This book is dedicated to study the inverse problem of ordinary differential equations, that is it focuses in finding all ordinary differential equations that satisfy a given set of properties. The Nambu bracket is the central tool in developing this approach. The authors start characterizing the ordinary differential equations in R^N which have a given set of partial integrals or first integrals. The results obtained are applied first to planar polynomial differential systems with a given set of such integrals, second to solve the 16th Hilbert problem restricted to generic algebraic limit cycles, third for solving the inverse problem for constrained Lagrangian and Hamiltonian mechanical systems, fourth for studying the integrability of a constrained rigid body. Finally the authors conclude with an analysis on nonholonomic mechanics, a generalization of the Hamiltonian principle, and the statement an solution of the inverse problem in vakonomic mechanics.
This book presents models written as partial differential equations and originating from various questions in population biology, such as physiologically structured equations, adaptive dynamics, and bacterial movement. Its purpose is to derive appropriate mathematical tools and qualitative properties of the solutions. The book further contains many original PDE problems originating in biosciences.
One of the world's foremost geometers, Alan Weinstein has made deep contributions to symplectic and differential geometry, Lie theory, mechanics, and related fields. Written in his honor, the invited papers in this volume reflect the active and vibrant research in these areas and are a tribute to Weinstein's ongoing influence.
The well-recognized contributors to this text cover a broad range of topics: Induction and reduction for systems with symmetry, symplectic geometry and topology, geometric quantization, the Weinstein Conjecture, Poisson algebra and geometry, Dirac structures, deformations for Lie group actions, Kähler geometry of moduli spaces, theory and applications of Lagrangian and Hamiltonian mechanics and dynamics, symplectic and Poisson groupoids, and quantum representations.
This volume uses a unified approach to representation theory and automorphic forms. It collects papers, written by leading mathematicians, that track recent progress in the expanding fields of representation theory and automorphic forms and their association with number theory and differential geometry. Topics include: Automorphic forms and distributions, modular forms, visible-actions, Dirac cohomology, holomorphic forms, harmonic analysis, self-dual representations, and Langlands Functoriality Conjecture, Both graduate students and researchers will find inspiration in this volume.
In this set of lecture notes, the author includes some of the latest research on the theory of Morrey Spaces associated with Harmonic Analysis. There are three main claims concerning these spaces that are covered: determining the integrability classes of the trace of Riesz potentials of an arbitrary Morrey function; determining the dimensions of singular sets of weak solutions of PDE (e.g. The Meyers-Elcart System); and determining whether there are any "full" interpolation results for linear operators between Morrey spaces. This book will serve as a useful reference to graduate students and researchers interested in Potential Theory, Harmonic Analysis, PDE, and/or Morrey Space Theory.
This highly readable book aims to ease the many challenges of starting undergraduate research. It accomplishes this by presenting a diverse series of self-contained, accessible articles which include specific open problems and prepare the reader to tackle them with ample background material and references. Each article also contains a carefully selected bibliography for further reading.The content spans the breadth of mathematics, including many topics that are not normally addressed by the undergraduate curriculum (such as matroid theory, mathematical biology, and operations research), yet have few enough prerequisites that the interested student can start exploring them under the guidance of a faculty member. Whether trying to start an undergraduate thesis, embarking on a summer REU, or preparing for graduate school, this book is appropriate for a variety of students and the faculty who guide them.
The work on Autonomic Road Transport Support (ARTS) presented here aims at meeting the challenge of engineering autonomic behavior in Intelligent Transportation Systems (ITS) by fusing research from the disciplines of traffic engineering and autonomic computing. Ideas and techniques from leading edge artificial intelligence research have been adapted for ITS over the last 30 years. Examples include adaptive control embedded in real time traffic control systems, heuristic algorithms (e.g. in SAT-NAV systems), image processing and computer vision (e.g. in automated surveillance interpretation). Autonomic computing which is inspired from the biological example of the body's autonomic nervous system is a more recent development. It allows for a more efficient management of heterogeneous distributed computing systems. In the area of computing, autonomic systems are endowed with a number of properties that are generally referred to as self-X properties, including self-configuration, self-healing, self-optimization, self-protection and more generally self-management. Some isolated examples of autonomic properties such as self-adaptation have found their way into ITS technology and have already proved beneficial. This edited volume provides a comprehensive introduction to Autonomic Road Transport Support (ARTS) and describes the development of ARTS systems. It starts out with the visions, opportunities and challenges, then presents the foundations of ARTS and the platforms and methods used and it closes with experiences from real-world applications and prototypes of emerging applications. This makes it suitable for researchers and practitioners in the fields of autonomic computing, traffic and transport management and engineering, AI, and software engineering. Graduate students will benefit from state-of-the-art description, the study of novel methods and the case studies provided.
