Rudolf Mathon

The Approximate Solution of Elliptic Boundary-Value Problems by Fundamental Solutions

The boundary conditions of an elliptic equation are approximated by using fundamental solutions with singularities located outside the region of interest as trial functions. By letting the singularities change their positions a highly adaptive though nonlinear approximation is achieved employing only a small number of trial functions. The method has been found to work well for problems with boundary data or boundaries of low continuity class. Of particular interest is the ease with which it can be applied to problems in three dimensions.

On Cyclic Steiner 2-Designs

Publisher Summary This chapter focuses on cyclic Steiner 2-Designs. It surveys the literature relevant to cyclic Steiner S (2, k, u) systems. A Steiner system S (t, k, u) is a collection of k-subsets of a u-set such that every t-subset of the u-set appears in exactly one of the k-subsets. The chapter discusses the existence, enumeration, and the determination of isomorphism. A brief historical overview and required definitions and notation are given.

Boundary methods for solving elliptic problems with singularities and interfaces

Boundary approximation techniques are described for solving homogeneous self-adjoint elliptic equations. Piecewise expansions into particular solutions are used which approximate both the boundary and interface conditions in a least squares sense. Convergence of such approximations is proved and error estimates are derived in a natural norm. Numerical experiments are reported for the singular Motz problem which yield extremely accurate solutions with only a modest computational effort.

CONSTRUCTIONS FOR CYCLIC STEINER 2-DESIGNS

Abstract This paper surveys direct and recursive constructions for cyclic Steiner 2-designs. A new method is presented for cyclic designs with blocks having a prime number of elements. Several new constructions are given for designs with block size 4 which are based on perfect systems of difference sets and additive sequences of permutations.

New Maximal Arcs in Desarguesian Planes

Constructions are described of maximal arcs in Desarguesian projective planes utilizing sets of conics on a common nucleus in PG(2, q). Several new infinite families of maximal arcs in PG(2, q) are presented and a complete enumeration is carried out for Desarguesian planes of order 16, 32, and 64. For each arc we list the order of its stabilizer and the numbers of subarcs it contains. Maximal arcs may be used to construct interesting new partial geometries, 2-weight codes, and resolvable Steiner 2-designs.

Tables of Parameters of BIBDs with r⩽41 including Existence, Enumeration, and Resolvability Results

Publisher Summary A balanced incomplete block design (BIBD) is a pair ( V,B ) where V is a ν set and B is a collection of b k- subsets of V called “blocks” such that each element of V is contained in exactly r blocks, and any 2-subset of V is contained in exactly X blocks. The numbers ν, b, r, k, λ are parameters of the BIBD. A BIBD ( V,B ) is resolvable if there exists a partition R of its set of blocks B into subsets called “parallel classes,” each of which in turn partitions the set V ; R is called a “resolution.” Given a symmetric BIBD (one with u = b , r = k ) , the residual design by deleting all elements of one block and the derived design by deleting all elements of the complement of one block can be obtained.

Construction methods for Bhaskar Rao and related designs

Mathematical and computational techniques are described for constructing and enumerating generalized Bhaskar Rao designs (GBRDs) . In particular, these methods are applied to GBRD(k + 1, k, 1(k − 1); G)s for 1 ≥ 1. Properties of the enumerated designs, such as automorphism groups, resolutions and contracted designs are tabulated. Also described are applications to group divisible designs, multi-dimensional Howell cubes, generalized Room squares, equidistant permutation arrays, and doubly resolvable two-fold triple systems.

Algorithmic Techniques for the Generation and Analysis of Strongly Regular Graphs and other Combinatorial Configurations

In this paper we examine computing techniques applicable to the construction and analysis of combinatorial configurations. These techniques are first described for arbitrary configurations and then elaborated for a particular example. The generation procedures which are reviewed are backtracking and hill-climbing. In order to analyse a configuration it is often necessary to employ heuristic algorithms since the properties to be determined constitute an NP-complete problem. Various heuristic strategies are discussed for such analyses. The example consists of the family of strongly regular graphs with parameter sets (25, 12, 5, 6) and (26, 10, 3, 4). For these strongly regular graphs we present some original generation techniques as well as new results on properties of these graphs.

On self-complementary strongly regular graphs

Abstract A complete enumeration is carried out of self-complementary strongly regular graphs with fewer than 53 vertices. New necessary conditions are derived for the cycles of complementing permutations and for the block valencies of the corresponding induced adjacency matrix partitions. Properties of the enumerated graphs such as automorphism groups, clique numbers and switching classes are tabulated.

Lower bounds for Ramsey numbers and association schemes

It is shown that if a cyclotomic m-class association scheme of order q contains no k-clique then it is possible to find an edge-coloring of Km(q+1) into m colors containing no monochromatic Kk+1. This leads to greatly improved lower bounds for the Ramsey numbers r(7, 7) ⩽ 205, r(9, 9) ⩽ 565, r(10, 10) ⩽ 798, and r(11, 11) ⩾ 1597.