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An Easy Path To Convex Analysis And Applications
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Book Synopsis An Easy Path to Convex Analysis and Applications by : Boris S. Mordukhovich
Download or read book An Easy Path to Convex Analysis and Applications written by Boris S. Mordukhovich and published by Morgan & Claypool Publishers. This book was released on 2013-12-01 with total page 219 pages. Available in PDF, EPUB and Kindle. Book excerpt: Convex optimization has an increasing impact on many areas of mathematics, applied sciences, and practical applications. It is now being taught at many universities and being used by researchers of different fields. As convex analysis is the mathematical f
Book Synopsis An Easy Path to Convex Analysis and Applications by : Boris Mordukhovich
Download or read book An Easy Path to Convex Analysis and Applications written by Boris Mordukhovich and published by Springer Nature. This book was released on 2023-06-16 with total page 313 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book examines the most fundamental parts of convex analysis and its applications to optimization and location problems. Accessible techniques of variational analysis are employed to clarify and simplify some basic proofs in convex analysis and to build a theory of generalized differentiation for convex functions and sets in finite dimensions. The book serves as a bridge for the readers who have just started using convex analysis to reach deeper topics in the field. Detailed proofs are presented for most of the results in the book and also included are many figures and exercises for better understanding the material. Applications provided include both the classical topics of convex optimization and important problems of modern convex optimization, convex geometry, and facility location.
Book Synopsis An Easy Path to Convex Analysis and Applications by : Boris Mordukhovich
Download or read book An Easy Path to Convex Analysis and Applications written by Boris Mordukhovich and published by Springer Nature. This book was released on 2022-05-31 with total page 202 pages. Available in PDF, EPUB and Kindle. Book excerpt: Convex optimization has an increasing impact on many areas of mathematics, applied sciences, and practical applications. It is now being taught at many universities and being used by researchers of different fields. As convex analysis is the mathematical foundation for convex optimization, having deep knowledge of convex analysis helps students and researchers apply its tools more effectively. The main goal of this book is to provide an easy access to the most fundamental parts of convex analysis and its applications to optimization. Modern techniques of variational analysis are employed to clarify and simplify some basic proofs in convex analysis and build the theory of generalized differentiation for convex functions and sets in finite dimensions. We also present new applications of convex analysis to location problems in connection with many interesting geometric problems such as the Fermat-Torricelli problem, the Heron problem, the Sylvester problem, and their generalizations. Of course, we do not expect to touch every aspect of convex analysis, but the book consists of sufficient material for a first course on this subject. It can also serve as supplemental reading material for a course on convex optimization and applications.
Book Synopsis Convex and Set-Valued Analysis by : Aram V. Arutyunov
Download or read book Convex and Set-Valued Analysis written by Aram V. Arutyunov and published by Walter de Gruyter GmbH & Co KG. This book was released on 2016-12-05 with total page 209 pages. Available in PDF, EPUB and Kindle. Book excerpt: This textbook is devoted to a compressed and self-contained exposition of two important parts of contemporary mathematics: convex and set-valued analysis. In the first part, properties of convex sets, the theory of separation, convex functions and their differentiability, properties of convex cones in finite- and infinite-dimensional spaces are discussed. The second part covers some important parts of set-valued analysis. There the properties of the Hausdorff metric and various continuity concepts of set-valued maps are considered. The great attention is paid also to measurable set-valued functions, continuous, Lipschitz and some special types of selections, fixed point and coincidence theorems, covering set-valued maps, topological degree theory and differential inclusions. Contents: Preface Part I: Convex analysis Convex sets and their properties The convex hull of a set. The interior of convex sets The affine hull of sets. The relative interior of convex sets Separation theorems for convex sets Convex functions Closedness, boundedness, continuity, and Lipschitz property of convex functions Conjugate functions Support functions Differentiability of convex functions and the subdifferential Convex cones A little more about convex cones in infinite-dimensional spaces A problem of linear programming More about convex sets and convex hulls Part II: Set-valued analysis Introduction to the theory of topological and metric spaces The Hausdorff metric and the distance between sets Some fine properties of the Hausdorff metric Set-valued maps. Upper semicontinuous and lower semicontinuous set-valued maps A base of topology of the spaceHc(X) Measurable set-valued maps. Measurable selections and measurable choice theorems The superposition set-valued operator The Michael theorem and continuous selections. Lipschitz selections. Single-valued approximations Special selections of set-valued maps Differential inclusions Fixed points and coincidences of maps in metric spaces Stability of coincidence points and properties of covering maps Topological degree and fixed points of set-valued maps in Banach spaces Existence results for differential inclusions via the fixed point method Notation Bibliography Index
Book Synopsis Convex Analysis and Monotone Operator Theory in Hilbert Spaces by : Heinz H. Bauschke
