7,274 research outputs found

    Local conditions for exponentially many subdivisions

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    Given a graph F, let st(F) be the number of subdivisions of F, each with a different vertex set, which one can guarantee in a graph G in which every edge lies in at least t copies of F. In 1990, Tuza asked for which graphs F and large t, one has that st(F) is exponential in a power of t. We show that, somewhat surprisingly, the only such F are complete graphs, and for every F which is not complete, st(F) is polynomial in t. Further, for a natural strengthening of the local condition above, we also characterize those F for which st(F) is exponential in a power of t

    Isotopic Equivalence from Bezier Curve Subdivision

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    We prove that the control polygon of a Bezier curve B becomes homeomorphic and ambient isotopic to B via subdivision, and we provide closed-form formulas to compute the number of iterations to ensure these topological characteristics. We first show that the exterior angles of control polygons converge exponentially to zero under subdivision.Comment: arXiv admin note: substantial text overlap with arXiv:1211.035

    How Close to Two Dimensions Does a Lennard-Jones System Need to Be to Produce a Hexatic Phase?

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    We report on a computer simulation study of a Lennard-Jones liquid confined in a narrow slit pore with tunable attractive walls. In order to investigate how freezing in this system occurs, we perform an analysis using different order parameters. Although some of the parameters indicate that the system goes through a hexatic phase, other parameters do not. This shows that to be certain whether a system has a hexatic phase, one needs to study not only a large system, but also several order parameters to check all necessary properties. We find that the Binder cumulant is the most reliable one to prove the existence of a hexatic phase. We observe an intermediate hexatic phase only in a monolayer of particles confined such that the fluctuations in the positions perpendicular to the walls are less then 0.15 particle diameters, i. e. if the system is practically perfectly 2d

    Tubular Neighborhoods of Nodal Sets and Diophantine Approximation

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    We give upper and lower bounds on the volume of a tubular neighborhood of the nodal set of an eigenfunction of the Laplacian on a real analytic closed Riemannian manifold M. As an application we consider the question of approximating points on M by nodal sets, and explore analogy with approximation by rational numbers.Comment: 22 pages; revised version containing full proof of lower bound; reference added; to appear in Amer. J. Math

    Digraphs and cycle polynomials for free-by-cyclic groups

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    Let \phi \in \mbox{Out}(F_n) be a free group outer automorphism that can be represented by an expanding, irreducible train-track map. The automorphism ϕ\phi determines a free-by-cyclic group Γ=Fn⋊ϕZ,\Gamma=F_n \rtimes_\phi \mathbb Z, and a homomorphism α∈H1(Γ;Z)\alpha \in H^1(\Gamma; \mathbb Z). By work of Neumann, Bieri-Neumann-Strebel and Dowdall-Kapovich-Leininger, α\alpha has an open cone neighborhood A\mathcal A in H1(Γ;R)H^1(\Gamma;\mathbb R) whose integral points correspond to other fibrations of Γ\Gamma whose associated outer automorphisms are themselves representable by expanding irreducible train-track maps. In this paper, we define an analog of McMullen's Teichm\"uller polynomial that computes the dilatations of all outer automorphism in A\mathcal A.Comment: 41 pages, 20 figure
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