Related papers: The L\"uroth problem
This is an extended version of my Nobel Lecture, delivered on December $8^{th}$ 2021. I will recall the genesis of the concept of multiple equilibria in natural sciences. I will then describe my contribution to the development of this…
The Stochastic Eulerian Tour Problem was introduced in 2008 as a stochastic variant of the well-known Eulerian Tour Problem. In a follow-up paper the same authors investigated some heuristics for solving the Stochastic Eulerian Tour…
Starting from an A-stable rational approximation to $\rm{e}^z$ of order $p$, $$r(z)= 1+ z+ \cdots + z^p/ p! + O(z^{p+1}),$$ families of stable methods are proposed to time discretize abstract IVP's of the type $u'(t) = A u(t) + f(t)$. These…
In 2007 an international conference engaged a reflection on the present conditions for sciences and scientific practice, to initiate a dialogue between science, philosophy, epistemology and sociology. Starting from many examples taken from…
Large Language Models (LLMs) have emerged as powerful tools for generating coherent text, understanding context, and performing reasoning tasks. However, they struggle with temporal reasoning, which requires processing time-related…
Introducing the notion of a rational system of measure preserving transformations and proving a recurrence result for such systems, we give sufficient conditions in order a subset of rational numbers to contain arbitrary long arithmetic…
In this series of lectures presented at the CIMPA Winter School on Discrete Integrable Systems in Pondicherry, India, in February, 2003 we give a review of the application of Lie point symmetries, and their generalizations to the study of…
An old problem due to J.-L. Lions going back to the 1960s asks whether the abstract Cauchy problem associated to non-autonomous forms has maximal regularity if the time dependence is merely assumed to be continuous or even measurable. We…
This paper surveys main and recent studies on temporal logics in a broad sense by presenting various logic systems, dealing with various time structures, and discussing important features, such as decidability (or undecidability) results,…
This paper explores the question of how accurately current large language models can perform logical reasoning in natural language, with an emphasis on whether these models exhibit reasoning biases similar to humans. Specifically, our study…
In the present study, we investigate and compare reasoning in large language models (LLM) and humans using a selection of cognitive psychology tools traditionally dedicated to the study of (bounded) rationality. To do so, we presented to…
We study a regularized variant of the Bayesian Persuasion problem, where the receiver's decision process includes a divergence-based penalty that accounts for deviations from perfect rationality. This modification smooths the underlying…
Since its introduction by P.L. Lions in his lectures and seminars at the College de France, see [9], and also the very helpful notes of Cardialaguet [4] on Lions' lectures, the Master Equation has attracted a lot of interest, and various…
In this paper, we study a general Syracuse problem. We give some necessary conditions concerning the existence of eventual non trivial cycles. Some properties based on linear logarithmic forms are established. New general conjectures are…
This note is based on a series of four lectures the author gave at University of Montpellier, September 28-30, 2015. He started with the notion of mass in general relativity, gave a brief review of some known constructions of quasi-local…
We review the status of different approaches to lattice quantum gravity indicating the successes and problems of each. Recent developments within the dynamical triangulation formulation are then described. Plenary talk at LATTICE 95 July…
The main result of [C. Morosi and L. Pizzocchero, Nonlinear Analysis, 2012] is presented in a variant, based on a C^infinity formulation of the Cauchy problem; in this approach, the a posteriori analysis of an approximate solution gives a…
Mechanized reasoning uses computers to verify proofs and to help discover new theorems. Computer scientists have applied mechanized reasoning to economic problems but -- to date -- this work has not yet been properly presented in economics…
Large Language Models (LLMs) have shown impressive performance on complex tasks through Chain-of-Thought (CoT) reasoning. However, conventional CoT relies on explicitly verbalized intermediate steps, which constrains its broader…
In my lectures at the Les Houches Summer School 2008, I discussed central concepts of computational statistical physics, which I felt would be accessible to the very cross-cultural audience at the school. I started with a discussion of…