Probability on Real Lie Algebras

Author: Uwe Franz,Nicolas Privault

Publisher: Cambridge University Press

ISBN: 110712865X

Category: Mathematics

Page: 302

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This monograph is a progressive introduction to non-commutativity in probability theory, summarizing and synthesizing recent results about classical and quantum stochastic processes on Lie algebras. In the early chapters, focus is placed on concrete examples of the links between algebraic relations and the moments of probability distributions. The subsequent chapters are more advanced and deal with Wigner densities for non-commutative couples of random variables, non-commutative stochastic processes with independent increments (quantum Lévy processes), and the quantum Malliavin calculus. This book will appeal to advanced undergraduate and graduate students interested in the relations between algebra, probability, and quantum theory. It also addresses a more advanced audience by covering other topics related to non-commutativity in stochastic calculus, Lévy processes, and the Malliavin calculus.

Analysis and Geometry on Groups

Author: Nicholas T. Varopoulos,L. Saloff-Coste,T. Coulhon

Publisher: Cambridge University Press

ISBN: 9780521088015

Category: Mathematics

Page: 172

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The geometry and analysis that is discussed in this book extends to classical results for general discrete or Lie groups, and the methods used are analytical, but are not concerned with what is described these days as real analysis. Most of the results described in this book have a dual formulation: they have a "discrete version" related to a finitely generated discrete group and a continuous version related to a Lie group. The authors chose to center this book around Lie groups, but could easily have pushed it in several other directions as it interacts with the theory of second order partial differential operators, and probability theory, as well as with group theory.

Dynamical Systems and Semisimple Groups

An Introduction

Author: Renato Feres,RENATO AUTOR FERES

Publisher: Cambridge University Press

ISBN: 9780521591621

Category: Mathematics

Page: 245

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This book comprises a systematic, self-contained introduction to the Margulis-Zimmer theory and provides an entry into current research. Taking as prerequisites only the standard first-year graduate courses in mathematics, the author develops in a detailed and self-contained way the main results on Lie groups, Lie algebras, and semisimple groups, including basic facts normally covered in first courses on manifolds and Lie groups plus topics such as integration of infinitesimal actions of Lie groups. He then derives the basic structure theorems for the real semisimple Lie groups, such as the Cartan and Iwasawa decompositions, and gives an extensive exposition of the general facts and concepts from topological dynamics and ergodic theory, including detailed proofs of the multiplicative ergodic theorem and Moore's ergodicity theorem.

The Mathieu Groups

Author: A. A. Ivanov

Publisher: Cambridge University Press

ISBN: 1108429785

Category: Mathematics

Page: 186

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The Mathieu Groups are presented in the context of finite geometry and the theory of group amalgams.

Representations of Elementary Abelian p-Groups and Vector Bundles

Author: David J. Benson

Publisher: Cambridge University Press

ISBN: 1316802736

Category: Mathematics

Page: N.A

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Questions about modular representation theory of finite groups can often be reduced to elementary abelian subgroups. This is the first book to offer a detailed study of the representation theory of elementary abelian groups, bringing together information from many papers and journals, as well as unpublished research. Special attention is given to recent work on modules of constant Jordan type, and the methods involve producing and examining vector bundles on projective space and their Chern classes. Extensive background material is provided, which will help the reader to understand vector bundles and their Chern classes from an algebraic point of view, and to apply this to modular representation theory of elementary abelian groups. The final section, addressing problems and directions for future research, will also help to stimulate further developments in the subject. With no similar books on the market, this will be an invaluable resource for graduate students and researchers working in representation theory.

Fourier Integrals in Classical Analysis

Author: Christopher D. Sogge

Publisher: Cambridge University Press

ISBN: 9780521434645

Category: Mathematics

Page: 236

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An advanced monograph concerned with modern treatments of central problems in harmonic analysis.

Understanding Markov Chains

Examples and Applications

Author: Nicolas Privault

Publisher: Springer Science & Business Media

ISBN: 9814451517

Category: Mathematics

Page: 354

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This book provides an undergraduate introduction to discrete and continuous-time Markov chains and their applications. A large focus is placed on the first step analysis technique and its applications to average hitting times and ruin probabilities. Classical topics such as recurrence and transience, stationary and limiting distributions, as well as branching processes, are also covered. Two major examples (gambling processes and random walks) are treated in detail from the beginning, before the general theory itself is presented in the subsequent chapters. An introduction to discrete-time martingales and their relation to ruin probabilities and mean exit times is also provided, and the book includes a chapter on spatial Poisson processes with some recent results on moment identities and deviation inequalities for Poisson stochastic integrals. The concepts presented are illustrated by examples and by 72 exercises and their complete solutions.

