Textbook
Nov 2025
Includes various (often real-world) applications and examples
Accessible to a wide audience
Has a clear presentation and good explanation
Access Source Code
Part of the book series: Probability Theory and Stochastic Modelling (PTSM, volume 108)
This book presents a broad range of computational techniques based on repeated random sampling, widely known as Monte Carlo methods and sometimes as stochastic simulation. These methods bring together ideas from probability theory, statistics, computer science, and statistical physics, providing tools for solving problems in fields such as operations research, biotechnology, and finance.
Topics include the generation and analysis of pseudorandom numbers (which are intended to imitate truly random numbers on a computer), the design and justification of Monte Carlo algorithms, and advanced approaches such as Markov chain Monte Carlo and stochastic optimization. In contrast to deterministic numerical methods, the outcome of a Monte Carlo algorithm is itself random – and one needs the tools of probability and statistics to interpret these results meaningfully. The theoretical foundations, particularly the law of large numbers and central limit theorem, are combined with practical algorithms that reveal both the strengths and subtleties of stochastic simulation.
The book includes numerous exercises, both theoretical and computational. Each chapter features step-by-step algorithms, illustrated examples, and results presented through numerical computations, tables, and a variety of plots and figures. All Python code used to produce these results is publicly available, allowing readers to reproduce and explore simulations on their own.
Intended primarily for graduate students and researchers, the exposition focuses on core concepts and intuitive understanding, avoiding excessive formalism. The book is suitable both for self-study and as a course text and offers a clear pathway from foundational principles to modern applications.
Textbook
Nov 2025
Builds a bridge from beginning number theory topics to modern advanced topics
Provides a self-contained path to a notoriously difficult area of mathematics, the Langlands program
Naturally transitions to modular forms
Part of the book series: Graduate Texts in Mathematics (GTM, volume 307)
The distinctive compilation of topics in this text provides readers with a smooth and leisurely transition from basic number theory to graduate topics courses on the Langlands program. It is a unique and self-contained resource for number theorists, instructors teaching basic analytic number theory, and the target readership of first and second year graduate students interested in number theory. Portions of the content are also accessible to mathematically mature advanced undergraduates. The copious number of exercises and examples throughout the text, aptly guide the reader. The prerequisite for using the book is a grounded understanding of number fields and local fields.
The book is well designed in its aims to build a bridge from beginning number theory topics to modern, advanced ones. Starting from scratch with the classical theory (Riemann's zeta function, Dirichlet L-functions, Dedekind zeta functions) it merges into a detailed account of Artin L-functions, Tate's thesis, and culminates in a discussion of the Deligne-Serre theorem and related results. These topics have not appeared together in book form.
Book
Oct 2025
Provides a comprehensive overview of the past decade's developments in the realm of non-standard function spaces
Covers an extensive range of results in the theory of integral operators, encompassing the linear and multilinear cases
Written for a broad audience, ranging from researchers to experts in applied mathematics and prospective students
Part of the book series: Operator Theory: Advances and Applications (OT, volume 310)
This volume, as a sequel to Volumes I-IV of “Integral Operators in Non-Standard Function Spaces”, is devoted to the authors’ most recent advances in harmonic analysis and their applications.
This volume focusses on Rellich inequalities in the variable exponent and multilinear settings, trace inequalities for linear and multilinear fractional integrals, sharp weighted estimates for norms of operators of harmonic analysis, criteria governing Sobolev-type inequalities for (generalized) fractional integrals associated with non-doubling measures, sharp Olsen-type inequalities, studies on Herz-type spaces, approximation in subspaces of Morrey spaces, introduction of variable exponent bounded variation spaces in the Riesz sense, and characterization of weighted Sobolev spaces via weighted Riesz bounded variation spaces.
The book is aimed at an audience ranging from researchers in operator theory and harmonic analysis to experts in applied mathematics and post graduate students. In particular, it is hoped that this book will serve as a source of inspiration for researchers in abstract harmonic analysis, function spaces, PDEs and boundary value problems.
Conference proceedings
Oct 2025
Each chapter is written by expert researchers in the field
Covers a broad area of topics in Applied Mathematics
Each chapter addresses a different topic of Applied Mathematics: biological modeling, optimal design of nuclear reactors
Part of the book series: Springer Proceedings in Mathematics & Statistics (PROMS, volume 513)
This book collects some contributions presented in the annual Congress "Mathematical Modeling & Human Behavior", held from 10 to 12 July 2024 at Universitat Politècnica de València, Valencia, Spain. These manuscripts deal with new mathematical methods and applications covering a wide range of areas of Mathematics, providing an overview of hot topics where the mathematical community is currently advancing. Mathematical modeling and numerical simulation play a key role in scientific advances in science and engineering. The combination of mathematical language and the power of computers to carry out rapid and accurate computations is currently permitting a better understanding of complex phenomena that are of great interest to our society. The unifying approach to success in dealing with such challenges falls within the realm of mathematical modeling. Thanks to the mathematical methods that are continuously developing, researchers can gain insights into biological processes and medical diseases, predict outcomes in climate phenomena, and optimize designs in engineering mechanisms with unprecedented accuracy and efficiency.
