Showing papers on "Matrix (mathematics) published in 1974"

Δ I = 1 2 Rule for Nonleptonic Decays in Asymptotically Free Field Theories

Abstract: The effective nonleptonic weak interaction is examined assuming the Weinberg-Salam theory of weak interactions and an exactly-conserved-color gauge symmetry for strong interactions. It is shown that the octet part of the nonleptonic weak interaction is more singular at short distances than the 27 part. The resulting enhancement of the octet term in the effective local weak Lagrangian, together with suggested mechanisms for the suppression of matrix elements of the 27 operator, may be sufficient to account for the observed $|\ensuremath{\Delta}I|=\frac{1}{2}$ rule.

An Antipodally Symmetric Distribution on the Sphere

Abstract: The distribution $\Psi(\mathbf{x}; Z, M) = \operatorname{const}. \exp(\mathrm{tr} (ZM^T \mathbf{xx}^T M))$ on the unit sphere in three-space is discussed. It is parametrized by the diagonal shape and concentration matrix $Z$ and the orthogonal orientation matrix $M. \Psi$ is applicable in the statistical analysis of measurements of random undirected axes. Exact and asymptotic sampling distributions are derived. Maximum likelihood estimators for $Z$ and $M$ are found and their asymptotic properties elucidated. Inference procedures, including tests for isotropy and circular symmetry, are proposed. The application of $\Psi$ is illustrated by a numerical example.

Newton-type methods for unconstrained and linearly constrained optimization

Abstract: This paper describes two numerically stable methods for unconstrained optimization and their generalization when linear inequality constraints are added. The difference between the two methods is simply that one requires the Hessian matrix explicitly and the other does not. The methods are intimately based on the recurrence of matrix factorizations and are linked to earlier work on quasi-Newton methods and quadratic programming.

Interaction between a circular inclusion and an arbitrarily oriented crack

Abstract: The plane interaction problem for a circular elastic inclusion embedded in an elastic matrix which contains an arbitrarily oriented crack is considered. Using the existing solutions for the edge dislocations as Green's functions, first the general problem of a through crack in the form of an arbitrary smooth arc located in the matrix in the vicinity of the inclusion is formulated. The integral equations for the line crack are then obtained as a system of singular integral equations with simple Cauchy kernels. The singular behavior of the stresses around the crack tips is examined and the expressions for the stress-intensity factors representing the strength of the stress singularities are obtained in terms of the asymptotic values of the density functions of the integral equations. The problem is solved for various typical crack orientations and the corresponding stress-intensity factors are given.

Optimization of stochastic linear systems with additive measurement and process noise using exponential performance criteria

Abstract: The expected value of a multiplicative performance criterion, represented by the exponential of a quadratic function of the state and control variables, is minimized subject to a discrete stochastic linear system with additive Gaussian measurement and process noise. This cost function, which is a generalization of the mean quadratic cost criterion, allows a degree of shaping of the probability density function of the quadratic cost criterion. In general, the control law depends upon a gain matrix which operates linearly on the smoothed history of the state vector from the initial to the current time. This gain matrix explicitly includes the covariance of the estimation errors of the entire state history. The separation theorem holds although the certainty equivalence principle does not. Two special cases are of importance. The first occurs when only the terminal state is costed. A feedback control law, linear in the current estimate of the state, results where the feedback gains are functionally dependent upon the error covariance of the current state estimate. The second occurs if all the intermediate states are costed but there is no process noise except for an initial condition uncertainty. A feedback law results which depends not only upon the current dynamical state estimate but also on an additional vector which is path dependent.

New wave‐operator identity applied to the study of persistent currents in 1D

Abstract: We show that a large class of backward‐scattering matrix elements involving Δk ∼ ± 2kF vanish for fermions interacting with two‐body attractive forces in one dimension. (These same matrix elements are finite for noninteracting particles and infinite for particles interacting with two‐body repulsive forces.) Our results demonstrate the possibility of persistent currents in one dimension at T = 0, and are a strong indication of a metal‐to‐insulator transition at T = 0 for repulsive forces. They are obtained by use of a convenient representation of the wave operator in terms of density‐fluctuation operators.

