49,383 research outputs found
Reversal of the circulation of a vortex by quantum tunneling in trapped Bose systems
We study the quantum dynamics of a model for a vortex in a Bose gas with
repulsive interactions in an anisotropic, harmonic trap. By solving the
Schr\"odinger equation numerically, we show that the circulation of the vortex
can undergo periodic reversals by quantum-mechanical tunneling. With increasing
interaction strength or particle number, vortices become increasingly stable,
and the period for reversals increases. Tunneling between vortex and antivortex
states is shown to be described to a good approximation by a superposition of
vortex and antivortex states (a Schr\"odinger cat state), rather than the
mean-field state, and we derive an analytical expression for the oscillation
period. The problem is shown to be equivalent to that of the two-site Bose
Hubbard model with attractive interactions.Comment: 5 pages, 5 figures; published in Phys. Rev. A, Rapid Communication
The central simple modules of Artinian Gorenstein algebras
Let A be a standard graded Artinian algebra over a field of characteristic
zero and let z be a linear form in A. We define the central simple modules for
each such pair (A, z). Assume that A is Gorenstein. Then we prove that A has
the strong Lefschetz property if and only if there exists a linear form z in A
such that all central simple modules of the pair (A,z) have the strong
Lefschetz property. In the course of proof we need to extend the definition of
the strong Lefschetz property to finite graded modules over graded Artinian
algebra, which previously was defined only for standard graded Artinian
algebras.Comment: 20 pages, To be published in Journal of Pure and Applied Algebr
Superfluid Density of Neutrons in the Inner Crust of Neutron Stars: New Life for Pulsar Glitch Models
Calculations of the effects of band structure on the neutron superfluid
density in the crust of neutron stars made under the assumption that the
effects of pairing are small [N. Chamel, Phys. Rev. C 85, 035801 (2012)] lead
to moments of inertia of superfluid neutrons so small that the crust alone is
insufficient to account for the magnitude of neutron star glitches. Inspired by
earlier work on ultracold atomic gases in an optical lattice, we investigate
fermions with attractive interactions in a periodic lattice in the mean-field
approximation. The effects of band structure are suppressed when the pairing
gap is of order or greater than the strength of the lattice potential. By
applying the results to the inner crust of neutron stars, we conclude that the
reduction of the neutron superfluid density is considerably less than
previously estimated and, consequently, it is premature to rule out models of
glitches based on neutron superfluidity in the crust.Comment: 5 pages, 3 figure
The Weak and Strong Lefschetz Properties for Artinian K-Algebras
Let A = bigoplus_{i >= 0} A_i be a standard graded Artinian K-algebra, where
char K = 0. Then A has the Weak Lefschetz property if there is an element ell
of degree 1 such that the multiplication times ell : A_i --> A_{i+1} has
maximal rank, for every i, and A has the Strong Lefschetz property if times
ell^d : A_i --> A_{i+d} has maximal rank for every i and d.
The main results obtained in this paper are the following.
1) EVERY height three complete intersection has the Weak Lefschetz property.
(Our method, surprisingly, uses rank two vector bundles on P^2 and the
Grauert-Mulich theorem.)
2) We give a complete characterization (including a concrete construction) of
the Hilbert functions that can occur for K-algebras with the Weak or Strong
Lefschetz property (and the characterization is the same one).
3) We give a sharp bound on the graded Betti numbers (achieved by our
construction) of Artinian K-algebras with the Weak or Strong Lefschetz property
and fixed Hilbert function. This bound is again the same for both properties.
Some Hilbert functions in fact FORCE the algebra to have the maximal Betti
numbers.
4) EVERY Artinian ideal in K[x,y] possesses the Strong Lefschetz property.
This is false in higher codimension.Comment: To appear in J. Algebr
Asymptotic Learning Curve and Renormalizable Condition in Statistical Learning Theory
Bayes statistics and statistical physics have the common mathematical
structure, where the log likelihood function corresponds to the random
Hamiltonian. Recently, it was discovered that the asymptotic learning curves in
Bayes estimation are subject to a universal law, even if the log likelihood
function can not be approximated by any quadratic form. However, it is left
unknown what mathematical property ensures such a universal law. In this paper,
we define a renormalizable condition of the statistical estimation problem, and
show that, under such a condition, the asymptotic learning curves are ensured
to be subject to the universal law, even if the true distribution is
unrealizable and singular for a statistical model. Also we study a
nonrenormalizable case, in which the learning curves have the different
asymptotic behaviors from the universal law
Electron screening in the liquid-gas mixed phases of nuclear matter
Screening effects of electrons on inhomogeneous nuclear matter, which
includes spherical, slablike, and rodlike nuclei as well as spherical and
rodlike nuclear bubbles, are investigated in view of possible application to
cold neutron star matter and supernova matter at subnuclear densities. Using a
compressible liquid-drop model incorporating uncertainties in the surface
tension, we find that the energy change due to the screening effects broadens
the density region in which bubbles and nonspherical nuclei appear in the phase
diagram delineating the energetically favorable shape of inhomogeneous nuclear
matter. This conclusion is considered to be general since it stems from a
model-independent feature that the electron screening acts to decrease the
density at which spherical nuclei become unstable against fission and to
increase the density at which uniform matter becomes unstable against proton
clustering.Comment: 12 pages, 8 figures, accepted for publication in Physical Review
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