Vicious random walkers and a discretization of Gaussian random matrix ensembles

The vicious random walker problem on a one-dimensional lattice is considered. Many walkers take simultaneous steps on the lattice and the configurations in which two of them arrive at the same site are prohibited. It is known that the probability distribution of N walkers after M steps can be written in a determinant form. Using an integration technique borrowed from the theory of random matrices, we show that arbitrary kth order correlation functions of the walkers can be expressed as quaternion determinants whose elements are compactly expressed in terms of symmetric Hahn polynomials.     

On the joint eigenvalue distribution for the matrix ensembles with non zero mean

Exact distributions are given for the two-dimensional case when the mean of the off-diagonal element is non-zero. The joint eigenvalue distribution for theN dimensional case, derived using the volume element in the space ofN ×N orthogonal matrices, is checked by rederiving the exact results forN=2. The smooth nature of theN-dimensional joint distribution supports the claim of the method of moments that the single eigenvalue distribution is a smooth function of the ratio of mean-to-mean square deviation.     

Eigenvalue density for ensemble of 2-body random hamiltonians with non-zero mean for matrix elements

We obtain an expression for the ensemble-averaged moments inm-particle space of 2-body random Hamiltonians with same non-zero mean for the matrix elements, in the limit ofN → ∞,m ≫ 2. The eigenvalue density function can then be immediately obtained in terms of the eigenvalue density (a Gaussian whenm ≫ 2) for zero mean ensembles. The results of Monte-Carlo calculations for iso-scalar rotationally invariant 2-body ensembles have also been given.     

On the Proof of Universality for Orthogonal and Symplectic Ensembles in Random Matrix Theory

We give a streamlined proof of a quantitative version of a result from P. Deift and D. Gioev, Universality in Random Matrix Theory for Orthogonal and Symplectic Ensembles. IMRP Int. Math. Res. Pap. (in press) which is crucial for the proof of universality in the bulk P. Deift and D. Gioev, Universality in Random Matrix Theory for Orthogonal and Symplectic Ensembles. IMRP Int. Math. Res. Pap. (in press) and also at the edge P. Deift and D. Gioev, {Universality at the edge of the spectrum for unitary, orthogonal and symplectic ensembles of random matrices. Comm. Pure Appl. Math. (in press) for orthogonal and symplectic ensembles of random matrices. As a byproduct, this result gives asymptotic information on a certain ratio of the β=1,2,4 partition functions for log gases.     

Moments of the Montroll-Weiss continuous-time random walk for arbitrary starting time

The generalized version of the Montroll-Weiss formalism for continuous-time random walks is employed to show that some of the asymptotic results for large times appropriate to the ordinary walk become exact when the start of the observations is arbitrary.     

Vicious random walkers in the limit of a large number of walkers

The vicious random walker problem on a line is studied in the limit of a large number of walkers. The multidimensional integral representing the probability that thep walkers will survive a timet (denotedP _t ^(p) ) is shown to be analogous to the partition function of a particular one-component Coulomb gas. By assuming the existence of the thermodynamic limit for the Coulomb gas, one can deduce asymptotic formulas forP _t ^(p) in the large-p, large-t limit. A straightforward analysis gives rigorous asymptotic formulas for the probability that after a timet the walkers are in their initial configuration (this event is termed a reunion). Consequently, asymptotic formulas for the conditional probability of a reunion, given that all walkers survive, are derived. Also, an asymptotic formula for the conditional probability density that any walker will arrive at a particular point in timet, given that allp walkers survive, is calculated in the limittp.     

Encryption by using matrix-added, or matrix-multiplied input images placed in the input plane of a double random phase encoding geometry

In this paper, we describe image encryption by combining the images with several matrices made with letters/numerals and placed in the input plane of a double random phase encoding (DRPE) system. Addition or multiplication of several such matrices with an input image provides a modified image pattern which is encrypted in a DRPE system utilizing the 4-f geometry. For gray scale images, the results of encryption in case of addition of matrices are found more reliable and secure as compared to the results of multiplication. Simulation results are presented in support of the idea. Reliability of the technique has been established by evaluating the mean square-error (MSE) between the decrypted and the original image.     

