Trilocal structures. III. Expansion in tree functions

An alternative set of expansion functions is described. The first 48 in the set are listed explicitly, and 16 generalized formulas are given, from which the entire rest-system set of functions can be constructed. Explicit and generalized matrix elements for the rest Hamiltonian are given. An auxiliary condition is introduced, leading to explicit formulas and generalized formulas for the expansion coefficients,C _j, accompanying the expansion functions, _j.     

Trilocal structures. V. Wave equations

In addition to the usual centroid-time wave equation, a trilocal structure will need to satisfy two relative-time wave equations. When the trilocal wave function is expanded in tree functions, each of the three wave equations becomes an infinite matrix equation, but when the four auxiliary conditions (defined in earlier articles in this series) are introduced, each wave equation reduces to a set of 16 linear homogeneous equations in 16 unknown expansion coefficients (the first 16 coefficients in the tree expansion). The 48 linear equations, in the 16 unknownC _j , are given explicitly. Every 16-by-16 determinant, formed from any 16 of these 48 linear homogeneous equations, must vanish if the trilocal structure is to be an acceptable solution; this requirement will be used in later calculations.     

Derivation of Effective Chiral Lagrangian Involving PSGB and Vector Bosons from QCD

The effective chiral Lagrangian for a matter field content consisting of pseudo-scalar Goldstone bosons and vector bosons (with hidden symmetry) is derived from the underlying QCD theory. No approximations are made. All the free parameters of the effective chiral Lagrangian are expressed in terms of QCD-based Green's functions. These may be regarded as the QCD definitions of these Lagrangian coefficients.     

Explicit Solutions of (2+1)-Dimensional Canonical Generalized KP,KdV, and(2+1)-Dimensional Burgers Equations with Variable Coefficients

In this paper, using the generalized G'/G-expansion method and the auxiliary differential equation method, we discuss the (2+1)-dimensional canonical generalized KP (CGKP), KdV, and (2+1)-dimensional Burgers equations with variable coefficients. Many exact solutions of the equations are obtained in terms of elliptic functions, hyperbolic functions, trigonometric functions, and rational functions.     

Solutions to Class of Linear and Nonlinear Fractional Differential Equations

In this paper, the fractional auxiliary sub-equation expansion method is proposed to solve nonlinear fractional differential equations. To illustrate the effectiveness of the method, we discuss the space-time fractional Kd V equation, the space-time fractional RLW equation, the space-time fractional Boussinesq equation, and the(3+1)-spacetime fractional ZK equation. The solutions are expressed in terms of fractional hyperbolic and fractional trigonometric functions. These solutions are useful to understand the mechanisms of the complicated nonlinear physical phenomena and fractional differential equations. Among these solutions, some are found for the first time. The analytical solution of homogenous linear FDEs with constant coefficients are obtained by using the series and the Mittag–Leffler function methods. The obtained results recover the well-know solutions when α = 1.     

Scattering from a Multi-Layered Sphere Located in a High-Order Hermite-Gaussian Beam

Scattering of a high-order Hermite-Gaussian beam by a multi-layered sphere is analyzed. The incident high- order Hermite-Gaussian beam field is expressed by the complex-source-point method and expanded in terms of spherical vector wave functions. The beam shape coefficients of the Hermite-Gaussian beam are obtained. Under electromagnetic field boundary conditions, coefficients in the expressions of scattering fields are derived. Results of the numerical calculation of scattering intensity are presented. The effects of the particle parameters and beam parameters on scattering intensity are discussed in detail.     

Analytical representations for the transient admittance of a plate vibrating in a medium

Flexural vibrations of a plate contacting on one side with an ideal compressible liquid are considered. The plate is driven by a harmonic force uniformly distributed along a straight line. The transient admittance of the plate as a function of the distance from the line of the force application is shown to be representable as a sum of an integer function and an integer function multiplied by a logarithmic function. A procedure for determining the power series expansions of these functions is described, and the initial terms of the expansions are derived. The approximations formed by these initial terms and the asymptotic expansion at infinity are compared with the results of numerical calculations for several particular values of the parameters. Vibrations of a liquid with an impedance load at its surface are considered as an auxiliary problem, and, in the framework of this problem, the initial terms of the power series expansions of the integer functions, which appear in the expression for the transient admittance, are determined. The expansions obtained make it possible to raise the speed of the admittance calculations near the points of application of the driving force.     

Optical properties of thin films on rough surfaces

A general theory, of second order in the film thickness and surface roughness over the wavelength, is developed of the optical properties of a thin film. These properties are described by a small number of electromagnetic constitutive coefficients. Formulae for these coefficients are derived in terms of the height-height correlation functions of the upper and lower surfaces of the film, and its average thickness. The reflectance, transmittance and ellipsometric coefficient are expressed in terms of the constitutive coefficients, for arbituary angles of incidence.     

Electromagnetic plane-wave scattering from a sectorial groove in a perfectly conducting plane

Scattering characteristics of plane waves by a sectorial groove in a perfectly conducting plane are investigated. Both the transverse magnetic (TM) and transverse electric (TE) polarizations of the incident wave are considered. Judicious use of the region-matching technique provides a rigorous series solution to the problem. The analyzed region is separated into two sub-regions by choosing a semi-circular auxiliary boundary. Thefield in each sub-region is expanded as a summationof proper wave functions with unknown coefficients. Enforcing the matching of conditions on the auxiliary boundary and of boundary condition on the circular-arc surface of the groove leads to a linear set of equations and the unknown coefficients are then determined. Numerical results demonstrate the influence of central angles of the sectorial groove on echo width, far-field pattern and near-field distribution. The presented geometry is easily applicable to the design and fabrication of a grating structure for optical switches and tunable filters.      

