Computer Aided Proofs In Analysis

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Computer Aided Proofs in Analysis

Author : Kenneth R. Meyer
Publisher : Springer Science & Business Media
ISBN 13 : 1461390923
Total Pages : 264 pages
Book Rating : 4.4/5 (613 download)

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Book Synopsis Computer Aided Proofs in Analysis by : Kenneth R. Meyer

Download or read book Computer Aided Proofs in Analysis written by Kenneth R. Meyer and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 264 pages. Available in PDF, EPUB and Kindle. Book excerpt: This IMA Volume in Mathematics and its Applications COMPUTER AIDED PROOFS IN ANALYSIS is based on the proceedings of an IMA Participating Institutions (PI) Conference held at the University of Cincinnati in April 1989. Each year the 19 Participating Institutions select, through a competitive process, several conferences proposals from the PIs, for partial funding. This conference brought together leading figures in a number of fields who were interested in finding exact answers to problems in analysis through computer methods. We thank Kenneth Meyer and Dieter Schmidt for organizing the meeting and editing the proceedings. A vner Friedman Willard Miller, Jr. PREFACE Since the dawn of the computer revolution the vast majority of scientific compu tation has dealt with finding approximate solutions of equations. However, during this time there has been a small cadre seeking precise solutions of equations and rigorous proofs of mathematical results. For example, number theory and combina torics have a long history of computer-assisted proofs; such methods are now well established in these fields. In analysis the use of computers to obtain exact results has been fragmented into several schools.

On Some Computer-aided Proofs in Analysis

Author : Tomas Johnson
Publisher :
ISBN 13 :
Total Pages : 14 pages
Book Rating : 4.:/5 (495 download)

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Book Synopsis On Some Computer-aided Proofs in Analysis by : Tomas Johnson

Download or read book On Some Computer-aided Proofs in Analysis written by Tomas Johnson and published by . This book was released on 2007 with total page 14 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Numerical Verification Methods and Computer-Assisted Proofs for Partial Differential Equations

Author : Mitsuhiro T. Nakao
Publisher : Springer Nature
ISBN 13 : 9811376697
Total Pages : 469 pages
Book Rating : 4.8/5 (113 download)

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Book Synopsis Numerical Verification Methods and Computer-Assisted Proofs for Partial Differential Equations by : Mitsuhiro T. Nakao

Download or read book Numerical Verification Methods and Computer-Assisted Proofs for Partial Differential Equations written by Mitsuhiro T. Nakao and published by Springer Nature. This book was released on 2019-11-11 with total page 469 pages. Available in PDF, EPUB and Kindle. Book excerpt: In the last decades, various mathematical problems have been solved by computer-assisted proofs, among them the Kepler conjecture, the existence of chaos, the existence of the Lorenz attractor, the famous four-color problem, and more. In many cases, computer-assisted proofs have the remarkable advantage (compared with a “theoretical” proof) of additionally providing accurate quantitative information. The authors have been working more than a quarter century to establish methods for the verified computation of solutions for partial differential equations, mainly for nonlinear elliptic problems of the form -∆u=f(x,u,∇u) with Dirichlet boundary conditions. Here, by “verified computation” is meant a computer-assisted numerical approach for proving the existence of a solution in a close and explicit neighborhood of an approximate solution. The quantitative information provided by these techniques is also significant from the viewpoint of a posteriori error estimates for approximate solutions of the concerned partial differential equations in a mathematically rigorous sense. In this monograph, the authors give a detailed description of the verified computations and computer-assisted proofs for partial differential equations that they developed. In Part I, the methods mainly studied by the authors Nakao and Watanabe are presented. These methods are based on a finite dimensional projection and constructive a priori error estimates for finite element approximations of the Poisson equation. In Part II, the computer-assisted approaches via eigenvalue bounds developed by the author Plum are explained in detail. The main task of this method consists of establishing eigenvalue bounds for the linearization of the corresponding nonlinear problem at the computed approximate solution. Some brief remarks on other approaches are also given in Part III. Each method in Parts I and II is accompanied by appropriate numerical examples that confirm the actual usefulness of the authors’ methods. Also in some examples practical computer algorithms are supplied so that readers can easily implement the verification programs by themselves.

