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Author: Carla D. Cotwright Publisher: ISBN: 9781109972139 Category : Languages : en Pages : 89
Book Description
Results that relate clones in a matroid to minors of that matroid are given. Also, matroids that contain few clonal-classes are characterized. An example of a result of the first type that is given is that if X is a four-element set in a 3-connected non-binary matroid M and X contains a clone-pair, then M has a U2.4-minor that uses X. This result generalizes several results from the literature such as Tutte's Excluded-Minor characterization of the binary matroids. An example of a result of the second type is Theorem 4.5.2 where the matroids having exactly two clonal-classes are characterized using certain truncations of the direct sum of two uniform matroids.
Author: Carla D. Cotwright Publisher: ISBN: 9781109972139 Category : Languages : en Pages : 89
Book Description
Results that relate clones in a matroid to minors of that matroid are given. Also, matroids that contain few clonal-classes are characterized. An example of a result of the first type that is given is that if X is a four-element set in a 3-connected non-binary matroid M and X contains a clone-pair, then M has a U2.4-minor that uses X. This result generalizes several results from the literature such as Tutte's Excluded-Minor characterization of the binary matroids. An example of a result of the second type is Theorem 4.5.2 where the matroids having exactly two clonal-classes are characterized using certain truncations of the direct sum of two uniform matroids.
Author: Joshua Adam Gray Publisher: ISBN: Category : Languages : en Pages : 96
Book Description
Two elements e and f in a matroid M are clones if and only if the mapping that interchanges e and f and fixes all other elements of M is an automorphism. In this dissertation, we have associated clone sets in representable matroids to arcs in projective spaces Chapter One gives a survey of basic topics in matroids and results on clone sets. In Chapter Two we give definitions of projective spaces and arcs and a survey of bounds on arc sizes. In Chapter Three we improve the bound on the size of a clone set in a 3-connected [Special characters omitted.] F -representable matroid using known bounds on the size of arcs in projective spaces. Furthermore in Chapter Four we extend a result of Reid and Zhou on clone sets in 3-connected matroids to clone sets in n -connected matroids and associate the representability of an n -connected matroid with a clone set of a specified size to the representability of a similar class of matroids. Using this result, we have used similar techniques to those in Chapter Three to obtain some preliminary bounds on the size of a clone set in an n -connected [Special characters omitted.] F -representable matroid and establish a conjecture on the size of a clone set in an n -connected [Special characters omitted.] F -representable matroid.
Author: Geoffrey Grimmett Publisher: Cambridge University Press ISBN: 1108542999 Category : Mathematics Languages : en Pages : 279
Book Description
This introduction to some of the principal models in the theory of disordered systems leads the reader through the basics, to the very edge of contemporary research, with the minimum of technical fuss. Topics covered include random walk, percolation, self-avoiding walk, interacting particle systems, uniform spanning tree, random graphs, as well as the Ising, Potts, and random-cluster models for ferromagnetism, and the Lorentz model for motion in a random medium. This new edition features accounts of major recent progress, including the exact value of the connective constant of the hexagonal lattice, and the critical point of the random-cluster model on the square lattice. The choice of topics is strongly motivated by modern applications, and focuses on areas that merit further research. Accessible to a wide audience of mathematicians and physicists, this book can be used as a graduate course text. Each chapter ends with a range of exercises.
Author: Geoffrey Grimmett Publisher: ISBN: Category : Mathematics Languages : en Pages : 330
Book Description
Professor Dominic Welsh has made significant contributions to the fields of combinatorics and discrete probability, including matroids, complexity, and percolation, and has taught, influenced and inspired generations of students and researchers in mathematics. This volume summarizes and reviews the consistent themes from his work through a series of articles written by renowned experts. These articles contain original research work, set in a broader context by the inclusion of review material. As a reference text in its own right, this book will be valuable to academic researchers, research students, and others seeking an introduction to the relevant contemporary aspects of these fields.
Author: Alexandre V. Borovik Publisher: Springer Science & Business Media ISBN: 9780817637644 Category : Mathematics Languages : en Pages : 292
Book Description
Matroids appear in diverse areas of mathematics, from combinatorics to algebraic topology and geometry, and "Coxeter Matroids" provides an intuitive and interdisciplinary treatment of their theory. In this text, matroids are examined in terms of symmetric and finite reflection groups; also, symplectic matroids and the more general coxeter matroids are carefully developed. The Gelfand-Serganova theorem, which allows for the geometric interpretation of matroids as convex polytopes with certain symmetry properties, is presented, and in the final chapter, matroid representations and combinatorial flag varieties are discussed. With its excellent bibliography and index and ample references to current research, this work will be useful for graduate students and research mathematicians.
Author: Bernhard Korte Publisher: Springer Science & Business Media ISBN: 3642581919 Category : Mathematics Languages : en Pages : 213
Book Description
With the advent of computers, algorithmic principles play an ever increasing role in mathematics. Algorithms have to exploit the structure of the underlying mathematical object, and properties exploited by algorithms are often closely tied to classical structural analysis in mathematics. This connection between algorithms and structure is in particular apparent in discrete mathematics, where proofs are often constructive, and can be turned into algorithms more directly. The principle of greediness plays a fundamental role both in the design of continuous algorithms (where it is called the steepest descent or gradient method) and of discrete algorithms. The discrete structure most closely related to greediness is a matroid; in fact, matroids may be characterized axiomatically as those independence systems for which the greedy solution is optimal for certain optimization problems (e.g. linear objective functions, bottleneck functions). This book is an attempt to unify different approaches and to lead the reader from fundamental results in matroid theory to the current borderline of open research problems. The monograph begins by reviewing classical concepts from matroid theory and extending them to greedoids. It then proceeds to the discussion of subclasses like interval greedoids, antimatroids or convex geometries, greedoids on partially ordered sets and greedoid intersections. Emphasis is placed on optimization problems in greedois. An algorithmic characterization of greedoids in terms of the greedy algorithm is derived, the behaviour with respect to linear functions is investigated, the shortest path problem for graphs is extended to a class of greedoids, linear descriptions of antimatroid polyhedra and complexity results are given and the Rado-Hall theorem on transversals is generalized. The self-contained volume which assumes only a basic familarity with combinatorial optimization ends with a chapter on topological results in connection with greedoids.
Author: Carla D. Cotwright
Author: Carla D. Cotwright
Author: Joshua Adam Gray
Author: American Mathematical Society
Author: Gary Gordon
Author: Geoffrey Grimmett
Author: Geoffrey Grimmett
Author: Alexandre V. Borovik
Author: Robert E. Bixby
Author: Bernhard Korte
Author: