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Soliton theory as a method for solving some classes of nonlinear evolution equations (soliton equations) is one of the most actively developing topics in mathematical physics.This book presents some spectral theory methods for the investigation of soliton equations ad the inverse scattering problems related to these equations.The authors give the theory of expansions for the Sturm-Liouville operator and the Dirac operator.On this basis, the spectral theory of recursion operators generating Korteweg-de Vries type equations is presented and the Ablowitz-Kaup-Newell-Segur scheme, through which the inverse scattering method could be understood as a Fourier-type transformation, is considered.Following these ideas, the authors investigate some of the questions related to inverse spectral problems, i.e. uniqueness theorems, construction of explicit solutions and approximative methods for solving inverse scattering problems.A rigorous investigation of the stability of soliton solutions including solitary waves for equations which do not allow integration within inverse scattering method is also presented. ... Read more

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Complex analytical methods are a powerful tool for special partial differential equations and systems. To make these methods applicable for a wider class, transformations and transmutations are used. ... Read more

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Providing a basic tool for studying nonlinear problems, Spectral Theory for Random and Nonautonomous Parabolic Equations and Applications focuses on the principal spectral theory for general time-dependent and random parabolic equations and systems. The text contains many new results and considers existing results from a fresh perspective.

Taking a clear, unified, and self-contained approach, the authors first develop the abstract general theory in the framework of weak solutions, before turning to cases of random and nonautonomous equations. They prove that time dependence and randomness do not reduce the principal spectrum and Lyapunov exponents of nonautonomous and random parabolic equations. The book also addresses classical Faber–Krahn inequalities for elliptic and time-periodic problems and extends the linear theory for scalar nonautonomous and random parabolic equations to cooperative systems. The final chapter presents applications to Kolmogorov systems of parabolic equations.

By thoroughly explaining the spectral theory for nonautonomous and random linear parabolic equations, this resource reveals the importance of the theory in examining nonlinear problems. ... Read more

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Investigates a wide range of problems where nonlinear wave phenomena play a crucial role, for instance in nonlinear acoustics, fluid dynamics, plasma physics and magnetohydrodynamics. Reviews the latest developments in the investigation of complex wave systems. ... Read more

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In a balanced presentation, this Monograph presents definitions and results on dynamical systems and chaos in a manner accessible to undergraduates in a range of disciplines with of theoretical results and practical applications. ... Read more

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This text provides a comprehensive treatment of representations on indefinite metric spaces, and their applications to the theory of *-derivations of C*-algebras.The book consists of two parts. The first studies the geometry of indefinite metric spaces (Krein and (Pi)(kappa)-spaces) and describes the theory of J-symmetric operator algebras and representations of *-algebras and groups on these spaces in a systematic form. For representations on (Pi)(kappa)-spaces, many significant new results are obtained; this establishes a possible approach to the general theory of representations.In the second part, different techniques of the theory of J-symmetric representations on Krein spaces are applied to the theory of *-derivations of C*-algebras implemented by skew-symmetric and dissipative operators. Various results are obtained, which establish a link between the deficiency indices of skew-symmetric operators implementing *-derivations of C*-algebras and dimensions of representations of these algebras. The problem of isomorphism of skew-symmetric operators is also touched upon.Numerous properties of the domains of *-derivations are investigated. These domains constitute an important subclass of differentiable Banach *-algebras, that is dense *-subalgebras of C*-algebras with properties in many respects similar to the properties of algebras of differentiable functions. The Weyl operator commutation relations are examined in the general context of *-derivations of C*-algebras. Powersí and Arvesonís indices of one-parameter semigroups of *-endomorphisms of the algebra B are considered, and various notions of the index of a *-derivation are introduced and studied.Application of the theory of J-symmetric representations on Krein spaces to the theory of *-derivations of C*-algebras is a new research area of growing interest and there are many exciting advances to be made in this field. The book covers a fairly large and complex body of material, and will serve as a stimulus to further research activity in this area. ... Read more

