Problems and Snapshots from the World of Probability

Problems and Snapshots from the World of Probability

Author: Gunnar Blom

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 252

ISBN-13: 1461243041

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We, the authors of this book, are three ardent devotees of chance, or some what more precisely, of discrete probability. When we were collecting the material, we felt that one special pleasure of the field lay in its evocation of an earlier age: many of our 'probabilistic forefathers' were dexterous solvers of discrete problems. We hope that this pleasure will be transmitted to the readers. The first problem-book of a similar kind as ours is perhaps Mosteller's well-known Fifty Challenging Problems in Probability (1965). Possibly, our book is the second. The book contains 125 problems and snapshots from the world of prob ability. A 'problem' generally leads to a question with a definite answer. A 'snapshot' is either a picture or a bird's-eye view of some probabilistic field. The selection is, of course, highly subjective, and we have not even tried to cover all parts of the subject systematically. Limit theorems appear only seldom, for otherwise the book would have become unduly large. We want to state emphatically that we have not written a textbook in probability, but rather a book for browsing through when occupying an easy-chair. Therefore, ideas and results are often put forth without a machinery of formulas and derivations; the conscientious readers, who want to penetrate the whole clockwork, will soon have to move to their desks and utilize appropriate tools.


Random Walks on Infinite Graphs and Groups

Random Walks on Infinite Graphs and Groups

Author: Wolfgang Woess

Publisher: Cambridge University Press

Published: 2000-02-13

Total Pages: 350

ISBN-13: 0521552923

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The main theme of this book is the interplay between the behaviour of a class of stochastic processes (random walks) and discrete structure theory. The author considers Markov chains whose state space is equipped with the structure of an infinite, locally finite graph, or as a particular case, of a finitely generated group. The transition probabilities are assumed to be adapted to the underlying structure in some way that must be specified precisely in each case. From the probabilistic viewpoint, the question is what impact the particular type of structure has on various aspects of the behaviour of the random walk. Vice-versa, random walks may also be seen as useful tools for classifying, or at least describing the structure of graphs and groups. Links with spectral theory and discrete potential theory are also discussed. This book will be essential reading for all researchers working in stochastic process and related topics.


Probability on Graphs

Probability on Graphs

Author: Geoffrey Grimmett

Publisher: Cambridge University Press

Published: 2018-01-25

Total Pages: 279

ISBN-13: 1108542999

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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.


Matroid Theory

Matroid Theory

Author: D. J. A. Welsh

Publisher: Courier Corporation

Published: 2010-01-01

Total Pages: 450

ISBN-13: 0486474399

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The theory of matroids connects disparate branches of combinatorial theory and algebra such as graph and lattice theory, combinatorial optimization, and linear algebra. This text describes standard examples and investigation results, and it uses elementary proofs to develop basic matroid properties before advancing to a more sophisticated treatment. 1976 edition.


Random Walks and Electric Networks

Random Walks and Electric Networks

Author: Peter G. Doyle

Publisher: American Mathematical Soc.

Published: 1984-12-31

Total Pages: 174

ISBN-13: 1614440220

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Probability theory, like much of mathematics, is indebted to physics as a source of problems and intuition for solving these problems. Unfortunately, the level of abstraction of current mathematics often makes it difficult for anyone but an expert to appreciate this fact. Random Walks and electric networks looks at the interplay of physics and mathematics in terms of an example—the relation between elementary electric network theory and random walks —where the mathematics involved is at the college level.


Random Graph Dynamics

Random Graph Dynamics

Author: Rick Durrett

Publisher: Cambridge University Press

Published: 2010-05-31

Total Pages: 203

ISBN-13: 1139460889

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The theory of random graphs began in the late 1950s in several papers by Erdos and Renyi. In the late twentieth century, the notion of six degrees of separation, meaning that any two people on the planet can be connected by a short chain of people who know each other, inspired Strogatz and Watts to define the small world random graph in which each site is connected to k close neighbors, but also has long-range connections. At a similar time, it was observed in human social and sexual networks and on the Internet that the number of neighbors of an individual or computer has a power law distribution. This inspired Barabasi and Albert to define the preferential attachment model, which has these properties. These two papers have led to an explosion of research. The purpose of this book is to use a wide variety of mathematical argument to obtain insights into the properties of these graphs. A unique feature is the interest in the dynamics of process taking place on the graph in addition to their geometric properties, such as connectedness and diameter.


Random Walks and Heat Kernels on Graphs

Random Walks and Heat Kernels on Graphs

Author: M. T. Barlow

Publisher: Cambridge University Press

Published: 2017-02-23

Total Pages: 239

ISBN-13: 1107674425

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Useful but hard-to-find results enrich this introduction to the analytic study of random walks on infinite graphs.


Inequalities: Theory of Majorization and Its Applications

Inequalities: Theory of Majorization and Its Applications

Author: Albert W. Marshall

Publisher: Springer Science & Business Media

Published: 2010-11-25

Total Pages: 919

ISBN-13: 0387682767

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This book’s first edition has been widely cited by researchers in diverse fields. The following are excerpts from reviews. “Inequalities: Theory of Majorization and its Applications” merits strong praise. It is innovative, coherent, well written and, most importantly, a pleasure to read. ... This work is a valuable resource!” (Mathematical Reviews). “The authors ... present an extremely rich collection of inequalities in a remarkably coherent and unified approach. The book is a major work on inequalities, rich in content and original in organization.” (Siam Review). “The appearance of ... Inequalities in 1979 had a great impact on the mathematical sciences. By showing how a single concept unified a staggering amount of material from widely diverse disciplines–probability, geometry, statistics, operations research, etc.–this work was a revelation to those of us who had been trying to make sense of his own corner of this material.” (Linear Algebra and its Applications). This greatly expanded new edition includes recent research on stochastic, multivariate and group majorization, Lorenz order, and applications in physics and chemistry, in economics and political science, in matrix inequalities, and in probability and statistics. The reference list has almost doubled.


Introduction to Random Graphs

Introduction to Random Graphs

Author: Alan Frieze

Publisher: Cambridge University Press

Published: 2016

Total Pages: 483

ISBN-13: 1107118506

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The text covers random graphs from the basic to the advanced, including numerous exercises and recommendations for further reading.