Equations In Matrix Form Research Articles

Double-shifted Chebyshev series for convolution integral and integral equations

Double-shifted Chebyshev polynomials are developed in this study to approximate the solutions of the convolution integral, Volterra integral equation, and Fredholm integral equation. This method simplifies the computations of integral equations to the successive solutions of a linear algebraic equation in matrix form. In addition, the computational complexity can be reduced remarkably. Three examples are illustrated. It is seen that the proposed approach is straightforward and convenient, and converges faster in finding approximations than other existing orthogonal function methods.

Equation for a particle with spin of 3/2 having anomolous magnetic moment

It is shown that it is possible to describe a particle with spin of 3/2 and anomolous magnetic moment within the theory of first-order relativistic wave equations in matrix form by introducing a minimum interaction with the external electromagnetic field. This is done by expanding the space of Lorentz group representations.

Equation matricielle de Hartree‐Fock permettant de déterminer la nième harmonique

AbstractThe system of time‐dependent Hartree‐Fock equations allowing the calculation the of nth harmonic generation is established by using the same method used in a former paper in the same conditions. These equations are still subjected to orthonormalization conditions. Then by using a recurrence method, we express these equations in matrix form, as well as the orthonormalization conditions; and, as in the former paper, we then replace the solution of the matrix equation subjected to a constraint by that of a free equation. Finally the elimination of the spin variables allows a further simplification. So we obtain an equation allowing the calculation of the nth harmonic generation.

MOVEMENT OF SPACE SPECKLE

The diffusing reflective surface illuminated by coherent light generates random speckle in space. The movement of the surface relative to the source causes corresponding movement of the speckle. When a moving speckle is tracked, it implies that the intensity of speckle is constant yet the phase of the speckle is allowed to vary. From the Fresnel-Kirchhoff Integration, three fundamental rules of speckle movement are derived, i.e. Equations (3), (4) and (5) in this paper. The first and second equations are similar to those of grating. The third equation is similar to the lens law. From these rules, a speckle movement formula, relating to the surface movement, is derived as equation (18), in which the coordinate system adopted is fixed in space. It is an equation in matrix form, which denotes six types of surface movements, i. e. three rotations and three translations. The experimental results are in good agreement with this formula. These fundamental rules can be used to analyse general space diffractive patterns.

Coupled Vertical-Lateral Dynamics of a Pneumatic Tired Vehicle: Part I—A Mathematical Model

A method is presented which permits the simulation of the coupled vertical and lateral rigid body vibration response of an automobile to roadway roughness inputs. A set of equations in matrix form is obtained for an assumed ten degree-of-freedom mathematical model of the vehicle-tire system using generalized linear and Euler angle coordinates. Kinematic relations for a rolling tire which treat it as an elastically supported string under tension are incorporated into the overall system model. Forces and moments that act in the tire-roadway contact interface are represented mathematically as a function of the parameters and response variables of the vehicle system. The nonlinear system equations are subsequently simplified in order to apply them to a vehicle moving along a straight roadway. The formulated model is shown to be adequate for predicting acceleration response in a frequency range of 0.1–10 Hz for the set of roadway test roadway sections considered in the present study.

Solution of integral equations using a set of block pulse functions

A set of the block pulse functions is applied to solve the Fredholm's and the Volterra's integral equations of the second kind. An algebraic equation in matrix form which is equivalent to the solution of the integral equation is developed. The approximate results are easily obtained by a few computations. An accurate solution canbe evaluated in a digital computer by solving the algebraic equation. Two examples are given.

Hydrodynamic behavior of short chain molecules. I. Extension of the Rouse–Zimm theory

The Rouse–Zimm formalism of hydrodynamic properties of long chain molecules (polymers) in solution is modified to extend its validity to the case of short chains. The modification introduces: (a) a realistic calculation of averages for the intrachain distances; (b) an exact solution of the eigenvalue equation in matrix form; (c) an approximate representation of short-range interactions (first and second order) by means of springlike forces between nonconsecutive chain segments. The theory is applied to the calculation of limiting viscosity numbers and the Flory–Fox constant of short polymethylene chains (number of segments ?50). Our results are compared with those from other theories.

Coupled channel T-operator equations with connected kernels : (II). N coupled two-body channels

A time-independent theory of rearrangement collisions involving transitions between two-body states is presented. It is assumed that the system of interest consists of particles that may be partitioned into two-body systems in N ways, including interchanges of particle labels without changing the kind of channel. An infinite family of sets of N coupled T-operator equations is derived by use of the channel coupling array, as in previous work on the three-body problem. Specialization to the channel-permuting arrays guaranteeing connected ( N−1)th iterates of the kernel of the coupled equations is made in the N-channel case ( N > 3) and the nature of the solutions to the coupled equations is discussed. Various approximation schemes to be used with numerical calculations are suggested. Since the transition operators for all rearrangement channels are coupled together, no problems concerning non-orthogonality of the eigenstates of different channel Hamiltonians are encountered; also the presence of the outgoing wave boundary condition in all channels is made explicit. The close resemblance of the equations in matrix form to those of one-channel scattering is exploited by introducing Møller wave operators and associated channel scattering states, an optical potential formalism that leads to rearrangement channel optical potential operators, and a variational formulation of the coupled equations using a Schwinger-like variational principle. A brief comparison with other many-body formalisms is also given.

