Presents an emphasis to the group-theoretical aspects of relativity
Designs for students and researchers with a basic background in SR and electrodynamics
Explores the essential role of symmetry in physics, with a focus on its dynamic presence in special relativity
Part of the book series: SpringerBriefs in Physics (SpringerBriefs in Physics)
This book explores the essential role of symmetry in physics, with a focus on its dynamic presence in special relativity (SR) and its implications for modern physical theories. By examining symmetry transformations and their connection to conservation laws and covariance, readers gain a comprehensive understanding of these foundational concepts.
The chapters cover a range of topics, including the Lorentz group, relativistic transformations, and the covariance of Maxwell's equations. Readers will delve into the mathematical structure of the Lorentz group, its integration with relativistic mechanics, and its application in electrodynamics. Special topics such as Lie groups, the homomorphism with SL(2,C), and the independence of Maxwell's equations are explored, offering a detailed view of the coherence and elegance of the covariant formulation of the Maxwell system.
Designed for those with a basic understanding of special relativity and electrodynamics, this book is an advanced supplement to introductory texts. It is particularly beneficial for undergraduate and graduate students, researchers, and academics in physics who wish to deepen their knowledge of symmetry and its significance in high-energy physics. The inclusion of detailed solutions to problems makes it suitable for self-study, providing a valuable resource for anyone interested in the mathematical and theoretical underpinnings of modern physics.
Presentation of different algorithms to solve the linear-quadratic optimal control problem with stochastic PDEs
Concise error analysis of the proposed algorithms
Comparison of complexities of proposed algorithms by computational studies
Part of the book series: SpringerBriefs on PDEs and Data Science (SBPDEDS)
This book is on the construction and convergence analysis of implementable algorithms to approximate the optimal control of a stochastic linear-quadratic optimal control problem (SLQ problem, for short) subject to a stochastic PDE. If compared to finite dimensional stochastic control theory, the increased complexity due to high-dimensionality requires new numerical concepts to approximate SLQ problems; likewise, well-established discretization and numerical optimization strategies from infinite dimensional deterministic control theory need fundamental changes to properly address the optimality system, where to approximate the solution of a backward stochastic PDE is conceptually new. The linear-quadratic structure of SLQ problems allows two equivalent analytical approaches to characterize its minimum: ‘open loop’ is based on Pontryagin’s maximum principle, and ‘closed loop’ utilizes the stochastic Riccati equation in combination with the feedback control law. The authors will discuss why, in general, complexities of related numerical schemes differ drastically, and when which direction should be given preference from an algorithmic viewpoint.
Provides new functional analytic results on spectral problems
Unifies and extends established results in different directions
Includes numerous open questions
Part of the book series: Lecture Notes in Mathematics (LNM, volume 2388)
This monograph provides new functional analytic developments on spectral problems in various applied fields such as jump equations, Kolmogorov differential equations, weighted graphs, neutron transport theory, population dynamics, linearized non-local Allen–Cahn equations and perturbed convolution semigroups. With an emphasis on spectral problems (in Lebesgue spaces) connected to positivity, it uses powerful tools developed over the last two decades to unify and extend established results in various directions, offering new perspectives in a diverse range of subjects. Many open questions are scattered throughout, in the hope of promoting further research.
Covering topics lying at the intersection between functional analysis and partial differential equations, the book is primarily targeted toward applied mathematicians working in kinetic theory, probability theory, mathematical biology and, more generally, partial differential equations, who are interested in peripheral spectral theory. It may also draw pure mathematicians working in semigroup theory towards very rich applied areas. Although mainly aimed at professional mathematicians, it will also be useful to PhD students and post-doctoral researchers. The long chapter devoted to transport theory will be of interest to physicists or engineers involved in neutron transport.
