2,388 research outputs found
Who's Afraid of the Hill Boundary?
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
Mader proved that for every integer there is a smallest real number
such that any graph with average degree at least must contain a
-minor. Fiorini, Joret, Theis and Wood conjectured that any graph with
vertices and average degree at least must contain a -minor
consisting of at most vertices. Shapira and Sudakov
subsequently proved that such a graph contains a -minor consisting of at
most vertices. Here we build on their method
using graph expansion to remove the factor and prove the
conjecture.
Mader also proved that for every integer there is a smallest real number
such that any graph with average degree larger than must contain
a -topological minor. We prove that, for sufficiently large , graphs
with average degree at least contain a -topological
minor consisting of at most vertices. Finally, we show
that, for sufficiently large , graphs with average degree at least
contain either a -minor consisting of at most
vertices or a -topological minor consisting of at most
vertices.Comment: 19 page
Oscillating about coplanarity in the 4 body problem
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 body problem in -space. The
proof, for , starts by identifying the center-of-mass zero configuration
space with real matrices, the coplanar configurations with
matrices whose determinant is zero, and the mass metric with the Frobenius
(standard Euclidean) norm. Let 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 along solutions. Bounds on
inter-body distances then yield an explicit lower bound for the
frequency of this oscillator, guaranteeing a degeneration within every time
interval of length . 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
When , the tree consists of a path containing
vertices, each of whose vertices has a disjoint path length
beginning at it. We show that, for any and , the binomial
random graph almost surely contains
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 disjoint bare paths length .
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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