Computer-assisted Proofs in Analysis and Programming in Logic

Author : Hans Koch
Publisher :
ISBN 13 :
Total Pages : 49 pages
Book Rating : 4.:/5 (66 download)

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Book Synopsis Computer-assisted Proofs in Analysis and Programming in Logic by : Hans Koch

Download or read book Computer-assisted Proofs in Analysis and Programming in Logic written by Hans Koch and published by . This book was released on 1994 with total page 49 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Computer Assisted Proofs in Analysis

Author : O. E. Lanford III
Publisher :
ISBN 13 :
Total Pages : 8 pages
Book Rating : 4.:/5 (897 download)

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Book Synopsis Computer Assisted Proofs in Analysis by : O. E. Lanford III

Download or read book Computer Assisted Proofs in Analysis written by O. E. Lanford III and published by . This book was released on 1987 with total page 8 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Fundamental Proof Methods in Computer Science

Author : Konstantine Arkoudas
Publisher : MIT Press
ISBN 13 : 0262342502
Total Pages : 1223 pages
Book Rating : 4.2/5 (623 download)

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Book Synopsis Fundamental Proof Methods in Computer Science by : Konstantine Arkoudas

Download or read book Fundamental Proof Methods in Computer Science written by Konstantine Arkoudas and published by MIT Press. This book was released on 2017-04-28 with total page 1223 pages. Available in PDF, EPUB and Kindle. Book excerpt: A textbook that teaches students to read and write proofs using Athena. Proof is the primary vehicle for knowledge generation in mathematics. In computer science, proof has found an additional use: verifying that a particular system (or component, or algorithm) has certain desirable properties. This book teaches students how to read and write proofs using Athena, a freely downloadable computer language. Athena proofs are machine-checkable and written in an intuitive natural-deduction style. The book contains more than 300 exercises, most with full solutions. By putting proofs into practice, it demonstrates the fundamental role of logic and proof in computer science as no other existing text does. Guided by examples and exercises, students are quickly immersed in the most useful high-level proof methods, including equational reasoning, several forms of induction, case analysis, proof by contradiction, and abstraction/specialization. The book includes auxiliary material on SAT and SMT solving, automated theorem proving, and logic programming. The book can be used by upper undergraduate or graduate computer science students with a basic level of programming and mathematical experience. Professional programmers, practitioners of formal methods, and researchers in logic-related branches of computer science will find it a valuable reference.

Computer Assisted Proof

Author : Fouad Sabry
Publisher : One Billion Knowledgeable
ISBN 13 :
Total Pages : 111 pages
Book Rating : 4.:/5 (661 download)

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Book Synopsis Computer Assisted Proof by : Fouad Sabry

Download or read book Computer Assisted Proof written by Fouad Sabry and published by One Billion Knowledgeable. This book was released on 2023-07-06 with total page 111 pages. Available in PDF, EPUB and Kindle. Book excerpt: What Is Computer Assisted Proof A mathematical proof is considered to be computer-assisted if it has been generated by the computer in some way, even if just in part. How You Will Benefit (I) Insights, and validations about the following topics: Chapter 1: Computer-assisted proof Chapter 2: Mathematical proof Chapter 3: Theorem Chapter 4: Metamath Chapter 5: Model checking Chapter 6: Computer algebra Chapter 7: Formal verification Chapter 8: Validated numerics Chapter 9: Logic Theorist Chapter 10: Seventeen or Bust (II) Answering the public top questions about computer assisted proof. (III) Real world examples for the usage of computer assisted proof in many fields. (IV) 17 appendices to explain, briefly, 266 emerging technologies in each industry to have 360-degree full understanding of computer assisted proof' technologies. Who This Book Is For Professionals, undergraduate and graduate students, enthusiasts, hobbyists, and those who want to go beyond basic knowledge or information for any kind of computer assisted proof.

