The aim of the present book is to give a systematic treatment of the inverse problem of the calculus of variations, i.e. how to recognize whether a system of differential equations can be treated as a system for extremals of a variational functional (the Euler-Lagrange equations), using contemporary geometric methods. Selected applications in geometry, physics, optimal control, and general relativity are also considered. The book includes the following chapters: - Helmholtz conditions and the method of controlled Lagrangians (Bloch, Krupka, Zenkov) - The Sonin-Douglas's problem (Krupka) - Inverse variational problem and symmetry in action: The Ostrogradskyj relativistic third order dynamics (Matsyuk.) - Source forms and their variational completion (Voicu) - First-order variational sequences and the inverse problem of the calculus of variations (Urban, Volna) - The inverse problem of the calculus of variations on Grassmann fibrations (Urban).

Classroom-tested and lucidly written, Multivariable Calculus gives a thorough and rigoroustreatment of differential and integral calculus of functions of several variables. Designed as ajunior-level textbook for an advanced calculus course, this book covers a variety of notions,including continuity , differentiation, multiple integrals, line and surface integrals, differentialforms, and infinite series. Numerous exercises and examples throughout the book facilitatethe student's understanding of important concepts.The level of rigor in this textbook is high; virtually every result is accompanied by a proof. Toaccommodate teachers' individual needs, the material is organized so that proofs can be deemphasizedor even omitted. Linear algebra for n-dimensional Euclidean space is developedwhen required for the calculus; for example, linear transformations are discussed for the treatmentof derivatives.Featuring a detailed discussion of differential forms and Stokes' theorem, Multivariable Calculusis an excellent textbook for junior-level advanced calculus courses and it is also usefulfor sophomores who have a strong background in single-variable calculus. A two-year calculussequence or a one-year honor calculus course is required for the most successful use of thistextbook. Students will benefit enormously from this book's systematic approach to mathematicalanalysis, which will ultimately prepare them for more advanced topics in the field.

This volume contains the Proceedings of the International Workshop "Variational Methods For Discontinuous Structures," held at Villa Erba Antica (Cernobbio) on the Lago di Como, July 4-6, 2001. The workshop was jointly organized by the Dipartimento di Matematica Francesco Brioschi of Milano Politecnico and the International School for Advanced Studies (SISSA) of Trieste. In past years the calculus of variations faced mainly the study of continuous structures, particularly problems with smooth solutions. One of the deepest and more delicate problems was the regularity of weak solutions. More recently, new sophisticated tools have been introduced in order to study discontinuities. In many variational problems solutions develop singularities, and sometimes the most interesting part of a solution is the singularity itself. The conference intended to focus on recent developments in this direction. Some of the talks were devoted to differential or variational modelling of image segmentation, occlusion and textures synthesizing in image analysis, variational description of micro-magnetic materials, dimension reduction and structured deformations in elasticity and plasticity, phase transitions, irrigation and drainage, evolution of crystalline shapes. In most cases theoretical and numerical analysis of these models were provided. Other talks were dedicated to specific problems of the calculus of variations: variational theory of weak or lower-dimensional structures, optimal transport problems with free Dirichlet regions, higher order variational problems, symmetrization in the BV framework. This volume contains contributions by 12 of the 16 speakers invited to deliver lectures in the workshop. Most of the contributions present original results in fields which are rapidly evolving at present.

Mechanics and mathematics have been complementary partners since Newton's time and the history of science shows much evidence of the beneficial influence of these disciplines on each other. Driven by increasingly elaborate modern technological applications the symbiotic relationship between mathematics and mechanics is continually growing. However, the increasingly large number of specialist journals has generated a duality gap between the two partners, and this gap is growing wider.
Advances in Mechanics and Mathematics (AMMA) is intended to bridge the gap by providing multi-disciplinary publications. This volume, AMMA 2002, includes two parts with three articles by four subject experts. Part 1 deals with nonsmooth static and dynamic systems. A systematic mathematical theory for multibody dynamics with unilateral and frictional constraints and a brief introduction to hemivariational inequalities together with some new developments in nonsmooth semi-linear elliptic boundary value problems are presented. Part 2 provides a comprehensive introduction and the latest research on dendritic growth in fluid mechanics, one of the most profound and fundamental subjects in the area of interfacial pattern formation, a commonly observed phenomenon in crystal growth and solidification processes.
Audience: Scientists and mathematicians, including advanced students (doctoral and post-doctoral level) at universities and in industry interested in mechanics and applied mathematics.

