Printed Physics

Printed Physics

The humanities, natural and technical sciences seemingly have little to say to each other despite all the trans-disciplinary efforts. The Applied Virtuality series will comprise four volumes that create and examine a discourse on the correlations between the larger contexts of ther present. Printed Physics, the first volume, begins with the discussion of developments in information technology that make the physical behavior of matter technologically programmable, allow for its factual construction, industrial production and its determination with symbols. Is it possible that a revitalization of the field of physics looms in the future similar to that which took place with geometry in the 19th century?

Printed Physics

Analysis, Geometry and Topology of Elliptic Operators

Analysis, Geometry and Topology of Elliptic Operators

Modern theory of elliptic operators, or simply elliptic theory, has been shaped by the Atiyah?Singer Index Theorem created 40 years ago. Reviewing elliptic theory over a broad range, 32 leading scientists from 14 different countries present recent developments in topology; heat kernel techniques; spectral invariants and cutting and pasting; noncommutative geometry; and theoretical particle, string and membrane physics, and Hamiltonian dynamics. The first of its kind, this volume is ideally suited to graduate students and researchers interested in careful expositions of newly-evolved achievements and perspectives in elliptic theory. The contributions are based on lectures presented at a workshop acknowledging Krzysztof P Wojciechowski’s work in the theory of elliptic operators.

Analysis, Geometry and Topology of Elliptic Operators

The Development of Arabic Mathematics

The Development of Arabic Mathematics

An understanding of developments in Arabic mathematics between the IXth and XVth century is vital to a full appreciation of the history of classical mathematics. This book draws together more than ten studies to highlight one of the major developments in Arabic mathematical thinking, provoked by the double fecondation between arithmetic and the algebra of al-Khwarizmi, which led to the foundation of diverse chapters of mathematics: polynomial algebra, combinatorial analysis, algebraic geometry, algebraic theory of numbers, diophantine analysis and numerical calculus. Thanks to epistemological analysis, and the discovery of hitherto unknown material, the author has brought these chapters into the light, proposes another periodization for classical mathematics, and questions current ideology in writing its history.

The Development of Arabic Mathematics

Stochastic Geometry, Spatial Statistics and Random Fields

Stochastic Geometry, Spatial Statistics and Random Fields

This volume provides a modern introduction to stochastic geometry, random fields and spatial statistics at a (post)graduate level. It is focused on asymptotic methods in geometric probability including weak and strong limit theorems for random spatial structures (point processes, sets, graphs, fields) with applications to statistics. Written as a contributed volume of lecture notes, it will be useful not only for students but also for lecturers and researchers interested in geometric probability and related subjects.

Stochastic Geometry, Spatial Statistics and Random Fields

Algebra, Geometry and Software Systems

Algebra, Geometry and Software Systems

The book contains surveys and research papers on mathematical software and algorithms. The common thread is that the field of mathematical applications lies on the border between algebra and geometry. Topics include polyhedral geometry, elimination theory, algebraic surfaces, Gr’obner bases, triangulations of point sets and the mutual relationship. This diversity is accompanied by the abundance of available software systems which often handle only special mathematical aspects. Therefore the volumes other focus is on solutions towards the integration of mathematical software systems. This includes low-level and XML based high-level communication channels as well as general frameworks for modular systems.

Algebra, Geometry and Software Systems

Teaching and Learning Geometry

Teaching and Learning Geometry

This fascinating title reviews the teaching and learning of school geometry from the perspective of both the new teacher and the more experienced teacher. It is designed to extend and deepen subject knowledge and to offer practical advice and ideas for the classroom in the context of current practice and research. Particular emphasis is given to the following elements: Understanding the key ideas of the geometry curriculum. Learning geometry effectively: lessons from research and current practice. Misconceptions and errors. Geometry reasoning: problem solving and proving. The role of technology in learning geometry.

Teaching and Learning Geometry

The W3 Algebra

The W3 Algebra

‘W’ algebras are nonlinear generalizations of Lie algebras that arise in the context of two-dimensional conformal field theories when one explores higher-spin extensions of the Virasoro algebra. They provide the underlying symmetry algebra of certain string generalizations which allow the extended world sheet gravity. This book presents such gravity theories, concentrating on the algebra of physical operators determined from an analysis of the corresponding BRST cohomology. It develops the representation theory of ‘W’ algebras needed to extend the standard techniques which were so successful in treating linear algebras. For certain strings corresponding to ‘W’N gravity we show that the operator cohomology has a natural geometric model. This result suggests new directions for the study of ‘W’ geometry.