Scattering States of Schrdinger Equation under the Modified Cusp Potential

We solve the one-dimensional time-independent Schrdinger equation in the presence of the modified Cusp potential and report the solutions in terms of the Whittaker functions. We obtain the reflection and transmission coefficients as well as the bound-state solutions in terms of the Whittaker functions. We comment on the solutions and discuss them in terms of the engaged parameters.     

Scattering States of Schrödinger Equation under the Modified Cusp Potential

We solve the one-dimensional time-independent Schrödinger equation in the presence of the modified Cusp potential and report the solutions in terms of the Whittaker functions. We obtain the reflection and transmission coefficients as well as the bound-state solutions in terms of the Whittaker functions. We comment on the solutions and discuss them in terms of the engaged parameters.     

On zero-field anisotropic static Kubo-Toyabe relaxation functions

For muons which have thermalized in the allowed stopping sites of a given crystal, zero-field static Kubo-Toyabe spin dynamics is expressed in terms of the crystal frame spherical harmonic coefficients of the local classical random magnetic field distributions associated with each stopping site. The resulting muon spin polarization involves nine observable relaxation functions which are the counter frame spherical tensor expansion coefficients of the second-rank dynamic motion tensor. These relaxation functions can be measured simultaneously using the skewed field technique for the same experimental conditions, that is, with a single apparatus and with a single crystal-counter frame orientation. The local field distributions of, in general, arbitrary symmetry are classical approximations to the magnetic field interactions between the spin of the muon and the spins of the nuclei associated with each site at which the muon has stopped. They are characterized by equating their moments with the quantal moments generated by the quantal magnetic fields. The observable consequences of the anisotropic second-order moments associated with the planar sites of hexagonal crystals are used as an illustration.     

Kinetic theory of dense fluids. II. The generalized Chapman-Enskog solution

In the second paper of this series we solve the kinetic equation proposed in the previous paper by a method following the spirit of Chapman and Enskog (generalized Chapman-Enskog method). The zeroth-order solution to the kinetic equation leads to the Euler equations in hydrodynamics for real fluids, and the first-order solution to the Navier-Stokes equations for real fluids. General formulas for transport coefficients such as viscosity and heat-conductivity coefficients are obtained for dense fluids, which are given in terms of time-correlation functions of fluxes conjugate to the thermodynamic forces. The results have the same formal structures as the time-correlation functions in linear response theory except for the collision operator appearing in place of the Liouville operator in the evolution operator for the system.     

A simple model potential description of the alkaline earth isoelectronic sequences

We have developed a simple model potential with a hard core and the correct large-r Coulombic behaviour, to describe the interaction of an electron with a closed shell. One has an exact, analytic ground state wave function for this potential. This potential is used to develop two-electron perturbed and unperturbed wave functions, with the correct asymptotic behaviour and cusp conditions. These wave functions allow us to obtain accurate values for the two-electron energies, polarisabilities, hyperpolarisabilities, and dispersion coefficients of alkaline earth sequences. Many of these results are the only ones available in the literature. Received 29 July 1999 and Received in final form 16 November 1999     

Bilinear forms and dark soliton behaviors for a higher-order variable-coefficient nonlinear Schrödinger equation in an inhomogeneous alpha helical protein

In this paper, a higher-order variable-coefficient nonlinear Schrödinger equation is studied, which describes an inhomogeneous alpha helical protein with higher-order excitation and interaction under the continuum approximation. With the aid of auxiliary function, we obtain the variable-coefficient Hirota’s bilinear equations under a set of integrable constraints. Using the Hirota’s method and symbolic computation, we derive the dark one-, two- and N-soliton solutions. Influences of the variable coefficients on the soliton velocity, amplitude, and shape are analyzed. For instance, when the variable coefficients are the linear and quadratic functions of time, since the pharmacological efficacy in specific sites of the alpha helical protein diffuses linearly and quadratically as time goes on, we obtain a parabolic and cubic soliton. Interactions between/among the two, three, and four solitons with different values of variable coefficients are also discussed with the results including the parabolic, cubic, periodical, and stationary solitons.     

Correlations of image quality metrics studied using systematically distorted videos

Several frequently used image quality metrics were selected to systematically study the correlations between them using many distorted videos, which were revised by gradual changes of usual types of distortion. A modified power-law equation was found as a general relation between the metrics. The parameters obtained by data fittings on the experimental results can be approximated as additions of independent components, which are functions of average, distortion and video. Hence, a metrics can be written as a product of the power-law functions of another metrics with individual coefficients corresponding to distortion, video, and the minor terms.     

Deuterium isotopic effect in 13C-N.M.R. spectra

We obtain the grand potential and the associated distribution functions for a system of hard rods on a line of finite length. These functions are shown to be related to those of a semi-infinite system bounded by one hard wall. It follows that the contact values of the density and distribution functions at the walls of the finite system are related to those at the wall of the semi-infinite system, and that the force on the walls is the density at contact. We obtain also the density and distribution functions for a system with repeating boundary conditions, which are also expressible in terms of those of the finite and semi-infinite systems.