This book provides an introduction to the analysis and the control mechanism of physical, chemical, biological, technological and economic models and their nonequilibrium evolution dynamics. Strong emphasis is placed on the foundation of variational principles, evolution and control equations, numerical methods, statistical concepts and techniques for solving or estimation of stochastic control problems for systems with a high degree of complexity. In particular, the central aim of this book is developing a synergetic connection between theoretical concepts and real applications. This book is a modern introduction and a helpful tool for researchers as well as for graduate students interested in econophysics and related topics.

Optimization in Science and Engineering is dedicated in honor of the 60th birthday of Distinguished Professor Panos M. Pardalos. Pardalos's past and ongoing work has made a significant impact on several theoretical and applied areas in modern optimization. As tribute to the diversity of Dr. Pardalos's work in Optimization, this book comprises a collection of contributions from experts in various fields of this rich and diverse area of science. Topics highlight recent developments and include: * Deterministic global optimization* Variational inequalities and equilibrium problems* Approximation and complexity in numerical optimization* Non-smooth optimization * Statistical models and data mining* Applications of optimization in medicine, energy systems, and complex network analysis This volume will be of great interest to graduate students, researchers, and practitioners, in the fields of optimization and engineering.

This book addresses remaining life prediction and predictive maintenance of equipment. It systematically summarizes the key research findings made by the author and his team and focuses on how to create equipment performance degradation and residual life prediction models based on the performance monitoring data produced by currently used and historical equipment. Some of the theoretical results covered here have been used to make remaining life predictions and maintenance-related decisions for aerospace products such as gyros and platforms. Given its scope, the book offers a valuable reference guide for those pursuing theoretical or applied research in the areas of fault diagnosis and fault-tolerant control, remaining life prediction, and maintenance decision-making.

This fourth edition of selecta of my work on the stability of matter contains recent work on two topics that continue to fascinate me: Quantum electrodynamics (QED) and the Bose gas. Three papers have been added to Part VII on QED. As I mentioned in the preface to the third edition, there must be a way to formulate a non-perturbative QED, presumably with an ultraviolet cutoff, that correctly describes low energy physics, i.e., ordinary matter and its interaction with the electromagnetic field. The new paper VII.5, which quantizes the results in V.9, shows that the elementary no-pair version of relativistic QED (using the Dirac operator) is unstable when many-body effects are taken into account. Stability can be restored, however, if the Dirac operator with the field, instead of the bare Dirac operator, is used to define an electron. Thus, the notion of a bare electron without its self-field is physically questionable."

This volume collects papers, based on invited talks given at the IMA workshop in Modeling, Stochastic Control, Optimization, and Related Applications, held at the Institute for Mathematics and Its Applications, University of Minnesota, during May and June, 2018. There were four week-long workshops during the conference. They are (1) stochastic control, computation methods, and applications, (2) queueing theory and networked systems, (3) ecological and biological applications, and (4) finance and economics applications. For broader impacts, researchers from different fields covering both theoretically oriented and application intensive areas were invited to participate in the conference. It brought together researchers from multi-disciplinary communities in applied mathematics, applied probability, engineering, biology, ecology, and networked science, to review, and substantially update most recent progress. As an archive, this volume presents some of the highlights of the workshops, and collect papers covering a broad range of topics.

Graduate students and researchers in applied mathematics, optimization, engineering, computer science, and management science will find this book a useful reference which provides an introduction to applications and fundamental theories in nonlinear combinatorial optimization. Nonlinear combinatorial optimization is a new research area within combinatorial optimization and includes numerous applications to technological developments, such as wireless communication, cloud computing, data science, and social networks. Theoretical developments including discrete Newton methods, primal-dual methods with convex relaxation, submodular optimization, discrete DC program, along with several applications are discussed and explored in this book through articles by leading experts.