Download or read book Convex Analysis and Monotone Operator Theory in Hilbert Spaces written by Heinz H. Bauschke and published by Springer. This book was released on 2017-02-28 with total page 624 pages. Available in PDF, EPUB and Kindle. Book excerpt: This reference text, now in its second edition, offers a modern unifying presentation of three basic areas of nonlinear analysis: convex analysis, monotone operator theory, and the fixed point theory of nonexpansive operators. Taking a unique comprehensive approach, the theory is developed from the ground up, with the rich connections and interactions between the areas as the central focus, and it is illustrated by a large number of examples. The Hilbert space setting of the material offers a wide range of applications while avoiding the technical difficulties of general Banach spaces. The authors have also drawn upon recent advances and modern tools to simplify the proofs of key results making the book more accessible to a broader range of scholars and users. Combining a strong emphasis on applications with exceptionally lucid writing and an abundance of exercises, this text is of great value to a large audience including pure and applied mathematicians as well as researchers in engineering, data science, machine learning, physics, decision sciences, economics, and inverse problems. The second edition of Convex Analysis and Monotone Operator Theory in Hilbert Spaces greatly expands on the first edition, containing over 140 pages of new material, over 270 new results, and more than 100 new exercises. It features a new chapter on proximity operators including two sections on proximity operators of matrix functions, in addition to several new sections distributed throughout the original chapters. Many existing results have been improved, and the list of references has been updated. Heinz H. Bauschke is a Full Professor of Mathematics at the Kelowna campus of the University of British Columbia, Canada. Patrick L. Combettes, IEEE Fellow, was on the faculty of the City University of New York and of Université Pierre et Marie Curie – Paris 6 before joining North Carolina State University as a Distinguished Professor of Mathematics in 2016.
Book Synopsis Analytical Techniques for Solving Nonlinear Partial Differential Equations by : Daniel J. Arrigo
Download or read book Analytical Techniques for Solving Nonlinear Partial Differential Equations written by Daniel J. Arrigo and published by Springer Nature. This book was released on 2022-06-01 with total page 151 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is an introduction to methods for solving nonlinear partial differential equations (NLPDEs). After the introduction of several PDEs drawn from science and engineering, the reader is introduced to techniques used to obtain exact solutions of NPDEs. The chapters include the following topics: Compatibility, Differential Substitutions, Point and Contact Transformations, First Integrals, and Functional Separability. The reader is guided through these chapters and is provided with several detailed examples. Each chapter ends with a series of exercises illustrating the material presented in each chapter. The book can be used as a textbook for a second course in PDEs (typically found in both science and engineering programs) and has been used at the University of Central Arkansas for more than ten years.
Book Synopsis A First Course in Complex Analysis by : Allan R. Willms
Download or read book A First Course in Complex Analysis written by Allan R. Willms and published by Morgan & Claypool Publishers. This book was released on 2022-04-20 with total page 237 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book introduces complex analysis and is appropriate for a first course in the subject at typically the third-year University level. It introduces the exponential function very early but does so rigorously. It covers the usual topics of functions, differentiation, analyticity, contour integration, the theorems of Cauchy and their many consequences, Taylor and Laurent series, residue theory, the computation of certain improper real integrals, and a brief introduction to conformal mapping. Throughout the text an emphasis is placed on geometric properties of complex numbers and visualization of complex mappings.
Book Synopsis Convex Analysis and Beyond by : Boris S. Mordukhovich
Download or read book Convex Analysis and Beyond written by Boris S. Mordukhovich and published by Springer Nature. This book was released on 2022-04-24 with total page 597 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents a unified theory of convex functions, sets, and set-valued mappings in topological vector spaces with its specifications to locally convex, Banach and finite-dimensional settings. These developments and expositions are based on the powerful geometric approach of variational analysis, which resides on set extremality with its characterizations and specifications in the presence of convexity. Using this approach, the text consolidates the device of fundamental facts of generalized differential calculus to obtain novel results for convex sets, functions, and set-valued mappings in finite and infinite dimensions. It also explores topics beyond convexity using the fundamental machinery of convex analysis to develop nonconvex generalized differentiation and its applications. The text utilizes an adaptable framework designed with researchers as well as multiple levels of students in mind. It includes many exercises and figures suited to graduate classes in mathematical sciences that are also accessible to advanced students in economics, engineering, and other applications. In addition, it includes chapters on convex analysis and optimization in finite-dimensional spaces that will be useful to upper undergraduate students, whereas the work as a whole provides an ample resource to mathematicians and applied scientists, particularly experts in convex and variational analysis, optimization, and their applications.
