Mixed Hodge structures and vector bundles on the projective Plane I

We describe an equivalence of categories between the category of mixed Hodge structures and a category of vector bundles on the toric complex projective plane which verify some semistability condition. We then apply this correspondence to define an invariant which generalises the notion of R-split mixed Hodge structure and compute extensions in the category of mixed Hodge structures in terms of extensions of the corresponding vector bundles. We also give a relative version of this correspondence and apply it to define stratifications of the bases of the variations of mixed Hodge structure.

The complexity of random ordered structures

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On the stability of cooperation structures

Qin [J. Eco. Th., 1996] recently showed that in a game of endogenous formation of cooperation structure, if the underlying TU-game is superadditive, then the full cooperation structure is stable. In this note, we characterize the class of games that ensure the stability of the full cooperation structure, and show that this class is much larger than that of superadditive TU-games.

Stable coalition structures with fixed decision scheme

This paper studies the stability of a finite local public goods economy in horizontal differentiation, where a jurisdiction's choice of the public good is given by an exogenous decision scheme. In this paper, we characterize the class of decision schemes that ensure the existence of an equilibrium with free mobility (that we call Tiebout equilibrium) for monotone distribution of players. This class contains all the decision schemes whose choice lies between the Rawlsian decision scheme and the median voter with mid-distance of the two median voters when there are ties. We show that for non-monotone distribution, there is no decision scheme that can ensure the stability of coalitions. In the last part of the paper, we prove the non-emptiness of the core of this coalition formation game

Contractive probability metrics and asymptotic behavior of dissipative kinetic equations

The present notes are intended to present a detailed review of the existing results in dissipative kinetic theory which make use of the contraction properties of two main families of probability metrics: optimal mass transport and Fourier-based metrics. The first part of the notes is devoted to a self-consistent summary and presentation of the properties of both probability metrics, including new aspects on the relationships between them and other metrics of wide use in probability theory. These results are of independent interest with potential use in other contexts in Partial Differential Equations and Probability Theory. The second part of the notes makes a different presentation of the asymptotic behavior of Inelastic Maxwell Models than the one presented in the literature and it shows a new example of application: particle's bath heating. We show how starting from the contraction properties in probability metrics, one can deduce the existence, uniqueness and asymptotic stability in classical spaces. A global strategy with this aim is set up and applied in two dissipative models.

Model structures on the category of small double categories

In this paper we obtain several model structures on DblCat, the category of small double categories. Our model structures have three sources. We first transfer across a categorification-nerve adjunction. Secondly, we view double categories as internal categories in Cat and take as our weak equivalences various internal equivalences defined via Grothendieck topologies. Thirdly, DblCat inherits a model structure as a category of algebras over a 2-monad. Some of these model structures coincide and the different points of view give us further results about cofibrant replacements and cofi brant objects. As part of this program we give explicit descriptions and discuss properties of free double categories, quotient double categories, colimits of double categories, and several nerves and categorifications.

Characterizations of Lojasiewicz inequalities and applications

The classical Lojasiewicz inequality and its extensions for partial differential equation problems (Simon) and to o-minimal structures (Kurdyka) have a considerable impact on the analysis of gradient-like methods and related problems: minimization methods, complexity theory, asymptotic analysis of dissipative partial differential equations, tame geometry. This paper provides alternative characterizations of this type of inequalities for nonsmooth lower semicontinuous functions defined on a metric or a real Hilbert space. In a metric context, we show that a generalized form of the Lojasiewicz inequality (hereby called the Kurdyka- Lojasiewicz inequality) relates to metric regularity and to the Lipschitz continuity of the sublevel mapping, yielding applications to discrete methods (strong convergence of the proximal algorithm). In a Hilbert setting we further establish that asymptotic properties of the semiflow generated by -∂f are strongly linked to this inequality. This is done by introducing the notion of a piecewise subgradient curve: such curves have uniformly bounded lengths if and only if the Kurdyka- Lojasiewicz inequality is satisfied. Further characterizations in terms of talweg lines -a concept linked to the location of the less steepest points at the level sets of f- and integrability conditions are given. In the convex case these results are significantly reinforced, allowing in particular to establish the asymptotic equivalence of discrete gradient methods and continuous gradient curves. On the other hand, a counterexample of a convex C2 function in R2 is constructed to illustrate the fact that, contrary to our intuition, and unless a specific growth condition is satisfied, convex functions may fail to fulfill the Kurdyka- Lojasiewicz inequality.

Formal deformations of Poisson structures in low dimensions

In this paper, we study formal deformations of Poisson structures, especially for three families of Poisson varieties in dimensions two and three. For these families of Poisson structures, using an explicit basis of the second Poisson cohomology space, we solve the deformation equations at each step and obtain a large family of formal deformations for each Poisson structure which we consider. With the help of an explicit formula, we show that this family contains, modulo equivalence, all possible formal eformations. We show moreover that, when the Poisson structure is generic, all members of the family are non-equivalent.

Free Mobility and Taste-Homogeneity of Jurisdiction Structures

We consider a population of agents distributed on the unit interval. Agents form jurisdictions in order to provide a public facility and share its costs equally. This creates an incentive to form large entities. Individuals also incur a transportation cost depending on their location and that of the facility which makes small jurisdictions advantageous. We consider a fairly general class of distributions of agents and generalize previous versions of this model by allowing for non-linear transportation costs. We show that, in general, jurisdictions are not necessarily homogeneous. However, they are if facilities are always intraterritory and transportation costs are superadditive. Superadditivity can be weakened to strictly increasing and strictly concave when agents are uniformly distributed. Keywords: Consecutiveness, stratification, local public goods, coalition formation, country formation. JEL Classification: C71 (Cooperative Games), D71 (Social Choice; Clubs; Committees; Associations), H73 (Interjurisdictional Differentials and Their Effects).

Moduli of flat conformal structures of hyperbolic type

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Condensation of polyhedric structures onto soap films

We study the existence of solutions to general measure-minimization problems over topological classes that are stable under localized Lipschitz homotopy, including the standard Plateau problem without the need for restrictive assumptions such as orientability or even rectifiability of surfaces. In case of problems over an open and bounded domain we establish the existence of a “minimal candidate”, obtained as the limit for the local Hausdorff convergence of a minimizing sequence for which the measure is lower-semicontinuous. Although we do not give a way to control the topological constraint when taking limit yet— except for some examples of topological classes preserving local separation or for periodic two-dimensional sets — we prove that this candidate is an Almgren-minimal set. Thus, using regularity results such as Jean Taylor’s theorem, this could be a way to find solutions to the above minimization problems under a generic setup in arbitrary dimension and codimension.

Isomorphism relations on computable structures

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Multi-stage oligopoly models with nested logit demand structures: A simplifying approach

Solving multi-stage oligopoly models by backward induction can easily become a com- plex task when rms are multi-product and demands are derived from a nested logit frame- work. This paper shows that under the assumption that within-segment rm shares are equal across segments, the analytical expression for equilibrium pro ts can be substantially simpli ed. The size of the error arising when this condition does not hold perfectly is also computed. Through numerical examples, it is shown that the error is rather small in general. Therefore, using this assumption allows to gain analytical tractability in a class of models that has been used to approach relevant policy questions, such as for example rm entry in an industry or the relation between competition and location. The simplifying approach proposed in this paper is aimed at helping improving these type of models for reaching more accurate recommendations.