Book Description

This book combines traditional mainstream calculus with the most flexible approach to new ideas and calculator/computer technology. It contains superb problem sets and a fresh conceptual emphasis flavored by new technological possibilities. Chapter topics cover functions, graphs, and models; prelude to calculus; the derivative; additional applications of the derivative; the integral; applications of the integral; calculus of transcendental functions; techniques of integration; differential equations; polar coordinates and parametric curves; infinite series; vectors, curves, and surfaces in space; partial differentiation; multiple integrals; and vector calculus. For individuals interested in the study of calculus.

Book Description

Diagram geometry, which introduces a range of techniques that enable an interaction between group theory and geometry, allows the mathematician to get information on a multi-dimensional geometric object from some knowledge of bi-dimensional properties.This exceptional introduction to the subject provides a wealth of descriptions and examples, explains basic concepts, and details the language of diagram geometrics. It provides an expert development of diagram geometry theory as well as more advanced concepts and techniques.Practical and rigorous, Diagram Geometries is the ideal introduction to the subject for advanced students and researchers in group theory and combinatorics. ... Read more

Book Description

This book is an English translation of the first part of the first textbook on Analytic Geometry, written in Latin by the Dutch statesman and mathematician Jan de Witt soon after Descartes invented the subject. De Witt (1625-1672) is best known for his work in actuarial mathematics ("Calculation of the Values of Annuities as Proportions of the Rents") and for his contributions to analytic geometry, including the focus-directrix definition of conics and the use of the discriminant to distinguish among them. In addition to the translation and annotations, this volume contains an introduction and commentary, including a discussion of the role of conics in Greek mathematics. ... Read more

Book Description

Book Description

This book brings together papers that cover a wide spectrum of areas and give an unsurpassed overview of research into differential geometry. ... Read more

Book Description

This text will acquaint the reader with the most recentadvances in Finslerian geometries, i.e. anisotropic geometries, andtheir applications by the Japanese, European and American schools.It contains three introductory articles, one from each of theseschools, giving a broad overview of basic ideas. Further papers treattopics from pure mathematics such as complex differential geometry,equivalence methods, Finslerian deformations, constant sprays,homogeneous contact transformations, Douglas spaces, submanifoldtheory, inverse problems, area theory, and more. This book completesthe Kluwer trilogy on Finslerian Geometry by P.L. Antonelli andhis associates. Audience: This volume will be of interest to physicists andmathematicians whose work involves quantum field theory, combinationtheory and relativity, programming and optimization. Mathematicalbiologists working in ecology and evolution will also find it useful. ... Read more

Reviews (1)

PETER L ANTONELLI ROCKS!
rock on!!! this book is rad, i never though finsler was so cool! everyone should buy this book! ... Read more

Reviews (1)

A fine overview
Following up on the prior book "Lectures on Riemann Surfaces", the author, in this book, makes heavy use of sheaf theory to study complex analytic vector bundles over Riemann surfaces. The use of sheaf theory will seem quite natural to the reader who has read the first book, and even those who have not but have a background in complex function theory will recognize sheaf theory as a generalization of the concept of analytic continuation. The power of sheaf theory is made very apparent in this book, and the reader can see clearly some notions from the 'classical' theory, such as the Riemann-Roch theorem, generalized to the case of vector bundles over Riemann surfaces.

In the first chapter, the author first considers sheaves over a commutative ring R with an identity element and sheaves of modules over these kinds of rings. A 'free sheaf' of rank m is defined as being one isomorphic to the direct sum of m R-modules. A 'locally free sheaf' of R-modules or rank m over a topological space M is defined, naturally, as one that is free when restricted to the open sets of a covering of M. The author then discusses carefully the notion of how to get a cohomology 'set' via the sheaf of (non-Abelian) groups defined by the presheaf of the general linear group. This discussion leads to a fiber bundle that represents the locally free sheaf. This is then used to obtain 'analytic sheaves', which are sheaves of modules over the sheaf of rings of germs of holomorphic functions. The elements of the cohomology set are called 'complex analytic vector bundles of rank m' over the Riemann surface M. Coherent analytic sheaves are then brought in order to make up for the fact that analytic sheaves are generally not locally free.

The local structure of coherent analytic sheaves is studied in chapter two, wherein it is proved first that every coherent analytic sheaf of a locally free sheaf is locally free. The author then show how to construct exact sequences of coherent analytic sheaves and shows explicitly how they are described in terms of locally free analytic sheaves for the case of the complex projective line. He then proves that every vector bundle over the projective line admits non-trivial meromorphic sections.

