3,155 research outputs found

    Finite Approximations to Quantum Physics: Quantum Points and their Bundles

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    There exists a physically well motivated method for approximating manifolds by certain topological spaces with a finite or a countable set of points. These spaces, which are partially ordered sets (posets) have the power to effectively reproduce important topological features of continuum physics like winding numbers and fractional statistics, and that too often with just a few points. In this work, we develop the essential tools for doing quantum physics on posets. The poset approach to covering space quantization, soliton physics, gauge theories and the Dirac equation are discussed with emphasis on physically important topological aspects. These ideas are illustrated by simple examples like the covering space quantization of a particle on a circle, and the sine-Gordon solitons.Comment: 24 pages, 8 figures on a uuencoded postscript file, DSF-T-29/93, INFN-NA-IV-29/93 and SU-4240-55

    On the Duality of Semiantichains and Unichain Coverings

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    We study a min-max relation conjectured by Saks and West: For any two posets PP and QQ the size of a maximum semiantichain and the size of a minimum unichain covering in the product P×QP\times Q are equal. For positive we state conditions on PP and QQ that imply the min-max relation. Based on these conditions we identify some new families of posets where the conjecture holds and get easy proofs for several instances where the conjecture had been verified before. However, we also have examples showing that in general the min-max relation is false, i.e., we disprove the Saks-West conjecture.Comment: 10 pages, 3 figure

    Neighborhood complexes and Kronecker double coverings

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    The neighborhood complex N(G)N(G) is a simplicial complex assigned to a graph GG whose connectivity gives a lower bound for the chromatic number of GG. We show that if the Kronecker double coverings of graphs are isomorphic, then their neighborhood complexes are isomorphic. As an application, for integers mm and nn greater than 2, we construct connected graphs GG and HH such that N(G)≅N(H)N(G) \cong N(H) but χ(G)=m\chi(G) = m and χ(H)=n\chi(H) = n. We also construct a graph KGn,k′KG_{n,k}' such that KGn,k′KG_{n,k}' and the Kneser graph KGn,kKG_{n,k} are not isomorphic but their Kronecker double coverings are isomorphic.Comment: 10 pages. Some results concerning box complexes are deleted. to appear in Osaka J. Mat
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