Transitive ensembles of random matrices related to orthogonal polynomials

Transitive correlations of eigenvalues for random matrix ensembles intermediate between real symmetric and hermitian, self-dual quaternion and hermitian, and antisymmetric and hermitian are studied. Expressions for exact n-point correlation functions are obtained for random matrix ensembles related to general orthogonal polynomials. The asymptotic formulas in the limit of large matrix dimension are evaluated at the spectrum edges for the ensembles related to the Legendre polynomials. The results interpolate known asymptotic formulas for random matrix eigenvalues.     

Formation of Superheavy Nuclei in Massive Fusion Reactions

We recently performed a series of improvement on evaluation of eigenvalues without complicated iterations.In this work we first discuss evaluation of the lowest eigenvalue for given systems,by which one conveniently obtains the value of the lowest eigenvalue based on the dimension and width of given matrix.We also discuss a strong correlation between eigenvalues and diagonal matrix elements for large matrices,by which one is able to predict eigenvalues approximately without iterations.     

A block Chebyshev–Davidson method with inner–outer restart for large eigenvalue problems

We propose a block Davidson-type subspace iteration using Chebyshev polynomial filters for large symmetric/hermitian eigenvalue problem. The method consists of three essential components. The first is an adaptive procedure for constructing efficient block Chebyshev polynomial filters; the second is an inner–outer restart technique inside a Chebyshev–Davidson iteration that reduces the computational costs related to using a large dimension subspace; and the third is a progressive filtering technique, which can fully employ a large number of good initial vectors if they are available, without using a large block size. Numerical experiments on several Hamiltonian matrices from density functional theory calculations show the efficiency and robustness of the proposed method.     

A family of poisson structures on hermitian symmetric spaces

We investigate the compatibility of symplectic Kirillov-Kostant-Souriau structure and Poisson-Lie structure on coadjoint orbits of semisimple Lie group. We prove that they are compatible for an orbit compact Lie group iff the orbit is hermitian symmetric space. We prove also the compatibility statement for non-compact hermitian symmetric space. As an example we describe a structure of symplectic leaves onCP ⁿ for this family. These leaves may be considered as a perturbation of Schubert cells. Possible applications to infinite-dimensional examples are discussed.     

Radiative temperature dissipation in a finite atmosphere—I. The homogeneous case

We have developed a new approach toward solving problems of linear radiative relaxation of LTE temperature perturbations in a plane-parallel atmosphere of finite extent. We show that the mathematical problem is one of solving an integral eigenvalue equation, for which non-trivial solutions exist only for discrete values of the radiative relaxation time. The solutions for the spatial part of the perturbation constitute a complete and orthogonal set of basis functions, making it possible to solve more general problems of temperature relaxation. In applying this method to radiative relaxation in the middle atmosphere of earth, we show how the additional influences of photochemical coupling, advection by winds, and eddy diffusion by small-scale turbulence may be easily included using matrix perturbation techniques. We have solved the homogeneous integral equation for a wide variety of vertical thicknesses in an idealized homogeneous slab medium. Adopting a number of different analytic line profiles (rectangular, Doupler, Voigt, and Lorentz) we have obtained numerical solutions using an exponential-kernel method for solving the integral equation. The discrete eigenvalue “spectrum” is presented for vertical optical depths (0–10³) at line-center, and is used in solving several initial-value problems for a decaying temperature perturbation. We find that the eigenvalue spectrum is bounded from above by the lowest-order eigenvalue, and bounded from below by the familiar transparent approximation. The dependence of the lowest even eigenvalue on optical depth and the relative separation of the higher eigenvalues are found to depend sensitively on the line profile.     

Some integrable systems related to affine lie algebras and homogeneous spaces

In an attempt to understand how a result on hamiltonian systems related to hermitian symmetric spaces can be seen as an application of the Adler-Kostant-Symes theorem, it is found that quite substantial extensions to this result can be made.     

Orthogonal and Symplectic Matrix Models: Universality and Other Properties

We study orthogonal and symplectic matrix models with polynomial potentials and multi interval supports of the equilibrium measures. For these models we find the bounds (similar to those for the hermitian matrix models) for the rate of convergence of linear eigenvalue statistics and for the variance of linear eigenvalue statistics and find the logarithms of partition functions up to the order O(1). We prove also the universality of local eigenvalue statistics in the bulk.     

Approximate vacuums in (:φ2m :g(x))2 field theories and perturbation series

A class of perturbation problems is considered, in which the Rayleigh-Schrödinger perturbation series for the ground state eigenvalue and eigenvector are presumed to diverge. This class includes the (:^2m :g(x))₂, (m=2, 3) quantum field theory models and the quantum mechanical anharmonic oscillator. It is shown that, using matrix elements and vectors which occur in the series coefficients, one may construct convergent approximants to the eigenvalue and eigenvector. Results of a calculation of the ground state energy of thex ⁴ anharmonic oscillator are given.Supported in part by the National Research Council of Canada.     

