Showing papers on "Matrix (mathematics) published in 1981"

Contaminant Transport in Fractured Porous Media: Analytical Solution for a Single Fracture

Abstract: A general analytical solution is developed for the problem of contaminant transport along a discrete fracture in a porous rock matrix. The solution takes into account advective transport in the fracture, longitudinal mechanical dispersion in the fracture, molecular diffusion in the fracture fluid along the fracture axis, molecular diffusion from the fracture into the matrix, adsorption into the face of the matrix, adsorption within the matrix, and radioactive decay. Certain assumptions are made which allow the problem to be formulated as two coupled, one-dimensional partial differential equations: one for the fracture and one for the porous matrix in a direction perpendicular to the fracture. The solution takes the form of an integral which is evaluated by Gaussian quadrature for each point in space and time. The general solution is compared to a simpler solution which assumes negligible longitudinal dispersion in the fracture. The comparison shows that in the lower ranges of groundwater velocities this assumption may lead to considerable error. Another comparison between the general solution and a numerical solution show excellent agreement under conditions of large diffusive loss. Since these are also the conditions under which the formulation of the general solution in two orthogonal directions is most subjectmore » to question, the results are strongly supportive of the validity of the formulation.« less

Computational methods of linear algebra

Abstract: The authors' survey paper is devoted to the present state of computational methods in linear algebra. Questions discussed are the means and methods of estimating the quality of numerical solution of computational problems, the generalized inverse of a matrix, the solution of systems with rectangular and poorly conditioned matrices, the inverse eigenvalue problem, and more traditional questions such as algebraic eigenvalue problems and the solution of systems with a square matrix (by direct and iterative methods).

Some matrix-variate distribution theory: Notational considerations and a Bayesian application

Abstract: SUMMARY We introduce and justify a convenient notation for certain matrix-variate distributions which, by its emphasis on the important underlying parameters, and the theory on which it is based, eases greatly the task of manipulating such distributions. Important examples include the matrix-variate normal, t, F and beta, and the Wishart and inverse Wishart distributions. The theory is applied to compound matrix distributions and to Bayesian prediction in the multivariate linear model.

Solvability, controllability, and observability of continuous descriptor systems

Abstract: In this paper, we investigate the properties of the continuous descriptor system E\dot{x}(t)=Ax(t)+Bu(t), 0\leqt\leqb where E, A, and B are complex and possibly singular matrices and u(t) is a complex function differentiable sufficiently many times. The traditional approach to such systems is to separate the state equations from the algebraic equations. However, such algorithms usually destroy the natural, physically-based sparsity and structure of the original system. Therefore, we consider descriptor systems in their original form. Such systems possess numerous properties not shared by the well-known state variable systems. First, we relate classical theories of matrix pencils to the solvability of descriptor systems. Then we extend the concepts of reachability, controllability, and observability of state variable systems to descriptor systems, and describe the set of reachable states for descriptor systems.

Anderson Localization in Two Dimensions

Abstract: The conductance for a two-dimensional tight-binding model with on-site disorder is calculated numerically with use of the Kubo formula. For weak disorder logarithmic localization is observed, in agreement with the scaling theory. The magnetoresistance is found to be negative in both the logarithmic and exponential localization regimes. Results for a model with random complex hopping matrix elements are also presented.

Parallel Matrix and Graph Algorithms

Abstract: Matrix multiplication algorithms for cube connected and perfect shuffle computers are presented. It is shown that in both these models two $n \times n$ matrices can be multiplied in $O(n/m + \log m)$ time when $n^2 m$, $1 \leqq m \leqq n$, processing elements (PEs) are available. When only $m^2 $, $1 \leqq m \leqq n$, PEs are available, two $n \times n$ matrices can be multiplied in $O(n^2/m + m(n/m)^{2.61} )$ time. It is shown that many graph problems can be solved efficiently using the matrix multiplication algorithms.

