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Structure and Power: an emerging landscape
No full textIn this paper, we give an overview of some recent work on applying tools fromcategory theory in finite model theory, descriptive complexity, constraintsatisfaction, and combinatorics. The motivations for this work come fromComputer Science, but there may also be something of interest for modeltheorists and other logicians. The basic setting involves studying the category of relational structures viaa resource-indexed family of adjunctions with some process category - whichunfolds relational structures into treelike forms, allowing natural resourceparameters to be assigned to these unfoldings. One basic instance of thisscheme allows us to recover, in a purely structural, syntax-free way: theEhrenfeucht-Fraisse~game; the quantifier rank fragments of first-order logic;the equivalences on structures induced by (i) the quantifier rank fragments,(ii) the restriction of this fragment to the existential positive part, and(iii) the extension with counting quantifiers; and the combinatorial parameterof tree-depth (Nesetril and Ossona de Mendez). Another instance recovers thek-pebble game, the finite-variable fragments, the corresponding equivalences,and the combinatorial parameter of treewidth. Other instances cover modal,guarded and hybrid fragments, generalized quantifiers, and a wide range ofcombinatorial parameters. This whole scheme has been axiomatized in a verygeneral setting, of arboreal categories and arboreal covers. Beyond this basic level, a landscape is beginning to emerge, in whichstructural features of the resource categories, adjunctions and comonads arereflected in degrees of logical and computational tractability of thecorresponding languages. Examples include semantic characterisation andpreservation theorems, and Lovasz-type results on counting homomorphisms.Comment: To appear in special issue for Trakhtenbrot centenary of Fundamenta Informaticae vol. 186 no 1-
Intégration par une approche implicite d’une loi de comportement elastoplastique dans le code de calcul par éléments finis Cast3M
No full textsoumission à EpisciencesThis paper is dedicated to the implementation of a law of mechanical behavior in the finite element software Cast3M using an open source code generator named Mfront. To do so, an elastoplastic behaviour model has been chosen from existing laws in the literature. Following an implicit discretization, a hardware library corresponding to the isotropic and kinematic strain-hardening model is generated using Mfront. The UMAT computer interface is used to build the library in Cast3M. A validation of the approach has been carried out by comparing the numerical results obtained with the generated hardware library and the equivalent pre-existing library in Cast3M. Simulations in the case of a tensile bar and a perforated plate show almost identical resultsCe papier est dédié à l’implantation d’une loi de comportement mécanique dans le logiciel éléments finis Cast3M à l’aide d’un générateur de code open source nommé Mfront. Pour ce faire, un modèle de comportement élasto-plastique a été choisi à partir des lois existantes dans la littérature. Suivant une discrétisation implicite, une bibliothèque matérielle correspondant au modèle de CHABOCHE à écrouissage isotrope et cinématique non linéaire est générée grâce à Mfront. L’interface informatique UMAT est utilisée pour construire la bibliothèque dans Cast3M. Une validation de l’approche a été menée en comparant les résultats numériques obtenus entre la bibliothèque matérielle générée et la bibliothèque équivalente préexistante dans Cast3M. Les simulations dans le cas d’une barre en traction et une plaque trouée montrent des résultats quasi identique
The use and impact of auditory stimulation in animals
No full textMusic can cause pleasant sensations in humans whereas some noises can cause discomfort. The effects of music and noise have also been somewhat studied in animals, showing different impacts. In this review we aim to illustrate the differences and similarities between animals, in terms of their sensitivity to auditory stimuli (noise or music), by first recalling some generalities about the physical characteristics of sound and the biological bases of hearing. Second, based on the studies reported in this review, we conclude that ambient noise is harmful and/or stressful, and that musical sounds can take many forms with a large range of impacts in animals. Finally, we present two practical examples of the use of music with animals (one in the context of a zoo and the other in cattle breeding) and an example of an experiment designed to understand the impact of music on neonate lambs. These three examples highlight how music can help to improve animal welfare
Mean field system of a two-layers neural model in a diffusive regime
No full textWe study a model of interacting neurons. The structure of this neural systemis composed of two layers of neurons such that the neurons of the first layersend their spikes to the neurons of the second one: if is the number ofneurons of the first layer, at each spiking time of the first layer, everyneuron of both layers receives an amount of potential of the form where is a centered random variable. This kind of structure of neurons canmodel a part of the structure of the visual cortex: the first layer representsthe primary visual cortex V1 and the second one the visual area V2. The modelconsists of two stochastic processes, one modelling the membrane potential ofthe neurons of the first layer, and the other the membrane potential of theneurons of the second one. We prove the convergence of these processes as thenumber of neurons~ goes to infinity and obtain a convergence speed. Theproofs rely on similar arguments as those used in [Erny, L\"ocherbach,Loukianova (2022)]: the convergence speed of the semigroups of the processes isobtained from the convergence speed of their infinitesimal generators using aTrotter-Kato formula, and from the regularity of the limit semigroup.Comment: 20 pages, 1 figur
Typability and Type Inference in Atomic Polymorphism
