Shape theory is an extension of homotopy theory from the realm of CW-complexes to arbitrary spaces. Besides applications in topology, it has interesting applications in various other areas of mathematics, especially in dynamical systems and C*-algebras. Strong shape is a refinement of ordinary shape with distinct advantages over the latter. Strong homology generalizes Steenrod homology and is an invariant of strong shape. The book gives a detailed account based on approximation of spaces by polyhedra (ANR's) using the technique of inverse systems. It is intended for researchers and graduate students. Special care is devoted to motivation and bibliographic notes.

This volume is a tribute to Maxim Kontsevich, one of the most original and influential mathematicians of our time. Maxim's vision has inspired major developments in many areas of mathematics, ranging all the way from probability theory to motives over finite fields, and has brought forth a paradigm shift at the interface of modern geometry and mathematical physics. Many of his papers have opened completely new directions of research and led to the solutions of many classical problems. This book collects papers by leading experts currently engaged in research on topics close to Maxim's heart. Contributors: S. Donaldson A. Goncharov D. Kaledin M. Kapranov A. Kapustin L. Katzarkov A. Noll P. Pandit S. Pimenov J. Ren P. Seidel C. Simpson Y. Soibelman R. Thorngren

Field Arithmetic explores Diophantine fields through their absolute Galois groups. This largely self-contained treatment starts with techniques from algebraic geometry, number theory, and profinite groups. Graduate students can effectively learn generalizations of finite field ideas. We use Haar measure on the absolute Galois group to replace counting arguments. New Chebotarev density variants interpret diophantine properties. Here we have the only complete treatment of Galois stratifications, used by Denef and Loeser, et al, to study Chow motives of Diophantine statements.

Progress from the first edition starts by characterizing the finite-field like P(seudo)A(lgebraically)C(losed) fields. We once believed PAC fields were rare. Now we know they include valuable Galois extensions of the rationals that present its absolute Galois group through known groups. PAC fields have projective absolute Galois group. Those that are Hilbertian are characterized by this group being pro-free. These last decade results are tools for studying fields by their relation to those with projective absolute group. There are still mysterious problems to guide a new generation: Is the solvable closure of the rationals PAC; and do projective Hilbertian fields have pro-free absolute Galois group (includes Shafarevich's conjecture)?

The third edition improves the second edition in two ways: First it removes many typos and mathematical inaccuracies that occur in the second edition (in particular in the references). Secondly, the third edition reports on five open problems (out of thirtyfour open problems of the second edition) that have been partially or fully solved since that edition appeared in 2005.

This collection documents the work of the Hyperuniverse Project which is a new approach to set-theoretic truth based on justifiable principles and which leads to the resolution of many questions independent from ZFC. The contributions give an overview of the program, illustrate its mathematical content and implications, and also discuss its philosophical assumptions. It will thus be of wide appeal among mathematicians and philosophers with an interest in the foundations of set theory. The Hyperuniverse Project was supported by the John Templeton Foundation from January 2013 until September 2015

"Categorical Perspectives" consists of introductory surveys as well as articles containing original research and complete proofs devoted mainly to the theoretical and foundational developments of category theory and its applications to other fields. A number of articles in the areas of topology, algebra and computer science reflect the varied interests of George Strecker to whom this work is dedicated. Notable also are an exposition of the contributions and importance of George Strecker's research and a survey chapter on general category theory. This work is an excellent reference text for researchers and graduate students in category theory and related areas.

Contributors: H.L. Bentley * G. Castellini * R. El Bashir * H. Herrlich * M. Husek * L. Janos * J. Koslowski * V.A. Lemin * A. Melton * G. Preua * Y.T. Rhineghost * B.S.W. Schroeder * L. Schr"der * G.E. Strecker * A. Zmrzlina"

