11,528 research outputs found
A More Precise Formulation of a Theorem of Landau in the Linear Case and Some Applications
AbstractWe propose a more precise statement for one of Landau's theorems about integral product-quotients of factorials of linear forms. We also study, in detail, the properties of the associated functions, and we propose an algorithm to find extrema
Magnetic fields in noncommutative quantum mechanics
We discuss various descriptions of a quantum particle on noncommutative space
in a (possibly non-constant) magnetic field. We have tried to present the basic
facts in a unified and synthetic manner, and to clarify the relationship
between various approaches and results that are scattered in the literature.Comment: Dedicated to the memory of Julius Wess. Work presented by F. Gieres
at the conference `Non-commutative Geometry and Physics' (Orsay, April 2007
Sharp interface limit for a phase field model in structural optimization
We formulate a general shape and topology optimization problem in structural
optimization by using a phase field approach. This problem is considered in
view of well-posedness and we derive optimality conditions. We relate the
diffuse interface problem to a perimeter penalized sharp interface shape
optimization problem in the sense of -convergence of the reduced
objective functional. Additionally, convergence of the equations of the first
variation can be shown. The limit equations can also be derived directly from
the problem in the sharp interface setting. Numerical computations demonstrate
that the approach can be applied for complex structural optimization problems
Adiabatic limit and the slow motion of vortices in a Chern-Simons-Schr\"odinger system
We study a nonlinear system of partial differential equations in which a
complex field (the Higgs field) evolves according to a nonlinear Schroedinger
equation, coupled to an electromagnetic field whose time evolution is
determined by a Chern-Simons term in the action. In two space dimensions, the
Chern-Simons dynamics is a Galileo invariant evolution for A, which is an
interesting alternative to the Lorentz invariant Maxwell evolution, and is
finding increasing numbers of applications in two dimensional condensed matter
field theory. The system we study, introduced by Manton, is a special case (for
constant external magnetic field, and a point interaction) of the effective
field theory of Zhang, Hansson and Kivelson arising in studies of the
fractional quantum Hall effect. From the mathematical perspective the system is
a natural gauge invariant generalization of the nonlinear Schroedinger
equation, which is also Galileo invariant and admits a self-dual structure with
a resulting large space of topological solitons (the moduli space of self-dual
Ginzburg-Landau vortices). We prove a theorem describing the adiabatic
approximation of this system by a Hamiltonian system on the moduli space. The
approximation holds for values of the Higgs self-coupling constant close to the
self-dual (Bogomolny) value of 1. The viability of the approximation scheme
depends upon the fact that self-dual vortices form a symplectic submanifold of
the phase space (modulo gauge invariance). The theorem provides a rigorous
description of slow vortex dynamics in the near self-dual limit.Comment: Minor typos corrected, one reference added and DOI give
Gevrey smoothing for weak solutions of the fully nonlinear homogeneous Boltzmann and Kac equations without cutoff for Maxwellian molecules
It has long been suspected that the non-cutoff Boltzmann operator has similar
coercivity properties as a fractional Laplacian. This has led to the hope that
the homogenous Boltzmann equation enjoys similar regularity properties as the
heat equation with a fractional Laplacian. In particular, the weak solution of
the fully nonlinear non-cutoff homogenous Boltzmann equation with initial datum
in , i.e., finite mass, energy
and entropy, should immediately become Gevrey regular for strictly positive
times. We prove this conjecture for Maxwellian molecules.Comment: 43 pages, 1 figur
- …