191 research outputs found

    Generating strings for bipartite Steinhaus graphs

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    AbstractLet b(n) be the number of bipartite Steinhaus graphs with n vertices. We show that b(n) satisfies the recurrence, b(2) = 2, b(3) = 4, and for k ⩾ 2, b(2k + 1) = 2b(k + 1) + 1, b(2k) = b(k) + b(k + 1). Thus b(n) ⩽ 52n − 72 with equality when n is one more than a power of two. To prove this recurrence, we describe the possible generating strings for these bipartite graphs

    Bipartite Steinhaus graphs

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    On skew loops, skew branes and quadratic hypersurfaces

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    A skew brane is an immersed codimension 2 submanifold in affine space, free from pairs of parallel tangent spaces. Using Morse theory, we prove that a skew brane cannot lie on a quadratic hypersurface. We also prove that there are no skew loops on embedded ruled developable discs in 3-space. The paper extends recent work by M. Ghomi and B. Solomon.Comment: 13 pages, 2 figure

    On the Hardness of Gray Code Problems for Combinatorial Objects

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    Can a list of binary strings be ordered so that consecutive strings differ in a single bit? Can a list of permutations be ordered so that consecutive permutations differ by a swap? Can a list of non-crossing set partitions be ordered so that consecutive partitions differ by refinement? These are examples of Gray coding problems: Can a list of combinatorial objects (of a particular type and size) be ordered so that consecutive objects differ by a flip (of a particular type)? For example, 000, 001, 010, 100 is a no instance of the first question, while 1234, 1324, 1243 is a yes instance of the second question due to the order 1243, 1234, 1324. We prove that a variety of Gray coding problems are NP-complete using a new tool we call a Gray code reduction.Comment: 15 pages, 5 figures, WALCOM 202

    Kernel Distribution Embeddings: Universal Kernels, Characteristic Kernels and Kernel Metrics on Distributions

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    Kernel mean embeddings have recently attracted the attention of the machine learning community. They map measures μ\mu from some set MM to functions in a reproducing kernel Hilbert space (RKHS) with kernel kk. The RKHS distance of two mapped measures is a semi-metric dkd_k over MM. We study three questions. (I) For a given kernel, what sets MM can be embedded? (II) When is the embedding injective over MM (in which case dkd_k is a metric)? (III) How does the dkd_k-induced topology compare to other topologies on MM? The existing machine learning literature has addressed these questions in cases where MM is (a subset of) the finite regular Borel measures. We unify, improve and generalise those results. Our approach naturally leads to continuous and possibly even injective embeddings of (Schwartz-) distributions, i.e., generalised measures, but the reader is free to focus on measures only. In particular, we systemise and extend various (partly known) equivalences between different notions of universal, characteristic and strictly positive definite kernels, and show that on an underlying locally compact Hausdorff space, dkd_k metrises the weak convergence of probability measures if and only if kk is continuous and characteristic.Comment: Old and longer version of the JMLR paper with same title (published 2018). Please start with the JMLR version. 55 pages (33 pages main text, 22 pages appendix), 2 tables, 1 figure (in appendix
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