Generalized Fourier Expansion Research Articles

Field quantization in 5D space-time withZ2parity and position/momentum propagator

Field quantization in 5D flat and warped space-times with ${Z}_{2}$ parity is comparatively examined. We carefully and closely derive 5D position/momentum propagators. Their characteristic behaviors depend on the 4D (real world) momentum in relation to the boundary parameter ($l$) and the bulk curvature ($\ensuremath{\omega}$). They also depend on whether the 4D momentum is spacelike or timelike. Their behaviors are graphically presented, and the ${Z}_{2}$ symmetry, the brane formation, and the singularities are examined. It is shown that the use of absolute functions is important for properly treating the singular behavior. The extra coordinate appears as a directed one like the temperature. The $\ensuremath{\delta}(0)$ problem, which is an important consistency check of the bulk-boundary system, is solved without the use of Kaluza-Klein (KK) expansion. The relation between the position/momentum propagator (a closed expression which takes into account all KK modes) and the KK-expansion-series propagator is clarified. In this process of comparison, two views on the extra space naturally come up: the orbifold picture and the interval (boundary) picture. Sturm-Liouville expansion (a generalized Fourier expansion) is essential there. Both 5D flat and warped quantum systems are formulated by Dirac's bra and ket vector formalism, which shows that the warped model can be regarded as a deformation of the flat one with the deformation parameter $\ensuremath{\omega}$. We examine the meaning of the position-dependent cutoff proposed by Randall and Schwartz.

Generalized Fourier expansion in kernels of convolution operators on Fourier hyperfunctions

We prove that the kernels of surjective convolution operators on Fourier hyperfunctions (and on Fourier ultra-hyperfunctions) admit a basis of exponential solutions. The corresponding coefficient spaces are explicitly determined.

Derivatives of generalized Fourier expansions on the sphere

We show that the term by term derivative of the Fourier expansion in spherical

Boundary eigensolutions in elasticity II. Application to computational mechanics

The theory of fundamental boundary eigensolutions for elastostatic problems, developed in Part I, is applied to formulate methods for computational mechanics. This theory shows that every elastic solution can be written as a linear combination of some fundamental boundary orthogonal deformations, thus providing a generalized Fourier expansion. One finds that traditional boundary element and finite element methods are largely consistent with this theory, but do not harness its full power. This theory shows that these computational methods are indirectly a generalized discrete Fourier analysis. Furthermore, by utilizing suitable boundary weight functions, boundary element and finite element formulations may be written exclusively in terms of bounded quantities, even for non-smooth problems involving notches, cracks, mixed boundary conditions and bi-material interfaces. The close relationship between the resulting boundary element and finite element methods also becomes evident. Both use displacement and surface traction as primary variables. A new degree-of-freedom concept is introduced, along with a stiffness tensor that enables one to visualize a finite element method via a boundary discretization process, just as in a boundary element approach. Global convergence characteristics of the traction-oriented finite element method are also developed. Comparisons with closed-form fundamental boundary eigensolutions for a circular elastic disc are presented in order to provide a means for assessing the numerical methods. Several other numerical examples are solved efficiently by using the concept of boundary eigensolutions in an indirect fashion. The results indicate that the algorithms follow the underlying theory and that solutions to non-smooth problems can be obtained in a systematic manner. Beyond this, the concept of boundary eigensolutions provides an alternative view of computational continuum mechanics that may lead to the development of other non-traditional approaches.

Basis for Breakup States of Three Identical Particles

A new basis for expanding three-body momentum-space states for three identical particles is studied. The basis states are simultaneously eigenstates of the total angular momentum and the total antisymmetrization operator. The total kinetic energy and two Dalitz-Fabri variables are chosen as the remaining three continuous variables. Zernike polynomials are used as a basis set for a generalized Fourier expansion in the Dalitz-Fabri variables. Born approximations to the nucleon-deuteron breakup amplitude (zero total orbital angular momentum) are calculated for Malfliet-Tjon I–III potentials and displayed in a Dalitz plot that shows the global structures of the reaction probabilities. Numerical results are presented, which indicate favorable convergence properties of the generalized Fourier expansion. These results suggest that the new basis set may be attractive in more realistic calculations.

Numerical Fourier Expansions of the Planetary Disturbing Function

Various Fourier expansions of the planetary disturbing function can be computed numerically with the use of numerical Fourier analysis. The task to compute the most general five-dimensional Fourier expansion of disturbing function has become feasible with typical server-class computers quite recently. In such an expansion two anomalies, two arguments of perihelions and two longitudes of the node are independent angular variables, while two semi-major axes, two eccentricities and two inclinations are fixed numerically. The semianalytical expansion of the disturbing function resulting from numerical Fourier analysis theoretically converges for any values of the parameters except for those sets of parameters which allow the bodies to collide. Various aspects of the numerical computation of the Fourier expansion are discussed. Theoretical and practical convergence of the Fourier series is discussed and illustrated.

