The articles collected in this volume are based on lectures given at the IMA Workshop, "Computational Radiology and Imaging: Therapy and Diagnostics," March 17-21, 1997. Introductory articles by the editors have been added. The focus is on inverse problems involving electromagnetic radiation and particle beams, with applications to X-ray tomography, nuclear medicine, near-infrared imaging, microwave imaging, electron microscopy, and radiation therapy planning. Mathematical and computational tools and models which play important roles in this volume include the X-ray transform and other integral transforms, the linear Boltzmann equation and, for near-infrared imaging, its diffusion approximation, iterative methods for large linear and non-linear least-squares problems, iterative methods for linear feasibility problems, and optimization methods. The volume is intended not only for mathematical scientists and engineers working on these and related problems, but also for non-specialists. It contains much introductory expository material, and a large number of references. Many unsolved computational and mathematical problems of substantial practical importance are pointed out.

This volume collects the edited and reviewed contribution presented in the 9th iTi Conference that took place virtually, covering fundamental and applied aspects in turbulence. In the spirit of the iTi conference, the volume is produced after the conference so that the authors had the opportunity to incorporate comments and discussions raised during the meeting. In the present book, the contributions have been structured according to the topics: I Experiments II Simulations and Modelling III Data Processing and Scaling IV Theory V Miscellaneous topics

Advanced Topics in Mathematical Analysis is aimed at researchers, graduate students, and educators with an interest in mathematical analysis, and in mathematics more generally. The book aims to present theory, methods, and applications of the selected topics that have significant, useful relevance to contemporary research.

This book paints a fresco of the field of extrapolation and rational approximation over the last several centuries to the present through the works of their primary contributors. It can serve as an introduction to the topics covered, including extrapolation methods, Pade approximation, orthogonal polynomials, continued fractions, Lanczos-type methods etc.; it also provides in depth discussion of the many links between these subjects. A highlight of this book is the presentation of the human side of the fields discussed via personal testimonies from contemporary researchers, their anecdotes, and their exclusive remembrances of some of the "actors." This book shows how research in this domain started and evolved. Biographies of other scholars encountered have also been included. An important branch of mathematics is described in its historical context, opening the way to new developments. After a mathematical introduction, the book contains a precise description of the mathematical landscape of these fields spanning from the 19th century to the first part of the 20th. After an analysis of the works produced after that period (in particular those of Richardson, Aitken, Shanks, Wynn, and others), the most recent developments and applications are reviewed.

In the part on Fourier analysis, we discuss pointwise convergence results, summability methods and, of course, convergence in the quadratic mean of Fourier series. More advanced topics include a first discussion of Hardy spaces. We also spend some time handling general orthogonal series expansions, in particular, related to orthogonal polynomials. Then we switch to the Fourier integral, i.e. the Fourier transform in Schwartz space, as well as in some Lebesgue spaces or of measures.Our treatment of ordinary differential equations starts with a discussion of some classical methods to obtain explicit integrals, followed by the existence theorems of Picard-Lindeloef and Peano which are proved by fixed point arguments. Linear systems are treated in great detail and we start a first discussion on boundary value problems. In particular, we look at Sturm-Liouville problems and orthogonal expansions. We also handle the hypergeometric differential equations (using complex methods) and their relations to special functions in mathematical physics. Some qualitative aspects are treated too, e.g. stability results (Ljapunov functions), phase diagrams, or flows.Our introduction to the calculus of variations includes a discussion of the Euler-Lagrange equations, the Legendre theory of necessary and sufficient conditions, and aspects of the Hamilton-Jacobi theory. Related first order partial differential equations are treated in more detail.The text serves as a companion to lecture courses, and it is also suitable for self-study. The text is complemented by ca. 260 problems with detailed solutions.

