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This book introduces the dilation theory of operators on Hilbert spaces and its relationship to complex geometry. Classical as well as very modern topics are covered in the book. On the one hand, it introduces the reader to the characteristic function, a classical object used by Sz.-Nagy and Foias and still a topic of current research. On the other hand, it describes the dilation theory of the symmetrized bidisc which has been developed mostly in the present century and is a very active topic of research. It also describes an abstract theory of dilation in the setting of set theory. This was developed very recently.A good portion of the book discusses various geometrical objects like the bidisc, the Euclidean unit ball, and the symmetrized bidisc. It shows the similarities and differences between the dilation theory in these domains. While completely positive maps play a big role in the dilation theory of the Euclidean unit ball, this is not so in the symmetrized bidisc for example. There, the central role is played by an operator equation. Targeted to graduate students and researchers, the book introduces the reader to different techniques applicable in different domains.

This book discusses the origin of graph theory from its humble beginnings in recreational mathematics to its modern setting or modeling communication networks, as is evidenced by the World Wide Web graph used by many Internet search engines. The second edition of the book includes recent developments in the theory of signed adjacency matrices involving the proof of sensitivity conjecture and the theory of Ramanujan graphs. In addition, the book discusses topics such as Pick's theorem on areas of lattice polygons and Graham-Pollak's work on addressing of graphs. The concept of graph is fundamental in mathematics and engineering, as it conveniently encodes diverse relations and facilitates combinatorial analysis of many theoretical and practical problems. The text is ideal for a one-semester course at the advanced undergraduate level or beginning graduate level.

This book develops the concepts of fundamental convex analysis and optimization by using advanced calculus and real analysis. Brief accounts of advanced calculus and real analysis are included within the book. The emphasis is on building a geometric intuition for the subject, which is aided further by supporting figures. Two distinguishing features of this book are the use of elementary alternative proofs of many results and an eclectic collection of useful concepts from optimization and convexity often needed by researchers in optimization, game theory, control theory, and mathematical economics. A full chapter on optimization algorithms gives an overview of the field, touching upon many current themes. The book is useful to advanced undergraduate and graduate students as well as researchers in the fields mentioned above and in various engineering disciplines.

This book discusses basic topics in the spectral theory of dynamical systems. Lastly, the second edition includes a new chapter "Calculus of Generalized Riesz Products", which discusses the recent work connecting generalized Riesz products, Hardy classes, Banach's problem of simple Lebesgue spectrum in ergodic theory and flat polynomials.

Provides an understanding of the various aspects of dynamical systems. Each chapter of the book specializes in one aspect of dynamical systems, and, thus, begins at an elementary level and goes on to cover advanced material.

Aims to convey 3 principal developments in the evolution of information theory, including Shannon's interpretation of Boltzmann entropy as a measure of information yielded by an elementary statistical experiment and basic coding theorems on storing messages and transmitting them through noisy communication channels in an optimal manner.

The material presented in this book is suited for a first course in Functional Analysis which can be followed by Masters students. The book includes a chapter on compact operators and the spectral theory for compact self-adjoint operators on a Hilbert space.

Including Affine and projective classification of Conics, 2 point homogeneity's of the planes, essential isometrics, non euclidean plan geometrics, in this book, the treatment of Geometry goes beyond the Kleinian views.

Contains the author's notes for a course that he taught at ETH, Zurich. The aim is to lead the reader to a proof of the Peter-Weyl theorem, the basic theorem in the representation theory of compact topological groups. The topological, analytical, and algebraic groundwork needed for the proof is provided as part of the course.

Treats some basic topics in the spectral theory of dynamical systems. The treatment is at a general level, but two more advanced theorems, one by H. Helson and W. Parry and the other by B. Host, are presented. Moreover, Ornstein's family of mixing rank one automorphisms is described with construction and proof.

Deals with topics usually studied in a masters or graduate level course on the theory of measure and integration. It starts with the Riemann integral and points out some of its shortcomings which motivate the theory of measure and the Lebesgue integral. There is a separate chapter on the change of variable formula and one on Lp- spaces.

Provides an introduction to basic topics in inequalities and their applications. These include the arithmetic mean-geometric mean inequality, Cauchy-Schwarz inequality, Chebyshev inequality, rearrangement inequality, convex and concave functions and Muirhead's theorem. More than 400 problems are included and their solutions are explained.

Introduces the first concepts of Algebraic Topology, such as general simplicial complexes, simplicial homology theory, fundamental groups, covering spaces and singular homology theory, in detail. The text has been designed for undergraduate and beginning graduate students of Mathematics. It assumes a minimal background of linear algebra, group theory and topological spaces.

Several generations of students of algebraic geometry have learned the subject from David Mumford's fabled "Red Book" containing notes of his lectures at Harvard University. This book contains what Mumford had intended to be Volume II. It covers the material in the "Red Book" in more depth with several more topics added.

Provides an introduction to what has come to be known as Standard Monomial Theory (SMT). SMT deals with the construction of nice bases of finite dimensional irreducible representations of semi-simple algebraic groups or, in geometric terms, nice bases of coordinate rings of flag varieties (and their Schubert subvarieties) associated to these groups.

The newly developed field of Seiberg-Witten gauge theory has become a well-established part of the differential topology of four-manifolds and three-manifolds. This book offers an introduction and an up-to-date review of the state of current research.

Combines the spirit of a textbook and of a monograph on the topic of Semigroups and their applications. It is expected to have potential users across a broad spectrum including operator theory, partial differential equations, harmonic analysis, probability and statistics and classical and quantum mechanics.