit differentiation) to finding periodic solutions of systems of differential equations ([1, Chapter 14] and [4], for example).

In 1961 W. S. Loud [3] studied the case of two equations in three unknowns. He considered only cases where up to third order derivatives were involved and only those cases where the derivative of the solutions at the critical point existed. Coddington and Levinson [1] consider a specific singular case involving n equations in n + m unknowns. In general the number of distinct critical cases involving up to third derivatives for such a general system is not known.

The K transformation (also called the Meijer transformation) has been extended by Zemanian [1; 2] to a class of generalized functions, For , he defined the K transform of f by

(1)

In [2, Section 6.6] the following inversion theorem for the K transform of f is proven:

(2)

in the sense of weak convergence in D'(I). Here, σ is any fixed real number greater than σf, μ is zero or a complex number with positive real part and D'f(I) is the space of Schwartz distributions on I = (0, ∞).

Let K0 and K1 be equational classes of algebras of the same type. The smallest equational class K containing K0 and K1 is the join of K0 and K1; in notation, K = K0 ∨ K1. The direct product K0 × K1 is the class of all algebras α which are isomorphic to an algebra of the form a0 × a1, a0 ∈ K1. Naturally, K0 × K1 ⊆ K0 ∨ K1, Our first theorem states a very simple condition under which K0 × K1 = K0 ∨ K1, and an additional condition under which the representation α ∨ a0 × a1 unique.

Let X be a compact metric space. By a packing in X we mean a subset S ⊆ X such that, for x, y ∈ S with x ≠ y, the distance d(x, y) ≥ 1. Since X is compact, any packing of X is finite. In fact, the set of numbers

{card(S): S is a packing in X}

is bounded. The cardinality of the largest packing in X will be called the packing number of X and will be denoted by ρ(X). If A(X) and P(X) denote the area and perimeter, respectively, of a compact convex subset X of the plane, then a special case of a result conjectured by H. Zassenhaus [6] and proved by N. Oler [l] is the following.

The theorem of Borsuk-Ulam states that n odd functions on the n-dimensional sphere always have a common zero. We have tried to obtain a similar theorem by "slightly" changing the conditions for the functions, but it turned out that only a very weak analogue can be expected in our case. Here we want to prove a few results and mention some of the questions which have remained unanswered.

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.

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].

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.

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.

The maximal ring of quotients (in the sense of Johnson [4] and Utumi [5[) of the ring C(X) of real valued continuous functions on a completely regular Hausdorff space X has been studied in [1] and [2]. The aim of the present paper is to provide some additional results to those, and to study the relevant extensions on the absence of the real maximal ideals.

In this note we show the existence of a spread which is not a dual spread, thus answering a question in [1]. We also obtain some related results on spreads and partial spreads.

Let ∑ = PG(2t-l, F) be a projective space of odd dimension (2t-l, ≥2) over the field F. In accordance with [1] we make the following definitions. A partial spread S of ∑ is a collection of (t-l)-dimensional projective subspaces of ∑ which are pairwise disjoint (skew).

Let A be an m × n matrix of complex numbers. Let Aτ and A* denote the transpose and conjugate transpose, respectively, of A. We say A is diagonal if A contains only zeros in all positions (i, j) with i ≠ j. In a recently published paper [4], M.H. Pearl established the following fact: There exist real orthogonal matrices O1 and O2 (O1 m-square, O2 n-square) such that O1AO2 is diagonal, if and only if both AA* and A*A are real. It is the purpose of this paper to show that a theorem substantially stronger than this result of Pearl's is included in the real case of a theorem of N.A. Wiegmann [2]. (For other papers related to Wiegmann's, see [l; 3].)

The purpose of this note is to give a new characterization of Prüfer domains using the concept of ring epimorphism, and to indicate some connections with well-known properties of Prüfer domains. All rings are commutative and have a unit element.

Let (X, d) be a metric space and f a mapping of X into itself. D.F. Bailey [l] considered a class of mappings f satisfying the condition: ∀x, y ∈ X, x ≠ y,

(1.1)

where I+ denotes the set of positive integers.

A family of sets is said to possess property if there exists a set such that and for every We consider the following question raised by P. Erdös |1|: let n and N be positive integers, n ≥ 2 and N ≥ 2n - 1 and let S be a set of N elements; what is the least integer (provided such an integer exists), for which there exists a family of subsets of S satisfying

1. (a) |F| = n for each

1. (b)

1. (c) does not have property

1. (d) if and then has property

The present paper seeks to put into a clearer focus the roles of certain axioms in axiomatic studies of finite dimensional Cartesian spaces over various classes of ordered fields. Following Tarski [3], we define a Cartesian space as follows.

Consider the functional equation

(1)

assumed valid for all real x, y and h. Notice that (1) can be written

(2)

a difference analogue of the wave equation, if we interpret etc., (i. e. symmetric h differences), and that (1) has an interesting geometric interpretation. The continuous solutions of (1) were found by Sakovič [5].

The André-Poincaré "probléme du scrutin" [9] can be stated as follows: In an election between two candidates A polls m votes, B polls n, m > n. If the votes are counted one by one what is the probability that A leads B throughout the counting? Many derivations and interpretations of the solution have been given and a convenient summary of methods till 1956 can be found in Feller [1]. So numerous are the generalizations of ballot problems and their applications since this date that we do not even attempt an enumeration here.

In [2, page 151], J. Lambek proposes the following exercise: With any maximal right ideal M of a ring R with 1 associate the ideal . Show that OM, is the right annihilator of the injective hull of the right R-module R/M. The purpose of this note is to show that the above statement is true for a much larger class of right ideals than that of maximal regular right ideals of a ring.