This volume contains revised papers that were presented at the international workshop entitled Computational Methods for Algebraic Spline Surfaces ("COMPASS"), which was held from September 29 to October 3, 2003, at Schloss Weinberg, Kefermarkt (A- tria). The workshop was mainly devoted to approximate algebraic geometry and its - plications. The organizers wanted to emphasize the novel idea of approximate implici- zation, that has strengthened the existing link between CAD / CAGD (Computer Aided Geometric Design) and classical algebraic geometry. The existing methods for exact implicitization (i. e., for conversion from the parametric to an implicit representation of a curve or surface) require exact arithmetic and are too slow and too expensive for industrial use. Thus the duality of an implicit representation and a parametric repres- tation is only used for low degree algebraic surfaces such as planes, spheres, cylinders, cones and toroidal surfaces. On the other hand, this duality is a very useful tool for - veloping ef?cient algorithms. Approximate implicitization makes this duality available for general curves and surfaces. The traditional exact implicitization of parametric surfaces produce global rep- sentations, which are exact everywhere. The surface patches used in CAD, however, are always de?ned within a small box only; they are obtained for a bounded parameter domain (typically a rectangle, or - in the case of "trimmed" surface patches - a subset of a rectangle). Consequently, a globally exact representation is not really needed in practice."

Feller Semigroups, Bernstein type Operators and Generalized Convexity Associated with Positive Projections.- Gregory's Rational Cubic Splines in Interpolation Subject to Derivative Obstacles.- Interpolation by Splines on Triangulations Oleg Davydov.- On the Use of Quasi-Newton Methods in DAE-Codes.- On the Regularity of Some Differential Operators.- Some Inequalities for Trigonometric Polynomials and their Derivatives.- Inf-Convolution and Radial Basis Functions.- On a Special Property of the Averaged Modulus for Functions of Bounded Variation.- A Simple Approach to the Variational Theory for Interpolation on Spheres.- Constants in Comonotone Polynomial Approximation - A Survey.- Will Ramanujan kill Baker-Gammel-Wills? (A Selective Survey of Pade Approximation).- Approximation Operators of Binomial Type.- Certain Results involving Gammaoperators.- Recent research at Cambridge on radial basis functions.- Representation of quasi-interpolants as differential operators and applications.- Native Hilbert Spaces for Radial Basis Functions I.- Adaptive Approximation with Walsh-similar Functions.- Dual Recurrence and Christoffel-Darboux-Type Formulas for Orthogonal Polynomials.- On Some Problems of Weighted Polynomial Approximation and Interpolation.- Asymptotics of derivatives of orthogonal polynomials based on generalized Jacobi weights. Some new theorems and applications.- List of participants.

This monograph presents a comprehensive introduction to timed automata (TA) and time Petri nets (TPNs) which belong to the most widely used models of real-time systems. Some of the existing methods of translating time Petri nets to timed automata are presented, with a focus on the translations that correspond to the semantics of time Petri nets, associating clocks with various components of the nets.

The Bachelier Society for Mathematical Finance, founded in 1996, held its 1st World Congress in Paris on June 28 to July 1, 2000, thus coinciding in time with the centenary of the thesis defence of Louis Bachelier. In his thesis Bachelier introduced Brownian motion as a tool for the analysis of financial markets as well as the exact definition of options, and this is widely considered the keystone for the emergence of mathematical finance as a scientific discipline. The prestigious list of plenary speakers in Paris included 2 Nobel laureates, Paul Samuelson and Robert Merton. Over 130 further selected talks were given in 3 parallel sessions, all well attended by the over 500 participants who registered from all continents.

For courses in Prealgebra. The Martin-Gay principle: Every student can succeed Elayn Martin-Gay's student-centric approach is woven seamlessly throughout her texts and MyLab courses, giving students the optimal amount of support through effective video resources, an accessible writing style, and study skills support built into the program. Elayn's legacy of innovations that support student success include Chapter Test Prep videos and a Video Organizer note-taking guide. Expanded resources in the latest revision bring even more updates to her program, all shaped by her focus on the student - a perspective that has made her course materials beloved by students and instructors alike. The Martin-Gay series offers market-leading content written by a preeminent author-educator, tightly integrated with the #1 choice in digital learning: MyLab Math. Also available with MyLab Math By combining trusted author content with digital tools and a flexible platform, MyLab personalizes the learning experience and improves results for each student. Bringing Elayn Martin-Gay's voice and approach into the MyLab course - though video resources, study skills support, and exercises refined with each edition - gives students the support to be successful in math. Note: You are purchasing a standalone product; MyLab Math does not come packaged with this content. Students, if interested in purchasing this title with MyLab Math, ask your instructor to confirm the correct package ISBN and Course ID. Instructors, contact your Pearson representative for more information. If you would like to purchase both the physical text and MyLab Math, search for: 0134674189 / 9780134674186 Prealgebra Plus MyLab Math with Pearson eText -- Access Card Package, 6/e Package consists of: 0134707648 / 9780134707648 Prealgebra 0135115795 / 9780135115794 MyLab Math with Pearson eText - Standalone Access Card - for Prealgebra