The W3 Algebra

Lectures on Algebra, Volume 1

Lectures on Algebra, Volume 1

This book is a timely survey of much of the algebra developed during the last several centuries including its applications to algebraic geometry and its potential use in geometric modeling. The present volume makes an ideal textbook for an abstract algebra course, while the forthcoming sequel. Lectures on Algebra II, will serve as a textbook for a linear algebra course. The author’s fondness for algebraic geometry shows up in both volumes, and his recent preoccupation with the applications of group theory to the calculation of Galois groups is evident in the second volume which contains more local rings and more algebraic geometry. Both books are based on the author’s lectures at Purdue University over the last few years.

Lectures on Algebra, Volume 1

From Quantum Cohomology to Integrable Systems

From Quantum Cohomology to Integrable Systems

Quantum cohomology has its origins in symplectic geometry and algebraic geometry, but is deeply related to differential equations and integrable systems. This text explains what is behind the extraordinary success of quantum cohomology, leading to its connections with many existing areas of mathematics as well as its appearance in new areas such as mirror symmetry. Certain kinds of differential equations (or D-modules) provide the key links between quantum cohomology and traditional mathematics; these links are the main focus of the book, and quantum cohomology and other integrable PDEs such as the KdV equation and the harmonic map equation are discussed within this unified framework. Aimed at graduate students in mathematics who want to learn about quantum cohomology in a broad context, and theoretical physicists who are interested in the mathematical setting, the text assumes basic familiarity with differential equations and cohomology.

From Quantum Cohomology to Integrable Systems

Applied Dimensional Analysis and Modeling

Applied Dimensional Analysis and Modeling

Applied Dimensional Analysis and Modeling provides the full mathematical background and step-by-step procedures for employing dimensional analyses, along with a wide range of applications to problems in engineering and applied science, such as fluid dynamics, heat flow, electromagnetics, astronomy and economics. This new edition offers additional worked-out examples in mechanics, physics, geometry, hydrodynamics, and biometry.

Applied Dimensional Analysis and Modeling

Geometry and Topology in Hamiltonian Dynamics and Statistical Mechanics

Geometry and Topology in Hamiltonian Dynamics and Statistical Mechanics

This book explores the foundations of hamiltonian dynamical systems and statistical mechanics, in particular phase transitions, from the point of view of geometry and topology. A broad participation of topology in these fields has been lacking and this book will provide a welcome overview of the current research in the area, in which the author himself is a pioneer. Using geometrical thinking to solve fundamental problems in these areas, compared to the purely analytical methods usually used in physics could be highly productive. The author skillfully guides the reader, whether mathematician or physicists through the background needed to understand and use these techniques.

Geometry and Topology in Hamiltonian Dynamics and Statistical Mechanics

Basic Theory of Algebraic Groups and Lie Algebras

Basic Theory of Algebraic Groups and Lie Algebras

he theory of algebraic groups results from the interaction of various basic techniques from field theory, multilinear algebra, commutative ring theory, algebraic geometry and general algebraic representation theory of groups and

Basic Theory of Algebraic Groups and Lie Algebras

Foundations of Hyperbolic Manifolds - Second Edition

Foundations of Hyperbolic Manifolds - Second Edition

This book is an exposition of the theoretical foundations of hyperbolic manifolds. It is intended to be used both as a textbook and as a reference. The reader is assumed to have a basic knowledge of algebra and topology at the first year graduate level of an American university. The book is divided into three parts. The first part is concerned with hyperbolic geometry and discrete groups. The second part is devoted to the theory of hyperbolic manifolds. The third part integrates the first two parts in a development of the theory of hyperbolic orbifolds. The second edition contains hundreds of changes and corrections, and new additions include: A more thorough discussion of polytopes; Discussion of Simplex Reflection groups has been expanded to give a complete classification of the Gram matrices of spherical, Euclidean and hyperbolic n-simplices; A new section on the volume of a simplex, in which a derivation of Schlaflis differential formula is presented; A new section with a proof of the n-dimensional Gauss-Bonnet theorem. The exercises have been thoroughly reworked, pruned, and upgraded, and over 100 new exercises have been added. The author has also prepared a solutions manual which is available to professors who choose to adopt this text for their course.

