Computational Oriented Matroids

Equivalence Classes of Matrices Within a Natural Framework

Author: Jürgen Bokowski

Publisher: Cambridge University Press

ISBN: 0521849306

Category: Computers

Page: 323

View: 8357

Oriented matroids play the role of matrices in discrete geometry, when metrical properties, such as angles or distances, are neither required nor available. Thus they are of great use in such areas as graph theory, combinatorial optimization and convex geometry. The variety of applications corresponds to the variety of ways they can be defined. Each of these definitions corresponds to a differing data structure for an oriented matroid, and handling them requires computational support, best realised through a functional language. Haskell is used here, and, for the benefit of readers, the book includes a primer on it. The combination of concrete applications and computation, the profusion of illustrations, many in colour, and the large number of examples and exercises make this an ideal introductory text on the subject. It will also be valuable for self-study for mathematicians and computer scientists working in discrete and computational geometry.

Pattern Recognition on Oriented Matroids

Author: Andrey O. Matveev

Publisher: Walter de Gruyter GmbH & Co KG

ISBN: 3110531143

Category: Mathematics

Page: 231

View: 6388

Pattern Recognition on Oriented Matroids covers a range of innovative problems in combinatorics, poset and graph theories, optimization, and number theory that constitute a far-reaching extension of the arsenal of committee methods in pattern recognition. The groundwork for the modern committee theory was laid in the mid-1960s, when it was shown that the familiar notion of solution to a feasible system of linear inequalities has ingenious analogues which can serve as collective solutions to infeasible systems. A hierarchy of dialects in the language of mathematics, for instance, open cones in the context of linear inequality systems, regions of hyperplane arrangements, and maximal covectors (or topes) of oriented matroids, provides an excellent opportunity to take a fresh look at the infeasible system of homogeneous strict linear inequalities – the standard working model for the contradictory two-class pattern recognition problem in its geometric setting. The universal language of oriented matroid theory considerably simplifies a structural and enumerative analysis of applied aspects of the infeasibility phenomenon. The present book is devoted to several selected topics in the emerging theory of pattern recognition on oriented matroids: the questions of existence and applicability of matroidal generalizations of committee decision rules and related graph-theoretic constructions to oriented matroids with very weak restrictions on their structural properties; a study (in which, in particular, interesting subsequences of the Farey sequence appear naturally) of the hierarchy of the corresponding tope committees; a description of the three-tope committees that are the most attractive approximation to the notion of solution to an infeasible system of linear constraints; an application of convexity in oriented matroids as well as blocker constructions in combinatorial optimization and in poset theory to enumerative problems on tope committees; an attempt to clarify how elementary changes (one-element reorientations) in an oriented matroid affect the family of its tope committees; a discrete Fourier analysis of the important family of critical tope committees through rank and distance relations in the tope poset and the tope graph; the characterization of a key combinatorial role played by the symmetric cycles in hypercube graphs. Contents Oriented Matroids, the Pattern Recognition Problem, and Tope Committees Boolean Intervals Dehn–Sommerville Type Relations Farey Subsequences Blocking Sets of Set Families, and Absolute Blocking Constructions in Posets Committees of Set Families, and Relative Blocking Constructions in Posets Layers of Tope Committees Three-Tope Committees Halfspaces, Convex Sets, and Tope Committees Tope Committees and Reorientations of Oriented Matroids Topes and Critical Committees Critical Committees and Distance Signals Symmetric Cycles in the Hypercube Graphs

Oriented Matroids

Author: Anders Björner

Publisher: Cambridge University Press

ISBN: 9780521777506

Category: Mathematics

Page: 548

View: 8956

First comprehensive, accessible account; second edition has expanded bibliography and a new appendix surveying recent research.

