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About 783 papers for “Computer-assisted proof”

Computer-Assisted Discovery and Proof

2007

With the advent of powerful, widely-available mathematical software, combined with ever-faster computer hardware, we are approaching a day when both the discovery and proof of mathematical facts can be done in a computer-assisted manner. This article presents several specific examples of this new paradigm in action. Kronecker said “In mathematics, I recognize true scientific value only in concrete mathematical truths, or to put it more pointedly, only in mathematical formulas. ” I would rather say “computations ” than “formulas, ” but my view is essentially the same. — Harold Edwards, Essays in Constructive Mathematics, 2004 All truths are easy to understand once they are discovered; the point is to discover them. — attributed to Galileo Galilei 1

Cited by 0

Computer-Assisted Discovery and Proof

David H. Bailey, Jonathan M. Borwein · OSTI OAI (U.S. Department of Energy Office of Scientific and Technical Information) · 2007

With the advent of powerful, widely-available mathematical software, combined with ever-faster computer hardware, we are approaching a day when both the discovery and proof of mathematical facts can be done in a computer-assisted manner. his article presents several specific examples of this new paradigm in action.

Cited by 18Open access

Computer-assisted proof of kernel inequalities

Cohn, Henry, Kumar, Abhinav, Miller, Stephen D. et al. · DSpace@MIT (Massachusetts Institute of Technology) · 2022

This data set provides a computer-assisted proof for the kernel inequalities needed to prove universal optimality in the paper "Universal optimality of the E_8 and Leech lattices and interpolation formulas" (by Cohn, Kumar, Miller, Radchenko, and Viazovska). It includes both our original proof using Mathematica and a revised proof using Sage.

Cited by 0Open access

Computer assisted proof for ordinary differential equations

Mário César Monteiro do Prado · Digital Library of Theses and Dissertations (Universidade de São Paulo) · 2015

Neste trabalho, apresentamos um método computacional rigoroso para a demonstração de existência de órbitas periódicas de alguns sistemas de equações diferenciais ordinárias com campo autônomo do tipo polinomial. Mostraremos que o problema de encontrar órbitas periódicas para esses sistemas de equações é equivalente a buscar por raízes de certas funções definidas no espaço de Banach das sequências com decaimento algébrico. O método pode ser dividido em duas etapas. Na primeira, buscamos numericamente por soluções periódicas aproximadas. Na segunda, mostraremos a existência de uma órbita periódica numa vizinhança da curva encontrada numericamente. O rigor das verificações computacionais é garantido pelo uso de aritimética intervalar.

Cited by 0Open access

A parallelized computer-assisted proof of chaos

Tibor Csendes, José A. Martínez, Leocadio Gonzlez Casado · 2007

We report on experiences with an adaptive subdivision method [1] supported by interval arithmetic and parallel techniques. Our parallel computing system is implemented for a computer, which has four processors and shared memory. The Branch and Bound technique shares the generated subintervals between the processor, hence the calculation is became a faster by more processor. This method enables us to prove subset relations of the form T (W) � U and thus to check certain sufficient conditions for chaotic behavior of dynamical systems in a rigorous way. The novelty is that the process of gaining the subset relations to be checked is, to a large extent, also automatized. The computational methods are demonstrated by finding planar subsets with chaotic behavior for the forced damped pendulum [3], and a Hnon map [1, 2]. This two dynamical systems need a large amount of calculation, so the parallelized technique is very useful. We give details on the parallelization techniques applied.

Cited by 0

A computer-assisted proof that e is rational

Rémi Garcia, Alexandre Goldsztejn · SPIRE - Sciences Po Institutional REpository · 2024

Computers are increasingly being used to help researchers write up mathematical proofs, sometimes by automatically writing out parts of it. In this paper we present different computer-assisted approaches and provide some use cases. Using the Mixed-Integer Linear Programming (MILP) approach, we demonstrate that $e$ is rational, contrary to what has already been proved by hand and with interactive proof assistants. This is the opportunity to dive into the numerical instabilities problems and to discuss limitations of MILP solvers. These tools have some advantages over other approaches, so we propose to explore methods to verify the solutions obtained. To this end, we develop a Julia package to automatically check whether solutions meet the initial requirements or not.

Cited by 0Open access

The Evolution of Computer-Assisted Proof In Analysis

Marek Rychlik · arXiv (Cornell University) · 2026

The intersection of numerical analysis and machine learning, particularly in the domain of Neural ODEs and Physics-Informed Neural Networks (PINNs), relies heavily on discrete approximations of continuous flows. However, in stiff systems, explicit discretization schemes can induce topological bifurcations, creating spurious attractors that do not exist in the underlying continuous dynamics. In this paper, we analyze a stiff 2D nonlinear system integrated via Heun's method, demonstrating how the discrete map undergoes a numerical bifurcation that renders the true equilibrium repelling along an invariant manifold. Adopting a literate programming paradigm where "the paper is the proof," we embed a Computer-Assisted Proof (CAP) directly within the manuscript. Utilizing rigorous complex interval arithmetic and a dimensionality-reducing "Snap-to-Axis" projection, we mathematically verify that a neighborhood of trajectories is permanently captured by a spurious period-4 sink. This work highlights the critical need for formal safety certifications in learned dynamical models and provides a framework for verifying their stability.