Generalized Fourier expansions for distributions and ultra distribution

Let Dw(Pa)' be a space of distributions or ultradistributions of Beurling type on the p-dimensional parallelepiped Pa:=Ppj=1[-aj,aj]IRp. We investigate the following problems: 1) When can any element of Dw(Pa)' be expanded in absolutely convergent series in a system of generalized exponentials (el(n))nINp with special exponents l(n), nINp. 2) When can a sequence of the coefficients (cn)nINp in an expansionu=SnINp cnel(n) be chosen so that it depends in a continuous and linear way on uIDw(Pa)', where 0<bj£ aj for all 1£j£ p.

Nonparametric Estimation of the Mode of A Distribution of Random Curves

Summary Motivated by the need to develop meaningful empirical approximations to a ‘typical’ data value, we introduce methods for density and mode estimation when data are in the form of random curves. Our approach is based on finite dimensional approximations via generalized Fourier expansions on an empirically chosen basis. The mode estimation problem is reduced to a problem of kernel-type multivariate estimation from vector data and is solved using a new recursive algorithm for finding the empirical mode. The algorithm may be used as an aid to the identification of clusters in a set of data curves. Bootstrap methods are employed to select the bandwidth.

A fourier-like method for the expectation problem in stochastic transport theory

Radiative transfer through a cloud with stochastic clumps is studied into an abstract Banach space setting. By using a generalized Fourier expansion, a sequence of evolution equations is deduced, which approximates the exact evolution equation for the expectation.

Analysis of Flow Injection Peaks with Orthogonal Polynomials

Digitized transient signals such as those acquired in flow injection analysis may be decomposed by a generalized Fourier expansion into a weighted linear combination of discrete orthogonal polynomials. Together, the coefficients from such an expansion form a spectrum analogous to that of the magnitude spectrum of a discrete Fourier transform and provide a useful alternative means of signal identification. This flexible method of representing peak shag in flow injection (and elsewhere) is not reliant upon any single mathematical model. Two families of functions, the Gram and Laguerre polynomials, were investigated. Both series were found to be sensitive to changes in peak shape and able to represent important features of flow injection time domains signals

Functional reconstruction and local prediction of chaotic time series.

A formalism for the global polynomial function approximation for the prediction of chaotic time series is introduced. It takes into account the ergodic invariant measure of the dynamical system. The method presented is based on the generalized Fourier expansion with respect to the polynomial system \ensuremath{\Pi} orthonormal to the invariant measure. The expansion coefficients, which are optimal in the ${\mathit{L}}_{2}$ sense with respect to the natural metrics ${\mathrm{\ensuremath{\rho}}}_{\mathit{I}}$, are expressed in terms of the hierarchy of moments scrM and the hierarchy of functional moments \ensuremath{\Gamma}. In this way, any kind of multivariable parameter optimization becomes unnecessary. The presented approach to reconstruction furnishes good results in the global reconstruction and prediction of time series arising from both discrete and continuous dynamical systems.

Generalized Fourier expansions for zero-solutions of surjective convolution operators onD {?} (?)?

Generalized Fourier expansions for zero-solutions of surjective convolution operators onD {?} (?)?

Density functional theory of freezing for hexagonal symmetry: Comparison with Landau theory

Density functional theory, studied recently by us [J. Chem. Phys. 87, 5449 (1987)] is used to study the freezing of hard disks and hard spheres into crystals with hexagonal symmetry. Two different numerical techniques are used, namely a Gaussian approximation to the crystal density and a more general Fourier expansion of the crystal density. The results from these methods are compared with each other, more approximate versions of density functional theory, and computer simulations. In addition, we compare density functional theory with Landau theories of first order transitions, in which the free energy is expanded as a power series, usually in just one order parameter. We find that traditional Landau theory has little validity when applied to the freezing transition.

Quantum theory of nondegenerate multiwave mixing: General formulation.

In this first paper of a series on the quantum theory of nondegenerate multiwave mixing applicable to traveling-wave interaction geometries, we describe our problem formulation. We consider the explicit dynamics of a subset of the field quantization modes interacting with a system of stationary two-level atoms contained in a volume much smaller than the field quantization volume. Because we make the realistic assumption of leaving the remaining infinite set of field modes as a common thermal-field reservoir, the resulting Langevin equations contain extra decay terms due to collective spontaneous emission or super-radiance. We show that all but one of these super-radiance terms are negligible in the following two limits: (a) when the number of atoms in a diffraction volume is small; (b) when the atoms are pumped far off resonance. There is, however, an anomalous decay term which does not appear in a classical model of super-radiance based upon coherently phased atomic dipoles. The magnitude of this anomalous term is neither dependent upon the number of atoms nor on the pump-frequency detuning and may not be negligible at a low pump intensity. Neglecting the super-radiance terms, we then present a general Fourier-expansion solution technique for obtaining the atomic polarization in the presence of any number of field modes. The expansion is shown to be convergent in a commonly occurring situation in which all the strong pump modes are frequency degenerate and the remaining nondegenerate modes are all weak compared to the atomic saturation intensity. In subsequent papers of this series, we will present methods to treat, with some rigor, the spatial propagation of an interacting multimode quantum field and apply these methods to traveling-wave squeezed-state generation experiments.