Bounds on the density of states in disordered systems

Abstract: For a class of tight-binding models governed by short-range one-particle Hamiltonians with site-diagonal and/or off-diagonal disorder and continuous distribution of the matrix elements it is proven that the averaged density of states does neither vanish nor diverge inside the band. This refutes for these models conjectures that the density of states vanishes or diverges at the mobility edge.

A method of integration over matrix variables

Abstract: The integral over twon ×n hermitan matricesZ(g, c)=∫dAdBexp{−tr[A 2+B 2−2cAB+g/n(A 4+B 4)]} is evaluated in the limit of largen. For this purpose use is made of the theory of diffusion equation and that of orthogonal polynomials with a non-local weight. The above integral arises in the study of the planar approximation to quantum field theory.

Estimation in a Multivariate "Errors in Variables" Regression Model: Large Sample Results

Abstract: In a multivariate "errors in variables" regression model, the unknown mean vectors $\mathbf{u}_{1i}: p \times 1, \mathbf{u}_{2i}: r \times 1$ of the vector observations $\mathbf{x}_{1i}, \mathbf{x}_{2i}$, rather than the observations themselves, are assumed to follow the linear relation: $\mathbf{u}_{2i} = \alpha + B\mathbf{u}_{1i}, i = 1,2,\cdots, n$. It is further assumed that the random errors $\mathbf{e}_i = \mathbf{x}_i - \mathbf{u}_i, \mathbf{x}'_i = (\mathbf{x}'_{1i}, \mathbf{x}'_{2i}), \mathbf{u}'_i = (\mathbf{u}'_{1i}, \mathbf{u}'_{2i})$, are i.i.d. random vectors with common covariance matrix $\Sigma_e$. Such a model is a generalization of the univariate $(r = 1)$ "errors in variables" regression model which has been of interest to statisticians for over a century. In the present paper, it is shown that when $\Sigma_e = \sigma^2I_{p+r}$, a wide class of least squares approaches to estimation of the intercept vector $\alpha$ and slope matrix $B$ all lead to identical estimators $\hat{\alpha}$ and $\hat{B}$ of these respective parameters, and that $\hat{\alpha}$ and $\hat{B}$ are also the maximum likelihood estimators (MLE's) of $\alpha$ and $B$ under the assumption of normally distributed errors $\mathbf{e}_i$. Formulas for $\hat{\alpha}, \hat{B}$ and also the MLE's $\hat{U}_1$ and $\hat{\sigma}^2$ of the parameters $U_1 = (\mathbf{u}_{11}, \cdots, \mathbf{u}_{1n})$ and $\sigma^2$ are given. Under reasonable assumptions concerning the unknown sequence $\{\mathbf{u}_{1i}, i = 1,2,\cdots\}, \hat{\alpha}, \hat{B}$ and $r^{-1}(r + p)\hat{\sigma}^2$ are shown to be strongly (with probability one) consistent estimators of $\alpha, B$ and $\sigma^2$, respectively, as $n \rightarrow \infty$, regardless of the common distribution of the errors $\mathbf{e}_i$. When this common error distribution has finite fourth moments, $\hat{\alpha}, \hat{B}$ and $r^{-1}(r + p)\hat{\sigma}^2$ are also shown to be asymptotically normally distributed. Finally large-sample approximate $100(1 - u){\tt\%}$ confidence regions for $\alpha, B$ and $\sigma^2$ are constructed.

Partial and Total Matrix Multiplication

Abstract: In 1979 considerable progress was made in estimating the complexity of matrix multiplication. Here the new techniques and recent results are presented, based upon the notion of approximate rank and the observation that certain patterns of partial matrix multiplication (some of the entries of the matrices may be zero) can efficiently be utilized to perform multiplication of large total matrices. By combining Pan’s trilinear technique with a strong version of our compression theorem for the case of several disjoint matrix multiplications it is shown that multiplication of $N \times N$ matrices (over arbitrary fields) is possible in time $O(N^\beta )$, where $\beta $ is a bit smaller than $3\ln 52/\ln 110 \approx 2.522$.

