The Handbook has three aims. One is to survey, for experts, convex geometry in its ramifications and its relations with other areas of mathematics. A second aim is to give future researchers in convex geometry a high-level introduction to most branches of convexity and its applications, showing the major ideas, methods, and results; The third aim is to prove useful for mathematicians working in other areas, as well as for econometrists, computer scientists, crystallographers, physicists, and engineers who are looking for geometric tools for their own work. Volume B covers discrete, analytic, and stochastic aspects of convexity.

The Handbook has three aims. One is to survey, for experts, convex geometry in its ramifications and its relations with other areas of mathematics. A second aim is to give future researchers in convex geometry a high-level introduction to most branches of convexity and its applications, showing the major ideas, methods, and results; The third aim is to prove useful for mathematicians working in other areas, as well as for econometrists, computer scientists, crystallographers, physicists, and engineers who are looking for geometric tools for their own work.

This text fits any course with the word “Manifold” in the title. It is a graduate level book.

Although multifractals are rooted in probability, much of the related literature comes from the physics and mathematics arena. Multifractals: Theory and Applications pulls together ideas from the both areas using a language that makes them accessible and useful to statistical scientists.Показать полностьюAlthough multifractals are rooted in probability, much of the related literature comes from the physics and mathematics arena. Multifractals: Theory and Applications pulls together ideas from the both areas using a language that makes them accessible and useful to statistical scientists. It provides a framework, in particular, for the evaluation of statistical properties of estimates of the Renyi fractal dimensions.The first section provides introductory material and different definitions of a multifractal measure. The author then examines some of the various constructions for describing multifractal measures. Building from the theory of large deviations, he focuses on constructions based on lattice coverings, covering by point-centered spheres, and cascades processes. The final section presents estimators of Renyi dimensions of integer order two and greater and discusses their properties. It also explores various applications of dimension estimation and provides a detailed case study of spatial point patterns of earthquake locations. Estimating fractal dimensions holds particular value in studies of nonlinear dynamical systems, time series, and spatial point patterns. With its careful yet practical blend of multifractals, estimation methods, and case studies, Multifractals: Theory and Applications provides a unique opportunity to explore the estimation methods from a statistical perspective.

The first three chapters of this book provide a short course on classical differential geometry and could be used at the junior level wih a little outside reading in linear algebra and advaced calculus. The first six chapters can be used for a one-semester course in differential geometry at the senior-graduate level. The entire book can be covered in a full year course.

The first two chapters are devoted to convexity in the classical sense, for functions of one and several real variables respectively. This gives a background for the study in the following chapters of related notions which occur in the theory of linear partial differential equations and complex analysis such as (pluri-) subharmonic functions, pseudoconvex sets, and sets which are convex for supports or singular supports with respect to a differential operator.

A Smarandache Geometry (1969) is a geometric space (i.e., one with points, lines) such that some “axiom” is false in at least two different ways, or is false and also sometimes true.Показать полностьюA Smarandache Geometry (1969) is a geometric space (i.e., one with points, lines) such that some “axiom” is false in at least two different ways, or is false and also sometimes true. Such axiom is said to be Smarandachely denied (or S-denied for short). In Smarandache geometry, the intent is to study non-uniformity, so we require it in a very general way. A manifold that supports a such geometry is called Smarandache manifold (or s-manifold for short). As a special case, in this book Dr. Howard Iseri studies the s-manifold formed by any collection of (equilateral) triangular disks joined together such that each edge is the identification of one edge each from two distinct disks and each vertex is the identification of one vertex each of five, six, or seven distinct disks. Thus, as a particular case, Euclidean, Lobacevsky-Bolyai-Gauss, and Riemann geometries may be united altogether, in the same space, by certain Smarandache geometries. These last geometries can be partially Euclidean and partially Non-Euclidean.

This is an introduction to the theory and applications of modern geometry. It differs from other books in its field in its emphasis on applications and its discussion of Special Relativity as a major example of a non-Euclidean geometry.Показать полностьюThis is an introduction to the theory and applications of modern geometry. It differs from other books in its field in its emphasis on applications and its discussion of Special Relativity as a major example of a non-Euclidean geometry. Besides Special Relativity, it covers two other important ares of non-Euclidean goemetry: spherical geometry (used in navigation and astronomy) and projective geometry (used in art). In addition, it reviews many useful topics from Euclidean geometry, emphasizing transformations, and includes a chapter on conics and planetary orbits. Applications are stressed throughout the book. Every topic is motivated by an application and many additional applications are given in the exercises. The book would be an excellent introduction to higher geometry for those students, especially prospective mathematics and teachers, who need to know how geometry is used in addition to its formal theory.