Application of the integral variation method to satellite orbit prediction

The Integral Variation (IV) method is a technique to generate an approximate solution to initial value problems involving systems of first-order ordinary differential equations. The technique makes use of generalized Fourier expansions in terms of shifted orthogonal polynomials. The IV method is briefly described and then applied to the problem of near Earth satellite orbit prediction. In particular, we will solve the Lagrange planetary equations including the first three zonal harmonics and drag. This is a highly nonlinear system of six coupled first-order differential equations. Comparison with direct numerical integration shows that the IV method indeed provides accurate analytical approximations to the orbit prediction problem.

Vibration of complex structures: The modal constraint method

A method is described for establishing the natural frequencies of an arbitrary structure with arbitrary supports. The method is based on the modal constraint technique described in a previous paper [1]. As shown in the present paper Weinstein's theory for the intermediate problem can be regarded as equivalent to the Lagrangian multiplier method: i.e., both methods result in the same eigenvalue equations. Weinstein's theory deals with modifications of base differential operators whereas the Lagrangian multiplier method deals with modifications of base energy functionals. The modal constraint technique is an extension of Weinstein's theory, or in energy terms the generalized Fourier expansion of the Lagrangian multiplier. The merits of this method lie in the fact that the eigenvalues and eigenfunctions of structures are used as base structures. The coupling of these structures are taken into account by Lagrangian generalized forces of the constraint acting on the base structures. Some examples are given and the results compared with known solutions.

Green's functions for composite media

A method is given on construction of Green's functions for finding field excited by a point (or line) source in a layered composite medium. The concept of Green's function for a composite medium as a whole is introduced in a rigorous manner by showing that it possesses all properties of the Green's function for an uniform medium. The appropriate Green's function for the whole composite medium is constructed through two solutions of the corresponding homogeneous equation over the whole multiregion. The method enables one to write down, directly, the explicit expression for the field in an arbitrary region produced by a point (or line) source located in another arbitrary region without resort to images, Fourier-Bessel integrals, integral transforms of quasi-orthogonal functions. It is an extension of the existing method, which constructs the Green's functions for uniform media through the generalized Fourier expansions and two solutions of the corresponding homogeneous equations, to cases heretofore beyond its scope.

Stability of a feedback controlled distributed system by modal representation

In this work a linear distributed parameter system controlled by a finite number of lumped linear feedback loops is considered. The spatial portion of the system is decomposed using generalized Fourier expansions, and a complex frequency domain transcendental characteristic determinant results. The theory developed is then applied to the automatic regulation of water flow in a ‘ reach ’ of an open irrigation canal. The dynamics of water flow in the transient state are described by a set of two first-order non-linear partial differential equations. The boundary conditions, governed by the water-flow dynamics under the gate, are also non-linear equations. For this particular system the feedback controller signal originates at the downstream boundary and controls the gates at the ‘ head ’ of the canal. The stability analysis of the complete closed-loop system is based on perturbation about some steady-state flow conditions which allows for the linearization of the system's non-linear partial differential equations and boundary conditions. Subsequent application of the Nyquist criterion to a modal representation of the distributed parameter portion of the control loop in cascade with the lumped parameter controller yields the stability conditions.

On the Convergence Rate of Generalized Fourier Expansions

On the Convergence Rate of Generalized Fourier Expansions

Higher Harmonics and Transport Coefficients of Plasmas in Circulary Polarized Magnetic Fields and Additional Electromagnetic Fields

AbstractWith the methods of kinetic theory on the basis of the Boltzmann and Fokker‐Planck kinetic equations the behaviour of a Lorentz plasma in a circularly polarized (rotating) magnetic field (rotation frequency ω), an alternating electric field (frequency ω′) and additional constant electromagnetic fields is investigated. By means of a generalized Fourier expansion it is shown that the above fields create in the plasma currents of the frequencies 0, ω′, ω–ω′,ω, ω+ω′, 2ω–ω′, 2ω, and 2ω+ω′. Transport coefficients are calculated explicitly and the validity of Onsager reciprocity relations and that of Kronig‐Kramers relations is discussed. The special case of the electric field induced by the rotating magnetic field is treated separately. Finally, problems of plasma containment are discussed.