Des perspectives rigoureuses pour la linguistique

A reading of many recent publications shows that theoretical concerns in the field of linguistics have declined, even though the founders of the discipline all emphasised the central importance of its role. This article aims to show how linguistic description has progressed thanks to the different theoretical tools that have been developed in the course of research. From this point of view, this article illustrates the following facts: lexicon, semantics and syntax are not separate instances but together form the units that are sentences. The elements of sentences must be described in terms of the set of properties that characterise them. This is the case for regular sequences. Finally, one of the promising objectives consists in finding a reasoned classification of fixed sequences. Linguistics, like any science, cannot do without theoretical tools.

Quantum computation of Groebner basis

In this essay, we examine the feasibility of quantum computation of Groebner basis which is a fundamental tool of algebraic geometry. The classical method for computing Groebner basis is based on Buchberger's algorithm, and our question is how to adopt quantum algorithm there. A Quantum algorithm for finding the maximum is usable for detecting head terms of polynomials, which are required for the computation of S-polynomials. The reduction of S-polynomials with respect to a Groebner basis could be done by the quantum version of Gauss-Jordan elimination of echelon which represents polynomials. However, the frequent occurrence of zero-reductions of polynomials is an obstacle to the effective application of quantum algorithms. This is because zero-reductions of polynomials occur in non-full-rank echelons, for which quantum linear systems algorithms (through the inversion of matrices) are inadequate, as ever-known quantum linear solvers (such as Harrow-Hassidim-Lloyd) require the clandestine computations of the inverses of eigenvalues. Hence, for the quantum computation of the Groebner basis, the schemes to suppress the zero-reductions are necessary. To this end, the F5 algorithm or its variant (F5C) would be the most promising, as these algorithms have countermeasures against the occurrence of zero-reductions and can construct full-rank echelons whenever the inputs are regular sequences. Between these two algorithms, the F5C is the better match for algorithms involving the inversion of matrices.

Surjectivity of the asymptotic Borel map in Carleman–Roumieu ultraholomorphic classes defined by regular sequences

We study the surjectivity of, and the existence of right inverses for, the asymptotic Borel map in Carleman–Roumieu ultraholomorphic classes defined by regular sequences in the sense of E. M. Dyn’kin. We extend previous results by J. Schmets and M. Valdivia, by V. Thilliez, and by the authors, and show the prominent role played by an index, associated with the sequence, that was introduced by V. Thilliez. The techniques involve regular variation, integral transforms and characterization results of A. Debrouwere in a half-plane, stemming from his study of the surjectivity of the moment mapping in general Gelfand–Shilov spaces.

Modeling of Astrocyte Networks: Toward Realistic Topology and Dynamics

Neuronal firing and neuron-to-neuron synaptic wiring are currently widely described as orchestrated by astrocytes—elaborately ramified glial cells tiling the cortical and hippocampal space into non-overlapping domains, each covering hundreds of individual dendrites and hundreds thousands synapses. A key component to astrocytic signaling is the dynamics of cytosolic Ca2+ which displays multiscale spatiotemporal patterns from short confined elemental Ca2+ events (puffs) to Ca2+ waves expanding through many cells. Here, we synthesize the current understanding of astrocyte morphology, coupling local synaptic activity to astrocytic Ca2+ in perisynaptic astrocytic processes and morphology-defined mechanisms of Ca2+ regulation in a distributed model. To this end, we build simplified realistic data-driven spatial network templates and compile model equations as defined by local cell morphology. The input to the model is spatially uncorrelated stochastic synaptic activity. The proposed modeling approach is validated by statistics of simulated Ca2+ transients at a single cell level. In multicellular templates we observe regular sequences of cell entrainment in Ca2+ waves, as a result of interplay between stochastic input and morphology variability between individual astrocytes. Our approach adds spatial dimension to the existing astrocyte models by employment of realistic morphology while retaining enough flexibility and scalability to be embedded in multiscale heterocellular models of neural tissue. We conclude that the proposed approach provides a useful description of neuron-driven Ca2+-activity in the astrocyte syncytium.