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2. Trends and Applications of Pure Mathematics to Mechanics : Invited and Contributed Papers Presented at a Symposium at Ecole Polytechnique, Palaiseau, France, November 28 – December 2, 1983
- Author
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P.G. Ciarlet, M. Roseau, P.G. Ciarlet, and M. Roseau
- Subjects
- Quantum physics, Mechanics, Mathematical analysis, System theory, Control theory, Mathematical optimization, Calculus of variations, Spintronics
- Published
- 2005
3. Harnack Inequalities and Nonlinear Operators : Proceedings of the INdAM Conference to Celebrate the 70th Birthday of Emmanuele DiBenedetto
- Author
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Vincenzo Vespri, Ugo Gianazza, Dario Daniele Monticelli, Fabio Punzo, Daniele Andreucci, Vincenzo Vespri, Ugo Gianazza, Dario Daniele Monticelli, Fabio Punzo, and Daniele Andreucci
- Subjects
- Mathematical analysis, Global analysis (Mathematics), Manifolds (Mathematics), Mathematical optimization, Calculus of variations, Potential theory (Mathematics)
- Abstract
The book contains two contributions about the work of Emmanuele DiBenedetto and a selection of original papers. The authors are some of the main experts in Harnack's inequalities and nonlinear operators. These papers are part of the contributions presented during the conference to celebrate the 70th birthday of Prof. Emmanuele DiBenedetto, which was held at “Il Palazzone” in Cortona from June 18th to 24th, 2017. The papers are focused on current research topics regarding the qualitative properties of solutions, connections with calculus of variations, Harnack inequality and regularity theory. Some papers are also related to various applications. Many of the authors have shared with Prof. DiBenedetto an intense scientific and personal collaboration, while many others have taken inspiration from and further developed his field of research. The topics of the conference are certainly of great interest for the international mathematical community.
- Published
- 2021
4. Direct and Inverse Problems of Mathematical Physics
- Author
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R.P. Gilbert, Joji Kajiwara, Yongzhi S. Xu, R.P. Gilbert, Joji Kajiwara, and Yongzhi S. Xu
- Subjects
- Differential equations, Mathematical analysis, Physics, Astronomy, Functions of complex variables, Mathematics, Mathematical optimization
- Abstract
This volume consists of papers presented in the special sessions on'Wave Phenomena and Related Topics', and'Asymptotics and Homogenization'of the ISAAC'97 Congress held at the University of Delaware, during June 2-7, 1997. The ISAAC Congress coincided with a U.S.-Japan Seminar also held at the University of Delaware. The latter was supported by the National Science Foundation through Grant INT -9603029 and the Japan Society for the Promotion of Science through Grant MTCS-134. It was natural that the'participants of both meetings should interact and consequently several persons attending the Congress also presented papers in the Seminar. The success of the ISAAC Congress and the U.S.-Japan Seminar has led to the ISAAC'99 Congress being held in Fukuoka, Japan during August 1999. Many of the same participants will return to this Seminar. Indeed, it appears that the spirit of the U.S.-Japan Seminar will be continued every second year as part of the ISAAC Congresses. We decided to include with the papers presented in the ISAAC Congress and the U.S.-Japan Seminar several very good papers by colleagues from the former Soviet Union. These participants in the ISAAC Congress attended at their own expense. This volume has the title Direct and Inverse Problems of Mathematical Physics which consists of the papers on scattering theory, coefficient identification, uniqueness and existence theorems, boundary controllability, wave propagation in stratified media, viscous flows, nonlinear acoustics, Sobolev spaces, singularity theory, pseudo differential operators, and semigroup theory.
- Published
- 2013
5. The Stability of Matter: From Atoms to Stars : Selecta of Elliott H. Lieb
- Author
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Elliott H. Lieb, Walter Thirring, Elliott H. Lieb, and Walter Thirring
- Subjects
- Mathematical physics, Condensed matter, Quantum physics, Mathematical analysis, Mathematical optimization, Calculus of variations
- Abstract
The second edition of this'selecta'of my work on the stability of matter was sold out and this presented an opportunity to add some newer work on the quantum mechanical many-body problem. In order to do so, and still keep the volume within manageable limits, it was necessary to delete a few papers that appeared in the previous editions. This was done without sacrificing content, however, since the material contained in the deleted papers still appears, in abbreviated form, at least, in other papers reprinted here. Seetions VII and VIII are new. The former is on quantum electrodynamics (QED), to which I was led by consideration of stability of the non-relativistic many-body Coulomb problem, as contained in the first and second editions. In particular, the fragility of stability of matter with c1assical magnetic fields, which requires abound on the fine-structure constant even in the non-relativistic case (item V.4), leads to the question of stability in a theory with quantized fields. There are many unresolved problems of QED if one attempts to develop a non perturbative theory - as everyone knows. A non-perturbative theory is essential, however, if one is going to understand the stability of the many-body problem, which is the stability of ordinary matter. Some physicists will say that a non perturbative QED does not exist - and this might be true in the absence of cutoffs - but an effective theory with cutoffs of a few Mev must exist since matter exists.
- Published
- 2013
6. Nonlinear Analysis and Variational Problems : In Honor of George Isac
- Author
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Panos M. Pardalos, Themistocles M. Rassias, Akhtar A. Khan, Panos M. Pardalos, Themistocles M. Rassias, and Akhtar A. Khan
- Subjects
- Mathematical analysis, Variational principles
- Abstract
The papers published in this volume focus on some of the most recent devel- ments in complementarity theory, variational principles, stability theory of fu- tional equations, nonsmooth optimization, and various other important topics of nonlinear analysis and optimization. This volume was initially planned to celebrate Professor George Isac's 70th birthday by bringing together research scientists from mathematical domains which have long bene ted from Isac's active research passion. Unfortunately, George Isac passed away in February 2009 at the age of 69. George Isac received his Ph. D. in 1973 from the Institute of Mathematics of the Romanian Academy of Sciences. He made outstanding contributions in s- eral branches of pure and applied mathematics, including complementarity theory, variational inequalities, xed point theory, scalar and vector optimization, theory of cones, eigenvalue problems, convex analysis, variational principles and regulari- tion methods, as well as a number of other topics. In his long and outstanding career, he wrote more than 200 papers and 13 books. Professor Isac was an avid traveler who visited more than 70 universities around the globe and delivered approximately 180 research presentations. He also authored seven books on poetry. During his s- enti c career he collaborated with numerous mathematicians. His research papers contain very deep, original and beautiful results. Through his signi cant contri- tions, he earned a distinguished position and became an internationally renowned leading scholar in his research elds.
- Published
- 2010
7. Topics in Mathematical Analysis and Applications
- Author
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Themistocles M. Rassias, László Tóth, Themistocles M. Rassias, and László Tóth
- Subjects
- Mathematical analysis
- Abstract
This volume presents significant advances in a number of theories and problems of Mathematical Analysis and its applications in disciplines such as Analytic Inequalities, Operator Theory, Functional Analysis, Approximation Theory, Functional Equations, Differential Equations, Wavelets, Discrete Mathematics and Mechanics. The contributions focus on recent developments and are written by eminent scientists from the international mathematical community. Special emphasis is given to new results that have been obtained in the above mentioned disciplines in which Nonlinear Analysis plays a central role.Some review papers published in this volume will be particularly useful for a broader readership in Mathematical Analysis, as well as for graduate students. An attempt is given to present all subjects in this volume in a unified and self-contained manner, to be particularly useful to the mathematical community.