Minimization of Multidimensional Linear Iterative Circuits

A model for two-dimensional linear iterative circuits is defined in the form of matrix equations. From the matrix equations, a two-dimensional characteristic function is defined. It is then proved that a matrix satisfies its two-dimensional characteristic function. This property is used to form a diagnostic matrix. Finally, the diagnostic matrix is used in a minimization technique.

Reduction of the X-ray intensity equation in matrix form for one-dimensionally disordered structures

The matrices in the X-ray intensity equation for one-dimensionally disordered structures are reduced by taking account of the symmetry character of the matrices.

Steady-state optimizing control of a group of chemical processes

This paper is devoted to steady-state optimizing control of a group of single-reactor continuous chemical processes. A generalized model has been developed, based on the heat and mass balance equations in matrix form. The least squares estimators of the process variables are presented. The optimization problem is formulated as a chance constrained programming problem under uncertainty and its “confidence equivalent” is proposed. An open-loop system has been developed for set-point control of a group of formaldehyde processes using a Minsk-32 computer, telegraph lines and remote terminals, and the application patterns are given.

Free-Field Correction for Laboratory Standard Microphones Mounted on a Semiinfinite Rod

A theoretical study has been made on the free-field correction of laboratory standard microphones mounted on a semiinfinite rod. In the analysis, we have discussed the diffraction phenomena of plane sound waves by the actual microphone system, taking into consideration the effect of a recess as well as the interaction between microphone diaphragm and medium. A new method of solving wave equations met with these boundary conditions, called the infinite-matrix method, is presented in this paper. Our whole problem is divided into three parts. The first is concerned with the external free field, which can be solved rigorously by Wiener-Hopf technique; the others refer to the solution of the sound field inside a recess and, finally, to motions of a diaphragm. After putting these solutions into the form of matrix equations with unknown variables, we obtain a system of simultaneous matrix equations by connecting solutions with each other. Numerical results are given for the free-field correction of the Western Electric Company type 640AA microphone. Comparison with the experiment, made at the National Bureau of Standards, indicates that sufficient agreement is obtained.

Convergent iterative solutions for the lowest state of a system with large perturbations

AbstractIterative methods for the calculation of the lowest eigenvalue and the corresponding eigenvector are derived from the partitioned Schrödinger equation in matrix form. While they have a formal similarity to the Brillouin‐Wigner perturbation expansion, they converge for a much wider class of matrices. The double iteration methods (DIM and DIMA) lead to a series of upper and lower bounds to the desired lowest eigenvalue in accordance with Löwdin's bracketing theorem and can be used in a strategy which insures convergence to the solution for arbitrarily large perturbations. A much more efficient single iteration method (SIMA) produces the proper solution even for numerically very ill‐conditioned matrices (large offdiagonal elements and small separation of the diagonal elements from the lowest). An important feature of the iterative methods is their insensitivity toward roundoff errors which makes them especially suited for very large matrices, where direct diagonalisation, for example by the Jacobi method, is impracticable.

A linearization tool for use with matrix formalisms of rotational dynamics

The dynamical equations of a rotating body whose motion is referred to an arbitrarily rotating reference frame are often desired in linearized form in connection with stability and orientation control studies. The method of this paper shows how to obtain these linearized equations in matrix form, saving considerable labor and promoting accuracy relative to the linearization process done at the sealer level. Illustrative applications are made to a librating orbiting body and a one-axis gyroscope on a rotating base.

Determination ofκ1(T)andκ2(T)for Type-II Superconductors with Arbitrary Impurity Concentration

Formulas are derived for the constants ${\ensuremath{\kappa}}_{1}$ and ${\ensuremath{\kappa}}_{2}$ of superconducting alloys for arbitrary temperatures and impurity concentrations. Beginning with Gorkov's equations in matrix form, we calculate the free-energy density up to fourth order in $|\ensuremath{\Delta}|$, using Abrikosov's assumption of a fluxoid lattice. The impurities are treated by the usual averaging technique, retaining only $s$ and $p$ scattering, and a special technique is developed to treat exactly the nonvanishing commutator of different components of the gauge-invariant derivative. The results contain the already known limiting cases. Intermediate values are obtained by performing machine calculations using the general formulas developed here. It is found that the values of ${\ensuremath{\kappa}}_{1}$ and ${\ensuremath{\kappa}}_{2}$ drop relatively strongly even for small impurity concentration, and also depend unexpectedly strongly on the ratio of $s$ and $p$ scattering. The recent result of Caroli, Cyrot, and de Gennes is confirmed according to which ${\ensuremath{\kappa}}_{2}$ is approximately equal to ${\ensuremath{\kappa}}_{1}$ in the dirty limit.

On relativistic wave equations

A method is given for obtaining relativistic wave equations in matrix form for particles with higher spins. Values are given for the constants of the β v matrices, the matrix of the inversion operator, and the scalar matrix which enters into the expression of the current for particles with spin 0, 1/2, 1, 3/2, or 2.

INVERSION PROPERTY OF THE FUNDAMENTAL MATRIX IN TRAJECTORY PERTURBATION PROBLEMS

Inversion of ordinary linear differential equation in matrix form for application to class of trajectory perturbation problems