Forcing with Random Variables and Proof Complexity

Author: Jan Krajíček
Publisher: Cambridge University Press
ISBN: 9781139493925
Release Date: 2010-12-23
Genre: Mathematics

This book introduces a new approach to building models of bounded arithmetic, with techniques drawn from recent results in computational complexity. Propositional proof systems and bounded arithmetics are closely related. In particular, proving lower bounds on the lengths of proofs in propositional proof systems is equivalent to constructing certain extensions of models of bounded arithmetic. This offers a clean and coherent framework for thinking about lower bounds for proof lengths, and it has proved quite successful in the past. This book outlines a brand new method for constructing models of bounded arithmetic, thus for proving independence results and establishing lower bounds for proof lengths. The models are built from random variables defined on a sample space which is a non-standard finite set and sampled by functions of some restricted computational complexity. It will appeal to anyone interested in logical approaches to fundamental problems in complexity theory.

Logical Approaches to Computational Barriers

Author: Arnold Beckmann
Publisher: Springer Science & Business Media
ISBN: 9783540354666
Release Date: 2006-06-26
Genre: Computers

This book constitutes the refereed proceedings of the Second International Conference on Computability in Europe, CiE 2006, held in Swansea, UK, in June/July 2006. The 31 revised full papers presented together with 30 invited papers were carefully reviewed and selected from about 80 submissions. Among them are papers corresponding to 8 plenary talks and papers of 6 special sessions entitled proofs and computation, computable analysis, challenges in complexity, foundations of programming, mathematical models of computers and hypercomputers, and G??del centenary: G??del's legacy for computability.

Complexity Science

Author: Robin Ball
Publisher: Cambridge University Press
ISBN: 9781107640566
Release Date: 2013-11-21
Genre: Mathematics

This book presents introductions to the essential mathematical aspects of complexity science, suitable for advanced undergraduate/masters-level students and researchers.

Economics for Mathematicians

Author: John William Scott Cassels
Publisher: Cambridge University Press
ISBN: 9780521286145
Release Date: 1981-12-10
Genre: Business & Economics

This is the expanded notes of a course intended to introduce students specializing in mathematics to some of the central ideas of traditional economics. The book should be readily accessible to anyone with some training in university mathematics; more advanced mathematical tools are explained in the appendices. Thus this text could be used for undergraduate mathematics courses or as supplementary reading for students of mathematical economics.

Advances in Elliptic Curve Cryptography

Author: Ian F. Blake
Publisher: Cambridge University Press
ISBN: 1139441221
Release Date: 2005-04-25
Genre: Mathematics

Since the appearance of the authors' first volume on elliptic curve cryptography in 1999 there has been tremendous progress in the field. In some topics, particularly point counting, the progress has been spectacular. Other topics such as the Weil and Tate pairings have been applied in new and important ways to cryptographic protocols that hold great promise. Notions such as provable security, side channel analysis and the Weil descent technique have also grown in importance. This second volume addresses these advances and brings the reader up to date. Prominent contributors to the research literature in these areas have provided articles that reflect the current state of these important topics. They are divided into the areas of protocols, implementation techniques, mathematical foundations and pairing based cryptography. Each of the topics is presented in an accessible, coherent and consistent manner for a wide audience that will include mathematicians, computer scientists and engineers.

Women in Ancient Greece and Rome

Author: Michael Massey
Publisher: Cambridge University Press
ISBN: 0521318076
Release Date: 1988-09-29
Genre: Juvenile Nonfiction

Michael Massey's thorough and provocative account is an ideal resource book for students following courses in classical studies and classical civilization. It draws on a wide range of contemporary quotations and is well illustrated with examples of original painting and sculpture. The book is divided into two sections--on Greece and on Rome--with material organized under thematic headings, ranging from religion to exploitation, entertainment to employment.

Martingales in Banach Spaces

Author: Gilles Pisier
Publisher: Cambridge University Press
ISBN: 9781316679463
Release Date: 2016-06-06
Genre: Mathematics

This book focuses on the major applications of martingales to the geometry of Banach spaces, and a substantial discussion of harmonic analysis in Banach space valued Hardy spaces is also presented. It covers exciting links between super-reflexivity and some metric spaces related to computer science, as well as an outline of the recently developed theory of non-commutative martingales, which has natural connections with quantum physics and quantum information theory. Requiring few prerequisites and providing fully detailed proofs for the main results, this self-contained study is accessible to graduate students with a basic knowledge of real and complex analysis and functional analysis. Chapters can be read independently, with each building from the introductory notes, and the diversity of topics included also means this book can serve as the basis for a variety of graduate courses.

