Series: Encyclopedia of Mathematics and its Applications
Published: October 2025
Format: Hardback
ISBN: 9781009562997
Addressing a significant gap in the study of number series, this book presents an in-depth theory of multiple number series and an exhaustive examination of one-dimensional series. It incorporates overlooked yet essential results alongside recent research advancements. Much of the text is based on the authors' original contributions, particularly in the development of relaxed monotonicity concepts, which have become fundamental tools in Fourier and functional analysis. Each chapter concludes with historical context, aiding readers in understanding the theory's evolution. The book is aimed at a wide audience, ranging from undergraduate students to experts in the field. It offers a modern perspective on the theory, along with detailed introductory chapters that make complex concepts accessible for students. The audience will find the novel contributions enriching and inspiring.
The first book to give a comprehensive account of the basic theory of number series in several dimensions
Thoroughly presents the one-dimensional theory of number series, combining classical results with the latest developments
Emphasizes generalized notions of monotonicity when studying convergence tests, which substantially extend the classical ones
Introduction
Part I. Single Series:
1. Generalized monotonicity
2. Number series in dimension one
3. Miscellaneous tests and results
Part II. Double Series:
4. Rudiments of double sequences and series
5. Tests for convergence of double series
6. Convergence tests involving generalizations of monotonicity
Part III. Multiple Series:
7. Rudiments of multiple sequences and series
8. Tests for convergence of multiple series
9. Multiple series with relaxed monotonicity
Table of correspondences
References ? Monographs and Textbooks
References
Symbol index
Index.
Series: Cambridge Monographs on Applied and Computational Mathematics
Published: November 2025
Format: Hardback
ISBN: 9781009556682
The burgeoning field of differential equations on graphs has experienced significant growth in the past decade, propelled by the use of variational methods in imaging and by its applications in machine learning. This text provides a detailed overview of the subject, serving as a reference for researchers and as an introduction for graduate students wishing to get up to speed.
The authors look through the lens of variational calculus and differential equations, with a particular focus on graph-Laplacian-based models and the graph Ginzburg-Landau functional. They explore the diverse applications, numerical challenges, and theoretical foundations of these models. A meticulously curated bibliography comprising approximately 800 references helps to contextualise this work within the broader academic landscape. While primarily a review, this text also incorporates some original research, extending or refining existing results and methods.
Provides a comprehensive overview of a rapidly developing field
Includes an exhaustive bibliography of 800 items
Features some original research, extending or refining existing results and methods
1. Introduction
2. Setup
3. Important models on graphs
4. Applications
5. Implementation
6. Connections between Allen?Cahn, MBO, and MCF
7. Discrete-to-continuum convergence
8. Connections with other fields and open questions
Appendix A. Γ-convergence
Appendix B. Steady states of two mass-conserving fidelity-forced diffusion equations
References
Index.
Series: Cambridge Monographs on Applied and Computational Mathematics
Published: November 2025
Format: Hardback
ISBN: 9781009663922
Cubature rules are indispensable tools in scientific computing and applied sciences whenever evaluating or discretizing integrals is needed. This monograph is the first comprehensive resource devoted to cubature rules in English since Stroud's classic 1971 book, and the first book about minimal cubature rules. The book explores the subject's theoretical side, which intersects with many branches of mathematics. Minimal cubature rules are intimately connected with common zeros of orthogonal polynomials, which can be described via the polynomial ideals and varieties. Many prominent or practical cubature rules are invariant under a finite group, and some involve symmetric functions and the discrete Fourier transform. Based on state-of-the-art research, the book systematically studies Gauss and minimal cubature rules, and includes a chapter on the practical aspects of construction cubature rules on triangles and simplexes. This comprehensive guide is ideal for researchers and advanced graduate students across the computational and applied mathematics community.
Presents state-of-the-art research on minimal cubature rules for the first time in book form
Introduces indispensable tools in scientific computing and applied sciences, whose study intersects many topics in mathematics, including approximation theory, computational algebraic geometry, discrete Fourier analysis, numerical analysis, orthogonal polynomials, and sampling theory
Includes a chapter on cubature rules on triangle and simplex, which play important roles in finite element methods for numerical solutions of differential equations
Preface
1. Quadrature rules and orthogonal polynomials
2. Cubature rules: basics
3. Orthogonal polynomials of several variables
4. Gauss cubature rules
5. Lower bounds for the number of nodes
6. First minimal cubature rules
7. Further minimal cubature rules
8. Discrete Fourier transform and cubature rules
9. Cubature rules and polynomial ideals
10. Epilogue: two open problems
11. Addendum: cubature rules on triangle and simplex
References
Index.
Series: London Mathematical Society Lecture Note Series
Published: December 2025
Format: Paperback
ISBN: 9781009664356
Written by leaders in the field, this text showcases some of the remarkable properties of the finite Toda lattice and applies this theory to establish universality for the associated Toda eigenvalue algorithm for random Hermitian matrices. The authors expand on a 2019 course at the Courant Institute to provide a comprehensive introduction to the area, including previously unpublished results. They begin with a brief overview of Hamiltonian mechanics and symplectic manifolds, then derive the action-angle variables for the Toda lattice on symmetric matrices. This text is one of the first to feature a new perspective on the Toda lattice that does not use the Hamiltonian structure to analyze its dynamics. Finally, portions of the above theory are combined with random matrix theory to establish universality for the runtime of the associated Toda algorithm for eigenvalue computation.
