Set Of Orthonormal Functions Research Articles

Magnetoplasma surface waves on the lateral surface of a semiconductor superlattice

The integral equation describing charge density fluctuations on the lateral surface of a semiconductor superlattice is transformed to a matrix equation by expanding the scalar potential in a complete orthonormal set of functions. Numerical results, obtained by keeping only a finite number of terms in the expansion, converge rapidly and display significant differences from the approximate analytic solution obtained earlier.

Application of the singular function expansion to an integral equation for scattering

The Dirichlet scattering problem is solved exactly for a surface of arbitrary smooth shape. The solution is given in terms of the complete, orthonormal set of functions defined on the surface and arising as singular functions of the integral operator. Since the singular values do not have a point of accumulation at zero, the method is of much greater practical value than the eigenmode expansion method (EEM). Since the terms in the infinite series are naturally ordered by size of singular value, the dominance of one or several terms in the series may be established without explicit evaluation of the coupling coefficients. Extentions of the method to other boundary value problems and applications to the singularity expansion method (SEM) are described.

Dynamics and Stability of support excited beam-colums with end mass

The dynamics and stability of beam-columns with support motion is discussed, and the general equations of motion for a translating beam-column are derived along with the associated boundary conditions. The response of a cantilevered beam-column supporting a large mass at its free end is considered where the support of the structure is excited harmonically in both the transverse and longitudinal directions. The free transverse vibration problem is solved yielding a frequency equation and set of mode shapes which are then used to generate an orthonormal set of functions corresponding to an equivalent problem, where the point mass is considered to be part of the beam-column. Employing an eigenfunction expansion, the stability of the structure is assessed, for the case of longitudinal excitation, by the determination of the boundedness of the solutions to the resulting system of differential equations governing the amplitudes of the eigenfunctions.

Spherical functions of the Lorentz group on the hyperboloids

By means of suitable coordinate systems spherical functions of the Lorentz group are derived on the double-sheeted, single-sheeted hyperboloids and similarly on the light cone. These form complete orthonormal sets of functions on each hyperboloid as well as on the light cone and transform according to the principal series of unitary representations of the Lorentz group. The spherical functions are given in a basis defined by the two-dimensional momentum (N 1+M 2,N 2 −M 1) of horospheric translations where N and M denote the generators of Lorentz transformations and spatial rotations. It is known that the upper sheet of the double-sheeted hyperboloid is a homogeneous space even under the subgroup\(\left( {\begin{array}{*{20}c} \alpha & \beta 0 & {\alpha ^{ - 1} } \end{array} } \right)\) ∈ SL(2, C). As 'a consequence of this the spherical functions have rather simple transformation properties under the Lorentz group.

Time-resolved fluorescence polarization from ordered biological assemblies

We calculate the time dependence of the polarized fluorescence signal from fluorescent-labeled elements of a biological assembly that are rotationally diffusing in an arbitrary three-dimensional angular potential. We have formulated this calculation using the model-independent description of the angular potential wherein the angular potential is described by an expansion in a complete set of orthonormal functions with the expansion coefficients (or order parameters) determined by time-independent methods (Burghardt, T. P., 1984, Biopolymers, 23:2382-2406). We have applied the calculation to fluorescent-labeled myosin cross-bridges in relaxed muscle fibers. In a related paper we describe the experimental observation of the myosin cross-bridges rotationally diffusing in an angular potential (Burghardt, T. P., and N. L. Thompson, 1985, Biochemistry, 24:3731-3735).