Mathematical Analysis of Evolution Information and Complexity

Author: Wolfgang Arendt
Publisher: John Wiley & Sons
ISBN: 9783527628032
Release Date: 2009-07-10
Genre: Science

Mathematical Analysis of Evolution, Information, and Complexity deals with the analysis of evolution, information and complexity. The time evolution of systems or processes is a central question in science, this text covers a broad range of problems including diffusion processes, neuronal networks, quantum theory and cosmology. Bringing together a wide collection of research in mathematics, information theory, physics and other scientific and technical areas, this new title offers elementary and thus easily accessible introductions to the various fields of research addressed in the book.

Lectures on the Coupling Method

Author: Torgny Lindvall
Publisher: Courier Corporation
ISBN: 9780486153247
Release Date: 2012-08-15
Genre: Mathematics

Practical and easy-to-use reference progresses from simple to advanced topics, covering, among other topics, renewal theory, Markov chains, Poisson approximation, ergodicity, and Strassen's theorem. 1992 edition.

Extremes and Recurrence in Dynamical Systems

Author: Valerio Lucarini
Publisher: John Wiley & Sons
ISBN: 9781118632192
Release Date: 2016-04-25
Genre: Mathematics

Written by a team of international experts, Extremes and Recurrence in Dynamical Systems presents a unique point of view on the mathematical theory of extremes and on its applications in the natural and social sciences. Featuring an interdisciplinary approach to new concepts in pure and applied mathematical research, the book skillfully combines the areas of statistical mechanics, probability theory, measure theory, dynamical systems, statistical inference, geophysics, and software application. Emphasizing the statistical mechanical point of view, the book introduces robust theoretical embedding for the application of extreme value theory in dynamical systems. Extremes and Recurrence in Dynamical Systems also features: • A careful examination of how a dynamical system can serve as a generator of stochastic processes • Discussions on the applications of statistical inference in the theoretical and heuristic use of extremes • Several examples of analysis of extremes in a physical and geophysical context • A final summary of the main results presented along with a guide to future research projects • An appendix with software in Matlab® programming language to help readers to develop further understanding of the presented concepts Extremes and Recurrence in Dynamical Systems is ideal for academics and practitioners in pure and applied mathematics, probability theory, statistics, chaos, theoretical and applied dynamical systems, statistical mechanics, geophysical fluid dynamics, geosciences and complexity science. VALERIO LUCARINI, PhD, is Professor of Theoretical Meteorology at the University of Hamburg, Germany and Professor of Statistical Mechanics at the University of Reading, UK. DAVIDE FARANDA, PhD, is Researcher at the Laboratoire des science du climat et de l’environnement, IPSL, CEA Saclay, Université Paris-Saclay, Gif-sur-Yvette, France. ANA CRISTINA GOMES MONTEIRO MOREIRA DE FREITAS, PhD, is Assistant Professor in the Faculty of Economics at the University of Porto, Portugal. JORGE MIGUEL MILHAZES DE FREITAS, PhD, is Assistant Professor in the Department of Mathematics of the Faculty of Sciences at the University of Porto, Portugal. MARK HOLLAND, PhD, is Senior Lecturer in Applied Mathematics in the College of Engineering, Mathematics and Physical Sciences at the University of Exeter, UK. TOBIAS KUNA, PhD, is Associate Professor in the Department of Mathematics and Statistics at the University of Reading, UK. MATTHEW NICOL, PhD, is Professor of Mathematics at the University of Houston, USA. MIKE TODD, PhD, is Lecturer in the School of Mathematics and Statistics at the University of St. Andrews, Scotland. SANDRO VAIENTI, PhD, is Professor of Mathematics at the University of Toulon and Researcher at the Centre de Physique Théorique, France.

Iterative Methods for Linear and Nonlinear Equations

Author: C. T. Kelley
Publisher: SIAM
ISBN: 1611970946
Release Date: 1995
Genre: Iterative methods (Mathematics)

Linear and nonlinear systems of equations are the basis for many, if not most, of the models of phenomena in science and engineering, and their efficient numerical solution is critical to progress in these areas. This is the first book to be published on nonlinear equations since the mid-1980s. Although it stresses recent developments in this area, such as Newton-Krylov methods, considerable material on linear equations has been incorporated. This book focuses on a small number of methods and treats them in depth. The author provides a complete analysis of the conjugate gradient and generalized minimum residual iterations as well as recent advances including Newton-Krylov methods, incorporation of inexactness and noise into the analysis, new proofs and implementations of Broyden's method, and globalization of inexact Newton methods. Examples, methods, and algorithmic choices are based on applications to infinite dimensional problems such as partial differential equations and integral equations. The analysis and proof techniques are constructed with the infinite dimensional setting in mind and the computational examples and exercises are based on the MATLAB environment.