Terms Of Eigenfunctions Research Articles

Modal method based on subsectional Gegenbauer polynomial expansion for lamellar gratings

A first approach of a modal method by Gegenbauer polynomial expansion (MMGE1) is presented for a plane wave diffraction by a lamellar grating. Modal methods are among the most popular methods that are used to solve the problem of lamellar gratings. They consist in describing the electromagnetic field in terms of eigenfunctions and eigenvalues of an operator. In the particular case of the Fourier modal method (FMM), the eigenfunctions are approximated by a finite Fourier sum, and this approximation can lead to a poor convergence of the FMM. The Wilbraham-Gibbs phenomenon may be one of the reasons for this poor convergence. Thus, it is interesting to investigate other basis functions that may represent the fields more accurately. The approach proposed in this paper consists in subdividing the pattern in homogeneous layers, according to the periodicity axis. The field is expanded, in each layer, on the basis of Gegenbauer's polynomials. Boundary conditions are rigorously written between adjacent layers; thus, an eigenvalue equation is obtained. The approach presented in this paper proves to describe the fields accurately. Finally, it is demonstrated that the results obtained with the MMGE1 are more accurate than several existing modal methods, such as the classical and the parametric FMM.

Characterization of self-adjoint ordinary differential operators

Symmetric differential expressions ℓ of order n = 2 k with real valued coefficients give rise to self adjoint operators in the space of weighted square integrable functions. Characterization theorems exist in the literature that describe such self-adjoint operators. All such characterizations begin by constructing the maximal domain of definition of the expression ℓ . The Glazman–Krein–Naimark theorem constructs the maximal domain in terms of eigenfunctions corresponding to a nonreal parameter λ . Representations in terms of certain functions related to a real parameter λ can also be found in the literature. In this paper we construct the maximal domain from two complementary self-adjoint realizations of ℓ . One operator is assumed to be known and the other one is computed explicitly. From these two operators we explicitly give all other self-adjoint operators associated with ℓ . A special class of operators associated with ℓ is what we call Type I operators. They arise in connection with a certain bilinear form that results from the weak formulation of the expression ℓ . Depending on the deficiency index of ℓ and the properties of the bilinear form we can have two complementary self-adjoint operators (two Type I operators) and, as it turns out, one of them is the celebrated Friedrich Extension. The other operator appears to be new. As in the general case, using these two operators we give an explicit characterization of all other operators of the same Type I.

Fock spaces, Landau operators and the time-harmonic Maxwell equations**This paper is dedicated to Professor Richard Delanghe on the occasion of his seventieth birthday.

We investigate the representations of the solutions to Maxwell's equations based on the combination of hypercomplex function-theoretical methods with quantum mechanical methods. Our approach provides us with a characterization for the solutions to the time-harmonic Maxwell system in terms of series expansions involving spherical harmonics resp. spherical monogenics. Also, a thorough investigation for the series representation of the solutions in terms of eigenfunctions of Landau operators that encode n-dimensional spinless electrons is given. This new insight should lead to important investigations in the study of regularity and hypo-ellipticity of the solutions to Schrödinger equations with natural applications in relativistic quantum mechanics concerning massive spinor fields.

Stress-function variational method for stress analysis of bonded joints under mechanical and thermal loads

High interfacial stresses near the ends of adherends are responsible for debonding failure of bonded joints used extensively in structural engineering and microelectronics packaging. This paper proposes a stress-function variational method for determination of the interfacial stresses in a single-sided strap joint subjected to mechanical and thermal loads. During the process, two interfacial shear and normal (peeling) stress functions are introduced, and the planar stresses of adherends of the joints are expressed in terms of the stress functions according to the static equilibrium equations. Two coupled governing ordinary differential equations (ODEs) of the stress functions are obtained through minimizing the complementary strain energy of the joints and solved explicitly in terms of eigenfunctions. The stress field of the joints based on this method can satisfy all the traction boundary conditions (BCs), especially the shear-free condition near the adherend ends. Compared to results based on finite element method (FEM) and other analytic methods in the literature, the present variational method is capable of predicting highly accurate interfacial stresses. Dependencies of the interfacial stresses upon the adherend geometries, moduli and temperature are examined. Results gained in this study are applicable to scaling analysis of joint strength and examination of solutions given by other methods. The present formalism can be extended conveniently to mechanical and thermomechanical stress analysis of other bonded structures such as adhesively bonded joints, composite joints, and recently developed flexible electronics, among others.