A general theory for matrix root-clustering in subregions of the complex plane

Abstract: We consider the general problem of root-clustering of a matrix in the complex plane: Let A \in C^{n \times n} and S \subset C . Find the largest class of S and an algebraic criterion which is necessary and sufficient for \lambda_{i}[A] \in S, i=1,2,..., n . We introduce two types of regions which constitute the largest class of S known to date. The criterion is presented both for open regions and closed ones. The results are used to define a design methodology for control systems. Moreover, all classical results are shown to be special cases of the present theory.

Calculation of ESR spectra and related Fokker–Planck forms by the use of the Lanczos algorithm

Abstract: The applicability of the Lanczos algorithm in the general ESR (and NMR) line shape problem is investigated in detail. This algorithm is generalized to permit tridiagonalization of complex symmetric matrices characteristic of this problem. It is found to yield very accurate numerical solutions with at least order of magnitude reductions in computation time compared to previous methods. It is shown that this great efficiency is a function of the sparsity of the matrix structure in these problems as well as the efficiency of selecting an approximation to the optimal basis set for representing the line shape problem as distinct from actually solving for the eigenvalues. Furthermore, it is found to aid in the analysis of truncation to minimize the basis set (MTS), which becomes nontrivial in complex problems, although the efficiency of the method is not very strongly dependent upon the MTS. It is also found that typical Fokker–Planck equations arising from stochastic modeling of molecular dynamics have the property of being representable by complex–symmetric matrices that are very sparse, so calculation of associated correlation functions can be very effectively implemented by the Lanczos algorithm. It is pointed out that large problems leading to matrices of very large dimension can be efficiently handled by the Lanczos algorithm.

Fatigue Failure Criteria for Unidirectional Fiber Composites

Abstract: : Three dimensional fatigue failure criteria for unidirectional fiber composites under states of cyclic stress are established in terms of quadratic stress polynomials which are expressed in terms of the transversely isotropic invariants of the cyclic stress. Two distinct fatigue failure modes, fiber mode and matrix mode, are modelled separately. Material information needed for the failure criteria are the S-N curves for single stress components. A preliminary approach to incorporate scatter into the failure criteria is presented. (Author)

Kronecker products and shuffle algebra

Abstract: Relates three classical concepts, viz. mixed radix number system, Kronecker product of matrices, and perfect shuffle. It presents an algebra which describes the hardware organization of the computation of a product Mv, where M is a matrix in Kronecker product form and v is a vector. The algebraic formalism describes both the blockwise structure of the computation and the various possible connection patterns.

Multi-channel sampling of low-pass signals

Abstract: A deterministic signal x(t) band limited to |\omega| is passed through m linear time-invariant filters (channels) to obtain the m outputs z_1(t),\cdots,Z_m(t) . If the filters are independent in a sense to be defined, then It Is shown that the common input x(t) may be reconstructed from samples of the outputs (Z_k) , each output being sampled at m \Pi samples per second or (1/m) th the rate associated with the Input signal. A rigorous derivation of this result Is given which proceeds from a minimum error energy criterion and leads to a system of linear algebraic equations for the optimal reconstruction filters. The system of equations derived here, which differs from the system given recently by Papoulis [1], has the advantage of depending on only one parameter \omega rather than on the two parameters \omega and t ; it also puts into evidence the fact that the spectra of the optimal reconstruction filters can be pieced together directly, without additional computation, from the elements of the system's inverse matrix. Lastly, the solutions of the system obtained in the Papoulis formulation are shown to be time-varying linear combinations of the simpler one-parameter solutions.

