A quadratic differential on aRiemann surface is locally represented by a ho lomorphic function element wh ich transforms like the square of a derivative under a conformal change of the parameter. More generally, one also allows for meromorphic function elements; however, in many considerations it is con venient to puncture the surface at the poles of the differential. One is then back at the holomorphic case. A quadratic differential defines, in a natural way, a field of line elements on the surface, with singularities at the critical points, i.e. the zeros and poles of the differential. The integral curves of this field are called the trajectories of the differential. A large part of this book is about the trajectory structure of quadratic differentials. There are of course local and global aspects to this structure. Be sides, there is the behaviour of an individual trajectory and the structure deter mined by entire subfamilies of trajectories. An Abelian or first order differential has an integral or primitive function is in general not single-valued. In the case of a quadratic on the surface, which differential, one first has to take the square root and then integrate. The local integrals are only determined up to their sign and arbitrary additive constants. However, it is this multivalued function which plays an important role in the theory; the trajectories are the images of the horizontals by single valued branches of its inverse."

The theory presented in this book is developed constructively, is based on a few axioms encapsulating the notion of objects (points and sets) being apart, and encompasses both point-set topology and the theory of uniform spaces. While the classical-logic-based theory of proximity spaces provides some guidance for the theory of apartness, the notion of nearness/proximity does not embody enough algorithmic information for a deep constructive development. The use of constructive (intuitionistic) logic in this book requires much more technical ingenuity than one finds in classical proximity theory - algorithmic information does not come cheaply - but it often reveals distinctions that are rendered invisible by classical logic.

In the first chapter the authors outline informal constructive logic and set theory, and, briefly, the basic notions and notations for metric and topological spaces. In the second they introduce axioms for a point-set apartness and then explore some of the consequences of those axioms. In particular, they examine a natural topology associated with an apartness space, and relations between various types of continuity of mappings. In the third chapter the authors extend the notion of point-set (pre-)apartness axiomatically to one of (pre-)apartness between subsets of an inhabited set. They then provide axioms for a quasiuniform space, perhaps the most important type of set-set apartness space. Quasiuniform spaces play a major role in the remainder of the chapter, which covers such topics as the connection between uniform and strong continuity (arguably the most technically difficult part of the book), apartness and convergence in function spaces, types of completeness, and neat compactness. Each chapter has a Notes section, in which are found comments on the definitions, results, and proofs, as well as occasional pointers to future work. The book ends with a Postlude that refers to other constructive approaches to topology, with emphasis on the relation between apartness spaces and formal topology.

Largely an exposition of the authors' own research, this is the first book dealing with the apartness approach to constructive topology, and is a valuable addition to the literature on constructive mathematics and on topology in computer science. It is aimed at graduate students and advanced researchers in theoretical computer science, mathematics, and logic who are interested in constructive/algorithmic aspects of topology.

Largely an exposition of the authors' own research, this is the first book dealing with the apartness approach to constructive topology, and is a valuable addition to the literature on constructive mathematics and on topology in computer science. It is aimed at graduate students and advanced researchers in theoretical computer science, mathematics, and logic who are interested in constructive/algorithmic aspects of topology.

This text, combining analysis and tools from mathematical probability, focuses on a systematic and novel exposition of a recent trend in pure and applied mathematics. The emphasis is on the unity of basis constructions and their expansions (bases which are computationally efficient), and on their use in several areas: from wavelets to fractals. The aim of this book is to show how to use processes from probability, random walks on branches, and their path-space measures in the study of convergence questions from harmonic analysis, with particular emphasis on the infinite products that arise in the analysis of wavelets. The book brings together tools from engineering (especially signal/image processing) and mathematics (harmonic analysis and operator theory). audience of students and workers in a variety of fields, meeting at the crossroads where they merge; hands-on approach with generous motivation; new pedagogical features to enhance teaching techniques and experience; includes more than 34 figures with detailed captions, illustrating the main ideas and visualizing the deeper connections in the subject; separate sections explain engineering terms to mathematicians and operator theory to engineers; and, interdisciplinary presentation and approach, combining central ideas from mathematical analysis (with a twist in the direction of operator theory and harmonic analysis), probability, computation, physics, and engineering. The presentation includes numerous exercises that are essential to reinforce fundamental concepts by helping both students and applied users practice sketching functions or iterative schemes, as well as to hone computational skills. Graduate students, researchers, applied mathematicians, engineers and physicists alike will benefit from this unique work in book form that fills a gap in the literature.