Permutation Group Algorithms

Author: Ákos Seress

Publisher: Cambridge University Press

ISBN: 9780521661034

Category: Computers

Page: 264

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Theory of permutation group algorithms for graduates and above. Exercises and hints for implementation throughout.

Induced Representations of Locally Compact Groups

Author: Eberhard Kaniuth,Keith F. Taylor

Publisher: Cambridge University Press

ISBN: 052176226X

Category: Mathematics

Page: 343

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"Locally compact groups arise in many diverse areas of mathematics, the physical sciences, and engineering and the presence of the group is usually felt through unitary representations of the group. This observation underlies the importance of understanding such representations and how they may be constructed, combined, or decomposed. Of particular importance are the irreducible unitary representations. In the middle of the last century, G.W. Mackey initiated a program to develop a systematic method for identifying all the irreducible unitary representations of a given locally compact group G. We denote the set of all unitary equivalence classes of irreducible unitary representations of G by G. Mackey's methods are only effective when G has certain restrictive structural characteristics; nevertheless, time has shown that many of the groups that arise in important problems are appropriate for Mackey's approach. The program Mackey initiated received contributions from many researchers with some of the mostsubstantial advances made by R.J. Blattner and J.M.G. Fell. Fell'swork is particularly important in studying Gas a topological space. At the core of this program is the inducing construction, which is a method of building a unitary representation of a group from a representation of a subgroup"--

Cambridge Tracts in Mathematics

Author: Jean Bertoin

Publisher: Cambridge University Press

ISBN: 9780521646321

Category: Mathematics

Page: 266

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This 1996 book is a comprehensive account of the theory of Lévy processes; aimed at probability theorists.

Probability and Computing

Randomized Algorithms and Probabilistic Analysis

Author: Michael Mitzenmacher,Eli Upfal

Publisher: Cambridge University Press

ISBN: 9780521835404

Category: Computers

Page: 352

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Randomization and probabilistic techniques play an important role in modern computer science, with applications ranging from combinatorial optimization and machine learning to communication networks and secure protocols.Assuming only an elementary background in discrete mathematics, this textbook is designed to accompany a one- or two-semester course for advanced undergraduates or beginning graduate students in computer science and applied mathematics. It gives an excellent introduction to the probabilistic techniques and paradigms used in the development of probabilistic algorithms and analyses, including random sampling, expectations, Markov's and Chevyshev's inequalities, Chernoff bounds, balls and bins models, the probabilistic method, Markov chains, MCMC, martingales, entropy, and other topics.

Rings and Factorization

Author: David Sharpe

Publisher: CUP Archive

ISBN: 9780521337182

Category: Mathematics

Page: 111

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This textbook is an introduction to the concept of factorization and its application to problems in algebra and number theory. With the minimum of prerequisites, the reader is introduced to the notion of rings, fields, prime elements and unique factorization. The author shows how concepts can be applied to a variety of examples such as factorizing polynomials, finding determinants of matrices and Fermat's 'two-squares theorem'. Based on an undergraduate course given at the University of Sheffield, Dr Sharpe has included numerous examples which demonstrate how frequently these ideas are useful in concrete, rather than abstract, settings. The book also contains many exercises of varying degrees of difficulty together with hints and solutions. Second and third year undergraduates will find this a readable and enjoyable account of a subject lying at the heart of much of mathematics.

The Taming of Chance

Author: Ian Hacking

Publisher: Cambridge University Press

ISBN: 9780521388849

Category: History

Page: 264

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This book combines detailed scientific historical research with characteristic philosophic breadth and verve.

An Elementary Introduction to Stochastic Interest Rate Modeling

Author: Nicolas Privault

Publisher: World Scientific

ISBN: 9814390860

Category: Business & Economics

Page: 228

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Interest rate modeling and the pricing of related derivatives remain subjects of increasing importance in financial mathematics and risk management. This book provides an accessible introduction to these topics by a step-by-step presentation of concepts with a focus on explicit calculations. Each chapter is accompanied with exercises and their complete solutions, making the book suitable for advanced undergraduate and graduate level students. This second edition retains the main features of the first edition while incorporating a complete revision of the text as well as additional exercises with their solutions, and a new introductory chapter on credit risk. The stochastic interest rate models considered range from standard short rate to forward rate models, with a treatment of the pricing of related derivatives such as caps and swaptions under forward measures. Some more advanced topics including the BGM model and an approach to its calibration are also covered.