Book Synopsis Discrete Distributions in Engineering and the Applied Sciences by : Rajan Chattamvelli
Download or read book Discrete Distributions in Engineering and the Applied Sciences written by Rajan Chattamvelli and published by Springer Nature. This book was released on 2022-06-01 with total page 205 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is an introductory book on discrete statistical distributions and its applications. It discusses only those that are widely used in the applications of probability and statistics in everyday life. The purpose is to give a self-contained introduction to classical discrete distributions in statistics. Instead of compiling the important formulas (which are available in many other textbooks), we focus on important applications of each distribution in various applied fields like bioinformatics, genomics, ecology, electronics, epidemiology, management, reliability, etc., making this book an indispensable resource for researchers and practitioners in several scientific fields. Examples are drawn from different fields. An up-to-date reference appears at the end of the book. Chapter 1 introduces the basic concepts on random variables, and gives a simple method to find the mean deviation (MD) of discrete distributions. The Bernoulli and binomial distributions are discussed in detail in Chapter 2. A short chapter on discrete uniform distribution appears next. The next two chapters are on geometric and negative binomial distributions. Chapter 6 discusses the Poisson distribution in-depth, including applications in various fields. Chapter 7 is on hypergeometric distribution. As most textbooks in the market either do not discuss, or contain only brief description of the negative hypergeometric distribution, we have included an entire chapter on it. A short chapter on logarithmic series distribution follows it, in which a theorem to find the kth moment of logarithmic distribution using (k-1)th moment of zero-truncated geometric distribution is presented. The last chapter is on multinomial distribution and its applications. The primary users of this book are professionals and practitioners in various fields of engineering and the applied sciences. It will also be of use to graduate students in statistics, research scholars in science disciplines, and teachers of statistics, biostatistics, biotechnology, education, and psychology.
Book Synopsis Variational Analysis and Applications by : Boris S. Mordukhovich
Download or read book Variational Analysis and Applications written by Boris S. Mordukhovich and published by Springer. This book was released on 2018-08-02 with total page 622 pages. Available in PDF, EPUB and Kindle. Book excerpt: Building on fundamental results in variational analysis, this monograph presents new and recent developments in the field as well as selected applications. Accessible to a broad spectrum of potential readers, the main material is presented in finite-dimensional spaces. Infinite-dimensional developments are discussed at the end of each chapter with comprehensive commentaries which emphasize the essence of major results, track the genesis of ideas, provide historical comments, and illuminate challenging open questions and directions for future research. The first half of the book (Chapters 1–6) gives a systematic exposition of key concepts and facts, containing basic material as well as some recent and new developments. These first chapters are particularly accessible to masters/doctoral students taking courses in modern optimization, variational analysis, applied analysis, variational inequalities, and variational methods. The reader’s development of skills will be facilitated as they work through each, or a portion of, the multitude of exercises of varying levels. Additionally, the reader may find hints and references to more difficult exercises and are encouraged to receive further inspiration from the gems in chapter commentaries. Chapters 7–10 focus on recent results and applications of variational analysis to advanced problems in modern optimization theory, including its hierarchical and multiobjective aspects, as well as microeconomics, and related areas. It will be of great use to researchers and professionals in applied and behavioral sciences and engineering.
