This volume originates from the INDAM Symposium on Trends on Applications of Mathematics to Mechanics (STAMM), which was held at the INDAM headquarters in Rome on 5-9 September 2016. It brings together original contributions at the interface of Mathematics and Mechanics. The focus is on mathematical models of phenomena issued from various applications. These include thermomechanics of solids and gases, nematic shells, thin films, dry friction, delamination, damage, and phase-field dynamics. The papers in the volume present novel results and identify possible future developments. The book is addressed to researchers involved in Mathematics and its applications to Mechanics.

Nonlinear Approaches in Engineering Applications focuses on nonlinear phenomena that are common in the engineering field. The nonlinear approaches described in this book provide a sound theoretical base and practical tools to design and analyze engineering systems with high efficiency and accuracy and with less energy and downtime. Presented here are nonlinear approaches in areas such as dynamic systems, optimal control and approaches in nonlinear dynamics and acoustics. Coverage encompasses a wide range of applications and fields including mathematical modeling and nonlinear behavior as applied to microresonators, nanotechnologies, nonlinear behavior in soil erosion,nonlinear population dynamics, and optimization in reducing vibration and noise as well as vibration in triple-walled carbon nanotubes.

This book gives senior undergraduate and beginning graduate students and researchers in computer vision, applied mathematics, computer graphics, and robotics a self-contained introduction to the geometry of 3D vision; that is the reconstruction of 3D models of objects from a collection of 2D images. Following a brief introduction, Part I provides background materials for the rest of the book. The two fundamental transformations, namely rigid body motion and perspective projection are introduced and image formation and feature extraction discussed. Part II covers the classic theory of two view geometry based on the so-called epipolar constraint. Part III shows that a more proper tool for studying the geometry of multiple views is the so- called rank considtion on the multiple view matrix. Part IV develops practical reconstruction algorithms step by step as well as discusses possible extensions of the theory. Exercises are provided at the end of each chapter. Software for examples and algorithms are available on the author's website.

This comprehensive textbook on combinatorial optimization places special emphasis on theoretical results and algorithms with provably good performance, in contrast to heuristics. It is based on numerous courses on combinatorial optimization and specialized topics, mostly at graduate level. This book reviews the fundamentals, covers the classical topics (paths, flows, matching, matroids, NP-completeness, approximation algorithms) in detail, and proceeds to advanced and recent topics, some of which have not appeared in a textbook before. Throughout, it contains complete but concise proofs, and also provides numerous exercises and references. This sixth edition has again been updated, revised, and significantly extended. Among other additions, there are new sections on shallow-light trees, submodular function maximization, smoothed analysis of the knapsack problem, the (ln 4+e)-approximation for Steiner trees, and the VPN theorem. Thus, this book continues to represent the state of the art of combinatorial optimization.

"Fixed-Point Algorithms for Inverse Problems in Science and Engineering" presents some of the most recent work from top-notch researchers studying projection and other first-order fixed-point algorithms in several areas of mathematics and the applied sciences. The material presented provides a survey of the state-of-the-art theory and practice in fixed-point algorithms, identifying emerging problems driven by applications, and discussing new approaches for solving these problems.

This book incorporates diverse perspectives from broad-ranging areas of research including, variational analysis, numerical linear algebra, biotechnology, materials science, computational solid-state physics, and chemistry.

Topics presented include:

Theory of Fixed-point algorithms: convex analysis, convex optimization, subdifferential calculus, nonsmooth analysis, proximal point methods, projection methods, resolvent and related fixed-point theoretic methods, and monotone operator theory.

Numerical analysis of fixed-point algorithms: choice of step lengths, of weights, of blocks for block-iterative and parallel methods, and of relaxation parameters; regularization of ill-posed problems; numerical comparison of various methods.

Areas of Applications: engineering (image and signal reconstruction and decompression problems), computer tomography and radiation treatment planning (convex feasibility problems), astronomy (adaptive optics), crystallography (molecular structure reconstruction), computational chemistry (molecular structure simulation) and other areas.

Because of the variety of applications presented, this book can easily serve as a basis for new and innovated research and collaboration.

The present monograph is the successor of Direct methods in the calculus of variations which was published in the Applied Mathematical Sciences series and is currently out of print. Although the core and the structure of the present book is similar to the old one, it is much more than a revised version. Fifteen years have passed since the publication of the Direct methods in the calculus of variations book and since the subject is a very active one, almost half of the book presently consists of new material. The perspective has also slightly changed, which is reflected in the change of the title. Indeed a new subject, quasiconvex analysis has now been developed. The present monograph, which is essentially a reference book on the subject of quasiconvex analysis, can be used, as was the earlier book for an advanced course on the calculus of variations.

