Depth, Crossings and Conflicts in Discrete Geometry
Depth, Crossings and Conflicts in Discrete Geometry

Author: Marek Sulovsk 'y

Publisher: Logos Verlag Berlin GmbH

Published: 2012

Total Pages: 180

ISBN-13: 383253119X

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Discrete geometry has been among the fastest growing fields of mathematics in the last decades. One of the most fascinating objects studied in discrete geometry are k-sets. Not only are they extremely difficult to understand but they also play an important role in estimating the running time of several geometric algorithms. This thesis presents developments in three areas related to k-sets. First, it examines the circle containment problem of Urrutia and Neumann-Lara and reveals its relationships to geometric partitioning problems and centre regions. Next, it investigates k-sets in low dimensions and generalises the k-edge crossing identity of Andrzejak et al. to the sphere. Last, it studies conflict-free colourings of geometric hypergraphs and extends many results on this topic to more restrictive list colouring variants.

Research Problems in Discrete Geometry
Research Problems in Discrete Geometry

Author: Peter Brass

Publisher: Springer Science & Business Media

Published: 2006-01-27

Total Pages: 507

ISBN-13: 0387299297

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This book is the result of a 25-year-old project and comprises a collection of more than 500 attractive open problems in the field. The largely self-contained chapters provide a broad overview of discrete geometry, along with historical details and the most important partial results related to these problems. This book is intended as a source book for both professional mathematicians and graduate students who love beautiful mathematical questions, are willing to spend sleepless nights thinking about them, and who would like to get involved in mathematical research.

Proceedings of the Twelfth Annual ACM-SIAM Symposium on Discrete Algorithms
Proceedings of the Twelfth Annual ACM-SIAM Symposium on Discrete Algorithms

Author: SIAM Activity Group on Discrete Mathematics

Publisher: SIAM

Published: 2001-01-01

Total Pages: 962

ISBN-13: 9780898714906

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Contains 130 papers, which were selected based on originality, technical contribution, and relevance. Although the papers were not formally refereed, every attempt was made to verify the main claims. It is expected that most will appear in more complete form in scientific journals. The proceedings also includes the paper presented by invited plenary speaker Ronald Graham, as well as a portion of the papers presented by invited plenary speakers Udi Manber and Christos Papadimitriou.

Forbidden Configurations in Discrete Geometry
Forbidden Configurations in Discrete Geometry

Author: David Eppstein

Publisher: Cambridge University Press

Published: 2018-05-17

Total Pages: 241

ISBN-13: 1108423914

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Unifies discrete and computational geometry by using forbidden patterns of points to characterize many of its problems.

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms
Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

Author:

Publisher: SIAM

Published: 2003-01-01

Total Pages: 896

ISBN-13: 9780898715385

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From the January 2003 symposium come just over 100 papers addressing a range of topics related to discrete algorithms. Examples of topics covered include packing Steiner trees, counting inversions in lists, directed scale-free graphs, quantum property testing, and improved results for directed multicut. The papers were not formally refereed, but attempts were made to verify major results. Annotation (c)2003 Book News, Inc., Portland, OR (booknews.com)

Discrete Geometry and Optimization
Discrete Geometry and Optimization

Author: Károly Bezdek

Publisher: Springer Science & Business Media

Published: 2013-07-09

Total Pages: 341

ISBN-13: 3319002007

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​Optimization has long been a source of both inspiration and applications for geometers, and conversely, discrete and convex geometry have provided the foundations for many optimization techniques, leading to a rich interplay between these subjects. The purpose of the Workshop on Discrete Geometry, the Conference on Discrete Geometry and Optimization, and the Workshop on Optimization, held in September 2011 at the Fields Institute, Toronto, was to further stimulate the interaction between geometers and optimizers. This volume reflects the interplay between these areas. The inspiring Fejes Tóth Lecture Series, delivered by Thomas Hales of the University of Pittsburgh, exemplified this approach. While these fields have recently witnessed a lot of activity and successes, many questions remain open. For example, Fields medalist Stephen Smale stated that the question of the existence of a strongly polynomial time algorithm for linear optimization is one of the most important unsolved problems at the beginning of the 21st century. The broad range of topics covered in this volume demonstrates the many recent and fruitful connections between different approaches, and features novel results and state-of-the-art surveys as well as open problems.