In the preface to Volume One I promised a second volume which would contain the theory of linear mappings and special classes of spaces important in analysis.Показать полностьюIn the preface to Volume One I promised a second volume which would contain the theory of linear mappings and special classes of spaces important in analysis. It took me nearly twenty years to fulfill this promise, at least to some extent. To the six chapters of Volume One I added two new chapters, one on linear mappings and duality (Chapter Seven), the second on spaces of linear mappings (Chapter Eight). A glance at the Contents and the short introductions to the two new chapters will give a fair impression of the material included in this volume. I regret that I had to give up my intention to write a third chapter on nuclear spaces. It seemed impossible to include the recent deep results in this field without creating a great further delay.

It is the intention of this monograph to provide an introduction to the special theory of relativity that is mathematically rigorous and yet spells out in considerable detail the physical significance of the mathematics.Показать полностьюIt is the intention of this monograph to provide an introduction to the special theory of relativity that is mathematically rigorous and yet spells out in considerable detail the physical significance of the mathematics. Particular care has been exercised in keeping clear the distinction between a physical phenomenon and the mathematical model which purports to describe that phenomenon so that, at any given point, it should be clear whether we are doing mathematics or appealing to physical arguments to interpret the mathematics.

Ce cours de topologie a ete dispense en licence a l'Universite de Rennes 1 de 1999 a 2002. Toutes les structures permettant de parler de limite et de continuite sont d'abord degagees, puis l'utilite de la compacite pour ramener des problemes de complexite infinie a l'etude d'un nombre fini de cas est explicitee.Показать полностьюCe cours de topologie a ete dispense en licence a l'Universite de Rennes 1 de 1999 a 2002. Toutes les structures permettant de parler de limite et de continuite sont d'abord degagees, puis l'utilite de la compacite pour ramener des problemes de complexite infinie a l'etude d'un nombre fini de cas est explicitee. Les premiers rudiments d'analyse fonctionnelle sont ensuite introduits : prediction de l'existence de la limite d'une suite dans un espace bien defini ; controle uniforme d'une regularite... Ce cours se termine sur la generalisation a la dimension infinie de la notion d'espace euclidien. Il comprend de nombreux exemples et environ 150 exercices non corriges.

The theory of toric varieties (also called torus embeddings) describes a fascinating interplay between algebraic geometry and the geometry of convex figures in real affine spaces.Показать полностьюThe theory of toric varieties (also called torus embeddings) describes a fascinating interplay between algebraic geometry and the geometry of convex figures in real affine spaces. This book is a unified up-to-date survey of the various results and interesting applications found since toric varieties were introduced in the early 1970’s. It is an updated and corrected English edition of the author’s book in Japanese published by Kinokuniya, Tokyo in 1985. Toric varieties are here treated as complex analytic spaces. Without assuming much prior knowledge of algebraic geometry, the author shows how elementary convex figures give rise to interesting complex analytic spaces. Easily visualized convex geometry is then used to describe algebraic geometry for these spaces, such as line bundles, projectivity, automorphism groups, birational transformations, differential forms and Mori’s theory. Hence this book might serve as an accessible introduction to current algebraic geometry. Conversely, the algebraic geometry of toric varieties gives new insight into continued fractions as well as their higher-dimensional analogues, the isoperimetric problem and other questions on convex bodies. Relevant results on convex geometry are collected together in the appendix.

This volume is devoted to the use of helices as a method for studying exceptional vector bundles, an important and natural concept in algebraic geometry.Показать полностьюThis volume is devoted to the use of helices as a method for studying exceptional vector bundles, an important and natural concept in algebraic geometry. The work arises out of a series of seminars organized in Moscow by A.N. Rudakov. The first article sets up the general machinery, and later ones explore its use in various contexts. As to be expected, the approach is concrete; the theory is considered for quadrics, ruled surfaces, K3 surfaces and PP^T3(C).

Symmetries and invariance principles play an important role in various branches of mathematics. This book deals with measures having weak symmetry properties.Показать полностьюSymmetries and invariance principles play an important role in various branches of mathematics. This book deals with measures having weak symmetry properties. Even mild conditions ensure that all invariant Borel measures on a second countable locally compact space can be expressed as images of specific product measures under a fixed mapping. The results derived in this book are interesting for their own and, moreover, a number of carefully investigated examples underline and illustrate their usefulness and applicability for integration problems, stochastic simulations and statistical applications.

