164,765 research outputs found

    Restricted density classification in one dimension

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    The density classification task is to determine which of the symbols appearing in an array has the majority. A cellular automaton solving this task is required to converge to a uniform configuration with the majority symbol at each site. It is not known whether a one-dimensional cellular automaton with binary alphabet can classify all Bernoulli random configurations almost surely according to their densities. We show that any cellular automaton that washes out finite islands in linear time classifies all Bernoulli random configurations with parameters close to 0 or 1 almost surely correctly. The proof is a direct application of a "percolation" argument which goes back to Gacs (1986).Comment: 13 pages, 5 figure

    Classification of eight dimensional perfect forms

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    In this paper, we classify the perfect lattices in dimension 8. There are 10916 of them. Our classification heavily relies on exploiting symmetry in polyhedral computations. Here we describe algorithms making the classification possible.Comment: 14 page

    On the Completeness of the Set of Classical W-Algebras Obtained from DS Reductions

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    We clarify the notion of the DS --- generalized Drinfeld-Sokolov --- reduction approach to classical W{\cal W}-algebras. We first strengthen an earlier theorem which showed that an sl(2)sl(2) embedding S⊂G{\cal S}\subset {\cal G} can be associated to every DS reduction. We then use the fact that a \W-algebra must have a quasi-primary basis to derive severe restrictions on the possible reductions corresponding to a given sl(2)sl(2) embedding. In the known DS reductions found to date, for which the \W-algebras are denoted by WSG{\cal W}_{\cal S}^{\cal G}-algebras and are called canonical, the quasi-primary basis corresponds to the highest weights of the sl(2)sl(2). Here we find some examples of noncanonical DS reductions leading to \W-algebras which are direct products of WSG{\cal W}_{\cal S}^{\cal G}-algebras and `free field' algebras with conformal weights Δ∈{0,12,1}\Delta \in \{0, {1\over 2}, 1\}. We also show that if the conformal weights of the generators of a W{\cal W}-algebra obtained from DS reduction are nonnegative Δ≥0\Delta \geq 0 (which isComment: 48 pages, plain TeX, BONN-HE-93-14, DIAS-STP-93-0
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