Foundations of Hyperbolic Manifolds - Second Edition

Introduction to Non-Euclidean Geometry

Introduction to Non-Euclidean Geometry

One of the first college-level texts for elementary courses in non-Euclidean geometry, this concise, readable volume is geared toward students familiar with calculus. A full treatment of the historical background explores the centuries-long efforts to prove Euclid’s parallel postulate and their triumphant conclusion. Numerous original exercises form an integral part of the book.

Introduction to Non-Euclidean Geometry

Riemannian Geometry - Second Edition

Riemannian Geometry - Second Edition

This book, based on a graduate course on Riemannian geometry and analysis on manifolds, held in Paris, covers the topics of differential manifolds, Riemannian metrics, connections, geodesics and curvature, with special emphasis on the intrinsic features of the subject. Classical results on the relations between curvature and topology are treated in detail. The book is quite self-contained, assuming of the reader only differential calculus in Euclidean space. It contains numerous exercises with full solutions and a series of detailed examples which are picked up repeatedly to illustrate each new definition or property introduced. For this third edition, some topics about the geodesic flow and Lorentzian geometry have been added and worked out in the same spirit.

Riemannian Geometry - Second Edition

Tensors and Their Applications

Tensors and Their Applications

The Book Is Written Is In Easy-To-Read Style With Corresponding Examples. The Main Aim Of This Book Is To Precisely Explain The Fundamentals Of Tensors And Their Applications To Mechanics, Elasticity, Theory Of Relativity, Electromagnetic, Riemannian Geometry And Many Other Disciplines Of Science And Engineering, In A Lucid Manner. The Text Has Been Explained Section Wise, Every Concept Has Been Narrated In The Form Of Definition, Examples And Questions Related To The Concept Taught. The Overall Package Of The Book Is Highly Useful And Interesting For The People Associated With The Field.

Tensors and Their Applications

Fractals and Universal Spaces in Dimension Theory

Fractals and Universal Spaces in Dimension Theory

Historically, for metric spaces the quest for universal spaces in dimension theory spanned approximately a century of mathematical research. The history breaks naturally into two periods – the classical (separable metric) and the modern (not-necessarily separable metric). The classical theory is now well documented in several books. This monograph is the first book to unify the modern theory from 1960-2007. Like the classical theory, the modern theory fundamentally involves the unit interval. Unique features include: * The use of graphics to illustrate the fractal view of these spaces; * Lucid coverage of a range of topics including point-set topology and mapping theory, fractal geometry, and algebraic topology; * A final chapter contains surveys and provides historical context for related research that includes other imbedding theorems, graph theory, and closed imbeddings; * Each chapter contains a comment section that provides historical context with references that serve as a bridge to the literature. This monograph will be useful to topologists, to mathematicians working in fractal geometry, and to historians of mathematics. Being the first monograph to focus on the connection between generalized fractals and universal spaces in dimension theory, it will be a natural text for graduate seminars or self-study – the interested reader will find many relevant open problems which will create further research into these topics.

Fractals and Universal Spaces in Dimension Theory

Current Research Topics on Galois Geometrics

Current Research Topics on Galois Geometrics

Galois geometry is the theory that deals with substructures living in projective spaces over finite fields, also called Galois fields.

Current Research Topics on Galois Geometrics

Algebraic Cycles, Sheaves, Shtukas and Moduli

Algebraic Cycles, Sheaves, Shtukas and Moduli

The articles in this volume are devoted to: – moduli of coherent sheaves; – principal bundles and sheaves and their moduli; – new insights into Geometric Invariant Theory; – stacks of shtukas and their compactifications; – algebraic cycles vs. commutative algebra; – Thom polynomials of singularities; – zero schemes of sections of vector bundles. The main purpose is to give ‘friendly’ introductions to the above topics through a series of comprehensive texts starting from a very elementary level and ending with a discussion of current research. In these texts, the reader will find classical results and methods as well as new ones. The book is addressed to researchers and graduate students in algebraic geometry, algebraic topology and singularity theory. Most of the material presented in the volume has not appeared in books before. Contributors: Jean-Marc Drzet, Toms L. Gmez, Adrian Langer, Piotr Pragacz, Alexander H. W. Schmitt, Vasudevan Srinivas, Ngo Dac Tuan, Andrzej Weber

Algebraic Cycles, Sheaves, Shtukas and Moduli