In the part on Fourier analysis, we discuss pointwise convergence results, summability methods and, of course, convergence in the quadratic mean of Fourier series. More advanced topics include a first discussion of Hardy spaces. We also spend some time handling general orthogonal series expansions, in particular, related to orthogonal polynomials. Then we switch to the Fourier integral, i.e. the Fourier transform in Schwartz space, as well as in some Lebesgue spaces or of measures.Our treatment of ordinary differential equations starts with a discussion of some classical methods to obtain explicit integrals, followed by the existence theorems of Picard-Lindeloef and Peano which are proved by fixed point arguments. Linear systems are treated in great detail and we start a first discussion on boundary value problems. In particular, we look at Sturm-Liouville problems and orthogonal expansions. We also handle the hypergeometric differential equations (using complex methods) and their relations to special functions in mathematical physics. Some qualitative aspects are treated too, e.g. stability results (Ljapunov functions), phase diagrams, or flows.Our introduction to the calculus of variations includes a discussion of the Euler-Lagrange equations, the Legendre theory of necessary and sufficient conditions, and aspects of the Hamilton-Jacobi theory. Related first order partial differential equations are treated in more detail.The text serves as a companion to lecture courses, and it is also suitable for self-study. The text is complemented by ca. 260 problems with detailed solutions.

This book provides a general introduction to applied analysis; vector analysis with physical motivation, calculus of variation, Fourier analysis, eigenfunction expansion, distribution, and so forth, including a catalogue of mathematical theories, such as basic analysis, topological spaces, complex function theory, real analysis, and abstract analysis. This book also uses fundamental ideas of applied mathematics to discuss recent developments in nonlinear science, such as mathematical modeling of reinforced random motion of particles, semiconductor device equation in applied physics, and chemotaxis in biology. Several tools in linear PDE theory, such as fundamental solutions, Perron's method, layer potentials, and iteration scheme, are described, as well as systematic descriptions on the recent study of the blowup of the solution.

The importance of mathematics in the study of problems arising from the real world, and the increasing success with which it has been used to model situations ranging from the purely deterministic to the stochastic, is well established. The purpose of the set of volumes to which the present one belongs is to make available authoritative, up to date, and self-contained accounts of some of the most important and useful of these analytical approaches and techniques. Each volume provides a detailed introduction to a specific subject area of current importance that is summarized below, and then goes beyond this by reviewing recent contributions, and so serving as a valuable reference source. The progress in applicable mathematics has been brought about by the extension and development of many important analytical approaches and techniques, in areas both old and new, frequently aided by the use of computers without which the solution of realistic problems would otherwise have been impossible.

Conventional methods of financial modeling are often overly exact, to the point that their purpose--to aid in financial decision making--is easily lost. Tarrazo's approach, the use of approximation, gives professionals in finance, economics, and portfolio management a sound and sophisticated way to improve their decision making, particularly in such tasks as economic prediction, financial planning, and portfolio management. Tarrazo reviews how to build models, especially those with simultaneous equation systems, then provides a simple way to use approximate equation systems to solve them. Down to earth, readable, and meticulously explained throughout, the book is not only an important tool in practical problem solving situations, but it also provides valuable methods and guidance for upper level students and their instructors.

Among the book's important contributions is its chapter on portfolio optimization. Tarrazo helps clarify the theory and application of modern portfolio theory, especially in regard to its implementation with commonly available information management tools (such as EXCEL). He also provides innovative ways to optimize portfolios under realistic conditions and a method to obtain optimal weights in interval form that does not rely on probability; instead, it relies on the mathematical quality of the matrix in the optimization. Another chapter shows that approximate equations are a general-purpose optimization tool, one that subsumes all other known optimization tools such as classical and mathematical programming. Tarrazo closes with an unusually full bibliography, containing more than 200 references spanning several areas of analysis and various disciplines.

This book is an introduction to the subject of mean curvature flow of hypersurfaces with special emphasis on the analysis of singularities. This flow occurs in the description of the evolution of numerous physical models where the energy is given by the area of the interfaces. These notes provide a detailed discussion of the classical parametric approach (mainly developed by R. Hamilton and G. Huisken). They are well suited for a course at PhD/PostDoc level and can be useful for any researcher interested in a solid introduction to the technical issues of the field. All the proofs are carefully written, often simplified, and contain several comments. Moreover, the author revisited and organized a large amount of material scattered around in literature in the last 25 years.

This book presents an extensive overview of logarithmic integral operators with kernels depending on one or several complex parameters. Solvability of corresponding boundary value problems and determination of characteristic numbers are analyzed by considering these operators as operator-value functions of appropriate complex (spectral) parameters. Therefore, the method serves as a useful addition to classical approaches. Special attention is given to the analysis of finite-meromorphic operator-valued functions, and explicit formulas for some inverse operators and characteristic numbers are developed, as well as the perturbation technique for the approximate solution of logarithmic integral equations. All essential properties of the generalized single- and double-layer potentials with logarithmic kernels and Green's potentials are considered. Fundamentals of the theory of infinite-matrix summation operators and operator-valued functions are presented, including applications to the solution of logarithmic integral equations. Many boundary value problems for the two-dimensional Helmholtz equation are discussed and explicit formulas for Green's function of canonical domains with separated logarithmic singularities are presented.

