Loeb Measures in Practice: Recent Advances
Loeb Measures in Practice: Recent Advances

Author: Nigel J. Cutland

Publisher: Springer

Published: 2004-10-11

Total Pages: 118

ISBN-13: 3540445315

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This expanded version of the 1997 European Mathematical Society Lectures given by the author in Helsinki, begins with a self-contained introduction to nonstandard analysis (NSA) and the construction of Loeb Measures, which are rich measures discovered in 1975 by Peter Loeb, using techniques from NSA. Subsequent chapters sketch a range of recent applications of Loeb measures due to the author and his collaborators, in such diverse fields as (stochastic) fluid mechanics, stochastic calculus of variations ("Malliavin" calculus) and the mathematical finance theory. The exposition is designed for a general audience, and no previous knowledge of either NSA or the various fields of applications is assumed.

Zeta Functions of Groups and Rings
Zeta Functions of Groups and Rings

Author: Marcus du Sautoy

Publisher: Springer Science & Business Media

Published: 2008

Total Pages: 217

ISBN-13: 354074701X

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Zeta functions have been a powerful tool in mathematics over the last two centuries. This book considers a new class of non-commutative zeta functions which encode the structure of the subgroup lattice in infinite groups. The book explores the analytic behaviour of these functions together with an investigation of functional equations. Many important examples of zeta functions are calculated and recorded providing an important data base of explicit examples and methods for calculation.

Tutorials in Mathematical Biosciences IV
Tutorials in Mathematical Biosciences IV

Author: Avner Friedman

Publisher: Springer

Published: 2008-04-26

Total Pages: 215

ISBN-13: 3540743316

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This book offers an introduction to fast growing research areas in evolution of species, population genetics, ecological models, and population dynamics. It reviews the concept and methodologies of phylogenetic trees, introduces ecological models, examines a broad range of ongoing research in population dynamics, and deals with gene frequencies under the action of migration and selection. The book features computational schemes, illustrations, and mathematical theorems.

Nonstandard Methods in Ramsey Theory and Combinatorial Number Theory
Nonstandard Methods in Ramsey Theory and Combinatorial Number Theory

Author: Mauro Di Nasso

Publisher: Springer

Published: 2019-05-23

Total Pages: 211

ISBN-13: 3030179567

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The goal of this monograph is to give an accessible introduction to nonstandard methods and their applications, with an emphasis on combinatorics and Ramsey theory. It includes both new nonstandard proofs of classical results and recent developments initially obtained in the nonstandard setting. This makes it the first combinatorics-focused account of nonstandard methods to be aimed at a general (graduate-level) mathematical audience. This book will provide a natural starting point for researchers interested in approaching the rapidly growing literature on combinatorial results obtained via nonstandard methods. The primary audience consists of graduate students and specialists in logic and combinatorics who wish to pursue research at the interface between these areas.

Laplacian Eigenvectors of Graphs
Laplacian Eigenvectors of Graphs

Author: Türker Biyikoglu

Publisher: Springer

Published: 2007-07-07

Total Pages: 121

ISBN-13: 3540735100

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This fascinating volume investigates the structure of eigenvectors and looks at the number of their sign graphs ("nodal domains"), Perron components, and graphs with extremal properties with respect to eigenvectors. The Rayleigh quotient and rearrangement of graphs form the main methodology. Eigenvectors of graph Laplacians may seem a surprising topic for a book, but the authors show that there are subtle differences between the properties of solutions of Schrödinger equations on manifolds on the one hand, and their discrete analogs on graphs.

Stochastic Calculus for Fractional Brownian Motion and Related Processes
Stochastic Calculus for Fractional Brownian Motion and Related Processes

Author: Yuliya Mishura

Publisher: Springer Science & Business Media

Published: 2008-01-02

Total Pages: 411

ISBN-13: 3540758720

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This volume examines the theory of fractional Brownian motion and other long-memory processes. Interesting topics for PhD students and specialists in probability theory, stochastic analysis and financial mathematics demonstrate the modern level of this field. It proves that the market with stock guided by the mixed model is arbitrage-free without any restriction on the dependence of the components and deduces different forms of the Black-Scholes equation for fractional market.

Simplicial Complexes of Graphs
Simplicial Complexes of Graphs

Author: Jakob Jonsson

Publisher: Springer

Published: 2007-12-10

Total Pages: 376

ISBN-13: 3540758593

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A graph complex is a finite family of graphs closed under deletion of edges. Graph complexes show up naturally in many different areas of mathematics. Identifying each graph with its edge set, one may view a graph complex as a simplicial complex and hence interpret it as a geometric object. This volume examines topological properties of graph complexes, focusing on homotopy type and homology. Many of the proofs are based on Robin Forman's discrete version of Morse theory.

Parameter Estimation in Stochastic Differential Equations
Parameter Estimation in Stochastic Differential Equations

Author: Jaya P. N. Bishwal

Publisher: Springer

Published: 2007-09-26

Total Pages: 271

ISBN-13: 3540744487

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Parameter estimation in stochastic differential equations and stochastic partial differential equations is the science, art and technology of modeling complex phenomena. The subject has attracted researchers from several areas of mathematics. This volume presents the estimation of the unknown parameters in the corresponding continuous models based on continuous and discrete observations and examines extensively maximum likelihood, minimum contrast and Bayesian methods.

Hypoelliptic Estimates and Spectral Theory for Fokker-Planck Operators and Witten Laplacians
Hypoelliptic Estimates and Spectral Theory for Fokker-Planck Operators and Witten Laplacians

Author: Francis Nier

Publisher: Springer

Published: 2005-01-17

Total Pages: 215

ISBN-13: 3540315535

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There has recently been a renewal of interest in Fokker-Planck operators, motivated by problems in statistical physics, in kinetic equations, and differential geometry. Compared to more standard problems in the spectral theory of partial differential operators, those operators are not self-adjoint and only hypoelliptic. The aim of the analysis is to give, as generally as possible, an accurate qualitative and quantitative description of the exponential return to the thermodynamical equilibrium. While exploring and improving recent results in this direction, this volume proposes a review of known techniques on: the hypoellipticity of polynomial of vector fields and its global counterpart, the global Weyl-Hörmander pseudo-differential calculus, the spectral theory of non-self-adjoint operators, the semi-classical analysis of Schrödinger-type operators, the Witten complexes, and the Morse inequalities.