At the heart of this monograph is the Brunn-Minkowski theory. It can be used to great effect in studying such ideas as volume and surface area and the generalizations of these.Показать полностьюAt the heart of this monograph is the Brunn-Minkowski theory. It can be used to great effect in studying such ideas as volume and surface area and the generalizations of these. In particular the notions of mixed volume and mixed area arise naturally and the fundamental inequalities that are satisfied by mixed volumes are considered in detail. The author presents a comprehensive introduction to convex bodies and gives full proofs for some deeper theorems which have never previously been brought together. Many hints and pointers to connections with other fields are given, and an exhaustive reference list is included.

In the present paper, a method of deformation is developed, which fits the requirements of the pattern and generalizes the method of [5(c)], for structures on manifolds defined by arbitrary transitive, continuous pseudo groups of transformations. A brief account of the methods and main results of this paper is to be found in [12(b)].

Minkowski geometry is a non-Euclidean geometry in a finite number of dimensions that is different from elliptic and hyperbolic geometry (and from the Minkowskian geometry of spacetime).Показать полностьюMinkowski geometry is a non-Euclidean geometry in a finite number of dimensions that is different from elliptic and hyperbolic geometry (and from the Minkowskian geometry of spacetime). Here the linear structure is the same as the Euclidean one but distance is not “uniform” in all directions. Instead of the usual sphere in Euclidean space, the unit ball is a general symmetric convex set. Therefore, although the parallel axiom is valid, Pythagoras' theorem is not. This book begins by presenting the topological properties of Minkowski spaces, including the existence and essential uniqueness of Haar measure, followed by the fundamental metric properties — the group of isometries, the existence of certain bases and the existence of the Lowner ellipsoid. This is followed by characterizations of Euclidean space among normed spaces and a full treatment of two-dimensional spaces. The three central chapters present the theory of area and volume in normed spaces. The author describes the fascinating geometric interplay among the isoperimetrix (the convex body which solves the isoperimetric problem), the unit ball and their duals, and the ways in which various roles of the ball in Euclidean space are divided among them. The next chapter deals with trigonometry in Minkowski spaces and the last one takes a brief look at a number of numerical parameters associated with a normed space, including J. J. Schaffer’s ideas on the intrinsic geometry of the unit sphere. Each chapter ends with a section of historical notes and the book ends with a list of 50 unsolved problems. Minkowski Geometry will appeal to students and researchers interested in geometry, convexity theory and functional analysis.

High-level study examines Einstein’s electrodynamics of moving bodies, Minkowski spacetime, gravitational geometry and other topics. High-level study discusses Newtonian principles and 19th-century views on electrodynamics and the aether, covers Einstein’s electrodynamics of moving bodies, Minkowski geometry and other topics.Показать полностьюHigh-level study examines Einstein’s electrodynamics of moving bodies, Minkowski spacetime, gravitational geometry and other topics. High-level study discusses Newtonian principles and 19th-century views on electrodynamics and the aether, covers Einstein’s electrodynamics of moving bodies, Minkowski geometry and other topics. A rich exposition of the elements of the Special and General Theory of Relativity.

This book is designed to be an introduction to some of the basic ideas in the field of algebraic topology. In particular, it is devoted to the foundations and applications of homology theory.Показать полностьюThis book is designed to be an introduction to some of the basic ideas in the field of algebraic topology. In particular, it is devoted to the foundations and applications of homology theory. The only prerequisite for the student is a basic knowledge of abelian groups and point set topology. The essentials of singular homology are given in the first chapter, along with some of the most important applications. In this way the student can quickly see the importance of the material. The successive topics include attaching spaces, finite CW complexes, the Eilenberg-Steenrod axioms, cohomology products, manifolds, Poincare duality, and fixed point theory. Throughout the book the approach is as illustrative as possible, with numerous examples and diagrams. Extremes of generality are sacrificed when they are likely to obscure the essential concepts involved. The book is intended to be easily read by students as a textbook for a course or as a source for individual study. The second edition has been substantially revised. It includes a new chapter on covering spaces in addition to illuminating new exercises.

By combining the power of Mathematica with an analytic geometry system called Descarta2D, the author has succeeded in meshing an ancient field of study with modern computational tools, the result being a simple, yet powerful, approach to studying analytic geometry.Показать полностьюBy combining the power of Mathematica with an analytic geometry system called Descarta2D, the author has succeeded in meshing an ancient field of study with modern computational tools, the result being a simple, yet powerful, approach to studying analytic geometry. Students, engineers and mathematicians alike who are interested in analytic geometry can use this book and software for the study, research or just plain enjoyment of analytic geometry.