Spectra of large diluted but bushy random graphs

We compute an asymptotic expansion in of the limit in of the empirical spectral measure of the adjacency matrix of an Erd?s‐Rényi random graph with vertices and parameter . We present two different methods, one of which is valid for the more general setting of locally tree‐like graphs. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 49, 160–184, 2016     

极大加代数的对称代数S上互补基本矩阵

主要研究了极大加代数的对称代数S上互补基本矩阵,给出本征积的概念,证明了S上的Laplace定理,由此推出所有互补基本矩阵的行列式相等,且任意两个互补基本矩阵的行列式中的非零项均一一对应相等.在一个互补基本矩阵的行列式中,对于确定非零项的任一置换,给出了在另一个互补基本矩阵的行列式中找到置换使其确定相同非零项的方法.     

Self‐adjoint Sturm–Liouville problems with an infinite number of boundary conditions

We study Sturm–Liouville (SL) problems on an infinite number of intervals, adjacent endpoints are linked by means of boundary conditions, and characterize the conditions which determine self‐adjoint operators in a Hilbert space which is the direct sum of the spaces for each interval. These conditions can be regular or singular, separated or coupled. Furthermore, the inner products of the summand spaces may be multiples of the usual inner products with different spaces having different multiples. We also extend the GKN Theorem to cover the infinite number of intervals theory with modified inner products and discuss the connection between our characterization and the classical one with the usual inner products. Our results include the finite number of intervals case.     

Continuous maps on triangular matrices that preserve commutativity

We investigate continuous injective maps defined on the space of upper triangular matrices over a field F that preserve commutativity of matrices in both directions. We show that every such map is a composition of either an inner automorphism and a locally polynomial map or the two latter and one more automorphism of upper triangular matrices.     

Spectra of infinite graphs with tails

We compute explicitly (modulo solutions of certain algebraic equations) the spectra of infinite graphs obtained by attaching one or several infinite paths to some vertices of given finite graphs. The main result concerns a canonical form for the adjacency matrix of such infinite graphs, and the algorithm of its calculation. The argument relies upon the spectral theory of eventually free Jacobi matrices. We also study some other couplings of infinite graphs (stars and Bethe–Caley trees).     

Polyoxometalates entrapped in sol–gel matrices for reducing electron exchange column applications

Electron exchange columns were developed by utilizing the redox properties of polyoxometalates (POMs) entrapped in silica matrices via the sol–gel route. The properties of the columns strongly depend on the composition of the precursors used to prepare the matrices. The columns exhibit good reversibility and are the first ‘reducing’ electron exchange columns ever prepared. They are also the first columns where both the matrix and the entrapped redox agent are inorganic compounds. This increases their stability. However, the redox properties of the entrapped POMs in the matrices are affected by the composition of the matrices.     

Invariant Quantum States of Quadratic Hamiltonians

The problem of finding covariance matrices that remain constant in time for arbitrary multi-dimensional quadratic Hamiltonians (including those with time-dependent coefficients) is considered. General solutions are obtained.