Elliptic Curves

Function Theory, Geometry, Arithmetic

Author: Henry McKean,Victor Moll

Publisher: Cambridge University Press

ISBN: 9780521658171

Category: Mathematics

Page: 280

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The subject of elliptic curves is one of the jewels of nineteenth-century mathematics, originated by Abel, Gauss, Jacobi, and Legendre. This 1997 book presents an introductory account of the subject in the style of the original discoverers, with references to and comments about more recent and modern developments. It combines three of the fundamental themes of mathematics: complex function theory, geometry, and arithmetic. After an informal preparatory chapter, the book follows an historical path, beginning with the work of Abel and Gauss on elliptic integrals and elliptic functions. This is followed by chapters on theta functions, modular groups and modular functions, the quintic, the imaginary quadratic field, and on elliptic curves. Requiring only a first acquaintance with complex function theory, this book is an ideal introduction to the subject for graduate students and researchers in mathematics and physics, with many exercises with hints scattered throughout the text.

Fractals in Probability and Analysis

Author: Christopher J. Bishop,Yuval Peres

Publisher: Cambridge University Press

ISBN: 1107134110

Category: Mathematics

Page: 412

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This is a mathematically rigorous introduction to fractals which emphasizes examples and fundamental ideas. Building up from basic techniques of geometric measure theory and probability, central topics such as Hausdorff dimension, self-similar sets and Brownian motion are introduced, as are more specialized topics, including Kakeya sets, capacity, percolation on trees and the traveling salesman theorem. The broad range of techniques presented enables key ideas to be highlighted, without the distraction of excessive technicalities. The authors incorporate some novel proofs which are simpler than those available elsewhere. Where possible, chapters are designed to be read independently so the book can be used to teach a variety of courses, with the clear structure offering students an accessible route into the topic.

Lie Groups, Physics, and Geometry

An Introduction for Physicists, Engineers and Chemists

Author: Robert Gilmore

Publisher: Cambridge University Press

ISBN: 113946907X

Category: Science

Page: N.A

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Describing many of the most important aspects of Lie group theory, this book presents the subject in a 'hands on' way. Rather than concentrating on theorems and proofs, the book shows the applications of the material to physical sciences and applied mathematics. Many examples of Lie groups and Lie algebras are given throughout the text. The relation between Lie group theory and algorithms for solving ordinary differential equations is presented and shown to be analogous to the relation between Galois groups and algorithms for solving polynomial equations. Other chapters are devoted to differential geometry, relativity, electrodynamics, and the hydrogen atom. Problems are given at the end of each chapter so readers can monitor their understanding of the materials. This is a fascinating introduction to Lie groups for graduate and undergraduate students in physics, mathematics and electrical engineering, as well as researchers in these fields.

Computer Vision

Algorithms and Applications

Author: Richard Szeliski

Publisher: Springer Science & Business Media

ISBN: 9781848829350

Category: Computers

Page: 812

View: 1393

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Computer Vision: Algorithms and Applications explores the variety of techniques commonly used to analyze and interpret images. It also describes challenging real-world applications where vision is being successfully used, both for specialized applications such as medical imaging, and for fun, consumer-level tasks such as image editing and stitching, which students can apply to their own personal photos and videos. More than just a source of “recipes,” this exceptionally authoritative and comprehensive textbook/reference also takes a scientific approach to basic vision problems, formulating physical models of the imaging process before inverting them to produce descriptions of a scene. These problems are also analyzed using statistical models and solved using rigorous engineering techniques. Topics and features: structured to support active curricula and project-oriented courses, with tips in the Introduction for using the book in a variety of customized courses; presents exercises at the end of each chapter with a heavy emphasis on testing algorithms and containing numerous suggestions for small mid-term projects; provides additional material and more detailed mathematical topics in the Appendices, which cover linear algebra, numerical techniques, and Bayesian estimation theory; suggests additional reading at the end of each chapter, including the latest research in each sub-field, in addition to a full Bibliography at the end of the book; supplies supplementary course material for students at the associated website, http://szeliski.org/Book/. Suitable for an upper-level undergraduate or graduate-level course in computer science or engineering, this textbook focuses on basic techniques that work under real-world conditions and encourages students to push their creative boundaries. Its design and exposition also make it eminently suitable as a unique reference to the fundamental techniques and current research literature in computer vision.