Format: Hardback, height x width: 235x155 mm, Approx. 400 p.
Series: Springer Monographs in Mathematics
Pub. Date: 10-Apr-2023
ISBN-13: 9783031103902
This book complements the authors' monograph Cellular Automata and Groups [ CAG] (Springer Monographs in Mathematics). It consists of more than 500 solved exercises in symbolic dynamics and geometric group theory with connections to geometry and topology, ring and module theory, automata theory and theoretical computer science. Each solution is detailed and entirely self-contained, in the sense that it only requires a standard undergraduate-level background in abstract algebra and general topology, together with results established in [ CAG] and in previous exercises. It includes a wealth of gradually worked out examples and counterexamples presented here for the first time in textbook form. Additional comments provide some historical and bibliographical information, including an account of related recent developments and suggestions for further reading. The eight-chapter division from [ CAG] is maintained. Each chapter begins with a summary of the main definitions and results contained in the corresponding chapter of [ CAG]. The book is suitable either for classroom or individual use.
1 Cellular Automata.- 2 Residually Finite Groups.- 3 Surjunctive Groups.- 4 Amenable Groups.- 5 The Garden of Eden Theorem.- 6 Finitely Generated Amenable Groups.- 7 Local Embeddability and Sofic Groups.- 8 Linear Cellular Automata.
Format: Hardback, height x width: 235x155 mm, Approx. 440 p., 1 Hardback
Series: Graduate Texts in Mathematics 296
Pub. Date: 03-Mar-2023
ISBN-13: 9783031197062
This textbook offers an introduction to the theory of Drinfeld modules, mathematical objects that are fundamental to modern number theory. After the first two chapters conveniently recalling prerequisites from abstract algebra and non-Archimedean analysis, Chapter 3 introduces Drinfeld modules and the key notions of isogenies and torsion points. Over the next four chapters, Drinfeld modules are studied in settings of various fields of arithmetic importance, culminating in the case of global fields. Throughout, numerous number-theoretic applications are discussed, and the analogies between classical and function field arithmetic are emphasized. Drinfeld Modules guides readers from the basics to research topics in function field arithmetic, assuming only familiarity with graduate-level abstract algebra as prerequisite. With exercises of varying difficulty included in each section, the book is designed to be used as the primary textbook for a graduate course on the topic, and may also provide a supplementary reference for courses in algebraic number theory, elliptic curves, and related fields. Furthermore, researchers in algebra and number theory will appreciate it as a self-contained reference on the topic.
Preface.- Acknowledgements.- Notation and Conventions.
Chapter 1. Algebraic Preliminaries.
Chapter 2. Non-Archimedean Fields.
Chapter 3. Basic Properties of Drinfeld Modules.
Chapter 4. Drinfeld Modules over Finite Fields.
Chapter 5. Analytic Theory of Drinfeld Modules.
Chapter 6. Drinfeld Modules over Local Fields.
Chapter 7. Drinfeld Modules over Global Fields.-
Appendix A. Drinfeld modules for general function rings.-
Appendix B. Notes on exercises.- Bibliography.- Index.
Bibliog. data: 1st ed. 2023. 2023. xii, 482 S. XII, 482 p. 174 illus., 161 illus. in color. 235 mm
Format: Hardcover
Series: Applied Mathematical Sciences 212
ISBN-13: 9783031204081
This monograph is devoted to the study of multiscale model reduction methods from the point of view of multiscale finite element methods. Multiscale numerical methods have become popular tools for modeling processes with multiple scales. These methods allow reducing the degrees of freedom based on local offline computations. Moreover, these methods allow deriving rigorous macroscopic equations for multiscale problems without scale separation and high contrast. Multiscale methods are also used to design efficient solvers. This book offers a combination of analytical and numerical methods designed for solving multiscale problems. The book mostly focuses on methods that are based on multiscale finite element methods. Both applications and theoretical developments in this field are presented. The book is suitable for graduate students and researchers, who are interested in this topic.