On guaranteed stability of uncertain linear systems via linear control

Abstract: This paper addresses the problem of stabilizing an uncertain linear system. The uncertaintyq(·) which enters the dynamics is nonstistical in nature. That is, noa priori statistics forq(·) are assumed; only boundsQ on the admissible variations ofq(·) are taken as given. The results given here applied to so-called matched systems differ from previous results in two ways. Firstly, the stabilizing control in this paper is linear; for this same class of problems, many of the existing results would require a nonlinear control. Furthermore, those results which do in fact yield linear controls are only valid when a certain matrix Ω(q) (formed using the given data) is negative definite for allq ∈Q. In contrast, the theory given here only requires compactness of the bounding setQ. Secondly, we show that the so-called matching conditions (used in earlier work) can be generalized so as to encompass a larger class of dynamical systems.

Turbulent Wind and Its Effect on Flight

Abstract: = components of (W— WA) in FB or FA = components of Wm FE = velocity of airplane mass centre relative toFA = reference steady value of V = local velocity of the air relative to the Earth = mean value of W, velocity of frame FA = state vector = dX/du, etc., classical stability derivatives = wavelength, 27T/Q, = rms value of stochastic variable indicated by subscript = vector separating two points in FA = angular velocity of airplane relative to FE = local angular velocity of air relative to E = wave number vector in FA = matrix of three-dimensional spectrum functions = matrix of two-dimensional spectrum functions = matrix of one-dimensional spectrum functions = Laplace transform = transpose of matrix = ensemble average = complex conjugate Reference Frames FE:(xE,yE,zE)

Survey of numerical methods for solution of large systems of linear equations for electromagnetic field problems

Abstract: Many of the popular methods for the solution of large matrix equations are surveyed with the hope of finding an efficient method suitable for both electromagnetic scattering and radiation problems and system identification problems.

Hand-held computer

Abstract: The invention relates to a hand-held computer comprising a peripheral frame (10) which contains all the connections and in which cards are removably plugged. A flat display (21), mounted on said frame, is topped by a transparent plate (22) which forms a matrix of touch-sensitive areas or points. The matrix is also mounted on the frame (10). The screen and the plate occupy the maximum of the upper surface of the casing and are electrically connected to the other circuits of the computer by contact with the frame on which they are mounted. The computer may be provided with phoneme recognition means, with hand-writing recognition means or with TV or radio receiving means by insertion of appropriate cards.

A Schur method for pole assignment

Abstract: In this paper a new numerical algorithm for pole assignment of linear time-invariant systems is presented. The proposed algorithm uses reliable numerical techniques based on the Schur form of the state matrix and on the use of orthogonal transformations. Only the "bad" eigenvalues of the system are modified by a sequential pole-shifting procedure, and the resulted gains in the feedback matrix are minimized.

The Unitary group for the evaluation of electronic energy matrix elements

Abstract: 1. Unitary Group Approach to Many-Electron Correlation Problem.- 2. The Graphical Unitary Group Approach and its Application to Direct Configuration Interaction Calculations.- 3. A Harmonic Level Approach to Unitary Group Methods in CI and Perturbation Theory Calculations.- 4. Many-Body Correlations Using Unitary Groups l.- 5. Factorization of the Direct CI Coupling Coefficients into Internal and External Parts.- 6. Multiconfiguration Self-Consistent-Field Wavefuntion for Excited States.- 7. Minicomputer Implementation of the Vector Coupling Approach to the Calculation of Unitary Group Generator Matrix Elements.- 8. New Directions for the Loop-Driven Graphical Unitary Group Approach: Analytic Gradients and an MCSCF Procedure.- 9. The Occupation-Branching-Number Representation.- 10. Review of Vector Coupling Methods in the Unitary Group Approach to Many-Electron Problems.- 11. Symmetric Group Graphical Approach to the Configuration Interaction Method.- 12. Orbital Description of Unitary Group Basis.- 13. On the Relation Between the Unitary Group Approach and the Conventional Approaches to the Correlation Problem.- 14. Unitary Bases for X-Ray Photoelectron Spectroscopy.- 15. Broken Unitary Tableaus, Itinerant Nuclear Spins, and Spontaneous Molecular Symmetry Collapse.- 16. CI-Energy Expressions in Terms of the Reduced Density Matrix Elements of a General Reference.- 17. The Unitary Group Formulation of Quantum Chemistry: Generator States.- 18. The Unitary Group Approach to Bonded Functions.