- Published
- 2014
8. Convex Functions and Optimization Methods on Riemannian Manifolds
- Author
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C. Udriste and C. Udriste
- Subjects
- Mathematical optimization, Calculus of variations, Geometry, Numerical analysis, Mathematical analysis, Mathematical models
- Abstract
The object of this book is to present the basic facts of convex functions, standard dynamical systems, descent numerical algorithms and some computer programs on Riemannian manifolds in a form suitable for applied mathematicians, scientists and engineers. It contains mathematical information on these subjects and applications distributed in seven chapters whose topics are close to my own areas of research: Metric properties of Riemannian manifolds, First and second variations of the p-energy of a curve; Convex functions on Riemannian manifolds; Geometric examples of convex functions; Flows, convexity and energies; Semidefinite Hessians and applications; Minimization of functions on Riemannian manifolds. All the numerical algorithms, computer programs and the appendices (Riemannian convexity of functions f:R ~ R, Descent methods on the Poincare plane, Descent methods on the sphere, Completeness and convexity on Finsler manifolds) constitute an attempt to make accesible to all users of this book some basic computational techniques and implementation of geometric structures. To further aid the readers,this book also contains a part of the folklore about Riemannian geometry, convex functions and dynamical systems because it is unfortunately'nowhere'to be found in the same context; existing textbooks on convex functions on Euclidean spaces or on dynamical systems do not mention what happens in Riemannian geometry, while the papers dealing with Riemannian manifolds usually avoid discussing elementary facts. Usually a convex function on a Riemannian manifold is a real valued function whose restriction to every geodesic arc is convex.
- Published
- 2013
9. Vector Optimization : Theory, Applications, and Extensions
- Author
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Johannes Jahn and Johannes Jahn
- Subjects
- Operations research, Mathematical optimization, Mathematical analysis, Management science
- Abstract
In vector optimization one investigates optimal elements such as min imal, strongly minimal, properly minimal or weakly minimal elements of a nonempty subset of a partially ordered linear space. The prob lem of determining at least one of these optimal elements, if they exist at all, is also called a vector optimization problem. Problems of this type can be found not only in mathematics but also in engineer ing and economics. Vector optimization problems arise, for exam ple, in functional analysis (the Hahn-Banach theorem, the lemma of Bishop-Phelps, Ekeland's variational principle), multiobjective pro gramming, multi-criteria decision making, statistics (Bayes solutions, theory of tests, minimal covariance matrices), approximation theory (location theory, simultaneous approximation, solution of boundary value problems) and cooperative game theory (cooperative n player differential games and, as a special case, optimal control problems). In the last decade vector optimization has been extended to problems with set-valued maps. This new field of research, called set optimiza tion, seems to have important applications to variational inequalities and optimization problems with multivalued data. The roots of vector optimization go back to F. Y. Edgeworth (1881) and V. Pareto (1896) who has already given the definition of the standard optimality concept in multiobjective optimization. But in mathematics this branch of optimization has started with the leg endary paper of H. W. Kuhn and A. W. Tucker (1951). Since about v Vl Preface the end of the 60's research is intensively made in vector optimization.
- Published
- 2013
10. Validation Numerics : Theory and Applications
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R. Albrecht, G. Alefeld, H.J. Stetter, R. Albrecht, G. Alefeld, and H.J. Stetter
- Subjects
- Numerical analysis, Algebras, Linear, Mathematical analysis, System theory, Control theory, Mathematical optimization, Calculus of variations
- Abstract
The articles in this book give a comprehensive overview on the whole field of validated numerics. The problems covered include simultaneous systems of linear and nonlinear equations, differential and integral equations and certain applications from technical sciences. Furthermore some papers which improve the tools are included. The book is a must for scientists working in numerical analysis, computer science and in technical fields.
- Published
- 2012
11. Variational Analysis and Applications
- Author
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Franco Giannessi, Antonino Maugeri, Franco Giannessi, and Antonino Maugeri
- Subjects
- Mathematical optimization, Calculus of variations, Mathematical analysis, Differential equations, Mathematics
- Abstract
This Volume contains the (refereed) papers presented at the 38th Conference of the School of Mathematics'G.Stampacchia'of the'E.Majorana'Centre for Scientific Culture of Erice (Sicily), held in Memory ofG. Stampacchia and J.-L. Lions in the period June 20 - July 2003. The presence of participants from Countries has greatly contributed to the success of the meeting. The School of Mathematics was dedicated to Stampacchia, not only for his great mathematical achievements, but also because He founded it. The core of the Conference has been the various features of the Variational Analysis and their motivations and applications to concrete problems. Variational Analysis encompasses a large area of modem Mathematics, such as the classical Calculus of Variations, the theories of perturbation, approximation, subgradient, subderivates, set convergence and Variational Inequalities, and all these topics have been deeply and intensely dealt during the Conference. In particular, Variational Inequalities, which have been initiated by Stampacchia, inspired by Signorini Problem and the related work of G. Fichera, have offered a very great possibility of applications to several fundamental problems of Mathematical Physics, Engineering, Statistics and Economics. The pioneer work of Stampacchia and Lions can be considered as the basic kernel around which Variational Analysis is going to be outlined and constructed. The Conference has dealt with both finite and infinite dimensional analysis, showing that to carry on these two aspects disjointly is unsuitable for both.
- Published
- 2007
12. Mathematical Modelling of Industrial Processes : Lectures Given at the 3rd Session of the Centro Internazionale Matematico Estivo (C.I.M.E.) Held in Bari, Italy, Sept. 24-29, 1990
- Author
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Stavros Busenberg, Bruno Forte, Hendrik K. Kuiken, Vincenzo Capasso, Antonio Fasano, Stavros Busenberg, Bruno Forte, Hendrik K. Kuiken, Vincenzo Capasso, and Antonio Fasano
- Subjects
- Mathematics, Business, Management science, Mathematical analysis, System theory, Control theory, Mathematical optimization, Calculus of variations, Mathematical physics
- Abstract
The 1990 CIME course on Mathematical Modelling of Industrial Processes set out to illustrate some advances in questions of industrial mathematics, i.e.of the applications of mathematics (with all its'academic'rigour) to real-life problems. The papers describe the genesis of the models and illustrate their relevant mathematical characteristics. Among the themesdealt with are: thermally controlled crystal growth, thermal behaviour of a high-pressure gas-discharge lamp, the sessile-drop problem, etching processes, the batch-coil- annealing process, inverse problems in classical dynamics, image representation and dynamical systems, scintillation in rear projections screens, identification of semiconductor properties,pattern recognition with neural networks. CONTENTS: H.K. Kuiken: Mathematical Modelling of Industrial Processes.- B. Forte: Inverse Problems in Mathematics for Industry.- S. Busenberg: Case Studies in Industrial Mathematics.
