Systematic Computer Assisted Proofs of periodic orbits of Hamiltonian systems

The numerical study of Dynamical Systems leads to obtain invariant objects of the systems such as periodic orbits, invariant tori, attractors and so on, that helps to the global understanding of the problem. In this paper we focus on the rigorous computation of periodic orbits and their distribution on the phase space, which configures the so called skeleton of the system. We use Computer Assisted Proof techniques to make a rigorous proof of the existence and the stability of families of periodic orbits in two-degrees of freedom Hamiltonian systems, which provide rigorous skeletons of periodic orbits. To that goal we show how to prove the existence and stability of a huge set of discrete initial conditions of periodic orbits, and later, how to prove the existence and stability of continuous families of periodic orbits. We illustrate the approach with two paradigmatic problems: the Hénon–Heiles Hamiltonian and the Diamagnetic Kepler problem.     

Reverse Mathematics and Recursive Graph Theory

We examine a number of results of infinite combinatorics using the techniques of reverse mathematics. Our results are inspired by similar results in recursive combinatorics. Theorems included concern colorings of graphs and bounded graphs, Euler paths, and Hamilton paths.     

Proof validation in real analysis: Inferring and checking warrants

In the study reported here, we investigate the skills needed to validate a proof in real analysis, i.e., to determine whether a proof is valid. We first argue that when one is validating a proof, it is not sufficient to make certain that each statement in the argument is true. One must also check that there is good reason to believe that each statement follows from the preceding statements or from other accepted knowledge, i.e., that there is a valid warrant for making that statement in the context of this argument. We then report an exploratory study in which we investigated the behavior of 13 undergraduates when they were asked to determine whether or not a particular flawed mathematical argument is a valid mathematical proof. The last line of this purported proof was true, but did not follow legitimately from the earlier assertions in the proof. Our findings were that only six of these undergraduates recognized that this argument was invalid and only two did so for legitimate mathematical reasons. On a more positive note, when asked to consider whether the last line of the proof followed from previous assertions, a total of 10 students concluded that the statement did not and rejected the proof as invalid.     

Non decomposable connectives of linear logic

This paper studies the so-called generalized multiplicative connectives of linear logic, focusing on the question of finding the “non-decomposable” ones, i.e., those that cannot be expressed as combinations of the default binary connectives of multiplicative linear logic, ⊗ (times) and ⅋ (par). In particular, we concentrate on generalized connectives of a surprisingly simple form, called “entangled connectives”, and prove a characterization theorem giving a criterion for identifying the undecomposable entangled ones.     

Variation on a theme of Schütte

Let ? be a primitive recursive well‐ordering on the natural numbers and assume that its order‐type is greater than or equal to the proof‐theoretic ordinal of the theory T. We show that the proof‐theoretic strength of T is not increased if we add the negation of the statement which formalizes transfinite induction along ?. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Vague numbers

If there are vague numbers, it would be easier to use numbers as semantic values in a treatment of vagueness while avoiding precise cut-off points. When we assign a particular statement a range of values (less than 1 and greater than 0) there is no precise sharp cut-off point that locates the greatest lower bound or the least upper bound of the interval, I should like to say. Is this possible? “Vague Numbers” stands for awareness of the problem. I do not present a serious theory of vague numbers. I sketch some reasons for using a many-value semantics. These reasons refer to my earlier treatments of determinacy and definitions of higher-order borderline cases. I also sketch how definitions of independence use the determinacy operator. The distinction between actually assigned values and values whose assignments are acceptable helps avoid unwanted precise cut-off points.     

The Hauptsatz for Stratified Comprehension: A Semantic Proof

We prove the cut-elimination theorem, Gentzen's Hauptsatz, for the system for stratified comprehension, i. e. Quine's NF minus extensionality. Mathematics Subject Classification: 03B15, 03F05.     

Realisability in weak systems of explicit mathematics

This paper is a direct successor to 12 . Its aim is to introduce a new realisability interpretation for weak systems of explicit mathematics and use it in order to analyze extensions of the theory PET in 12 by the so‐called join axiom of explicit mathematics.