Book Synopsis PDE Models for Atherosclerosis Computer Implementation in R by : William E. Schiesser
Download or read book PDE Models for Atherosclerosis Computer Implementation in R written by William E. Schiesser and published by Springer Nature. This book was released on 2022-06-01 with total page 136 pages. Available in PDF, EPUB and Kindle. Book excerpt: Atherosclerosis is a pathological condition of the arteries in which plaque buildup and stiffening (hardening) can lead to stroke, myocardial infarction (heart attacks), and even death. Cholesterol in the blood is a key marker for atherosclerosis, with two forms: (1) LDL - low density lipoproteins and (2) HDL - high density lipoproteins. Low LDL and high HDL concentrations are generally considered essential for limited atherosclerosis and good health. This book pertains to a mathematical model for the spatiotemporal distribution of LDL and HDL in the arterial endothelial inner layer (EIL, intima). The model consists of a system of six partial differential equations (PDEs) with the dependent variables 1. (,): concentration of modified LDL 2. h(,): concentration of HDL 3. (,): concentration of chemoattractants 4. (,): concentration of ES cytokines 5. (,): density of monocytes/macrophages 6. (,): density of foam cells and independent variables 1. : distance from the inner arterial wall 2. : time The focus of this book is a discussion of the methodology for placing the model on modest computers for study of the numerical solutions. The foam cell density (,) as a function of the bloodstream LDL and HDL concentrations is of particular interest as a precursor for arterial plaque formation and stiffening. The numerical algorithm for the solution of the model PDEs is the method of lines (MOL), a general procedure for the computer-based numerical solution of PDEs. The MOL coding (programming) is in R, a quality, open-source scientific computing system that is readily available from the Internet. The R routines for the PDE model are discussed in detail, and are available from a download link so that the reader/analyst/researcher can execute the model to duplicate the solutions reported in the book, then experiment with the model, for example, by changing the parameters (constants) and extending the model with additional equations. /div
Book Synopsis An Introduction to Convexity, Optimization, and Algorithms by : Heinz H. Bauschke
Download or read book An Introduction to Convexity, Optimization, and Algorithms written by Heinz H. Bauschke and published by SIAM. This book was released on 2023-12-20 with total page 192 pages. Available in PDF, EPUB and Kindle. Book excerpt: This concise, self-contained volume introduces convex analysis and optimization algorithms, with an emphasis on bridging the two areas. It explores cutting-edge algorithms—such as the proximal gradient, Douglas–Rachford, Peaceman–Rachford, and FISTA—that have applications in machine learning, signal processing, image reconstruction, and other fields. An Introduction to Convexity, Optimization, and Algorithms contains algorithms illustrated by Julia examples and more than 200 exercises that enhance the reader’s understanding of the topic. Clear explanations and step-by-step algorithmic descriptions facilitate self-study for individuals looking to enhance their expertise in convex analysis and optimization. Designed for courses in convex analysis, numerical optimization, and related subjects, this volume is intended for undergraduate and graduate students in mathematics, computer science, and engineering. Its concise length makes it ideal for a one-semester course. Researchers and professionals in applied areas, such as data science and machine learning, will find insights relevant to their work.
Book Synopsis Continuous Distributions in Engineering and the Applied Sciences -- Part I by : Rajan Chattamvelli
Download or read book Continuous Distributions in Engineering and the Applied Sciences -- Part I written by Rajan Chattamvelli and published by Springer Nature. This book was released on 2022-06-01 with total page 151 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is an introductory book on continuous statistical distributions and its applications. It is primarily written for graduate students in engineering, undergraduate students in statistics, econometrics, and researchers in various fields. The purpose is to give a self-contained introduction to most commonly used classical continuous distributions in two parts. Important applications of each distribution in various applied fields are explored at the end of each chapter. A brief overview of the chapters is as follows. Chapter 1 discusses important concepts on continuous distributions like location-and-scale distributions, truncated, size-biased, and transmuted distributions. A theorem on finding the mean deviation of continuous distributions, and its applications are also discussed. Chapter 2 is on continuous uniform distribution, which is used in generating random numbers from other distributions. Exponential distribution is discussed in Chapter 3, and its applications briefly mentioned. Chapter 4 discusses both Beta-I and Beta-II distributions and their generalizations, as well as applications in geotechnical engineering, PERT, control charts, etc. The arcsine distribution and its variants are discussed in Chapter 5, along with arcsine transforms and Brownian motion. This is followed by gamma distribution and its applications in civil engineering, metallurgy, and reliability. Chapter 7 is on cosine distribution and its applications in signal processing, antenna design, and robotics path planning. Chapter 8 discusses the normal distribution and its variants like lognormal, and skew-normal distributions. The last chapter of Part I is on Cauchy distribution, its variants and applications in thermodynamics, interferometer design, and carbon-nanotube strain sensing. A new volume (Part II) covers inverse Gaussian, Laplace, Pareto, 2, T, F, Weibull, Rayleigh, Maxwell, and Gumbel distributions.
Book Synopsis Aspects of Differential Geometry III by : Esteban Calviño-Louzao
Download or read book Aspects of Differential Geometry III written by Esteban Calviño-Louzao and published by Morgan & Claypool Publishers. This book was released on 2017-05-25 with total page 169 pages. Available in PDF, EPUB and Kindle. Book excerpt: Differential Geometry is a wide field. We have chosen to concentrate upon certain aspects that are appropriate for an introduction to the subject; we have not attempted an encyclopedic treatment. Book III is aimed at the first-year graduate level but is certainly accessible to advanced undergraduates. It deals with invariance theory and discusses invariants both of Weyl and not of Weyl type; the Chern‒Gauss‒Bonnet formula is treated from this point of view. Homothety homogeneity, local homogeneity, stability theorems, and Walker geometry are discussed. Ricci solitons are presented in the contexts of Riemannian, Lorentzian, and affine geometry.