This is the second of three volumes containing the proceedings of the International Colloquium 'Free Boundary Problems: Theory and Applications', held in Montreal from June 13 to June 22, 1990. The main theme of this volume is the concept of free boundary problems associated with solids. The first free boundary problem, the freezing of water - the Stefan problem - is the prototype of solidification problems which form the main part of this volume. The two sections treting this subject cover a large variety of topics and procedures, ranging from a theoretical mathematical treatment of solvability to numerical procedures for practical problems. Some new and interesting problems in solid mechanics are discussed in the first section while in the last section the important new subject of solid-solid-phase transition is examined.

Asymptotic analysis of stochastic stock price models is the central topic of the present volume. Special examples of such models are stochastic volatility models, that have been developed as an answer to certain imperfections in a celebrated Black-Scholes model of option pricing. In a stock price model with stochastic volatility, the random behavior of the volatility is described by a stochastic process. For instance, in the Hull-White model the volatility process is a geometric Brownian motion, the Stein-Stein model uses an Ornstein-Uhlenbeck process as the stochastic volatility, and in the Heston model a Cox-Ingersoll-Ross process governs the behavior of the volatility. One of the author's main goals is to provide sharp asymptotic formulas with error estimates for distribution densities of stock prices, option pricing functions, and implied volatilities in various stochastic volatility models. The author also establishes sharp asymptotic formulas for the implied volatility at extreme strikes in general stochastic stock price models. The present volume is addressed to researchers and graduate students working in the area of financial mathematics, analysis, or probability theory. The reader is expected to be familiar with elements of classical analysis, stochastic analysis and probability theory.

Alternating current (AC) induction and synchronous machines are frequently used in variable speed drives with applications ranging from computer peripherals, robotics, and machine tools to railway traction, ship propulsion, and rolling mills. The notable impact of vector control of AC drives on most traditional and new technologies, the multitude of practical configurations proposed, and the absence of books treating this subject as a whole with a unified approach were the driving forces behind the creation of this book.

Vector Control of AC Drives examines the remarkable progress achieved worldwide in vector control from its introduction in 1969 to the current technology. The book unifies the treatment of vector control of induction and synchronous motor drives using the concepts of general flux orientation and the feed-forward (indirect) and feedback (direct) voltage and current vector control. The concept of torque vector control is also introduced and applied to all AC motors. AC models for drive applications developed in complex variables (space phasors), both for induction and synchronous motors, are used throughout the book. Numerous practical implementations of vector control are described in considerable detail, followed by representative digital simulations and test results taken from the recent literature.

Vector Control of AC Drives will be a welcome addition to the reference collections of electrical and mechanical engineers involved with machine and system design.

This reference/text desribes the basic elements of the integral, finite, and discrete transforms - emphasizing their use for solving boundary and initial value problems as well as facilitating the representations of signals and systems.;Proceeding to the final solution in the same setting of Fourier analysis without interruption, Integral and Discrete Transforms with Applications and Error Analysis: presents the background of the FFT and explains how to choose the appropriate transform for solving a boundary value problem; discusses modelling of the basic partial differential equations, as well as the solutions in terms of the main special functions; considers the Laplace, Fourier, and Hankel transforms and their variations, offering a more logical continuation of the operational method; covers integral, discrete, and finite transforms and trigonometric Fourier and general orthogonal series expansion, providing an application to signal analysis and boundary-value problems; and examines the practical approximation of computing the resulting Fourier series or integral representation of the final solution and treats the errors incurred.;Containing many detailed examples and numerous end-of-chapter exercises of varying difficulty for each section with answers, Integral and Discrete Transforms with Applications and Error Analysis is a thorough reference for analysts; industrial and applied mathematicians; electrical, electronics, and other engineers; and physicists and an informative text for upper-level undergraduate and graduate students in these disciplines.