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I wonder if I could understand........
I teach Math high school students at home. Secretly, being a schoolgirl, I was bored with my calculating capacity. In many Math subjects, I was interested in infinitude only. Though nobody explained me about it, A. Einstein's book was my best friend. Because he said, "Math is only a tool." above all he was humorous and romantic. I wonder if I could understand Algebras by V.S. Edward Kissin........ Keep the idea to myself, for we make sincere efforts to........ ... Read more

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A theory of generalized Cauchy-Riemann systems with polar singularities of order not less than one is presented and its application to study of infinitesimal bending of surfaces having positive curvature and an isolated flat point is given. The book contains results of investigations obtained by the author and his collaborators. ... Read more

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Geometric ideas and techniques play an important role in operator theory and the theory of operator algebras. Smooth Homogeneous Structures in Operator Theory builds the background needed to understand this circle of ideas and reports on recent developments in this fruitful field of research.

Requiring only a moderate familiarity with functional analysis and general topology, the author begins with an introduction to infinite dimensional Lie theory with emphasis on the relationship between Lie groups and Lie algebras. A detailed examination of smooth homogeneous spaces follows. This study is illustrated by familiar examples from operator theory and develops methods that allow endowing such spaces with structures of complex manifolds. The final section of the book explores equivariant monotone operators and Kähler structures. It examines certain symmetry properties of abstract reproducing kernels and arrives at a very general version of the construction of restricted Grassmann manifolds from the theory of loop groups.

The author provides complete arguments for nearly every result. An extensive list of references and bibliographic notes provide a clear picture of the applicability of geometric methods in functional analysis, and the open questions presented throughout the text highlight interesting new research opportunities.

Daniel Beltitâ is a Principal Researcher at the Institute of Mathematics "Simion Stoilow" of the Romanian Academy, Bucharest, Romania. ... Read more

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Discusses mainly some nonlinear boundary value problems for elliptic equations & systems, various boundary value problems for degenerate and higher order elliptic equations and systems, for functions of several complex variables and functions in Clifford analysis. ... Read more

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Elastic plates form a class of very important mechanical structures that appear in a wide range of practical applications, from building bodies to microchip production. As the sophistication of industrial designs has increased, so has the demand for greater accuracy in analysis. This in turn has led modelers away from Kirchoff's classical theory for thin plates and toward increasingly refined models that yield not only the deflection of the middle section, but also account for transverse shear deformation. The improved performance of these models is achieved, however, at the expense of a much more complicated system of governing equations and boundary conditions.In this Monograph, the authors conduct a rigorous mathematical study of a number of boundary value problems for the system of partial differential equations that describe the equilibrium bending of an elastic plate with transverse shear deformation. Specifically, the authors explore the existence, uniqueness, and continuous dependence of the solution on the data. In each case, they give the variational formulation of the problems and discuss their solvability in Sobolev spaces. They then seek the solution in the form of plate potentials and reduce the problems to integral equations on the contour of the domain.This treatment covers an extensive range of problems and presents the variational method and the boundary integral equation method applied side-by-side. Readers will find that this feature of the book, along with its clear exposition, will lead to a firm and useful understanding of both the model and the methods. ... Read more

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Presents in a systematic and unified manner the ray method, in its various forms, for studying nonlinear wave propagation in situations of physical interest, essentially fluid dynamics and plasma physics. ... Read more

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Some amazing stories of applied mathematics, including the creation of significant profits
All mathematicians know that math can be used for a great many things. Educators, who have to wear a generalist's hat most of the time, rarely get the opportunity to demonstrate that fact. As a group, they are largely overworked, forced to deal with their classes, mountains of paperwork and yet somehow find time to keep up in their field. With all of this happening, it is hard to find opportunities for students to engage with industry and be involved in solving some of their problems. However, despite all of these obstacles, some teachers have been able to develop such collaborations, and this book is a collection of reports on their success.
I was very impressed by what the people were able to do in these projects. The most interesting ones were projects where the industrial partner reaped substantial monetary benefits. I was amazed to read that in one project the work of one student saved the sponsoring company $17,000,000 (this is NOT a typo). As I read through this book, I began getting ideas for developing relationships between my school and local businesses. We currently have some relationships in the area of computing but nothing in the area of mathematics.
These papers describe true success stories in the area of applied mathematics and that success is multi-faceted. Of course, the companies gain financial advantage, I often wondered if they had the decency to at least contribute money back to the school of origin. The students and school gain valuable experience and bragging rights in the mathematical world that is substantial albeit immeasurable. This is certainly one of the most interesting math books that I have read recently.

