Previous publications on the generalization of the Thomae formulae to "Zn" curves have emphasized the theory's implications in mathematical physics and depended heavily on applied mathematical techniques. This book redevelops these previous results demonstrating how they can be derived directly from the basic properties of theta functions as functions on compact Riemann surfaces.

"Generalizations of Thomae's Formulafor "Zn" Curves" includes several refocused proofs developed in a generalized context that is more accessible to researchers in related mathematical fields such as algebraic geometry, complex analysis, and number theory.

This book is intended for mathematicians with an interest in complex analysis, algebraic geometry or number theory as well as physicists studying conformal field theory."

The aim of this book is to construct categories of spaces which contain all the C?-manifolds, but in addition infinitesimal spaces and arbitrary function spaces. To this end, the techniques of Grothendieck toposes (and the logic inherent to them) are explained at a leisurely pace and applied. By discussing topics such as integration, cohomology and vector bundles in the new context, the adequacy of these new spaces for analysis and geometry will be illustrated and the connection to the classical approach to C?-manifolds will be explained.

As is known, the book named "Multivariate spline functions and their applications" has been published by the Science Press in 1994. This book is an English edition based on the original book mentioned 1 above with many changes, including that of the structure of a cubic - interpolation in n-dimensional spline spaces, and more detail on triangu- lations have been added in this book. Special cases of multivariate spline functions (such as step functions, polygonal functions, and piecewise polynomials) have been examined math- ematically for a long time. I. J. Schoenberg (Contribution to the problem of application of equidistant data by analytic functions, Quart. Appl. Math., 4(1946), 45 - 99; 112 - 141) and W. Quade & L. Collatz (Zur Interpo- lations theories der reellen periodischen function, Press. Akad. Wiss. (PhysMath. KL), 30(1938), 383- 429) systematically established the the- ory of the spline functions. W. Quade & L. Collatz mainly discussed the periodic functions, while I. J. Schoenberg's work was systematic and com- plete. I. J. Schoenberg outlined three viewpoints for studing univariate splines: Fourier transformations, truncated polynomials and Taylor ex- pansions. Based on the first two viewpoints, I. J. Schoenberg deduced the B-spline function and its basic properties, especially the basis func- tions. Based on the latter viewpoint, he represented the spline functions in terms of truncated polynomials. These viewpoints and methods had significantly effected on the development of the spline functions.

'Et moi ..... si j'avait su comment en revenir, One service mathematics has rendered the je n'y serais point aIle.' human race. It has put common sense back Jules Verne where it belongs. on the topmost shelf next to the dusty canister labelled 'discarded non. The series is divergent; therefore we may be sense'. able to do something with it. Eric T. Bell 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."

Convexity is an ancient idea going back to Archimedes. Used sporadically in the mathematical literature over the centuries, today it is a flourishing area of research and a mathematical subject in its own right. Convexity is used in optimization theory, functional analysis, complex analysis, and other parts of mathematics. Convex Analysis introduces analytic tools for studying convexity and provides analytical applications of the concept. The book includes a general background on classical geometric theory which allows readers to obtain a glimpse of how modern mathematics is developed and how geometric ideas may be studied analytically. Featuring a user-friendly approach, the book contains copious examples and plenty of figures to illustrate the ideas presented. It also includes an appendix with the technical tools needed to understand certain arguments in the book, a tale of notation, and a thorough glossary to help readers with unfamiliar terms. This book is a definitive introductory text to the concept of convexity in the context of mathematical analysis and a suitable resource for students and faculty alike.