Model-independent electron spin resonance for measuring order of immobile components in a biological assembly

A model-independent description of the angular orientation distribution of elements in an ordered biological assembly is applied to the electron spin resonance (ESR) technique. As in a previous model-independent treatment of fluorescence polarization (Burghardt, T.P., 1984, Biopolymers, 23:2383-2406) the elemental order is described by an angular distribution of molecular frames with one frame fixed in each element of the assembly. The distribution is expanded in a complete orthonormal set of functions. The coefficients of the series expansion (the order parameters) describe the orientation distribution of the elements in the assembly without reference to a model and can be obtained from the observed spectrum. The method establishes the limitations of ESR in detecting order in the assembly by determining which distribution coefficients the technique can detect. A method of determining the order parameters from an ESR spectra, using a set of ESR basis spectra, is developed. We also describe a treatment that incorporates the actual line shape measured from randomly oriented, immobile elements. In this treatment, no model-dependent assumptions about the line shape are required. We have applied the model-independent analysis to ESR spectra from spin-labeled myosin cross-bridges in muscle fibers. The results contain detailed information on the spin-probe angular distribution and differ in interesting ways from previous model-dependent interpretations of the spectra.

Source depth estimation in waveguides

In the present paper an approach to source depth estimation in a waveguide is proposed. The method is based on the mode filtering technique. By using a mode filter, the field sampling data of a vertical array are transformed to a vector which consists of two parts: (1) the eigenfunction of each mode and (2) the modal phase corresponding to the source range. The latter part can be compensated by means of range information extraction proposed in our previous paper [Shang et al., J. Acoust. Soc. Am. Suppl. 1 74, S78 (1983)]. The data vector consisting of a finite number of mode eigenfunctions then becomes ‘‘almost orthogonal’’ and can be used to construct a source depth power estimator. It is well known that the ‘‘closure’’ is a property of a complete, orthonormal set of functions, and because the mathematical expression of ‘‘closure’’ is a correlation type, a correlation operator is a reasonable one for identifying source depth. The resolution is the magnitude of −(H/M), where H is the water depth and M is the effective mode number. Some numerical examples and scale model experimental results are shown.

Weakly convergent expansions of a plane wave and their use in Fourier integrals

The Fourier transform of an irreducible spherical tensor is normally computed with the help of the Rayleigh expansion of a plane wave in terms of spherical Bessel functions and spherical harmonics. The angular integrations are then trivial. However, the remaining radial integral containing a spherical Bessel function may be so complicated that the applicability of Fourier transformation is severely restricted. As an alternative, the use of weakly convergent expansions of a plane wave in terms of complete orthonormal sets of functions is suggested. The weakly convergent expansions of a plane wave are constructed in such a way that their application in Fourier integrals leads to expansions of the Fourier or inverse Fourier transform that converge with respect to the norm of either the Hilbert space L2(R3) or the Sobolev space W(1)2(R3). Accordingly, these weakly convergent expansions may be viewed as distributions that are defined on either L2(R3) or W(1)2(R3). The properties of some complete orthonormal sets of functions, in particular their Fourier transforms, are also studied. Shibuya and Wulfman [Proc. R. Soc. London Ser. A 286, 376 (1965)] derived an expansion of a plane wave involving the four-dimensional spherical harmonics. It is shown that this Shibuya–Wulfman expansion is also a distribution which is defined on the Sobolev space W(1)2(R3). Finally, as an application it is shown how weakly convergent expansions can be used profitably for the construction of addition theorems.

Comment on "Hydrogenlike orbitals of bound states referred to another coordinate"

A recently published addition theorem for the bound-state hydrogenlike atomic orbitals [T. Yamaguchi, Phys. Rev. A 27, 646 (1983)] is critically analyzed. It is shown that the convergence of the expansion derived by Yamaguchi is not guaranteed since the continuum states of the hydrogen spectrum are not included properly in the expansion. Alternative complete orthonormal sets of functions for which the problems with the continuum states do not occur are suggested. Overlap integrals of bound-state hydrogenlike functions which serve as Fourier coefficients in the addition theorem are also investigated. It is demonstrated that in all cases explicit expressions for the overlap integrals can be derived.

