2,388 research outputs found

    Who's Afraid of the Hill Boundary?

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    The Jacobi-Maupertuis metric allows one to reformulate Newton's equations as geodesic equations for a Riemannian metric which degenerates at the Hill boundary. We prove that a JM geodesic which comes sufficiently close to a regular point of the boundary contains pairs of conjugate points close to the boundary. We prove the conjugate locus of any point near enough to the boundary is a hypersurface tangent to the boundary. Our method of proof is to reduce analysis of geodesics near the boundary to that of solutions to Newton's equations in the simplest model case: a constant force. This model case is equivalent to the beginning physics problem of throwing balls upward from a fixed point at fixed speeds and describing the resulting arcs, see Fig. 2

    Logarithmically-small Minors and Topological Minors

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    Mader proved that for every integer tt there is a smallest real number c(t)c(t) such that any graph with average degree at least c(t)c(t) must contain a KtK_t-minor. Fiorini, Joret, Theis and Wood conjectured that any graph with nn vertices and average degree at least c(t)+ϵc(t)+\epsilon must contain a KtK_t-minor consisting of at most C(ϵ,t)lognC(\epsilon,t)\log n vertices. Shapira and Sudakov subsequently proved that such a graph contains a KtK_t-minor consisting of at most C(ϵ,t)lognloglognC(\epsilon,t)\log n \log\log n vertices. Here we build on their method using graph expansion to remove the loglogn\log\log n factor and prove the conjecture. Mader also proved that for every integer tt there is a smallest real number s(t)s(t) such that any graph with average degree larger than s(t)s(t) must contain a KtK_t-topological minor. We prove that, for sufficiently large tt, graphs with average degree at least (1+ϵ)s(t)(1+\epsilon)s(t) contain a KtK_t-topological minor consisting of at most C(ϵ,t)lognC(\epsilon,t)\log n vertices. Finally, we show that, for sufficiently large tt, graphs with average degree at least (1+ϵ)c(t)(1+\epsilon)c(t) contain either a KtK_t-minor consisting of at most C(ϵ,t)C(\epsilon,t) vertices or a KtK_t-topological minor consisting of at most C(ϵ,t)lognC(\epsilon,t)\log n vertices.Comment: 19 page

    Oscillating about coplanarity in the 4 body problem

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    For the Newtonian 4-body problem in space we prove that any zero angular momentum bounded solution suffers infinitely many coplanar instants, that is, times at which all 4 bodies lie in the same plane. This result generalizes a known result for collinear instants ("syzygies") in the zero angular momentum planar 3-body problem, and extends to the d+1d+1 body problem in dd-space. The proof, for d=3d=3, starts by identifying the center-of-mass zero configuration space with real 3×33 \times 3 matrices, the coplanar configurations with matrices whose determinant is zero, and the mass metric with the Frobenius (standard Euclidean) norm. Let SS denote the signed distance from a matrix to the hypersurface of matrices with determinant zero. The proof hinges on establishing a harmonic oscillator type ODE for SS along solutions. Bounds on inter-body distances then yield an explicit lower bound ω\omega for the frequency of this oscillator, guaranteeing a degeneration within every time interval of length π/ω\pi/\omega. The non-negativity of the curvature of oriented shape space (the quotient of configuration space by the rotation group) plays a crucial role in the proof.Comment: 26 pages, 5 figure

    Sharp threshold for embedding combs and other spanning trees in random graphs

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    When knk|n, the tree Combn,k\mathrm{Comb}_{n,k} consists of a path containing n/kn/k vertices, each of whose vertices has a disjoint path length k1k-1 beginning at it. We show that, for any k=k(n)k=k(n) and ϵ>0\epsilon>0, the binomial random graph G(n,(1+ϵ)logn/n)\mathcal{G}(n,(1+\epsilon)\log n/ n) almost surely contains Combn,k\mathrm{Comb}_{n,k} as a subgraph. This improves a recent result of Kahn, Lubetzky and Wormald. We prove a similar statement for a more general class of trees containing both these combs and all bounded degree spanning trees which have at least ϵn/log9n\epsilon n/ \log^9n disjoint bare paths length log9n\lceil\log^9 n\rceil. We also give an efficient method for finding large expander subgraphs in a binomial random graph. This allows us to improve a result on almost spanning trees by Balogh, Csaba, Pei and Samotij.Comment: 20 page
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