Spectral theory of discontinuous functions of self-adjoint operators and scattering theory

In the smooth scattering theory framework, we consider a pair of self-adjoint operators H₀, H and discuss the spectral projections of these operators corresponding to the interval (−∞,λ). The purpose of the paper is to study the spectral properties of the difference D(λ) of these spectral projections. We completely describe the absolutely continuous spectrum of the operator D(λ) in terms of the eigenvalues of the scattering matrix S(λ) for the operators H₀ and H. We also prove that the singular continuous spectrum of the operator D(λ) is empty and that its eigenvalues may accumulate only at “thresholds” in the absolutely continuous spectrum.     

Weakly convex sets and modulus of nonconvexity

We consider a definition of a weakly convex set which is a generalization of the notion of a weakly convex set in the sense of Vial and a proximally smooth set in the sense of Clarke, from the case of the Hilbert space to a class of Banach spaces with the modulus of convexity of the second order. Using the new definition of the weakly convex set with the given modulus of nonconvexity we prove a new retraction theorem and we obtain new results about continuity of the intersection of two continuous set-valued mappings (one of which has nonconvex images) and new affirmative solutions of the splitting problem for selections. We also investigate relationship between the new definition and the definition of a proximally smooth set and a smooth set.     

Normed lattices with discrete ultrapowers

The paper is devoted to investigations of the order structure of ultrapowers. A characterization of discrete normed lattices, all of whose ultrapowers are discrete, is presented (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The improved linear multistep methods for differential equations with piecewise continuous arguments

This paper deals with the convergence of the linear multistep methods for the equation x′(t) = ax(t) + a₀x([t]). Numerical experiments demonstrate that the 2-step Adams-Bashforth method is only of order p = 0 when applied to the given equation. An improved linear multistep methods is constructed. It is proved that these methods preserve their original convergence order for ordinary differential equations (ODEs) and some numerical experiments are given.     

Function spaces and product topologies on powers of spaces

A study is made of two classes of product topologies on powers of spaces: the general box product topologies, and the general uniform product topologies. Some examples are given and some results are shown about the properties of these general product spaces. This is applied to show that certain spaces of continuous functions with the fine topology are homeomorphic to box product spaces, and certain spaces of continuous functions with the uniform topology are homeomorphic to uniform product spaces.     

连续设施选址:模型、方法与应用

给定度量空间和该空间中的若干顾客,设施选址为在该度量空间中确定新设施的位置使得某种目标达到最优。连续设施选址是设施选址中的一类重要问题,其中的设施可在度量空间的某连续区域上进行选址。本文对连续设施选址的模型、算法和应用方面的工作进行了综述。文章首先讨论了连续设施选址中几个重要元素,包括新设施个数、距离度量函数、目标函数;然后介绍了连续选址中的几种经典模型和拓展模型;接着概述了求解连续选址问题的常用优化方法和技术,包括共轭对偶、全局优化、不确定优化、变分不等式方法、维诺图;最后介绍了连续设施选址的重要应用并给出了研究展望。     

λ-Kantorovich算子的逼近性质

首先在无穷空间上构造了一类新的λ-Szász-Kantorovich算子,通过分析计算得到了该类算子矩的估计及Korovkin型逼近性质;其次,利用连续模和K-泛函的等价关系给出了收敛速度的刻画;最后,借助于Holder不等式建立了Lipschitz连续函数的收敛定理.     

Sensor calibration for off-the-grid spectral estimation

This paper studies sensor calibration in spectral estimation where the true frequencies are located on a continuous domain. We consider a uniform array of sensors that collects measurements whose spectrum is composed of a finite number of frequencies, where each sensor has an unknown calibration parameter. Our goal is to recover the spectrum and the calibration parameters simultaneously from multiple snapshots of the measurements. In the noiseless case with an infinite number of snapshots, we prove uniqueness of this problem up to certain trivial, inevitable ambiguities based on an algebraic method, as long as there are more sensors than frequencies. We then analyze the sensitivity of this algebraic technique with respect to the number of snapshots and noise.We next propose an optimization approach that makes full use of the measurements by minimizing a non-convex objective which is non-negative and continuously differentiable over all calibration parameters and Toeplitz matrices. We prove that, in the case of infinite snapshots and noiseless measurements, the objective vanishes only at equivalent solutions to the true calibration parameters and the measurement covariance matrix. The objective is minimized using Wirtinger gradient descent which is proven to converge to a critical point. We show empirically that this critical point provides a good approximation of the true calibration parameters and the underlying frequencies.     

一类含高阶Riemann-Liouville型分数阶导数脉冲微分方程的通解

本文首先运用迭代法获得一类含多项Riemann-Liouville型分数阶导数的微分方程的连续通解,然后应用数学归纳法得到这类脉冲微分方程的分片连续通解. 所得结果归结于脉冲分数阶微分方程领域,对分数阶微分方程研究者有参考意义.