- Published
- 2006
13. Functional Analysis, Sobolev Spaces, and Calculus of Variations
- Author
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Pablo Pedregal and Pablo Pedregal
- Subjects
- Mathematical analysis, Mathematical optimization, Calculus of variations
- Abstract
This book aims at introducing students into the modern analytical foundations to treat problems and situations in the Calculus of Variations solidly and rigorously. Since no background is taken for granted or assumed, as the textbook pretends to be self-contained, areas like basic Functional Analysis and Sobolev spaces are studied to the point that chapters devoted to these topics can be utilized by themselves as an introduction to these important parts of Analysis. The material in this regard has been selected to serve the needs of classical variational problems, leaving broader treatments for more advanced and specialized courses in those areas. It should not be forgotten that problems in the Calculus of Variations historically played a crucial role in pushing Functional Analysis as a discipline on its own right. The style is intentionally didactic. After a first general chapter to place optimization problems in infinite-dimensional spaces in perspective, the first part of the book focuses on the initial important concepts in Functional Analysis and introduces Sobolev spaces in dimension one as a preliminary, simpler case (much in the same way as in the successful book of H. Brezis). Once the analytical framework is covered, one-dimensional variational problems are examined in detail including numerous examples and exercises. The second part dwells, again as a first-round, on another important chapter of Functional Analysis that students should be exposed to, and that eventually will find some applications in subsequent chapters. The first chapter of this part examines continuous operators and the important principles associated with mappings between functional spaces; and another one focuses on compact operators and their fundamental and remarkable properties for Analysis. Finally, the third part advances to multi-dimensional Sobolev spaces and the corresponding problems in the Calculus of Variations. In this setting, problems become much more involved and, for this same reason, much more interesting and appealing. In particular, the final chapter dives into a number of advanced topics, some of which reflect a personal taste. Other possibilities stressing other kinds of problems are possible. In summary, the text pretends to help students with their first exposure to the modern calculus of variations and the analytical foundation associated with it. In particular, it covers an extended introduction to basic functional analysis and to Sobolev spaces. The tone of the text and the set of proposed exercises will facilitate progressive understanding until the need for further challenges beyond the topics addressed here will push students to more advanced horizons.
- Published
- 2024
14. Calculus III : Practice Problems, Methods, and Solutions
- Author
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Mehdi Rahmani-Andebili and Mehdi Rahmani-Andebili
- Subjects
- Engineering mathematics, Mathematics, Mathematical analysis, Mathematical optimization, Calculus of variations, Differential equations
- Abstract
This study guide is designed for students taking a Calculus III course. The textbook includes examples, questions, and practice problems that will help students to review and sharpen their knowledge of the subject and enhance their performance in the classroom. The material covered in the book includes linear algebra and analytical geometry; lines, surfaces, and vector functions in three-dimensional coordinate systems; multiple-variable functions; multiple integrals and their applications; line integrals and their applications. Offering detailed solutions, multiple methods for solving problems, and clear explanations of concepts, this hands-on guide will improve students'problem-solving skills and foster a solid understanding of calculus, which will benefit them in all of their calculus-based courses.
- Published
- 2023
15. Calculus II : Practice Problems, Methods, and Solutions
- Author
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Mehdi Rahmani-Andebili and Mehdi Rahmani-Andebili
- Subjects
- Engineering mathematics, Mathematical analysis, Mathematical optimization, Calculus of variations, Differential equations
- Abstract
This study guide is designed for students taking a Calculus II course. The textbook includes examples, questions, and practice problems that will help students to review and sharpen their knowledge of the subject and enhance their performance in the classroom. The material covered in the book includes applications of integration, sequences and series and their applications, polar coordinate systems, and complex numbers. Offering detailed solutions, multiple methods for solving problems, and clear explanations of concepts, this hands-on guide will improve students'problem-solving skills and foster a solid understanding of calculus, which will benefit them in all of their calculus-based courses
- Published
- 2023
16. Near Extensions and Alignment of Data in R(superscript)n : Whitney Extensions of Near Isometries, Shortest Paths, Equidistribution, Clustering and Non-rigid Alignment of Data in Euclidean Space
- Author
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Steven B. Damelin and Steven B. Damelin
- Subjects
- Nomography (Mathematics), Euclidean algorithm, Isometrics (Mathematics), Mathematical analysis, Geometry, Analytic, Rigidity (Geometry)
- Abstract
Near Extensions and Alignment of Data in Rn Comprehensive resource illustrating the mathematical richness of Whitney Extension Problems, enabling readers to develop new insights, tools, and mathematical techniques Near Extensions and Alignment of Data in Rn demonstrates a range of hitherto unknown connections between current research problems in engineering, mathematics, and data science, exploring the mathematical richness of near Whitney Extension Problems, and presenting a new nexus of applied, pure and computational harmonic analysis, approximation theory, data science, and real algebraic geometry. For example, the book uncovers connections between near Whitney Extension Problems and the problem of alignment of data in Euclidean space, an area of considerable interest in computer vision. Written by a highly qualified author, Near Extensions and Alignment of Data in Rn includes information on: Areas of mathematics and statistics, such as harmonic analysis, functional analysis, and approximation theory, that have driven significant advances in the field Development of algorithms to enable the processing and analysis of huge amounts of data and data sets Why and how the mathematical underpinning of many current data science tools needs to be better developed to be useful New insights, potential tools, and mathematical techniques to solve problems in Whitney extensions, signal processing, shortest paths, clustering, computer vision, optimal transport, manifold learning, minimal energy, and equidistribution Providing comprehensive coverage of several subjects, Near Extensions and Alignment of Data in Rn is an essential resource for mathematicians, applied mathematicians, and engineers working on problems related to data science, signal processing, computer vision, manifold learning, and optimal transport.
- Published
- 2023
17. Applied Analysis, Optimization and Soft Computing : ICNAAO-2021, Varanasi, India, December 21–23
- Author
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Tanmoy Som, Debdas Ghosh, Oscar Castillo, Adrian Petrusel, Dayaram Sahu, Tanmoy Som, Debdas Ghosh, Oscar Castillo, Adrian Petrusel, and Dayaram Sahu
- Subjects
- Mathematical optimization, Mathematical analysis, Differential equations, Mathematical models, Computer science
- Abstract
This book contains select contributions presented at the International Conference on Nonlinear Applied Analysis and Optimization (ICNAAO-2021), held at the Department of Mathematics Sciences, Indian Institute of Technology (BHU) Varanasi, India, from 21–23 December 2021. The book discusses topics in the areas of nonlinear analysis, fixed point theory, dynamical systems, optimization, fractals, applications to differential/integral equations, signal and image processing, and soft computing, and exposes the young talents with the newer dimensions in these areas with their practical approaches and to tackle the real-life problems in engineering, medical and social sciences. Scientists from the U.S.A., Austria, France, Mexico, Romania, and India have contributed their research. All the submissions are peer reviewed by experts in their fields.