Published in Journal of Recreational Mathematics, reprinted with permission
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A survey of some problems of current interest in the realm of classical nonlinear electromagnetic theory. ... Read more

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Inverse boundary problems are a rapidly developing area of applied mathematics with applications throughout physics and the engineering sciences. However, the mathematical theory of inverse problems remains incomplete and needs further development to aid in the solution of many important practical problems.Inverse Boundary Spectral Problems develop a rigorous theory for solving several types of inverse problems exactly. In it, the authors consider the following: "Can the unknown coefficients of an elliptic partial differential equation be determined from the eigenvalues and the boundary values of the eigenfunctions?"Along with this problem, many inverse problems for heat and wave equations are solved.The authors approach inverse problems in a coordinate invariant way, that is, by applying ideas drawn from differential geometry. To solve them, they apply methods of Riemannian geometry, modern control theory, and the theory of localized wave packets, also known as Gaussian beams. The treatment includes the relevant background of each of these areas. Although the theory of inverse boundary spectral problems has been in development for at least 10 years, until now the literature has been scattered throughout various journals. This self-contained monograph summarizes the relevant concepts and the techniques useful for dealing with them. ... Read more

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Extremality results proved in this Monograph for an abstract operator equation provide the theoretical framework for developing new methods that allow the treatment of a variety of discontinuous initial and boundary value problems for both ordinary and partial differential equations, in explicit and implicit forms. By means of these extremality results, the authors prove the existence of extremal solutions between appropriate upper and lower solutions of first and second order discontinuous implicit and explicit ordinary and functional differential equations. They then study the dependence of these extremal solutions on the data.The authors begin by developing an existence theory for an abstract operator equation in ordered spaces and offer new tools for dealing with different kinds of discontinuous implicit and explicit differential equation problems. They present a unified approach to the existence of extremal solutions of quasilinear elliptic and parabolic problems and extend the upper and lower solution method to elliptic and parabolic inclusion of hemivariation type using variational and nonvariational methods.Nonlinear Differential Equations in Ordered Spaces includes research that appears for the first time in book form and is designed as a source book for pure and applied mathematicians. Its self-contained presentation along with numerous worked examples and complete, detailed proofs also make it accessible to researchers in engineering as well as advanced students in these fields. ... Read more

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This book presents selected papers of the proceedings of the First HERMIS (Hellenic Research on Mathematics and Informatics) Conference, held in Athens in 1992. The main themes within the HERMIS '92 Conference were computer mathematics, operational research and statistics, and mathematics in economic science. Contributors presented their recent studies and results, and more than 50 papers were contributed. Subjects covered included: numerical solution of differential equations; parallel processing and parallel algorithms; optimization and approximation; algorithms in operational research and control theory; statistical methods and analysis; mathematics in economic science; artificial intelligence; and data bases technology. In addition, this book contains a number of selected research articles published recently in the Hellenic Mathematical Society Bulletin in the form of special issues on computer mathematics. This book is intended for applied mathematicians, computer scientists and mathematicians. ... Read more

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This volume gives an account of the current state of weight theory for integral operators, such as maximal functions, Riesz potential, singular integrals and their generalization in Lorentz and Orlicz spaces. Starting with the crucial concept of a space of homogeneous type, it continues with general criteria for the boundedness of the integral operators considered, then address special settings and applications to classical operators in Euclidean spaces. ... Read more

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The International Society for the Interaction of Mechanics and Mathematics has a long-standing and respected tradition of hosting symposia that provide a forum for disseminating new developments and methods. Trends in Applications of Mathematics to Mechanics represents the proceedings of the eleventh such symposium, held at the University of Nice in May 1998.Comprising invited lectures and refereed papers, this volume includes recent results that open perspectives on fields in mechanics and their methodological counterparts in mathematics. It also surveys important advances in the areas where mathematics and mechanics interact. The applications addressed include: ... Read more