4,702 research outputs found
About the Algebraic Solutions of Smallest Enclosing Cylinders Problems
Given n points in Euclidean space E^d, we propose an algebraic algorithm to
compute the best fitting (d-1)-cylinder. This algorithm computes the unknown
direction of the axis of the cylinder. The location of the axis and the radius
of the cylinder are deduced analytically from this direction. Special attention
is paid to the case d=3 when n=4 and n=5. For the former, the minimal radius
enclosing cylinder is computed algebrically from constrained minimization of a
quartic form of the unknown direction of the axis. For the latter, an
analytical condition of existence of the circumscribed cylinder is given, and
the algorithm reduces to find the zeroes of an one unknown polynomial of degree
at most 6. In both cases, the other parameters of the cylinder are deduced
analytically. The minimal radius enclosing cylinder is computed analytically
for the regular tetrahedron and for a trigonal bipyramids family with a
symmetry axis of order 3.Comment: 13 pages, 0 figure; revised version submitted to publication
(previous version is a copy of the original one of 2010
Radii minimal projections of polytopes and constrained optimization of symmetric polynomials
We provide a characterization of the radii minimal projections of polytopes
onto -dimensional subspaces in Euclidean space \E^n. Applied on simplices
this characterization allows to reduce the computation of an outer radius to a
computation in the circumscribing case or to the computation of an outer radius
of a lower-dimensional simplex. In the second part of the paper, we use this
characterization to determine the sequence of outer -radii of regular
simplices (which are the radii of smallest enclosing cylinders). This settles a
question which arose from the incidence that a paper by Wei{\ss}bach (1983) on
this determination was erroneous. In the proof, we first reduce the problem to
a constrained optimization problem of symmetric polynomials and then to an
optimization problem in a fixed number of variables with additional integer
constraints.Comment: Minor revisions. To appear in Advances in Geometr
The Ekman-Hartmann layer in MHD Taylor-Couette flow
We study magnetic effects induced by rigidly rotating plates enclosing a
cylindrical MHD Taylor-Couette flow at the finite aspect ratio . The
fluid confined between the cylinders is assumed to be liquid metal
characterized by small magnetic Prandtl number, the cylinders are perfectly
conducting, an axial magnetic field is imposed \Ha \approx 10, the rotation
rates correspond to \Rey of order . We show that the end-plates
introduce, besides the well known Ekman circulation, similar magnetic effects
which arise for infinite, rotating plates, horizontally unbounded by any walls.
In particular there exists the Hartmann current which penetrates the fluid,
turns into the radial direction and together with the applied magnetic field
gives rise to a force. Consequently the flow can be compared with a Taylor-Dean
flow driven by an azimuthal pressure gradient. We analyze stability of such
flows and show that the currents induced by the plates can give rise to
instability for the considered parameters. When designing an MHD Taylor-Couette
experiment, a special care must be taken concerning the vertical magnetic
boundaries so they do not significantly alter the rotational profile.Comment: 9 pages, 6 figures; accepted to PR
Non-equilibrium electromagnetic fluctuations: Heat transfer and interactions
The Casimir force between arbitrary objects in equilibrium is related to
scattering from individual bodies. We extend this approach to heat transfer and
Casimir forces in non-equilibrium cases where each body, and the environment,
is at a different temperature. The formalism tracks the radiation from each
body and its scatterings by the other objects. We discuss the radiation from a
cylinder, emphasizing its polarized nature, and obtain the heat transfer
between a sphere and a plate, demonstrating the validity of proximity transfer
approximation at close separations and arbitrary temperatures.Comment: 4 pages, 2 figures, published version, minor changes (e.g. typos
Analysis of Incomplete Data and an Intrinsic-Dimension Helly Theorem
The analysis of incomplete data is a long-standing challenge in practical statistics. When, as is typical, data objects are represented by points in R^d , incomplete data objects correspond to affine subspaces (lines or Î-flats).With this motivation we study the problem of finding the minimum intersection radius r(L) of a set of lines or Î-flats L: the least r such that there is a ball of radius r intersecting every flat in L. Known algorithms for finding the minimum enclosing ball for a point set (or clustering by several balls) do not easily extend to higher dimensional flats, primarily because âdistancesâ between flats do not satisfy the triangle inequality. In this paper we show how to restore geometry (i.e., a substitute for the triangle inequality) to the problem, through a new analog of Hellyâs theorem. This âintrinsic-dimensionâ Helly theorem states: for any family L of Î-dimensional convex sets in a Hilbert space, there exist Î + 2 sets L' â L such that r(L) †2r(L'). Based upon this we present
an algorithm that computes a (1+Δ)-core set L' â L, |L'| = O(Î^4/Δ), such that the ball centered at a point c with radius (1 +Δ)r(L') intersects every element of L. The running time of the algorithm is O(n^(Î+1)dpoly(Î/Δ)). For the case of lines or line segments (Î = 1), the (expected) running time of the algorithm can be improved to O(ndpoly(1/Δ)).We note that the size of the core set depends only on the dimension of the input objects and is independent of the input size n and the dimension d of the ambient space
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