Current research and applications in nonlinear analysis influenced by Haim Brezis and Louis Nirenberg are presented in this book by leading mathematicians. Each contribution aims to broaden reader's understanding of theories, methods, and techniques utilized to solve significant problems. Topics include: Sobolev Spaces Maximal monotone operators A theorem of Brezis-Nirenberg Operator-norm convergence of the Trotter product formula Elliptic operators with infinitely many variables Pseudo-and quasiconvexities for nonsmooth function Anisotropic surface measures Eulerian and Lagrangian variables Multiple periodic solutions of Lagrangian systems Porous medium equation Nondiscrete Lassonde-Revalski principle Graduate students and researchers in mathematics, physics, engineering, and economics will find this book a useful reference for new techniques and research areas. Haim Brezis and Louis Nirenberg's fundamental research in nonlinear functional analysis and nonlinear partial differential equations along with their years of teaching and training students have had a notable impact in the field.

This book consolidates decades of knowledge on space flight navigation theory, which has thus far been spread across various research articles. By gathering this research into a single text, it will be more accessible to students curious about the study of space flight navigation. Books on optimal control theory and orbital mechanics have not adequately explored the field of space flight navigation theory until this point. The opening chapters introduce essential concepts within optimal control theory, such as the optimization of static systems, special boundary conditions, and dynamic equality constraints. An analytical approach is focused on throughout, as opposed to computational. The result is a book that emphasizes simplicity and practicability, which makes it accessible and engaging. This holds true in later chapters that involve orbital mechanics, two-body maneuvers, bounded inputs, and flight in non-spherical gravity fields. The intended audience is primarily upper-undergraduate students, graduate students, and researchers of aerospace, mechanical, and/or electrical engineering. It will be especially valuable to those with interests in spacecraft dynamics and control. Readers should be familiar with basic dynamics and modern control theory. Additionally, a knowledge of linear algebra, variational methods, and ordinary differential equations is recommended.

In 1781, Gaspard Monge defined the problem of ""optimal transportation"", or the transferring of mass with the least possible amount of work, with applications to engineering in mind. In 1942, Leonid Kantorovich applied the newborn machinery of linear programming to Monge's problem, with applications to economics in mind. In 1987, Yann Brenier used optimal transportation to prove a new projection theorem on the set of measure preserving maps, with applications to fluid mechanics in mind. Each of these contributions marked the beginning of a whole mathematical theory, with many unexpected ramifications. Nowadays, the Monge-Kantorovich problem is used and studied by researchers from extremely diverse horizons, including probability theory, functional analysis, isoperimetry, partial differential equations, and even meteorology. Originating from a graduate course, the present volume is at once an introduction to the field of optimal transportation and a survey of the research on the topic over the last 15 years. The book is intended for graduate students and researchers, and it covers both theory and applications. Readers are only assumed to be familiar with the basics of measure theory and functional analysis.

This book includes a review of mathematical tools like modelling, analysis of stochastic processes, calculus of variations and stochastic differential equations which are applied to solve financial problems like modern portfolio theory and option pricing. Every chapter presents exercises which help the reader to deepen his understanding. The target audience comprises research experts in the field of finance engineering, but the book may also be beneficial for graduate students alike.

An introduction to the variational methods used to formulate and solve mathematical and physical problems, allowing the reader an insight into the systematic use of elementary (partial) convexity of differentiable functions in Euclidian space. By helping students directly characterize the solutions for many minimization problems, the text serves as a prelude to the field theory for sufficiency, laying as it does the groundwork for further explorations in mathematics, physics, mechanical and electrical engineering, as well as computer science.

Non-scalar variational problems appear in different fields. In geometry, for in stance, we encounter the basic problems of harmonic maps between Riemannian manifolds and of minimal immersions; related questions appear in physics, for example in the classical theory of a-models. Non linear elasticity is another example in continuum mechanics, while Oseen-Frank theory of liquid crystals and Ginzburg-Landau theory of superconductivity require to treat variational problems in order to model quite complicated phenomena. Typically one is interested in finding energy minimizing representatives in homology or homotopy classes of maps, minimizers with prescribed topological singularities, topological charges, stable deformations i. e. minimizers in classes of diffeomorphisms or extremal fields. In the last two or three decades there has been growing interest, knowledge, and understanding of the general theory for this kind of problems, often referred to as geometric variational problems. Due to the lack of a regularity theory in the non scalar case, in contrast to the scalar one - or in other words to the occurrence of singularities in vector valued minimizers, often related with concentration phenomena for the energy density - and because of the particular relevance of those singularities for the problem being considered the question of singling out a weak formulation, or completely understanding the significance of various weak formulations becames non trivial."