Computer-Aided Reasoning

Author : Matt Kaufmann
Publisher : Springer Science & Business Media
ISBN 13 : 1475731884
Total Pages : 337 pages
Book Rating : 4.4/5 (757 download)

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Book Synopsis Computer-Aided Reasoning by : Matt Kaufmann

Download or read book Computer-Aided Reasoning written by Matt Kaufmann and published by Springer Science & Business Media. This book was released on 2013-04-17 with total page 337 pages. Available in PDF, EPUB and Kindle. Book excerpt: Computer-Aided Reasoning: ACL2 Case Studies illustrates how the computer-aided reasoning system ACL2 can be used in productive and innovative ways to design, build, and maintain hardware and software systems. Included here are technical papers written by twenty-one contributors that report on self-contained case studies, some of which are sanitized industrial projects. The papers deal with a wide variety of ideas, including floating-point arithmetic, microprocessor simulation, model checking, symbolic trajectory evaluation, compilation, proof checking, real analysis, and several others. Computer-Aided Reasoning: ACL2 Case Studies is meant for two audiences: those looking for innovative ways to design, build, and maintain hardware and software systems faster and more reliably, and those wishing to learn how to do this. The former audience includes project managers and students in survey-oriented courses. The latter audience includes students and professionals pursuing rigorous approaches to hardware and software engineering or formal methods. Computer-Aided Reasoning: ACL2 Case Studies can be used in graduate and upper-division undergraduate courses on Software Engineering, Formal Methods, Hardware Design, Theory of Computation, Artificial Intelligence, and Automated Reasoning. The book is divided into two parts. Part I begins with a discussion of the effort involved in using ACL2. It also contains a brief introduction to the ACL2 logic and its mechanization, which is intended to give the reader sufficient background to read the case studies. A more thorough, textbook introduction to ACL2 may be found in the companion book, Computer-Aided Reasoning: An Approach. The heart of the book is Part II, where the case studies are presented. The case studies contain exercises whose solutions are on the Web. In addition, the complete ACL2 scripts necessary to formalize the models and prove all the properties discussed are on the Web. For example, when we say that one of the case studies formalizes a floating-point multiplier and proves it correct, we mean that not only can you read an English description of the model and how it was proved correct, but you can obtain the entire formal content of the project and replay the proofs, if you wish, with your copy of ACL2. ACL2 may be obtained from its home page. The results reported in each case study, as ACL2 input scripts, as well as exercise solutions for both books, are available from this page.

Computer Arithmetic and Formal Proofs

Author : Sylvie Boldo
Publisher : Elsevier
ISBN 13 : 0081011709
Total Pages : 326 pages
Book Rating : 4.0/5 (81 download)

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Book Synopsis Computer Arithmetic and Formal Proofs by : Sylvie Boldo

Download or read book Computer Arithmetic and Formal Proofs written by Sylvie Boldo and published by Elsevier. This book was released on 2017-11-17 with total page 326 pages. Available in PDF, EPUB and Kindle. Book excerpt: Floating-point arithmetic is ubiquitous in modern computing, as it is the tool of choice to approximate real numbers. Due to its limited range and precision, its use can become quite involved and potentially lead to numerous failures. One way to greatly increase confidence in floating-point software is by computer-assisted verification of its correctness proofs. This book provides a comprehensive view of how to formally specify and verify tricky floating-point algorithms with the Coq proof assistant. It describes the Flocq formalization of floating-point arithmetic and some methods to automate theorem proofs. It then presents the specification and verification of various algorithms, from error-free transformations to a numerical scheme for a partial differential equation. The examples cover not only mathematical algorithms but also C programs as well as issues related to compilation. Describes the notions of specification and weakest precondition computation and their practical use Shows how to tackle algorithms that extend beyond the realm of simple floating-point arithmetic Includes real analysis and a case study about numerical analysis

Lectures on Finite Precision Computations

Author : Francoise Chaitin-Chatelin
Publisher : SIAM
ISBN 13 : 9780898719673
Total Pages : 251 pages
Book Rating : 4.7/5 (196 download)

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Book Synopsis Lectures on Finite Precision Computations by : Francoise Chaitin-Chatelin

Download or read book Lectures on Finite Precision Computations written by Francoise Chaitin-Chatelin and published by SIAM. This book was released on 1996-01-01 with total page 251 pages. Available in PDF, EPUB and Kindle. Book excerpt: Finite precision computations are at the heart of the daily activities of many engineers and researchers in all branches of applied mathematics. Written in an informal style, the book combines techniques from engineering and mathematics to describe the rigorous and novel theory of computability in finite precision. In the challenging cases of nonlinear problems, theoretical analysis is supplemented by software tools to explore the stability on the computer.

