9,080 research outputs found

    Involutive Division Technique: Some Generalizations and Optimizations

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    In this paper, in addition to the earlier introduced involutive divisions, we consider a new class of divisions induced by admissible monomial orderings. We prove that these divisions are noetherian and constructive. Thereby each of them allows one to compute an involutive Groebner basis of a polynomial ideal by sequentially examining multiplicative reductions of nonmultiplicative prolongations. We study dependence of involutive algorithms on the completion ordering. Based on properties of particular involutive divisions two computational optimizations are suggested. One of them consists in a special choice of the completion ordering. Another optimization is related to recomputing multiplicative and nonmultiplicative variables in the course of the algorithm.Comment: 19 page

    Generalized constructive tree weights

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    The Loop Vertex Expansion (LVE) is a quantum field theory (QFT) method which explicitly computes the Borel sum of Feynman perturbation series. This LVE relies in a crucial way on symmetric tree weights which define a measure on the set of spanning trees of any connected graph. In this paper we generalize this method by defining new tree weights. They depend on the choice of a partition of a set of vertices of the graph, and when the partition is non-trivial, they are no longer symmetric under permutation of vertices. Nevertheless we prove they have the required positivity property to lead to a convergent LVE; in fact, we formulate this positivity property precisely for the first time. Our generalized tree weights are inspired by the Brydges-Battle-Federbush work on cluster expansions and could be particularly suited to the computation of connected functions in QFT. Several concrete examples are explicitly given.Comment: 22 pages, 2 figure

    Understanding Algorithm Performance on an Oversubscribed Scheduling Application

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    The best performing algorithms for a particular oversubscribed scheduling application, Air Force Satellite Control Network (AFSCN) scheduling, appear to have little in common. Yet, through careful experimentation and modeling of performance in real problem instances, we can relate characteristics of the best algorithms to characteristics of the application. In particular, we find that plateaus dominate the search spaces (thus favoring algorithms that make larger changes to solutions) and that some randomization in exploration is critical to good performance (due to the lack of gradient information on the plateaus). Based on our explanations of algorithm performance, we develop a new algorithm that combines characteristics of the best performers; the new algorithms performance is better than the previous best. We show how hypothesis driven experimentation and search modeling can both explain algorithm performance and motivate the design of a new algorithm

    Sunspots do matter: a simple disproof of Mas-Colell

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    Mas-Colell conjectured that, for economies admitting a unique deterministic equilibrium, sunspot equilibria that 'matter'' would not exist. I prove an abundance of such equilibria using simple linear algebra.Animal Spirits
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