Showing papers in "Periodica Mathematica Hungarica in 1982"

Transformations of linear connections, II

Abstract: Combining several results on related (or conjugate) connections, defined on banachable fibre bundles, we set up a machinery, which permits to study various transformations of linear connections. Global and local methods are applied throughout. As an application, we get an extension of the classical affine transformations to the context of infinite-dimensional vector bundles. Another application shows that, realising the ordinary linear differential equations (in Banach spaces) as connections, we get the usual transformations of (equivalent) equations. Thus, some classical results on differential equations, such as the Theorem of Floquet, can have a “geometric” interpretation.

Upper bound for transversals of tripartite hypergraphs

Abstract: We proveτ ≤ 2 7/9v for 3-partite hypergraphs. (This is an improvement of the trivial boundτ ≤ 3v.)

The space of minimal prime ideals in a 0-distributive semilattice

Abstract: LetS be a 0-distributive semilattice and\(\mathfrak{M}\) be its minimal spectrum. It is shown that\(\mathfrak{M}\) is Hausdorff. The compactness of\(\mathfrak{M}\) has been characterized in several ways. A representation theorem (like Stone's theorem for Boolean algebras) for disjunctive, 0-distributive semilattices is obtained.

Product spaces and semi-separation axioms

Abstract: In this paper new characterizations of semi-R0 and semi-R1 spaces are obtained and used to prove that the product space of semi-R0, semi-T1, and semi-T0 spaces is, respectively, semi-R0, semi-T1, and semi-T0; and that the product space of semi-R1 space need not be semi-R1. An example is given where the product space is semi-T2 and one of the factor spaces is not semi-T0 or semi-R0.

Ōtsukische Räume mit Einem Zweifach Rekurrenten Metrischen Grundtensor

Abstract: In this note necessary and sufficient conditions are determined for Weyl—Ōtsukispaces to have a birecurrent metric, i.e.,∇ m ∇ k g ij =γ km g ij . It is proved that in this space the metric tensor is an eigen-tensor. The special caseP = ϱ(x)δ is examined and we prove that in this case the recurrent metric tensor is likewise birecurrent.

A note onR1-topological spaces

Abstract: The purpose of the present note is to give a number of characterizations of theR1-axiom and to show that theR1-axiom is equivalent to the weakly Hausdorff axiom introduced byB. Banaschewski andJ. M. Maranda [2]. In anR1-space it is shown that the locally compactness property is also open hereditary and that the closure of an almost compact set is the union of the closures of its points. A necessary and sufficient condition is obtained under which a locally compact set dense in anR1-space is open. Finally a variant of a well-known theorem regarding two continuous functions of a topological space into aT2-space is formulated forR1-spaces.

Maximal elements in topological and generalized metric spaces

Abstract: In this note, some problems concerning existence of maximal elements in a topological as well as in a generalized metric space, equipped with an ordering, are studied. The results presented here may be considered as a partial refinement of those established in [2] for uniform structures.

Ein Einschliessungsverfahren für Lösungen Fehlerbehafteter Linearer Gleichungen

Abstract: In a partially ordered space, the method xn+1 = L+xn+ − N+xn- − L−y+ + N− yn- + r, yn+1 = N+y+ − L+yn- − N−xn+ + L−x− + t of successive approximation is developed in order to enclose the solutions of a set of linear fixed point equations monotonously. The method works if only the inequalities x0 ≤ y0, x0 ≤ x1, y1 ≤ y0 related to the starting elements are satisfied. In finite-dimensional spaces suitable starting vectors can be computed if a sufficiently good approximation for the fixed points is known.

Spectral measures, I: General theory in a topological vector space

Abstract: Part I of this paper is devoted to the general theory of spectral measures in topological vector spaces. We extend the Hilbert space theory to this setting and generalize the notion of spectral measure in some useful ways to provide a framework for Part II, etc.

Spectral measures, II: Characterization of scalar operators

Abstract: We continue the development of part I. The Riesz representation theorem is proved without assuming local convexity. This theorem is applied to give sufficient conditions for an operator (continuous or otherwise) to be “spectral”. A uniqueness problem is pointed out and the function calculus is extended to the case of several variables. A Radon—Nikodym theorem is proved.

Polynomials and finite sets connected with an identity

Abstract: It is shown that the polynomials satisfying the identityf(x) f(x + 1) = f(x 2 +x − a), wherea either belongs to a field of characteristic zero or is transcendental over a prime field of characteristic exceeding 2, are precisely those of the form(x 2 −a) n ; thus extending a result proved by Nathanson in the complex case. The result is not, in general, true in characteristic 2. Additionally, a class of finite sets, considered by Nathanson in connection with the identity, is completely determined.

Locally constant dyadic derivatives

Abstract: We show that in order for a Walsh series to be locally constant it is necessary for certain blocks of that series to sum to zero. As a consequence, we show that a functionf with a somewhat sparse Walsh—Fourier series is necessarily a Walsh polynomial if its strong dyadic derivative is constant on an interval. In particular, if a Rademacher seriesR is strongly dyadically differentiable and if that derivative is constant on any open subset of [0, 1], thenR is a Rademacher polynomial.

Bridges of longest circuits and path coverings of labelled trees

Abstract: Let H be a subgraph of a graph G. We say that two edges el, e 2 E G, el, e 2 ~ H are equivalent iff e 1 = e 2 or there exists a path p with the end-edges e 1 and e 2 and no inner vertex of p belongs to H. A class of equivalent edges together with all vertices incident to edges of this class is called a bridge S of H in G. The vertices of S ffl H are called the attaching vertices of S; the other vertices of S are called inner vertices. In [4] I investigated bridges of maximal circuits and paths in graphs with a certain minimal valency. In this paper properties of maximal circuits, maximal paths, and maximal odd circuits are studied. K. HAUSCHILD, H. I-IERRE and W. I~AUTENBERG [2] proved in 1971: (1) Let S denote a bridge o[ a maximal circuit L of length l in a 2-connected graph G. If S contains at least one circuit, then each maximal circuit L' i n S has length l' ~ l -- 1.

Some notes on the upper and lower radicals

Abstract: H(I) fqM~ ~, for every O~I<~A~M where I <~ A means I is an ideal of A. Note: we write I for the class {1} containing I as its member. A regular class may not contain the ring 0, for the sake of short statement we shall assume that regular classes contain the ring 0. It is well-known that if the class M is regular then ~(M) = {A E H(A) N M = 0}

On the tensor operators in parallelizable spaces

Abstract: The tensor-operators over parallelizable spaces are defined. Their Hermitian conjugation is defined by introducing an inner product based on the tensor-integral Some properties of the ordinary operators are generalized for the tensor-operators