Showing papers on "Matrix (mathematics) published in 1979"

Electromagnetic propagation in birefringent layered media

Abstract: The propagation of electromagnetic radiation in birefringent layered media is considered. A general formulation of the plane-wave propagation in an arbitrarily birefringent layered medium is presented. The concepts of dynamical matrix and propagation matrix are introduced. A 4 × 4 transfer matrix method is used to relate the field amplitudes in different layers. Our general theory is then applied to the special case of periodic birefringent layered media, especially the Solc birefringent layered media [ I. Solc , Cesk. Casopis Fẏs.3, 366 ( 1953);Cesk. Casopis Fẏs.10, 16 ( 1960)]. The unit cell translation operator is derived. The band structures as well as the Bloch waves are obtained by diagonalizing the translation operator. Coupled mode theory is extended to the case of birefringent periodic perturbation to explain the exchange Bragg scattering. A general mode dispersion relation for guided waves is also obtained in terms of the transfer matrix elements.

The Commutation Matrix: Some Properties and Applications

Abstract: The commutation matrix $K$ is defined as a square matrix containing only zeroes and ones. Its main properties are that it transforms vecA into vecA', and that it reverses the order of a Kronecker product. An analytic expression for $K$ is given and many further properties are derived. Subsequently, these properties are applied to some problems connected with the normal distribution. The expectation is derived of $\varepsilon' A\varepsilon\cdot\varepsilon' B\varepsilon\cdot\varepsilon'C\varepsilon$, where $\varepsilon \sim N(0, V)$, and $A, B, C$ are symmetric. Further, the expectation and covariance matrix of $x \otimes y$ are found, where $x$ and $y$ are normally distributed dependent variables. Finally, the variance matrix of the (noncentral) Wishart distribution is derived.

Wigner distribution function and its application to first-order optics

Abstract: The Wigner distribution function of optical signals and systems has been introduced. The concept of such functions is not restricted to deterministic signals, but can be applied to partially coherent light as well. Although derived from Fourier optics, the description of signals and systems by means of Wigner distribution functions can be interpreted directly in terms of geometrical optics: (i) for quadratic-phase signals (and, if complex rays are allowed to appear, for Gaussian signals, too), it leads immediately to the curvature matrix of the signal; (ii) for Luneburg’s first-order system, it directly yields the ray transformation matrix of the system; (iii) for the propagation of quadratic-phase signals through first-order systems, it results in the well-known bilinear transformation of the signal’s curvature matrix. The zeroth-, first-, and second-order moments of the Wigner distribution function have been interpreted in terms of the energy, the center of gravity, and the effective width of the signal, respectively. The propagation of these moments through first-order systems has been derived. Since a Gaussian signal is completely described by its three lowest-order moments, the propagation of such a signal through first-order systems is known as well.

Resistivity modeling for arbitrarily shaped three-dimensional structures

Abstract: A numerical technique has been developed to solve the three‐dimensional (3-D) potential distribution about a point source of current located in or on the surface of a half‐space containing an arbitrary 3-D conductivity distribution. Self‐adjoint difference equations are obtained for Poisson’s equation using finite‐difference approximations in conjunction with an elemental volume discretization of the lower half‐space. Potential distribution at all points in the set defining the subsurface are simultaneously solved for multiple point sources of current. Accurate and stable solutions are obtained using full, banded, Cholesky decomposition of the capacitance matrix as well as the recently developed incomplete Cholesky‐conjugate gradient iterative method. A comparison of the 2-D and 3-D simple block‐shaped models, for the collinear dipole‐dipole array, indicates substantially lower anomaly indices for inhomogeneities of finite strike‐extent. In general, the strike‐extents of inhomogeneities have to be approxi...

Amorphous-silicon field-effect device and possible application

Abstract: The characteristics of an insulated-gate field-effect transistor made from amorphous silicon (a-Si) deposited in a glow discharge are discussed. It is suggested that the a-Si device could be applied with advantage in an addressable matrix of a liquid-crystal display panel.

Statistical Characterization of Altitude Matrices by Computer. Report 6. An Integrated System of Terrain Analysis and Slope Mapping.