Book Synopsis Aspects of Differential Geometry IV by : Esteban Calviño-Louzao
Download or read book Aspects of Differential Geometry IV written by Esteban Calviño-Louzao and published by Springer Nature. This book was released on 2022-06-01 with total page 149 pages. Available in PDF, EPUB and Kindle. Book excerpt: Book IV continues the discussion begun in the first three volumes. Although it is aimed at first-year graduate students, it is also intended to serve as a basic reference for people working in affine differential geometry. It also should be accessible to undergraduates interested in affine differential geometry. We are primarily concerned with the study of affine surfaces {which} are locally homogeneous. We discuss affine gradient Ricci solitons, affine Killing vector fields, and geodesic completeness. Opozda has classified the affine surface geometries which are locally homogeneous; we follow her classification. Up to isomorphism, there are two simply connected Lie groups of dimension 2. The translation group R2 is Abelian and the + group\index{ax+b group} is non-Abelian. The first chapter presents foundational material. The second chapter deals with Type surfaces. These are the left-invariant affine geometries on R2. Associating to each Type surface the space of solutions to the quasi-Einstein equation corresponding to the eigenvalue =-1$ turns out to be a very powerful technique and plays a central role in our study as it links an analytic invariant with the underlying geometry of the surface. The third chapter deals with Type surfaces; these are the left-invariant affine geometries on the + group. These geometries form a very rich family which is only partially understood. The only remaining homogeneous geometry is that of the sphere 2. The fourth chapter presents relations between the geometry of an affine surface and the geometry of the cotangent bundle equipped with the neutral signature metric of the modified Riemannian extension.
Book Synopsis Inverse Obstacle Scattering with Non-Over-Determined Scattering Data by : Alexander G. Ramm
Download or read book Inverse Obstacle Scattering with Non-Over-Determined Scattering Data written by Alexander G. Ramm and published by Springer Nature. This book was released on 2022-06-01 with total page 53 pages. Available in PDF, EPUB and Kindle. Book excerpt: The inverse obstacle scattering problem consists of finding the unknown surface of a body (obstacle) from the scattering (;;), where (;;) is the scattering amplitude, ; 2 is the direction of the scattered, incident wave, respectively, 2 is the unit sphere in the R3 and k > 0 is the modulus of the wave vector. The scattering data is called non-over-determined if its dimensionality is the same as the one of the unknown object. By the dimensionality one understands the minimal number of variables of a function describing the data or an object. In an inverse obstacle scattering problem this number is 2, and an example of non-over-determined data is () := (;0;0). By sub-index 0 a fixed value of a variable is denoted. It is proved in this book that the data (), known for all in an open subset of 2, determines uniquely the surface and the boundary condition on . This condition can be the Dirichlet, or the Neumann, or the impedance type. The above uniqueness theorem is of principal importance because the non-over-determined data are the minimal data determining uniquely the unknown . There were no such results in the literature, therefore the need for this book arose. This book contains a self-contained proof of the existence and uniqueness of the scattering solution for rough surfaces.
Book Synopsis Monte Carlo Methods by : Sujaul Chowdhury
Download or read book Monte Carlo Methods written by Sujaul Chowdhury and published by Springer Nature. This book was released on 2022-06-01 with total page 123 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is intended for undergraduate students of Mathematics, Statistics, and Physics who know nothing about Monte Carlo Methods but wish to know how they work. All treatments have been done as much manually as is practicable. The treatments are deliberately manual to let the readers get the real feel of how Monte Carlo Methods work. Definite integrals of a total of five functions (), namely Sin(), Cos(), e, loge(), and 1/(1+2), have been evaluated using constant, linear, Gaussian, and exponential probability density functions (). It is shown that results agree with known exact values better if () is proportional to (). Deviation from the proportionality results in worse agreement. This book is on Monte Carlo Methods which are numerical methods for Computational Physics. These are parts of a syllabus for undergraduate students of Mathematics and Physics for the course titled "Computational Physics." Need for the book: Besides the three referenced books, this is the only book that teaches how basic Monte Carlo methods work. This book is much more explicit and easier to follow than the three referenced books. The two chapters on the Variational Quantum Monte Carlo method are additional contributions of the book. Pedagogical features: After a thorough acquaintance with background knowledge in Chapter 1, five thoroughly worked out examples on how to carry out Monte Carlo integration is included in Chapter 2. Moreover, the book contains two chapters on the Variational Quantum Monte Carlo method applied to a simple harmonic oscillator and a hydrogen atom. The book is a good read; it is intended to make readers adept at using the method. The book is intended to aid in hands-on learning of the Monte Carlo methods.
