Isotropic matrix elements of the collision integral for the Boltzmann equation

We have proposed an algorithm for constructing matrix elements of the collision integral for the nonlinear Boltzmann equation isotropic in velocities. These matrix elements have been used to start the recurrent procedure for calculating matrix elements of the velocity-nonisotropic collision integral described in our previous publication. In addition, isotropic matrix elements are of independent interest for calculating isotropic relaxation in a number of physical kinetics problems. It has been shown that the coefficients of expansion of isotropic matrix elements in Ω integrals are connected by the recurrent relations that make it possible to construct the procedure of their sequential determination.     

An algorithm for the efficient acoustic analysis of silencers of any general geometry

Transfer matrix analysis provides a very efficient means to analyse the linear plane-wave acoustic performance of silencer systems in the frequency domain. However, the nature of the algorithm for combination and reduction of the matrices is different for different combinations of elements. This paper describes an efficient algorithm for acoustic analysis of any general silencer system. The basic format of the algorithm is the identification of sub-systems of two-port acoustic elements. Computational time is also reduced by recording the order in which all of the elements are analysed and the sub-systems are reduced. Examples of the analysis of several complex silencer systems are presented. The gain in efficiency over a general global matrix approach is exceptional.     

Automated computation of spin- and colour-correlatedBorn matrix elements

I report on an implementation of an algorithm for the automated numerical calculation of spin- and colour-correlated Born matrix elements in QCD. These spin- and colour-correlated matrix elements are needed for NLO calculations in combination with the subtraction method. Both massless and massive quarks are considered. There are no restrictions on the number of external particles. As a trivial sub-case, the algorithm also applies to Born matrix elements without any correlations. These are sufficient for leading order calculations.     

Matrix elements of vibration kinetic energy operator of tetrahedral molecules in non-orthogonal-dependent coordinates

In this work, we propose an algorithm for calculating the matrix elements of the kinetic energy operator for tetrahedral molecules. This algorithm uses the dependent six-angle coordinates (6A) and takes into account the full symmetry of molecules. Unlike A.V. Nikitin, M. Rey, and Vl. G. Tyuterev who operate with the kinetic energy operator only in Radau orthogonal coordinates, we consider a general case. The matrix elements are shown to be a sum of products of one-dimensional integrals.     

An optimal algorithm for single-mode close-loop excitation in shallow water

An optimal algorithm for single-mode close-loop excitation in shallow water is presented.By analyzing the covariance of estimation value of Green’s function matrix,an optimal source array weights matrix is presented to estimate Green’s function matrix.The weights matrix is a unitary matrix,and absolute values of the matrix elements are equal. Algorithm based on the weights matrix makes single-mode excitation converge at maximum speed and be steady.Advantages of the algorithm are confirmed by numerical simulations. Finally,results of shallow water experiment are presented,and the energy ratio of single mode is higher than 97 percent.     

Revisit to the symmetry relations in diffusely backscattered polarization patterns of turbid media

As there exists an inconsistency in claiming the symmetrical relations in the 16 Mueller matrix elements used to describe a turbid medium, the author restudies the symmetrical relationships between diffusely backscattered polarization patterns in isotropic turbid media and simulates all two-dimensional elements of diffusely backscattered Mueller matrix in both cases of Rayleigh and Mie scatterings using the doublescattering approximation and the Monte Carlo algorithm, respectively. The previous experimental observatious are compared with the numerically determined matrix elements, showing a good agreement in both double-scattering model and Monte Carlo simulation. The symmetrical relations between the Mueller matrix elements are clarified.     

Stable reformulation of transfer matrix method for wave propagation in layered anisotropic media.

The numerical instability problem in the standard transfer matrix method has been resolved by introducing the layer stiffness matrix and using an efficient recursive algorithm to calculate the global stiffness matrix for an arbitrary anisotropic layered structure. For general anisotropy the computational algorithm is formulated in matrix form. In the plane of symmetry of an orthotropic layer the layer stiffness matrix is represented analytically. It is shown that the elements of the stiffness matrix are as simple as those of the transfer matrix and only six of them are independent. Reflection and transmission coefficients for layered media bounded by liquid or solid semi-spaces are formulated as functions of the total stiffness matrix elements. It has been demonstrated that this algorithm is unconditionally stable and more efficient than the standard transfer matrix method. The stiffness matrix formulation is convenient in satisfying boundary conditions for different layered media cases and in obtaining modal solutions. Based on this method characteristic equations for Lamb and surface waves in multilayered orthotropic media have been obtained. Due to the stability of the stiffness matrix method, the solutions of the characteristic equations are numerically stable and efficient. Numerical examples are given.     