- Published
- 2023
18. Partial Differential Equations and Applications : Colloquium in Honor of Hamidou Touré, Ouagadougou, Burkina Faso, November 5–9, 2018
- Author
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Toka Diagana, Khalil Ezzinbi, Stanislas Ouaro, Toka Diagana, Khalil Ezzinbi, and Stanislas Ouaro
- Subjects
- Differential equations, Difference equations, Functional equations, Mathematical analysis, Mathematical optimization, Calculus of variations
- Abstract
This volume convenes selected, peer-reviewed works presented at the Partial Differential Equations and Applications Colloquium in Honor of Prof. Hamidou Toure that was held at the University Ouaga 1, Ouagadougou, Burkina Faso, November 5–9, 2018.Topics covered in this volume include boundary value problems for difference equations, differential forms in global analysis, functional differential equations, and stability in the context of PDEs. Studies on SIR and SIRS epidemic models, of special interest to researchers in epidemiology, are also included.This volume is dedicated to Dr. Hamidou Touré, a Research Professor at the University of Ouaga 1. Dr. Touré has made important scientific contributions in many fields of mathematical sciences. Dr. Touré got his PhD (1994) from the University of Franche-Comté of Besançon, France, and is one of the key leaders and mentor of several generations of mathematicians in French-speaking Africa.This conference was purposely held in Ouagadougou in reverence of Dr. Touré's efforts for the development of mathematics in Africa since the beginning of his career in early 1982 to the current days.
- Published
- 2023
19. Introduction to Scientific Computing and Data Analysis
- Author
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Mark H. Holmes and Mark H. Holmes
- Subjects
- Mathematics—Data processing, Mathematical optimization, Mathematical analysis
- Abstract
This textbook provides an introduction to numerical computing and its applications in science and engineering. The topics covered include those usually found in an introductory course, as well as those that arise in data analysis. This includes optimization and regression-based methods using a singular value decomposition. The emphasis is on problem solving, and there are numerous exercises throughout the text concerning applications in engineering and science. The essential role of the mathematical theory underlying the methods is also considered, both for understanding how the method works, as well as how the error in the computation depends on the method being used. The codes used for most of the computational examples in the text are available on GitHub. This new edition includes material necessary for an upper division course in computational linear algebra.
- Published
- 2023
20. Handbook of Mathematical Models and Algorithms in Computer Vision and Imaging : Mathematical Imaging and Vision
- Author
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Ke Chen, Carola-Bibiane Schönlieb, Xue-Cheng Tai, Laurent Younes, Ke Chen, Carola-Bibiane Schönlieb, Xue-Cheng Tai, and Laurent Younes
- Subjects
- Mathematics—Data processing, Image processing—Digital techniques, Computer vision, Mathematical optimization, Mathematical analysis, Neural networks (Computer science)
- Abstract
This handbook gathers together the state of the art on mathematical models and algorithms for imaging and vision. Its emphasis lies on rigorous mathematical methods, which represent the optimal solutions to a class of imaging and vision problems, and on effective algorithms, which are necessary for the methods to be translated to practical use in various applications. Viewing discrete images as data sampled from functional surfaces enables the use of advanced tools from calculus, functions and calculus of variations, and nonlinear optimization, and provides the basis of high-resolution imaging through geometry and variational models. Besides, optimization naturally connects traditional model-driven approaches to the emerging data-driven approaches of machine and deep learning. No other framework can provide comparable accuracy and precision to imaging and vision. Written by leading researchers in imaging and vision, the chapters in this handbook all start with gentle introductions, which make this work accessible to graduate students. For newcomers to the field, the book provides a comprehensive and fast-track introduction to the content, to save time and get on with tackling new and emerging challenges. For researchers, exposure to the state of the art of research works leads to an overall view of the entire field so as to guide new research directions and avoid pitfalls in moving the field forward and looking into the next decades of imaging and information services. This work can greatly benefit graduate students, researchers, and practitioners in imaging and vision; applied mathematicians; medical imagers; engineers; and computer scientists.
- Published
- 2023
21. Applications of Lie Groups to Differential Equations
- Author
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Peter J. Olver and Peter J. Olver
- Subjects
- Mathematical analysis, Group theory, Mathematical optimization, Calculus of variations, System theory, Control theory
- Abstract
Symmetry methods have long been recognized to be of great importance for the study of the differential equations. This book provides a solid introduction to those applications of Lie groups to differential equations which have proved to be useful in practice. The computational methods are presented so that graduate students and researchers can readily learn to use them. Following an exposition of the applications, the book develops the underlying theory. Many of the topics are presented in a novel way, with an emphasis on explicit examples and computations. Further examples, as well as new theoretical developments, appear in the exercises at the end of each chapter.
- Published
- 2022
22. Modeling, Simulation and Optimization in the Health- and Energy-Sector
- Author
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René Pinnau, Nicolas R. Gauger, Axel Klar, René Pinnau, Nicolas R. Gauger, and Axel Klar
- Subjects
- Mathematics, Mathematics—Data processing, Mathematical analysis, Mathematical optimization, Medical sciences
- Abstract
This volume is addressed to people who are interested in modern mathematical solutions for real life applications. In particular, mathematical modeling, simulation and optimization is nowadays successfully used in various fields of application, like the energy- or health-sector. Here, mathematics is often the driving force for new innovations and most relevant for the success of many interdisciplinary projects. The presented chapters demonstrate the power of this emerging research field and show how society can benefit from applied mathematics.
- Published
- 2022
23. Optimal Control of Dynamic Systems Driven by Vector Measures : Theory and Applications
- Author
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N. U. Ahmed, Shian Wang, N. U. Ahmed, and Shian Wang
- Subjects
- Differential equations, Stochastic processes, System theory, Control theory, Mathematical optimization, Calculus of variations, Mathematical analysis
- Abstract
This book is devoted to the development of optimal control theory for finite dimensional systems governed by deterministic and stochastic differential equations driven by vector measures. The book deals with a broad class of controls, including regular controls (vector-valued measurable functions), relaxed controls (measure-valued functions) and controls determined by vector measures, where both fully and partially observed control problems are considered. In the past few decades, there have been remarkable advances in the field of systems and control theory thanks to the unprecedented interaction between mathematics and the physical and engineering sciences. Recently, optimal control theory for dynamic systems driven by vector measures has attracted increasing interest. This book presents this theory for dynamic systems governed by both ordinary and stochastic differential equations, including extensive results on the existence of optimal controls and necessary conditions foroptimality. Computational algorithms are developed based on the optimality conditions, with numerical results presented to demonstrate the applicability of the theoretical results developed in the book. This book will be of interest to researchers in optimal control or applied functional analysis interested in applications of vector measures to control theory, stochastic systems driven by vector measures, and related topics. In particular, this self-contained account can be a starting point for further advances in the theory and applications of dynamic systems driven and controlled by vector measures.
- Published
- 2021
24. Control of Degenerate and Singular Parabolic Equations : Carleman Estimates and Observability
- Author
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Genni Fragnelli, Dimitri Mugnai, Genni Fragnelli, and Dimitri Mugnai
- Subjects
- Mathematical analysis, Mathematical optimization, Calculus of variations
- Abstract
This book collects some basic results on the null controllability for degenerate and singular parabolic problems. It aims to provide postgraduate students and senior researchers with a useful text, where they can find the desired statements and the related bibliography. For these reasons, the authors will not give all the detailed proofs of the given theorems, but just some of them, in order to show the underlying strategy in this area.