This IMA Volume in Mathematics and its Applications PATTERN FORMATION IN CONTINUOUS AND COUPLED SYSTEMS is based on the proceedings of a workshop with the same title, but goes be yond the proceedings by presenting a series of mini-review articles that sur vey, and provide an introduction to, interesting problems in the field. The workshop was an integral part of the 1997-98 IMA program on "EMERG ING APPLICATIONS OF DYNAMICAL SYSTEMS." I would like to thank Martin Golubitsky, University of Houston (Math ematics) Dan Luss, University of Houston (Chemical Engineering), and Steven H. Strogatz, Cornell University (Theoretical and Applied Mechan ics) for their excellent work as organizers of the meeting and for editing the proceedings. I also take this opportunity to thank the National Science Foundation (NSF), and the Army Research Office (ARO), whose financial support made the workshop possible. Willard Miller, Jr., Professor and Director v PREFACE Pattern formation has been studied intensively for most of this cen tury by both experimentalists and theoreticians, and there have been many workshops and conferences devoted to the subject. In the IMA workshop on Pattern Formation in Continuous and Coupled Systems held May 11-15, 1998 we attempted to focus on new directions in the patterns literature."

This book presents applications of Newton-like and other similar methods to solve abstract functional equations involving fractional derivatives. It focuses on Banach space-valued functions of a real domain - studied for the first time in the literature. Various issues related to the modeling and analysis of fractional order systems continue to grow in popularity, and the book provides a deeper and more formal analysis of selected issues that are relevant to many areas - including decision-making, complex processes, systems modeling and control - and deeply embedded in the fields of engineering, computer science, physics, economics, and the social and life sciences. The book offers a valuable resource for researchers and graduate students, and can also be used as a textbook for seminars on the above-mentioned subjects. All chapters are self-contained and can be read independently. Further, each chapter includes an extensive list of references.

Incomplete second order linear differential equations in Banach spaces as well as first order equations have become a classical part of functional analysis. This monograph is an attempt to present a unified systematic theory of second order equations y" (t) ] Ay' (t) + By (t) = 0 including well-posedness of the Cauchy problem as well as the Dirichlet and Neumann problems. Exhaustive yet clear answers to all posed questions are given. Special emphasis is placed on new surprising effects arising for complete second order equations which do not take place for first order and incomplete second order equations. For this purpose, some new results in the spectral theory of pairs of operators and the boundary behavior of integral transforms have been developed. The book serves as a self-contained introductory course and a reference book on this subject for undergraduate and post- graduate students and research mathematicians in analysis. Moreover, users will welcome having a comprehensive study of the equations at hand, and it gives insight into the theory of complete second order linear differential equations in a general context - a theory which is far from being fully understood.

For many, modern functional analysis dates back to Banach's book [Ba32]. Here, such powerful results as the Hahn-Banach theorem, the open-mapping theorem and the uniform boundedness principle were developed in the setting of complete normed and complete metrizable spaces. When analysts realized the power and applicability of these methods, they sought to generalize the concept of a metric space and to broaden the scope of these theorems. Topological methods had been generally available since the appearance of Hausdorff's book in 1914. So it is surprising that it took so long to recognize that they could provide the means for this generalization. Indeed, the theory of topo- logical vector spaces was developed systematically only after 1950 by a great many different people, induding Bourbaki, Dieudonne, Grothendieck, Kothe, Mackey, Schwartz and Treves. The resulting body of work produced a whole new area of mathematics and generalized Banach's results. One of the great successes here was the development of the theory of distributions. While the not ion of a convergent sequence is very old, that of a convergent fil- ter dates back only to Cartan [Ca]. And while sequential convergence structures date back to Frechet [Fr], filter convergence structures are much more recent: [Ch], [Ko] and [Fi]. Initially, convergence spaces and convergence vector spaces were used by [Ko], [Wl], [Ba], [Ke64], [Ke65], [Ke74], [FB] and in particular [Bz] for topology and analysis.

It isn't that they can't see the solution. It is Approach your problems from the right end and begin with the answers. Then one day, that they can't see the problem. perhaps you will find the final question. G. K. Chesterton. The Scandal of Father The Hermit Gad in Crane Feathers' in R. Brown The point of a Pin'. van GuIik's The Chinese Maze Murders. Growing speciaIization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "experimental mathematics," "CFD," "completely integrable systems," "chaos, synergetics and large-scale order," which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics."

