2,960 research outputs found
Sharpening Geometric Inequalities using Computable Symmetry Measures
Many classical geometric inequalities on functionals of convex bodies depend
on the dimension of the ambient space. We show that this dimension dependence
may often be replaced (totally or partially) by different symmetry measures of
the convex body. Since these coefficients are bounded by the dimension but
possibly smaller, our inequalities sharpen the original ones. Since they can
often be computed efficiently, the improved bounds may also be used to obtain
better bounds in approximation algorithms.Comment: This is a preprint. The proper publication in final form is available
at journals.cambridge.org, DOI 10.1112/S002557931400029
Convex bodies of states and maps
We give a general solution to the question when the convex hulls of orbits of
quantum states on a finite-dimensional Hilbert space under unitary actions of a
compact group have a non-empty interior in the surrounding space of all density
states. The same approach can be applied to study convex combinations of
quantum channels. The importance of both problems stems from the fact that,
usually, only sets with non-vanishing volumes in the embedding spaces of all
states or channels are of practical importance. For the group of local
transformations on a bipartite system we characterize maximally entangled
states by properties of a convex hull of orbits through them. We also compare
two partial characteristics of convex bodies in terms of largest balls and
maximum volume ellipsoids contained in them and show that, in general, they do
not coincide. Separable states, mixed-unitary channels and k-entangled states
are also considered as examples of our techniques.Comment: 18 pages, 1 figur
A new Lenstra-type Algorithm for Quasiconvex Polynomial Integer Minimization with Complexity 2^O(n log n)
We study the integer minimization of a quasiconvex polynomial with
quasiconvex polynomial constraints. We propose a new algorithm that is an
improvement upon the best known algorithm due to Heinz (Journal of Complexity,
2005). This improvement is achieved by applying a new modern Lenstra-type
algorithm, finding optimal ellipsoid roundings, and considering sparse
encodings of polynomials. For the bounded case, our algorithm attains a
time-complexity of s (r l M d)^{O(1)} 2^{2n log_2(n) + O(n)} when M is a bound
on the number of monomials in each polynomial and r is the binary encoding
length of a bound on the feasible region. In the general case, s l^{O(1)}
d^{O(n)} 2^{2n log_2(n) +O(n)}. In each we assume d>= 2 is a bound on the total
degree of the polynomials and l bounds the maximum binary encoding size of the
input.Comment: 28 pages, 10 figure
Equations, inequations and inequalities characterizing the configurations of two real projective conics
Couples of proper, non-empty real projective conics can be classified modulo
rigid isotopy and ambient isotopy.
We characterize the classes by equations, inequations and inequalities in the
coefficients of the quadratic forms defining the conics.
The results are well--adapted to the study of the relative position of two
conics defined by equations depending on parameters.Comment: 31 pages. See also
http://emmanuel.jean.briand.free.fr/publications/twoconics/ Added references
to important prior work on the subject. The title changed accordingly. Some
typos and imprecisions corrected. To be published in Applicable Algebra in
Engineering, Communication and Computin
Equilibrium Configurations of Synchronous Binaries: Numerical Solutions and Application to Kuiper-Belt Binary 2001 QG298
We present numerical computations of the equilibrium configurations of
tidally-locked homogeneous binaries, rotating in circular orbits. Unlike the
classical Roche approximations, we self-consistently account for the tidal and
rotational deformations of both components, and relax the assumptions of
ellipsoidal configurations and Keplerian rotation. We find numerical solutions
for mass ratios q between 1e-3 and 1, starting at a small angular velocity for
which tidal and rotational deformations are small, and following a sequence of
increasing angular velocities. Each series terminates at an appropriate ``Roche
limit'', above which no equilibrium solution can be found. Even though the
Roche limit is crossed before the ``Roche lobe'' is filled, any further
increase in the angular velocity will result in mass-loss. For close,
comparable-mass binaries, we find that local deviations from ellipsoidal forms
may be as large as 10-20%, and departures from Keplerian rotation are
significant. We compute the light curves that arise from our equilibrium
configurations, assuming their distance is >>1 AU (e.g. in the Kuiper Belt). We
consider both backscatter (proportional to the projected area) and diffuse
(Lambert) reflections. Backscatter reflection always yields two minima of equal
depths. Diffuse reflection, which is sensitive to the surface curvature,
generally gives rise to unequal minima. We find detectable intensity
differences of up to 10% between our light curves and those arising from the
Roche approximations. Finally, we apply our models to Kuiper Belt binary 2001
QG298, and find a nearly edge-on binary with a mass ratio q = 0.93
^{+0.07}_{-0.03}, angular velocity Omega^2/G rho = 0.333+/-0.001 (statistical
errors only), and pure diffuse reflection. For the observed period of 2001
QG298, these parameters imply a bulk density, rho = 0.72 +/- 0.04 g cm^-3.Comment: Accepted to Ap
Data depth and floating body
Little known relations of the renown concept of the halfspace depth for
multivariate data with notions from convex and affine geometry are discussed.
Halfspace depth may be regarded as a measure of symmetry for random vectors. As
such, the depth stands as a generalization of a measure of symmetry for convex
sets, well studied in geometry. Under a mild assumption, the upper level sets
of the halfspace depth coincide with the convex floating bodies used in the
definition of the affine surface area for convex bodies in Euclidean spaces.
These connections enable us to partially resolve some persistent open problems
regarding theoretical properties of the depth
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