Nonlinear Optimization is an intriguing area of study where mathematical theory, algorithms and applications converge to calculate the optimal values of continuous functions. Within this subject, Global Optimization aims at finding global optima for difficult problems in which many local optima might exist.

This book provides a compelling introduction to global and non-linear optimization providing interdisciplinary readers with a strong background to continue their studies into these and other related fields. The book offers insight in relevant concepts such as "region of attraction" and "Branch-and-Bound" by elaborating small numerical examples and exercises for the reader to follow.

The emphasis throughout the present volume is on the practical application of theoretical mathematical models helping to unravel the underlying mechanisms involved in processes from mathematical physics and biosciences. It has been conceived as a unique collection of abstract methods dealing especially with nonlinear partial differential equations (either stationary or evolutionary) that are applied to understand concrete processes involving some important applications related to phenomena such as: boundary layer phenomena for viscous fluids, population dynamics,, dead core phenomena, etc. It addresses researchers and post-graduate students working at the interplay between mathematics and other fields of science and technology and is a comprehensive introduction to the theory of nonlinear partial differential equations and its main principles also presents their real-life applications in various contexts: mathematical physics, chemistry, mathematical biology, and population genetics. Based on the authors' original work, this volume provides an overview of the field, with examples suitable for researchers but also for graduate students entering research. The method of presentation appeals to readers with diverse backgrounds in partial differential equations and functional analysis. Each chapter includes detailed heuristic arguments, providing thorough motivation for the material developed later in the text. The content demonstrates in a firm way that partial differential equations can be used to address a large variety of phenomena occurring in and influencing our daily lives. The extensive reference list and index make this book a valuable resource for researchers working in a variety of fields and who are interested in phenomena modeled by nonlinear partial differential equations.

This monograph explores the early development of the calculus of variations in continental Europe during the Eighteenth Century by illustrating the mathematics of its founders. Closely following the original papers and correspondences of Euler, Lagrange, the Bernoullis, and others, the reader is immersed in the challenge of theory building. We see what the founders were doing, the difficulties they faced, the mistakes they made, and their triumphs. The authors guide the reader through these works with instructive commentaries and complements to the original proofs, as well as offering a modern perspective where useful. The authors begin in 1697 with Johann Bernoulli's work on the brachystochrone problem and the events leading up to it, marking the dawn of the calculus of variations. From there, they cover key advances in the theory up to the development of Lagrange's -calculus, including: * The isoperimetrical problems * Shortest lines and geodesics * Euler's Methodus Inveniendi and the two Additamenta Finally, the authors give the readers a sense of how vast the calculus of variations has become in centuries hence, providing some idea of what lies outside the scope of the book as well as the current state of affairs in the field. This book will be of interest to anyone studying the calculus of variations who wants a deeper intuition for the techniques and ideas that are used, as well as historians of science and mathematics interested in the development and evolution of modern calculus and analysis.

From its origins in the minimization of integral functionals, the notion of 'variations' has evolved greatly in connection with applications in optimization, equilibrium, and control. It refers not only to constrained movement away from a point, but also to modes of perturbation and approximation that are best describable by 'set convergence', variational convergence of functions' and the like. This book develops a unified framework and, in finite dimensions, provides a detailed exposition of variational geometry and subdifferential calculus in their current forms beyond classical and convex analysis. Also covered are set-convergence, set-valued mappings, epi-convergence, duality, maximal monotone mappings, second-order subderivatives, measurable selections and normal integrands.

This book is intended to provide a systematic overview of so-called smart techniques, such as nature-inspired algorithms, machine learning and metaheuristics. Despite their ubiquitous presence and widespread application to different scientific problems, such as searching, optimization and /or classification, a systematic study is missing in the current literature. Here, the editors collected a set of chapters on key topics, paying attention to provide an equal balance of theory and practice, and to outline similarities between the different techniques and applications. All in all, the book provides an unified view on the field on intelligent methods, with their current perspective and future challenges.

Menahem Max Schiffer, a mathematician of many interests, produced a body of work including topics on geometric function theory, Riemann surfaces, and partial differential equations, with a focus on applications and mathematical physics. Perhaps his best known work is that in the calculus of variations, especially extremal problem, s which find application in many scientific areas.

This two volume set presents over 50 of the most groundbreaking contributions of this beloved mathematician. All of the reprints of Schiffer s works herein have extensive annotation and invited commentaries, giving new clarity and insight into the impact and legacy of Schiffer's works. A complete bibliography and brief biography make this a rounded and invaluable reference."

