On uniform approximation of abstract functions on convex polygons

Approximation aggregates of summatory type are constructed for functions f: Q → X, where Q is a convex polygon in ℝ² and X is a real Banach space. An assertion providing an accuracy estimate for an uniform approximation in terms of the modulus of continuity of the second order for f is proved. Differential properties of such aggregates are studied. Bibliography: 7 titles. __________ Translated from Problemy Matematicheskogo Analiza, No. 35, 2007, pp. 59–77     

Hypergraphs and hypergroups

We associate to every hypergraph a commutative quasi-hypergroupH _qG and find a necessary and sufficient condition on so thatH is associative. For certain, any finite included, we determine a sequence= ₀, ₁,, _n of hypergraphs such that ifH ₀ ,H ₁ ,H, H _n is the sequence of the associated quasi-hypergroups,H _n is a join space.Presented by I. Rosenberg.     

Wrapping polygons in polygons

This paper is developed toI ₂(2g).c-geometries, namely, point-line-plane structures where planes are generalized 2g-gons with exactly two lines on every point and any two intersecting lines belong to a unique plane.I ₂(2g).c-geometries appear in several contexts, sometimes in connection with sporadic simple groups. Many of them are homomorphic images of truncations of geometries belonging to Coxeter diagrams. TheI ₂(2g).c-geometries obtained in this way may be regarded as the standard ones. We characterize them in this paper. For everyI ₂(2g).c-geometry , we define a numberw(), which counts the number of times we need to walk around a 2g-gon contained in a plane of , building up a wall of planes around it, before closing the wall. We prove thatw()=1 if and only if is standard and we apply that result to a number of special cases.     

G-Graphs and Algebraic Hypergraphs

G-graphs have been introduced in [A. Bretto, A. Faisant. A new way for associating a graph to a group, Mathematica Slovaca 55, No.1, (2005), 1-8], and have some applications in both symmetric and semi-symmetric graphs, cage graphs, and expander graphs [A. Bretto, A. Faisant and L. Gillibert, G-graphs: A new representation of groups. Journal of Symbolic Computation, Volume 42, Issue 5, May 2007, Pages 549-560], [A. Bretto, L. Gillibert and B. Laget, Symmetric and Semisymmetric Graphs Construction Using G-graphs, Proceedings of ISSAC'05, July 24-27 2005, Beijing, China. ACM Proceedings (2005), 61-67], [A. Bretto, L. Gillibert, A New Upper Bound for the (p,6) cages, Proceedings of ISSAC'07, July 29-August 1st 2007, Waterloo Canada. ACM Proceedings (2007)]. In this paper we present a characterization of G-graphs, introduce the notion of principal clique hypergraphs, and present some of their basic properties.     

超图中的着色问题

本文是近三十年来有关超图中涉及的着色问题的综述。它包含了有关超图着色中的基本结果,临界可着色性,２－可着色性,非２－可着色性以及在超图中与顶点着色、边着色和其它着色相关的极值问题。     

Self-Complementary Non-Uniform Hypergraphs

Let V be a finite set. For a nonempty subset K of positive integers, a K-hypergraph on V is a hypergraph with vertex set V and edge set ${E=\bigcup_{k\in K}E_k}$ , where E _k is a nonempty set of k-subsets of V. We define the complement of a K-hypergraph (V, E) to be the K-hypergraph on V whose edge set consists of the subsets of V with cardinality in K which do not lie in E. A K-hypergraph is called self-complementary if it is isomorphic to its complement. The two extreme classes of self-complementary K-hypergraphs have been studied previously. When |K|?=?1 these are the self-complementary uniform hypergraphs, and when |K|?=?|V| ? 1, these are the so called ‘self-complementary hypergraphs’ studied by Zwonek. In this paper we determine necessary conditions on the order of self-complementary K-hypergraphs, and on the order of regular or vertex-transitive self-complementary K-hypergraphs, for various sets of positive integers K. We also present several constructions for K-hypergraphs to show that these necessary conditions are sufficient for certain sets K. In the language of design theory, the t-subset-regular self-complementary K-hypergraphs correspond to large sets of two isomorphic t-wise balanced designs, or t-partitions, in which the block sizes lie in the set K. Hence the results of this paper imply results in design theory.     

Heavy Transversals and Indecomposable Hypergraphs

A weighted hypergraph is a hypergraph H = (V, E) with a weighting function , where R is the set of reals. A multiset S V generates a partial hypergraph H ^S with edges , where both the cardinality and the total weight w(S) are counted with multiplicities of vertices in S. The transversal number of H is represented by (H). We prove the following: there exists a function f(n) such that, for any weighted n-Helly hypergraph H, (H ^B ) 1, for all multisets B V if and only if (H ^A ) 1, for all multisets A V with . We provide lower and upper bounds for f(n) using a link between indecomposable hypergraphs and critical weighted n-Helly hypergraphs.* On leave from Computing and Automation Research Institute, Hungarian Academy of Sciences.     

Betti Numbers of Hypergraphs

In this article, we study some algebraic properties of hypergraphs, in particular their Betti numbers. We define some different types of complete hypergraphs, which to the best of our knowledge are not previously considered in the literature. Also, in a natural way, we define a product on hypergraphs, which in a sense is dual to the join operation on simplicial complexes. For such product, we give a general formula for the Betti numbers, which specializes neatly in case of linear resolutions.     

Hypergraphs do not jump

The number α, 0≦α≦1, is a jump forr if for any positive ε and any integerm,m≧r, anyr-uniform hypergraph withn>n _o (ε,m) vertices and at least (α+ε) \(\left( {\begin{array}{*{20}c} n \\ r \\ \end{array} } \right)\) edges contains a subgraph withm vertices and at least (α+c) \(\left( {\begin{array}{*{20}c} m \\ r \\ \end{array} } \right)\) edges, wherec=c(α) does not depend on ε andm. It follows from a theorem of Erdös, Stone and Simonovits that forr=2 every α is a jump. Erdös asked whether the same is true forr≧3. He offered $ 1000 for answering this question. In this paper we give a negative answer by showing that \(1 - \frac{1}{{l^{r - 1} }}\) is not a jump ifr≧3,l>2r.     

On Equitable Colorings of Hypergraphs

Mathematical Notes - A two-coloring is said to be equitable if, on the one hand, there are no monochromatic edges (the coloring is regular) and, on the other hand, the cardinalities of color...     

Untangling polygons and graphs

Untangling is a process in which some vertices of a plane graph are moved to obtain a straight-line plane drawing. The aim is to move as few vertices as possible. We present an algorithm that untangles the cycle graph $C_n$ while keeping at least $\mathrm{\Omega }(n^{2/3})$ vertices fixed. For any graph G, we also present an upper bound for the number of fixed vertices in the worst case. The bound is a function of the number of vertices, maximum degree and diameter of G. One of its consequences is the upper bound $O({(n\log n)}^{2/3})$ for all 3-vertex-connected planar graphs.     

Supra-Hereditary Properties of Hypergraphs

A hypergraph-property P is supra-hereditary if whenever a hypergraph H has property P and K is a hypergraph generated by a nonempty subset A of X(H), then K also has the property P. In this paper we give a characterization of irreducible supra-hereditary hypergraphs thereby doubly extending a similar result in graph theory. Hence, we raise some fundamental questions in hypergraph theory.