Energy loss dependent transversal etching rates of heavy ion tracks in plastic

By the method of electrolytical etching track etching rates V_t and corresponding transversal track etching rates V_trans of single heavy ion tracks in thin Makrofol KG foils have been measured at ion energies from 10–480 MeV/u. Makrofol KG foils of 8 μm thickness were irradiated perpendicular to the surface with ⁷⁹Au and ⁵⁴Xe ions at specific energies with energy loss values of REL=(10–90) ^*10³ MeVcm²/g at GSI Darmstadt, Germany, and Lawrence Berkeley Lab., Cal., USA. Using the electrolytical etching method by measuring the resistance of the foil during the etching process (etching conditions: 6n NaOH, room temperature and controlled 50° C) the breakthrough time and track etching rates V_t, V_trans and V_m (bulk etching rate) were analysed. Response curves (V_t/V_m)-1 as a function of Restricted Energy Loss (REL), the maximum extension of the ion induced damage perpendicular to the ion path and the dimension of the ion track core depending on the deposited energy can be estimated.     

一类abundant半群的构造

本文讨论ａｂｕｎｄａｎｔ半群上中间幂等元的性质，研究具有正规中间幂等元的ｑｕａｓｉ－ａｄｅｑｕａｔｅ半群及若干极端情形，并分别给出各类半群的构造．     

Quasi-Type δ Semigroups with an Adequate Transversal

In this paper,some properties of quasi-type δ semigroups with an adequate transversal are explored.In particular,abundant semigroups with a cancellative transversal are characterized.Our results genera...     

Hitting all maximum cliques with a stable set using lopsided independent transversals

Rabern recently proved that any graph with contains a stable set meeting all maximum cliques. We strengthen this result, proving that such a stable set exists for any graph with . This is tight, i.e. the inequality in the statement must be strict. The proof relies on finding an independent transversal in a graph partitioned into vertex sets of unequal size. © 2010 Wiley Periodicals, Inc. J Graph Theory 67:300‐305, 2011     

Multidimensional Permanents and an Upper Bound on the Number of Transversals in Latin Squares

The permanent of a multidimensional matrix is the sum of products of entries over all diagonals. A nonnegative matrix whose every 1‐dimensional plane sums to 1 is called polystochastic. A latin square of order n is an array of n symbols in which each symbol occurs exactly once in each row and each column. A transversal of such a square is a set of n entries such that no two entries share the same row, column, or symbol. Let T(n) be the maximum number of transversals over all latin squares of order n. Here, we prove that over the set of multidimensional polystochastic matrices of order n the permanent has a local extremum at the uniform matrix for whose every entry is equal to . Also, we obtain an asymptotic value of the maximal permanent for a certain set of nonnegative multidimensional matrices. In particular, we get that the maximal permanent of polystochastic matrices is asymptotically equal to the permanent of the uniform matrix, whence as a corollary we have an upper bound on the number of transversals in latin squares      

Halving transversal designs

A factor H of a transversal design TD(k,n) = (V,𝒢, ℬ︁), where V is the set of points, 𝒢 the set of groups of size n and ℬ︁ the set of blocks of size k, is a triple (V,𝒢, 𝒟) such that 𝒟 is a subset of ℬ︁. A halving of a TD (k, n) is a pair of factors H_i = (V, 𝒢, 𝒟_i), i = 1,2 such that 𝒟₁ ∪ 𝒟₂ = ℬ︁, 𝒟₁ ∩ 𝒟₂ = ∅︁ and H₁ is isomorphic to H₂. A path of length q is a sequence x₀, x₁,…,x_q of points such that for each i = 1, 2,…, q the points x_i‐1 and x_i belong to a block B_i and no point appears more than once. The distance between points x and y in a factor H is the length of the shortest path from x to y. The diameter of a connected factor H is the maximum of the set of distances among all pairs of points of H. We prove that a TD (3, n) is halvable into isomorphic factors of diameter d only if d = 2,3,4, or ∞ and we completely determine for which values of n there exists such a halvable TD (3, n). We also show that if any group divisible design with block size at least 3 is decomposed into two factors with the same finite diameter d, then d≤ 4. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 83–99, 2000     

On simple and supersimple transversal designs

In this paper, the existence of a transversal design TD_λ (4, g) is proved for all indices λ satisfying 2 ≤ λ ≤ g such that any two of its blocks intersect in at most two elements. Similar results are obtained for transversal designs without repeated blocks. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 311–320, 2000     

New recursive methods for transversal covers

A transversal cover is a set of gk points in k disjoint groups of size g and a minimum collection of transversal subset s, called blocks, such that any pair of points not contained in the same group appear in at least one block. The case g = 2 was investigated and completely solved by Sperner, Renyi, Katona, Kleitman, and Spencer. For all g, asymptotic results are known, but little is understood for small values of k. Sloane and others have initiated the investigation of g = 3. The present article is concerned with constructive techniques for all g and k. One of the principal constructions generalizes Wilson's theorem for transversal designs. This article also discusses a simulated annealing algorithm for finding transversal covers and gives a table of the best known transversal covers at this time. © 1999 John Wiley & Sons, Inc. J Combin Designs 7: 185–203, 1999