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La modélisation mathématique comme domaine de recherche : avancées dans l’analyse écologique
No full textThis paper focuses on the recent evolution of the field of mathematical modelling research and how different research questions have been formulated and addressed by international theoretical frameworks. This paper focuses on the “epistemological”, “economic” and “ecological” dimensions of these different theoretical approaches to mathematical modelling. We will see that the ecological issue is still largely absent from many approaches. This will allow us, in a second step, to show some tools of the anthropological theory of the didactic (ATD) such as the study and research paths (SRP) that we have been proposed to facilitate the design, implementation and analysis of modelling practices in different school levels and in teacher education.Este artículo se centra en la evolución reciente del campo de la investigación en modelización matemática y en cómo se han formulado y abordado diferentes problemas de investigación en los distintos enfoques teóricos en el ámbito internacional. Nos centramos más concretamente en cómo los distintos enfoques teóricos a la modelización matemática abordan las dimensiones “epistemológica”, “económica” y “ecológica”. Mostraremos cómo la problemática ecológica sigue estando bastante ausente en muchos de estos enfoques. Esto nos permitirá, en la segunda parte, mostrar algunas herramientas de la teoría antropológico de lo didáctico (TAD), como los recorridos de estudio e investigación (REI) que han sido propuestos para facilitar el diseño, implementación y análisis de prácticas de modelización en distintos niveles escolares y en la formación de profesores.Cet article s’intéresse à l’évolution récente du domaine de recherche sur la modélisation mathématique et la manière dont différentes questions de recherche ont été formulées et abordées par des cadres théoriques internationaux. Ce travail se focalise plus particulièrement sur les dimensions « épistémologique », « économique » et « écologique » de ces différentes approches théoriques de la modélisation mathématique. Nous verrons que la problématique écologique est encore en grande partie absente de nombreuses approches. Ceci permettra, dans un second temps, de montrer certains outils de la théorie anthologique du didactique (TAD) comme les parcours d’étude et de recherche (PER) que nous avons proposés pour faciliter la conception, la mise en œuvre et l’analyse des pratiques de modélisation dans différents niveaux scolaires et dans la formation des enseignants
L’enseignant de mathématiques aux temps des technologies numériques : un cadre théorique adaptant la double approche pour étudier son activité
No full textArticle à paraître dans un numéro spécial de la revue Recherches en Didactique des MathématiquesThis text deals with the mathematics teacher’s activity in digital technologies-based sessions. It is composed of two parts. The first part outlines recent studies and theoretical models that have been developed to study the teaching dimension in learning and teaching within environments integrating these technologies. The second presents a theoretical framework for the study of the teacher's activity in ordinary classroom situations: the DAaT (Double Approach applied to Technology). This framework is derived from the Double Didactic and Ergonomic Approach (Robert and Rogalski, 2002) while introducing adaptations and adding complementary constructs in order to better consider the specificity of the studied environments. The theoretical concepts thus developed are presented and discussed. The conclusion returns to the usefulness of the theoretical and methodological tools provided by the research in this field for the enrichment of teacher practices or for teacher educationCe texte porte sur l’activité de l'enseignant de mathématiques utilisant des technologies numériques. Il est composé de deux parties. La première esquisse un panorama des études récentes et des modèles théoriques qui ont été développés pour étudier la dimension enseignante dans des environnements d’apprentissage et d’enseignement intégrant ces technologies. La deuxième présente un cadre théorique pour l’étude de l’activité de l’enseignant dans des situations ordinaires de classe : la DAaT (Double Approche adaptée aux Technologies). Ce cadre prend appui sur la Double Approche didactique et ergonomique des pratiques (Robert et Rogalski, 2002) tout en y introduisant des adaptations et y apportant des compléments et des prolongements afin de rendre compte de la spécificité des environnements étudiés. Les concepts théoriques ainsi développés sont présentés et discutés. La conclusion revient sur l’utilité des outils et résultats issus de la recherche dans ce domaine pour l’enrichissement des pratiques ou pour leur formatio
Reduction for asynchronous Boolean networks: elimination of negatively autoregulated components
No full textTo simplify the analysis of Boolean networks, a reduction in the number ofcomponents is often considered. A popular reduction method consists ineliminating components that are not autoregulated, using variable substitution.In this work, we show how this method can be extended, for asynchronousdynamics of Boolean networks, to the elimination of vertices that have anegative autoregulation, and study the effects on the dynamics and interactionstructure. For elimination of non-autoregulated variables, the preservation ofattractors is in general guaranteed only for fixed points. Here we givesufficient conditions for the preservation of complex attractors. The removalof so called mediator nodes (i.e. vertices with indegree and outdegree one) isoften considered, and frequently does not affect the attractor landscape. Weclarify that this is not always the case, and in some situations even subtlechanges in the interaction structure can lead to a different asymptoticbehaviour. Finally, we use properties of the more general elimination methodintroduced here to give an alternative proof for a bound on the number ofattractors of asynchronous Boolean networks in terms of the cardinality ofpositive feedback vertex sets of the interaction graph.Comment: 16 page
On the statistic and valley-hopping