Self-contained, and collating for the first time material that has until now only been published in journals - often in Russian - this book will be of interest to functional analysts, especially those with interests in topological vector spaces, and to algebraists concerned with category theory. The closed graph theorem is one of the corner stones of functional analysis, both as a tool for applications and as an object for research. However, some of the spaces which arise in applications and for which one wants closed graph theorems are not of the type covered by the classical closed graph theorem of Banach or its immediate extensions. To remedy this, mathematicians such as Schwartz and De Wilde (in the West) and Rajkov (in the East) have introduced new ideas which have allowed them to establish closed graph theorems suitable for some of the desired applications. In this book, Professor Smirnov uses category theory to provide a very general framework, including the situations discussed by De Wilde, Rajkov and others. General properties of the spaces involved are discussed and applications are provided in measure theory, global analysis and differential equations.

This monograph provides a definitive overview of recent advances in the stability and oscillation of autonomous delay differential equations. Topics include linear and nonlinear delay and integrodifferential equations, which have potential applications to both biological and physical dynamic processes. Chapter 1 deals with an analysis of the dynamical characteristics of the delay logistic equation, and a number of techniques and results relating to stability, oscillation and comparison of scalar delay and integrodifferential equations are presented. Chapter 2 provides a tutorial-style introduction to the study of delay-induced Hopf bifurcation to periodicity and the related computations for the analysis of the stability of bifurcating periodic solutions. Chapter 3 is devoted to local analyses of nonlinear model systems and discusses many methods applicable to linear equations and their perturbations. Chapter 4 considers global convergence to equilibrium states of nonlinear systems, and includes oscillations of nonlinear systems about their equilibria. Qualitative analyses of both competitive and cooperative systems with time delays feature in both Chapters 3 and 4. Finally, Chapter 5 deals with recent developments in models of neutral differential equations and their applications to population dynamics. Each chapter concludes with a number of exercises and the overall exposition recommends this volume as a good supplementary text for graduate courses. For mathematicians whose work involves functional differential equations, and whose interest extends beyond the boundaries of linear stability analysis.

This book has been designed to deal with the topics which are indispensable in the advanced age of computer science. The first three chapters cover mathematical logic, sets, relations and function. Next come the chapters on ordered sets, Boolean albegra and switching circuits and matrices. Finally there are individual chapters on combinatorics, discrete numeric functions, generating functinos, recurrence relations, algebraic structures and graph theory; Graphs are binary trees. The purpose of this book is to present principles and concepts of discrete structures as relevant to student learning. The matter has been presented in as simple and lucid manner as possible and a large number of solved examples to understand the concept and principle of the theory have been introduced.

Since 1990 the German Research Society (Deutsche Forschungsgemeinschaft, DFG) has been funding PhD courses (Graduiertenkollegs) at selected universi- ties in the Federal Republic of Germany. TU Berlin has been one of the first universities joining that new funding program of DFG. The PhD courses have been funded over aperiod of 9 years. The grant for the nine years sums up to approximately 5 million DM. Our Grnduiertenkolleg on Communication-based Systems has been assigned to the Computer Science Department of TU Berlin although it is a joined effort of all three universities in Berlin, Technische Uni- versitat (TU), Freie Universitat (FU), and Humboldt Universitat (HU). The Graduiertenkolleg has been started its program in October 1991. The professors responsible for the program are: Hartmut Ehrig (TU), Gunter Hommel (TU), Stefan Jahnichen (TU), Peter Lohr (FU), Miroslaw Malek (RU), Peter Pep- per (TU), Radu Popescu-Zeletin (TU), Herbert Weber (TU), and Adam Wolisz (TU). The Graduiertenkolleg is a PhD program for highly qualified persons in the field of computer science. Twenty scholarships have been granted to fellows of the Graduiertenkolleg for a maximal period of three years. During this time the fellows take part in a selected educational program and work on their PhD thesis.

