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Geometric spectral theory of quantum graphs
No full textThese are lecture notes from a course given at the summer school "Heatkernels and spectral geometry: from manifolds to graphs" in Bregenz, Austria,2022. They are designed to be accessible to doctoral level students, andinclude background chapters on Laplacians on domains and quantum graphs beforemoving on to specialised topics involving the dependence and optimisation ofoperator eigenvalues on a metric graph in function of the graph geometry, drawnin part from the recent literature.Comment: Final version, to appear in Communications in Mathematic
Ng\^o support theorem and polarizability of quasi-projective commutative group schemes
No full textWe prove that any commutative group scheme over an arbitrary base scheme offinite type over a field with connected fibers and admitting a relatively ampleline bundle is polarizable in the sense of Ng\^o. This extends theapplicability of Ng\^o's support theorem to new cases, for example toLagrangian fibrations with integral fibers and has consequences to theconstruction of algebraic classes.Comment: comments welcom
Construction de Réseaux d'Ordre Supérieur à partir de Traces : Méthodes et Outils
No full textHigher-order networks are a class of graphs that incorporate “memory no-des” in order to take into account the indirect interactions that can existin sequential data. They differ from so-called “order 1” networks, whichonly take direct relationships into account. In this article, we provide anoverview of this concept, detailing their construction and the mining tech-niques that can be employed. We present the Python package honyx, whichcontains algorithms already available in the literature. We propose a tuto-rial on its use through a case study of commercial flight itineraries in theUnited States. We also discuss some of the challenges and future directionsin the field.Les réseaux d'ordre supérieur sont une classe de réseaux qui intègrent des "noeuds-mémoires" afin de prendre en compte les interactions pouvant exister dans des données séquentielles, par opposition aux réseaux dits d'"ordre 1" qui ne prennent en compte que les relations directes. Dans cet article, nous donnons un aperçu de ce concept en détaillant leur construction et les techniques de fouille qui peuvent être employées. Nous proposons un didacticiel sur un cas d'étude utilisant une implémentation de notre part des algorithmes présents dans la littérature. Nous abordons également certains des défis et des orientations futures dans ce domaine
Sparse juntas on the biased hypercube
No full textWe give a structure theorem for Boolean functions on the -biased hypercubewhich are -close to degree in , showing that they are closeto sparse juntas. Our structure theorem implies that such functions are-close to constant functions. We pinpoint the exactvalue of the constant . We also give an analogous result for monotoneBoolean functions on the biased hypercube which are -close to degree in , showing that they are close to sparse DNFs. Our structuretheorems are optimal in the following sense: for every , weidentify a class of degree sparse juntas whichare -close to Boolean (in the monotone case, width sparseDNFs) such that a Boolean function on the -biased hypercube is-close to degree in iff it is -close to afunction in .Comment: 44 pages. TheoretiCS journal articl
Interpolation and moduli spaces of vector bundles on very general blowups of the projective plane
No full textIn this paper, we study certain moduli spaces of vector bundles on the blowupof the projective plane in at least 10 very general points. Moduli spaces ofsheaves on general type surfaces may be nonreduced, reducible and evendisconnected. In contrast, moduli spaces of sheaves on minimal rationalsurfaces and certain del Pezzo surfaces are irreducible and smooth along thelocus of stable bundles. We find examples of moduli spaces of vector bundles onmore general blowups of the projective plane that are disconnected and havecomponents of different dimensions. In fact, assuming the SHGH Conjecture, wecan find moduli spaces with arbitrarily many components of arbitrarily largedimension
Compte-rendu de la deuxième édition de la conférence Frognet
No full textThis account reviews the presentations made at the second Frognet conference, a French-speaking conference on graphs and social net- works. The conference was held in Montpellier on 6 and 7 April 2023. We detail the main themes addressed, the major sociometric surveys used, and the diversity of methodological approaches, sources and issues represented at the conference.Ce compte rendu revient sur les présentations de la deuxième conférence Frognet : conférence francophone sur les graphes et les réseaux sociaux. Cette édition s’est tenue à Montpellier les 6 et 7 avril 2023. Le compte-rendu détaille les grandes thématiques abor- dées, les grandes enquêtes sociométriques mobilisées, la diversité des approches méthodologiques, des sources et des questionnements représentés dans la conférence
Potentialisations of a class of fully-nonlinear symmetry-integrable evolution equations
No full textWe consider here the class of fully-nonlinear symmetry-integrable third-orderevolution equations in 1+1 dimensions that were proposed recently in thejournal Open Communications in Nonlinear Mathematical Physics, vol. 2, 216--228(2022). In particular, we report all zero-order and higher-orderpotentialisations for this class of equations using their integrating factors(or multipliers) up to order four. Chains of connecting evolution equations arealso obtained by multi-potentialisations.Comment: 34 page
The Economic Cultures of Fear and Love
No full textIn earlier work, the author has studied the economic role of planning horizons in making a case for complementarity as the predominant feature of social interdependence. This paper compares the different choice strategies implied by substitution, opposition and conflicts of interest in an economics of fear with those arising from horizon effects, economic complementarity and concerts of interest in an economics based on love. The contrasting implications of a psychological literature on negative vs. positive emotions and their health effects, along with the findings in neurophysiological research about how humans are hard-wired for empathy and compassion leads to some fundamental changes in how we might address and revise social problems through economic analysis. The aim of this paper is to extend a horizonal case for complementarity in the author's previous work into its psychological links to research findings on healthy cognitive function and its emotional basis. An economics of substitution yields quite different conclusions about optimal institutional forms and how we address and frame social relations than are implied by an explicitly horizonal economics of complementary social relations. Recent psychological and neurophysiological studies support a horizonal case for complementarity in social relations, showing that our orthodox substitution assumptions and models of competitive equilibrium should be rejected for a renewed economic analysis based on horizonal models of complementarity and cooperation as a means to achieve greater social well-being in a healthier and more integrative form of social organization. The issues to be explored are not entertained within the currently existing frame of orthodox economics, so we must step beyond our standard habitual assumptions
Weakly toll convexity and proper interval graphs
No full textA walk is a \textit{weakly toll walk} if implies and implies . A set of vertices of is {\it weakly toll convex} if for any two non-adjacentvertices any vertex in a weakly toll walk between and isalso in . The {\em weakly toll convexity} is the graph convexity spacedefined over weakly toll convex sets. Many studies are devoted to determine ifa graph equipped with a convexity space is a {\em convex geometry}. An\emph{extreme vertex} is an element of a convex set such that the set is also convex. A graph convexity space is said to be aconvex geometry if it satisfies the Minkowski-Krein-Milman property, whichstates that every convex set is the convex hull of its extreme vertices. It isknown that chordal, Ptolemaic, weakly polarizable, and interval graphs can becharacterized as convex geometries with respect to the monophonic, geodesic,, and toll convexities, respectively. Other important classes of graphscan also be characterized in this way. In this paper, we prove that a graph isa convex geometry with respect to the weakly toll convexity if and only if itis a proper interval graph. Furthermore, some well-known graph invariants arestudied with respect to the weakly toll convexity