The study of qualitative aspects of PDE's has always attracted much attention from the early beginnings. More recently, once basic issues about PDE's, such as existence, uniqueness and stability of solutions, have been understood quite well, research on topological and/or geometric properties of their solutions has become more intense. The study of these issues is attracting the interest of an increasing number of researchers and is now a broad and well-established research area, with contributions that often come from experts from disparate areas of mathematics, such as differential and convex geometry, functional analysis, calculus of variations, mathematical physics, to name a few. This volume collects a selection of original results and informative surveys by a group of international specialists in the field, analyzes new trends and techniques and aims at promoting scientific collaboration and stimulating future developments and perspectives in this very active area of research.

This book treats Modelling of CFD problems, Numerical tools for PDE, and Scientific Computing and Systems of ODE for Epidemiology, topics that are closely related to the scientific activities and interests of Prof. William Fitzgibbon, Prof. Yuri Kuznetsov, and Prof. O. Pironneau, whose outstanding achievements are recognised in this volume. It contains 20 contributions from leading scientists in applied mathematics dealing with partial differential equations and their applications to engineering, ab-initio chemistry and life sciences. It includes the mathematical and numerical contributions to PDE for applications presented at the ECCOMAS thematic conference "Contributions to PDE for Applications" held at Laboratoire Jacques Louis Lions in Paris, France, August 31- September 1, 2015, and at the Department of Mathematics, University of Houston, Texas, USA, February 26-27, 2016. This event brought together specialists from universities and research institutions who are developing or applying numerical PDE or ODE methods with an emphasis on industrial and societal applications. This volume is of interest to researchers and practitioners as well as advanced students or engineers in applied and computational mathematics. All contributions are written at an advanced scientific level with no effort made by the editors to make this volume self-contained. It is assumed that the reader is a specialist already who knows the basis of this field of research and has the capability of understanding and appreciating the latest developments in this field.

This book has been written in a frankly partisian spirit-we believe that singularity theory offers an extremely useful approach to bifurcation prob lems and we hope to convert the reader to this view. In this preface we will discuss what we feel are the strengths of the singularity theory approach. This discussion then Ieads naturally into a discussion of the contents of the book and the prerequisites for reading it. Let us emphasize that our principal contribution in this area has been to apply pre-existing techniques from singularity theory, especially unfolding theory and classification theory, to bifurcation problems. Many ofthe ideas in this part of singularity theory were originally proposed by Rene Thom; the subject was then developed rigorously by John Matherand extended by V. I. Arnold. In applying this material to bifurcation problems, we were greatly encouraged by how weil the mathematical ideas of singularity theory meshed with the questions addressed by bifurcation theory. Concerning our title, Singularities and Groups in Bifurcation Theory, it should be mentioned that the present text is the first volume in a two-volume sequence. In this volume our emphasis is on singularity theory, with group theory playing a subordinate role. In Volume II the emphasis will be more balanced. Having made these remarks, Iet us set the context for the discussion of the strengths of the singularity theory approach to bifurcation. As we use the term, bifurcation theory is the study of equations with multiple solutions."

Cardiovascular diseases have a major impact in Western countries. Mathematical models and numerical simulations can aid the understanding of physiological and pathological processes, complementing the information provided to medical doctors by medical imaging and other non-invasive means, and opening the possibility of a better diagnosis and more in-depth surgical planning.This book offers a mathematically sound and up-to-date foundation to the training of researchers, and serves as a useful reference for the development of mathematical models and numerical simulation codes. It is structured into different chapters, written by recognized experts in the field, but it features a common thread with consistency of notation and expressions and systematic cross-referencing. Many fundamental issues are faced, such as: the mathematical representation of vascular geometries extracted from medical images, modelling blood rheology and the complex multilayer structure of the vascular tissue, and its possible pathologies, the mechanical and chemical interaction between blood and vascular walls; the different scales coupling local and systemic dynamics. All these topics introduce challenging mathematical and numerical problems, demanding for advanced analysis and simulation techniques. This book is addressed to graduate students and researchers in the field of bioengineering, applied mathematics and medicine, wishing to engage themselves in the fascinating task of modeling how the cardiovascular system works.

