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 .     

Hierarchical Model-Based Clustering for Large Datasets

In recent years, hierarchical model-based clustering has provided promising results in a variety of applications. However, its use with large datasets has been hindered by a time and memory complexity that are at least quadratic in the number of observations. To overcome this difficulty, this article proposes to start the hierarchical agglomeration from an efficient classification of the data in many classes rather than from the usual set of singleton clusters. This initial partition is derived from a subgraph of the minimum spanning tree associated with the data. To this end, we develop graphical tools that assess the presence of clusters in the data and uncover observations difficult to classify. We use this approach to analyze two large, real datasets: a multiband MRI image of the human brain and data on global precipitation climatology. We use the real datasets to discuss ways of integrating the spatial information in the clustering analysis. We focus on two-stage methods, in which a second stage of processing using established methods is applied to the output from the algorithm presented in this article, viewed as a first stage.     

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.     

气驱油油气混相过程的界面传质特性及其分子机制

油气混相过程的界面传质特性对气驱提高原油采收率技术非常重要。本文针对吉林某油田的实际油组分,采用分子动力学模拟研究了气驱油过程,分析了不同气体和驱替压力下油气两相的状态变化以及界面特性,获得不同驱替气体的最小混相压力(MMP)。结果表明,随着驱替气体压力的升高,气相的密度逐渐增大,油相膨胀密度降低,气相与油相的混合程度增强,油气两相界面厚度增加,界面张力随之减小。同时发现,驱替相中二氧化碳浓度越高,在同等气体压力下,油气界面更厚,油气混合程度更高。纯CO₂驱油得到的MMP远远小于纯N₂驱油,当这两种气体摩尔比为1 : 1混合时MMP介于两种纯气体之间,说明要达到同样的驱油效果二氧化碳需要的压力更小。最后,本文从分子微观作用力角度解释了驱替气体不同时影响油气混相程度的机制,通过分子平均作用势曲线发现油相分子对CO₂的吸引力要大于N₂分子,因此CO₂分子更容易与油相混合,驱替效果更明显。     

Fluidization characteristics of sawdust in a cold flow fluidized bed

India has abundance of biomass such as rice husk, bagasse, wheat straw, sawdust etc. which is used as a main or auxiliary fuel in the fluidized bed combustor, gasifier and pyrolizer. Design of such fluidized bed equipments require the knowledge of minimum fluidization velocity (U_mf), complete fluidization velocity (U_cf) and transport disengagement height (TDH). The present work reports the fluidization characteristic, U_mf, U_cf and TDH of the individual size groups of sawdust and mixture thereof. The results indicate that the U_mf and U_cf have a tendency to increase with increase in particle diameter, however the TDH shows the reverse trend. The sawdust particle size of 925 and 1200 ​μm showed significant difference between their U_mf and U_cf, an essential parameter for controlled fluidization. Based on the experimental work new correlations for the prediction of U_mf, U_cf and TDH for sawdust sample are proposed. The proposed correlations of U_mf, U_cf and TDH are in good agreement with experimental values and the deviations found within the range of nearly ±10% for all the samples.