Matrix isolation spectroscopy

Abstract: 1 The history of matrix isolation spectroscopy- Section A - Techniques- 2 Infrared and Raman matrix isolation spectroscopy- 3 Electronic spectroscopy of matrix isolated solutes- 4 Magnetic circular dichroism - matrix isolation spectroscopy- 5 Electron spin resonance studies of radicals trapped in rare-gas matrices- 6 Moessbauer spectroscopy on matrix-isolated species- 7 Time and frequency resolved vibrational spectroscopy of matrix isolated molecules: Population and phase relaxation processes- 8 Stable molecules- 9 Generation and trapping of unstable solutes in low temperature matrices- 10 The characterisation of high temperature molecules using matrix isolation and vibrational spectroscopy- 11 High pressure studies- 12 Non-traditional matrix isolation: adducts- Section B - Matrix Effects- 13 Interpretation of infrared and Raman spectra of trapped molecular impurities from interaction potential calculations- 14 Matrix induced changes in the electronic spectra of isolated atoms and molecules- 15 Matrix effects studied by electron spin resonance spectroscopy- 16 Molecular motion in matrices- 17 Vibrational band intensities in matrices- Section C - Applications- 18 Matrix isolation spectroscopy of metal atoms and small clusters- 19 Vibrational spectra of matrix isolated gaseous ternary oxides- 20 Matrix isolation spectra (IR, Raman) of transition metal compounds- 21 Metal carbonyls - structure, photochemistry, and IR lasers- 22 Matrix isolation vibrational spectroscopy on organic molecules- 23 Conformational isomerism studied by matrix isolation vibrational spectroscopy- 24 Hydrogen bonding in matrices- Author Index

Row Stochastic Matrices Similar to Doubly Stochastic Matrices

Abstract: The problem of determining which row stochastic n-by-n matrices are similar to doubly stochastic matrices is considered. That not all are is indicated by example, and an abstract characterization as well as various explicit sufficient conditions are given. For example, if a row stochastic matrix has no entry smaller than (n+1)−1 it is similar to a doubly stochastic matrix. Relaxing the nonnegativity requirement, the real matrices which are similar to real matrices with row and column sums one are then characterized, and it is observed that all row stochastic matrices have this property. Some remarks are then made on the nonnegative eigenvalue problem with respect to i) a necessary trace inequality and ii) removing zeroes from the spectrum.

