29,784 research outputs found

    Periodicity in the cohomology of symmetric groups via divided powers

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    A famous theorem of Nakaoka asserts that the cohomology of the symmetric group stabilizes. The first author generalized this theorem to non-trivial coefficient systems, in the form of FI\mathrm{FI}-modules over a field, though one now obtains periodicity of the cohomology instead of stability. In this paper, we further refine these results. Our main theorem states that if MM is a finitely generated FI\mathrm{FI}-module over a noetherian ring k\mathbf{k} then ⚁n≄0Ht(Sn,Mn)\bigoplus_{n \ge 0} \mathrm{H}^t(S_n, M_n) admits the structure of a D\mathbf{D}-module, where D\mathbf{D} is the divided power algebra over k\mathbf{k} in a single variable, and moreover, this D\mathbf{D}-module is "nearly" finitely presented. This immediately recovers the periodicity result when k\mathbf{k} is a field, but also shows, for example, how the torsion varies with nn when k=Z\mathbf{k}=\mathbf{Z}. Using the theory of connections on D\mathbf{D}-modules, we establish sharp bounds on the period in the case where k\mathbf{k} is a field. We apply our theory to obtain results on the modular cohomology of Specht modules and the integral cohomology of unordered configuration spaces of manifolds.Comment: Fixed some minor mistakes and expanded the section on configuration space

    Structural operational semantics for stochastic and weighted transition systems

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    We introduce weighted GSOS, a general syntactic framework to specify well-behaved transition systems where transitions are equipped with weights coming from a commutative monoid. We prove that weighted bisimilarity is a congruence on systems defined by weighted GSOS specifications. We illustrate the flexibility of the framework by instantiating it to handle some special cases, most notably that of stochastic transition systems. Through examples we provide weighted-GSOS definitions for common stochastic operators in the literature

    Only rational homology spheres admit Ω(f)\Omega(f) to be union of DE attractors

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    If there exists a diffeomorphism ff on a closed, orientable nn-manifold MM such that the non-wandering set Ω(f)\Omega(f) consists of finitely many orientable (±)(\pm) attractors derived from expanding maps, then MM must be a rational homology sphere; moreover all those attractors are of topological dimension n−2n-2. Expanding maps are expanding on (co)homologies.Comment: 23 pages, 2 figure

    The Constructive method for query containment checking (extended version)

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    We present a new method that checks Query Containment for queries with negated derived atoms and/or integrity constraints. Existing methods for Query Containment checking that deal with these cases do not check actually containment but another related property called uniform containment, which is a sufficient but not necessary condition for containment. Our method can be seen as an extension of the canonical databases approach beyond the class of conjunctive queries.Postprint (published version

    Discrete orbits, recurrence and solvable subgroups of Diff(C^2,0)

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    We discuss the local dynamics of a subgroup of Diff(C^2,0) possessing locally discrete orbits as well as the structure of the recurrent set for more general groups. It is proved, in particular, that a subgroup of Diff(C^2,0) possessing locally discrete orbits must be virtually solvable. These results are of considerable interest in problems concerning integrable systems.Comment: The first version of this paper and "A note on integrability and finite orbits for subgroups of Diff(C^n,0)" are an expanded version of our paper "Discrete orbits and special subgroups of Diff(C^n,0)". An intermediate version re-submitted to the journal on March 2015 is available at http://www.fep.up.pt/docentes/hreis/publications.htm where there is also a comparison between these 3 version
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