- Published
- 2021
25. Multi-Valued Variational Inequalities and Inclusions
- Author
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Siegfried Carl, Vy Khoi Le, Siegfried Carl, and Vy Khoi Le
- Subjects
- Mathematical analysis, Differential equations, Operator theory, Mathematical optimization, Calculus of variations, Mathematics
- Abstract
This book focuses on a large class of multi-valued variational differential inequalities and inclusions of stationary and evolutionary types with constraints reflected by subdifferentials of convex functionals. Its main goal is to provide a systematic, unified, and relatively self-contained exposition of existence, comparison and enclosure principles, together with other qualitative properties of multi-valued variational inequalities and inclusions. The problems under consideration are studied in different function spaces such as Sobolev spaces, Orlicz-Sobolev spaces, Sobolev spaces with variable exponents, and Beppo-Levi spaces. A general and comprehensive sub-supersolution method (lattice method) is developed for both stationary and evolutionary multi-valued variational inequalities, which preserves the characteristic features of the commonly known sub-supersolution method for single-valued, quasilinear elliptic and parabolic problems. This method provides a powerful tool forstudying existence and enclosure properties of solutions when the coercivity of the problems under consideration fails. It can also be used to investigate qualitative properties such as the multiplicity and location of solutions or the existence of extremal solutions. This is the first in-depth treatise on the sub-supersolution (lattice) method for multi-valued variational inequalities without any variational structures, together with related topics. The choice of the included materials and their organization in the book also makes it useful and accessible to a large audience consisting of graduate students and researchers in various areas of Mathematical Analysis and Theoretical Physics.
- Published
- 2021
26. Geometric Properties for Parabolic and Elliptic PDE's
- Author
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Vincenzo Ferone, Tatsuki Kawakami, Paolo Salani, Futoshi Takahashi, Vincenzo Ferone, Tatsuki Kawakami, Paolo Salani, and Futoshi Takahashi
- Subjects
- Mathematical analysis, Mathematical optimization, Calculus of variations, Convex geometry, Discrete geometry, Mathematics, Functional analysis
- Abstract
This book contains the contributions resulting from the 6th Italian-Japanese workshop on Geometric Properties for Parabolic and Elliptic PDEs, which was held in Cortona (Italy) during the week of May 20–24, 2019. This book will be of great interest for the mathematical community and in particular for researchers studying parabolic and elliptic PDEs. It covers many different fields of current research as follows: convexity of solutions to PDEs, qualitative properties of solutions to parabolic equations, overdetermined problems, inverse problems, Brunn-Minkowski inequalities, Sobolev inequalities, and isoperimetric inequalities.
- Published
- 2021
27. Mathematical Control Theory for Stochastic Partial Differential Equations
- Author
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Qi Lü, Xu Zhang, Qi Lü, and Xu Zhang
- Subjects
- System theory, Control theory, Mathematical optimization, Calculus of variations, Probabilities, Mathematical analysis
- Abstract
This is the first book to systematically present control theory for stochastic distributed parameter systems, a comparatively new branch of mathematical control theory. The new phenomena and difficulties arising in the study of controllability and optimal control problems for this type of system are explained in detail. Interestingly enough, one has to develop new mathematical tools to solve some problems in this field, such as the global Carleman estimate for stochastic partial differential equations and the stochastic transposition method for backward stochastic evolution equations. In a certain sense, the stochastic distributed parameter control system is the most general control system in the context of classical physics. Accordingly, studying this field may also yield valuable insights into quantum control systems.A basic grasp of functional analysis, partial differential equations, and control theory for deterministic systems is the only prerequisite for reading this book.
- Published
- 2021
28. Mathematical Analysis and Optimization for Economists
- Author
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Michael J. Panik and Michael J. Panik
- Subjects
- Economics, Mathematical, Mathematical analysis, Mathematical optimization
- Abstract
In Mathematical Analysis and Optimization for Economists, the author aims to introduce students of economics to the power and versatility of traditional as well as contemporary methodologies in mathematics and optimization theory; and, illustrates how these techniques can be applied in solving microeconomic problems.This book combines the areas of intermediate to advanced mathematics, optimization, and microeconomic decision making, and is suitable for advanced undergraduates and first-year graduate students. This text is highly readable, with all concepts fully defined, and contains numerous detailed example problems in both mathematics and microeconomic applications. Each section contains some standard, as well as more thoughtful and challenging, exercises. Solutions can be downloaded from the CRC Press website. All solutions are detailed and complete.Features Contains a whole spectrum of modern applicable mathematical techniques, many of which are not found in other books of this type. Comprehensive and contains numerous and detailed example problems in both mathematics and economic analysis. Suitable for economists and economics students with only a minimal mathematical background. Classroom-tested over the years when the author was actively teaching at the University of Hartford. Serves as a beginner text in optimization for applied mathematics students. Accompanied by several electronic chapters on linear algebra and matrix theory, nonsmooth optimization, economic efficiency, and distance functions available for free on www.routledge.com/9780367759018.
- Published
- 2021
29. Lectures on Optimal Transport
- Author
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Luigi Ambrosio, Elia Brué, Daniele Semola, Luigi Ambrosio, Elia Brué, and Daniele Semola
- Subjects
- Mathematical analysis, Mathematical optimization, Calculus of variations, Measure theory
- Abstract
This textbook is addressed to PhD or senior undergraduate students in mathematics, with interests in analysis, calculus of variations, probability and optimal transport. It originated from the teaching experience of the first author in the Scuola Normale Superiore, where a course on optimal transport and its applications has been given many times during the last 20 years. The topics and the tools were chosen at a sufficiently general and advanced level so that the student or scholar interested in a more specific theme would gain from the book the necessary background to explore it. After a large and detailed introduction to classical theory, more specific attention is devoted to applications to geometric and functional inequalities and to partial differential equations.
- Published
- 2021
30. Examples and Problems in Advanced Calculus: Real-Valued Functions
- Author
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Bijan Davvaz and Bijan Davvaz
- Subjects
- Mathematical analysis, Mathematical optimization
- Abstract
This book includes over 500 most challenging exercises and problems in calculus. Topical problems and exercises are discussed on set theory, numbers, functions, limits and continuity, derivative, integral calculus, Rolle's theorem, mean value theorem, optimization problems, sequences and series. All the seven chapters recall important definitions, theorems and concepts, making this book immensely valuable to undergraduate students of engineering, mathematics, statistics, computer science and basic sciences.
- Published
- 2020
31. The GLOBAL Optimization Algorithm : Newly Updated with Java Implementation and Parallelization
- Author
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Balázs Bánhelyi, Tibor Csendes, Balázs Lévai, László Pál, Dániel Zombori, Balázs Bánhelyi, Tibor Csendes, Balázs Lévai, László Pál, and Dániel Zombori
- Subjects
- Mathematical optimization, Calculus of variations, Computer science—Mathematics, Operations research, Management science, Mathematical analysis
- Abstract
This book explores the updated version of the GLOBAL algorithm which contains improvements for a local search algorithm and new Java implementations. Efficiency comparisons to earlier versions and on the increased speed achieved by the parallelization, are detailed. Examples are provided for students as well as researchers and practitioners in optimization, operations research, and mathematics to compose their own scripts with ease. A GLOBAL manual is presented in the appendix to assist new users with modules and test functions. GLOBAL is a successful stochastic multistart global optimization algorithm that has passed several computational tests, and is efficient and reliable for small to medium dimensional global optimization problems. The algorithm uses clustering to ensure efficiency and is modular in regard to the two local search methods it starts with, but it can also easily apply other local techniques. The strength of this algorithm lies in its reliability and adaptive algorithm parameters. The GLOBAL algorithm is free to download also in the earlier Fortran, C, and MATLAB implementations.