Photoelastic and Electro-Optic Properties of Crystals

Abstract: 1. Photoelasticity of Crystals. Introduction.- 1.1. Discovery of the Phenomenon of Photoelasticity.- 1.2. Mathematical Formulation and Neumann's Constants. Pockels' Contribution.- 1.3. A Brief Historical Survey.- 1.3.1. Amorphous Solids.- 1.3.2. Cubic Crystals.- 1.3.3. Uniaxial and Biaxial Crystals.- 2. Mathematical Tools, Tensor Properties of Crystals, and Geometrical Crystallography.- 2.1. Linear Transformations.- 2.1.1. Coordinate Transformations.- 2.1.2. Orthogonality Relations.- 2.1.3. The Determinant of the Matrix [?ij] of the Direction-Cosine Scheme.- 2.2. Matrix Algebra.- 2.2.1. Introduction.- 2.2.2. Matrix Algebra and Coordinate Transformations.- 2.2.3. Some Common Types of Matrices.- 2.2.4. Orthogonal Matrix.- 2.2.5. Matrix Operators and Transformation of Tensor Components.- 2.2.6. The Diagonalization of a Matrix.- 2.3. Vectors and Their Transformation Laws.- 2.3.1. Vector Components and Coordinate Transformations.- 2.3.2. Transformations of Coordinate Differences.- 2.3.3. Transformation Law of Vectors.- 2.4. Tensor Nature of Physical Properties of Crystals and the Laws of Transformation of Cartesian Tensors.- 2.4.1. Concept of a Tensor Property and Some Examples of Tensor Properties.- 2.4.2. Transformation Law of Cartesian Tensors.- 2.4.3. Physical Properties and Crystal Symmetry.- 2.5. Crystal Symmetry and Geometrical Crystallography. The 32 Point Groups.- 2.5.1. The 32 Crystallographic Point Groups: Their Symmetry Elements and Some Examples of Crystals.- 2.5.2. Some Symmetry Operations and Their Representation by Symbols.- 2.5.3. The 32 Crystallographic Point Groups in the Schonflies Notation. Geometric Derivation.- 2.6. Symmetry Operations and Their Transformation Matrices.- 2.7. Symmetry Elements of the 32 Point Groups.- 2.7.1. Symmetry Elements of the 32 Point Groups.- 2.7.2. Comments on the 32 Crystallographic Point Groups and Their Symmetry Elements as Listed in Tables 2.3 and 2.5a.- 2.8. Neumann's Principle and Effect of Crystal Symmetry on Physical Properties.- 3. Pockels' Phenomenological Theory of Photoelasticity of Crystals.- 3.1. Introduction.- 3.2. Phenomenological Theory, Stress-Optical and Strain-Optical Constants in Four- and Two-Suffix Notations qij and pij Matrices for the 32 Crystallographic Point Groups.- 3.2.1. The Assumptions Forming the Basis of Pockels' Theory.- 3.2.2. Mathematical Formulation of Photoelasticity in Terms of qijkl and pijkl.- 3.2.3. Mathematical Formulation of Photoelasticity in Terms of qij and pij.- 3.2.4. Crystal Symmetry and the Number of Photoelastic Constants.- 3.3. Derivation of the Nonvanishing and Independent Photoelastic Constants for the Various Crystal Classes by Different Methods.- 3.3.1. Classical Method.- 3.3.2. Tensor Method.- 3.3.3. Group Theoretical Method.- 4. Elasticity of Crystals.- 4.1. Introduction.- 4.2. Stress and Strain as Tensors.- 4.2.1. Stress as a Second-Rank Tensor.- 4.2.2. Strain as a Second-Rank Tensor.- 4.3. Hooke's Law.- 4.3.1. Generalized Form of Hooke's Law with Elastic Constants cij and sij and the Matrices of cij and sij for the 32 Point Groups.- 4.3.2. Generalized Form of Hooke's Law with Elastic Constants cijkl and sijkl.- 4.3.3. Interrelation between cijkl and cmn and between sijkl and smn.- 4.4. Experimental Methods of Determining cij and sij Christoffel's Equation and Its Use in Determining cij of Crystals.- 4.5. Ultrasonics.- 4.5.1. Introduction.- 4.5.2. Diffraction of Light by Liquids Excited Ultrasonically.- 4.5.3. Optical Methods of Determining the Ultrasonic Velocities and Elastic Constants of Transparent Solids Employing the Schaefer-Bergmann Pattern, the Hiedemann Pattern, and the Lucas-Biquard Effect.