This open access book explores the amazing similarity between paths taken by people and many other things in life, and its impact on the way we live, teach and learn.Offering insights into the new scientific field of paths as part of the science of networks, it entertainingly describes the universal nature of paths in large networked structures. It also shows the amazing similarity in the ways humans and other - even nonliving - things navigate in a complex environment, to allow readers to easily grasp how paths emerge in many walks of life, and how they are navigated.Paths is based on the authors recent research in the area of paths on networks, which points to the possible birth of the new science of "paths" as a natural consequence `and extension) of the science of "networks."The approach is essentially story-based, supported by scientific findings, interdisciplinary approaches, and at times, even philosophical points of view. It alsoincludes short illustrative anecdotes showing the amazing similarities between real-world paths and discusses their applications in science and everyday life.Paths will appeal to network scientists and to anyone interested in popular science. By helping readers to step away from the "networked" view of many recent popular scientific books and start to think of longer paths instead of individual links, it sheds light on these problems from a genuinely new perspective.--------------------------------------------------------------------------------- The path is the goal. The essence behind this short sentence is known to many people around the world, expressed through the interpretations of some of the greatest thinkers like Lao-Tze and Gandhi. It means that it is the journey that counts, not the destination. When speaking about such subjective and intangible things, philosophy and religion are some of the only approaches that are addressed. In this book, the authors address this conventional wisdom from the perspective of natural science. They explore a sequence of steps that leads the reader closer to the nature of paths and accompany him on the search for "the path to paths".
This book provides a concise survey of the theory of zero product-determined algebras, which has been developed over the last 15 years. It is divided into three parts. The first part presents the purely algebraic branch of the theory, the second part presents the functional analytic branch, and the third part discusses various applications.
The book is intended for researchers and graduate students in ring theory, Banach algebra theory, and nonassociative algebra.
The theory of real-valued Sobolev functions is a classical part of analysis and has a wide range of applications in pure and applied mathematics. By contrast, the study of manifold-valued Sobolev maps is relatively new. The incentive to explore these spaces arose in the last forty years from geometry and physics. This monograph is the first to provide a unified, comprehensive treatment of Sobolev maps to the circle, presenting numerous results obtained by the authors and others. Many surprising connections to other areas of mathematics are explored, including the Monge-Kantorovich theory in optimal transport, items in geometric measure theory, Fourier series, and non-local functionals occurring, for example, as denoising filters in image processing. Numerous digressions provide a glimpse of the theory of sphere-valued Sobolev maps.
Each chapter focuses on a single topic and starts with a detailed overview, followed by the most significant results, and rather complete proofs. The "Complements and Open Problems" sections provide short introductions to various subsequent developments or related topics, and suggest newdirections of research. Historical perspectives and a comprehensive list of references close out each chapter. Topics covered include lifting, point and line singularities, minimal connections and minimal surfaces, uniqueness spaces, factorization, density, Dirichlet problems, trace theory, and gap phenomena.
Sobolev Maps to the Circle will appeal to mathematicians working in various areas, such as nonlinear analysis, PDEs, geometric analysis, minimal surfaces, optimal transport, and topology. It will also be of interest to physicists working on liquid crystals and the Ginzburg-Landau theory of superconductors.
This research monograph develops the theory of relative nonhomogeneous Koszul duality. Koszul duality is a fundamental phenomenon in homological algebra and related areas of mathematics, such as algebraic topology, algebraic geometry, and representation theory. Koszul duality is a popular subject of contemporary research.
This book, written by one of the world's leading experts in the area, includes the homogeneous and nonhomogeneous quadratic duality theory over a nonsemisimple, noncommutative base ring, the Poincare-Birkhoff-Witt theorem generalized to this context, and triangulated equivalences between suitable exotic derived categories of modules, curved DG comodules, and curved DG contramodules. The thematic example, meaning the classical duality between the ring of differential operators and the de Rham DG algebra of differential forms, involves some of the most important objects of study in the contemporary algebraic and differential geometry. For the first timein the history of Koszul duality the derived D-Omega duality is included into a general framework. Examples highly relevant for algebraic and differential geometry are discussed in detail.