No full textThis paper studies the relationship between the modifiedFoata\unicode{x2013}Strehl action (a.k.a. valley-hopping)\unicode{x2014}agroup action on permutations used to demonstrate the -positivity of theEulerian polynomials\unicode{x2014}and the number of rixed points\unicode{x2014}a recursively-defined permutationstatistic introduced by Lin in the context of an equidistribution problem. Wegive a linear-time iterative algorithm for computing the set of rixed points,and prove that the statistic is homomesic undervalley-hopping. We also demonstrate that a bijection introduced by Linand Zeng in the study of the statistic sends orbits of thevalley-hopping action to orbits of a cyclic version of valley-hopping, whichimplies that the number of fixed points is homomesic undercyclic valley-hopping.Comment: 27 page
Corrigendum to "On the monophonic rank of a graph" [Discrete Math. Theor. Comput. Sci. 24:2 (2022) #3]
No full textIn this corrigendum, we give a counterexample to Theorem 5.2 in "On themonophonic rank of a graph" [Discrete Math. Theor. Comput. Sci. 24:2 (2022)#3]. We also present a polynomial-time algorithm for computing the monophonicrank of a starlike graph.Comment: 6 pages, 2 figure
Realizable Learning is All You Need
No full textThe equivalence of realizable and agnostic learnability is a fundamentalphenomenon in learning theory. With variants ranging from classical settingslike PAC learning and regression to recent trends such as adversarially robustlearning, it's surprising that we still lack a unified theory; traditionalproofs of the equivalence tend to be disparate, and rely on strongmodel-specific assumptions like uniform convergence and sample compression. In this work, we give the first model-independent framework explaining theequivalence of realizable and agnostic learnability: a three-line blackboxreduction that simplifies, unifies, and extends our understanding across a widevariety of settings. This includes models with no known characterization oflearnability such as learning with arbitrary distributional assumptions andmore general loss functions, as well as a host of other popular settings suchas robust learning, partial learning, fair learning, and the statistical querymodel. More generally, we argue that the equivalence of realizable and agnosticlearning is actually a special case of a broader phenomenon we call propertygeneralization: any desirable property of a learning algorithm (e.g. noisetolerance, privacy, stability) that can be satisfied over finite hypothesisclasses extends (possibly in some variation) to any learnable hypothesis class
On the dual positive cones and the algebraicity of a compact K\"ahler manifold
No full textWe investigate the algebraicity of compact K\"ahler manifolds admitting apositive rational Hodge class of bidimension . We prove that if the dualK\"ahler cone of a compact K\"ahler manifold contains a rational class asan interior point, then its Albanese variety is projective. As a consequence,we answer the Oguiso--Peternell problem for Ricci-flat compact K\"ahlermanifolds. We also study related algebraicity problems for threefolds.Comment: Published versio
Invariant conservative finite-difference schemes for the one-dimensional shallow water magnetohydrodynamics equations in Lagrangian coordinates
No full textInvariant finite-difference schemes for the one-dimensional shallow waterequations in the presence of a magnetic field for various bottom topographiesare constructed. Based on the results of the group classification recentlycarried out by the authors, finite-difference analogues of the conservationlaws of the original differential model are obtained. Some typical problems areconsidered numerically, for which a comparison is made between the cases of amagnetic field presence and when it is absent (the standard shallow watermodel). The invariance of difference schemes in Lagrangian coordinates and theenergy preservation on the obtained numerical solutions are also discussed.Comment: 22 pages, 6 figures. This version (v7) has been specially prepared for publication in the Open Commun. Nonlinear Math. Phys., Special Issue in Memory of Professor Decio Levi. Notice that in the previous version (v5) of the paper an appendix was added with an informal discussion on stability issues. By the decision of the authors, this appendix was not included in v6 and v
Kontsevich's star-product up to order 7 for affine Poisson brackets: where are the Riemann zeta values?
No full textThe Kontsevich star-product admits a well-defined restriction to the class ofaffine -- in particular, linear -- Poisson brackets; its graph expansionconsists only of Kontsevich's graphs with in-degree for aerialvertices. We obtain the formula with harmonic propagators for the graph weights (over aerial vertices); we verify that all these weights satisfy thecyclic weight relations by Shoikhet--Felder--Willwacher, that they match thecomputations using the software by Panzer, and theresulting affine star-product is associative modulo . We discover that the Riemann zeta value , which enters theharmonic graph weights (up to rationals), actually disappears from the analyticformula of \textit{because}all the -linear combinations of Kontsevich graphs near represent differential consequences of the Jacobi identityfor the affine Poisson bracket, hence their contribution vanishes. We thusderive a ready-to-use shorter formula mod~ with only rational coefficients.Comment: This paper is an extract from the PhD thesis of R.B. and is based on a series of talks and posters in Banff and Strasbourg; contains 44 pages, 2 tables, 2 appendices. Old chapter 3 and old appendices 3--6 were extracted from original version 1 and, in expanded form, published separately, see arXiv:2309.16664 [math.QA
A mean-field model of Integrate-and-Fire neurons: non-linear stability of the stationary solutions
No full textWe investigate a stochastic network composed of Integrate-and-Fire spikingneurons, focusing on its mean-field asymptotics. We consider an invariantprobability measure of the McKean-Vlasov equation and establish an explicitsufficient condition to ensure the local stability of this invariantdistribution. Furthermore, we prove a conjecture proposed initially by J.Touboul and P. Robert regarding the bistable nature of a specific instance ofthis neuronal model