Mathematics Faculty Book Series

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  Beyond the Triangle - Brownian Motion, Ito Stochastic Calculus, and Fokker-Planck Equation: Fractional Generalizations

  Sabir Umarov, Marjorie Hahn, and Kei Kobayashi

  (From the publisher): The book is devoted to the fundamental relationship between three objects: a stochastic process, stochastic differential equations driven by that process and their associated Fokker–Planck–Kolmogorov equations. This book discusses wide fractional generalizations of this fundamental triple relationship, where the driving process represents a time-changed stochastic process; the Fokker–Planck–Kolmogorov equation involves time-fractional order derivatives and spatial pseudo-differential operators; and the associated stochastic differential equation describes the stochastic behavior of the solution process. It contains recent results obtained in this direction.

  This book is important since the latest developments in the field, including the role of driving processes and their scaling limits, the forms of corresponding stochastic differential equations, and associated FPK equations, are systematically presented. Examples and important applications to various scientific, engineering, and economics problems make the book attractive for all interested researchers, educators, and graduate students.

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  Introduction to Fractional and Pseudo-Differential Equations with Singular Symbols

  Sabir Umarov

  Publisher's description: The book systematically presents the theories of pseudo-differential operators with symbols singular in dual variables, fractional order derivatives, distributed and variable order fractional derivatives, random walk approximants, and applications of these theories to various initial and multi-point boundary value problems for pseudo-differential equations. Fractional Fokker-Planck-Kolmogorov equations associated with a large class of stochastic processes are presented. A complex version of the theory of pseudo-differential operators with meromorphic symbols based on the recently introduced complex Fourier transform is developed and applied for initial and boundary value problems for systems of complex differential and pseudo-differential equations.

  Contents: Function Spaces and Distributions.- Pseudo-Differential Operators with Singular Symbols (DOSS).- Fractional Calculus and Fractional Order Operators.- Boundary Value Problems for Pseudo-Differential Equations with Singular Symbols.- Initial and Boundary Value Problems for Fractional Order Differential Equations.- Distributed and Variable Order Differential-Operator Equations.- Fractional Fokker-Planck-Kolmogorov Equations.- Random Walk Approximants of Mixed and Time-Changed Levy Processes.- Complex DOSS and Systems of Complex Differential Equations.- References.

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  Recent Advances in Riemannian and Lorentzian Geometries

  Krishan L. Duggal and Ramesh Sharma

  This volume covers the proceedings of a special session “Recent advances in Riemannian and Lorentzian geometries” of the annual meeting of American Mathematical Society, held at Baltimore, January 15-18, 2003. The speakers presented their research on Riemannian, Lorentzian, and pseudo-Riemannian manifolds.

  The topics covered included classification of curvature-related operators, curvature-homogeneous Einstein 4-manifolds, linear stability/instability, singularity and hyperbolic operators of spacetimes, spectral geometry, cut loci of nilpotent Lie groups, conformal geometry of almost Hermitian manifolds and also submanifolds of complex and contact submanifolds.

  This special session presented a great setting for differential geometers to interact among themselves and to expose the interplay/exchange between Riemannian and Lorentzian geometries. All the topics published in this volume were formally refereed.

  This volume can serve as a good reference source and provide indications for further research. It is suitable for graduate students and research mathematicians interested in differential geometry.

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  Symmetries of Spacetimes and Riemannian Manifolds

  Krishan L. Duggal and Ramesh Sharma

  This book provides an upto date information on metric, connection and curva­ ture symmetries used in geometry and physics. More specifically, we present the characterizations and classifications of Riemannian and Lorentzian manifolds (in particular, the spacetimes of general relativity) admitting metric (i.e., Killing, ho­ mothetic and conformal), connection (i.e., affine conformal and projective) and curvature symmetries. Our approach, in this book, has the following outstanding features: (a) It is the first-ever attempt of a comprehensive collection of the works of a very large number of researchers on all the above mentioned symmetries. (b) We have aimed at bringing together the researchers interested in differential geometry and the mathematical physics of general relativity by giving an invariant as well as the index form of the main formulas and results. (c) Attempt has been made to support several main mathematical results by citing physical example(s) as applied to general relativity. (d) Overall the presentation is self contained, fairly accessible and in some special cases supported by an extensive list of cited references. (e) The material covered should stimulate future research on symmetries. Chapters 1 and 2 contain most of the prerequisites for reading the rest of the book. We present the language of semi-Euclidean spaces, manifolds, their tensor calculus; geometry of null curves, non-degenerate and degenerate (light like) hypersurfaces. All this is described in invariant as well as the index form.