Provides a comprehensive current state of the field Convex Geometric Analysis
Gives an overview of the works exposed during the onference
Shows new lines of interest in the field
Part of the book series: RSME Springer Series (RSME, volume 17)
These proceedings result from the International Conference 'Geometry, Analysis & Convexity' (OLE 2022) held from 20th to 24th June 2022 at the Instituto de Matemáticas de la Universidad de Sevilla (IMUS), Spain and they include some of the contributions presented at this conference. This book is addressed to any researcher interested in convex geometric analysis and asymptotic analysis as well as integral geometry and discrete geometry and their applications in convexity, and related topics. Convex geometric analysis was born from the increasing interaction between classical (convex) geometry and asymptotic (convex) analysis. During the last three decades, the study of the integral geometry of convex bodies has been fuelled by the introduction of methods, results and new points of view coming from other branches of mathematics such as probability, harmonic analysis, geometry of finite dimensional normed spaces, integral geometry and discrete geometry. These recent advances have revealed fruitful connections between geometric inequalities, transport theory and information theory.
Asymptotic convex analysis is mainly concerned with geometric properties of convex bodies in finite dimensional normed spaces, focused when the dimension tends to infinity. The understanding of high dimensional phenomena becomes an important point since high dimensional problems are frequently encountered in mathematics and applied sciences. Concentration of measure phenomenon can be viewed as an isoperimetric problem, which lies at the heart of classical geometry and calculus of variation. Besides convex geometry, geometric analysis has been developed using techniques and deep theorems from integral geometry, where the notion of measure is generalized to the concept of the so-called valuation, and it has developed from a simple technique to a fundamental area, the theory of valuations. The underlying structure of the valuation space (invariant under translations) is intrinsically connected with affine or analytic isoperimetric inequalities, among others. It is addressed to researchers in this field.
Extends optimal control theory beyond finite horizons while handling state constraints
Focuses on optimal control problems involving nonsmooth cost functions, dynamics, and differential inclusions
Develops solution concepts for Hamilton-Jacobi-Bellman equations over infinite horizons
Structured as an accessible brief course for graduate students while maintaining depth for researchers in the field
Part of the book series: SpringerBriefs in Mathematics (BRIEFSMATH)
This book provides an introduction to nonsmooth constrained optimal control theory over infinite horizon, tackling the mathematical challenges that arise when classical finite-horizon methods prove inadequate. The work focuses on recent advances in handling nonsmooth time-dependent data and state constraints—scenarios that commonly arise in fields such as engineering, machine learning, and artificial intelligence.
At its core, the book establishes foundational contributions, including viability results, extensions of Pontryagin's maximum principle to infinite horizons, regularity analysis of value functions, and necessary optimality conditions, with particular attention to transversality conditions and sensitivity relations. The analysis culminates in studying Hamilton-Jacobi-Bellman equations through weak solution notions suited to the nonsmooth framework.
Written as a brief course, the book aims to provide graduate students and researchers with the mathematical tools needed to analyze optimal control problems over infinite time horizons, where standard approaches may not apply.
Very few publications on random distributions
Provides a fast entrance into the theory of generalized functions
Includes the interconnections of four types of distributions
Shows distributions can be incorporated into your research work
The aim of the present work is to give a unifying treatment of the four faces of the theory of generalized functions – linear, nonlinear, random and infinite dimensional distributions. The book deals with all of these categories in a comparative and interconnected way, giving a broad overview without getting lost in too many details.
This original presentation touches upon a number of mathematical areas, such as functional analysis, theory of functions, measure theory, operator theory, differentiable manifolds, probability theory, stochastic processes and stochastic analysis. For example, in one of the chapters it builds a bridge from Gaussian measures on Hilbert spaces, Malliavin calculus, Wiener chaos, Meyer-Watanabe distributions, Hida distributions and White Noise Analysis up to Kondratiev spaces, ending with Colombeau versions.
Any researcher who is looking for an overview on the state-of-the art, or to specialize in the theory of distributions, will find this book a useful resource.