Conference proceedings
Dec 2025
Addresses a broad range of mathematical methods and their usage in a wide variety of disciplines
Highlights some very impactful research on inverse problems and computer tomography
Promotes versatile mathematical methods in view of advancing applied disciplines
Part of the book series: Springer Proceedings in Mathematics & Statistics (PROMS, volume 515)
This book addresses a broad range of mathematical methods and their usage in a wide variety of disciplines ranging from physics, electrical engineering, chemistry, biology and computer sciences to society and economics. Papers are contributed by an international group of authors and focus on unique problems from, e.g., game theory, algebraic topology for cognitive maps, Fermat-Weber methodology, modeling and simulation in insurance sectors, forecasting thunderstorms or wind gusts, volcanic modeling, machine learning in disease monitoring, simulations in battery research and optical microresonators. This book additionally highlights some very impactful research on inverse problems and computer tomography as well as highly relevant societal topics of protests management and their mathematical formulation. This book promotes versatile mathematical methods in view of advancing applied disciplines with emphasis on mutual benefit. This book serves as a reference guide for researchers for advancing their own research. Moreover, this book serves as reference source for students of mathematics, sciences, engineering, or economics who are interested in interdisciplinary applications of mathematics.
Book
Oct 2025
Provides a complete study of monotone and relative rearrangement
Includes a comprehensive and full study of Sobolev embeddings and PDEs
Covers a vast array of special topics and applications of Banach function spaces
Part of the book series: Lecture Notes in Mathematics (LNM, volume 2376)
This book develops the properties of monotone rearrangement and relative rearrangement (sometimes called pseudo-rearrangement). It introduces applications to variational problems involving monotone rearrangements, a priori estimates for partial differential equations, and stationary or evolution problems associated with variable exponents. The properties of Sobolev embeddings for non-standard spaces such as BMO, VMO, Zygmung spaces and more general spaces invariant under rearrangement are also reviewed. The book is relatively self-contained – elementary details for non-specialists are covered in the first chapter, including, among other things, some punctual inequalities for the Sobolev embeddings and Pólya-Szegő type inequalities, which lead, for instance, to explicit and even precise estimates. The final chapter includes numerous exercises, with solutions. Based on the author’s Réarrangement relatif: un instrument d'estimations dans les problèmes aux limites (Springer, 2008), this edition contains additional recent results and new exercises concerning interpolation theory.
Book
Oct 2025
Provides an in-depth discussion of numerics for differential equations on metric graphs
Discusses a finite element and spectral discretization with ready to use code and detailed numerical examples
Presents an innovative method for solving quantum graph eigenvalue problems
Part of the book series: Lecture Notes in Mathematics (LNM, volume 2382)
This book discusses the fundamentals of the numerics of parabolic partial differential equations posed on network structures interpreted as metric spaces. These so-called metric graphs frequently occur in the context of quantum graphs, where they are studied together with a differential operator and coupling conditions at the vertices. The two central methods covered here are a Galerkin discretization with linear finite elements and a spectral Galerkin discretization with basis functions obtained from an eigenvalue problem on the metric graph. The solution of the latter eigenvalue problems, i.e., the computation of quantum graph spectra, is therefore an important aspect of the method, and is treated in depth. Further, a real-world application of metric graphs to the modeling of the human connectome (brain network) is included as a major motivation for the investigated problems. Aimed at researchers and graduate students with a practical interest in diffusion-type and eigenvalue problems on metric graphs, the book is largely self-contained; it provides the relevant background on metric (and quantum) graphs as well as the discussed numerical methods. Numerous detailed numerical examples are given, supplemented by the publicly available Julia package MeGraPDE.jl.
Book
Nov 2025
The first book on beta-type simplices
Essentially self-contained
Provides a thorough introduction to beta-type distributions
Part of the book series: Lecture Notes in Mathematics (LNM, volume 2383)
This book provides an introduction to the theory of random beta-type simplices and polytopes, exploring their connections to key research areas in stochastic and convex geometry. The random points defining the beta-type simplices, a class of random simplices introduced by Ruben and Miles, follow beta, beta-prime, or Gaussian distributions in the Euclidean space, and need not be identically distributed. A key tool in the analysis of these simplices, the so-called canonical decomposition, is presented here in a generalized form and is employed to derive explicit formulas for the moments of the volumes of beta-type simplices and to prove distributional representations for these volumes. Three independent approaches are described, including the original Ruben–Miles method. In addition, a version of the canonical decomposition for beta-type polytopes is provided, characterizing their typical faces as volume-weighted beta-type simplices. This is then applied to compute various expected functionals of beta-type polytopes, such as their volume, surface area and number of facets. The formulas for the moments of the volumes are also used to investigate several high-dimensional phenomena. Among these, a central limit theorem is established for the logarithmic volume of beta-type simplices in the high-dimensional limit. The canonical decomposition further motivates the study of beta-type distributions on affine Grassmannians, a subject to which the last chapter is dedicated.
Largely self-contained, requiring minimal prior knowledge, the book connects these topics to a broad range of past and current research, serving as an excellent resource for graduate students and researchers seeking to engage with the field of stochastic and integral geometry.