Showing papers in "Periodica Mathematica Hungarica in 1978"

The size ramsey number

Abstract: Let denote the class of all graphsG which satisfyG→(G 1,G 2). As a way of measuring minimality for members of , we define thesize Ramsey number ř(G 1,G 2) by . We then investigate various questions concerned with the asymptotic behaviour ofř.

Mengerian theorems for paths of bounded length

Abstract: Let u and v be non-adjacent points in a connected graph G . A classical result known to all graph theorists is that called MENGER's theorem . The point version of this result says that the maximum number of point-disjoint paths joining u and v is equal to the minimum number of points whose deletion destroys all paths joining u and v . The theorem may be proved purely in the language of graphs (probably the best known proof is indirect, and is due to DiRAC [3] while a more neglected, but direct, proof may be found in ORE [7]) . One may also prove the theorem by appealing to flow theory (e.g . BERGE [1], p. 167) . In many real-world situations which can be modeled by graphs certain paths joining two non-adjacent points may well exist, but may prove essentially useless because they are too long . Such considerations led the authors to study the following two parameters . Let n be any positive integer and let u and v be any two non-adjacent points in a graph G . Denote by A„(u, v) the maximum number of point-disjoint paths joining u and v whose length (i .e ., number of lines) does not exceed n. Analogously, let V„(u, v) be the minimum number of points in G the deletion of which destroys all paths joining u and v which do not exceed n in length., A special case would obtain when n = p = I V(G)I, and we have by Monger's theorem, the equality A„(u, v) = V,,(u, v) .

Embedding theorems for graphs establishing negative partition relations

Abstract: We started to work on this paper with the following observation . If 4 is a graph on J~ 1 vertices and establishing the negative partition relation tt l ]) 2 then 4 is universal for countable graphs . This last statement means that every countable graph N is isomorphic to a spanned subgraph of q . We have stated a number of results and problems of the above type in our paper [4] written in 1971 . (See Problems VIII, IX, X on pp . 285-286 .) Quite a few of these problems will be solved or modified by the results of this paper . S. Shelah has made an important remark concerning our problem. He has shown that our starting result cannot be generalized for higher cardinals .

A strong law of the empirical density function

Abstract: LetX 1,X 2,... be a sequence of i.i.d.r.v's with a twice differentiable density functionf. Suppose thatf is vanishing outside of the interval [0, 1], strictly positive inside and |f″| is bounded. Further let λ be an arbitrary density function satisfying some regularity conditions and define the empirical density functionf n as $$f_n (x) = (nh_n )^{ - 1} \sum\limits_{k = 1}^n {\lambda ((x - X_k )h_n ^{ - 1} )} $$ where {h n } is a decreasing sequence of positive numbers tending to 0 and satisfying some further restrictions. Then for any e(>0) we have $$\mathop {lim}\limits_{n \to \infty } \left( {\frac{{nh_n }}{{2\Lambda ^2 \log h_n ^{ - 1} }}} \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern- ulldelimiterspace} 2}} \mathop {sup}\limits_{\varepsilon< x< 1 - \varepsilon } \left| {\frac{{f_n (x) - f(x)}}{{f^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern- ulldelimiterspace} 2}} (x)}}} \right| = 1 a.s.$$ where $$\Lambda ^2 = \int\limits_{ - \infty }^{ + \infty } {\lambda ^2 (x)dx} .$$

Solving equations in groups: A survey of Frobenius' theorem

Abstract: A fundamental result of Frobenius states that in a finite group the number of elements which satisfy the equationxn=1, wheren divides the order of the group, is divisible byn. Here 1 denotes the identity of the group. This theorem and several generalizations were obtained by Frobenius at the turn of the century. These results have stimulated a great amount of interest in counting solutions of equations in groups. This article discusses these results and traces the various developments which these fundamental papers have generated.

Boolean algebras of projections

Abstract: Bade, in (1), studied Boolean algebras of projections on Banach spaces and showed that a er-complete Boolean algebra of projections on a Banach space enjoys properties formally similar to those of a Boolean algebra of projections on Hilbert space. (His exposition is reproduced in (7: XVII).) Edwards and Ionescu Tulcea showed that the weakly closed algebra generated by a tr-complete Boolean algebra of projections can be represented as a von Neumann algebra; and that the representation isomorphism can be chosen to be norm, weakly, and strongly bicontinuous on bounded sets (8): Bade's results were then seen to follow from their Hilbert space counterparts. I show here that it is natural to relax the condition of ^-completeness to weak relative compactness; indeed, a Boolean algebra of projections has a

Some localized separation axioms and their implications

Abstract: In a topological spaceX, a T2-distinct pointx means that for anyy∈X x≠y, there exist disjoint open neighbourhoods ofx andy. Similarly, T0-distinct points and T1distinct points are defined. In a Ti-distinct point-setA, we assume that eachx∈A is a Ti-distinct point (i=0, 1, 2). In the present paper some implications of these notions which ‘localize’ the Ti-separation axioms (i=0, 1, 2) requirement, are studied. Suitable variants of regularity and normality in terms of T2-distinct points are shown hold in a paracompact space (without the assumption of any separation axioms). Later T0-distinct points are used to give two characterizations of the RD-axiom.1 In the end, some simple results are presented including a condition under which an almost compact set is closed and a result regarding two continuous functions from a topological space into a Hausdorff space is sharpened. A result which relates a limit pointv to an ω-limit point is stated.

Characterization of the normal distribution by constant regression of quadratic statistics on linear statistics

Abstract: This paper contains applications of theorems of [1] for quadratic statistics which have constant regression on linear statistics. Two theorems are proved. The first is a sufficient condition which assumes that the characteristic function of a sample is an entire function. The second gives a new characterization of the normal distribution.

On the divergence of certain Hermite-Fejér interpolation

Abstract: Summary In his paper [1]P. Turan discovers the interesting behaviour of Hermite-Fejer interpolation (based on the Cebysev roots) not describing the derivative values at “exceptional” nodes {ηn}n=1∞. Answering to his question we construct such exceptional node-sequence for which the mentioned process is bounded for bounded functions whenever −1