3,396 research outputs found

    Logspace computations in graph products

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    We consider three important and well-studied algorithmic problems in group theory: the word, geodesic, and conjugacy problem. We show transfer results from individual groups to graph products. We concentrate on logspace complexity because the challenge is actually in small complexity classes, only. The most difficult transfer result is for the conjugacy problem. We have a general result for graph products, but even in the special case of a graph group the result is new. Graph groups are closely linked to the theory of Mazurkiewicz traces which form an algebraic model for concurrent processes. Our proofs are combinatorial and based on well-known concepts in trace theory. We also use rewriting techniques over traces. For the group-theoretical part we apply Bass-Serre theory. But as we need explicit formulae and as we design concrete algorithms all our group-theoretical calculations are completely explicit and accessible to non-specialists

    Superposition as a logical glue

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    The typical mathematical language systematically exploits notational and logical abuses whose resolution requires not just the knowledge of domain specific notation and conventions, but not trivial skills in the given mathematical discipline. A large part of this background knowledge is expressed in form of equalities and isomorphisms, allowing mathematicians to freely move between different incarnations of the same entity without even mentioning the transformation. Providing ITP-systems with similar capabilities seems to be a major way to improve their intelligence, and to ease the communication between the user and the machine. The present paper discusses our experience of integration of a superposition calculus within the Matita interactive prover, providing in particular a very flexible, "smart" application tactic, and a simple, innovative approach to automation.Comment: In Proceedings TYPES 2009, arXiv:1103.311

    On the enumeration of closures and environments with an application to random generation

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    Environments and closures are two of the main ingredients of evaluation in lambda-calculus. A closure is a pair consisting of a lambda-term and an environment, whereas an environment is a list of lambda-terms assigned to free variables. In this paper we investigate some dynamic aspects of evaluation in lambda-calculus considering the quantitative, combinatorial properties of environments and closures. Focusing on two classes of environments and closures, namely the so-called plain and closed ones, we consider the problem of their asymptotic counting and effective random generation. We provide an asymptotic approximation of the number of both plain environments and closures of size nn. Using the associated generating functions, we construct effective samplers for both classes of combinatorial structures. Finally, we discuss the related problem of asymptotic counting and random generation of closed environemnts and closures

    Graph products of spheres, associative graded algebras and Hilbert series

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    Given a finite, simple, vertex-weighted graph, we construct a graded associative (non-commutative) algebra, whose generators correspond to vertices and whose ideal of relations has generators that are graded commutators corresponding to edges. We show that the Hilbert series of this algebra is the inverse of the clique polynomial of the graph. Using this result it easy to recognize if the ideal is inert, from which strong results on the algebra follow. Non-commutative Grobner bases play an important role in our proof. There is an interesting application to toric topology. This algebra arises naturally from a partial product of spheres, which is a special case of a generalized moment-angle complex. We apply our result to the loop-space homology of this space.Comment: 19 pages, v3: elaborated on connections to related work, added more citations, to appear in Mathematische Zeitschrif

    VC-saturated set systems

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    The well-known Sauer lemma states that a family F⊆2[n]\mathcal{F}\subseteq 2^{[n]} of VC-dimension at most dd has size at most ∑i=0d(ni)\sum_{i=0}^d\binom{n}{i}. We obtain both random and explicit constructions to prove that the corresponding saturation number, i.e., the size of the smallest maximal family with VC-dimension d≥2d\ge 2, is at most 4d+14^{d+1}, and thus is independent of nn

    Chiral expansion and Macdonald deformation of two-dimensional Yang-Mills theory

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    We derive the analog of the large NN Gross-Taylor holomorphic string expansion for the refinement of qq-deformed U(N)U(N) Yang-Mills theory on a compact oriented Riemann surface. The derivation combines Schur-Weyl duality for quantum groups with the Etingof-Kirillov theory of generalized quantum characters which are related to Macdonald polynomials. In the unrefined limit we reproduce the chiral expansion of qq-deformed Yang-Mills theory derived by de Haro, Ramgoolam and Torrielli. In the classical limit q=1q=1, the expansion defines a new β\beta-deformation of Hurwitz theory wherein the refined partition function is a generating function for certain parameterized Euler characters, which reduce in the unrefined limit β=1\beta=1 to the orbifold Euler characteristics of Hurwitz spaces of holomorphic maps. We discuss the geometrical meaning of our expansions in relation to quantum spectral curves and β\beta-ensembles of matrix models arising in refined topological string theory.Comment: 45 pages; v2: References adde

    Computing Adapted Bases for Conformal Automorphism Groups of Riemann Surfaces

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    The concept of an adapted homology basis for a prime order conformal automorphism of a compact Riemann surface extends to arbitrary finite groups of conformal automorphisms. Here we compute some examples of adapted homology bases for some groups of automorphisms. The method is to begin by apply the Schreier-Reidemeister rewriting process along with the Schreier-Reidemeister Theorem and then to eliminate generators and relations until there is one single large defining relation for the fundamental group in which every generator and its inverse occurs. We are then able to compute the action of the group on the homology image of these generators in the first homology group. The matrix of the action is in a simple form. This has applications to the representation variety.Comment: Typos and spelling error fixed; 17 pages; to appear AMS Conn Math; Proc. Linkopoing Conference, 201
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