Symmetry of the nonlinear collision operator matrix and new prospects in the moment method for solving the Boltzmann equation

Traditionally, the moment method has been used in kinetic theory to calculate transport coefficients. Its application to the solution of more complicated problems runs into enormous difficulties associated with calculating the matrix elements of the collision operator. The corresponding formulas for large values of the indices are either lacking or are very cumbersome. In this paper relations between matrix elements are derived from very general principles, and these can be employed as simple recurrence relations for calculating all the nonlinear and linear anisotropic matrix elements from assigned linear isotropic matrix elements. Efficient programs which implement this algorithm are developed. The possibility of calculating the distribution function out to 8–10 thermal velocities is demonstrated. The results obtained open up prospects for solving many topical problems in kinetic theory. Zh. Tekh. Fiz. 69, 6–9 (September 1999)     

浅海声场单模闭环发射的最佳估计算法

提出了一种浅海声场单模闭环发射的最佳估计算法。通过对声场格林函数矩阵估计协方差的分析,设计了一种对格林函数矩阵作最佳估计的最佳发射阵权系数矩阵,此权系数矩阵为酉矩阵且各元素绝对值相等。基于此矩阵的单模发射算法可以使得单模激发过程以最快的速度收敛,并且发射稳定。算法的优势在模拟计算中得到了验证。最后给出了海上实验的结果,单模含量高达97%。      

一类块三对角矩阵求逆的算法

讨论了一类块三对角矩阵的求逆问题.由块三对角矩阵的LU分解,得到了其逆矩阵块元素的显式表达式.当考虑该表达式的结构特征时,可得到块元素的递推关系式,由此得到一个求逆矩阵的新算法.该算法比已有的块三对角矩阵求逆算法的计算复杂度和计算时间低.     

A generalized scheme for analysis of multifarious commercially used mufflers

Commercial automotive mufflers are often too complex to be broken into a cascade of one-dimensional elements with predetermined transfer matrices. The one-dimensional (1-D) scheme presented in this paper is based on an algorithm that uses user-friendly visual volume elements along with the theory of transfer matrix based muffler analysis. This work attempts to exploit the speed of the one-dimensional analysis with the flexibility, generality and user-friendliness of three-dimensional analysis using geometric modeling. A code based on the developed algorithm has been employed to demonstrate the generality of the proposed method in analyzing commercial mufflers by considering three very diverse classes of mufflers with different kinds of combinations of reactive, perforated and absorptive elements. Though the examples used in the paper are not very complex for they are meant to be just representative cases of certain classes of mufflers, yet the algorithm can handle a large domain of commercial mufflers of high degree of complexity. Results from the present algorithm have been validated through comparisons with both the analytical (plane wave based) and the more general, three-dimensional FEM based results. The forte of the proposed method is its power to construct the system matrix consistent with the boundary conditions from the geometrical model to evaluate the four-pole parameters of the entire muffler and thence its transmission loss, etc. Thus, the algorithm can be used in conjunction with the transfer matrix based muffler programs to analyze the entire exhaust system of an automobile.     

How to calculate decoherence matrices numerically

A central object in the interpretation of quantum mechanics of closed systems is the decoherence matrix. But only for a very small number of models one is able to give explicit expressions for its elements. So numerical methods are required. Unfortunately the dimensions of these matrices are usually very high, which makes also a direct numerical calculation impossible. In this paper I will develop an algorithm that combines the numerical calculation of the elements of the decoherence matrix with a permanent estimation. If the chosen histories are almost decoherent, the algorithm will provide the diagonal and maybe some near to diagonal elements only. In this case, within numerical errors, we can assume, that all not calculated elements are smaller than a given threshold. If the histories are coherent, the algorithm has to calculate too much data (there exist too much non-vanishing elements) and the success depends mainly on the available computer time.As an example I apply this procedure to the Caldeira—Leggett-model.     

Efficient Finite Difference Solutions to the Time-Dependent Schrödinger Equation

The matrix elements of the exponential of a finite difference realization of the one-dimensional Laplacian are found exactly. This matrix is used to formulate an efficient algorithm for the numerical solution to the time-dependent quantum mechanical scattering of a single particle from a time-independent potential in one-space and one-time dimension. The method generalizes to higher spatial dimensions, as well as to multiparticle problems.     