This volume collects chapters that examine representation theory as connected with affine Lie algebras and their quantum analogues, in celebration of the impact Vyjayanthi Chari has had on this area. The opening chapters are based on mini-courses given at the conference "Interactions of Quantum Affine Algebras with Cluster Algebras, Current Algebras and Categorification", held on the occasion of Chari's 60th birthday at the Catholic University of America in Washington D.C., June 2018. The chapters that follow present a broad view of the area, featuring surveys, original research, and an overview of Vyjayanthi Chari's significant contributions. Written by distinguished experts in representation theory, a range of topics are covered, including:String diagrams and categorificationQuantum affine algebras and cluster algebrasSteinberg groups for Jordan pairsDynamical quantum determinants and PfaffiansInteractions of Quantum Affine Algebras with Cluster Algebras, Current Algebras and Categorification will be an ideal resource for researchers in the fields of representation theory and mathematical physics.
The chapters in this volume explore the influence of the Russian school on the development of algebraic geometry and representation theory, particularly the pioneering work of two of its illustrious members, Alexander Beilinson and Victor Ginzburg, in celebration of their 60th birthdays. Based on the work of speakers and invited participants at the conference "Interactions Between Representation Theory and Algebraic Geometry", held at the University of Chicago, August 21-25, 2017, this volume illustrates the impact of their research and how it has shaped the development of various branches of mathematics through the use of D-modules, the affine Grassmannian, symplectic algebraic geometry, and other topics. All authors have been deeply influenced by their ideas and present here cutting-edge developments on modern topics. Chapters are organized around three distinct themes:Groups, algebras, categories, and representation theoryD-modules and perverse sheavesAnalogous varieties defined by quivers
Representation Theory and Algebraic Geometry will be an ideal resource for researchers who work in the area, particularly those interested in exploring the impact of the Russian school.
This monograph provides an exhaustive treatment of several classes of Noetherian rings and morphisms of Noetherian local rings. Chapters carefully examine some of the most important topics in the area, including Nagata, F-finite and excellent rings, Bertini's Theorem, and Cohen factorizations. Of particular interest is the presentation of Popescu's Theorem on Neron Desingularization and the structure of regular morphisms, with a complete proof. Classes of Good Noetherian Rings will be an invaluable resource for researchers in commutative algebra, algebraic and arithmetic geometry, and number theory.
This volume, originally published in China and translated into four other languages, presents a fascinating and unique account of the history of mathematics, divided into eight chronologically organized chapters. Tracing the development of mathematics across disparate regions and peoples, with particular emphasis on the relationship between mathematics and civilization, it examines mathematical sources and inspirations leading from Egypt, Babylon and ancient Greece and expanding to include Chinese, Indian and Arabic mathematics, the European Renaissance and the French revolution up through the Nineteenth and Twentieth Centuries. Each chapter explores connections among mathematics and cultural elements of the time and place treated, accompanying the reader in a varied and exciting journey through human civilizations. The book contemplates the intersections of mathematics with other disciplines, including the relationship between modern mathematics and modern art, and the resultingapplications, with the aid of images and photographs, often taken by the author, which further enhance the enjoyment for the reader.
Written for a general audience, this book will be of interest to anyone who's studied mathematics in university or even high school, while also benefiting researchers in mathematics and the humanities.
This book introduces several powerful techniques and fundamental ideas involving block matrices of operators, as well as matrices with elements in a C*-algebra. These techniques allow for the solution of problems that may be difficult to treat. Specifically, 2×2 operator matrices yield significant mathematical inequalities in various fields of operator theory and matrix analysis. The authors employ block matrices to simplify complicated problems. Operator matrices have garnered attention for their applications in quantum information and computing theories.
Each chapter concludes with a diverse set of exercises and problems for readers, along with references to relevant literature. Some problems pose open questions, while others challenge readers and provide suggestions for future research. This book is suitable for an advanced undergraduate or graduate course and can be used in the classroom. It also serves as a valuable resource for researchers and students in mathematics and physics who have a basic understanding of linear algebra, functional analysis, and operator theory.