Determination of the instant of glottal closure from the speech wave

Abstract: For vowels excited by vigorous glottal vibrations, the instant of glottal closure is tentatively identified with the moment of strongest excitation (at not too low frequencies) and worst linear predictability. Some predictor methods for its determination are reviewed, which do not always yield reliable and unequivocal results. Then Sobakin's method using the determinant of the autocovariance matrix is examined critically and reinterpreted such that the determinant is maximum if the beginning of the interval on which the autocovariance matrix is calculated coincides with the glottal closure. This hypothesis is tested by comparison with the predictor methods and by looking at the inversely filtered waveforms and the formants obtained from predictors determined on a shifted interval. The determinant method seems to be very reliable even for otherwise difficult cases, such as the vowel /u/.

A note on stability of linear time-varying systems

Abstract: An example is given to show that even if the eigenvalues of the system A -matrix A(t) of a linear time-varying system \dot{x}(t) = A(t)x(t) are independent of t and some of them have positive real parts, the system is asymptotically stable.

Matrix Derivatives with an Application to an Adaptive Linear Decision Problem

Abstract: A theory of matrix differentiation is presented which uses the concept of a matrix of derivative operators. This theory allows matrix techniques to be used in both the derivation and the description of results. Several new operations and identities are presented which facilitate the process of matrix differentiation. The derivative theorems and new operations are then applied to the problem of determining optimal policies in a linear decision model with unknown coefficients, a problem which would be cumbersome if not impossible to solve without these theorems and operations.

The Solution of a Toeplitz Set of Linear Equations

Abstract: The solution of a set of m linear equations with a non-Hermitian Toeplitz associated matrix is considered. Presently available fast algorithms solve this set with 4m2 “operations” (an “operation” is defined here as a set of one addition and one multiplication). An improved algorithm requiring only 3m2 “operations” is presented.

Structure and reactions of matrix-isolated tetracarbonyliron(0)

Abstract: U.v. photolysis of 13CO enriched pentacarbonyliron in SF6 and Ar matrices at 20 K indicates that tetracarbonyliron has a C2v structure, with bond angles ca. 145 and 120°, as estimated from i.r. intensities. Photolysis of [Fe(CO)5] in an N2 matrix produces [Fe(CO)4] which can be reacted reversibly with the matrix to form [Fe(CO)4(N2)]. Equivalent experiments in CH4 or Xe matrices produce two [Fe(CO)4] species; one is [Fe(CO)4] with the same structure as in other matrices, and there is strong circumstantial evidence for the other being [Fe(CO)4(CH4)] or [Fe(CO)4Xe]. The behaviour of [Os(CO)5] is similar. The results are compared with those obtained in frozen hydrocarbon glasses at 77 K. In matrices with high concentrations of [Fe(CO)5](>1 : 1 000 carbonyl compound : matrix), polynuclear iron carbonyls, [Fe2(CO)8], [Fe2(CO)9], and [Fe3(CO)12], are formed.

An Unbalanced Jackknife

Abstract: It is proved that the jackknife estimate $\tilde{\theta} = n\hat{\theta} - (n - 1)(\sum \hat{\theta}_{-i}/n)$ of a function $\theta = f(\beta)$ of the regression parameters in a general linear model $\mathbf{Y} = \mathbf{X\beta} + \mathbf{e}$ is asymptotically normally distributed under conditions that do not require $\mathbf{e}$ to be normally distributed. The jackknife is applied by deleting in succession each row of the $\mathbf{X}$ matrix and $\mathbf{Y}$ vector in order to compute $\hat{\mathbf{\beta}}_{-i}$, which is the least squares estimate with the $i$th row deleted, and $\hat{\theta}_{-i} = f(\hat\mathbf{\beta}_{-i})$. The standard error of the pseudo-values $\tilde{\theta}_i = n\hat{\theta} - (n - 1)\hat{\theta}_{-i}$ is a consistent estimate of the asymptotic standard deviation of $\tilde{\theta}$ under similar conditions. Generalizations and applications are discussed.

Spectroscopy and collision theory. III. Atomic eigenchannel calculation by a Hartree-Fock-Roothaan method

Abstract: Previous papers have expressed various experimental spectral data in terms of a single set of parameters (eigen-quantum-defects, transformation matrix, and excitation dipole moments). The parameters pertain to eigenstates of an electron-ion scattering matrix which represents only the effect of short-range interactions. This paper presents a method for calculating the same parameters by solving the many-electron Schr\"odinger equation for an atom within a limited spherical volume. Quantitative results, for Ar states with $J=1$ and odd parity, are compared with data extracted earlier from an analysis of spectral data, and are also used to reproduce the first 65 discrete line positions and their intensities, the positions and profiles of autoionization lines, and the branching ratio of the photoionization.