Our book gives the complex counterpart of Klein's classic book on the icosahedron. We show that the following four apparently disjoint theories: the symmetries of the Hessian polyhedra (geometry), the resolution of some system of algebraic equations (algebra), the system of partial differential equations of Appell hypergeometric functions (analysis) and the modular equation of Picard modular functions (arithmetic) are in fact dominated by the structure of a single object, the Hessian group $mathfrak{G}'_{216}$. It provides another beautiful example on the fundamental unity of mathematics.

A collection of problems and solutions in real analysis based on the major textbook, "Principles of Real Analysis" (also by Aliprantis and Burkinshaw), "Problems in Real Analysis" is the ideal companion for senior science and engineering undergraduates and first-year graduate courses in real analysis. It is intended for use as an independent source, and is an invaluable tool for students who wish to develop a deep understanding and proficiency in the use of integration methods.
"Problems in Real Analysis" teaches the basic methods of proof and problem-solving by presenting the complete solutions to over 600 problems that appear in "Principles of Real Analysis, Third Edition." The problems are distributed in forty sections, and cover the entire spectrum of difficulty.

This book provides a systematic account of several breakthroughs in the modern theory of zeta functions. It contains two different approaches to introduce and study genuine zeta functions for reductive groups (and their maximal parabolic subgroups) defined over number fields. Namely, the geometric one, built up from stability of principal lattices and an arithmetic cohomology theory, and the analytic one, from Langlands' theory of Eisenstein systems and some techniques used in trace formula, respectively. Apparently different, they are unified via a Lafforgue type relation between Arthur's analytic truncations and parabolic reductions of Harder-Narasimhan and Atiyah-Bott. Dominated by the stability condition and/or the Lie structures embedded in, these zeta functions have a standard form of the functional equation, admit much more refined symmetric structures, and most surprisingly, satisfy a weak Riemann hypothesis. In addition, two levels of the distributions for their zeros are exposed, i.e. a classical one giving the Dirac symbol, and a secondary one conjecturally related to GUE.This book is written not only for experts, but for graduate students as well. For example, it offers a summary of basic theories on Eisenstein series and stability of lattices and arithmetic principal torsors. The second part on rank two zeta functions can be used as an introduction course, containing a Siegel type treatment of cusps and fundamental domains, and an elementary approach to the trace formula involved. Being in the junctions of several branches and advanced topics of mathematics, these works are very complicated, the results are fundamental, and the theory exposes a fertile area for further research.

This text is an introduction to the use of vectors in a wide range of undergraduate disciplines. It is written specifically to match the level of experience and mathematical qualifications of students entering undergraduate and Higher National programmes and it assumes only a minimum of mathematical background on the part of the reader. Basic mathematics underlying the use of vectors is covered, and the text goes from fundamental concepts up to the level of first-year examination questions in engineering and physics. The material treated includes electromagnetic waves, alternating current, rotating fields, mechanisms, simple harmonic motion and vibrating systems. There are examples and exercises and the book contains many clear diagrams to complement the text. The provision of examples allows the student to become proficient in problem solving and the application of the material to a range of applications from science and engineering demonstrates the versatility of vector algebra as an analytical tool.

This book introduces the reader to important concepts in modern applied analysis, such as homogenization, gradient flows on metric spaces, geometric evolution, Gamma-convergence tools, applications of geometric measure theory, properties of interfacial energies, etc. This is done by tackling a prototypical problem of interfacial evolution in heterogeneous media, where these concepts are introduced and elaborated in a natural and constructive way. At the same time, the analysis introduces open issues of a general and fundamental nature, at the core of important applications. The focus on two-dimensional lattices as a prototype of heterogeneous media allows visual descriptions of concepts and methods through a large amount of illustrations.

Concave analysis deals mainly with concave and quasi-concave functions, although convex and quasi-convex functions are considered because of their mutual inherent relationship. The aim of Elements of Concave Analysis and Applications is to provide a basic and self-contained introduction to concepts and detailed study of concave and convex functions. It is written in the style of a textbook, designed for courses in mathematical economics, finance, and manufacturing design. The suggested prerequisites are multivariate calculus, ordinary and elementary PDEs, and elementary probability theory.

