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1	Of Proofs, Mathematicians, and Computers Yepremyan, Astrik 01 January 2015 (has links) As computers become a more prevalent commodity in mathematical research and mathematical proof, the question of whether or not a computer assisted proof can be considered a mathematical proof has become an ongoing topic of discussion in the mathematics community. The use of the computer in mathematical research leads to several implications about mathematics in the present day including the notion that mathematical proof can be based on empirical evidence, and that some mathematical conclusions can be achieved a posteriori instead of a priori, as most mathematicians have done before. While some mathematicians are open to the idea of a computer-assisted proof, others are skeptical and would feel more comfortable if presented with a more traditional proof, as it is more surveyable. A surveyable proof enables mathematicians to see the validity of a proof, which is paramount for mathematical growth, and offer critique. In my thesis, I will present the role that the mathematical proof plays within the mathematical community, and thereby conclude that because of the dynamics of the mathematical community and the constant activity of proving, the risks that are associated with a mistake that stems from a computer-assisted proof can be caught by the scrupulous activity of peer review in the mathematics community. Eventually, as the following generations of mathematicians become more trained in using computers and in computer programming, they will be able to better use computers in producing evidence, and in turn, other mathematicians will be able to both understand and trust the resultant proof. Therefore, it remains that whether or not a proof was achieved by a priori or a posteriori, the validity of a proof will be determined by the correct logic behind it, as well as its ability to convince the members of the mathematical community—not on whether the result was reached a priori with a traditional proof, or a posteriori with a computer-assisted proof. mathematical proof computer-assisted proof four-color theorem philosophy of mathematics mathematics community Other Mathematics Philosophy of Science
2	Lineární kódy a projektivní rovina řádu 10 / Linear codes and a projective plane of order 10 Liška, Ondřej January 2013 (has links) Projective plane of order 10 does not exist. Proof of this assertion was finished in 1989 and is based on the nonexistence of a binary code C generated by the incidence vectors of the plane's lines. As part of the proof of the nonexistence of code C, the coefficients of its weight enumerator were studied. It was shown that coefficients A12, A15, A16 and A19 have to be equal to zero, which contradicted other findings about the relationship among the coefficients. Presented diploma thesis elaborately analyses the phases of the proof and, in several places, enhances them with new observations and simplifications. Part of the proof is generalized for projective planes of order 8m + 2. 1
3	Multiplicidade exata de soluções de equações diferenciais via um método assistido por computador / Computer assisted proof for ordinary differential equations Prado, Mário César Monteiro do 15 May 2019 (has links) 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. / In this work, we present a rigorous computational method for proving the existence of periodic orbits of some systems of ordinary differential equations with autonomous vector field of polynomial type. We show that the problem of finding periodic orbits for these systems is equivalent to check for roots of certain functions defined in the Banach space of sequences with algebraic decay. The method can be divided into two steps. First, we seek, numerically, to approximated periodic solutions. Then, we show the existence of a periodic orbit in a neighborhood of the curve numerically found in the previous stage. The accuracy of the computational verifications is guaranteed by the use of interval arithmetic. Computer assisted proof Differential equations Equações diferenciais Método de Newton Newton's method Órbitas periódicas Periodic orbits Polinômios radiais Radii Polinomials
4	Demonstrações assistidas por computador para equações diferenciais ordinárias / Computer assisted proof for ordinary differential equations Prado, Mário César Monteiro do 23 February 2015 (has links) 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. / In this work, we present a rigorous computational method for proving the existence of periodic orbits of some systems of ordinary differential equations with autonomous vector field of polynomial type. We show that the problem of finding periodic orbits for these systems is equivalent to check for roots of certain functions defined in the Banach space of sequences with algebraic decay. The method can be divided into two steps. First, we seek, numerically, to approximated periodic solutions. Then, we show the existence of a periodic orbit in a neighborhood of the curve numerically found in the previous stage. The accuracy of the computational verifications is guaranteed by the use of interval arithmetic. Computer assisted proof Differential equations Equações diferenciais Método de Newton Newton's method Órbitas periódicas Periodic orbits Polinômios radiais Radii polinomiais
