One of the most challenging and fascinating problems of the theory of neural nets is that of asymptotic behavior, of how a system behaves as time proceeds. This is of particular relevance to many practical applications. Here we focus on association, generalization, and representation. We turn to the last topic first. The introductory chapter, "Global Analysis of Recurrent Neural Net works," by Andreas Herz presents an in-depth analysis of how to construct a Lyapunov function for various types of dynamics and neural coding. It includes a review of the recent work with John Hopfield on...
One of the most challenging and fascinating problems of the theory of neural nets is that of asymptotic behavior, of how a system behaves as time proc...
Many connections have been found between the theory of analytic functions of one or more complex variables and the study of commutative Banach algebras. While function theory has often been employed to answer algebraic questions such as the existence of idempotents in a Banach algebra, concepts arising from the study of Banach algebras including the maximal ideal space, the Silov boundary, Geason parts, etc. have led to new questions and to new methods of proofs in function theory. This book is concerned with developing some of the principal applications of function theory in several complex...
Many connections have been found between the theory of analytic functions of one or more complex variables and the study of commutative Banach algebra...
At present, roughly half of the world's population lives in urban centers. There are now more than 20 cities with a population of over 10 million inhabitants, compared to less than 5 about 50 years ago. This tendency toward urbanization is expected to continue, particularly in the developing world. A consequence of this growing trend is that millions of people are being exposed to harmful levels of urban air pollutants caused mainly by emissions from motor vehicles and from industrial and domestic activities involving the combustion of fossil fuels. The driving force for the design and...
At present, roughly half of the world's population lives in urban centers. There are now more than 20 cities with a population of over 10 million inha...
1. The ?rst edition of this book was published in 1977. The text has been well received and is still used, although it has been out of print for some time. In the intervening three decades, a lot of interesting things have happened to mathematical logic: (i) Model theory has shown that insights acquired in the study of formal languages could be used fruitfully in solving old problems of conventional mathematics. (ii) Mathematics has been and is moving with growing acceleration from the set-theoretic language of structures to the language and intuition of (higher) categories, leaving behind...
1. The ?rst edition of this book was published in 1977. The text has been well received and is still used, although it has been out of print for some ...
For most practicing analysts who use functional analysis, the restriction to Banach spaces seen in most real analysis graduate texts is not enough for their research. This graduate text, while focusing on locally convex topological vector spaces, is intended to cover most of the general theory needed for application to other areas of analysis. Normed vector spaces, Banach spaces, and Hilbert spaces are all examples of classes of locally convex spaces, which is why this is an important topic in functional analysis.
While this graduate text focuses on what is needed for applications, it...
For most practicing analysts who use functional analysis, the restriction to Banach spaces seen in most real analysis graduate texts is not enough ...
This book demonstrates current trends in research on combinatorial and computational commutative algebra with a primary emphasis on topics related to monomial ideals.
Providing a useful and quick introduction to areas of research spanning these fields, Monomial Ideals is split into three parts. Part I offers a quick introduction to the modern theory of Grobner bases as well as the detailed study of generic initial ideals. Part II supplies Hilbert functions and resolutions and some of the combinatorics related to monomial ideals including the Kruskal--Katona theorem and algebraic...
This book demonstrates current trends in research on combinatorial and computational commutative algebra with a primary emphasis on topics related ...
TheclassicaltheoryofFourierseriesandintegrals, aswellasLaplacetra- forms, is of great importance for physical and technical applications, and its mathematical beauty makes it an interesting study for pure mathema- cians as well. I have taught courses on these subjects for decades to civil engineeringstudents, andalsomathematicsmajors, andthepresentvolume can be regarded as my collected experiences from this work. There is, of course, an unsurpassable book on Fourier analysis, the tr- tise by Katznelson from 1970. That book is, however, aimed at mathem- ically very mature students and can...
TheclassicaltheoryofFourierseriesandintegrals, aswellasLaplacetra- forms, is of great importance for physical and technical applications, and its math...
This text is an elementary introduction to differential geometry. Although it was written for a graduate-level audience, the only requisite is a solid back- ground in calculus, linear algebra, and basic point-set topology. The first chapter covers the fundamentals of differentiable manifolds that are the bread and butter of differential geometry. All the usual topics are cov- ered, culminating in Stokes' theorem together with some applications. The stu- dents' first contact with the subject can be overwhelming because of the wealth of abstract definitions involved, so examples have been...
This text is an elementary introduction to differential geometry. Although it was written for a graduate-level audience, the only requisite is a solid...
This book introduces the theory of modular forms, from which all rational elliptic curves arise, with an eye toward the Modularity Theorem. Discussion covers elliptic curves as complex tori and as algebraic curves; modular curves as Riemann surfaces and as algebraic curves; Hecke operators and Atkin-Lehner theory; Hecke eigenforms and their arithmetic properties; the Jacobians of modular curves and the Abelian varieties associated to Hecke eigenforms. As it presents these ideas, the book states the Modularity Theorem in various forms, relating them to each other and touching on their...
This book introduces the theory of modular forms, from which all rational elliptic curves arise, with an eye toward the Modularity Theorem. Discuss...