This classic is an ideal introduction for students into the methodology and thinking of higher mathematics. It covers material not usually taught in the more technically-oriented introductory classes and will give students a well-rounded foundation for future studies.

This, the fourth volume of the handbook "Integrals and Series", contains tables of the direct Laplace transforms and includes results set forth in books of a similar kind and in periodical literature. All the tables are arranged in two columns - originals f(x) and corresponding images F(p). The Laplace transformation is extensively used in various problems of pure and applied mathematics. Particularly widespread and effective is its application to problems arising in the theory of operational calculus and its applications, embracing the most diverse branches of science and technology. An important advantage of methods using the Laplace transformation lies in the possibility of compiling tables of various elementary and special functions commonly encountered in applications. A number of Laplace transforms are expressed in terms of Meijer G-function. When combined with the table of special cases of the G-function, these formulae make it possible to obtain Laplace transforms of various elementary and special functions of mathematical physics.

This volume presents an accessible overview of mathematical control theory and analysis of PDEs, providing young researchers a snapshot of these active and rapidly developing areas. The chapters are based on two mini-courses and additional talks given at the spring school "Trends in PDEs and Related Fields" held at the University of Sidi Bel Abbes, Algeria from 8-10 April 2019. In addition to providing an in-depth summary of these two areas, chapters also highlight breakthroughs on more specific topics such as: Sobolev spaces and elliptic boundary value problems Local energy solutions of the nonlinear wave equation Geometric control of eigenfunctions of Schroedinger operators Research in PDEs and Related Fields will be a valuable resource to graduate students and more junior members of the research community interested in control theory and analysis of PDEs.

One service mathematics has rendered the 'Et moi, ..., si j'avait su comment en revenir, human race. It has put common sense back je n'y serais point alle.' where it belongs, on the topmost shelf next Jules Verne to the dusty canister labelled 'discarded non sense'. The series is divergent; therefore we may be Eric 1'. Bell able to do something with it. O. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered com puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series."

The interest in inverse problems of spectral analysis has increased considerably in recent years due to the applications to important non-linear equations in mathematical physics. This monograph is devoted to the detailed theory of inverse problems and methods of their solution for the Sturm-Liouville case. Chapters 1--6 contain proofs which are, in many cases, very different from those known earlier. Chapters 4--6 are devoted to inverse problems of quantum scattering theory with attention being focused on physical applications. Chapters 7--11 are based on the author's recent research on the theory of finite- and infinite-zone potentials. A chapter discussing the applications to the Korteweg--de Vries problem is also included. This monograph is important reading for all researchers in the field of mathematics and physics.

This unified, self-contained book examines the mathematical tools used for decomposing and analyzing functions, specifically, the application of the [discrete] Fourier transform to finite Abelian groups. With countless examples and unique exercise sets at the end of each section, Fourier Analysis on Finite Abelian Groups is a perfect companion to a first course in Fourier analysis. This text introduces mathematics students to subjects that are within their reach, but it also has powerful applications that may appeal to advanced researchers and mathematicians. The only prerequisites necessary are group theory, linear algebra, and complex analysis.

This is the third Volume in a series of books devoted to the interdisciplinary area between mathematics and physics, all ema nating from the Advanced Study Institutes held in Istanbul in 1970, 1972 and 1977. We believe that physics and mathematics can develop best in harmony and in close communication and cooper ation with each other and are sometimes inseparable. With this goal in mind we tried to bring mathematicians and physicists together to talk and lecture to each other-this time in the area of nonlinear equations. The recent progress and surge of interest in nonlinear ordi nary and partial differential equations has been impressive. At the same time, novel and interesting physical applications mul tiply. There is a unifying element brought about by the same characteristic nonlinear behavior occurring in very widely differ ent physical situations, as in the case of "solitons," for exam ple. This Volume gives, we believe, a very good indication over all of this recent progress both in theory and applications, and over current research activity and problems. The 1977 Advanced Study Institute was sponsored by the NATO Scientific Affairs Division, The University of the Bosphorus and the Turkish Scientific and Technical Research Council. We are deeply grateful to these Institutions for their support, and to lecturers and participants for their hard work and enthusiasm which created an atmosphere of lively scientific discussions."

