Computer Assisted Proof For Mathematics
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Computer Assisted Proof
Author: Fouad Sabry
language: en
Publisher: One Billion Knowledgeable
Release Date: 2023-07-06
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 Assisted Proof for Mathematics
Abstract: "We give brief account of the use of computers to help us develop mathematical proofs, acting as a clerical assistant with knowledge of logical rules. The paper then focusses on one such system, Pollack's LEGO, based on the Calculus of Constructions, and it shows how this may be used to define mathematical concepts and express proofs. We aim at a gentle introduction, rather than a technical exposition."
Numerical Verification Methods and Computer-Assisted Proofs for Partial Differential Equations
Author: Mitsuhiro T. Nakao
language: en
Publisher: Springer Nature
Release Date: 2019-11-11
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.