- 4.5.4. Mayer and Hiedemann's Experiments.- 4.5.5. Raman-Nath Theory of Diffraction of Light by Ultrasonic Waves.- 4.5.6. Doppler Effect and Coherence Phenomenon.- 4.6. Brillouin Effect and Crystal Elasticity.- 4.6.1. Introduction.- 4.6.2. Theory of Light Scattering in Birefringent Crystals.- 4.6.3. Concluding Remarks.- 5. Experimental Methods of Determining the Photoelastic Constants.- 5.1. Optical Behavior of a Solid under a Mechanical Stress, and Neumann's Constants.- 5.2. Derivation of Expressions for the Stress Birefringence in Terms of qij for Cubic and Noncubic Crystals.- 5.2.1. Stress Birefringence in Cubic Crystals.- 5.2.2. Stress Birefringence in Noncubic Crystals.- 5.2.3. Tensor Method of Deriving q?ijkl in Terms of qmnop.- 5.2.4. Expression for the Change of Thickness in Terms of sij for an Orthorhombic Crystal for a Specific Orientation.- 5.3. Experimental Determination of qij and pij by Optical Methods.- 5.3.1. Measurement of Stress Birefringence, and Relative Path Retardation.- 5.3.2. Measurement of Absolute Path Retardation by Interferometric Methods.- 5.3.3. Photoelastic Studies of Optically Active Crystals.- 5.4. Dispersion of qij by Spectroscopic Methods.- 5.4.1. Birefringent Compensator for Studying Very Small Changes in Double Refraction.- 5.4.2. Dispersion of the Individual Stress-Optical Coefficients q11 and q12 of Vitreous Silica.- 5.4.3. Interference-Spectroscope Method of Studying the Absolute Photoelastic Coefficients of Glasses and Their Variation with Wavelength.- 5.5. Elliptic Vibrations and Elliptically Polarized Light.- 5.5.1. Composition of Two Rectangular Vibrations Giving an Ellipse: Use of the Senarmont Compensator.- 5.5.2. Photometric Method for the Measurement of Photoelastic Birefringence.- 5.5.3. The Poincare Sphere and Its Application to the Study of the Photoelastic Behavior of Optically Active Crystals.- 5.6. Ultrasonic Methods of Studying the Elasto-Optic Behavior of Crystals.- 5.6.1. Introduction.- 5.6.2. Mueller's Theory.- 5.6.3. Experimental Determination of Pij/pkl by Three Different Methods Due to Mueller.- 5.6.4. Pettersen's Method of Determining Pij/pkl.- 5.6.5. Bragg Diffraction Method of Determining the Individual Values of pij.- 5.6.6. Borrelli and Miller's Method of Determining the pij of Glass.- 5.6.7. Technological Applications of the Acousto-Optic Effect.- 5.7. Brillouin Scattering and Photoelasticity of Crystals.- 6. Atomistic Theory of Photoelasticity of Cubic Crystals.- 6.1. Introduction.- 6.2. Mueller's Theory-A Brief Survey.- 6.3. Effect of Hydrostatic Pressure on the Index of Refraction n The Strain Polarizability Constant ?0.- 6.4. Anisotropy of Rj and ?itj.- 6.5. Thermo-Optic Behavior of Crystals and Photoelastic behavior.- 6.6. Pockels' Photoelastic Groups in Cubic Crystals and Mueller's Theory.- 6.7. Photoelastic Dispersion in Cubic Crystals ?0 as a Function of Crystalline Material, Wavelength of Light, and Temperature.- 6.8. Effect of Elastic Deformation on the Oscillator Strengths and Dispersion Frequencies of Optical Electrons.- 6.9. Temperature Dependence of Stress-Optical Dispersion.- 6.10. Reversal of the Sign of Stress Birefringence in Pure and Mixed Crystals.- 6.10.1. Pure Crystals.- 6.10.2. Mixed Crystals of KCl and KBr.- 6.11. Stress-Optical and Strain-Optical Isotropy in Cubic Crystals.- 6.12. Optic Axial Angle and Its Dispersion in Stressed Cubic Crystals of T and Th Classes.- 7. Piezoelectricity.- 7.1. Introduction.- 7.2. Direct and Converse Piezoelectric Effects.- 7.3. Mathematical Formulation, Piezoelectric Constants dijk in Tensor Notation, and dij in Two-Suffix Notation Relation between dijl and dij.