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Author: Richard F. Bass
Publisher: American Mathematical Soc.
ISBN: 0821809687
Category : Mathematics
Languages : en
Pages : 113
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Book Description
A long open problem in probability theory has been the following: Can the graph of planar Brownian motion be split by a straight line? In this volume, the authors provide a solution, discuss related works, and present a number of open problems.
Author: Richard F. Bass
Publisher: American Mathematical Soc.
ISBN: 0821809687
Category : Mathematics
Languages : en
Pages : 113
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Book Description
A long open problem in probability theory has been the following: Can the graph of planar Brownian motion be split by a straight line? In this volume, the authors provide a solution, discuss related works, and present a number of open problems.
Author: Peter Mörters
Publisher: Cambridge University Press
ISBN: 1139486578
Category : Mathematics
Languages : en
Pages :
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Book Description
This eagerly awaited textbook covers everything the graduate student in probability wants to know about Brownian motion, as well as the latest research in the area. Starting with the construction of Brownian motion, the book then proceeds to sample path properties like continuity and nowhere differentiability. Notions of fractal dimension are introduced early and are used throughout the book to describe fine properties of Brownian paths. The relation of Brownian motion and random walk is explored from several viewpoints, including a development of the theory of Brownian local times from random walk embeddings. Stochastic integration is introduced as a tool and an accessible treatment of the potential theory of Brownian motion clears the path for an extensive treatment of intersections of Brownian paths. An investigation of exceptional points on the Brownian path and an appendix on SLE processes, by Oded Schramm and Wendelin Werner, lead directly to recent research themes.
Author: Donald J. Estep
Publisher: American Mathematical Soc.
ISBN: 0821820729
Category : Mathematics
Languages : en
Pages : 125
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Book Description
This paper is concerned with the computational estimation of the error of numerical solutions of potentially degenerate reaction-diffusion equations. The underlying motivation is a desire to compute accurate estimates as opposed to deriving inaccurate analytic upper bounds. In this paper, we outline, analyze, and test an approach to obtain computational error estimates based on the introduction of the residual error of the numerical solution and in which the effects of the accumulation of errors are estimated computationally. We begin by deriving an a posteriori relationship between the error of a numerical solution and its residual error using a variational argument. This leads to the introduction of stability factors, which measure the sensitivity of solutions to various kinds of perturbations. Next, we perform some general analysis on the residual errors and stability factors to determine when they are defined and to bound their size. Then we describe the practical use of the theory to estimate the errors of numerical solutions computationally. Several key issues arise in the implementation that remain unresolved and we present partial results and numerical experiments about these points. We use this approach to estimate the error of numerical solutions of nine standard reaction-diffusion models and make a systematic comparison of the time scale over which accurate numerical solutions can be computed for these problems. We also perform a numerical test of the accuracy and reliability of the computational error estimate using the bistable equation. Finally, we apply the general theory to the class of problems that admit invariant regions for the solutions, which includes seven of the main examples. Under this additional stability assumption, we obtain a convergence result in the form of an upper bound on the error from the a posteriori error estimate. We conclude by discussing the preservation of invariant regions under discretization.
Author: Yael Karshon
Publisher: American Mathematical Soc.
ISBN: 0821811819
Category : Mathematics
Languages : en
Pages : 87
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Book Description
This book is intended for graduate students and research mathematicians interested in global analysis, analysis on manifolds, and symplectic geometry.
Author: Brian Marcus
Publisher: American Mathematical Soc.
ISBN: 0821826468
Category : Biography & Autobiography
Languages : en
Pages : 114
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Book Description
The two parts of this monograph contain two separate but related papers. The longer paper in Part A obtains necessary and sufficient conditions for several types of codings of Markov chains onto Bernoulli shifts. It proceeds by replacing the defining stochastic matrix of each Markov chain by a matrix whose entries are polynomials with positive coefficients in several variables; a Bernoulli shift is represented by a single polynomial with positive coefficients, $p$. This transforms jointly topological and measure-theoretic coding problems into combinatorial ones. In solving the combinatorial problems in Part A, the work states and makes use of facts from Part B concerning $p DEGREESn$ and its coefficients. Part B contains the shorter paper on $p DEGREESn$ and its coefficients, and is independ
Author: Roger D. Nussbaum
Publisher: American Mathematical Soc.
ISBN: 0821809695
Category : Mathematics
Languages : en
Pages : 113
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Book Description
The classical Frobenius-Perron Theorem establishes the existence of periodic points of certain linear maps in ${\mathbb R} DEGREESn$. The authors present generalizations of this theorem to nonlinea
Author: Mara D. Neusel
Publisher: American Mathematical Soc.
ISBN: 0821820915
Category : Mathematics
Languages : en
Pages : 175
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Book Description
This book is intended for researchers and graduate students in commutative algebra, algebraic topology and invariant theory.
Author: Jonathan Brundan
Publisher: American Mathematical Soc.
ISBN: 0821826166
Category : Mathematics
Languages : en
Pages : 127
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Book Description
We give a self-contained account of the results originating in the work of James and the second author in the 1980s relating the representation theory of GL[n(F[q) over fields of characteristic coprime to q to the representation theory of "quantum GL[n" at roots of unity. The new treatment allows us to extend the theory in several directions. First, we prove a precise functorial connection between the operations of tensor product in quantum GL[n and Harish-Chandra induction in finite GL[n. This allows us to obtain a version of the recent Morita theorem of Cline, Parshall and Scott valid in addition for p-singular classes. From that we obtain simplified treatments of various basic known facts, such as the computation of decomposition numbers and blocks of GL[n(F[q) from knowledge of the same for the quantum group, and the non-defining analogue of Steinberg's tensor product theorem. We also easily obtain a new double centralizer property between GL[n(F[[q) and quantum GL[n, generalizing a result of Takeuchi. Finally, we apply the theory to study the affine general linear group, following ideas of Zelevinsky in characteristic zero. We prove results that can be regarded as the modular analogues of Zelevinsky's and Thoma's branching rules. Using these, we obtain a new dimension formula for the irreducible cross-characteristic representations of GL[n(F[q), expressing their dimensions in terms of the characters of irreducible modules over the quantum group.
Author: Ross Lawther
Publisher: American Mathematical Soc.
ISBN: 0821819666
Category : Mathematics
Languages : en
Pages : 146
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Book Description
This book is intended for graduate students and research mathematicians interested in group theory and genralizations
Author: Guy David
Publisher: American Mathematical Soc.
ISBN: 0821820486
Category : Mathematics
Languages : en
Pages : 146
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Book Description
This book is intended for graduate students and research mathematicians interested in calculus of variations and optimal control; optimization.
