A note on the eigenvalues of p‐fractional Hardy–Sobolev operator with indefinite weight

In this article, we study the eigenvalues of p‐fractional Hardy operator where , , , and Ω is an unbounded domain in with Lipschitz boundary containing 0. The weight function V may change sign and may have singular points. We also show that the least positive eigenvalue is simple and it is uniquely associated to a nonnegative eigenfunction. Moreover, we proved that there exists a sequence of eigenvalues as .     

Oscillation theorems for symplectic difference systems

We consider symplectic difference systems involving a spectral parameter, together with the Dirichlet boundary conditions. The main result of the paper is a discrete version of the so-called oscillation theorem which relates the number of finite eigenvalues less than a given number to the number of focal points of the principal solution of the symplectic system. In two recent papers the same problem was treated and an essential ingredient was to establish the concept of the multiplicity of a focal point. But there was still a rather restrictive condition needed, which is eliminated here by using the concept of finite eigenvalues (or zeros) from the theory of matrix pencils.     

On weighted spectral radius of unraveled balls and normalized Laplacian eigenvalues

For a graph G, the unraveled ball of radius r centered at a vertex v is the ball of radius r centered at v in the universal cover of G. We obtain a lower bound on the weighted spectral radius of unraveled balls of fixed radius in a graph with positive weights on edges, which is used to present an upper bound on the $s^{\text{th}}$ (where $s\ge 2$) smallest normalized Laplacian eigenvalue of irregular graphs under minor assumptions. Moreover, when $s=2$, the result may be regarded as an Alon–Boppana type bound for a class of irregular graphs.     

On the inverse spectral theory in a non-homogeneous interior transmission problem

We study the inverse spectral problem in an interior transmission eigenvalue problem. The Cartwright’s theory in value distribution theory gives a connection between the distributional structure of the eigenvalues and the asymptotic behaviours of its defining functional determinants. Given a sufficient quantity of transmission eigenvalues, we prove a uniqueness of the refraction index in inhomogeneous medium as an uniqueness problem in entire function theory. The asymptotically periodical structure of the zero set of the solutions helps to locate infinitely many eigenvalues of infinite degree of freedom.     

Search of truncation of (N−1) electron basis containing full connected triple excitations in computing main and satellite ionization potentials via Fock‐space coupled cluster approach

A valence‐universal multireference coupled cluster (VUMRCC) theory, realized via the eigenvalue independent partitioning (EIP) route, has been implemented with full inclusion of triples excitations for computing and analyzing the entire main and several satellite peaks in the ionization potential spectra of several molecules. The EIP‐VUMRCC method, unlike the traditional VUMRCC theory, allows divergence‐free homing‐in to satellite roots which would otherwise have been plagued by intruders, and is thus numerically more robust to obtain more efficient and dependable computational schemes allowing more extensive use of the approach. The computed ionization potentials (IPs) as a result of truncation of the (N−1) electron basis manifold involving virtual functions such as 2h‐p and 3h‐2p by different energy thresholds varying from 5 to 15 a.u. with 1 a.u. intervals as well as thresholds such as 20, 25, and 30 a.u. have been carefully looked into. Cutoff at around 25 a.u. turns out to be an optimal threshold. Molecules such as C₂H₄ and C₂H₂ (X = D,T), and N₂ and CO (X = D,T,Q) with Dunning's cc‐pVXZ bases have been investigated to determine all main and 2h‐p shake‐up and 3h‐2p double shake‐up satellite IPs. We believe that the present work will pave the way to a wider application of the method by providing main and satellite IPs for some problematic N‐electron closed shell systems. © 2013 Wiley Periodicals, Inc.     

一类Hermitian鞍点矩阵的特征值估计

本文研究了一类大型稀疏Hermitian鞍点线性系统Az=(B E E* 0)(x y)=(f g)=b系数矩阵的特征值,其中B∈C~(p×p)是Hermitian正定阵矩阵,E∈C~(p×q)是列降秩.本文分别给出了该系数矩阵正特征值与负特征值界的一个估计式,同时通过数值算例验证本文所给出的特征值界的估计是合理且有效的.     

A new framework for implicit restarting of the Krylov–Schur algorithm

This paper introduces a new framework for implicit restarting of the Krylov–Schur algorithm. It is shown that restarting with arbitrary polynomial filter is possible by reassigning some of the eigenvalues of the Rayleigh quotient through a rank‐one correction, implemented using only the elementary transformations (translation and similarity) of the Krylov decomposition. This framework includes the implicitly restarted Arnoldi (IRA) algorithm and the Krylov–Schur algorithm with implicit harmonic restart as special cases. Further, it reveals that the IRA algorithm can be turned into an eigenvalue assignment method. Copyright © 2014 John Wiley & Sons, Ltd.     

化学工程:从基础研究到工业应用

本文介绍了我国中长期发展规划中所制定的化学化工学科发展目标,综述了化学工程学科近十余年来在队伍建设、平台建设和国家级奖励等方面所取得的成果,列举了若干化学工程在传递与过程强化、化工分离过程、精细与药物化工、能源化工和材料化工领域从基础研究到工业化应用的例子和成果,展望了化学工程学科今后发展的方向及目标.     

Frequency ratios of Sr,Yb, and Hg based optical lattice clocks and their applications

This article describes the recent progress of optical lattice clocks with neutral strontium (⁸⁷Sr), ytterbium (¹⁷¹Yb) and mercury (¹⁹⁹Hg) atoms. In particular, we present frequency comparison between the clocks locally via an optical frequency comb and between two Sr clocks at remote sites using a phase-stabilized fibre link. We first review cryogenic Sr optical lattice clocks that reduce the room-temperature blackbody radiation shift by two orders of magnitude and serve as a reference in the following clock comparisons. Similar physical properties of Sr and Yb atoms, such as transition wavelengths and vapour pressure, have allowed our development of a compatible clock for both species. A cryogenic Yb clock is evaluated by referencing a Sr clock. We also report on an Hg clock, which shows one order of magnitude less sensitivity to blackbody radiation, while its large nuclear charge makes the clock sensitive to the variation of fine-structure constant. Connecting all three types of clocks by an optical frequency comb, the ratios of the clock frequencies are determined with uncertainties smaller than possible through absolute frequency measurements. Finally, we describe a synchronous frequency comparison between two Sr-based remote clocks over a distance of 15 km between RIKEN and the University of Tokyo, as a step towards relativistic geodesy.