- 7.4. Deduction of the Surviving dijk for Some Crystal Classes by Tensor Method, and the dij Matrices for the 21 Noncentrosymmetric Classes.- 7.5. Concluding Remarks.- 8. Electro-Optic Effects in Crystals: Pockels Linear Electro-Optic and Kerr Quadratic Electro-Optic Effects.- 8.1. Introduction.- 8.2. Demonstration of the Electro-Optic Effects, Linear and Quadratic.- 8.3. Historical Survey.- 8.3.1. Earlier Work.- 8.3.2. More Recent Work.- 8.4. Pockels' Phenomenological Theory of the Linear Electro-Optic Effect in Three- and Two-Suffix Notations, Rijk and rij.- 8.5. Derivation of the Relation between the Linear Electro-Optic Constants of a Crystal: Free and Clamped Constants.- 8.5.1. Discussion: Primary and Secondary Electro-Optic Effects, and Clamped and Unclamped Electro-Optic Coefficients.- 8.5.2. Methods of Obtaining the Primary and Secondary Linear Electro-Optic Effects.- 8.6. Kerr Quadratic Electro-Optic Effect: Pockels' Phenomenological Theory.- 8.7. Crystal Symmetry and the Number of Surviving Linear Electro-Optic Coefficients Rijk and rij and Their Deduction by Tensor Method: rij Matrices for the 21 Noncentrosymmetric Classes.- 8.7.1. Crystal Symmetry and the Surviving Linear Electro-Optic Constants.- 8.7.2. Tensor Method of Deducing the Nonvanishing Independent Rijk.- 8.8. Derivation of the Expressions for ? = f(rij) for Some Typical Crystal Classes and Orientations.- 8.8.1. Cubic System: Classes 23(T) and $$\bar 43m$$ (Td).- 8.8.2. Tetragonal System: Class $$\bar 42m$$ (D2d).- 8.8.3. Trigonal System: Class 32 (D3).- 8.9. Experimental Methods of Determining rij.- 8.9.1. General Description and Application to Some Typical Crystal Classes.- 8.9.2. Some Experimental Methods.- 8.9.3. Methods of Applying the Electric Field to the Crystal Prism.- 8.9.4. Experimental Determination of rij in Some Specific Cases of Crystals.- 8.10. Some Points of Interest on the Use of the Pockels Effect in Crystals, and Half-Wave Voltage V?/2.- 8.11. Some Technological Applications of Pockels Cells (Linear Electro-Optic Devices).- 8.11.1. Use of the Electro-Optic Effect in Technology.- 8.11.2. Some Applications of Electro-Optic Devices.- Author Index.

On the eigenvectors of symmetric Toeplitz matrices

Abstract: This paper presents a number of results concerning the eigenvectors of a symmetric Toeplitz matrix and the location of the zeros of the filters (eigenfilters) whose coefficients are the elements of the eigenvectors. One of the results is that the eigenfilters corresponding to the maximum and minimum eigenvalues, if distinct, have their zeros on the unit circle, while the zeros of the other eigenfilters may or may not have their zeros on the unit circle. Even if the zeros of the eigenfilters of a matrix are all on the unit circle, the matrix need not be Toeplitz. Examples are given to illustrate the different properties.

The coupled‐cluster method with a multiconfiguration reference state

Abstract: The coupled-cluster approach to obtaining the bond-state wave functions of many-electron systems is extended, with a set of physically reasonable approximations, to admit a multiconfiguration reference state. This extension permits electronic structure calculations to be performed on correlated closed- or open-shell systems with potentially uniform precision for all molecular geometries. Explicit coupled cluster working equations are derived using a multiconfiguration reference state for the case in which the so-called cluster operator is approximated by its one- and two-particle components. The evaluation of the requisite matrix elements is facilitated by use of the unitary group generators which have recently received wide attention and use in the quantum chemistry community.