5	Configurations centrales en toile d'araignée Hénot, Olivier 10 1900 (has links) No description available. problème des N corps configuration centrale configuration en toile d'araignée preuve assistée par ordinateur N-body problem central configuration spiderweb configuration computer-assisted proof
6	Demonstrações assistidas por computador para equações diferenciais ordinárias / Computer assisted proof for ordinary differential equations Mário César Monteiro do Prado 23 February 2015 (has links) 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. / In this work, we present a rigorous computational method for proving the existence of periodic orbits of some systems of ordinary differential equations with autonomous vector field of polynomial type. We show that the problem of finding periodic orbits for these systems is equivalent to check for roots of certain functions defined in the Banach space of sequences with algebraic decay. The method can be divided into two steps. First, we seek, numerically, to approximated periodic solutions. Then, we show the existence of a periodic orbit in a neighborhood of the curve numerically found in the previous stage. The accuracy of the computational verifications is guaranteed by the use of interval arithmetic. Equações diferenciais Método de Newton Órbitas periódicas Polinômios radiais Computer assisted proof Differential equations Newton's method Periodic orbits Radii polinomiais
7	Computer-Assisted Proofs and Other Methods for Problems Regarding Nonlinear Differential Equations Fogelklou, Oswald January 2012 (has links) This PhD thesis treats some problems concerning nonlinear differential equations. In the first two papers computer-assisted proofs are used. The differential equations there are rewritten as fixed point problems, and the existence of solutions are proved. The problem in the first paper is one-dimensional; with one boundary condition given by an integral. The problem in the second paper is three-dimensional, and Dirichlet boundary conditions are used. Both problems have their origins in fluid dynamics. Paper III describes an inverse problem for the heat equation. Given the solution, a solution dependent diffusion coefficient is estimated by intervals at a finite set of points. The method includes the construction of set-valued level curves and two-dimensional splines. In paper IV we prove that there exists a unique, globally attracting fixed point for a differential equation system. The differential equation system arises as the number of peers in a peer-to-peer network, which is described by a suitably scaled Markov chain, goes to infinity. In the proof linearization and Dulac's criterion are used. computer-assisted proof numerical verification viscous Burgers’ equation enclosure existence nonlinear boundary value problems Euler equations inverse problem bicubic spline interval analysis heat equation fluid limit peer-to-peer networks fixed points stability global stability
8	Provably Sound and Secure Automatic Proving and Generation of Verification Conditions / Tillförlitligt sund och säker automatisk generering och bevisning av verifieringsvillkor Lundberg, Didrik January 2018 (has links) Formal verification of programs can be done with the aid of an interactive theorem prover. The program to be verified is represented in an intermediate language representation inside the interactive theorem prover, after which statements and their proofs can be constructed. This is a process that can be automated to a high degree. This thesis presents a proof procedure to efficiently generate a theorem stating the weakest precondition for a program to terminate successfully in a state upon which a certain postcondition is placed. Specifically, the Poly/ML implementation of the SML metalanguage is used to generate a theorem in the HOL4 interactive theorem prover regarding the properties of a program written in BIR, an abstract intermediate representation of machine code used in the PROSPER project. / Bevis av säkerhetsegenskaper hos program genom formell verifiering kan göras med hjälp av interaktiva teorembevisare. Det program som skall verifieras representeras i en mellanliggande språkrepresentation inuti den interaktiva teorembevisaren, varefter påståenden kan konstrueras, som sedan bevisas. Detta är en process som kan automatiseras i hög grad. Här presenterar vi en metod för att effektivt skapa och bevisa ett teorem som visar sundheten hos den svagaste förutsättningen för att ett program avslutas framgångsrikt under ett givet postvillkor. Specifikt använder vi Poly/ML-implementationen av SML för att generera ett teorem i den interaktiva teorembevisaren HOL4 som beskriver egenskaper hos ett program i BIR, en abstrakt mellanrepresentation av maskinkod som används i PROSPER-projektet. HOL4 HOL Higher-order logic SML Poly/ML Formal methods Axiomatic semantics Formal verification Static verification Program verification Hoare logic Floyd-Hoare logic ITP Interactive theorem prover Theorem prover Proof assistant BIR Automated theorem proving ATP Automated deduction Computer-assisted proof Automated reasoning Computer Sciences Datavetenskap (datalogi) Software Engineering Programvaruteknik

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