Abstract: : A system is described which: (a) replaces existing sets of diverse terrain indices with a group of statistics for point-characteristics; (b) calculates all of these statistics in a single computer run from a single data set; and (c) utilizes available altitude matrix data The procedures are applicable to Altitude matrix data at any grid mesh From altitudes in each 3 x 3 submatrix, a quadratic surface is fitted and solved for its first and second horizontal and vertical derivatives at the central point This yields the slope gradient, slope aspect, profile convexity, and plan convexity at every point in the original matrix, except for the peripheral rows and columns These 'point' descriptors are presented as: (1) line-printer shaded maps; (2) histograms; (3) scatter plots of each pair; (4) matrix of pair-wise correlations, plus circular regressions on aspect, and several multiple regressions, and (5) summary (moment-based) statistics In general, the five basic descriptors have little relation to each other, except that gradient is usually a quadratic function of altitude A comparison is made with other approaches, such as spectral analysis and fractal modelling The long-distance persistence properties of terrain mean that considerable extra variance at long wavelengths is usually incorporated when the study area is extended Hence the auto correlation function varies with the length of series or size of area Variance of derivatives are also affected, but means, skews, and kurtoses are not

An estimate for the condition number of a matrix

Abstract: It is important in practice when solving linear systems to have an economical method for estimating the condition number $\kappa (A)$ of the matrix of coefficients. An algorithm involving $O(n^2 )$ arithmetic operations is described; it gives a reliable indication of the order of magnitude of $\kappa (A)$.

Vec and vech operators for matrices, with some uses in jacobians and multivariate statistics

Abstract: The vec of a matrix X stacks columns of X one under another in a single column; the vech of a square matrix X does the same thing but starting each column at its diagonal element. The Jacobian of a one-to-one transformation X → Y is then ∣∣∂(vecX)/∂(vecY) ∣∣ when X and Y each have functionally independent elements; it is ∣∣ ∂(vechX)/∂(vechY) ∣∣ when X and Y are symmetric; and there is a general form for when X and Y are other patterned matrices. Kronecker product properties of vec(ABC) permit easy evaluation of this determinant in many cases. The vec and vech operators are also very convenient in developing results in multivariate statistics.

The Nevanlinna–Pick Problem for Matrix-Valued Functions

Abstract: This paper contains a detailed treatment of the Nevanlinna–Pick interpolation problem for matrix-valued functions. A close relationship with the trigonometric moment problem is put into light. Matrix extensions of Pick’s solvability criterion and Nevanlinna’s iterative algorithm are presented. Necessary and sufficient conditions are obtained for uniqueness of the solution of the infinite interpolation problem. The approach is essentially based upon the theories of J-contractive transformations and of Weyl matrix circles.

An Efficient SS/TDMA Time Slot Assignment Algorithm

Abstract: This paper presents an efficient time slot assignment algorithm for an SS/TDMA system. The technique utilized in the algorithm is a systematic method of finding distinct representatives from the row sets of a traffic matrix. The assignment efficiency resulting from the algorithm is 100% for any traffic matrix. The number of switching modes generated by the algorithm is bounded by n^{2} - 2n + 2 for an n \times n traffic matrix. The computational procedures are illustrated by an example for the Advanced WESTAR system. Also included in the paper are the computer simulation results on the numbers of required switching modes for various simulated traffic matrices.

