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Topology of Random Simplicial Complexes and Phase Transitions for Homology

2007 Complexes
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Release : 2007
Genre : Complexes
Kind : eBook
Book Rating : 665/5 ( reviews)

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Book Synopsis Topology of Random Simplicial Complexes and Phase Transitions for Homology by : Matthew Kahle

Download or read book Topology of Random Simplicial Complexes and Phase Transitions for Homology written by Matthew Kahle. This book was released on 2007. Available in PDF, EPUB and Kindle. Book excerpt: In this dissertation we discuss different models of random simplicial complexes and compute facts about their expected topological features. Each of the models is based in some way on the Erdős-Renyi random graph G(n, p).

Handbook of Discrete and Computational Geometry

2017-11-22 Computers
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Release : 2017-11-22
Genre : Computers
Kind : eBook
Book Rating : 421/5 ( reviews)

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Book Synopsis Handbook of Discrete and Computational Geometry by : Csaba D. Toth

Download or read book Handbook of Discrete and Computational Geometry written by Csaba D. Toth. This book was released on 2017-11-22. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Torsion in Homology of Random Simplicial Complexes

2018 Homology theory
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Release : 2018
Genre : Homology theory
Kind : eBook
Book Rating : /5 ( reviews)

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Book Synopsis Torsion in Homology of Random Simplicial Complexes by : J. Andrew Newman

Download or read book Torsion in Homology of Random Simplicial Complexes written by J. Andrew Newman. This book was released on 2018. Available in PDF, EPUB and Kindle. Book excerpt: During the mid-twentieth century, Paul Erdos and Alfréd Rényi developed their now-standard random graph model. Beyond being practical in graph theory to nonconstructively prove the existence of graphs with certain interesting properties, the Erdős–Rényi model is also a model for generating random (one-dimensional) topological spaces. Within the last fifteen years, this model has been generalized to the higher-dimensional simplicial complex model of Nati Linial and Roy Meshulam. As in the case of the probabilistic method more generally, there are (at least) two reasons why one might apply random methods in topology: to understand what a "typical" topological space looks like and to give nonconstructive proofs of the existence of topological spaces with certain properties. Here we consider both of these applications of randomness in topology in considering the properties of torsion in homology of simplicial complexes. For the former, we discuss experimental results that strongly suggest torsion in homology of random Linial–Meshulam complexes is distributed according to Cohen–Lenstra heuristics. For the latter, we use the probabilistic method to give an upper bound on the number of vertices required to construct d-dimensional simplicial complexes with prescribed torsion in homology. This upper bound is optimal in the sense that it is a constant multiple of a known lower bound.

Simplicial Structures in Topology

2010-09-30 Mathematics
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Release : 2010-09-30
Genre : Mathematics
Kind : eBook
Book Rating : 366/5 ( reviews)

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Book Synopsis Simplicial Structures in Topology by : Davide L. Ferrario

Download or read book Simplicial Structures in Topology written by Davide L. Ferrario. This book was released on 2010-09-30. Available in PDF, EPUB and Kindle. Book excerpt: Simplicial Structures in Topology provides a clear and comprehensive introduction to the subject. Ideas are developed in the first four chapters. The fifth chapter studies closed surfaces and gives their classification. The last chapter of the book is devoted to homotopy groups, which are used in short introduction on obstruction theory. The text is more in tune with the original development of algebraic topology as given by Henry Poincaré (singular homology is discussed). Illustrative examples throughout and extensive exercises at the end of each chapter for practice enhance the text. Advanced undergraduate and beginning graduate students will benefit from this book. Researchers and professionals interested in topology and applications of mathematics will also find this book useful.

Fractal Geometry and Stochastics VI

2021-03-23 Mathematics
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Release : 2021-03-23
Genre : Mathematics
Kind : eBook
Book Rating : 494/5 ( reviews)

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Book Synopsis Fractal Geometry and Stochastics VI by : Uta Freiberg

Download or read book Fractal Geometry and Stochastics VI written by Uta Freiberg. This book was released on 2021-03-23. Available in PDF, EPUB and Kindle. Book excerpt: This collection of contributions originates from the well-established conference series "Fractal Geometry and Stochastics" which brings together researchers from different fields using concepts and methods from fractal geometry. Carefully selected papers from keynote and invited speakers are included, both discussing exciting new trends and results and giving a gentle introduction to some recent developments. The topics covered include Assouad dimensions and their connection to analysis, multifractal properties of functions and measures, renewal theorems in dynamics, dimensions and topology of random discrete structures, self-similar trees, p-hyperbolicity, phase transitions from continuous to discrete scale invariance, scaling limits of stochastic processes, stemi-stable distributions and fractional differential equations, and diffusion limited aggregation. Representing a rich source of ideas and a good starting point for more advanced topics in fractal geometry, the volume will appeal to both established experts and newcomers.

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