Format: Paperback / softback, 277 pages, height x width: 235x155 mm, X, 277 p.,
Series: Lecture Notes in Mathematics 2338
Pub. Date: 20-Nov-2023
ISBN-13: 9783031410192
This book studies the potential functions of one-dimensional recurrent random walks on the lattice of integers with step distribution of infinite variance. The central focus is on obtaining reasonably nice estimates of the potential function. These estimates are then applied to various situations, yielding precise asymptotic results on, among other things, hitting probabilities of finite sets, overshoot distributions, Green functions on long finite intervals and the half-line, and absorption probabilities of two-sided exit problems.
The potential function of a random walk is a central object in fluctuation theory. If the variance of the step distribution is finite, the potential function has a simple asymptotic form, which enables the theory of recurrent random walks to be described in a unified way with rather explicit formulae. On the other hand, if the variance is infinite, the potential function behaves in a wide range of ways depending on the step distribution, which the asymptotic behaviour of many functionals of the random walk closely reflects.
In the case when the step distribution is attracted to a strictly stable law, aspects of the random walk have been intensively studied and remarkable results have been established by many authors. However, these results generally do not involve the potential function, and important questions still need to be answered. In the case where the random walk is relatively stable, or if one tail of the step distribution is negligible in comparison to the other on average, there has been much less work. Some of these unsettled problems have scarcely been addressed in the last half-century. As revealed in this treatise, the potential function often turns out to play a significant role in their resolution.
Aimed at advanced graduate students specialising in probability theory, this book will also be of interest to researchers and engineers working with random walks and stochastic systems.
Preface.- Introduction.- Preliminaries.- Bounds of the Potential
Function.- Some Explicit Asymptotic Forms of a(x).- Applications Under m+/m
0.- The Two-Sided Exit Problem - General Case.- The Two-Sided Exit Problem
for Relatively Stable Walks.- Absorption Problems for Asymptotically Stable
Random Walks.- Asymptotically Stable RandomWalks Killed Upon Hitting a Finite
Set.- Appendix.- References.- Notation Index.- Subject Index.
Format: Paperback / softback, 120 pages, height x width: 235x155 mm, X, 120 p.,
Series: SpringerBriefs in Mathematics
Pub. Date: 29-Sep-2023
ISBN-13: 9783031433313
This book develops limit theorems for a natural class of long range random walks on finitely generated torsion free nilpotent groups. The limits in these limit theorems are Levy processes on some simply connected nilpotent Lie groups. Both the limit Levy process and the limit Lie group carrying this process are determined by and depend on the law of the original random walk. The book offers the first systematic study of such limit theorems involving stable-like random walks and stable limit Levy processes in the context of (non-commutative) nilpotent groups.
Setting the stage.- Introduction.- Polynomial coordinates and
approximate dilations.- Vague convergence and change of group law.- Weak
convergence of the processes.- Local limit theorem.- Symmetric Levy processes
on nilpotent groups.- Measures in SM( ) and their geometries.- Adapted
approximate group dilations.- The main results for random walks driven by
measures in SM( ).
Format: Paperback / softback, 119 pages, height x width: 235x155 mm, 7 Illustrations, black and white; X, 119 p. 7 illus.,
Series: SpringerBriefs in Mathematics
Pub. Date: 11-Oct-2023
ISBN-13: 9783031423536
This book presents an enticing introduction to tropical circuits and their use as a rigorous mathematical model for dynamic programming (DP), which is one of the most fundamental algorithmic paradigms for solving combinatorial, discrete optimization problems.
In DP, an optimization problem is broken up into smaller subproblems that are solved recursively. Many classical DP algorithms are pure in that they only use the basic (min,+) or (max,+) operations in their recursion equations. In tropical circuits, these operations are used as gates. Thanks to the rigorous combinatorial nature of tropical circuits, elements from the Boolean and arithmetic circuit complexity can be used to obtain lower bounds for tropical circuits, which play a crucial role in understanding the limitations and capabilities of these computational models. This book aims to offer a toolbox for proving lower bounds on the size of tropical circuits.
In this work, the reader will find lower-bound ideas and methods that have emerged in the last few years, with detailed proofs. Largely self-contained, this book is meant to be approachable by graduate students in mathematics and computer science with a special interest in circuit complexity.
Chapter.