Stability of dynamical systems: A constructive approach

Abstract: A set A of n \times n complex matrices is stable if for every neighborhood of the origin U \subset C^{n} , there exists another neighborhood of the origin V , such that for each M \in A' (the set of finite products of matrices in A), MV\subset U . Matrix and Liapunov stability are related. Theorem: A set of matrices A is stable if and only if there exists a norm, |\cdot | , such that for all M \in A , and all z \in C^{n} , |Mz| . The norm is a Liapunov function for the set A . It need not be smooth; using smooth norms to prove stability can be inadequate. A novel central result is a constructive algorithm which can determine the stability of A based on the following. Theorem: A,={M0,Mj,. .,Mmi) is a set of m distinct complex matrices. Let Wo be a bounded neighborhood of the origin. Define for k > 0 , Wk =convexhbull ~ Mk'Wk - I where k'=(k- 1) mod m . Then A isstableifand only if V-U Q,_ is bounded. W* is the norm of the first theorem. The constructive algorithm represents a convex set by its extreme points and uses linear programming to construct the successive W_k . Sufficient conditions for the finiteness of constructing W_k from W_{k-1} , and for stopping the algorithm when either the set is proved stable or unstable are presented. A is generalized to be any convex set of matrices. A dynamical system of differential equations is stable if a corresponding set of matrices --associated with the logarithmic norms of the matrices of the linearized equations--is stable. The concept of the stability of a set of matrices is related to the existence of a matrix norm. Such a norm is an induced matrix norm if and only if the set of matrices is maximally stable (ie., it cannot be enlarged and remain stable).

Axial Currents in Nuclei: The Gamow-Teller Matrix Element

Abstract: We propose that the quenching of Gamow-Teller matrix elements in light nuclei can be effectively described by a mechanism that governs the $p$-wave pion-nucleus scattering. This may be providing a solution to the long-standing problem. The resulting Landau-Migdal parameter ${g}^{\ensuremath{'}}$ turns out to be quite consistent with the values extracted from the spectra of pionlike excitations and places a stringent constraint on pion condensation in nuclear matter.

Matrix formulation of the reconstruction of phase values from phase differences

Abstract: Methods in speckle imaging and adapative optics, as well as a new technique in digital image restoration, require the calculation of the Fourier phase spectrum from measurements of the differences on a two-dimensional grid of the phase spectrum. The calculation of phases from phase differences has been analyzed in the literature and relaxation mechanisms for computing the phase have been derived by least-squares analysis. In the following paper we formulate the phase reconstruction problem in terms of a vector-matrix multiplication, and we then show that previous solution methods are equivalent to this general description. We also analyze the errors in reconstruction and reconcile previously published error results based on simulations with an analytical error expression derived from Parseval’s theorem. Finally, we comment upon the rate of convergence of phase reconstructions, and discuss numerical analysis literature which indicates that the methods previously published for phase reconstruction can be made to converge much faster.

Construction of Green's functions from an exact S matrix

Abstract: The bootstrap program for determining Green's functions from an exact $S$ matrix is carried out for the simplest soliton field theory of a scalar field with $S$-matrix operator $S={(\ensuremath{-}1)}^{\frac{N(N\ensuremath{-}1)}{2}}$, where $N$ is the total number operator. Despite the formal simplicity of the $S$ matrix, the Green's functions derived have a rich structure. The results can be checked since this field theory is none other than that of the order variable of the Ising model in the scaling limit above the critical temperature.

Approximating the inverse of a matrix for use in iterative algorithms on vector processors

Abstract: Most iterative techniques for solving the symmetric positive-definite systemAx=b involve approximating the matrixA by another symmetric positive-definite matrixM and then solving a system of the formMz=d at each iteration. On a vector machine such as the CDC-STAR-100, the solution of this new system can be very time consuming. If, however, an approximationM−1 can be given toA−1, the solutionz=M−1d can be computed rapidly by matrix multiplication, a fast operation on the STAR. Approximations using the Neumann expansion of the inverse ofA give reasonable forms forM−1 and are presented. Computational results using the conjugate gradient method for the “5-point” matrixA are given.

Radiative lifetimes of OH(A2Σ) and Einstein coefficients for the A-X system of OH and OD

Abstract: Radiative lifetimes of individual rotational states of OH(A2Σ) were observed using a delayed coincidence technique. The values obtained were scaled to a theoretical functional form to give τrad(A2Σ, v = 0, N = 1, J = 32 = 686 ± 14 ns. Einstein A and B coefficients for the A-X system of OH and OD were computed, using numerical integrals of a transition moment of the form Re(r) = 3.75 × 10-30 (1-0.75 r) C … m and vibrational wavefunctions obtained from numerical solutions of the radial Schrodinger equations (including centrifugal terms) for RKR potentials for the two electronic states. Matrix elements in Hund's case b were transformed to intermediate coupling for the X state to yield proper oscillator strengths for computation of the Einstein coefficients. Tables of the A and B coefficients for the 0-0 vibrational transitions are included. Extended tables are available on microfiche or microfilm from the authors.