Introduction.- Homogenization and Numerical Homogenization of Linear Equations.- Local Model Reduction: Introduction to Multiscale Finite Element Methods.- Generalized Multiscale Finite Element Methods: Main Concepts and Overview.- Adaptive Strategies.- Selected Global Formulations for GMsFEM and Energy Stable Oversampling.- GMsFEM Using Sparsity in the Snapshot Spaces.- Space-time GMsFEM.- Constraint Energy Minimizing Concepts.- Non-local Multicontinua Upscaling.- Space-time GMsFEM.- Multiscale Methods for Perforated Domains.- Multiscale Stabilization.- GMsFEM for Selected Applications.- Homogenization and Numerical Homogenization of Nonlinear Equations.- GMsFEM for Nonlinear Problems.- Nonlinear Non-local Multicontinua Upscaling.- Global-local Multiscale Model Reduction Using GMsFEM.- Multiscale Methods in Temporal Splitting. Efficient Implicit-explicit Methods for Multiscale Problems.- References.- Index.
Eric Chung is a Professor in the Department of Mathematics and an Outstanding Fellow of the Faculty of Science at the Chinese University of Hong Kong. His research focuses on numerical discretizations of partial differential equations and the development of computational multiscale methods for challenging applications.Yalchin Efendiev is a Professor in the Department of Mathematics at the Texas A&M University.Thomas Y. Hou is the Charles Lee Powell Professor of Applied and Computational Mathematics at the California Institute of Technology. His research focuses on multiscale analysis and computation, fluid interface problems, and singularity formation of 3D Euler and Navier-Stokes equations.
Format: Paperback / softback, height x width: 235x155 mm, Approx. 350 p
Series: Lecture Notes in Mathematics 2314
Pub. Date: 06-Apr-2023
ISBN-13: 9789811976698
This book provides an accessible overview concerning the stochastic numerical methods inheriting long-time dynamical behaviours of finite and infinite-dimensional stochastic Hamiltonian systems. The long-time dynamical behaviours under study include symplectic structure, invariants, ergodicity and invariant measure. The emphasis is placed on the systematic construction and the probabilistic superiority of stochastic symplectic methods, which preserve the geometric structure of the stochastic flow of stochastic Hamiltonian systems. The problems considered in this book are related to several fascinating research hotspots: numerical analysis, stochastic analysis, ergodic theory, stochastic ordinary and partial differential equations, and rough path theory. This book will appeal to researchers who are interested in these topics.
1 Deterministic Hamiltonian System 1.1 Geometric Structure 1.2 Generating Function 1.2.1 Hamilton-Jacobi Partial Differential Equation 1.2.2 Series Expansion 1.3 Geometric Structure Preserving Scheme 2 Stochastic Hamiltonian System 2.1 Stochastic Differential Equation 2.2 Variational Equation 2.3 Stochastic First Integral 2.4 Stochastic Hamiltonian System 2.4.1 Stochastic Symplectic Transformation 2.4.2 Stochastic Generating Function 2.5 Non-Canonical Stochastic Hamiltonian System 2.5.1 Dissipative Stochastic Hamiltonian System 2.5.2 Stochastic Poisson System 3 Stochastic Structure Preserving Numerical Integrators 3.1 Stochastic Numerical Integrator 3.1.1 Stochastic Runge-Kutta Method 3.1.2 Stochastic Splitting Method 3.1.3 Strong and Weak Convergence 3.2 Stochastic Symplectic Integrator 3.2.1 Stochastic Symplectic Method Based on Pade Approximation 3.2.2 Stochastic Runge-Kutta Method 3.2.3 Stochastic Symplectic Method Based on Generating Function 3.2.4 Stochastic Volume Preserving Method 3.3 Stochastic Method Conserving First Integral 4 Stochastic Modified Equation and Its Applications 4.1 Kolmogorov Equation 4.2 Stochastic Modified Equation 4.2.1 Modified Equation Based on Kolmogorov Equation 4.2.2 Modified Equation for Symplectic Scheme Based on Generating Function 4.3 Stochastic Numerical Integrator Based on Modified Equation 4.3.1 Stochastic Symplectic Scheme Based on Modified Equation via Generating Function 4.3.2 Conformal Symplectic Scheme for Stochastic Langevin Equation 5 Stochastic Hamiltonian Partial Differential Equation 5.1 Stochastic Hamiltonian system in infinite dimension 5.1.1 Stochastic Geometry and Hamiltonian Structure 5.2 Stochastic Maxwell Equation 5.2.1 Well-posedness and Longtime Behavior 5.2.2 Stochastic Symplectic and Multi-Symplectic Scheme 5.2.3 Physical Properties Preserving Numerical Approximation 5.3 Damped Stochastic Schrodinger Equation with Potential 5.3.1 Well-posedness and Geometric Structure 5.3.2 Stochastic Conformal Multi-Symplectic Scheme Based on Generating Function
Format: Paperback / softback, height x width: 235x155 mm, Approx. 570 p.
