Rigorous mean-field model for coherent-potential approximation: Anderson model with free random variables

A model of a randomly disordered system with site-diagonal random energy fluctuations is introduced. It is an extension of Wegner'sn-orbital model to arbitrary eigenvalue distribution in the electronic level space. The new feature is that the random energy values are not assumed to be independent at different sites, but free. Freeness of random variables is an analog of the concept of independence for noncommuting random operators. A possible realization is the ensemble of randomly rotated matrices at different lattice sites. The one- and two-particle Green functions of the proposed Hamiltonian are calculated exactly. The eigenstates are extended and the conductivity is novanishing everywhere inside the band. The long-range behavior and the zero-frequency limit of the two-particle Green function are universal with respect to the eigenvalue distribution in the electronic level space. The solutions solve the CPA equation for the one- and two-particle Green function of the corresponding Anderson model. Thus our (multisite) model is a rigorous mean-field model for the (single-site) CPA. We show how the Lloyd model is included in our model and treat various kinds of noises.     

双反对称矩阵反问题的最小二乘解

1 引 言Rn×m表示所有n×m阶实矩阵集合,Rrn×m表示Rn×m中秩为r的子集;ORn×m表示所有n阶正交阵的集合;A+表示A的Moore-Penrose广义逆;Iκ表示κ阶单位阵;||·||表示Frobenius范数;ASRn×m表示n阶实反对称阵的全体;A＊B表示A与B的Hadamard乘     

Parallel Dixon Matrices by Bracket

It is known that the Dixon matrix can be constructed in parallel either by entry or by diagonal. This paper presents another parallel matrix construction, this time by bracket. The parallel by bracket algorithm is the fastest among the three, but not surprisingly it requires the highest number of processors. The method also shows analytically that the Dixon matrix has a total of m(m+1)²(m+2)n(n+1)²(n+2)/36 brackets but only mn(m+1)(n+1)(mn+2m+2n+1)/6 of them are distinct.     

Spectroscopic studies of CO2 trapped in rare gas matrices

Low temperature (5 K) high resolution (0.15 and 0.03 cm⁻¹) absorption spectra of ¹³CO₂ have been recorded in neon, argon, krypton, and xenon matrices, in the ν₃ and ν₂ regions. Diffusion experiments have been performed in krypton and xenon in order to identify vibrational traps which could be responsible for the decrease and shortening of the emission observed after laser excitation: high-frequency structures in the ν₃ region are assigned to dimers and a doubling of the monomer line is due to a site effect. In neon, only a double substitutional site, with a splitting of the degenerate ν₂ vibration, is observed. In argon, as previously reported, a single and a double site are characterized. In krypton and xenon, where ν₂ is not split, only single sites would be predicted. As one of them exhibits a ν₃ line highly sensitive to temperature, we expect a large coupling with the lattice and a fast vibrational relaxation. This site is very likely the vibrational trap we are looking for.     

Nested regular polygon solutions for planar 2N-body problems

In this paper we study the necessary conditions for the masses of the nested regular polygon solutions of the planar 2N-body problem.We prove that the masses at the vertices of each regular polygon must be equal to each other.     

A language-driven generalized numerical database translator

Numerical databases arise in many scientific applications to keep track of large dense and sparse matrices, residing on secondary devices in matrix compact data representation. This paper describes a language-driven generalized data translator for translating any numerical database from one matrix compact data representation to another. Our approach is to describe any matrix compact data representation by a physical schema and any numerical database and its mapping to storage by data language facilities. The languages are processed by a Generalized Syntax-Directed Translation Scheme (GSDTS) to automatically generate FORTRAN conversion programs which become the major modules of the translator.     

A constructive characterization ofQ o-matrices with nonnegative principal minors

In [2] we characterized the class of matrices with nonnegative principla minors for which the linear-complementarity problem always has a solution. That class is contained in the one we study here. Our main result gives a finitely testable set of necessary and sufficient conditions under which a matrix with nonnegative principal minors has the property that if a corresponding linear complementarity problem is feasible then it is solvable. In short, we constructively characterize the matrix class known asQ _o ∩P _o . Research partially supported by the National Science Foundation Grant DMS-8420623 and U.S. Department of Energy Contract DE-AA03-76SF00326, PA # DE-AS03-76ER72018.     

Structural properties and recognition of restricted and strongly unimodular matrices

A (0, ±1) matrix A is restricted unimodular if every matrix obtained from A by setting to zero any subset of its entries is totally unimodular. Restricted unimodular matrices are also known as matrices without odd cycles. They have been studied by Commoner and recently Yannakakis has given a polynomial algorithm to recognize when a matrix belongs to this class. A matrix A is strongly unimodular if any matrix obtained from A by setting at most one of its entries to zero is totally unimodular. Crama et al. have shown that (0,1) matrix A is strongly unimodular if and only if any basis of (A, 1) is triangular, whereI is an identity matrix of suitable dimensions. In this paper we give a very simple algorithm to test whether a matrix is restricted unimodular and we show that all strongly unimodular matrices can be obtained by composing restricted unimodular matrices with a simple operation. Partially supported by a New York University Research Challenge Fund Grant.     

Eigenfunctions of expected value operators in the Wishart distribution,II

Let V = (v_ij) denote the k × k symmetric scatter matrix following the Wishart distribution W(k, n, Σ). The problem posed is to characterize the eigenfunctions of the expectation operators of the Wishart distribution, i.e., those scalar-valued functions f(V) such that (E_nf)(V) = λ_n,kf(V). A finite sequence of polynomial eigenspaces, EP spaces, exists whose direct sum is the space of all homogeneous polynomials. These EP subspaces are invariant and irreducible under the action of the congruence transformation V → T′VT. Each of these EP subspaces contains an orthogonally invariant subspace of dimension one. The number of EP subspaces is determined and eigenvalues are computed. Bi-linear expansions of $\text{|I + VA|}{}^{\mathrm{<mtext>?n</mtext><mtext>2</mtext>}}$ and (tr VA)^r into eigenfunctions are given. When f(V) is an EP polynomial, then f(V^?1) is an EP function. These EP subspaces are identical to the more abstractly defined polynomial subspaces studied by James.     

Jacobi matrices for measures modified by a rational factor

This paper describes how, given the Jacobi matrixJ for the measure d(t), it is possible to produce the Jacobi matrix for the measurer(t)d(t) wherer(t) is a quotient of polynomials. The method uses a new factoring algorithm to generate the Jacobi matrices associated with the partial fraction decomposition ofr(t) and then applies a previously developed summing technique to merge these Jacobi matrices. The factoring method performs best just where Gautschi's minimal solution method for this problem is weakest and vice versa. This suggests a hybrid strategy which is believed to be the most powerful yet for solving this problem. The method is demonstrated on a simple example and some numerical tests illustrate its performance characteristics.