Validated Numerics

Author : Warwick Tucker
Publisher : Princeton University Press
ISBN 13 : 069124765X
Total Pages : 152 pages
Book Rating : 4.6/5 (912 download)

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Book Synopsis Validated Numerics by : Warwick Tucker

Download or read book Validated Numerics written by Warwick Tucker and published by Princeton University Press. This book was released on 2023-06-06 with total page 152 pages. Available in PDF, EPUB and Kindle. Book excerpt: A comprehensive, self-contained primer on validated numerics This textbook provides a comprehensive introduction to the theory and practice of validated numerics, an emerging new field that combines the strengths of scientific computing and pure mathematics. In numerous fields ranging from pharmaceutics and engineering to weather prediction and robotics, fast and precise computations are essential. Based on the theory of set-valued analysis, a new suite of numerical methods is developed, producing efficient and reliable solvers for numerous problems in nonlinear analysis. Validated numerics yields rigorous computations that can find all possible solutions to a problem while taking into account all possible sources of error—fast, and with guaranteed accuracy. Validated Numerics offers a self-contained primer on the subject, guiding readers from the basics to more advanced concepts and techniques. This book is an essential resource for those entering this fast-developing field, and it is also the ideal textbook for graduate students and advanced undergraduates needing an accessible introduction to the subject. Validated Numerics features many examples, exercises, and computer labs using MATLAB/C++, as well as detailed appendixes and an extensive bibliography for further reading. Provides a comprehensive, self-contained introduction to validated numerics Requires no advanced mathematics or programming skills Features many examples, exercises, and computer labs Includes code snippets that illustrate implementation Suitable as a textbook for graduate students and advanced undergraduates

Accuracy and Reliability in Scientific Computing

Author : Bo Einarsson
Publisher : SIAM
ISBN 13 : 0898715849
Total Pages : 348 pages
Book Rating : 4.8/5 (987 download)

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Book Synopsis Accuracy and Reliability in Scientific Computing by : Bo Einarsson

Download or read book Accuracy and Reliability in Scientific Computing written by Bo Einarsson and published by SIAM. This book was released on 2005-08-01 with total page 348 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book investigates some of the difficulties related to scientific computing, describing how these can be overcome.

A Computer-Assisted Proof of Universality for Area-Preserving Maps

Author : Jean Pierre Eckmann
Publisher : American Mathematical Soc.
ISBN 13 : 0821822896
Total Pages : 131 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis A Computer-Assisted Proof of Universality for Area-Preserving Maps by : Jean Pierre Eckmann

Download or read book A Computer-Assisted Proof of Universality for Area-Preserving Maps written by Jean Pierre Eckmann and published by American Mathematical Soc.. This book was released on 1984 with total page 131 pages. Available in PDF, EPUB and Kindle. Book excerpt: We study iterates of area-preserving maps as the simplest examples of conservative dynamical systems.

The Princeton Companion to Applied Mathematics

Author : Nicholas J. Higham
Publisher : Princeton University Press
ISBN 13 : 1400874475
Total Pages : 1031 pages
Book Rating : 4.4/5 (8 download)

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Book Synopsis The Princeton Companion to Applied Mathematics by : Nicholas J. Higham

Download or read book The Princeton Companion to Applied Mathematics written by Nicholas J. Higham and published by Princeton University Press. This book was released on 2015-09-15 with total page 1031 pages. Available in PDF, EPUB and Kindle. Book excerpt: The must-have compendium on applied mathematics This is the most authoritative and accessible single-volume reference book on applied mathematics. Featuring numerous entries by leading experts and organized thematically, it introduces readers to applied mathematics and its uses; explains key concepts; describes important equations, laws, and functions; looks at exciting areas of research; covers modeling and simulation; explores areas of application; and more. Modeled on the popular Princeton Companion to Mathematics, this volume is an indispensable resource for undergraduate and graduate students, researchers, and practitioners in other disciplines seeking a user-friendly reference book on applied mathematics. Features nearly 200 entries organized thematically and written by an international team of distinguished contributors Presents the major ideas and branches of applied mathematics in a clear and accessible way Explains important mathematical concepts, methods, equations, and applications Introduces the language of applied mathematics and the goals of applied mathematical research Gives a wide range of examples of mathematical modeling Covers continuum mechanics, dynamical systems, numerical analysis, discrete and combinatorial mathematics, mathematical physics, and much more Explores the connections between applied mathematics and other disciplines Includes suggestions for further reading, cross-references, and a comprehensive index