There has been an increase in attention toward systems involving large numbers of small players, giving rise to the theory of mean field games, mean field type control and nonlinear Markov games. Exhibiting various real world problems involving major and minor agents, this book presents a systematic continuous-space approximation approach for mean-field interacting agents models and mean-field games models. After describing Markov-chain methodology and a modeling of mean-field interacting systems, the text presents various structural conditions on the chain to yield respective socio-economic models, focusing on migration models via binary interactions. The specific applications are wide-ranging - including inspection and corruption, cyber-security, counterterrorism, coalition building and network growth, minority games, and investment policies and optimal allocation - making this book relevant to a wide audience of applied mathematicians interested in operations research, computer science, national security, economics, and finance.

During the last century, we have witnessed the birth and evolution of sport as an economic activity, which has created jobs on the one hand, but also problems of management on the other. This process has not been immune from the parti- lar characteristics associated with sport, typically united here more than in other activities: technique, physical effort, entertainment and passion. And all this within a framework of ever-increasing consumption of ?nancial resources. It is not s- prising, therefore, that commonly-used economic models, based on mechanistic approaches, do not provide a viable solution to increasingly complex and incre- ingly frequent problems. Any attempt to apply such an approach in this technical, economic and ?nancial context can only result in failure. The high degree of subj- tivity inherent in sporting activity requires new tools, in which remodeled conc- tual, theoretical and technical elements should play an important role. Complexity, uncertainty and subjectivity are therefore basic to understand, and deal with, the phenomenon of sport. The necessity of resorting to these elements was identi?ed over a quarter of a century ago by a small group of professors and researchers at the University of Barcelona. Together we started the ?rst postgraduate courses and organized se- nars to alert sports centre managers, as well as to make private and public organi- tions aware of the increasing importance of a proper, speci?c management for sports organizations.

The contributions included in the volume are drawn from presentations at ODS2019 - International Conference on Optimization and Decision Science, which was the 49th annual meeting of the Italian Operations Research Society (AIRO) held at Genoa, Italy, on 4-7 September 2019. This book presents very recent results in the field of Optimization and Decision Science. While the book is addressed primarily to the Operations Research (OR) community, the interdisciplinary contents ensure that it will also be of very high interest for scholars and researchers from many scientific disciplines, including computer sciences, economics, mathematics, and engineering. Operations Research is known as the discipline of optimization applied to real-world problems and to complex decision-making fields. The focus is on mathematical and quantitative methods aimed at determining optimal or near-optimal solutions in acceptable computation times. This volume not only presents theoretical results but also covers real industrial applications, making it interesting for practitioners facing decision problems in logistics, manufacturing production, and services. Readers will accordingly find innovative ideas from both a methodological and an applied perspective.

This book focuses on the development of approximation-related algorithms and their relevant applications. Individual contributions are written by leading experts and reflect emerging directions and connections in data approximation and optimization. Chapters discuss state of the art topics with highly relevant applications throughout science, engineering, technology and social sciences. Academics, researchers, data science practitioners, business analysts, social sciences investigators and graduate students will find the number of illustrations, applications, and examples provided useful. This volume is based on the conference Approximation and Optimization: Algorithms, Complexity, and Applications, which was held in the National and Kapodistrian University of Athens, Greece, June 29-30, 2017. The mix of survey and research content includes topics in approximations to discrete noisy data; binary sequences; design of networks and energy systems; fuzzy control; large scale optimization; noisy data; data-dependent approximation; networked control systems; machine learning ; optimal design; no free lunch theorem; non-linearly constrained optimization; spectroscopy.