Handbook of Discrete and Computational Geometry, Third Edition

Author: Csaba D. Toth,Joseph O'Rourke,Jacob E. Goodman

Publisher: CRC Press

ISBN: 1498711421

Category: Computers

Page: 1928

View: 5831

The Handbook of Discrete and Computational Geometry is intended as a reference book fully accessible to nonspecialists as well as specialists, covering all major aspects of both fields. The book offers the most important results and methods in discrete and computational geometry to those who use them in their work, both in the academic world—as researchers in mathematics and computer science—and in the professional world—as practitioners in ?elds as diverse as operations research, molecular biology, and robotics. Discrete geometry has contributed signi?cantly to the growth of discrete mathematics in recent years. This has been fueled partly by the advent of powerful computers and by the recent explosion of activity in the relatively young ?eld of computational geometry. This synthesis between discrete and computational geometry lies at the heart of this Handbook. A growing list of application fields includes combinatorial optimization, computer-aided design, computer graphics, crystallography, data analysis, error-correcting codes, geographic information systems, motion planning, operations research, pattern recognition, robotics, solid modeling, and tomography.

Handbook of Discrete and Computational Geometry, Third Edition

Author: Csaba D. Toth,Joseph O'Rourke,Jacob E. Goodman

Publisher: CRC Press

ISBN: 1351645919

Category: Computers

Page: 1928

View: 8738

The Handbook of Discrete and Computational Geometry is intended as a reference book fully accessible to nonspecialists as well as specialists, covering all major aspects of both fields. The book offers the most important results and methods in discrete and computational geometry to those who use them in their work, both in the academic world—as researchers in mathematics and computer science—and in the professional world—as practitioners in ?elds as diverse as operations research, molecular biology, and robotics. Discrete geometry has contributed signi?cantly to the growth of discrete mathematics in recent years. This has been fueled partly by the advent of powerful computers and by the recent explosion of activity in the relatively young ?eld of computational geometry. This synthesis between discrete and computational geometry lies at the heart of this Handbook. A growing list of application fields includes combinatorial optimization, computer-aided design, computer graphics, crystallography, data analysis, error-correcting codes, geographic information systems, motion planning, operations research, pattern recognition, robotics, solid modeling, and tomography.

Mathematical Software - ICMS 2006

Second International Congress on Mathematical Software, Castro Urdiales, Spain, September 1-3, 2006, Proceedings

Author: Andres Iglesias,Nobuki Takayama

Publisher: Springer Science & Business Media

ISBN: 3540380841

Category: Computers

Page: 452

View: 2653

This volume contains the outstanding collection of invited papers and refereed papers selected for the Second International Congress on Mathematical Software, ICMS 2006, held in Castro Urdiales, Spain, September 1-3, 2006. We cordially invite you to visit the ICMS 2006 website http://www.icms2006.unican.es where you can find all relevant information about this interesting event. ICMS 2006 was the second edition of this congress, which follows up the successful ICMS 2002 held in Beijing, China. Since its inception, this congress has been a satellite event of the International Congress of Mathematicians - ICM, the world’s largest conference on mathematics, celebrated every four years since the edition of 1900 in Paris, where David Hilbert presented his 23 famous problems. For the first time, this 2006 edition of ICM is held in Spain (see: http://www.icm2006.org for details), and so is ICMS 2006. This congress was devoted to all aspects of mathematical software, whose appearance is — in our opinion — one of the most important events in mathematics. Mathematical software systems are used to construct examples, to prove theorems, and to find new mathematical phenomena. Conversely, mathematical research often motivates developments of new algorithms and new systems. Beyond mathematics, mathematical software systems are becoming indispensable tools in many branches of science and technology.

Foundations of Computational Mathematics

Proceedings of the Smalefest 2000, Hong Kong, 13-17, 2000

Author: Felipe Cucker

Publisher: World Scientific

ISBN: 9789812778031

Category: Mathematics

Page: 480

View: 2349

This invaluable book contains 19 papers selected from those submitted to a conference held in Hong Kong in July 2000 to celebrate the 70th birthday of Professor Steve Smale. It may be regarded as a continuation of the proceedings of SMALEFEST 1990 ("From Topology to Computation") held in Berkeley, USA, 10 years before, but with the focus on the area in which Smale worked more intensively during the '90's, namely the foundations of computational mathematics.