Given a sheaf over one Riemann surface M, an analytic mapping from M to another Riemann surface induces a sheaf on the other, and the author studies these induced sheaves in chapter 3. The author then generalizes the projective line results from chapter 2 to the case of a coherent analytic sheaf over an arbitary compact Riemann surface.

In chapter 4 the author proves the famous Riemann-Roch theorem for the case of vector bundles over compact Riemann surfaces of genus g. This leads straightforwardly, as the author shows, to a form of the Riemann-Roch theorem for coherent analytic sheaves. Then, after defining the notion of a dual bundle to a complex vector bundle, the author proves a version of Serre duality for vector bundles.

After a lengthy discussion of how to extend a complex analytic line bundle to a complex analytic vector bundle of rank 2, the author in chapter 5 discusses how to determine which line bundles can be subbundles of a given vector bundle. This leads to an extension of the notion of a divisor, and the author then gives a classification of rank 2 vector bundles.

For line bundles, the vanishing of its Chern class guarantees that it has a 'flat' representative. The situation for higher rank complex vector bundles is more complicated, due to the general linear group not being abelian. The author shows how to find conditions for the admission of flat representatives in these bundles in chapter 6. This is followed naturally by a discussion of flat vector bundles in chapter 7, wherein a notion of a 'flat sheaf' of rank n is formulated. Flat vector bundles are shown to have a 'characteristic representation', which is viewed as the fundamental group of the Riemann surface acting as operators on complex n-space. This is then used to obtain a "sheafified" deRham complex for a flat vector bundle. The notion of a 'period' of a differential form makes its first appearance here.

Chapter 8 covers basically the same subject as chapter 7, but from an analytic viewpoint. The DeRham sequence in this viewpoint involves, as expected, the sheaf of germs of flat sections of the bundle and the sheaf of germs of holomorphic sections of the bundle. The author shows how the 'Prym differentials' arise as differential forms that generalize the Abelian differentials on a compact Riemann surface. He also discusses how the analytic properties of flat sheaves can be studied by considering an exact sequence involving instead the sheaf of germs of meromorphic functions on the Riemann surface. Sections of the flat vector bundle in this case give rise to the meromorphic Prym differentials, which intuitively can be thought of as meromorphic differential forms which have zero residues at each point of the Riemann surface.

By considering for a Riemann surface the mapping which associates to a flat vector bundle its characteristic representation, the author studies families of flat vector bundles in chapter 9. He shows how to obtain a complex analytic structure associated to this family. Techniques from the theory of several complex variables are utilized without review to study the complex analytic equivalence of flat vector bundles, and he shows that every flat vector bundle is analytically equivalent to an 'irreducible' flat vector bundle. Here 'irreducible' is an algebraic geometry notion, and refers to the fact that the homomorphisms from the fundamental group of the Riemann surface to the complex general linear group has the structure of a complex analytic variety. The subset of irreducible representations then is a complex analytic manifold. ... Read more

Book Description

This text gives the most recent results in Finsler and related geometries from the Miron school.Both pure and applied topics are covered. For example Higher-Order geometry, Hamilton and Cartan spaces, Legendre transformations, self-duality in Gauge fields, constant curvature spaces, Electromagnetics, Gravity theory, Black Holes, complex Finsler geometry and Finsler-Lagrange-Hamilton structures in control and optimization. There is also an article on Finsler Seismic ray theory which uses the software FINSLER based on MAPLE. ... Read more

Book Description

Excellent for engineering and technology students, this text goes far beyond instruction in standard orthographic projection to clarify all the tools of descriptive geometry--and how they apply to individual fields. The text places special emphasis on applications in all the various engineering disciplines: mechanical, plastics, industrial, piping, aerospace, marine, civil, and structural. As a result, students quickly grasp the value of descriptive geometry as they apply the tools and techniques to practical problems. By organizing information around the field's central concept--line of sight--the presentation facilitates understanding in a way unmatched by any other text. The worktext format provides students with all the resources they need--text and workbook--under one cover. ALSO AVAILABLE INSTRUCTOR SUPPLEMENTS CALL CUSTOMER SUPPORT TO ORDERInstructor's Guide, ISBN: 0-7668-0118-7Keywords: Descriptive Geometry ... Read more

Book Description

The book collects results about realization spaces of polytopes. It gives a presentation of the author's "Universality Theorem for 4-polytopes". It is a comprehensive survey of the important results that have been obtained in that direction. The approaches chosen are direct and very geometric in nature. The book is addressed to researchers and to graduate students. The former will find a comprehensive source for the above mentioned results. The latter will find a readable introduction to the field. The reader is assumed to be familiar with basic concepts of linear algebra. ... Read more