Abstract

“If a theorem about graphs can be expressed in terms of edges and circuits only it probably exemplifies a more general theorem about matroids.” This assertion, made by Tutte more than twenty years ago, will be the theme of this paper. In particular, a number of examples will be given of the two-way interaction between graph theory and matroid theory that enriches both subjects.

Introduction

This paper aims to be accessible to those with no previous experience of matroids; only some basic familiarity with graph theory and linear algebra will be assumed. In particular, the next section introduces matroids by showing how such objects arise from graphs. It then presents a minimal amount of theory to make the rest of the paper comprehensible. Throughout, the emphasis is on the links between graphs and matroids.

Section 3 begins by showing how 2-connectedness for graphs extends naturally to matroids. It then indicates how the number of edges in a 2-connected loopless graph can be bounded in terms of the circumference and the size of a largest bond. The main result of the section extends this graph result to matroids. The results in this section provide an excellent example of the two-way interaction between graph theory and matroid theory.

In order to increase the accessibility of this paper, the matroid technicalities have been kept to a minimum. Most of those that do arise have been separated from the rest of the paper and appear in two separate sections, 4 and 10, which deal primarily with proofs. The first of these sections outlines the proofs of the main results from Section 3.

Abstract

This paper connects the practice of wireless communication with the mathematics of quadratic forms developed by Radon and Hurwitz about a hundred years ago. Orthogonal designs, known as space-time block codes in the communications literature, provide the bridge between the two subjects. The columns of the design represent different time slots, the rows represent different transmit antennas, and the entries are the symbols to be transmitted. Multiple transmit antennas provide independent paths from the base station to the mobile terminal, and in effect this creates a single channel that is more reliable than any constituent path. The mathematics developed by Hurwitz and Radon is used to derive fundamental limits on transmission rates. The algebraic structure of the 2 × 2 space-time block code (a representation of Hamilton's biquaternions) is used to suppress interference from a second space-time user, when a second antenna is available at the mobile terminal.

Introduction

Classical coding theory is concerned with the representation of information that is to be transmitted over some noisy channel. This general framework includes the algebraic theory of error correcting codes, where codewords are strings of symbols taken from some finite field, and it includes data transmission over Gaussian channels, where codewords are vectors in Euclidean space. Fifty years of information theory and coding has led to a number of consumer products that make essential use of coding to improve reliability; for example, compact disk players, hard disk drives and wireline modems. The discovery of turbo codes by Berrou, Glavieux, and Thitmajshima [3] has led to the construction of codes that essentially achieve the Shannon capacity of the Gaussian channel.

Abstract

A survey of the most important results on partial m-systems and m-systems of finite classical polar spaces will be given. Also, the paper contains several recent results on the topic. Finally, many applications of m-systems to strongly regular graphs, linear projective two-weight codes, maximal arcs, generalized quadrangles and semi-partial geometries are mentioned.

Introduction

Let P be a finite polar space of rank r ≥ 2. An ovoid O of P is a pointset of P, which has exactly one point in common with each generator of P, that is, with each maximal totally singular subspace of P. A spread S of P is a set of generators, which constitutes a partition of the pointset. It appears that ∣O∣ = ∣S∣ for any ovoid O and any spread S of any given polar space P; this common number will be denoted by μP. Ovoids and spreads have many connections with and applications to projective planes, circle geometries, generalized polygons, strongly regular graphs, partial geometries, semi-partial geometries, codes, designs.

A partial msystem of P, with 0 ≤ m ≤ r − 1, is any set {π1, π2,…,πk} of (k≠0) totally singular m-spaces of P such that no generator containing πi has a point in common with (π1 ∪ π2 ∪ … ∪ πk) − πi,πk, with i = 1, 2,…, k. For any partial m-system M of P the bound ∣M∣ ≤ μP holds. If ∣M∣ = μP, then the partial m-system M of P is called an m-system of P.

On the occasion of the 18th British Combinatorial Conference at the University of Sussex, 1 to 6 July, 2001, this book comprises the survey papers by the nine invited speakers and a memoire of Crispin Nash-Williams, past chairman of the British Combinatorial Committee.

The survey papers range across many parts of modern combinatorics.

Martin Aigner discusses the ideas of Penrose on the 4-colour problem, as well as the application of Penrose polynomials to other combinatorial structures.

Ian Anderson surveys some of the key ideas in the study of cyclic designs, including some of the classical results of the past 150 years as well as some very recent developments.

Robert Calderbank and Ayman Naguib show the connection between the practice of wireless communication with the mathematics of quadratic forms developed by Radon and Hurwitz about a hundred years ago. This occurs through orthogonal designs, known as space-time block codes in the communications literature.

Leslie Goldberg surveys the computational problems of randomly sampling unlabelled combinatorial structures, and of counting and approximately counting unlabelled structures.