Schur Parametrization of Positive Definite Block-Toeplitz Systems

Abstract: The parameters occurring in Szego’s recurrence relations associated with a class C function are known to be the same as the Schur parameters of the corresponding class S function The present paper contains a matrix extension of this result Certain important questions about the matrix classes S and C, related to spectral factorization, are studied by means of their Schur–Szego parameters

Z(4) symmetric factorized s matrix in two space-time dimensions

Abstract: The factorizedS-matrix with internal symmetryZ4 is constructed in two space-time dimensions. The two-particle amplitudes are obtained by means of solving the factorization, unitarity and analyticity equations. The solution of factorization equations can be expressed in terms of elliptic functions. TheS-matrix contains the resonance poles naturally. The simple formal relation between the general factorizedS-matrices and the Baxter-type lattice transfer matrices is found. In the sense of this relation theZ4-symmetricS-matrix corresponds to the Baxter transfer matrix itself.

On the Estimation of Sparse Hessian Matrices

Abstract: This paper studies automatic procedures for estimating second derivatives of a real valued function of several variables. The estimates are obtained from differences in first derivative vectors, and it is supposed that the required matrix is sparse and that its sparsity structure is known. Our main purpose is to find ways of taking advantage of the sparsity structure and the symmetry of the second derivative matrix, in order to make small the number of first derivative vectors that have to be calculated. Two new algorithms are proposed, which seem to be very successful in practice and which do not require much computer arithmetic. One is a direct method and the other is a substitution method, these terms being explained in the paper. Some examples show, however, that the given methods may not minimize the number of first derivative vector calculations.

A study of extrapolation methods based on multistep schemes without parasitic solutions

Abstract: The paper presents a theoretical approach to the construction of extrapolation methods for systems of the kind. $$L[y] = \varepsilon f(t,y,),$$ whereL is a general linear differential operator of orderk. For e=0, the discretization schemes are required to beexact and to contain only solutions in the nullspace ofL. For e≠0, the paper studies the construction of methods that permitquadratic extrapolation. In the special casek=2, a new two-step method is obtained that applies to systems of the type $$y'' + Ay = \varepsilon f(t,y,y')$$ whereA is a real, symmetric, positive semi-definite matrix. This algorithm might be of use inregular celestial mechanics-apart from any other possible applications.

Evidence for massless modes in the 'solvable model' of a spin glass

Abstract: Mean-field equations for the long-range spin-glass model are solved numerically for T

The general structure of integrable evolution equations

Abstract: This paper presents some new results in connection with the structure of integrable evolution equations. It is found that the most general integrable evolution equations in one spatial dimension which is solvable using the inverse scattering transform (i. s. t.) associated with the n th order eigenvalue problem V x = ( ξR 0 + P ( x , t )) V has the simple and elegant form G ( D R , t ) P t – F ( D R , t ) x [ R 0 , P ] = Ω ( D R , t ) [ C , P ], where G , F and Ω are entire functions of an integro-differential operatos D R and the bracket refers to the commutator. The list provided by this form is not exhaustive but contains most of the known integrable equations and many new ones of both mathematical and physical significance. The simple structure allows the identification in a straightforward manner of the equation in this class which is closest to a given equation of interest. The x dependent coefficients enable the inclusion of the effects of field gradients. Furthermore when the partial derivative with respect to t is zero, the remaining equation class contains many nonlinear ordinary differential equation of importance, such as the Painleve equations of the second and third kind. The properties of the scattering matrix A( ξ , t ) corresponding to the potential P( x , t ) are investigated and in particular the time evolution of A ( ξ , t ) is found to be G ( ξ , t ) A t + F ( ξ , t ) A ξ = Ω ( ξ , t )[ C , A ], The role of the diagonal entries and the principal corner minors in providing the Hamiltonian structure and constants of the motion is discussed. The central role that certain quadratic products of the eigenfunctions play in the theory is briefly described and the necessary groundwork from a singular perturbation theory is given when n = 2 or 3.