An infinite element and a formula for numerical quadrature over an infinite interval

AbstractIn this publication we present a new infinite element and discuss a formula for numerical quadrature over infinite regions. Both the element and the formula are based on the properties of a set of orthonormal functions, obtained by mapping the Legendre polynomials on the infinite interval. The element can represent any (physical) function over the entire infinite region, and the formula can integrate most integrands over the infinite interval. They are thus very convenient for implementation in a general computer program, which could then be used to solve different problems with no restrictions concerning the asymptotic behaviour of the solutions or the form of integrands to be integrated over the infinite interval. However, the element and the formula will perform at their best when used in conjunction, and for problems of electrostatic, magnetostatic, etc., potential.The formula was compared with other existing formulae and found to be either superior or satisfactory in practically all cases, as shown by the results of many numerical experiments.Expressions of shape functions are given for the element. A test problem was solved using the element and the formula, and the results are shown to be more accurate than those obtained when solving the same problem using infinite elements with exponential decay and Gauss‐Laguerre numerical quadrature.

On the Vlasov-Poisson system of equations for an inhomogeneous cylindrical plasma

The formal solution of the linearized Vlasov-Poisson system of equations is carried out for an inhomogeneous strongly magnetized plasma column. The system considered is closely related to several experiments in magnetically confined cylindrical plasmas. A generalization of the Fourier-Laplace method of solution is presented using expansion in the appropriate set of orthonormal functions to handle the boundary value problem in finite systems. The Fourier-Bessel components of the dielectric constant epsilon /sub //m/ll '(k, omega ) for the cylindrical inhomogeneous plasma and a matrix dispersion relation are derived to study the propagation of waves in the bounded system. In the case of neutral plasmas in equilibrium, the problem of finding the roots of the dispersion relation is equivalent to solving an eigenvalue equation. The method is applied to study a typical case in a Q-machine.

Solidification of a low peclet number fluid flow in a round pipe with the boundary condition of the third kind

AbstractThis paper presents a theoretical investigation on the solidification of a low Peclet number fluid flow in the thermal entrance region of a round pipe. The velocity is assumed laminar and fully developed throughout the pipe and the fluid temperature is taken to be uniform at X = −∞. The pipe wall is adiabatic at X ≤ 0 and is cooled convectively at X ≥ 0. The exact solution in the liquid and solid phase are obtained by using the method of separation of variables and constructing two sets of orthonormal functions from the nonorthogonal eigenfunctious at X = ± 0.The solutions including temperature distributions, liquid‐solid interfaces and Nusselt numbers for the parameters, Bi = 20, 4, 0.4, 0.04, Pe = 1, 3, 5, 10, 20, 30, ∞ and the superheat ratios, ξ = (1 − θf)/θf, θf= 0.1, 0.2, …, 0.9 are presented in this paper. The length of ice‐free zone xf corresponding to Pe = ∞ are in excellent agreement with the existing solution.

Reactivity effects and flux perturbations due to spatial random dispersal in the number densities of core materials

An analysis was made on the reactivity and the flux profile modulation caused by spatial random perturbations in the number densities of core materials of a bare, uniform, critical, reference reactor. Using an orthonormal function set of the Helmholtz modes, a one-group diffusion equation with the multiplication factor k is transformed into an infinite dimensional vector equation with the reactivity ϱ = 1 − 1 k . The problem is then analyzed as the perturbations in the smallest eigenvalue in magnitude and the corresponding eigenvector of an infinite dimensional matrix. This approach leads to the conclusion that spatial random perturbations cause the positive reactivity effect and a flattened flux profile. The results are exact up to the second order of perturbations when zero ensemble average of perturbations is assumed, and are exact up to the third order of perturbations when the Gaussian distribution is assumed in addition.

Continuous charge distribution models of ions in polar media. Part 6.—Reltationship between quantum and classical charge distributions

A formal relationship between the classical phenomenological charge distribution representations of ions and the quantum mechanical treatment is presented. This relationship is established by means of an expansion of the molecular electronic quantal charge density in terms of a complete set of orthonormal functions. Such an expansion is similar in form to a classical multipole expansion for an arbitrary charge distribution. The analysis reported provides a rigorous mathematical and an enhanced physical interpretation of the “effective Bohr radial” quantities introduced with the earlier phenomenological models.