Cross-correlation dynamics in financial time series

The dynamics of the equal-time cross-correlation matrix of multivariate financial time series is explored by examination of the eigenvalue spectrum over sliding time windows. Empirical results for the S&P 500 and the Dow Jones Euro Stoxx 50 indices reveal that the dynamics of the small eigenvalues of the cross-correlation matrix, over these time windows, oppose those of the largest eigenvalue. This behaviour is shown to be independent of the size of the time window and the number of stocks examined.A basic one-factor model is then proposed, which captures the main dynamical features of the eigenvalue spectrum of the empirical data. Through the addition of perturbations to the one-factor model, (leading to a ‘market plus sectors’ model), additional sectoral features are added, resulting in an Inverse Participation Ratio comparable to that found for empirical data. By partitioning the eigenvalue time series, we then show that negative index returns, (drawdowns), are associated with periods where the largest eigenvalue is greatest, while positive index returns, (drawups), are associated with periods where the largest eigenvalue is smallest. The study of correlation dynamics provides some insight on the collective behaviour of traders with varying strategies.     

Globally conformal invariant gauge field theory with rational correlation functions

Operator product expansions (OPE) for the product of a scalar field with its conjugate are presented as infinite sums of bilocal fields V_κ(x₁,x₂) of dimension (κ,κ). For a globally conformal invariant (GCI) theory we write down the OPE of V_κ into a series of twist (dimension minus rank) 2κ symmetric traceless tensor fields with coefficients computed from the (rational) 4-point function of the scalar field.

We argue that the theory of a GCI hermitian scalar field of dimension 4 in D=4 Minkowski space such that the 3-point functions of a pair of 's and a scalar field of dimension 2 or 4 vanish can be interpreted as the theory of local observables of a conformally invariant fixed point in a gauge theory with Lagrangian density .     

Pfaffian Expressions for Random Matrix Correlation Functions

It is well known that Pfaffian formulas for eigenvalue correlations are useful in the analysis of real and quaternion random matrices. Moreover the parametric correlations in the crossover to complex random matrices are evaluated in the forms of Pfaffians. In this article, we review the formulations and applications of Pfaffian formulas. For that purpose, we first present the general Pfaffian expressions in terms of the corresponding skew orthogonal polynomials. Then we clarify the relation to Eynard and Mehta’s determinant formula for hermitian matrix models and explain how the evaluation is simplified in the cases related to the classical orthogonal polynomials. Applications of Pfaffian formulas to random matrix theory and other fields are also mentioned.     

Simplification of the Gram Matrix Eigenvalue Problem for Quadrature Amplitude Modulation Signals

In quantum information science, it is very important to solve the eigenvalue problem of the Gram matrix for quantum signals. This allows various quantities to be calculated, such as the error probability, mutual information, channel capacity, and the upper and lower bounds of the reliability function. Solving the eigenvalue problem also provides a matrix representation of quantum signals, which is useful for simulating quantum systems. In the case of symmetric signals, analytic solutions to the eigenvalue problem of the Gram matrix have been obtained, and efficient computations are possible. However, for asymmetric signals, there is no analytic solution and universal numerical algorithms that must be used, rendering the computations inefficient. Recently, we have shown that, for asymmetric signals such as amplitude-shift keying coherent-state signals, the Gram matrix eigenvalue problem can be simplified by exploiting its partial symmetry. In this paper, we clarify a method for simplifying the eigenvalue problem of the Gram matrix for quadrature amplitude modulation (QAM) signals, which are extremely important for applications in quantum communication and quantum ciphers. The results presented in this paper are applicable to ordinary QAM signals as well as modified QAM signals, which enhance the security of quantum cryptography.     

Comparing the structure of an emerging market with a mature one under global perturbation

In this paper we investigate the Tehran stock exchange (TSE) and Dow Jones Industrial Average (DJIA) in terms of perturbed correlation matrices. To perturb a stock market, there are two methods, namely local and global perturbation. In the local method, we replace a correlation coefficient of the cross-correlation matrix with one calculated from two Gaussian-distributed time series, whereas in the global method, we reconstruct the correlation matrix after replacing the original return series with Gaussian-distributed time series. The local perturbation is just a technical study. We analyze these markets through two statistical approaches, random matrix theory (RMT) and the correlation coefficient distribution. By using RMT, we find that the largest eigenvalue is an influence that is common to all stocks and this eigenvalue has a peak during financial shocks. We find there are a few correlated stocks that make the essential robustness of the stock market but we see that by replacing these return time series with Gaussian-distributed time series, the mean values of correlation coefficients, the largest eigenvalues of the stock markets and the fraction of eigenvalues that deviate from the RMT prediction fall sharply in both markets. By comparing these two markets, we can see that the DJIA is more sensitive to global perturbations. These findings are crucial for risk management and portfolio selection.     