一类扰动发展方程近似解

采用了一个简单而有效的技巧, 研究了一类扰动发展方程. 首先引入求解一个相应典型方程的行波孤波解. 然后利用渐近方法得到了原扰动发展方程的近似解. 利用泛函分析的不动点定理, 指出了近似解级数的收敛性, 并讨论了近似解的精度.     

Mean-field cage theory for the random close packed state of a metastable hard-sphere glass

In 2005, we developed a mean-field cage theory for the freezing of a stable hard-sphere fluid using the character of a stable hard-sphere fluid, some observations and the mean-configuration approximation [X.Z. Wang, J. Chem. Phys. 122 (2005) 044515]. It was found that near the freezing point, a thermal fluctuation of a cage causes the hard sphere in this cage to exchange positions with one of its nearest neighbors. In this paper, we extend the theory to the random close packed state of a metastable hard-sphere glass. It is found that near the random close packing point, a thermal fluctuation of a cage sets three of the hard spheres in this cage and its nearest cages into the local circulatory motion, resulting in indirect position exchanges among these three hard spheres. We obtain an analytical formula for the random close packing density. The predicted values are in good agreement with the experimental and simulation results for spatial dimensions d=2–7$d=2\text{–}7$.     

Effect of local structure on electron paramagnetic resonance spectra for trigonal [Cr(H2O)6]3+coordination complex in the sulfate alums series: a ligand field theory study

A simple theoretical method is introduced for studying the interrelation between electronic and molecular structures.By diagonalizing the 120 × 120 complete energy matrices,the relationships between zero-field splitting(ZFS) parameter D and local distortion parameter △θ for Cr 3+ ions doped,separately,in α-and β-alums are investigated.Our results indicate that there exists an approximately linear relationship between D and △θ in a temperature range 4.2-297 K and the signs of D and △θ are opposite to each other.Moreover,in order to understand the contribution of spin-orbit coupling coefficient ζ to ZFS parameter D,the relation between D and ζ is also discussed.     

An asymptotic solving method for the periodic solution of a class of disturbed nonlinear evolution equation

<正>A class of disturbed evolution equation is considered using a simple and valid technique.We first introduce the periodic traveling-wave solution of a corresponding typical evolution equation.Then the approximate solution for an original disturbed evolution equation is obtained using the asymptotic method.We point out that the series of approximate solution is convergent and the accuracy of the asymptotic solution is studied using the fixed point theorem for the functional analysis.     

Identification of a dynamical model for an acoustic enclosure using the wavelet transform

This paper presents results of research work in the identification of a dynamical model for an acoustic enclosure, a duct with rectangular cross-section, closed ends, and side-mounted speaker enclosures. An acoustic enclosure is excited randomly and random decrement functions are built to convert the random responses to free acoustic responses. It is shown that the estimation of resonance frequencies is possible using the wavelet transform of the system’s free response. Using a particular form of the son wavelet function, results are improved compared to those obtained with the traditionally Morlet wavelet function. An optimal value of a parameter of the son wavelet function is obtained by minimization of the wavelet entropy. The accuracy of this new technique is confirmed by applying it to a numerical example, and to an acoustic enclosure. The advantage of using the wavelet transform method over the Fourier-based modal analysis that would normally be used for the enclosure modes problem is established.     

Existence of the transfer matrix formalism for a class of classical continuous gases

For classical gases of particles interacting through nonnegative, many-body interactions of short range it is verified that the corresponding grand canonical Gibbs measures have the global Markov property for sufficiently low values of the chemical activity. This yields the existence of a (nonsymmetric in general) transfer matrix formalism for such systems.     

On the spectrum of the dynamical matrix for a class of disordered harmonic systems

We study some aspects of the effect of mass disorder on the spectrum of the dynamical matrix of an infinite crystal in the harmonic approximation. Under suitable conditions on the masses, it is shown that the spectrum contains an absolutely continuous part and a nonempty set of isolated point eigenvalues of finite multiplicity whose number is smaller than or equal to the number of impurity atoms if the latter is finite. These conditions are satisfied only in the limiting case of zero concentration of each species of impurity. We draw some conjectures and make remarks on the spectrum under less restrictive conditions on the masses and briefly compare them with known results for random harmonic systems.Supported in part by the Swiss National Fund (to M.R.) and the Eidgenössische Stipendienkommission für ausländische Studierende (to W. F. W.).