The present lecture note is dedicated to the study of the optimality conditions and the duality results for nonlinear vector optimization problems, in ?nite and in?nite dimensions. The problems include are nonlinear vector optimization problems, s- metric dual problems, continuous-time vector optimization problems, relationships between vector optimization and variational inequality problems. Nonlinear vector optimization problems arise in several contexts such as in the building and interpretation of economic models; the study of various technolo- cal processes; the development of optimal choices in ?nance; management science; production processes; transportation problems and statistical decisions, etc. In preparing this lecture note a special effort has been made to obtain a se- contained treatment of the subjects; so we hope that this may be a suitable source for a beginner in this fast growing area of research, a semester graduate course in nonlinear programing, and a good reference book. This book may be useful to theoretical economists, engineers, and applied researchers involved in this area of active research. The lecture note is divided into eight chapters: Chapter 1 brie?y deals with the notion of nonlinear programing problems with basic notations and preliminaries. Chapter 2 deals with various concepts of convex sets, convex functions, invex set, invex functions, quasiinvex functions, pseudoinvex functions, type I and generalized type I functions, V-invex functions, and univex functions.

In a unified form, this monograph presents fundamental results on the approximation of centralized and decentralized stochastic control problems, with uncountable state, measurement, and action spaces. It demonstrates how quantization provides a system-independent and constructive method for the reduction of a system with Borel spaces to one with finite state, measurement, and action spaces. In addition to this constructive view, the book considers both the information transmission approach for discretization of actions, and the computational approach for discretization of states and actions. Part I of the text discusses Markov decision processes and their finite-state or finite-action approximations, while Part II builds from there to finite approximations in decentralized stochastic control problems. This volume is perfect for researchers and graduate students interested in stochastic controls. With the tools presented, readers will be able to establish the convergence of approximation models to original models and the methods are general enough that researchers can build corresponding approximation results, typically with no additional assumptions.

This book reflects a significant part of authors' research activity dur ing the last ten years. The present monograph is constructed on the results obtained by the authors through their direct cooperation or due to the authors separately or in cooperation with other mathematicians. All these results fit in a unitary scheme giving the structure of this work. The book is mainly addressed to researchers and scholars in Pure and Applied Mathematics, Mechanics, Physics and Engineering. We are greatly indebted to Viorica Venera Motreanu for the careful reading of the manuscript and helpful comments on important issues. We are also grateful to our Editors of Kluwer Academic Publishers for their professional assistance. Our deepest thanks go to our numerous scientific collaborators and friends, whose work was so important for us. D. Motreanu and V. Radulescu IX Introduction The present monograph is based on original results obtained by the authors in the last decade. This book provides a comprehensive expo sition of some modern topics in nonlinear analysis with applications to the study of several classes of boundary value problems. Our framework includes multivalued elliptic problems with discontinuities, variational inequalities, hemivariational inequalities and evolution problems. The treatment relies on variational methods, monotonicity principles, topo logical arguments and optimization techniques. Excepting Sections 1 and 3 in Chapter 1 and Sections 1 and 3 in Chapter 2, the material is new in comparison with any other book, representing research topics where the authors contributed. The outline of our work is the following."

Like norms, translation invariant functions are a natural and powerful tool for the separation of sets and scalarization. This book provides an extensive foundation for their application. It presents in a unified way new results as well as results which are scattered throughout the literature. The functions are defined on linear spaces and can be applied to nonconvex problems. Fundamental theorems for the function class are proved, with implications for arbitrary extended real-valued functions. The scope of applications is illustrated by chapters related to vector optimization, set-valued optimization, and optimization under uncertainty, by fundamental statements in nonlinear functional analysis and by examples from mathematical finance as well as from consumer and production theory. The book is written for students and researchers in mathematics and mathematical economics. Engineers and researchers from other disciplines can benefit from the applications, for example from scalarization methods for multiobjective optimization and optimal control problems.

As is well known, The Great Divide (a.k.a. The Continental Divide) is formed by the Rocky Mountains stretching from north to south across North America. It creates a virtual "stone wall" so high that wind, rain, snow, etc. cannot cross it. This keeps the weather distinct on both sides. Since railroad trains cannot climb steep grades and tunnels through these mountains are almost formidable, the Canadian Pacific Railroad searched for a mountain pass providing the lowest grade for its tracks. Employees discovered a suitable mountain pass, called the Kicking Horse Pass, el. 5404 ft., near Banff, Alberta. (One can speculate as to the reason for the name.) This pass is also used by the Trans-Canada Highway. At the highest point of the pass the railroad tracks are horizontal with mountains rising on both sides. A mountain stream divides into two branches, one flowing into the Atlantic Ocean and the other into the Pacific. One can literally stand (as the author did) with one foot in the Atlantic Ocean and the other in the Pacific. The author has observed many mountain passes in the Rocky Mountains and Alps. What connections do mountain passes have with nonlinear partial dif ferential equations? To find out, read on ..."