This book provides a systematic and comprehensive account of asymptotic sets and functions from which a broad and useful theory emerges in the areas of optimization and variational inequalities. A variety of motivations leads mathematicians to study questions about attainment of the infimum in a minimization problem and its stability, duality and minmax theorems, convexification of sets and functions, and maximal monotone maps. For each there is the central problem of handling unbounded situations. Such problems arise in theory but also within the development of numerical methods. The book focuses on the notions of asymptotic cones and associated asymptotic functions that provide a natural and unifying framework for the resolution of these types of problems. These notions have been used largely and traditionally in convex analysis, yet these concepts play a prominent and independent role in both convex and nonconvex analysis. This book covers convex and nonconvex problems, offering detailed analysis and techniques that go beyond traditional approaches. The book will serve as a useful reference and self-contained text for researchers and graduate students in the fields of modern optimization theory and nonlinear analysis.

Many problems for partial difference and integro-difference equations can be written as difference equations in a normed space. This book is devoted to linear and nonlinear difference equations in a normed space. Our aim in this monograph is to initiate systematic investigations of the global behavior of solutions of difference equations in a normed space. Our primary concern is to study the asymptotic stability of the equilibrium solution. We are also interested in the existence of periodic and positive solutions. There are many books dealing with the theory of ordinary difference equations. However there are no books dealing systematically with difference equations in a normed space. It is our hope that this book will stimulate interest among mathematicians to develop the stability theory of abstract difference equations.
Note that even for ordinary difference equations, the problem of stability analysis continues to attract the attention of many specialists despite its long history. It is still one of the most burning problems, because of the absence of its complete solution,
but many general results available for ordinary difference equations
(for example, stability by linear approximation) may be easily proved for abstract difference equations.
The main methodology presented in this publication is based on a combined use of recent norm estimates for operator-valued functions with the following
methods and results:
a) the freezing method;
b) the Liapunov type equation;
c) the method of majorants;
d) the multiplicative representation of solutions.
In addition, we present stability results for abstract Volterra discrete equations.
The bookconsists of 22 chapters and an appendix. In Chapter 1, some definitions and preliminary results are collected. They are systematically used in the next chapters.
In, particular, we recall very briefly some basic notions and results of the theory of operators in Banach and ordered spaces. In addition, stability concepts are presented and Liapunov's functions are introduced. In Chapter 2 we review various classes of linear operators and their spectral properties. As examples, infinite matrices are considered. In Chapters 3 and 4, estimates for the norms of operator-valued and matrix-valued functions are suggested. In particular, we consider Hilbert-Schmidt, Neumann-Schatten, quasi-Hermitian and quasiunitary operators. These classes contain numerous infinite matrices arising in applications. In Chapter 5, some perturbation results for linear operators in a Hilbert space are presented. These results are then used in the next chapters to derive bounds for the spectral radiuses. Chapters 6-14 are devoted to asymptotic and exponential stabilities, as well as boundedness of solutions of linear and nonlinear difference equations. In Chapter 6 we investigate the linear equation with a bounded constant operator acting in a Banach space. Chapter 7 is concerned with the Liapunov type operator equation. Chapter 8 deals with estimates for the spectral radiuses of concrete operators, in particular, for infinite matrices. These bounds enable the formulation of explicit stability conditions. In Chapters 9 and 10 we consider nonautonomous (time-variant) linear equations. An essential role in this chapter is played by the evolution operator. In addition, we use the "freezing" method and multiplicativerepresentations of solutions to construct the majorants for linear equations. Chapters 11 and 12 are devoted to semilinear autonomous and nonautonomous equations. Chapters 13 and 14 are concerned with linear and nonlinear higher order difference equations. Chapter 15 is devoted to the input-to-state stability. In Chapter 16 we study periodic solutions of linear and nonlinear difference equations in a Banach space, as well as the global orbital stability of solutions of vector difference equations. Chapters 17 and 18 deal with linear and nonlinear Volterra discrete equations in a Banach space. An important role in these chapter is played by operator pencils. Chapter 19 deals with a class of the Stieltjes differential equations.
These equations generalize difference and differential equations. We apply estimates for norms of operator valued functions and properties of the multiplicative integral to certain classes of linear and nonlinear Stieltjes differential equations to obtain solution estimates that allow us to study the stability and boundedness of solutions. We also show the existence and uniqueness of solutions as well as the continuous dependence of the solutions on the time integrator. Chapter 20 provides some results regarding the Volterra--Stieltjes equations. The Volterra--Stieltjes equations include Volterra difference and Volterra integral equations. We obtain estimates for the norms of solutions of the Volterra--Stieltjes equation. Chapter 21 is devoted to difference equations with continuous time. In Chapter 22, we suggest some conditions for the existence of nontrivial and positive steady states of difference equations, as well as bounds for the stationary solutions.
- Deals systematically with difference equations in normed spaces
- Considers new classes of equations that could not be studied in the frameworks of ordinary and partial difference equations
- Develops the freezing method and presents recent results on Volterra discrete equations
- Contains an approach based on the estimates for norms of operator functions