Matrix quadratic equations

Abstract: Matrix quadratic equations have found the most diverse applications. The present article gives a connected account of their theory, and contains some new results and new proofs of known results.

Bidiagonalization of Matrices and Solution of Linear Equations

Abstract: An algorithm given by Golub and Kahan [2] for reducing a general matrix to bidiagonal form is shown to be very important for large sparse matrices. The singular values of the matrix are those of the bidiagonal form, and these can be easily computed. The bidiagonalization algorithm is shown to be the basis of important methods for solving the linear least squares problem for large sparse matrices. Eigenvalues of certain 2-cyclic matrices can also be efficiently computed using this bidiagonalization.

Circuit theory for a class of anisotropic and gyrotropic thin‐film optical waveguides and design of nonreciprocal devices for integrated optics

Abstract: A circuit‐theoretic treatment is presented for thin‐film optical waveguides using a class of real anisotropic and gyrotropic materials. The analysis is based on the two‐mode approximation in normal‐mode theory. The terminal behavior of those guides is described by the 2 × 2 transmission matrix and expressions for matrix elements of six systems that are introduced as canonical elements in circuit synthesis are derived. Properties of canonical elements are discussed with particular attention on their reciprocity. Using the transmission matrix, we can treat the design of thin‐film optical devices by a simple matrix operation familiar in conventional transmission‐line circuit synthesis. As a typical application the design of various nonreciprocal integrated‐optical devices is treated in detail. The desired response is synthesized by cascading selected canonical elements in an appropriate order. Examples include the gyrator, unidirectional mode converter, differential phase shifter, isolator, and circulator.

Infrared spectra and structure of some matrix‐isolated lanthanide and actinide oxides

Abstract: The midinfrared spectra of the oxides of Ce, Pr, Sm, Eu, Tb, and Th have been obtained by vaporizing the solid oxides and trapping the vapor species in argon matrices. On the basis of oxygen‐18 isotope shifts, a linear structure has been determined for PrO2, and nonlinear structures for CeO2, TbO2, and ThO2. Values of ωe and ωeχe have been determined for the monoxides of Pr, Eu, Tb, and Th from the small deviations between the observed and the theoretical harmonic isotope shifts.

Discontinuous cellulose fiber treated with plastic polymer and lubricant

Abstract: Discontinuous cellulose fiber is treated with both plastic polymer and lubricant to produce improved treated fiber for reinforcing a plastic matrix.

Model reduction of multivariable control systems by means of matrix continued fractions

Abstract: A unified continued fraction theory which can be considered as a generalized feedback theory is established. A multiple cycle model consisting of many feedback constant matrices and many feed forward integral matrices is constructed. The outer feedback constant matrix corresponds to the first partial quotient matrix of the overall matrix continued fraction, whereas the outer forward integral matrix is corresponding to the second quotient matrix. Therefore, the partial quotient matrices of the continued fraction and the feedback and feed forward matrices of the system diagram are one-to-one correspondence in order. The influence of each matrix on the performance of the entire system depends on its position. The outer ones are much more important than the inner ones. A reduction model can be obtained by discarding certain inner matrices.

On the inverses of some patterned matrices arising in the theory of stationary time series

Abstract: Expressions are obtained for the determinant and inverse of the covariance matrix of a set of n consecutive observations on a mixed autoregressive moving average process. Explicit formulae for the inverse of this matrix are given for the general autoregressive process of order p (n ? p), and for the first order mixed autoregressive moving average process.

Parallel solution of recurrence problems

Abstract: An mth-order recurrence problem is defined as the computation of the sequence x1,, xN, where x1 = f(ai, xi-1,,xi-m), and ai, is some vector of parameters This paper investigates general algorithms for solving such problems on highly parallel computers We show that if the recurrence function f has associated with it two other functions that satisfy certain composition properties, then we can construct elegant and efficient parallel algorithms that can compute all N elements of the series in time proportional to ⌈log2N⌉ The class of problems having this property includes linear recurrences of all orders- both homogeneous and inhomogeneous, recurrences involving matrix or binary quantities, and various nonlinear problemsin volving operations such as computation with matrix inverses, exponentiation, and modulo division