1. BasicsChapter.
2. Combinatorial BoundsChapter.
3. Rectangle
BoundsChapter.
4. Bounds for Approximating CircuitsChapter.
5. Tropical
Branching ProgramsChapter.
6. Extended Tropical Circuits
Format: Hardback, 322 pages, height x width: 235x155 mm, 15 Illustrations, black and white; IV, 322 p. 15 illus.,
Series: Applied Mathematical Sciences 219
Pub. Date: 30-Oct-2023
ISBN-13: 9783031425240
This monograph provides a comprehensive study of the Riemann problem for systems of conservation laws arising in continuum physics. It presents the state-of-the-art on the dynamics of compressible fluids and mixtures that undergo phase changes, while remaining accessible to applied mathematicians and engineers interested in shock waves, phase boundary propagation, and nozzle flows. A large selection of nonlinear hyperbolic systems is treated here, including the Saint-Venant, van der Waals, and Baer-Nunziato models. A central theme is the role of the kinetic relation for the selection of under-compressible interfaces in complex fluid flows. This book is recommended to graduate students and researchers who seek new mathematical perspectives on shock waves and phase dynamics.
1 Overview of this monograph.- 2 Models arising in fluid and solid
dynamics.- 3 Nonlinear hyperbolic systems of balance laws.- 4 Riemann problem
for ideal fluids.- 5 Compressible fluids governed by a general equation of
state.- 6 Nonclassical Riemann solver with prescribed kinetics. The
hyperbolic regime.- 7 Nonclassical Riemann solver with prescribed kinetics.
The hyperbolic-elliptic regime.- 8 Compressible fluids in a nozzle with
discontinuous cross-section. Isentropic flows.- 9 Compressible fluids in a
nozzle with discontinuous cross-section. General flows.- 10 Shallow water
flows with discontinuous topography.- 11 Shallow water flows with temperature
gradient.- 12 Baer-Nunziato model of two-phase flows.- References.- Index.
Format: Paperback / softback, 100 pages, height x width: 235x155 mm, 50 Tables, color; 50 Illustrations, color; 20 Illustrations, black and white; X, 100 p. 70 illus., 50 illus. in color
Series: SpringerBriefs in Mathematics
Pub. Date: 21-Oct-2023
ISBN-13: 9783031423321
This book is an introduction to the use of machine learning and data-driven approaches in fluid simulation and animation, as an alternative to traditional modeling techniques based on partial differential equations and numerical methods ? and at a lower computational cost.
This work starts with a brief review of computability theory, aimed to convince the reader ? more specifically, researchers of more traditional areas of mathematical modeling ? about the power of neural computing in fluid animations. In these initial chapters, fluid modeling through Navier-Stokes equations and numerical methods are also discussed.
The following chapters explore the advantages of the neural networks approach and show the building blocks of neural networks for fluid simulation. They cover aspects related to training data, data augmentation, and testing.
The volume completes with two case studies, one involving Lagrangian simulation of fluids using convolutional neural networks and the other using Generative Adversarial Networks (GANs) approaches.
Introduction.- Fluids and Deep Learning: A Brief Review.- Fluid Modeling through Navier-Stokes Equations and Numerical Methods.- Why Use Neural Networks for Fluid Animation.- Modeling Fluids through Neural Networks.- Fluid Rendering.- Traditional Techniques.- Advanced Techniques.- Deep Learning in Rendering.- Case Studies.- Perspectives.- Discussion and Final Remarks.- References.
Bibliog. data: 1st ed. 2023. 2023. xx, 317 S. XX, 317 p. 235 mm
ISBN-13: 9783031427909
This is the first comprehensive book on Trotter-Kato approximations of stochastic differential equations (SDEs) in infinite dimensions and applications. This research monograph brings together the varied literature on this topic since 1985 when such a study was initiated. The author provides a clear and systematic introduction to the theory of Trotter-Kato approximations of SDEs and also presents its applications to practical topics such as stochastic stability and stochastic optimal control. The theory assimilated here is developed slowly and methodically in digestive pieces.The book begins with a motivational chapter introducing several different models that highlight the importance of the theory on abstract SDEs that will be considered in the subsequent chapters. The author next introduces the necessary mathematical background and then leads the reader into the main discussion of the monograph, namely, the Trotter-Kato approximations of many classes of SDEs in Hilbe
rt spaces, Trotter-Kato approximations of SDEs in UMD Banach spaces and some of their applications. Most of the results presented in the main chapters appear for the first time in a book form. The monograph also contains many illustrative examples on stochastic partial differential equations and one in finance as an application of the Trotter-Kato formula. The key steps are included in all proofs which will help the reader to get a real insight into the theory of Trotter-Kato approximations and its use. This book is intended for researchers and graduate students in mathematics specializing in probability theory. It will also be useful to numerical analysts, engineers, physicists and practitioners who are interested in applying the theory of stochastic evolution equations. Since the approach is based mainly in semigroup theory, it is accessible to a wider audience including non-specialists in stochastic processes.