No full textIt is well-known that typability, type inhabitation and type inference areundecidable in the Girard-Reynolds polymorphic system F. It has recently beenproven that type inhabitation remains undecidable even in the predicativefragment of system F in which all universal instantiations have an atomicwitness (system Fat). In this paper we analyze typability and type inference inCurry style variants of system Fat and show that typability is decidable andthat there is an algorithm for type inference which is capable of dealing withnon-redundancy constraints
Search for integrable two-component versions of the lattice equations in the ABS-list
No full textWe search and classify two-component versions of the quad equations in theABS list, under certain assumptions. The independent variables will be called and in addition to multilinearity and irreducibility the equation pair isrequired to have the following specific properties: (1) The two equationsforming the pair are related by exchange. (2) When both equations reduce to one of the equations in the ABS list. (3) Evolution inany corner direction is by a multilinear equation pair. One straightforward wayto construct such two-component pairs is by taking some particular equation inthe ABS list (in terms of ), using replacement forsome particular shifts, after which the other equation of the pair is obtainedby property (1). This way we can get 8 pairs for each starting equation. One ofour main results is that due to condition (3) this is in fact complete for H1,H3, Q1, Q3. (For H2 we have a further case, Q2, Q4 we did not check.) As forthe CAC integrability test, for each choice of the bottom equations we could inprinciple have possible side-equations. However, we find that onlyequations constructed with an even number of replacementsare possible, and for each such equation there are two sets of "side" equationpairs that produce (the same) genuine B\"acklund transformation and Lax pair.Comment: 14 pages, final versio
Meeting the challenges of teaching mathematics in higher education today
No full textEditorial to the first issue of ÉpiDEMES.Among the research in Mathematics Education devoted to higher education, two types of initiativescan be distinguished: those promoting the development of research results and those promotingreflections on mathematics teaching. In both cases, mathematics educators and Mathematicsresearchers have played – sometimes jointly – a leading role. The creation of the journalÉpiDEMES (Epijournal de Didactique et Epistémologie des Mathématiques pour l'EnseignementSupérieur in French) is positioned in the continuity of these efforts
A Coalgebraic Approach to Dualities for Neighborhood Frames
No full textWe develop a uniform coalgebraic approach to J\'onsson-Tarski and Thomasontype dualities for various classes of neighborhood frames and neighborhoodalgebras. In the first part of the paper we construct an endofunctor on thecategory of complete and atomic Boolean algebras that is dual to the doublepowerset functor on . This allows us to show that Thomasonduality for neighborhood frames can be viewed as an algebra-coalgebra duality.We generalize this approach to any class of algebras for an endofunctorpresented by one-step axioms in the language of infinitary modal logic. As aconsequence, we obtain a uniform approach to dualities for various classes ofneighborhood frames, including monotone neighborhood frames, pretopologicalspaces, and topological spaces. In the second part of the paper we develop a coalgebraic approach toJ\'{o}nsson-Tarski duality for neighborhood algebras and descriptiveneighborhood frames. We introduce an analogue of the Vietoris endofunctor onthe category of Stone spaces and show that descriptive neighborhood frames areisomorphic to coalgebras for this endofunctor. This allows us to obtain acoalgebraic proof of the duality between descriptive neighborhood frames andneighborhood algebras. Using one-step axioms in the language of finitary modallogic, we restrict this duality to other classes of neighborhood algebrasstudied in the literature, including monotone modal algebras and contingencyalgebras. We conclude the paper by connecting the two types of dualities via canonicalextensions, and discuss when these extensions are functorial
Why Does Propositional Quantification Make Modal and Temporal Logics on Trees Robustly Hard?
No full textAdding propositional quantification to the modal logics K, T or S4 is knownto lead to undecidability but CTL with propositional quantification under thetree semantics (tQCTL) admits a non-elementary Tower-complete satisfiabilityproblem. We investigate the complexity of strict fragments of tQCTL as well asof the modal logic K with propositional quantification under the treesemantics. More specifically, we show that tQCTL restricted to the temporaloperator EX is already Tower-hard, which is unexpected as EX can only enforcelocal properties. When tQCTL restricted to EX is interpreted on N-bounded treesfor some N >= 2, we prove that the satisfiability problem is AExpPol-complete;AExpPol-hardness is established by reduction from a recently introduced tilingproblem, instrumental for studying the model-checking problem for intervaltemporal logics. As consequences of our proof method, we prove Tower-hardnessof tQCTL restricted to EF or to EXEF and of the well-known modal logics such asK, KD, GL, K4 and S4 with propositional quantification under a semantics basedon classes of trees
Average-case algorithms for testing isomorphism of polynomials, algebras, and multilinear forms
No full textWe study the problems of testing isomorphism of polynomials, algebras, andmultilinear forms. Our first main results are average-case algorithms for theseproblems. For example, we develop an algorithm that takes two cubic forms , and decides whether and areisomorphic in time for most . This average-case setting hasdirect practical implications, having been studied in multivariate cryptographysince the 1990s. Our second result concerns the complexity of testingequivalence of alternating trilinear forms. This problem is of interest in bothmathematics and cryptography. We show that this problem is polynomial-timeequivalent to testing equivalence of symmetric trilinear forms, by showing thatthey are both Tensor Isomorphism-complete (Grochow-Qiao, ITCS, 2021), thereforeis equivalent to testing isomorphism of cubic forms over most fields.Comment: Journal version at the journal of Groups, Complexity, Cryptolog