A distributed-mass approach for dynamic analysis of Bernoulli-Euler plane frames

The exact dynamic analysis of plane frames should consider the effect of mass distribution in beam elements, which can be achieved by using the dynamic stiffness method. Solving for the natural frequencies and mode shapes from the dynamic stiffness matrix is a nonlinear eigenproblem. The Wittrick-Williams algorithm is a reliable tool to identify the natural frequencies. A deflated matrix method to determine the mode shapes is presented. The dynamic stiffness matrix may create some null modes in which the joints of beam elements have null deformation. Adding an interior node at the middle of beam elements can eliminate the null modes of flexural vibration, but does not eliminate the null modes of axial vibration. A force equilibrium approach to solve for the null modes of axial vibration is presented. Orthogonal conditions of vibration modes in the Bernoulli-Euler plane frames, which are required in solving the transient response, are theoretically derived. The decoupling process for the vibration modes of the same natural frequency is also presented.     

Theoretical influence coefficients for correction of matrix effects in x‐ray fluorescence analysis of intermediate‐thickness samples

Theoretical influence coefficients for x‐ray fluorescence analysis of intermediate‐thickness samples is proposed. The algorithm, derived from the Sherman equation, corrects simultaneously both absorption and enhancement effects in specimens with various thicknesses. The coefficients take into account all matrix elements, hence they have to be calculated using an iterative approach. Nevertheless, if matrix diversification is limited, the coefficients can be treated as constants for given mass per unit area of the specimen. Then the iteration can be avoided and calculations are simple and fast. The dependence of the influence coefficients on matrix composition and sample thickness is discussed. The validity of the proposed algorithm was checked with certified geological materials, four‐element polycrystals and ferroelectric ceramics using wavelength‐dispersive x‐ray spectrometry. Copyright © 2005 John Wiley & Sons, Ltd.     

Decomposition algorithm of an experimental Mueller matrix

This study deals with the interpretation of experimental Mueller matrices. The understanding of such a matrix is not straightforward in the case, in particular, of a strongly depolarizing medium, which is therefore disturbed and where relevant pieces of information are often distributed among its various elements. As a result, information data need to be extracted by a decomposition of any Mueller matrix into simple elements to uncouple the existing polarimetric effects. This led us to develop an algorithm in order to characterize any depolarizing, or not, polarimetric system. In addition to differentiating the experimental noise from the intrinsic depolarization of the optical system under study, this algorithm proved to: (i) separate depolarization from birefringence and dichroism and (ii) characterize the isotropic or anisotropic nature of the depolarization. At last, this algorithm was validated through the study of several optical systems with different polarimetric properties.     

一种新的自定标算法--图像变换法

提出了一种新的基于图像变换的自定标算法,由于绝对二次曲线的像的各个元素不在一个数量级上,导致自定标过程中对噪音极其敏感.该方法针对于这一缺点,首先估计一个内参矩阵,然后通过图像变换,使绝对二次曲线的像的各个元素在一个数量级上.理论分析及模拟图像序列和真实图像序列的实验结果表明,该自定标算法能够提高定标准确度及鲁棒性.     

Calculation of higher-order approximations of the coefficients of the Herman-Wallis factor. Test for hydrogen halides

The higher-order approximations of the perturbation theory for the Herman-Wallis factor, which describes the effect of the vibrational-rotational interaction on the matrix elements of dipole moments of diatomic molecules, are calculated within the formalism of quantum number polynomials. An algorithm for taking into account higher orders of the perturbation theory for calculating random coefficients is found; the algorithm has a better convergence than previous theoretical methods. As applied to hydrogen halides, the results obtained agree well with current experimental data.     

Field form of perturbation theory for calculating the matrix elements of operators between the states of two degenerate atomic levels. I

It is shown that the calculation of the matrix elements between degenerate states of arbitrary atomicmolecular systems can be reduced to a calculation of the matrix elements for some auxiliary operator. The perturbation theory for this calculation can be given a field form, and its matrix elements can be expressed in terms of the contributions from special Feynman diagrams. The exact rules are formulated for constructing the contributions from these diagrams.     

Asymptotics of collision integral matrix elements in the isotropic case

The method of nonlinear moments, when used to solve the Boltzmann equation, necessitates the calculation of collision integral matrix elements. The matrix elements are hard to calculate numerically, especially at large indices. The asymptotics of the matrix elements are constructed. In terms of the model of pseudopower particle interaction, a formula free of summation is derived. This makes it possible to find the asymptotic behavior of linear and nonlinear elements when two indices are large. For an arbitrary interaction cross section, asymptotic expansions of linear and nonlinear matrix elements in one index are obtained. For Maxwellian molecules, asymptotic formulas are derived for three large indices.