1 Introduction and Motivating Examples.- 2 Mathematical Machinery.- 3 Trotter-Kato Approximations of Stochastic Differential Equations.- 4 Trotter-Kato Approximations of Stochastic Differential Equations in UMD Banach Spaces.- 6 Applications to Stochastic Optimal Control.- Appendix A: Nuclear and Hilbert-Schmidt Operators.- Appendix B: Convergence of Analytic Semigroups.- Appendix C: The Pettis Measurability Theorem.- Appendix D: R-Boundedness and Gamma-Boundedness.- Appendix E: The Feynman-Kac Formula.- Bibliographical Notes and Remarks.- Bibliography.
T. E. Govindan earned a Master"s degree in Statistics from the Indian Institute of Technology, Kanpur, India, followed by a Ph.D. in Mathematics in 1991 from the Indian Institute of Technology, Bombay, India. After serving the University of Bombay, India for over eight years, he moved to Mexico in 1999. Currently, he is a Professor of Mathematics at the National Polytechnic Institute in Mexico. He specializes in stochastic analysis, particularly on stochastic differential equations.
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ISBN: 978-981-12-7958-4 (hardcover)
ISBN: 978-981-12-8191-4 (softcover)
Our physical world is embedded in a geometric environment. Plane geometry has many amazing wonders beyond those that are briefly touched on in school curriculums. The triangle, one of the basic instruments in geometry, has a plethora of unexpected curiosities. Geometric Gems presents one of the largest collections of triangle curiosities currently available, which the authors discuss in an easily understood fashion, requiring nothing more of readers other than the very basics of school geometry to appreciate these curiosities and their justifications or proofs.
The book is intended to be widely appreciated by a general audience, and their love for geometry should be greatly enhanced through exploring these many unexpected relationships in geometry. Geometric Gems is also suitable for mathematics teachers, to enhance the education of their students with these highly motivating triangle properties.
Introduction
Geometric Curiosities
Proofs of the Geometric Curiosities
Toolbox
The book is intended the general readership, mathematics teachers, and mathematicians interested in extending their appreciation for geometry.
ISBN: 978-981-12-8014-6 (hardcover)
Corresponding to the link of Ito's stochastic differential equations (SDEs) and linear parabolic equations, distribution dependent SDEs (DDSDEs) characterize nonlinear Fokker?Planck equations. This type of SDEs is named after McKean?Vlasov due to the pioneering work of H P McKean (1966), where an expectation dependent SDE is proposed to characterize nonlinear PDEs for Maxwellian gas. Moreover, by using the propagation of chaos for Kac particle systems, weak solutions of DDSDEs are constructed as weak limits of mean field particle systems when the number of particles goes to infinity, so that DDSDEs are also called mean-field SDEs. To restrict a DDSDE in a domain, we consider the reflection boundary by following the line of A V Skorohod (1961).
This book provides a self-contained account on singular SDEs and DDSDEs with or without reflection. It covers well-posedness and regularities for singular stochastic differential equations; well-posedness for singular reflected SDEs; well-posedness of singular DDSDEs; Harnack inequalities and derivative formulas for singular DDSDEs; long time behaviors for DDSDEs; DDSDEs with reflecting boundary; and killed DDSDEs.
Singular Stochastic Differential Stochastic Differential Equations
Singular Reflected SDEs
DDSDEs: Well-Posedness
DDSDEs: Harnack Inequality and Derivative Estimates
DDSDEs: Long Time Behaviors
DDSDEs with Reflecting Boundary
Killed DDSDEs
Readership:
Postgraduate students and researchers in the areas of stochastic analysis, stochastic differential equations, stochastic partial differential equations, and stochastic dynamics. The book may be used as textbook for advanced courses on stochastic analysis.