3,395 research outputs found
On Khachiyan's algorithm for the computation of minimum-volume enclosing ellipsoids
Cataloged from PDF version of article.Given A := {a(1),..., a(m)} subset of R(d) whose affine hull is R(d), we study the problems of computing an approximate rounding of the convex hull of A and an approximation to the minimum-volume enclosing ellipsoid of V. In the case of centrally symmetric sets, we first establish that Khachiyan's barycentric coordinate descent (BCD) method is exactly the polar of the deepest cut ellipsoid method using two-sided symmetric cuts. This observation gives further insight into the efficient implementation of the BCD method. We then propose a variant algorithm which computes an approximate rounding of the convex hull of,91, and which can also be used to compute an approximation to the minimum-volume enclosing ellipsoid of A.. Our algorithm is a modification of the algorithm of Kumar and Yildirim, which combines Khachiyan's BCD method with a simple initialization scheme to achieve a slightly improved polynomial complexity result, and which returns a small "core set." We establish that our algorithm computes an approximate solution to the dual optimization formulation of the minimum-volume enclosing ellipsoid problem that satisfies a more complete set of approximate optimality conditions than either of the two previous algorithms. Furthermore, this added benefit is achieved without any increase in the improved asymptotic complexity bound of the algorithm of Kumar and Yildirim or any increase in the bound on the size of the computed core set. In addition, the "dropping idea" used in our algorithm has the potential of computing smaller core sets in practice. We also discuss several possible variants of this dropping technique. (C) 2007 Elsevier B.V. All rights reserved
On Khachiyan's Algorithm for the Computation of Minimum Volume Enclosing Ellipsoids
On Khachiyan's Algorithm for the Computation of Minimum Volume Enclosing Ellipsoid
Towards a Microscopic Model of Magnetoelectric Interactions in Ni3V2O8
We develop a microscopic magnetoelectric coupling in NiVO (NVO)
which gives rise to the trilinear phenomenological coupling used previously to
explain the phase transition in which magnetic and ferroelectric order
parameters appear simultaneously. Using combined neutron scattering
measurements and first-principles calculations of the phonons in NVO, we
determine eleven phonons which can induce the observed spontaneous
polarization. Among these eleven phonons, we find that a few of them can
actually induce a significant dipole moment. Using the calculated atomic
charges, we find that the required distortion to induce the observed dipole
moment is very small (~0.001 \AA) and therefore it would be very difficult to
observe the distortion by neutron-powder diffraction. Finally, we identify the
derivatives of the exchange tensor with respect to atomic displacements which
are needed for a microscopic model of a spin-phonon coupling in NVO and which
we hope will be obtained from a fundamental quantum calculation such as LDA+U.
We also analyze two toy models to illustrate that the Dzyaloskinskii-Moriya
interaction is very important for coexisting of magnetic and ferroelectric
order but it is not the only mechanism when the local site symmetry of the
system is low enough.Comment: 20 pages, 10 figure
Anisotropic Spin Hamiltonians due to Spin-Orbit and Coulomb Exchange Interactions
This paper contains the details of Phys. Rev. Lett. 73, 2919 (1994) and, to a
lesser extent, Phys. Rev. Lett. 72, 3710 (1994). We treat a Hubbard model which
includes all the 3d states of the Cu ions and the 2p states of the O ions. We
also include spin-orbit interactions, hopping between ground and excited
crystal field states of the Cu ions, and rather general Coulomb interactions.
Our analytic results for the spin Hamiltonian, H, are corroborated by numerical
evaluations of the energy splitting of the ground manifold for two holes on
either a pair of Cu ions or a Cu-O-Cu complex. In the tetragonal symmetry case
and for the model considered, we prove that H is rotationally invariant in the
absence of Coulomb exchange. When Coulomb exchange is present, each bond
Hamiltonian has full biaxial anisotropy, as expected for this symmetry. For
lower symmetry situations, the single bond spin Hamiltonian is anisotropic at
order t**6 for constant U and at order t**2 for nonconstant U. (Constant U
means that the Coulomb interaction between orbitals does not depend on which
orbitals are involved.)Comment: 50 pages, ILATEX Version 2.09 <13 Jun 1989
New approximations for the cone of copositive matrices and its dual
We provide convergent hierarchies for the cone C of copositive matrices and
its dual, the cone of completely positive matrices. In both cases the
corresponding hierarchy consists of nested spectrahedra and provide outer
(resp. inner) approximations for C (resp. for its dual), thus complementing
previous inner (resp. outer) approximations for C (for the dual). In
particular, both inner and outer approximations have a very simple
interpretation. Finally, extension to K-copositivity and K-complete positivity
for a closed convex cone K, is straightforward.Comment: 8
A New Method of Probing the Phonon Mechanism in Superconductors including MgB
Weak localization has a strong influence on both the normal and
superconducting properties of metals. In particular, since weak localization
leads to the decoupling of electrons and phonons, the temperature dependence of
resistance (i.e., ) is decreasing with increasing disorder, as
manifested by Mooij's empirical rule. In addition, Testardi's universal
correlation of (i.e., ) and the resistance ratio (i.e.,
) follows. This understanding provides a new means to probe the
phonon mechanism in superconductors including MgB. The merits of this
method are its applicability to any superconductors and its reliability because
the McMillan's electron-phonon coupling constant and
change in a broad range, from finite values to zero, due to weak localization.
Karkin et al's preliminary data of irradiated MgB show the Testardi
correlation, indicating that the dominant pairing mechanism in MgB is the
phonon-mediated interaction.Comment: 9 pages, latex, 3 figure
Magnetic anisotropies and general on--site Coulomb interactions in the cuprates
This paper derives the anisotropic superexchange interactions from a Hubbard
model for excitations within the copper 3d band and the oxygen 2p band of the
undoped insulating cuprates. We extend the recent calculation of Yildirim et
al. [Phys. Rev. B {\bf VV}, pp, 1995] in order to include the most general
on--site Coulomb interactions (including those which involve more than two
orbitals) when two holes occupy the same site. Our general results apply when
the oxygen ions surrounding the copper ions form an octahedron which has
tetragonal symmetry (but may be rotated as in lanthanum cuprate). For the
tetragonal cuprates we obtain an easy--plane anisotropy in good agreement with
experimental values. We predict the magnitude of the small in--plane anisoComment: 25 pages, revte
Magnetic Excitations of Undoped Iron Oxypnictides
We study the magnetic excitations of undoped iron oxypnictides using a
three-dimensional Heisenberg model with single-ion anisotropy. Analytic forms
of the spin wave dispersion, velocities, and structure factor are given. Aside
from quantitative comparisons which can be made to inelastic neutron scattering
experiments, we also give qualitative criteria which can distinguish various
regimes of coupling strength. The magnetization reduction due to quantum zero
point fluctuations shows clear dependence on the c-axis coupling.Comment: 4 pages, 5 figures, to appear in Frontiers of Physics in China: a
special issue on Iron-based superconductor
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