Modern achievements in the intensively developing field of applied mathematics as applied to physical chemistry are presented in this monograph. In particular, it proposes a new approach to extremal problem theory for nonlinear operators, differential-operator equations and inclusions, and for variational inequalities in Banach spaces. An axiomatic study of nonlinear maps (including multi-valued ones) is given, and the properties of resolving operators for systems, consisting of operator and differential-operator equations, are stated in nonlinear-map terms. The solvability conditions and the properties of extremal problem solutions are obtained, while their weak expansions and necessary conditions of optimality in variational inequality form are formulated. In addition. the monograph proposes regularization methods and approximation schemes. This book is adressed to scientists, graduates and undergraduates who are interested in nonlinear analysis, control theory, system analysis and differential equations.

Focusing on the study of nonsmooth vector functions, this book presents a comprehensive account of the calculus of generalized Jacobian matrices and their applications to continuous nonsmooth optimization problems, as well as variational inequalities in finite dimensions. The treatment is motivated by a desire to expose an elementary approach to nonsmooth calculus, using a set of matrices to replace the nonexistent Jacobian matrix of a continuous vector function.

This volume presents the proceedings of a colloquium inspired by the former President of the French Mathematical Society, Michel Herve. The aim was to promote the development of mathematics through applications. Since the ancient supports the new, it seemed appropriate to center the theoretical conferences on new subjects. Since the world is movement and creation, the theoretical conferences were planned on mechanics (movement) and bifurcation theory (creation). Five aspects of mechanics were to be presented, but, unfortunately, it has not been possible to include the statis- tical mechanics aspect. So that only four aspects are presented: Classical mechanics (Hamiltonian, Lagrangian, Poisson) (W.N. Tulczyjew, J .E. l-lhite, C.M. MarIe). - Quantum mechanics (in particular the passage from the classi- cal to the quantum approach and the problem of finding the explicit solution of Schrodinger's equation)(M. Cahen and S. Gutt, J. Leray). Fluid mechanics (meaning problems involving partial differ- ential equations. One of the speakers we hoped would attend the conference was in Japan at the time, however his lecture is presented in these proceedings.) (J.F. Pommaret, H.I-l. Shi) - Mathematical "information" theory (S. Guiasll) Traditional physical arguments are characterized by their great homogeneity, and mathematically expressed by the compactness prop- erty. In such cases, there is a kind of duality between locality and globality, which allows the use of the infinitesimal in global considerations.