- Published
- 2018
32. Free Boundary Problems : Regularity Properties Near the Fixed Boundary
- Author
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Darya Apushkinskaya and Darya Apushkinskaya
- Subjects
- Mathematical analysis, Mathematical optimization, Calculus of variations
- Abstract
This book is concerned with several elliptic and parabolic obstacle-type problems with a focus on the cases where the free and fixed boundaries meet. The results presented complement those found in existing books in the subject, which mainly treat regularity properties away from the fixed boundary.The topics include optimal regularity, analysis of global solutions, tangential touch of the free and fixed boundaries, as well as Lipschitz- and $C^1$-regularity of the free boundary. Special attention is given to local versions of various monotonicity formulas.The intended audience includes research mathematicians and advanced graduate students interested in problems with free boundaries.
- Published
- 2018
33. Handbook of Heuristics
- Author
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Rafael Martí, Panos M. Pardalos, Mauricio G. C. Resende, Rafael Martí, Panos M. Pardalos, and Mauricio G. C. Resende
- Subjects
- Algorithms, Mathematical optimization, Mathematical analysis, Engineering mathematics, Engineering—Data processing, Computer science—Mathematics, Computer software
- Abstract
Heuristics are strategies using readily accessible, loosely applicable information to control problem solving. Algorithms, for example, are a type of heuristic. By contrast, Metaheuristics are methods used to design Heuristics and may coordinate the usage of several Heuristics toward the formulation of a single method. GRASP (Greedy Randomized Adaptive Search Procedures) is an example of a Metaheuristic. To the layman, heuristics may be thought of as ‘rules of thumb'but despite its imprecision, heuristics is a very rich field that refers to experience-based techniques for problem-solving, learning, and discovery. Any given solution/heuristic is not guaranteed to be optimal but heuristic methodologies are used to speed up the process of finding satisfactory solutions where optimal solutions are impractical. The introduction to this Handbook provides an overview of the history of Heuristics along with main issues regarding the methodologies covered. This is followed by Chapters containing various examples of local searches, search strategies and Metaheuristics, leading to an analyses of Heuristics and search algorithms. The reference concludes with numerous illustrations of the highly applicable nature and implementation of Heuristics in our daily life. Each chapter of this work includes an abstract/introduction with a short description of the methodology. Key words are also necessary as part of top-matter to each chapter to enable maximum search engine optimization. Next, chapters will include discussion of the adaptation of this methodology to solve a difficult optimization problem, and experiments on a set of representative problems.
- Published
- 2018
34. Nonlinear Algebra In An Acorn: With Applications To Deep Learning
- Author
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Martin J Lee, Ken Kang Too Tsang, Martin J Lee, and Ken Kang Too Tsang
- Subjects
- Mathematical analysis, Nonlinear theories
- Abstract
A simple algorithm for solving a set of nonlinear equations by matrix algebra has been discovered recently — first by transforming them into an equivalent matrix equation and then finding the solution analytically in terms of the inverse matrix of this equation. With this newly developed ACORN (Adaptive Constrained Optimal Robust Nonlinear) algorithm, it is possible to minimize the objective function [constructed from the functions in the nonlinear set of equations] without computing its derivatives.This book will present the details of ACORN algorithm and how it is used to solve large scale nonlinear equations with an innovative approach ACORN Magic [minimization algorithms gathered in a cloud].The ultimate motivation of this work is its application to optimization. In recent years, with the advances in big-data, optimization becomes an even more powerful tool in knowledge discovery. ACORN Magic is the perfect choice in this kind of application because of that fact that it is fast, robust and simple enough to be embedded in any type of machine learning program.
- Published
- 2018
35. Fractional and Multivariable Calculus : Model Building and Optimization Problems
- Author
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A.M. Mathai, H.J. Haubold, A.M. Mathai, and H.J. Haubold
- Subjects
- Mathematical models, Mathematical optimization, Special functions, Mathematical analysis
- Abstract
This textbook presents a rigorous approach to multivariable calculus in the context of model building and optimization problems. This comprehensive overview is based on lectures given at five SERC Schools from 2008 to 2012 and covers a broad range of topics that will enable readers to understand and create deterministic and nondeterministic models. Researchers, advanced undergraduate, and graduate students in mathematics, statistics, physics, engineering, and biological sciences will find this book to be a valuable resource for finding appropriate models to describe real-life situations.The first chapter begins with an introduction to fractional calculus moving on to discuss fractional integrals, fractional derivatives, fractional differential equations and their solutions. Multivariable calculus is covered in the second chapter and introduces the fundamentals of multivariable calculus (multivariable functions, limits and continuity, differentiability, directionalderivatives and expansions of multivariable functions). Illustrative examples, input-output process, optimal recovery of functions and approximations are given; each section lists an ample number of exercises to heighten understanding of the material. Chapter three discusses deterministic/mathematical and optimization models evolving from differential equations, difference equations, algebraic models, power function models, input-output models and pathway models. Fractional integral and derivative models are examined. Chapter four covers non-deterministic/stochastic models. The random walk model, branching process model, birth and death process model, time series models, and regression type models are examined. The fifth chapter covers optimal design. General linear models from a statistical point of view are introduced; the Gauss–Markov theorem, quadratic forms, and generalized inverses of matrices are covered. Pathway, symmetric, and asymmetric models are covered in chapter six, the concepts are illustrated with graphs.
- Published
- 2017
36. Positive Trigonometric Polynomials and Signal Processing Applications
- Author
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Bogdan Dumitrescu and Bogdan Dumitrescu
- Subjects
- Geometry, Mathematical optimization, Engineering, Mathematical analysis, Analysis (Mathematics)
- Abstract
This revised edition is made up of two parts: theory and applications. Though many of the fundamental results are still valid and used, new and revised material is woven throughout the text. As with the original book, the theory of sum-of-squares trigonometric polynomials is presented unitarily based on the concept of Gram matrix (extended to Gram pair or Gram set). The programming environment has also evolved, and the book's examples are changed accordingly. The applications section is organized as a collection of related problems that use systematically the theoretical results. All the problems are brought to a semi-definite programming form, ready to be solved with algorithms freely available, like those from the libraries SeDuMi, CVX and Pos3Poly. A new chapter discusses applications in super-resolution theory, where Bounded Real Lemma for trigonometric polynomials is an important tool. This revision is written to be more appealing and easier to use for new readers.>
- Published
- 2017
37. Lecture Notes On Calculus Of Variations
- Author
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Kung-ching Chang and Kung-ching Chang
- Subjects
- Mathematical analysis, Calculus of variations, Functionals
- Abstract
This is based on the course'Calculus of Variations'taught at Peking University from 2006 to 2010 for advanced undergraduate to graduate students majoring in mathematics. The book contains 20 lectures covering both the theoretical background material as well as an abundant collection of applications. Lectures 1-8 focus on the classical theory of calculus of variations. Lectures 9-14 introduce direct methods along with their theoretical foundations. Lectures 15-20 showcase a broad collection of applications. The book offers a panoramic view of the very important topic on calculus of variations. This is a valuable resource not only to mathematicians, but also to those students in engineering, economics, and management, etc.