The goal of this book is to investigate the behavior of weak solutions to the elliptic interface problem in a neighborhood of boundary singularities: angular and conic points or edges. This problem is considered both for linear and quasi-linear equations, which are among the less studied varieties. As a second edition of Transmission Problems for Elliptic Second-Order Equations for Non-Smooth Domains (Birkhäuser, 2010), this volume includes two entirely new chapters: one about the oblique derivative problems for the perturbed p(x)-Laplacian equation in a bounded n-dimensional cone, and another about the existence of bounded weak solutions.
Researchers and advanced graduate students will appreciate this compact compilation of new material in the field.
This book expounds on the recent developments in applications of holomorphic functions in the theory of hypercomplex and anti-Hermitian manifolds as well as in the geometry of bundles. It provides detailed information about holomorphic functions in algebras and discusses some of the areas in geometry with applications. The book proves the existence of a one-to-one correspondence between hyper-complex anti-Kähler manifolds and anti-Hermitian manifolds with holomorphic metrics, and also a deformed lifting to bundles. Researchers and students of geometry, algebra, topology and physics may find the book useful as a self-study guide.
Wave phenomena are ubiquitous in nature. Their mathematical modeling, simulation and analysis lead to fascinating and challenging problems in both analysis and numerical mathematics. These challenges and their impact on significant applications have inspired major results and methods about wave-type equations in both fields of mathematics.
The Conference on Mathematics of Wave Phenomena 2018 held in Karlsruhe, Germany, was devoted to these topics and attracted internationally renowned experts from a broad range of fields. These conference proceedings present new ideas, results, and techniques from this exciting research area.
This volume features a collection of contributed articles and lecture notes from the XIII Symposium on Probability and Stochastic Processes, held at UNAM, Mexico, in December 2017.It is split into two main parts: the first one presents lecture notes of the course provided by Mauricio Duarte, followed by its second part which contains research contributions of some of the participants.
This collection of contributions originates from the well-established conference series "Fractal Geometry and Stochastics" which brings together researchers from different fields using concepts and methods from fractal geometry.
Carefully selected papers from keynote and invited speakers are included, both discussing exciting new trends and results and giving a gentle introduction to some recent developments. The topics covered include Assouad dimensions and their connection to analysis, multifractal properties of functions and measures, renewal theorems in dynamics, dimensions and topology of random discrete structures, self-similar trees, p-hyperbolicity, phase transitions from continuous to discrete scale invariance, scaling limits of stochastic processes, stemi-stable distributions and fractional differential equations, and diffusion limited aggregation.
Representing a rich source of ideas and a good starting point for more advanced topics in fractal geometry, the volume will appeal to both established experts and newcomers.
This book introduces the reader to important concepts in modern applied analysis, such as homogenization, gradient flows on metric spaces, geometric evolution, Gamma-convergence tools, applications of geometric measure theory, properties of interfacial energies, etc. This is done by tackling a prototypical problem of interfacial evolution in heterogeneous media, where these concepts are introduced and elaborated in a natural and constructive way. At the same time, the analysis introduces open issues of a general and fundamental nature, at the core of important applications. The focus on two-dimensional lattices as a prototype of heterogeneous media allows visual descriptions of concepts and methods through a large amount of illustrations.
This monograph presents recent results concerning nonlinear fractional elliptic problems in the whole space. More precisely, it investigates the existence, multiplicity and qualitative properties of solutions for fractional Schrdinger equations by applying suitable variational and topological methods.The book is mainly intended for researchers in pure and applied mathematics, physics, mechanics, and engineering. However, the material will also be useful for students in higher semesters and young researchers, as well as experienced specialists working in the field of nonlocal PDEs. This is the first book to approach fractional nonlinear Schrdinger equations by applying variational and topological methods.
This volume presents lectures given at the Wisla 19 Summer School: Differential Geometry, Differential Equations, and Mathematical Physics, which took place from August 19 - 29th, 2019 in Wisla, Poland, and was organized by the Baltic Institute of Mathematics. The lectures were dedicated to symplectic and Poisson geometry, tractor calculus, and the integration of ordinary differential equations, and are included here as lecture notes comprising the first three chapters. Following this, chapters combine theoretical and applied perspectives to explore topics at the intersection of differential geometry, differential equations, and mathematical physics. Specific topics covered include:Parabolic geometryGeometric methods for solving PDEs in physics, mathematical biology, and mathematical financeDarcy and Euler flows of real gasesDifferential invariants for fluid and gas flowDifferential Geometry, Differential Equations, and Mathematical Physics is ideal for graduate students and researchers working in these areas. A basic understanding of differential geometry is assumed.