Series: Lecture Notes in Mathematics 2320
Pub. Date: 25-Dec-2022
ISBN-13: 9789811967870
The real-variable theory of function spaces has always been at the core of harmonic analysis. In particular, the real-variable theory of the Hardy space is a fundamental tool of harmonic analysis, with applications and connections to complex analysis, partial differential equations, and functional analysis. This book is devoted to exploring properties of generalized Herz spaces and establishing a complete real-variable theory of Hardy spaces associated with local and global generalized Herz spaces via a totally fresh perspective. This means that the authors view these generalized Herz spaces as special cases of ball quasi-Banach function spaces. In this book, the authors first give some basic properties of generalized Herz spaces and obtain the boundedness and the compactness characterizations of commutators on them. Then the authors introduce the associated Herz-Hardy spaces, localized Herz-Hardy spaces, and weak Herz-Hardy spaces, and develop a complete real-variable theory of these Herz-Hardy spaces, including their various maximal function, atomic, molecular as well as various Littlewood-Paley function characterizations. As applications, the authors establish the boundedness of some important operators arising from harmonic analysis on these Herz-Hardy spaces. Finally, the inhomogeneous Herz-Hardy spaces and their complete real-variable theory are also investigated. With the fresh perspective and the improved conclusions on the real-variable theory of Hardy spaces associated with ball quasi-Banach function spaces, all the obtained results of this book are new and their related exponents are sharp. This book will be appealing to researchers and graduate students who are interested in function spaces and their applications.
Preface i1 Generalized Herz Spaces of Rafeiro and Samko 1.1 Matuszewska-Orlicz Indices 1.2 Generalized Herz Spaces 1.3 Convexities 1.4 Absolutely Continuous Quasi-Norms 1.5 Boundedness of Sublinear Operators 1.6 Fefferman-Stein Vector-Valued Inequalities 1.7 Dual and Associate Spaces of Local Generalized Herz Spaces 1.8 Extrapolation Theorems 2 Block Spaces and Their Applications 2.1 Block Spaces 2.2 Duality 2.3 Boundedness of Sublinear Operators 3 Boundedness and Compactness Characterizations of Commutators on Generalized Herz Spaces 3.1 Boundedness Characterizations 3.2 Compactness Characterizations 4 Generalized Herz-Hardy Spaces 4.1 Maximal Function Characterizations 4.2 Relations with Generalized Herz Spaces 4.3 Atomic Characterizations 4.4 Generalized Finite Atomic Herz-Hardy Spaces 4.5 Molecular Characterizations 4.6 Littlewood-Paley Function Characterizations 4.7 Dual Space of HK p,q ,0(Rn) 4.8 Boundedness of CalderALon-Zygmund Operators 4.9 Fourier Transform 5 Localized Generalized Herz-Hardy Spaces 5.1 Maximal Function Characterizations 5.2 Relations with Generalized Herz-Hardy Spaces 5.3 Atomic Characterizations 5.4 Molecular Characterizations 5.5 Littlewood-Paley Function Characterizations 5.6 Boundedness of Pseudo-Differential Operators 6 Weak Generalized Herz-Hardy Spaces 6.1 Maximal Function Characterizations 6.2 Relations with Weak Generalized Herz Spaces 6.3 Atomic Characterizations 6.4 Molecular Characterizations 6.5 Littlewood-Paley Function Characterizations 6.6 Boundedness of CalderALon-Zygmund Operators 6.7 Real Interpolations 7 Inhomogeneous Generalized Herz Spaces and Inhomogeneous Block Spaces 7.1 Inhomogeneous Generalized Herz Spaces 7.1.1 Convexities 7.1.2 Absolutely Continuous Quasi-Norms 7.1.3 Boundedness of Sublinear Operators and Fefferman-Stein Vector-Valued Inequalities 7.1.4 Dual and Associate Spaces of Inhomogeneous Local Generalized Herz Spaces 7.1.5 Extrapolation Theorems 7.2 Inhomogeneous Block Spaces and Their Applications 7.2.1 Inhomogeneous