Cubic harmonic analysis of magnetic anisotropy measurements on single crystal Ho xTb 1−xFe 2 laves phase compounds

Room temperature magnetic anisotropy measurements have been made on a series of low anisotropy Ho xTb 1−xFe 2 compounds, with 0.826 < x < 0.908, using torque magnetometry techniques. An analysis of the data using the conventional expansion of anisotropy energy in terms of the direction cosines proved inadequate because of the presence of higher order contributions to the anisotropy. When these contributions were included in the expansion, non unique anisotropy constants were obtained which were not consistent with the predictions of the single ion model. An analysis of the data using an expansion of the anisotropy in terms of an orthonormal set of functions, namely the cubic harmonics developed by Mueller and Priestly, proved more successful. The results showed that the two lowest order anisotropy constants, κ 4 and κ 6, both varied linearly with holmium concentrations as predicted by the single ion model; however, κ 8, the next higher order term, was not consistent with the predicted behavior. The origin of this contribution is believed to be related to the magnetostriction.

Some properties of spin-weighted spheroidal harmonics

We analyse the angular eigenfunctions - spin-weighted spheroidal harmonics-and eigenvalues of Teukolsky’s equation. This equation describes infinitesimal scalar, electromagnetic and gravitational perturbations of rotating (Kerr) black holes. We derive analytic expressions for the eigenvalues up to sixth order in the expansion parameter for low frequencies and an analogous expansion in the high-frequency limit. Spinweighted spheroidal harmonics form a complete and orthonormal set of functions on a prolate spheroid. They are, however, not the eigenfunctions of the natural Laplace operator on a spheroid and thus do not allow an obvious geometrical interpretation as the corresponding spin-weighted spherical harmonics.

Schauder-type expansions of continuous functions on the unit interval

Let the space of continuous functions on [0, 1] which vanish at 0 be denoted by C. It will be shown that for any complete orthonormal set of functions { α i ( s)} of bounded variation and such that α i (1) = 0, there is a simply described linear combination of the continuous functions {∝ 0 t α i ( s) ds} which converges uniformly to x( t) for almost all x ϵ C (“almost all” in the sense of Wiener measure).

Optimal point-wise discrete control and controllers' allocation strategies for stochastic distributed systems

The design of point-wise discrete controllers for a class of stochastic distributed-parameter systems is considered. Assuming a fixed set of controllers' positions, the optimal feedback control is derived using a direct approach in which the infinite dimensional space is approximated using a set of orthonormal functions. The resulting optimal cost is minimized again w.r.t. this set of positions, using gradient techniques, to get the optimal locations for the controller. A one-dimensional diffusion process is used to demonstrate the algorithm.

Expansion technique for crystal surface problems

An expansion technique for representing functions localized at crystal surfaces is developed by making use of the Gottlieb polynomials to construct a basis set of orthonormal functions of a discrete variable (the integers from zero to infinity). Using these functions, it is shown that a large class of surface problems can be recast in the form of an eigenvalue problem involving a matrix whose elements can be obtained in closed form. The matrix size can be minimized by varying a parameter upon which the basis functions depend. As an example, the dispersion curve of a ferromagnetic spin wave localized at the (001) surface of an fcc crystal is calculated.

Correlation tensors of the blackbody radiation in rectangular cavities

Electromagnetic equilibrium fluctuations in finite cavities filled with a dissipative medium (dielectric functione (ω)=e′+ie″) and bounded by walls of infinite conductivity are considered. Expanding the fields in terms of a complete and orthonormal set of functions and solving the Maxwell equations the response of the EM field to external forces (polarization and magnetization) is obtained. With the aid of the fluctuation dissipation theorem and the linear response functions the 2nd order correlation tensors of the EM field are derived.