Evolution equations of hyperbolic or more general p-evolution type form an active field of current research. This volume aims to collect some recent advances in the area in order to allow a quick overview of ongoing research. The contributors are first rate mathematicians. This collection of research papers is centred around parametrix constructions and microlocal analysis; asymptotic constructions of solutions; energy and dispersive estimates; and associated spectral transforms. Applications concerning elasticity and general relativity complement the volume. The book gives an overview of a variety of ongoing current research in the field and, therefore, allows researchers as well as students to grasp new aspects and broaden their understanding of the area. "

This two volume set presents over 50 of the most groundbreaking contributions of Menahem M Schiffer. All of the reprints of Schiffer's works herein have extensive annotation and invited commentaries, giving new clarity and insight into the impact and legacy of Schiffer's work. A complete bibliography and brief biography make this a rounded and invaluable reference.

This comprehensive book presents a rigorous and state-of-the-art treatment of variational inequalities and complementarity problems in finite dimensions. This class of mathematical programming problems provides a powerful framework for the unified analysis and development of efficient solution algorithms for a wide range of equilibrium problems in economics, engineering, finance, and applied sciences. New research material and recent results, not otherwise easily accessible, are presented in a self-contained and consistent manner. The book is published in two volumes, with the first volume concentrating on the basic theory and the second on iterative algorithms. Both volumes contain abundant exercises and feature extensive bibliographies. Written with a wide range of readers in mind, including graduate students and researchers in applied mathematics, optimization, and operations research as well as computational economists and engineers, this book will be an enduring reference on the subject and provide the foundation for its sustained growth.

This edited book aims at presenting current research activities in the field of robust variable-structure systems. The scope equally comprises highlighting novel methodological aspects as well as presenting the use of variable-structure techniques in industrial applications including their efficient implementation on hardware for real-time control. The target audience primarily comprises research experts in the field of control theory and nonlinear dynamics but the book may also be beneficial for graduate students.

This comprehensive book presents a rigorous and state-of-the-art treatment of variational inequalities and complementarity problems in finite dimensions. This class of mathematical programming problems provides a powerful framework for the unified analysis and development of efficient solution algorithms for a wide range of equilibrium problems in economics, engineering, finance, and applied sciences. New research material and recent results, not otherwise easily accessible, are presented in a self-contained and consistent manner. The book is published in two volumes, with the first volume concentrating on the basic theory and the second on iterative algorithms. Both volumes contain abundant exercises and feature extensive bibliographies. Written with a wide range of readers in mind, including graduate students and researchers in applied mathematics, optimization, and operations research as well as computational economists and engineers, this book will be an enduring reference on the subject and provide the foundation for its sustained growth.

Optimization is a rich and thriving mathematical discipline. The theory underlying current computational optimization techniques grows ever more sophisticated. The powerful and elegant language of convex analysis unifies much of this theory. The aim of this book is to provide a concise, accessible account of convex analysis and its applications and extensions, for a broad audience. It can serve as a teaching text, at roughly the level of first year graduate students. While the main body of the text is self-contained, each section concludes with an often extensive set of optional exercises. The new edition adds material on semismooth optimization, as well as several new proofs that will make this book even more self-contained.

2 Radiant sets 236 3 Co-radiant sets 239 4 Radiative and co-radiative sets 241 5 Radiant sets with Lipschitz continuous Minkowski gauges 245 6 Star-shaped sets and their kernels 249 7 Separation 251 8 Abstract convex star-shaped sets 255 References 260 11 DIFFERENCES OF CONVEX COMPACTA AND METRIC SPACES OF CON- 263 VEX COMPACTA WITH APPLICATIONS: A SURVEY A. M. Rubinov, A. A. Vladimirov 1 Introduction 264 2 Preliminaries 264 3 Differences of convex compact sets: general approach 266 4 Metric projections and corresponding differences (one-dimensional case) 267 5 The *-difference 269 6 The Demyanov difference 271 7 Geometric and inductive definitions of the D-difference 273 8 Applications to DC and quasidifferentiable functions 276 9 Differences of pairs of set-valued mappings with applications to quasidiff- entiability 278 10 Applications to approximate subdifferentials 280 11 Applications to the approximation of linear set-valued mappings 281 12 The Demyanov metric 282 13 The Bartels-Pallaschke metric 284 14 Hierarchy of the three norms on Qn 285 15 Derivatives 287 16 Distances from convex polyhedra and convergence of convex polyhedra 289 17 Normality of convex sets 290 18 D-regular sets 291 19 Variable D-regular sets 292 20 Optimization 293 References 294 12 CONVEX APPROXIMATORS.

Covering some of the key areas of optimal control theory (OCT), a rapidly expanding field, the authors use new methods to set out a version of OCT's more refined'maximum principle.' The results obtainedhave applicationsin production planning, reinsurance-dividend management, multi-model sliding mode control, and multi-model differential games.

This book" "explores material that will be of great interest to post-graduate students, researchers, and practitioners in applied mathematics and engineering, particularly in the area of systems and control."