768 research outputs found

    Disjointly representing set systems

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    AbstractA family F of sets is s-disjointly representable if there is a family S of disjoint sets each of size s such that every F∈F contains some S∈S. Let f(r,s) be the minimum size of a family F of r-sets which is not s-disjointly representable. We give upper and lower bounds on f(r,s) which are within a constant factor when s is fixed

    Homotopy versus isotopy: spheres with duals in 4-manifolds

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    David Gabai recently proved a smooth 4-dimensional "Light Bulb Theorem" in the absence of 2-torsion in the fundamental group. We extend his result to 4-manifolds with arbitrary fundamental group by showing that an invariant of Mike Freedman and Frank Quinn gives the complete obstruction to "homotopy implies isotopy" for embedded 2-spheres which have a common geometric dual. The invariant takes values in an Z/2Z-vector space generated by elements of order 2 in the fundamental group and has applications to unknotting numbers and pseudo-isotopy classes of self-diffeomorphisms. Our methods also give an alternative approach to Gabai's theorem using various maneuvers with Whitney disks and a fundamental isotopy between surgeries along dual circles in an orientable surface.Comment: Included into section 2 of this version is a proof that the operation of `sliding a Whitney disk over itself' preserves the isotopy class of the resulting Whitney move in the current setting. Some expository clarifications have also been added. Main results and proofs are unchanged from the previous version. 39 pages, 25 figure

    The sleep cycle: a mathematical analysis from a global workspace perspective

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    Dretske's technique of invoking necessary conditions from information theory to describe mental process can be used to derive a version of Hobson's AIM treatment of the sleep/wake cycle from a mathematical formulation of Baars' Global Workspace model of consciousness. One implication of the analysis is that some sleep disorders may be recognizably similar to many other chronic, developmental dysfunctions, including autoimmune and coronary heart disease, obesity, hypertension, and anxiety disorder, in that these afflictions often have roots in utero or adverse early childhood experiences or exposures to systematic patterns of structured stress. Identification and alteration of such factors might have considerable impact on population-level patterns of sleep disorders, suggesting the possibility of a public health approach rather than current exorbitantly expensive case-by-case medical intervention

    Improved Cheeger's Inequality: Analysis of Spectral Partitioning Algorithms through Higher Order Spectral Gap

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    Let \phi(G) be the minimum conductance of an undirected graph G, and let 0=\lambda_1 <= \lambda_2 <=... <= \lambda_n <= 2 be the eigenvalues of the normalized Laplacian matrix of G. We prove that for any graph G and any k >= 2, \phi(G) = O(k) \lambda_2 / \sqrt{\lambda_k}, and this performance guarantee is achieved by the spectral partitioning algorithm. This improves Cheeger's inequality, and the bound is optimal up to a constant factor for any k. Our result shows that the spectral partitioning algorithm is a constant factor approximation algorithm for finding a sparse cut if \lambda_k$ is a constant for some constant k. This provides some theoretical justification to its empirical performance in image segmentation and clustering problems. We extend the analysis to other graph partitioning problems, including multi-way partition, balanced separator, and maximum cut

    Algebraic and Geometric intersection numbers for free groups

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    We show that the algebraic intersection number of Scott and Swarup for splittings of free groups coincides with the geometric intersection number for the sphere complex of the connected sum of copies of S2×S1S^2\times S^1.Comment: 7 page
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