Cited by 0Open access

The Evolution of Computer-Assisted Proof In Analysis

Marek Rychlik · arXiv (Cornell University) · 2026

The intersection of numerical analysis and machine learning, particularly in the domain of Neural ODEs and Physics-Informed Neural Networks (PINNs), relies heavily on discrete approximations of continuous flows. However, in stiff systems, explicit discretization schemes can induce topological bifurcations, creating spurious attractors that do not exist in the underlying continuous dynamics. In this paper, we analyze a stiff 2D nonlinear system integrated via Heun's method, demonstrating how the discrete map undergoes a numerical bifurcation that renders the true equilibrium repelling along an invariant manifold. Adopting a literate programming paradigm where "the paper is the proof," we embed a Computer-Assisted Proof (CAP) directly within the manuscript. Utilizing rigorous complex interval arithmetic and a dimensionality-reducing "Snap-to-Axis" projection, we mathematically verify that a neighborhood of trajectories is permanently captured by a spurious period-4 sink. This work highlights the critical need for formal safety certifications in learned dynamical models and provides a framework for verifying their stability.

Cited by 0Open access

A computer-assisted proof of the Feigenbaum conjectures

Oscar E. Lanford · Bulletin of the American Mathematical Society · 1982

even mappings \\p of the interval [-1, 1] into itself such that Ml. i//(0) = 1, M2. JC^'(JC)<0 forx^O. M2 says that \\p is strictly increasing on [-1, 0) and strictly decreasing on (0, 1],

Cited by 358Open access

Computer Assisted Proof for Normally Hyperbolic Invariant Manifolds

Capinski, Maciej J., Simo, Carles · arXiv (Cornell University) · 2011

We present a topological proof of the existence of a normally hyperbolic invariant manifold for maps. In our approach we do not require that the map is a perturbation of some other map for which we already have an invariant manifold. But a non-rigorous, good enough, guess is necessary. The required assumptions are formulated in a way which allows for rigorous computer assisted verification. We apply our method for a driven logistic map, for which non-rigorous numerical simulation in plain double precision suggests the existence of a chaotic attractor. We prove that this numerical evidence is false and that the attractor is a normally hyperbolic invariant curve.

Cited by 0Open access

A Computer-Assisted Proof of the Bellman-Ford Lemma

Peter H. Schmitt · Repository KITopen (Karlsruhe Institute of Technology) · 2011

These notes serve (at least) two purposes. First, they document a proof done with the KeY system of a purely mathematical statement, Lemma 1 below, within the context of Dijkstra’s Shortest Path Algorithm. This is an unusual application of the KeY system that is designed to verify Java programs. The verification of a Java implementation of Dijkstra’s algorithm itself is the topic of the Diploma thesis [10]. Secondly, we use this simple proof exercise to review the widely practiced method to handle partial functions via underspecification that is also used in the KeY system. Little can be found in the literature on the theoretical foundations of this approach. This report proposes a first step towards a theory of underspecification. Particular emphasis is devoted to the axiomatisation of abstract data types with partial functions. As a side issue we also include some comments on conservative extensions.

Cited by 1Open access

Chaos in the Lorenz Equations: a Computer Assisted Proof

Konstantin Mischaikowy · 1994

Abstract: A new technique for obtaining rigorous results concerning the global dynamics of nonlinear systems is described. The technique combines abstract existence results based on the Conley index theory with computer assisted computations. As an application of these methods it is proven that for a explicit parameter value the Lorenz equations exhibit chaotic dynamics.

Cited by 0

COMPUTER ASSISTED PROOF OF SPURIOUS EIGENSOLUTION FOR ANNULAR AND

2013

problem, null-field integral equation. Spurious eigensolutions in the boundary integral equation (BIE) or boundary element method (BEM) for doublyconnected domain problems, the eccentric and annular membranes, are studied analytically and numerically in this paper. For the mathematical analysis, we employ the null-field integral equation, the degenerate kernels and the Fourier series to prove the existence of spurious eigensolutions in the continuous system. Examples of eccentric case, annular membrane and general shape are solved by using the null-field equation approach and BEM, respectively. Base on the numerical experiment, computer-assisted proof for the existence of spurious eigenvalues in comparison with the trivial outer boundary data is given. The spurious eigenvalue is found to be the true eigenvalue of a circular membrane with the inner radius. The SVD structure for the four influence matrices is examined. Also, the trivial outer boundary densities are found in case of spurious eigenvalues and are shown in the bar chart. I.

Cited by 0

Period doubling in the Rossler system - a computer assisted proof

Daniel Wilczak, Piotr Zgliczyński · ArXiv.org · 2007

The goal of this paper is to show how to produce a piece of rigorous bifurcation diagram of periodic orbits for an ODE. We study the Rossler system, one of the textbook examples of ODEs generating nontrivial dynamics, for the parameter range containing two period doubling bifurcations.

Cited by 1Open access