Discrete and Computational Geometry

The Goodman-Pollack Festschrift

Author: Boris Aronov,Saugata Basu,Janos Pach,Micha Sharir

Publisher: Springer Science & Business Media

ISBN: 3642555667

Category: Mathematics

Page: 853

View: 2537

An impressive collection of original research papers in discrete and computational geometry, contributed by many leading researchers in these fields, as a tribute to Jacob E. Goodman and Richard Pollack, two of the ‘founding fathers’ of the area, on the occasion of their 2/3 x 100 birthdays. The topics covered by the 41 papers provide professionals and graduate students with a comprehensive presentation of the state of the art in most aspects of discrete and computational geometry, including geometric algorithms, study of arrangements, geometric graph theory, quantitative and algorithmic real algebraic geometry, with important connections to algebraic geometry, convexity, polyhedral combinatorics, the theory of packing, covering, and tiling. The book serves as an invaluable source of reference in this discipline.

Match

Author: N.A

Publisher: N.A

ISBN: N.A

Category: Chemistry

Page: N.A

View: 831


Axioms and hulls

Author: Donald Ervin Knuth

Publisher: Springer

ISBN: N.A

Category: Mathematics

Page: 109

View: 9227

This monograph, by one of the greatest computer scientists of our time, attempts a comprehensive investigation of a basic computational problem: to find a formal mathematical characterization of which pairs of cities on a map are neighbors of each other.

Automated Deduction in Geometry

Third International Workshop, ADG 2000, Zurich, Switzerland, September 25-27, 2000, Revised Papers

Author: Jürgen Richter-Gebert,Dongming Wang

Publisher: Springer

ISBN: 3540454101

Category: Computers

Page: 328

View: 1893

This book constitutes the thoroughly refereed post-proceedings of the Third International Workshop on Automated Deduction in Geometry, ADG 2000, held in Zurich, Switzerland, in September 2000.The 16 revised full papers and two invited papers presented were carefully selected for publication during two rounds of reviewing and revision from a total of initially 31 submissions. Among the issues addressed are spatial constraint solving, automated proving of geometric inequalities, algebraic proof, semi-algebraic proofs, geometrical reasoning, computational synthetic geometry, incidence geometry, and nonstandard geometric proofs.