Block Spaces 7.2.2 Duality Between Inhomogeneous Block Spaces and Global Generalized Herz Spaces 7.2.3 Boundedness of Sublinear Operators 7.3 Boundedness and Compactness Characterizations of Commutators 7.3.1 Boundedness Characterizations 7.3.2 Compactness Characterizations 8 Hardy Spaces Associated with Inhomogeneous Generalized Herz Spaces 8.1 Inhomogeneous Generalized Herz-Hardy Spaces 8.1.1 Maximal Function Characterizations 8.1.2 Relations with Inhomogeneous Generalized Herz Spaces 8.1.3 Atomic Characterizations 8.1.4 Inhomogeneous Generalized Finite Atomic Herz-Hardy Spaces 8.1.5 Molecular Characterizations 8.1.6 Littlewood-Paley Function Characterizations 8.1.7 Dual Space of HKp,q ,0(Rn) 8.1.8 Boundedness of CalderALon-Zygmund Operators 8.1.9 Fourier Transform 8.2 Inhomogeneous Localized Generalized Herz-Hardy Spaces 8.2.1 Maximal Function Characterizations 8.2.2 Relations with Inhomogeneous Generalized Herz-Hardy Spaces 8.2.3 Atomic Characterizations 8.2.4 Molecular Characterizations 8.2.5 Littlewood-Paley Function Characterizations 8.2.6 Boundedness of Pseudo-Differential Operators 8.3 Inhomogeneous Weak Generalized Herz-Hardy Spaces 8.3.1 Maximal Function Characterizations 8.3.2 Relations with Inhomogeneous Weak Generalized Herz Spaces 8.3.3 Atomic Characterizations 8.3.4 Molecular Characterizations 8.3.5 Littlewood-Paley Function Characterizations 8.3.6 Boundedness of CalderALon-Zygmund Operators 8.3.7 Real Interpolations Bibliography Index Abstract
Pages: 492
ISBN: 978-981-126-348-4 (hardcover)
Professor Xihua Cao (1920?2005) was a leading scholar at East China Normal University (ECNU) and a famous algebraist in China. His contribution to the Chinese academic circle is particularly the formation of a world-renowned "ECNU School" in algebra, covering research areas include algebraic groups, quantum groups, algebraic geometry, Lie algebra, algebraic number theory, representation theory and other hot fields. In January 2020, in order to commemorate Professor Xihua Cao's centenary birthday, East China Normal University held a three-day academic conference. Scholars at home and abroad gave dedications or delivered lectures in the conference. This volume originates from the memorial conference, collecting the dedications of scholars, reminiscences of family members, and 16 academic articles written based on the lectures in the conference, covering a wide range of research hot topics in algebra. The book shows not only scholars' respect and memory for Professor Xihua Cao, but also the research achievements of Chinese scholars at home and abroad.
Dedications and Reminiscences:
Professor Xihua Cao and His School
Dedications
Reminiscences of Family Members
Research Papers:
A Particular Class of Properly Stratified Algebras
Quantized Cohomological Hall Algebra of the d-Loop Quiver Revisited
Generating Algebraic Groups by Small Subgroups and Equivariant Representations
On Quadri-Bialgebras
Partial Orderings on Affine Weyl Groups
Surveys and Overviews:
On Toroidal Vertex Algebras
A Survey on Infinite-Dimensional Representations of Reductive Algebraic Groups with Frobenius Maps
Inductive Conditions of Alperin Weight
A Survey on Quantum Linear and Queer Supergroups
BGG Category Į and Z-Graded Representation Theory
Moduli Spaces of Vector Bundles on a Node Curve
Automorphic Descents
The Mazur Conjecture, Lang-Trotter Conjecture and the Hardy-Littlewood Conjecture
A Survey on the Cell Theory of Weighted Coxeter Groups
On Bases of Quantum Affine Algebras
Lecture Notes:
Lectures on Dualities ABC in Representation Theory
Appendix: A Brief Biography of Xihua Cao
Readership: Researchers and postgraduate students interested in algebras, especially in algebraic groups, quantum groups, Lie algebras, algebraic geometry and algebraic number theory; General readers of mathematics and mathematical history.