Sparse grids are a popular tool for the numerical treatment of high-dimensional problems. Where classical numerical discretization schemes fail in more than three or four dimensions, sparse grids, in their different flavors, are frequently the method of choice. This volume of LNCSE presents selected papers from the proceedings of the fourth workshop on sparse grids and applications, and demonstrates once again the importance of this numerical discretization scheme. The articles present recent advances in the numerical analysis of sparse grids in connection with a range of applications including computational chemistry, computational fluid dynamics, and big data analytics, to name but a few.

The study of shape optimization problems encompasses a wide spectrum of academic research with numerous applications to the real world. In this work these problems are treated from both the classical and modern perspectives and target a broad audience of graduate students in pure and applied mathematics, as well as engineers requiring a solid mathematical basis for the solution of practical problems.

Key topics and features:

* Presents foundational introduction to shape optimization theory

* Studies certain classical problems: the isoperimetric problem and the Newton problem involving the best aerodynamical shape, and optimization problems over classes of convex domains

* Treats optimal control problems under a general scheme, giving a topological framework, a survey of "gamma"-convergence, and problems governed by ODE

* Examines shape optimization problems with Dirichlet and Neumann conditions on the free boundary, along with the existence of classical solutions

* Studies optimization problems for obstacles and eigenvalues of elliptic operators

* Poses several open problems for further research

* Substantial bibliography and index

Driven by good examples and illustrations and requiring only a standard knowledge in the calculus of variations, differential equations, and functional analysis, the book can serve as a text for a graduate course in computational methods of optimal design and optimization, as well as an excellent reference for applied mathematicians addressing functional shape optimization problems.

Generalized Measure Theory examines the relatively new mathematical area of generalized measure theory. The exposition unfolds systematically, beginning with preliminaries and new concepts, followed by a detailed treatment of important new results regarding various types of nonadditive measures and the associated integration theory. The latter involves several types of integrals: Sugeno integrals, Choquet integrals, pan-integrals, and lower and upper integrals. All of the topics are motivated by numerous examples, culminating in a final chapter on applications of generalized measure theory. Some key features of the book include: many exercises at the end of each chapter along with relevant historical and bibliographical notes, an extensive bibliography, and name and subject indices. The work is suitable for a classroom setting at the graduate level in courses or seminars in applied mathematics, computer science, engineering, and some areas of science. A sound background in mathematical analysis is required. Since the book contains many original results by the authors, it will also appeal to researchers working in the emerging area of generalized measure theory.

This volume is dedicated to Leonid Lerer on the occasion of his seventieth birthday. The main part presents recent results in Lerer's research area of interest, which includes Toeplitz, Toeplitz plus Hankel, and Wiener-Hopf operators, Bezout equations, inertia type results, matrix polynomials, and related areas in operator and matrix theory. Biographical material and Lerer's list of publications complete the volume.

This book is about normal forms--the simplest form into which a dynamical system can be put for the purpose of studying its behavior in the neighborhood of a rest point--and about unfoldings--used to study the local bifurcations that the system can exhibit under perturbation. The book presents the advanced theory of normal forms, showing their interaction with representation theory, invariant theory, Groebner basis theory, and structure theory of rings and modules. A complete treatment is given both for the popular "inner product style" of normal forms and the less well known "sl(2) style" due to Cushman and Sanders, as well as the author's own "simplified" style. In addition, this book includes algorithms suitable for use with computer algebra systems for computing normal forms. The interaction between the algebraic structure of normal forms and their geometrical consequences is emphasized. The book contains previously unpublished results in both areas (algebraic and geometrical) and includes suggestions for further research. The book begins with two nonlinear examples--one semisimple, one nilpotent--for which normal forms and unfoldings are computed by a variety of elementary methods. After treating some required topics in linear algebra, more advanced normal form methods are introduced, first in the context of linear normal forms for matrix perturbation theory, and then for nonlinear dynamical systems. Then the emphasis shifts to applications: geometric structures in normal forms, computation of unfoldings, and related topics in bifurcation theory. This book will be useful to researchers and advanced students in dynamical systems, theoretical physics, and engineering.