Presents a full development of the theory of the finite Toda lattice
Includes a self-contained discussion of how to use properties of random matrices to analyze the runtime of numerical algorithms, such as the Toda algorithm
Provides a new perspective by discussing the Toda lattice without using Hamiltonian structure to analyze its dynamics
1. Introduction
2. Hamiltonian mechanics and integrable systems
3. The Toda lattice
4. Toda without Hamiltonian structure
5. Random matrix ensembles
6. Universality for the Toda algorithm
References
Notation and Abbreviations
Index.
ISBN 9781032967653
216 Pages 75 B/W Illustrations
October 21, 2025 by Chapman & Hall
Convex Analysis in Polynomial Spaces with Applications is intended to serve a broad audience of undergraduate and graduate students, junior and senior researchers, and as a general self-study guide for anyone who wishes to get acquainted with geometry of Banach spaces of polynomials with applications. This text is specifically designed to be appealing and accessible to the reader, and provides a general overview on the topic together with new and interesting directions of research. The text also contains original results and material never published before.
・ Comprehensive review on the geometry of spaces of polynomials.
・ Visually attractive and accessible presentation, with over 75 explanatory figures.
・ Contains many examples illustrating the results and techniques appearing in the book.
・ Open (and deep!) questions within the area are provided so that the interested reader can begin doing independent research using the techniques presented in the text.
・ It also features original results by the authors.
Preface Author Biographies I Polynomials on Unbalanced Convex Bodies 1 Preliminaries II The Geometry of Homogeneous Trinomials on the Unit Square 2 Preliminaries III The Krein-Milman Approach, Classical Inequalities and Applications 3 Preliminaries Bibliography
List of Figures Index
Hardback
ISBN 9781041091257
784 Pages
October 13, 2025 by Chapman & Hall
Near Vector Spaces and Related Topics provides a systematic treatment of the introductory theory of near vector spaces, as well as a range of associated areas.
Since many topics in nonlinear analysis rely on the properties established in topological vector space, the concepts and topics presented in this book may stir up the interest of some researchers working in mathematical analysis, especially nonlinear analysis, and thus may potentially open a new avenue of research. The main prerequisites for most of the material in this book are basic concepts of functional analysis, including the basic tools of topology.
This book is accessible to senior undergraduate students in mathematics and may also be used as a graduate level text, or as a reference for researchers who work on the applications of nonlinear analysis.
・ Valuable resource for researchers and postgraduate students interested in the foundation of fuzzy sets and nonlinear analysis
・ Presents new, previously unpublished material on near vector spaces
・ Well-organized and comprehensive treatment of the subject.
Preface Ch 1 Near Vector Spaces Ch 2 Near Metric Spaces Ch 3 Near Normed Spaces Ch 4 Near Hilbert Spaces Ch 5 Hahn-Banach Extension Theorems Ch 6 Dual Spaces Ch 7 Topological Near Vector Spaces Ch 8 Near Fixed Point Bibliography Index
Copyright 2026
Hardback
ISBN 9781032997766
280 Pages 7 B/W Illustrations
November 3, 2025 by Chapman & Hall
Statistics of Survey Sampling offers a comprehensive and rigorous introduction to the principles and practices of survey sampling. Bridging the gap between statistical theory and real-world data collection, this textbook presents both classical methods and modern developments, equipping readers with the tools to design effective surveys and make reliable inferences from sample data.
With a strong foundation in design-based inference and frequentist methodology, the book emphasizes representativeness, efficiency, and the integration of auxiliary information in estimation procedures. It also introduces emerging research topics that reflect the evolving landscape of data collection and analysis.
Rigorous treatment of statistical theory for design-based inference in probability sampling
Thorough exploration of model-assisted estimation techniques using auxiliary data
Coverage of modern topics including data integration, analytic inference, predictive inference, and voluntary sample analysis
Detailed examples illustrate the methods throughout the book
Focused development within the frequentist framework, with limited emphasis on Bayesian or nonparametric methods
Exercises in all chapters enable use as a course text or for self-study
Includes appendices on key background topics such as asymptotic theory and projection techniques
This textbook is ideal for graduate students in statistics with prior courses in statistical theory and linear models. It is also a valuable reference for researchers and practitioners engaged in survey design, public policy evaluation, official statistics, and data science applications involving sample-based inference.
1. Introduction
2. Horvitz-Thompson Estimation
3. Simple and Systematic Sampling Designs
4. Stratified Sampling
5. Sampling with Unequal Probabilities
6. Cluster Sampling: Single stage cluster sampling
7. Cluster Sampling: Two-stage cluster sampling
8. Estimation: Part 1
9. Estimation: Part 2
10. Variance Estimation
11. Two-phase Sampling
12. Unit Nonresponse
13. Imputation
14. Analytic Inference
15. Analysis of Voluntary Samples