Computer Aided Verification

Author : Tayssir Touili
Publisher : Springer
ISBN 13 : 3642142958
Total Pages : 691 pages
Book Rating : 4.6/5 (421 download)

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Book Synopsis Computer Aided Verification by : Tayssir Touili

Download or read book Computer Aided Verification written by Tayssir Touili and published by Springer. This book was released on 2010-07-09 with total page 691 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the refereed proceedings of the 22nd International Conference on Computer Aided Verification, CAV 2010, held in Edinburgh, UK, in July 2010 as part of the Federated Logic Conference, FLoC 2010. The 34 revised full papers presented together with 17 tool papers, 4 invited talks and 3 invited tutorials were carefully reviewed and selected from 101 regular paper and 44 tool paper submissions. The papers are dedicated to the advancement of the theory and practice of computer-assisted formal analysis methods for hardware and software systems. They are organized in topical sections on software model checking; model checking and automata; tools; counter and hybrid systems verification; memory consistency; verification of hardware and low level code; synthesis; concurrent program verification; compositional reasoning; and decision procedures.

Rigorous Numerics in Dynamics

Author : Jan Bouwe van den Berg
Publisher : American Mathematical Soc.
ISBN 13 : 1470428148
Total Pages : 224 pages
Book Rating : 4.4/5 (74 download)

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Book Synopsis Rigorous Numerics in Dynamics by : Jan Bouwe van den Berg

Download or read book Rigorous Numerics in Dynamics written by Jan Bouwe van den Berg and published by American Mathematical Soc.. This book was released on 2018-07-12 with total page 224 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume is based on lectures delivered at the 2016 AMS Short Course “Rigorous Numerics in Dynamics”, held January 4–5, 2016, in Seattle, Washington. Nonlinear dynamics shapes the world around us, from the harmonious movements of celestial bodies, via the swirling motions in fluid flows, to the complicated biochemistry in the living cell. Mathematically these phenomena are modeled by nonlinear dynamical systems, in the form of ODEs, PDEs and delay equations. The presence of nonlinearities complicates the analysis, and the difficulties are even greater for PDEs and delay equations, which are naturally defined on infinite dimensional function spaces. With the availability of powerful computers and sophisticated software, numerical simulations have quickly become the primary tool to study the models. However, while the pace of progress increases, one may ask: just how reliable are our computations? Even for finite dimensional ODEs, this question naturally arises if the system under study is chaotic, as small differences in initial conditions (such as those due to rounding errors in numerical computations) yield wildly diverging outcomes. These issues have motivated the development of the field of rigorous numerics in dynamics, which draws inspiration from ideas in scientific computing, numerical analysis and approximation theory. The articles included in this volume present novel techniques for the rigorous study of the dynamics of maps via the Conley-index theory; periodic orbits of delay differential equations via continuation methods; invariant manifolds and connecting orbits; the dynamics of models with unknown nonlinearities; and bifurcations diagrams.

Computer Aided Verification

Author : Ed Brinksma
Publisher : Springer Science & Business Media
ISBN 13 : 3540439978
Total Pages : 645 pages
Book Rating : 4.5/5 (44 download)

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Book Synopsis Computer Aided Verification by : Ed Brinksma

Download or read book Computer Aided Verification written by Ed Brinksma and published by Springer Science & Business Media. This book was released on 2002-07-19 with total page 645 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains the proceedings of the conference on Computer Aided V- i?cation (CAV 2002), held in Copenhagen, Denmark on July 27-31, 2002. CAV 2002 was the 14th in a series of conferences dedicated to the advancement of the theory and practice of computer-assisted formal analysis methods for software and hardware systems. The conference covers the spectrum from theoretical - sults to concrete applications, with an emphasis on practical veri?cation tools, including algorithms and techniques needed for their implementation. The c- ference has traditionally drawn contributions from researchers as well as prac- tioners in both academia and industry. This year we received 94 regular paper submissions out of which 35 were selected. Each submission received an average of 4 referee reviews. In addition, the CAV program contained 11 tool presentations selected from 16 submissions. For each tool presentation, a demo was given at the conference. The large number of tool submissions and presentations testi?es to the liveliness of the ?eld and its applied ?avor.