Bojan Mohar considers the interplay between graph minors and graphs embedded in surfaces.

Michael Molloy surveys the progress on two fundamental problems in random graphs and random boolean formulae. The first is the question of how many edges must be added to a random graph until it is not almost surely k-colourable.

1. Introduction

Using random matrices to count combinatorial objects is not a new idea. It stems from the pioneering work [Brezin et al. 1978], which showed how the perturbative expansion of a simple nongaussian matrix integral led, using standard Feynman diagram techniques, to the counting of discretized surfaces. It has resulted in many applications: from the physical side, it allowed to define a discretized version of 2D quantum gravity [Di Francesco et al. 1995] and to study various statistical models on random lattices [Kazakov 1986; Rostov 1989; Gaudin and Rostov 1989; Rostov and Staudacher 1992]. From the mathematical side, let us cite the Rontsevitch integral [Rontsevich 1991; Witten 1991; Itzykson and Zuber 1992], and the counting of meanders and foldings [Makeenko 1996; Di Francesco et al. 1997; 1998].

Here we shall try to apply this idea to the field of knot theory. Our basic aim will be to count knots or related objects. The next section defines these objects, and is followed by a brief overview of matrix models and how they can be related to knots.

1. Introduction

This is a survey article dealing with connections between orthogonal polynomials, functional equations they satisfy, and some extremal problems. Although the results surveyed are not new we believe that we are putting together results from different sources which appear together for the first time, many of them are of recent vintage. One question in the theory of orthogonal polynomials is how the zeros of a parameter dependent sequence of orthogonal polynomials change with the parameters involved. Stieltjes [1885a; 1885b] proved that the zeros of Jacobi polynomials increase with β and decrease with a for a > -1 and β > -1. The Jacobi polynomials satisfy the following orthogonality relation [Szego 1975, (4.3.3)]:

In Section 2 we state Stieltjes's results and describe the circle of ideas around them. Section 3 surveys the Coulomb Fluid method of Freeman Dyson and its potential theoretic set-up. We shall use the shifted factorial notion [Andrews et al. 1999]

1. Introduction

A class of problems — important for their applications to computer science and computational biology as well as for their inherent mathematical interest — is the statistical analysis of a string of random symbols. The symbols, called letters, are assumed to belong to an alphabet A of fixed size k. The set of all such strings (or words) of length N, W(A,N), forms the sample space in the statistical analysis of these strings. A natural measure on W is to assign each letter equal probability, namely 1/ k, and to define the probability measure on words by the product measure. Thus each letter in a word occurs independently and with equal probability. We call such random word models homogeneous. Of course, for some applications, each letter in the alphabet does not occur with the same frequency and it is therefore natural to assign to each letter i a probability pi. If we again use the product measure for the words (letters in a word occur independently), then the resulting random word models are called inhomogeneous. Fixing an ordering of the alphabet A, a weakly increasing subsequence of a Word.

1. Introduction la. Isomonodromic Deformation Equations. We consider rational covariant derivative operators on the punctured Riemann sphere, having the form They have regular singular points at and an irregular singularity at with Poincare index 1. If the residue matrices are deformed differentiably with respect to the parameters and the monodromy (including Stokes parameters and connection matrices) of the operator will be invariant under such deformations, as was shown in [Jimbo et al. 1980; Jimbo et al. 1981], provided the differential equations implied by the commutativity conditions.

1. Introduction

A well-known theme in random matrix theory (RMT), zeta functions, quantum chaos, and statistical mechanics, is the universality of scaling limits of correlation functions. In RMT, the relevant correlation functions are for eigenvalues of random matrices (see [De; TW; BZ; BK; So] and their references). In the case of zeta functions, the correlations are between the zeros [KS]. In quantum dynamics, they are between eigenvalues of ‘typical’ quantum maps whose underlying classical maps have a specified dynamics. In the ‘chaotic case’ it is conjectured that the correlations should belong to the universality class of RMT, while in integrable cases they should belong to that of Poisson processes. The latter has been confirmed for certain families of integrable quantum maps, scattering matrices and Hamiltonians (see [Ze2; RS; Sa; ZZ] and their references). In statistical mechanics, there is a large literature on universality of critical exponents [Car]; other rigorous results include analysis of universal scaling limits of Gibbs measures at critical points [Sin]. In this article we are concerned with a somewhat new arena for scaling and universality, namely that of RPT (random polynomial theory) and its algebro-geometric generalizations [Han; Hal; BBL; BD; BSZ1; BSZ2; SZ1; NV]. The focus of these articles is on the configurations and correlations of zeros of random polynomials and their generalizations, which we discuss below. Random polynomials can also be used to define random holomorphic maps to projective space, but we leave that for the future. Our purpose here is partly to review the results of [SZ1; BSZ1; BSZ2] on universality of scaling limits of correlations between zeros of random holomorphic sections on complex manifolds.