- Published
- 2017
38. Multivariable Calculus with MATLAB® : With Applications to Geometry and Physics
- Author
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Ronald L. Lipsman, Jonathan M. Rosenberg, Ronald L. Lipsman, and Jonathan M. Rosenberg
- Subjects
- Mathematical optimization, Calculus of variations, Mathematical analysis
- Abstract
This comprehensive treatment of multivariable calculus focuses on the numerous tools that MATLAB® brings to the subject, as it presents introductions to geometry, mathematical physics, and kinematics. Covering simple calculations with MATLAB®, relevant plots, integration, and optimization, the numerous problem sets encourage practice with newly learned skills that cultivate the reader's understanding of the material. Significant examples illustrate each topic, and fundamental physical applications such as Kepler's Law, electromagnetism, fluid flow, and energy estimation are brought to prominent position. Perfect for use as a supplement to any standard multivariable calculus text, a “mathematical methods in physics or engineering” class, for independent study, or even as the class text in an “honors” multivariable calculus course, this textbook will appeal to mathematics, engineering, and physical science students. MATLAB® is tightly integrated into every portion of this book, and its graphical capabilities are used to present vibrant pictures of curves and surfaces. Readers benefit from the deep connections made between mathematics and science while learning more about the intrinsic geometry of curves and surfaces. With serious yet elementary explanation of various numerical algorithms, this textbook enlivens the teaching of multivariable calculus and mathematical methods courses for scientists and engineers.
- Published
- 2017
39. Mathematical Tapas : Volume 1 (for Undergraduates)
- Author
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Jean-Baptiste Hiriart-Urruty and Jean-Baptiste Hiriart-Urruty
- Subjects
- Algebras, Linear, Mathematical analysis, Mathematical optimization
- Abstract
This book contains a collection of exercises (called “tapas”) at undergraduate level, mainly from the fields of real analysis, calculus, matrices, convexity, and optimization.Most of the problems presented here are non-standard and some require broad knowledge of different mathematical subjects in order to be solved. The author provides some hints and (partial) answers and also puts these carefully chosen exercises into context, presents information on their origins, and comments on possible extensions. With stars marking the levels of difficulty, these tapas show or prove something interesting, challenge the reader to solve and learn, and may have surprising results.This first volume of Mathematical Tapas will appeal to mathematicians, motivated undergraduate students from science-based areas, and those generally interested in mathematics.
- Published
- 2016
40. Mathematical Analysis, Approximation Theory and Their Applications
- Author
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Themistocles M. Rassias, Vijay Gupta, Themistocles M. Rassias, and Vijay Gupta
- Subjects
- Mathematical analysis, Approximation theory
- Abstract
Designed for graduate students, researchers, and engineers in mathematics, optimization, and economics, this self-contained volume presents theory, methods, and applications in mathematical analysis and approximation theory. Specific topics include: approximation of functions by linear positive operators with applications to computer aided geometric design, numerical analysis, optimization theory, and solutions of differential equations. Recent and significant developments in approximation theory, special functions and q-calculus along with their applications to mathematics, engineering, and social sciences are discussed and analyzed. Each chapter enriches the understanding of current research problems and theories in pure and applied research.
- Published
- 2016
41. Calculus Problems
- Author
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Marco Baronti, Filippo De Mari, Robertus van der Putten, Irene Venturi, Marco Baronti, Filippo De Mari, Robertus van der Putten, and Irene Venturi
- Subjects
- Mathematical analysis, Mathematical optimization, Calculus of variations
- Abstract
This book, intended as a practical working guide for calculus students, includes 450 exercises. It is designed for undergraduate students in Engineering, Mathematics, Physics, or any other field where rigorous calculus is needed, and will greatly benefit anyone seeking a problem-solving approach to calculus. Each chapter starts with a summary of the main definitions and results, which is followed by a selection of solved exercises accompanied by brief, illustrative comments. A selection of problems with indicated solutions rounds out each chapter.A final chapter explores problems that are not designed with a single issue in mind but instead call for the combination of a variety of techniques, rounding out the book's coverage. Though the book's primary focus is on functions of one real variable, basic ordinary differential equations (separation of variables, linear first order and constant coefficients ODEs) are also discussed. The material is taken from actual written tests that have been delivered at the Engineering School of the University of Genoa. Literally thousands of students have worked on these problems, ensuring their real-world applicability.
- Published
- 2016
42. Mean Field Theories and Dual Variation - Mathematical Structures of the Mesoscopic Model
- Author
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Takashi Suzuki and Takashi Suzuki
- Subjects
- Mathematical analysis, Mathematical optimization, Calculus of variations, Mathematical physics, Population genetics, Biomathematics
- Abstract
Mean field approximation has been adopted to describe macroscopic phenomena from microscopic overviews. It is still in progress; fluid mechanics, gauge theory, plasma physics, quantum chemistry, mathematical oncology, non-equilibirum thermodynamics. spite of such a wide range of scientific areas that are concerned with the mean field theory, a unified study of its mathematical structure has not been discussed explicitly in the open literature. The benefit of this point of view on nonlinear problems should have significant impact on future research, as will be seen from the underlying features of self-assembly or bottom-up self-organization which is to be illustrated in a unified way. The aim of this book is to formulate the variational and hierarchical aspects of the equations that arise in the mean field theory from macroscopic profiles to microscopic principles, from dynamics to equilibrium, and from biological models to models that arise from chemistry and physics.
- Published
- 2015
43. Local Minimization, Variational Evolution and Γ-Convergence
- Author
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Andrea Braides and Andrea Braides
- Subjects
- Mathematics, Differential equations, Mathematical optimization, Calculus of variations, Approximation theory, Mathematical analysis, Functional analysis
- Abstract
This book addresses new questions related to the asymptotic description of converging energies from the standpoint of local minimization and variational evolution. It explores the links between Gamma-limits, quasistatic evolution, gradient flows and stable points, raising new questions and proposing new techniques. These include the definition of effective energies that maintain the pattern of local minima, the introduction of notions of convergence of energies compatible with stable points, the computation of homogenized motions at critical time-scales through the definition of minimizing movement along a sequence of energies, the use of scaled energies to study long-term behavior or backward motion for variational evolutions. The notions explored in the book are linked to existing findings for gradient flows, energetic solutions and local minimizers, for which some generalizations are also proposed.