This book discusses, develops and applies the theory of Vilenkin-Fourier series connected to modern harmonic analysis. The classical theory of Fourier series deals with decomposition of a function into sinusoidal waves. Unlike these continuous waves the Vilenkin (Walsh) functions are rectangular waves. Such waves have already been used frequently in the theory of signal transmission, multiplexing, filtering, image enhancement, code theory, digital signal processing and pattern recognition. The development of the theory of Vilenkin-Fourier series has been strongly influenced by the classical theory of trigonometric series. Because of this it is inevitable to compare results of Vilenkin-Fourier series to those on trigonometric series. There are many similarities between these theories, but there exist differences also. Much of these can be explained by modern abstract harmonic analysis, which studies orthonormal systems from the point of view of the structure of a topological group. The first part of the book can be used as an introduction to the subject, and the following chapters summarize the most recent research in this fascinating area and can be read independently. Each chapter concludes with historical remarks and open questions. The book will appeal to researchers working in Fourier and more broad harmonic analysis and will inspire them for their own and their students' research. Moreover, researchers in applied fields will appreciate it as a sourcebook far beyond the traditional mathematical domains.

This text on measure theory with applications to partial differential equations covers general measure theory, Lebesgue spaces of real-valued and vector-valued functions, different notions of measurability for the latter, weak convergence of functions and measures, Radon and Young measures, capacity. A comprehensive discussion of applications to quasilinear parabolic and hyperbolic problems is provided.

The idea of organising a colloquium on turbulence emerged during the sabbatical leave of Prof. Arkady Tsinober in Zurich. New experimental observations and the insight gained through direct numerical simulations have been stimulating research in turbulence and are leading to the developments of new concepts. The organisers felt the necessity to bring together researchers who have contributed significantly to the advances in this field in a colloquium in which the current achievements and the future development in the theoretical, numerical and experimental approaches would be discussed. The main emphasis of the colloquium was put on discussions. These discussions led to an interesting and exciting exchange of ideas, but also involved its very laborious transcription onto paper. It was due to the personal efforts of Mrs. A. Vyskocil, Dr. N. Malik and Dr. X. Studerus that this work could be completed. The colloquium was held in the relaxed atmosphere of the Centro Stefano Franscini in Monte Verita near Ascona, a locality of exceptional natural beauty, which was put at our disposal by the Swiss Federal Institute of Technology. We would like to express our gratitude for this generous financial and logistic support, which contributed considerably to the success of the colloquium. Zurich, April 1993 Th. Dracos, A. Tsinober Participants Adrian, R. J. Kambe, T. Antonia, R. A. Kit,E. Aref, H. Landahl, M. T. Betchov, R. Lesieur, M. Bewersdorff, H. -W. Malik, N. Castaing, B. Moffatt, H. K. Chen, J. Moin,P. Dracos, T. Mullin, T. Frisch, U. Novikov, E. A.

In addition to expanding and clarifying a number of sections of the first edition, it generalizes the analysis that eliminates the noncausal pre-acceleration so that it applies to removing any pre-deceleration as well. It also introduces a robust power series solution to the equation of motion that produces an extremely accurate solution to problems such as the motion of electrons in uniform magnetic fields.

This monograph explores applications of Carleman estimates in the study of stabilization and controllability properties of partial differential equations, including the stabilization property of the damped wave equation and the null-controllability of the heat equation. All analysis is performed in the case of open sets in the Euclidean space; a second volume will extend this treatment to Riemannian manifolds. The first three chapters illustrate the derivation of Carleman estimates using pseudo-differential calculus with a large parameter. Continuation issues are then addressed, followed by a proof of the logarithmic stabilization of the damped wave equation by means of two alternative proofs of the resolvent estimate for the generator of a damped wave semigroup. The authors then discuss null-controllability of the heat equation, its equivalence with observability, and how the spectral inequality allows one to either construct a control function or prove the observability inequality. The final part of the book is devoted to the exposition of some necessary background material: the theory of distributions, invariance under change of variables, elliptic operators with Dirichlet data and associated semigroup, and some elements from functional analysis and semigroup theory.