Algorithmen - Eine Einführung

Author: Thomas H. Cormen,Charles E. Leiserson,Ronald Rivest,Clifford Stein

Publisher: Walter de Gruyter GmbH & Co KG

ISBN: 3110522012

Category: Computers

Page: 1339

View: 3982

Der "Cormen" bietet eine umfassende und vielseitige Einführung in das moderne Studium von Algorithmen. Es stellt viele Algorithmen Schritt für Schritt vor, behandelt sie detailliert und macht deren Entwurf und deren Analyse allen Leserschichten zugänglich. Sorgfältige Erklärungen zur notwendigen Mathematik helfen, die Analyse der Algorithmen zu verstehen. Den Autoren ist es dabei geglückt, Erklärungen elementar zu halten, ohne auf Tiefe oder mathematische Exaktheit zu verzichten. Jedes der weitgehend eigenständig gestalteten Kapitel stellt einen Algorithmus, eine Entwurfstechnik, ein Anwendungsgebiet oder ein verwandtes Thema vor. Algorithmen werden beschrieben und in Pseudocode entworfen, der für jeden lesbar sein sollte, der schon selbst ein wenig programmiert hat. Zahlreiche Abbildungen verdeutlichen, wie die Algorithmen arbeiten. Ebenfalls angesprochen werden Belange der Implementierung und andere technische Fragen, wobei, da Effizienz als Entwurfskriterium betont wird, die Ausführungen eine sorgfältige Analyse der Laufzeiten der Programme mit ein schließen. Über 1000 Übungen und Problemstellungen und ein umfangreiches Quellen- und Literaturverzeichnis komplettieren das Lehrbuch, dass durch das ganze Studium, aber auch noch danach als mathematisches Nachschlagewerk oder als technisches Handbuch nützlich ist. Für die dritte Auflage wurde das gesamte Buch aktualisiert. Die Änderungen sind vielfältig und umfassen insbesondere neue Kapitel, überarbeiteten Pseudocode, didaktische Verbesserungen und einen lebhafteren Schreibstil. So wurden etwa - neue Kapitel zu van-Emde-Boas-Bäume und mehrfädigen (engl.: multithreaded) Algorithmen aufgenommen, - das Kapitel zu Rekursionsgleichungen überarbeitet, sodass es nunmehr die Teile-und-Beherrsche-Methode besser abdeckt, - die Betrachtungen zu dynamischer Programmierung und Greedy-Algorithmen überarbeitet; Memoisation und der Begriff des Teilproblem-Graphen als eine Möglichkeit, die Laufzeit eines auf dynamischer Programmierung beruhender Algorithmus zu verstehen, werden eingeführt. - 100 neue Übungsaufgaben und 28 neue Problemstellungen ergänzt. Umfangreiches Dozentenmaterial (auf englisch) ist über die Website des US-Verlags verfügbar.

Approximation and Optimization

Proceedings of the International Seminar, held in Havana, Cuba, January 12-16, 1987

Author: Juan A. Gomez-Fernandez,Francisco Guerra-Vazquez,Miguel A. Jimenez-Pozo,Guillermo Lopez-Lagomasino

Publisher: Springer

ISBN: 9783540504436

Category: Mathematics

Page: 284

View: 9028


Discrete and Computational Geometry

Papers from the DIMACS Special Year

Author: Jacob E. Goodman,Richard D. Pollack,William L. Steiger

Publisher: American Mathematical Soc.

ISBN: 9780821871010

Category: Mathematics

Page: 378

View: 8026

The first DIMACS special year, held during 1989-1990, was devoted to discrete and computational geometry. More than 200 scientists, both long- and short-term visitors, came to DIMACS to participate in the special year activities. Among the highlights were six workshops at Rutgers and Princeton Universities that defined the focus for much of the special year. The workshops addressed the following topics: geometric complexity, probabilistic methods in discrete and computational geometry, polytopes and convex sets, arrangements, and algebraic and practical issues in geometric computation. This volume presents some of the results growing out of the workshops and the special year activities. Containing both survey articles and research papers, this collection presents an excellent overview of significant recent progress in discrete and computational geometry. The diversity of these papers demonstrate how geometry continues to provide a vital source of ideas in theoretical computer science and discrete mathematics as well as fertile ground for interaction and simulation between the two disciplines.

Learning and Geometry: Computational Approaches

Author: David Kueker,Carl Smith

Publisher: Springer Science & Business Media

ISBN: 1461240883

Category: Computers

Page: 212

View: 6260

The field of computational learning theory arose out of the desire to for mally understand the process of learning. As potential applications to artificial intelligence became apparent, the new field grew rapidly. The learning of geo metric objects became a natural area of study. The possibility of using learning techniques to compensate for unsolvability provided an attraction for individ uals with an immediate need to solve such difficult problems. Researchers at the Center for Night Vision were interested in solving the problem of interpreting data produced by a variety of sensors. Current vision techniques, which have a strong geometric component, can be used to extract features. However, these techniques fall short of useful recognition of the sensed objects. One potential solution is to incorporate learning techniques into the geometric manipulation of sensor data. As a first step toward realizing such a solution, the Systems Research Center at the University of Maryland, in conjunction with the Center for Night Vision, hosted a Workshop on Learning and Geometry in January of 1991. Scholars in both fields came together to learn about each others' field and to look for common ground, with the ultimate goal of providing a new model of learning from geometrical examples that would be useful in computer vision. The papers in the volume are a partial record of that meeting.