This book is not a research monograph about Malliavin calculus with the latest results and the most sophisticated proofs. It does not contain all the results which are known even for the basic subjects which are addressed here. The goal was to give the largest possible variety of proof techniques. For instance, we did not focus on the proof of concentration inequality for functionals of the Brownian motion, as it closely follows the lines of the analog result for Poisson functionals. This book grew from the graduate courses I gave at Paris-Sorbonne and Paris-Saclay universities, during the last few years. It is supposed to be as accessible as possible for students who have knowledge of Ito calculus and some rudiments of functional analysis.

This collection covers all papers and partial talks given by Prof Weiyue Ding, who was a member of the Chinese Academy of Sciences. Prof Weiyue Ding devoted his academic career to the research in the field of ordinary differential equations and geometric analysis, e.g. Poincare-Birkhoff fixed point theorems, blow-up analysis for heat flow of harmonic maps.

Computational Analysis of Structured Media presents a systematical approach to analytical formulae for the effective properties of deterministic and random composites. Schwarz's method and functional equations yield for use in symbolic-numeric computations relevant to the effective properties. The work is primarily concerned with constructive topics of boundary value problems, complex analysis, and their applications to composites. Symbolic-numerical computations are widely used to deduce new formulae interesting for applied mathematicians and engineers. The main line of presentation is the investigation of two-phase 2D composites with non-overlapping inclusions randomly embedded in matrices.

The series is aimed specifically at publishing peer reviewed reviews and contributions presented at workshops and conferences. Each volume is associated with a particular conference, symposium or workshop. These events cover various topics within pure and applied mathematics and provide up-to-date coverage of new developments, methods and applications.

With this fun romp through the world of equations we encounter in our everyday lives, you'll find yourself flipping through the stories of fifty-two formulas faster than a deck of cards. John M. Henshaw's intriguing true accounts, each inspired by a different mathematical equation, are both succinct and easy to read. His tales come from the spheres of sports, business, history, the arts, science, and technology. Anecdotes about famous equations, like E=mc 2, appear alongside tales of not-so-famous-but equally fascinating-equations, such as the one used to determine the SPF number for sunscreen. Drawn from the breadth of human endeavor, Henshaw's stories demonstrate the power and utility of math. He entertains us by exploring the ways that equations can be used to explain, among other things, Ponzi schemes, the placebo effect, "dog years," IQ, the wave mechanics of tsunamis, the troubled modern beekeeping industry, and the Challenger disaster. Smartly conceived and fast paced, his book offers something for anyone curious about math and its impacts.

Fractional calculus in terms of mathematics and statistics and its applications to problems in natural sciences is NOT yet part of university teaching curricula. This book is one attempt to provide an approach to include topics of fractional calculus into university curricula. Additionally the material is useful for people who do research work in the areas of special functions, fractional calculus, applications of fractional calculus, and mathematical statistics.

Means in Mathematical Analysis addresses developments in global analysis, non-linear analysis, and the many problems of associated fields, including dynamical systems, ergodic theory, combinatorics, differential equations, approximation theory, analytic inequalities, functional equations and probability theory. The series comprises highly specialized research monographs written by eminent scientists, handbooks and selected multi-contributor reference works (edited volumes), bringing together an extensive body of information. It deals with the fundamental interplay of nonlinear analysis with other headline domains, particularly geometry and analytic number theory, within the mathematical sciences.

An engineer's guide to numerical analysis To properly function in today's work environment, engineers require a working familiarity with numerical analysis. This book provides that necessary background, striking a balance between analytical rigor and an applied approach focusing on methods particular to the solving of engineering problems. An Introduction to Numerical Analysis for Electrical and Computer Engineers gives electrical and computer engineering students their first exposure to numerical analysis and serves as a refresher for professionals as well. Emphasizing the earlier stages of numerical analysis for engineers with real-life solutions for computing and engineering applications, the book: Forms a logical bridge between first courses in matrix/linear algebra and the more sophisticated methods of signal processing and control system courses Includes MATLAB(r)-oriented examples, with a quick introduction to MATLAB for those who need it Provides detailed proofs and derivations for many key results Specifically tailored to the needs of computer and electrical engineers, this is the resource engineers have long needed in order to master an area of mathematics critical to the