Mathematics and Statistics

This volume collects papers based on lectures given at the XL Workshop on Geometric Methods in Physics, held in Białowieża, Poland in July 2023. These chapters provide readers an overview of cutting-edge research in infinite-dimensional groups, integrable systems, quantum groups, Lie algebras and their generalizations and a wide variety of other areas. Specific topics include:

• Yang-Baxter equation
• The restricted Siegel disc and restricted Grassmannian
• Geometric and deformation  quantization
• Degenerate integrability
• Lie algebroids and groupoids
• Skew braces

Geometric Methods in Physics XL will be a valuable resource for mathematicians and physicists interested in recent developments at the intersection of these areas.

This book covers the basics of numerical methods. Avoiding the definition-theorem-proof style, it instead focuses on numerical examples and simple pseudo-codes.

The text begins with a chapter on floating point arithmetic before moving on to discuss norms, conditions numbers, solutions of systems of equations, the least squares problem, eigenvalue problems, interpolation, numerical integration, ordinary differential equations, optimization (including a detailed case study), and practical error estimations. Exercises (partly in MATLAB) are provided at the end of each chapter. Suitable for readers with minimal mathematical knowledge, the book not only offers an elementary introduction to numerical mathematics for programmers and engineers but also provides supporting material for students and teachers of mathematics.

This edited volume introduces undergraduate students to several diverse and emerging research topics in three major areas of risk management: actuarial science, mathematical finance, and sports analytics. Each chapter is self-contained and provides students with the required background knowledge, context, and relevant references to motivate students and to enable them to successfully undertake research in these areas.  A variety of accessible research projects are included that can be conducted independently or in collaboration with a mentor. Throughout, emphasis is placed on the importance and advantages of interdisciplinary thinking in developing innovative approaches to problem-solving. This volume will also serve as a valuable resource to faculty mentors, graduate students, and others who are interested in supervising undergraduate research.

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.

This is an open access book. Probabilistic tsunami hazard and risk assessment (abbreviated as PTHA and PTRA, respectively) has evolved quickly over the past 10 to 15 years and now forms the basis of many risk mitigation efforts. Methods are largely based on those used for probabilistic hazard and risk assessment for earthquakes. However, while seismic risk can be represented by a single workflow, tsunami risk is characterized by a number of different and cascading workflows. Funded in parts by the COST Action CA18109 AGITHAR (2019-2023), a group of more than 120 experts gathered to advance the scientific basis for PTHA and PTRA. Amongst them were over 50 volunteering to contribute to this so-called “cookbook” for PTHA and PTRA.

The aim of our cookbook is to give an overview of existing methods for PTHA and PTRA, to unify the description of named workflows, and to make best practices examples available to a wider community. We use the analogy of a cookbook, because we strive to describe the ingredients, i.e., the basic building blocks of PTHA and PTRA, and give examples of good recipes, i.e., well-designed workflows for concrete application fields. Furthermore, a cookbook gives inspiration but does not necessarily standardize methods. So, while this effort aims at some standardization of methodology, it gives enough freedom to build on this knowledge to further develop the field. The book is intended for scientists, but also practitioners and decision-makers in tsunami hazard and risk mitigation.

This book grew out of a series of three Simons Symposia in 2019, 2022, and 2024, that focused on the interactions between three types of dynamics—algebraic, complex, and arithmetic—and their connections to the closely related areas of algebraic, complex, and arithmetic geometry. Dynamical systems play a key role both in pure mathematics and its applications. These meetings brought experts together from across these fields for discussions on the most significant recent developments and directions for further research.

This monograph, the second of the two volumes forming the third edition, is an enlarged, completely updated, and extensively revised version of the corresponding material from the authoritative second edition. It is completely self-contained.

This volume contains chapters on tensors, three-dimensional continuum mechanics, constitutive equations and their physically natural restrictions for three-dimensional elasticity and strain-rate viscoelasticity, steady-state and dynamical problems for these theories, general geometrically and mechanically exact theories of rods and shells, methods for treating dynamical problems of strain-rate viscoelasticity, and the role of material response for quasilinear hyperbolic systems of elasticity.

Each chapter contains a wealth of interesting, challenging, and tractable exercises.

From the reviews of the second edition:

"Antman’s impressive work is … a comprehensivetreatise on nonlinear elasticity and a quintessential example of applied nonlinear analysis. … The text has been revised and updated, Several new sections have been added … This book is a ‘must’ for researchers and graduate students interested in nonlinear continuum mechanics and applied analysis. The work is scholarly and well written. … ‘This book is directed toward scientists, engineers, and mathematicians who wish to see careful treatments of uncompromised problems.’" (Timothy J. Healey, SIAM Review, Vol. 49 (2), 2007)

This graduate text presents the elegant and profound theory of continuous parameter Markov processes and many of its applications.  The authors focus on developing context and intuition before formalizing the theory of each topic, illustrated with examples.

After a review of some background material, the reader is introduced to semigroup theory, including the Hille–Yosida Theorem,  used to construct continuous parameter Markov processes.  Illustrated with examples, it is a cornerstone of Feller’s seminal theory of the most general one-dimensional diffusions studied in a later chapter. This is followed by two chapters with probabilistic constructions of jump Markov processes,  and   processes with independent increments, or Lévy processes. The greater part of the book is devoted to  Itô’s fascinating theory of stochastic differential equations,  and to the study of  asymptotic properties of diffusions  in all dimensions, such as   explosion, transience, recurrence,  existence of steady states,  and the speed of convergence to equilibrium.  A broadly applicable functional central limit theorem for ergodic Markov processes is presented with important examples. Intimate connections between diffusions  and linear second order elliptic and parabolic partial differential equations are laid out in two chapters, and are used for computational purposes.  Among Special Topics chapters, two study anomalous diffusions: one on  skew Brownian motion, and the other on an intriguing multi-phase homogenization of solute transport in porous media.

This book presents contributions related to new research results presented at the 27th International Conference on Difference Equations and Applications, ICDEA 2022, that was held at CentraleSupélec, Université Paris-Saclay, France, under the auspices of the International Society of Difference Equations (ISDE), July 18–22, 2022. The book aims not only to disseminate these results but to foster further advances in the fields of difference equations and discrete dynamical systems. Also included are applications to economic growth modeling, population dynamics, epidemic modeling, game theory, control systems, and network analysis. The target audience for the book includes Ph.D. students, researchers, educators, and practitioners in these fields.