- Published
- 2014
44. Finite Element and Boundary Element Techniques From Mathematical and Engineering Point of View
- Author
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E. Stein, W. Wendland, E. Stein, and W. Wendland
- Subjects
- Numerical analysis, Mathematical analysis, System theory, Control theory, Mathematical optimization, Calculus of variations, Computer simulation, Mechanics
- Abstract
Traditional FEM and the more recent BEM underlie many engineering computational methods and corresponding software. Both methods have their merits and also their limitations. The combination of both methods will provide an improved numerical tool in the future. The aim of this book is to present significant basic formulations of FEM and BEM and to show their common practical and mathematical foundations, their differences and possibilities for their combination. These include variational foundations, FEM and BEM for linear and non-linear elasticity and potential problems, the combination of FEM-BEM asymptotic error analysis, modifications due to corner and crack singularities and corresponding improvement of convergence, plastic analysis, numerical algorithms and engineering applications.
- Published
- 2014
45. Implicit Functions and Solution Mappings : A View From Variational Analysis
- Author
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Asen L. Dontchev, R. Tyrrell Rockafellar, Asen L. Dontchev, and R. Tyrrell Rockafellar
- Subjects
- Mathematical optimization, Mathematical analysis, Numerical analysis
- Abstract
The implicit function theorem is one of the most important theorems in analysis and its many variants are basic tools in partial differential equations and numerical analysis.This second edition of Implicit Functions and Solution Mappings presents an updated and more complete picture of the field by including solutions of problems that have been solved since the first edition was published, and places old and new results in a broader perspective. The purpose of this self-contained work is to provide a reference on the topic and to provide a unified collection of a number of results which are currently scattered throughout the literature. Updates to this edition include new sections in almost all chapters, new exercises and examples, updated commentaries to chapters and an enlarged index and references section.
- Published
- 2014
46. Global Optimization : Deterministic Approaches
- Author
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Reiner Horst, Hoang Tuy, Reiner Horst, and Hoang Tuy
- Subjects
- Operations research, Mathematics, Mathematical analysis, Econometrics, System theory, Control theory, Mathematical optimization, Calculus of variations
- Abstract
The main contents and character of the monograph did not change with respect to the first edition. However, within most chapters we incorporated quite a number of modifications which take into account the recent development of the field, the very valuable suggestions and comments that we received from numerous colleagues and students as well as our own experience while using the book. Some errors and misprints in the first edition are also corrected. Reiner Horst May 1992 Hoang Tuy PREFACE TO THE FIRST EDITION The enormous practical need for solving global optimization problems coupled with a rapidly advancing computer technology has allowed one to consider problems which a few years aga would have been considered computationally intractable. As a consequence, we are seeing the creation of a large and increasing number of diverse algorithms for solving a wide variety of multiextremal global optimization problems. The goal of this book is to systematically clarify and unify these diverse approaches in order to provide insight into the underlying concepts and their pro perties. Aside from a coherent view of the field much new material is presented.
- Published
- 2013
47. The Stability of Matter: From Atoms to Stars : Selecta of Elliot H. Lieb
- Author
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Elliott H. Lieb, Walter Thirring, Elliott H. Lieb, and Walter Thirring
- Subjects
- Condensed matter, Spintronics, Quantum physics, Mathematical analysis, System theory, Control theory, Mathematical optimization, Calculus of variations
- Abstract
The first edition of'The Stability of Matter: From Atoms to Stars'was sold out after a time unusually short for a selecta collection and we thought it ap propriate not just to make a reprinting but to include eight new contributionso They demonstrate that this field is still lively and keeps revealing unexpected featureso Of course, we restricted ourselves to developments in which Elliott Lieb participated and thus the heroic struggle in Thomas-Fermi theory where 7 3 5 3 the accuracy has been pushed from Z 1 to Z 1 is not includedo A rich landscape opened up after Jakob Yngvason's observation that atoms in magnetic fields also are described in suitable limits by a Thomas-Fermi-type theoryo Together with Elliott Lieb and Jan Philip Solovej it was eventually worked out that one has to distinguish 5 regionso If one takes as a dimensionless measure of the magnetic field strength B the ratio Larmor radius/Bohr radius one can compare it with N''Z and for each of the domains 4 3 (i) B « N 1, 4 3 (ii) B''N 1, 4 3 3 (iii) N 1« B « N, 3 (iv) B''N, 3 (v) B » N a different version ofmagnetic Thomas-Fermi theory becomes exact in the limit N --+ ooo In two dimensions and a confining potential ('quantum dots') the situation is somewhat simpler, one has to distinguish only (i) B « N, (ii) B''N,
- Published
- 2013
48. An Introduction to Nonlinear Analysis: Theory
- Author
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Zdzislaw Denkowski, Stanislaw Migórski, Nikolaos S. Papageorgiou, Zdzislaw Denkowski, Stanislaw Migórski, and Nikolaos S. Papageorgiou
- Subjects
- Mathematical analysis, Mathematical optimization, Calculus of variations, Geometry, Mathematical models, Mathematics
- Abstract
An Introduction to Nonlinear Analysis: Theory is an overview of some basic, important aspects of Nonlinear Analysis, with an emphasis on those not included in the classical treatment of the field. Today Nonlinear Analysis is a very prolific part of modern mathematical analysis, with fascinating theory and many different applications ranging from mathematical physics and engineering to social sciences and economics. Topics covered in this book include the necessary background material from topology, measure theory and functional analysis (Banach space theory). The text also deals with multivalued analysis and basic features of nonsmooth analysis, providing a solid background for the more applications-oriented material of the book An Introduction to Nonlinear Analysis: Applications by the same authors. The book is self-contained and accessible to the newcomer, complete with numerous examples, exercises and solutions. It is a valuable tool, not only for specialists in the field interested in technical details, but also for scientists entering Nonlinear Analysis in search of promising directions for research.
- Published
- 2013
49. Cartesian Currents in the Calculus of Variations II : Variational Integrals
- Author
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Mariano Giaquinta, Guiseppe Modica, Jiri Soucek, Mariano Giaquinta, Guiseppe Modica, and Jiri Soucek
- Subjects
- Mathematical optimization, Calculus of variations, Mathematical analysis, Mathematical physics, Geometry
- Abstract
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.
- Published
- 2013
50. The Theory of Anisotropic Elastic Plates
- Author
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T.S. Vashakmadze and T.S. Vashakmadze
- Subjects
- Mechanics, Mathematical analysis, Numerical analysis, Mathematical models, Mathematical optimization, Calculus of variations
- Abstract
The main purpose of this work is construction of the mathematical theory of elastic plates and shells, by means of which the investigation of basic boundary value problems of the spatial theory of elasticity in the case of cylindrical do mains reduces to the study of two-dimensional boundary value problems (BVP) of comparatively simple structure. In this respect in sections 2-5 after the introductory material, methods of re duction, known in the literature as usually being based on simplifying hypotheses, are studied. Here, in contradiction to classical methods, the problems, connected with construction of refined theories of anisotropic nonhomogeneous plates with variable thickness without the assumption of any physical and geometrical re strictions, are investigated. The comparative analysis of such reduction methods was carried out, and, in particular, in section 5, the following fact was established: the error transition, occuring with substitution of a two-dimensional model for the initial problem on the class of assumed solutions is restricted from below. Further, in section 6, Vekua's method of reduction, containing regular pro cess of study of three-dimensional problem, is investigated. In this direction, the problems, connected with solvability, convergence of processes, and construction of effective algorithms of approximate solutions are studied.
- Published
- 2013
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