Explore related subjects
Presents a comprehensive and up-to-date survey of recent developments in geometric inequalities and their applications
Includes chapters on a broad spectrum of topics contributed by leading scientists from across the world
Discusses Riemannian manifolds, statistical maps and their inequalities, and Ricci–Yamabe solitons
Part of the book series: Infosys Science Foundation Series (ISFS)
Part of the book sub series: Infosys Science Foundation Series in Mathematical Sciences (ISFM)
This contributed volume includes chapters written by leading experts from around the world and provides a thorough and up-to-date exploration of geometric inequalities and their far-reaching applications. Covering a broad spectrum of topics, it discusses the intricacies of geometric solitons, generalized Ricci–Yamabe solitons on three-dimensional Lie groups, and Riemannian invariants in submanifold theory. Readers will find in-depth discussions on B.Y. Chen inequalities for submanifolds of Kenmotsu space forms, refined Chen–Ricci inequalities for submersions from Sasakian space forms, and essential characterizations of perfect fluid and generalized Robertson–Walker space-times admitting k-almost Ricci–Yamabe solitons. The book also investigates Riemannian concircular structure manifolds, statistical maps and their inequalities, as well as hyperbolic and η-hyperbolic Ricci–Yamabe solitons.
Easily accessible to both math and non-math majors who are taking an introductory course on Stochastic Processes
Filled with numerous exercises to test students' understanding of key concepts
A gentle introduction to help students ease into later chapters, also suitable for self-study
Accompanied with computer simulation codes in R and Python
Part of the book series: Springer Undergraduate Mathematics Series (SUMS)
This book provides an undergraduate-level introduction to discrete and continuous-time Markov chains and their applications, with a particular focus on the first step analysis technique and its applications to the computation of average hitting times and ruin probabilities. It also discusses classical topics such as recurrence and transience, stationary and limiting distributions, as well as branching processes. It starts by examining in detail two important examples (gambling processes and random walks) before presenting the general theory in the subsequent chapters. It also provides an introduction to discrete-time martingales and their relation to ruin probabilities and mean exit times, together with a chapter on spatial Poisson processes. The concepts presented are illustrated by examples, 150 exercises and 22 problems with their solutions.
This book is a revised and expanded version of the previous edition, and includes additional exercises and problems with complete solutions. As in the previous book, all exercises and problems are solved in detail, with many graphs and explanatory figures.
**
The fifteen articles in this proceedings volume consist of expanded and peer-reviewed versions of expert talks
The articles represent an illuminating cross section of their agendas
The research and survey articles on certain recent developments in analysis, geometry, statistics and stochastics
Part of the book series: Tohoku Series in Mathematical Sciences (TSMC, volume 1)
This proceedings volume presents a collection of fifteen peer-reviewed and expanded articles based on expert talks delivered at two workshops held in September 2023: the RIMS Symposium on Analysis, Geometry and Stochastics on Metric Spaces at the Research Institute of Mathematical Sciences (Kyoto), organized by S. Shimizu and W. Tuschmann, and Metrics and Measures 2023 at the Mathematical Institute of Tohoku University (Sendai), organized by T. Shioya. These events were part of the 2023 activities of the HeKKSaGOn Mathematics Group, a collaborative network founded in 2010 that brings together six leading universities: Heidelberg, Karlsruhe (KIT), Kyoto, Sendai (Tohoku), Göttingen, and Osaka.
The volume reflects recent developments in analysis, geometry, non-Euclidean statistics, and stochastics, as well as their applications. The articles offer a rich and diverse cross section of the themes explored during the workshops. Topics include long-time limits of diffusion means, Helmholtz–Weyl decompositions in exterior domains, Green functions in nonnegative Ricci curvature, locally homogeneous RCD spaces, Wald spaces for phylogenetic trees, Korevaar–Schoen energy forms on fractals, gluing constructions under lower curvature bounds, combinatorial Riemannian manifolds, the string topology coproduct and intersection multiplicity, probabilistic morphisms and geometric methods in statistical, manifold, and machine learning, gradient flows of convex functions, open problems in metric measure geometry, dynamical aspects of Hadamard spaces, obstacle problems for mean curvature flows, and moduli spaces of nonnegatively curved metrics on Riemannian manifolds and RCD spaces.
Together, these contributions aim to inspire both experts and newcomers in the field, offering insights into cutting-edge research and fostering further exploration across disciplines.
*