von Neumann’s trace inequality for Hilbert–Schmidt operators

von Neumann’s inequality in matrix theory refers to the fact that the Frobenius scalar product of two matrices is less than or equal to the scalar product of the respective singular values. Moreover, equality can only happen if the two matrices share a joint set of singular vectors, and this latter part is hard to find in the literature. We extend these facts to the separable Hilbert space setting, and provide a self-contained proof of the “latter part”.     

Convergence of solutions for the fractional Cahn–Hilliard system

This paper deals with the Cauchy–Dirichlet problem for the fractional Cahn–Hilliard equation. The main results consist of global (in time) existence of weak solutions, characterization of parabolic smoothing effects (implying under proper condition eventual boundedness of trajectories), and convergence of each solution to a (single) equilibrium. In particular, to prove the convergence result, a variant of the so-called ?ojasiewicz–Simon inequality is provided for the fractional Dirichlet Laplacian and (possibly) non-analytic (but $C^1$) nonlinearities.     

On the Local Uniqueness of Solutions of Variational Inequalities Under H-Differentiability

In this paper, we give some sufficient conditions for the local uniqueness of solutions to nonsmooth variational inequalities where the underlying functions are H-differentiable and the underlying set is a closed convex set/polyhedral set/box/polyhedral cone. We show how the solution of a linearized variational inequality is related to the solution of the variational inequality. These results extend/unify various similar results proved for C ¹ and locally Lipschitzian variational inequality problems. When specialized to the nonlinear complementarity problem, our results extend/unify those of C ² and C ¹ nonlinear complementarity problems.     

Uniform bounds under increment conditions

We apply a majorizing measure theorem of Talagrand to obtain uniform bounds for sums of random variables satisfying increment conditions of the type considered in Gál-Koksma Theorems. We give some applications.

     

鞅差序列的Bernstein型不等式及其应用

本文将独立随机变量序列的Bernstein型不等式推广到鞅差序列情形,给出该不等式的一个应用,并在一定条件下证明了非参数回归中函数估计的强相合性.     

Integral Inequalities and Global Solutions of Semilinear Evolution Equations

A criterion for the nonexplosion of solutions to semilinear evolution equations on Banach spaces is proved. The result is obtained by applying a modification of the Bihari type inequality to the case of a weakly singular nonlinear integral inequality.     

高斯型空间光孤子相互作用的数值研究

 通过数值模拟的方法，对高斯孤子在对数型饱和非线性介质中的相互作用进行了研究，考查了两光束间的相对振幅和相对相位对其相互作用的影响。结果表明：高斯孤子之间的相互作用敏感地依赖于两光束间的相对振幅和相对相位。在不同的振幅差异范围内，光束间的主要作用交替地表现为相互排斥和相互吸引，并由于高斯孤子的不稳定性，导致了光束在碰撞后以一种尺寸周期性变化的呼吸模式传输。随着相对相位的增大，两光束间始终持续地表现出强烈的排斥作用，直到相对相位增加到一个2π周期之后。而且碰撞之后，光束也都以呼吸模式进行传输，其分离的角度越大，呼吸就越明显。     

On Inequalities with Measures of Sobolev Type Embedding Theorems on Open Sets of the Real Axis

We consider some Sobolev-type spaces and obtain a necessary and sufficient condition for their embedding in a Lebesgue space.     

弱(L_1,L_2)-BLD映射的正则性

本文首先将文［１］中的ＢＬＤ映射推广为弱（Ｌ１，Ｌ２）－ＢＬＤ映射，并证明了如下正则性结果：存在两个可积指数 Ｐ１＝Ｐ１（ｎ，Ｌ１，Ｌ２）＜ｎ＜ｑ１＝ｑ１（ｎ，Ｌ１，Ｌ２），使得对任意弱（Ｌ１，Ｌ２）－ＢＬＤ映射ｆ∈（Ω，Ｒｎ），都有ｆ∈（Ω，Ｒｎ），即ｆ为（Ｌ１，Ｌ２）－ＢＬＤ映射．