This book proposes, for the first time, a basic formulation for structural control that takes into account the stochastic dynamics induced by engineering excitations in the nature of non-stationary and non-Gaussian processes. Further, it establishes the theory of and methods for stochastic optimal control of randomly-excited engineering structures in the context of probability density evolution methods, such as physically-based stochastic optimal (PSO) control. By logically integrating randomness into control gain, the book helps readers design elegant control systems, mitigate risks in civil engineering structures, and avoid the dilemmas posed by the methods predominantly applied in current practice, such as deterministic control and classical linear quadratic Gaussian (LQG) control associated with nominal white noises.

Quadratic programming (QP) is one advanced mathematical technique that allows for the optimization of a quadratic function in several variables in the presence of linear constraints. This book presents recently developed algorithms for solving large QP problems and focuses on algorithms which are, in a sense optimal, i.e., they can solve important classes of problems at a cost proportional to the number of unknowns. For each algorithm presented, the book details its classical predecessor, describes its drawbacks, introduces modifications that improve its performance, and demonstrates these improvements through numerical experiments.

This self-contained monograph can serve as an introductory text on quadratic programming for graduate students and researchers. Additionally, since the solution of many nonlinear problems can be reduced to the solution of a sequence of QP problems, it can also be used as a convenient introduction to nonlinear programming.

This book is devoted to the mathematical analysis of the numerical solution of boundary integral equations treating boundary value, transmission and contact problems arising in elasticity, acoustic and electromagnetic scattering. It serves as the mathematical foundation of the boundary element methods (BEM) both for static and dynamic problems. The book presents a systematic approach to the variational methods for boundary integral equations including the treatment with variational inequalities for contact problems. It also features adaptive BEM, hp-version BEM, coupling of finite and boundary element methods - efficient computational tools that have become extremely popular in applications. Familiarizing readers with tools like Mellin transformation and pseudodifferential operators as well as convex and nonsmooth analysis for variational inequalities, it concisely presents efficient, state-of-the-art boundary element approximations and points to up-to-date research. The authors are well known for their fundamental work on boundary elements and related topics, and this book is a major contribution to the modern theory of the BEM (especially for error controlled adaptive methods and for unilateral contact and dynamic problems) and is a valuable resource for applied mathematicians, engineers, scientists and graduate students.

Optimal Impulsive Control explores the class of impulsive dynamic optimization problems-problems that stem from the fact that many conventional optimal control problems do not have a solution in the classical setting-which is highly relevant with regard to engineering applications. The absence of a classical solution naturally invokes the so-called extension, or relaxation, of a problem, and leads to the notion of generalized solution which encompasses the notions of generalized control and trajectory; in this book several extensions of optimal control problems are considered within the framework of optimal impulsive control theory. In this framework, the feasible arcs are permitted to have jumps, while the conventional absolutely continuous trajectories may fail to exist. The authors draw together various types of their own results, centered on the necessary conditions of optimality in the form of Pontryagin's maximum principle and the existence theorems, which shape a substantial body of optimal impulsive control theory. At the same time, they present optimal impulsive control theory in a unified framework, introducing the different paradigmatic problems in increasing order of complexity. The rationale underlying the book involves addressing extensions increasing in complexity from the simplest case provided by linear control systems and ending with the most general case of a totally nonlinear differential control system with state constraints. The mathematical models presented in Optimal Impulsive Control being encountered in various engineering applications, this book will be of interest to both academic researchers and practising engineers.

This book provides researchers and graduate students with a thorough introduction to the variational analysis of nonlinear problems described by nonlocal operators. The authors give a systematic treatment of the basic mathematical theory and constructive methods for these classes of nonlinear equations, plus their application to various processes arising in the applied sciences. The equations are examined from several viewpoints, with the calculus of variations as the unifying theme. Part I begins the book with some basic facts about fractional Sobolev spaces. Part II is dedicated to the analysis of fractional elliptic problems involving subcritical nonlinearities, via classical variational methods and other novel approaches. Finally, Part III contains a selection of recent results on critical fractional equations. A careful balance is struck between rigorous mathematics and physical applications, allowing readers to see how these diverse topics relate to other important areas, including topology, functional analysis, mathematical physics, and potential theory.