Weak disorder expansion of Lyapunov exponents of products of random matrices: A degenerate theory

The weak disorder expansion of Lyapunov exponents of products of random matrices is derived by a new method. Our treatment can be easily generalized to the problem when in the limit of zero randomness two eigenvalues of the matrices are equal. For real degenerate matrices, the formula for the leading term of the Lyapunov exponent is derived. It has the form of a continuous fraction, which converges quickly to the exact value.     

Transfer matrices of dipoles with bending radius variation

With the increasing demand of high brightness in light source, the uniform dipole can not meet the needs of low emittance, and thus the dipole with bending radius variation is introduced in this paper. The transfer matrix of a non-uniform dipole whose bending radius is linearly changed is chosen as an example and a very simple calculation formula of non-uniform dipole transfer matrices is given. The transfer matrices of some common profile non-uniform dipoles are also listed. The comparison of these transfer matrices and the matrices calculated with slices method verifies the numerical accuracy of this formula. This method can make the non-uniform beam dynamic problem simpler, very helpful for emittance research and lattice design with non-uniform dipoles.     

First order formulation of resonance testing

An efficient formula for determining the matrix of frequency response functions is derived for the linear system of ordinary differential equations of structural dynamics having constant coefficients. The eigenvalues and eigenvectors of the system associated with the known mass, stiffness and damping matrices are used to accomplish this without recourse to the inversion of complex matrices at each excitation frequency. The result may be applied to single or multi-point excitation techniques and the matrices need not be symmetric. The eigenvalues are assumed to occur in complex conjugate pairs with non-positive real parts and the Jordan canonical form of the system matrix is presumed to be diagonal. Expressions are given for the sensitivity of the response and an example of an eight storey building is used to demonstrate the computational efficiency of the formula.     

On the Largest Lyapunov Exponent for Products of Gaussian Matrices

The paper provides a new integral formula for the largest Lyapunov exponent of Gaussian matrices, which is valid in the real, complex and quaternion-valued cases. This formula is applied to derive asymptotic expressions for the largest Lyapunov exponent when the size of the matrix is large and compare the Lyapunov exponents in models with a spike and no spikes.     

Weak disorder expansion of Liapunov exponents in a degenerate case

It is shown how the weak disorder expansion of the Liapunov exponents of a product of random matrices can be derived when the unperturbed matrices have two degenerate eigenvalues. The general expression of the Liapunov exponents at the lowest nontrivial order in disorder is given.     

A formula for the determinant of a sum of matrices

We give a formula, involving circular words and symmetric functions of the eigenvalues, for the determinant of a sum of matrices. Theorem of Hamilton-Cayley is deduced from this formula.UQAM and LITPUniversité Paris 7 and LITP.     

Fixed point resolution in extended WZW models

A formula is derived for the fixed point resolution matrices of simple current extended WZW models and coset conformal field theories. Unlike the analogous matrices for unextended WZW models, these matrices are in general not symmetric, and they may have field-dependent twists. They thus provide non-trivial realizations of the general conditions presented in earlier work with Fuchs and Schweigert.     

On quantization of r matrices for Belavin-Drinfeld triples

We suggest a formula for quantum universal R matrices corresponding to quasitriangular classical r matrices classified by Belavin and Drinfeld for all simple Lie algebras. The R matrices are obtained by twisting the standard universal R matrix.     

On the BCH formula of Rezek and Kosloff

The BCH formula of Rezek and Kosloff (2006) [10] is a convenient tool to handle a family of density matrices, which occurs in the study of quantum heat engines. We prove the formula using a known argument from Lie theory.     

A Finite Dimensional Analog of the Krein Formula

Abstract

I offer a simple and useful formula for the resolvent of a small rank perturbation of large matrices. I discuss applications of this formula, in particular, to analytical and numerical solving of difference boundary value problems. I present examples connected with such problems for the difference Laplacian and estimate numerical efficiency of the corresponding algorithms.     