Eigenfunction entropy and spectral compressibility for critical random matrix ensembles

Based on numerical and perturbation series arguments we conjecture that for certain critical random matrix models the information dimension of eigenfunctions D(1) and the spectral compressibility χ are related by the simple equation χ+D(1)/d=1, where d is system dimensionality.     

Effects of the Scattering Length on the Yrast Spectrum for a Two-Component Bose-Einstein Condensates

We study the energy eigenvalue and the yrast states for a harmonically two-component weak-interacting Bose-Einstein condensate (BEC) when the intra-species and interspecies scattering length are different. The energy shift for different energy eigenvalues related to intra-and interspecies scattering lengths is calculated with the perturbation method. The actual yrast spectrum is more complicated than that when intra-species and interspecies scattering length are equal. The degenerated features disappear and so do the perfect symmetric features.     

Universality of the Local Spacing Distribution¶in Certain Ensembles of Hermitian Wigner Matrices

Consider an N×N hermitian random matrix with independent entries, not necessarily Gaussian, a so-called Wigner matrix. It has been conjectured that the local spacing distribution, i.e. the distribution of the distance between nearest neighbour eigenvalues in some part of the spectrum is, in the limit as N→∞, the same as that of hermitian random matrices from GUE. We prove this conjecture for a certain subclass of hermitian Wigner matrices. Received: 21 June 2000 / Accepted: 26 July 2000     

Finite diffraction models for electron bandstructures

Truncations of the infinite determinant resulting from the plane wave expansion method for an electron in a periodic potential are analysed to determine how well they can reproduce the low-lying eigenvalues or energy bands. The availability of rigorous expansions for the solutions of the Mathieu equation, essentially the Schrödinger equation for an electron in a one-dimensional cosine potential, suggests that problem for definitive comparisons. Since the only models which can reproduce the fundamental quadratic behaviour of bands at the zone centre have symmetric (odd) sets of reciprocal lattice vectors, the lowest order candidate has an odd determinant of size 3 × 3 through the reciprocal lattice vectors which define the 1st. Brillouin zone of the one-dimensional lattice. As zone centred determinants are not symmetric about zone boundaries they will not give a vanishing group velocity at that point and the effects of truncation will be at their worst. The 3 × 3 or cubic model factorises at the zone centre and a detailed analysis in closed form is straightforward. Agreement with the rigorous results is better than 1% everywhere. The next largest model, with a 5 × 5 determinant, gives errors which are several orders of magnitude smaller throughout the 1st. Brillouin zone. In addition it is found that even the 3 × 3 model gives much better results for the lowest eigenvalue than does second order perturbation theory. Numerical comparisons are also made for the group velocity and (inverse) effective mass using a Hellman-Feynmann approach to calculate the derivatives. Extensions to complex periodic potentials and higher dimensions are briefly discussed.     

Aeroelastic flutter of a disk rotating in an unbounded acoustic medium

The linear aeroelastic stability of an unbaffled flexible disk rotating in an unbounded fluid is investigated by modeling the disk-fluid system as a rotating Kirchhoff plate coupled to the irrotational motions of a compressible inviscid fluid. A perturbed eigenvalue formulation is used to compute systematically the coupled system eigenvalues. Both a semi-analytical and a numerical method are employed to solve the fluid boundary value problem. The semi-analytical approach involves a perturbation series solution of the dual integral equations arising from the fluid boundary value problem. The numerical approach is a boundary element method based on the Hadamard finite part. Unlike previous works, it is found that a disk with zero material damping destabilizes immediately beyond its lowest critical speed. Upon the inclusion of small disk material damping, the flutter speeds become supercritical and increase with decreasing fluid density. The competing effects of radiation damping into the surrounding fluid and disk material damping control the onset of flutter at supercritical speed. The results are expected to be relevant for the design of rotating disk systems in data storage, turbomachinery and manufacturing applications.     

势阱中粒子能级与波函数微扰计算的代数递推公式

利用超位力定理（HVT）和Hellmann-Feynman定理（HFT）,导出了由有精确解的势阱的能级值用微扰法直接计算一维势阱的各级近似能级的普遍代数公式,并导出由能级近似值计算定态波函数近似表达式的代数公式,给出了代数公式具体应用的几个典型一维势阱实例,此法可推广到二维势阱与三维势阱的情形。     

Perturbation theory for two-dimensional gauge models

Euclidean symmetric integration, previously proposed for the SU(N) Yang-Mills theory in the limit of large N, is used to study the propagators in the Schwinger model and the massive vector-meson model in two dimensions. The result of summing the perturbation series agrees with the exact solution in each case. Therefore, perturbation theory is here capable of dealing with non-analytic behaviour in the coupling constant.