Crossing Numbers of Graphs
Crossing Numbers of Graphs

Author: Marcus Schaefer

Publisher: CRC Press

Published: 2018-01-02

Total Pages: 377

ISBN-13: 1498750508

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Crossing Numbers of Graphs is the first book devoted to the crossing number, an increasingly popular object of study with surprising connections. The field has matured into a large body of work, which includes identifiable core results and techniques. The book presents a wide variety of ideas and techniques in topological graph theory, discrete geometry, and computer science. The first part of the text deals with traditional crossing number, crossing number values, crossing lemma, related parameters, computational complexity, and algorithms. The second part includes the rich history of alternative crossing numbers, the rectilinear crossing number, the pair crossing number, and the independent odd crossing number.It also includes applications of the crossing number outside topological graph theory. Aimed at graduate students and professionals in both mathematics and computer science The first book of its kind devoted to the topic Authored by a noted authority in crossing numbers

Handbook of Discrete and Computational Geometry
Handbook of Discrete and Computational Geometry

Author: Csaba D. Toth

Publisher: CRC Press

Published: 2017-11-22

Total Pages: 1928

ISBN-13: 1498711421

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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 fields as diverse as operations research, molecular biology, and robotics. Discrete geometry has contributed significantly 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 field 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.

Front-End Vision and Multi-Scale Image Analysis
Front-End Vision and Multi-Scale Image Analysis

Author: Bart M. Haar Romeny

Publisher: Springer Science & Business Media

Published: 2008-10-24

Total Pages: 470

ISBN-13: 140208840X

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Many approaches have been proposed to solve the problem of finding the optic flow field of an image sequence. Three major classes of optic flow computation techniques can discriminated (see for a good overview Beauchemin and Barron IBeauchemin19951): gradient based (or differential) methods; phase based (or frequency domain) methods; correlation based (or area) methods; feature point (or sparse data) tracking methods; In this chapter we compute the optic flow as a dense optic flow field with a multi scale differential method. The method, originally proposed by Florack and Nielsen [Florack1998a] is known as the Multiscale Optic Flow Constrain Equation (MOFCE). This is a scale space version of the well known computer vision implementation of the optic flow constraint equation, as originally proposed by Horn and Schunck [Horn1981]. This scale space variation, as usual, consists of the introduction of the aperture of the observation in the process. The application to stereo has been described by Maas et al. [Maas 1995a, Maas 1996a]. Of course, difficulties arise when structure emerges or disappears, such as with occlusion, cloud formation etc. Then knowledge is needed about the processes and objects involved. In this chapter we focus on the scale space approach to the local measurement of optic flow, as we may expect the visual front end to do. 17. 2 Motion detection with pairs of receptive fields As a biologically motivated start, we begin with discussing some neurophysiological findings in the visual system with respect to motion detection.

Surveys on Discrete and Computational Geometry
Surveys on Discrete and Computational Geometry

Author: Jacob E. Goodman

Publisher: American Mathematical Soc.

Published: 2008

Total Pages: 568

ISBN-13: 0821842390

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This volume contains nineteen survey papers describing the state of current research in discrete and computational geometry as well as a set of open problems presented at the 2006 AMS-IMS-SIAM Summer Research Conference Discrete and Computational Geometry--Twenty Years Later, held in Snowbird, Utah, in June 2006. Topics surveyed include metric graph theory, lattice polytopes, the combinatorial complexity of unions of geometric objects, line and pseudoline arrangements, algorithmic semialgebraic geometry, persistent homology, unfolding polyhedra, pseudo-triangulations, nonlinear computational geometry, $k$-sets, and the computational complexity of convex bodies.