Explicit integration of the full symmetric Toda hierarchy and the sorting property

We give an explicit formula for the solution to the initial-value problem of the full symmetric Toda hierarchy. The formula is obtained by the orthogonalization procedure of Szegö, and is also interpreted as a consequence of the QR factorization method of Symes. The sorting property of the dynamics is also proved for the case of a generic symmetric matrix in the sense described in the text, and generalizations of tridagonal formulae are given for the case of matrices with 2M+1 nonzero diagonals.     

Eigenvalue densities of real and complex Wishart correlation matrices

Wishart correlation matrices are the standard model for the statistical analysis of time series. The ensemble averaged eigenvalue density is of considerable practical and theoretical interest. For complex time series and correlation matrices, the eigenvalue density is known exactly. In the real case, a fundamental mathematical obstacle made it forbiddingly complicated to obtain exact results. We use the supersymmetry method to fully circumvent this problem. We present an exact formula for the eigenvalue density in the real case in terms of twofold integrals and finite sums.     

一维三原子分子的振动

考虑三原子分子的振动问题,利用线性代数的化标准型理论,获得该振动系统简正频率的数学表达式,并讨论了几种特殊的情况.     

The scattering matrices for a non-spherical scalar-relativistic potential

The scattering matrices for a non-spherical scalar-relativistic potential are written down. A new formula is derived that makes it possible to evaluate quite easily a necessary angular integration.     

2-Matrix versus Complex Matrix Model, Integrals over the Unitary Group as Triangular Integrals

We prove that the 2-hermitian matrix model and the complex-matrix model obey the same loop equations, and as a byproduct, we find a formula for Itzykzon-Zuber type integrals over the unitary group. Integrals over U(n) are rewritten as gaussian integrals over triangular matrices and then computed explicitly. That formula is an efficient alternative to the former Shatashvili's formula. An erratum to this article is available at .     

Entanglement of formation for a class of quantum states

Entanglement of formation for a class of higher-dimensional quantum mixed states is studied in terms of a generalized formula of concurrence for N-dimensional quantum systems. As applications, the entanglement of formation for a class of 16×16 density matrices are calculated.     

Random matrix theory of singular values of rectangular complex matrices I: Exact formula of one-body distribution function in fixed-trace ensemble

The fixed-trace ensemble of random complex matrices is the fundamental model that excellently describes the entanglement in the quantum states realized in a coupled system by its strongly chaotic dynamical evolution [see H. Kubotani, S. Adachi, M. Toda, Phys. Rev. Lett. 100 (2008) 240501]. The fixed-trace ensemble fully takes into account the conservation of probability for quantum states. The present paper derives for the first time the exact analytical formula of the one-body distribution function of singular values of random complex matrices in the fixed-trace ensemble. The distribution function of singular values (i.e. Schmidt eigenvalues) of a quantum state is so important since it describes characteristics of the entanglement in the state. The derivation of the exact analytical formula utilizes two recent achievements in mathematics, which appeared in 1990s. The first is the Kaneko theory that extends the famous Selberg integral by inserting a hypergeometric type weight factor into the integrand to obtain an analytical formula for the extended integral. The second is the Petkovšek–Wilf–Zeilberger theory that calculates definite hypergeometric sums in a closed form.     

Supersymmetry Approach to Wishart Correlation Matrices: Exact Results

We calculate the ‘one-point function’, meaning the marginal probability density function for any single eigenvalue, of real and complex Wishart correlation matrices. No explicit expression had been obtained for the real case so far. We succeed in doing so by using supersymmetry techniques to express the one-point function of real Wishart correlation matrices as a twofold integral. The result can be viewed as a resummation of a series of Jack polynomials in a non-trivial case. We illustrate our formula by numerical simulations. We also rederive a known expression for the one-point function of complex Wishart correlation matrices.     

正交曲线坐标系中加速度的矩阵求法

利用矩阵和一个微商公式，把变量替换法求正交曲线坐标系中加速度运虎的繁琐程度大为降低。