Reductions and deviations for stochastic partial differential equations under fast dynamical boundary conditions

In order to understand the impact of random influences at physical boundary on the evolution of multiscale systems, a stochastic partial differential equation model under a fast random dynamical boundary condition is investigated. The noises in the model and in the boundary condition are both additive. An effective equation is derived and justified by reducing the random \emph{dynamical} boundary condition to a simpler one. The effective system is still a stochastic partial differential equation. Furthermore, the quantitative comparison between the solution of the original stochastic system and the effective solution is provided by establishing normal deviations and large deviations principles. Namely, the normal deviations are asymptotically characterized, while the rate and speed of the large deviations are estimated.Comment: This is a revised version with 29 pages. To appear in Stochastic Analysis and Applications, 200

Low-Complexity QL-QR Decomposition Based Beamforming Design for Two-Way MIMO Relay Networks

In this paper, we investigate the optimization problem of joint source and relay beamforming matrices for a twoway amplify-and-forward (AF) multi-input multi-output (MIMO) relay system. The system consisting of two source nodes and two relay nodes is considered and the linear minimum meansquare- error (MMSE) is employed at both receivers. We assume individual relay power constraints and study an important design problem, a so-called determinant maximization (DM) problem. Since this DM problem is nonconvex, we consider an efficient iterative algorithm by using an MSE balancing result to obtain at least a locally optimal solution. The proposed algorithm is developed based on QL, QR and Choleskey decompositions which differ in the complexity and performance. Analytical and simulation results show that the proposed algorithm can significantly reduce computational complexity compared with their existing two-way relay systems and have equivalent bit-error-rate (BER) performance to the singular value decomposition (SVD) based on a regular block diagonal (RBD) scheme

Finite temperature phase diagram of trapped Fermi gases with population imbalance

We consider a trapped Fermi gas with population imbalance at finite temperatures and map out the detailed phase diagram across a wide Feshbach resonance. We take the Larkin-Ovchinnikov-Fulde-Ferrel (LOFF) state into consideration and minimize the thermodynamical potential to ensure stability. Under the local density approximation, we conclude that a stable LOFF state is present only on the BCS side of the Feshbach resonance, but not on the BEC side or at unitarity. Furthermore, even on the BCS side, a LOFF state is restricted at low temperatures and in a small region of the trap, which makes a direct observation of LOFF state a challenging task.Comment: 9 pages, 7 figure

Positivity restrictions to the transverse polarization of the inclusively detected spin-half baryons in unpolarized electron-positron annihilation

The positivity constraints to the structure functions for the inclusive spin-half baryon production by a time-like photon fragmentation are investigated. One conclusion is that $\hat F$, which arises from the hadronic final-state interactions, is subjected to an inequality between its absolute value and the two spin-independent structure functions. On the basis of this finding, we derive a formula through which the upper limits can be given for the transverse polarization of the inclusively detected spin-half baryons in unpolarized electron-positron annihilation. The derived upper bound supplies a consistency check for the judgement of reliability of experimental data and model calculations.Comment: final version to appear in Z. Phys. C, references update

Tomography of correlation functions for ultracold atoms via time-of-flight images

We propose to utilize density distributions from a series of time-of-flight images of an expanding cloud to reconstruct single-particle correlation functions of trapped ultra-cold atoms. In particular, we show how this technique can be used to detect off-diagonal correlations of atoms in a quasi-one-dimensional trap, where both real- and momentum- space correlations are extracted at a quantitative level. The feasibility of this method is analyzed with specific examples, taking into account finite temporal and spatial resolutions in experiments.Comment: 7 pages, 4 figure

Hamiltonian Systems with L\'evy Noise: Symplecticity, Hamilton's Principle and Averaging Principle

This work focuses on topics related to Hamiltonian stochastic differential equations with L\'{e}vy noise. We first show that the phase flow of the stochastic system preserves symplectic structure, and propose a stochastic version of Hamilton's principle by the corresponding formulation of the stochastic action integral and the Euler-Lagrange equation. Based on these properties, we further investigate the effective behaviour of a small transversal perturbation to a completely integrable stochastic Hamiltonian system with L\'{e}vy noise. We establish an averaging principle in the sense that the action component of solution converges to the solution of a stochastic differential equation when the scale parameter goes to zero. Furthermore, we obtain the estimation for the rate of this convergence. Finally, we present an example to illustrate these results

Highly Accurate Nystr\"{o}m Volume Integral Equation Method for the Maxwell equations for 3-D Scatters

In this paper, we develop highly accurate Nystr\"{o}m methods for the volume integral equation (VIE) of the Maxwell equation for 3-D scatters. The method is based on a formulation of the VIE equation where the Cauchy principal value of the dyadic Green's function can be computed accurately for a finite size exclusion volume with some explicit corrective integrals of removable singularities. Then, an effective interpolated quadrature formula for tensor product Gauss quadrature nodes in a cube is proposed to handle the hyper-singularity of integrals of the dyadic Green's function. The proposed high order Nystr\"{o}m VIE method is shown to have high accuracy and demonstrates $p$-convergence for computing the electromagnetic scattering of cubes in $R^3$

A new criterion for finite non-cyclic groups

Let $H$ be a subgroup of a group $G$. We say that $H$ satisfies the power condition with respect to $G$, or $H$ is a power subgroup of $G$, if there exists a non-negative integer $m$ such that $H=G^{m}=$. In this note, the following theorem is proved: Let $G$ be a group and $k$ the number of non-power subgroups of $G$. Then (1) $k=0$ if and only if $G$ is a cyclic group(theorem of F. Sz$\acute{a}$sz) ;(2) $0 < k <\infty$ if and only if $G$ is a finite non-cyclic group; (3) $k=\infty$ if and only if $G$ is a infinte non-cyclic group. Thus we get a new criterion for the finite non-cyclic groups.Comment: 6 page

Kinetic solutions for nonlocal scalar conservation laws

This work is devoted to examine the uniqueness and existence of kinetic solutions for a class of scalar conservation laws involving a nonlocal super-critical diffusion operator. Our proof for uniqueness is based upon the analysis on a microscopic contraction functional and the existence is enabled by a parabolic approximation. As an illustration, we obtain the existence and uniqueness of kinetic solutions for the generalized fractional Burgers-Fisher type equations. Moreover, we demonstrate the kinetic solutions' Lipschitz continuity in time, and continuous dependence on nonlinearities and L\'{e}vy measures.Comment: 22 page

Schauder estimates for stochastic transport-diffusion equations with L\'{e}vy processes

We consider a transport-diffusion equation with L\'{e}vy noises and H\"{o}lder continuous coefficients. By using the heat kernel estimates, we derive the Schauder estimates for the mild solutions. Moreover, when the transport term vanishes and $p=2$, we show that the H\"{o}lder index in space variable is optimal.Comment: 25 page