2. The Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3. Convergence Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ., . . . . 60 4. Complexity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 5. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 A Simple Proof for a Result of Ollerenshaw on Steiner Trees . . . . . . . . . . 68 Xiufeng Du, Ding-Zhu Du, Biao Gao, and Lixue Qii 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 2. In the Euclidean Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3. In the Rectilinear Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4. Discussion . . . . . . . . . . . . -. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 Optimization Algorithms for the Satisfiability (SAT) Problem . . . . . . . . . 72 Jun Gu 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 2. A Classification of SAT Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7:3 3. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV 4. Complete Algorithms and Incomplete Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . 81 5. Optimization: An Iterative Refinement Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 6. Local Search Algorithms for SAT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 7. Global Optimization Algorithms for SAT Problem . . . . . . . . . . . . . . . . . . . . . . . . 106 8. Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 9. Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 10. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Ergodic Convergence in Proximal Point Algorithms with Bregman Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 Osman Guier 1. Introduction . . .: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 2. Convergence for Function Minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 3. Convergence for Arbitrary Maximal Monotone Operators . . . . . . . . . . . .

This essential companion volume to Chaitin's highly successful "The Limits of Mathematics", also published by Springer, gives a brilliant historical survey of the work of this century on the foundations of mathematics, in which the author was a major participant. The Unknowable is a very readable and concrete introduction to Chaitin's ideas, and it includes a detailed explanation of the programming language used by Chaitin in both volumes. It will enable computer users to interact with the author's proofs and discover for themselves how they work. The software for The Unknowable can be downloaded from the author's Web site.

As is known, the book named "Multivariate spline functions and their applications" has been published by the Science Press in 1994. This book is an English edition based on the original book mentioned 1 above with many changes, including that of the structure of a cubic - interpolation in n-dimensional spline spaces, and more detail on triangu- lations have been added in this book. Special cases of multivariate spline functions (such as step functions, polygonal functions, and piecewise polynomials) have been examined math- ematically for a long time. I. J. Schoenberg (Contribution to the problem of application of equidistant data by analytic functions, Quart. Appl. Math., 4(1946), 45 - 99; 112 - 141) and W. Quade & L. Collatz (Zur Interpo- lations theories der reellen periodischen function, Press. Akad. Wiss. (PhysMath. KL), 30(1938), 383- 429) systematically established the the- ory of the spline functions. W. Quade & L. Collatz mainly discussed the periodic functions, while I. J. Schoenberg's work was systematic and com- plete. I. J. Schoenberg outlined three viewpoints for studing univariate splines: Fourier transformations, truncated polynomials and Taylor ex- pansions. Based on the first two viewpoints, I. J. Schoenberg deduced the B-spline function and its basic properties, especially the basis func- tions. Based on the latter viewpoint, he represented the spline functions in terms of truncated polynomials. These viewpoints and methods had significantly effected on the development of the spline functions.

Studies in Logic and the Foundations of Mathematics, Volume 123: Constructivism in Mathematics: An Introduction, Vol. II focuses on various studies in mathematics and logic, including metric spaces, polynomial rings, and Heyting algebras. The publication first takes a look at the topology of metric spaces, algebra, and finite-type arithmetic and theories of operators. Discussions focus on intuitionistic finite-type arithmetic, theories of operators and classes, rings and modules, linear algebra, polynomial rings, fields and local rings, complete separable metric spaces, and located sets. The text then examines proof theory of intuitionistic logic, theory of types and constructive set theory, and choice sequences. The book elaborates on semantical completeness, sheaves, sites, and higher-order logic, and applications of sheaf models. Topics include a derived rule of local continuity, axiom of countable choice, forcing over sites, sheaf models for higher-order logic, and complete Heyting algebras. The publication is a valuable reference for mathematicians and researchers interested in mathematics and logic.

This is a two-volume collection presenting the selected works of Herbert Busemann, one of the leading geometers of the twentieth century and one of the main founders of metric geometry, convexity theory and convexity in metric spaces. Busemann also did substantial work (probably the most important) on Hilbert's Problem IV. These collected works include Busemann's most important published articles on these topics. Volume I of the collection features Busemann's papers on the foundations of geodesic spaces and on the metric geometry of Finsler spaces. Volume II includes Busemann's papers on convexity and integral geometry, on Hilbert's Problem IV, and other papers on miscellaneous subjects. Each volume offers biographical documents and introductory essays on Busemann's work, documents from his correspondence and introductory essays written by leading specialists on Busemann's work. They are a valuable resource for researchers in synthetic and metric geometry, convexity theory and the foundations of geometry.