This introductory course on the classical Boundary Element Method also contains advanced topics such as the Dual Reciprocity and the Hybrid Boundary Element Methods. The latter methods are extensions that permit the application of BME to anisotropic materials, as well as multi-field problems and fluid-structure interaction. The class-tested textbook offers a clear and easy-to-understand introduction to the subject, including worked-out examples that describe all the basic features of the method. The first two chapters not only establish the mathematical basis for BEM but also review the basics of continuum mechanics for field problems, perhaps a unique feature for a text on numerical methods. This helps the reader to understand the physical principles of the field problems, to apply the method judiciously, and toe critically evaluate the results.

These notes are the result of a course in dynamical systems given at Orsay during the 1976-77 academic year. I had given a similar course at the Gradu ate Center of the City University of New York the previous year and came to France equipped with the class notes of two of my students there, Carol Hurwitz and Michael Maller. My goal was to present Smale's n-Stability Theorem as completely and compactly as possible and in such a way that the students would have easy access to the literature. I was not confident that I could do all this in lectures in French, so I decided to distribute lecture notes. I wrote these notes in English and Remi Langevin translated them into French. His work involved much more than translation. He consistently corrected for style, clarity, and accuracy. Albert Fathi got involved in reading the manuscript. His role quickly expanded to extensive rewriting and writing. Fathi wrote (5. 1) and (5. 2) and rewrote Theorem 7. 8 when I was in despair of ever getting it right with all the details. He kept me honest at all points and played a large role in the final form of the manuscript. He also did the main work in getting the manuscript ready when I had left France and Langevin was unfortunately unavailable. I ran out of steam by the time it came to Chapter 10. M."

This long-awaited book by two of the foremost researchers and writers in the field is the first part of a treatise that will cover the subject in breadth and depth, paying special attention to the historical origins, partly in applications, e.g. from geometrical optics, of parts of the theory. A variety of aids to the reader are provided: the detailed table of contents, an introduction to each chapter, section and subsection, an overview of the relevant liter- ature (in Vol. 2) plus the references in the Scholia to each chapter, in the (historical) footnotes, and in the biblio- graphy, and finally an index of the examples used throughout the book. Later volumes will deal with direct methods and regularity theory. Both individually and collectively these volumes will undoubtedly become standard references.

Statistical inferential methods are widely used in the study of various physical, biological, social, and other phenomena. Parametric estimation is one such method. Although there are many books which consider problems of statistical point estimation, this volume is the first to be devoted solely to the problem of unbiased estimation. It contains three chapters dealing, respectively, with the theory of point statistical estimation, techniques for constructing unbiased estimators, and applications of unbiased estimation theory. These chapters are followed by a comprehensive appendix which classifies and lists, in the form of tables, all known results relating to unbiased estimators of parameters for univariate distributions. About one thousand minimum variance unbiased estimators are listed. The volume also contains numerous examples and exercises. This volume will serve as a handbook on point unbiased estimation for researchers whose work involves statistics. It can also be recommended as a supplementary text for graduate students.

Asymptotic Characteristics of Entire Functions and Their Applications in Mathematics and Biophysics is the second edition of the same book in Russian, revised and enlarged. It is devoted to asymptotical questions of the theory of entire and plurisubharmonic functions. The new and traditional asymptotical characteristics of entire functions of one and many variables are studied. Applications of these indices in different fields of complex analysis are considered, for example Borel-Laplace transformations and their modifications, Mittag-Leffler function and its natural generalizations, integral methods of summation of power series and Riemann surfaces.

In the second edition, a new appendix is devoted to the consideration of those questions for a class of entire functions of proximate order. A separate chapter is devoted to applications in biophysics, where the algorithms of mathematical analysis of homeostasis system behaviour, dynamics under external influence are investigated, which may be used in different fields of natural science and technique.

This book is of interest to research specialists in theoretical and applied mathematics, postgraduates and students of universities who are interested in complex and real analysis and its applications.