Inverse analysis with integral transformed temperature fields: Identification of thermophysical properties in heterogeneous media

The objective of this work is to introduce the use of integral transformed temperature measured data for the solution of inverse heat transfer problems, instead of the common local transient temperature measurements. The proposed approach is capable of significantly compressing the measured data through the integral transformation, without losing the information contained in the measurements and required for the solution of the inverse problem. The data compression is of special interest for modern measurement techniques, such as the infrared thermography, that allows for fine spatial resolutions and large frequencies, possibly resulting on a very large amount of measured data. In order to critically address the use of integral transformed measurements, we examine in this paper the simultaneous estimation of spatially variable thermal conductivity and thermal diffusivity in one-dimensional heat conduction within heterogeneous media. The direct problem solution is analytically obtained via integral transforms and the related eigenvalue problem is solved by the Generalized Integral Transform Technique (GITT). The inverse problem is handled with Bayesian inference by employing a Markov Chain Monte Carlo (MCMC) method. The unknown functions appearing in the formulation are expanded in terms of eigenfunctions as well, so that the unknown parameters become the corresponding series coefficients. Such projection of the functions in an infinite dimensional space onto a parametric space of finite dimension also permits that several quantities appearing in the solution of the direct problem be analytically computed. Simulated measurements are used in the inverse analysis; they are assumed to be additive, uncorrelated, normally distributed, with zero means and known covariances. Both Gaussian and non-informative uniform distributions are used as priors for demonstrating the robustness of the estimation procedure.

Normal modes of oscillations of a rotating membrane with a rigid central insert (an application of Heun functions)

The eigenvalues problem eigenvalues for the equation of the transverse oscillations of a homogeneous annular membrane with a rigid insert, rotating with a constant angular velocity about its central axis, is considered. Exact analytical expressions for the eigenfunctions in terms of special functions (local Heun functions), as well as normalization integrals, are found. An explicit expression for the time-invariant shape of the membrane during regular precession of its rotation axis is obtained.

On low eigenvalues of the Laplacian with mixed boundary conditions

By means of the so-called α-symmetrization we study the eigenvalue problem for the Laplace operator with mixed boundary conditions. We obtain various bounds for combinations of the low eigenvalues and some sharp comparison results for the first eigenfunction in terms of Bessel functions.

Analytic treatment of vortex states in cylindrical superconductors in applied axial magnetic field

We solve the linear Ginzburg–Landau (GL) equation in the presence of a uniform magnetic field with cylindrical symmetry and we find analytic expressions for the eigenfunctions in terms of the confluent hypergeometric functions. The discrete spectrum results from an implicit equation associated to the boundary conditions and it is resolved in analytic form using the continued fractions formalism. We study the dependence of the spectrum and the eigenfunctions on the sample size and the surface conditions for solid and hollow cylindrical superconductors. Finally, the solutions of the nonlinear GL formalism are constructed as expansions in the linear GL eigenfunction basis and selected by minimization of the free energy. We present examples of vortex states and their energies for different samples in enhancing/suppressing superconductivity surroundings.

Renormalization flow in extreme value statistics

The renormalization group transformation for extreme value statistics of independent,identically distributed variables, recently introduced to describe finite-size effects, ispresented here in terms of a partial differential equation (PDE). This yields a flow infunction space and gives rise to the known family of Fisher–Tippett limit distributions asfixed points, together with the universal eigenfunctions around them. The PDE turns outto handle correctly distributions even having discontinuities. Remarkably, the PDE admitsexact solutions in terms of eigenfunctions even farther from the fixed points. In particular,such are unstable manifolds emanating from and returning to the Gumbel fixed point,when the running eigenvalue and the perturbation strength parameter obey a pairof coupled ordinary differential equations. Exact renormalization trajectoriescorresponding to linear combinations of eigenfunctions can also be given, and it isshown that such are all solutions of the PDE. Explicit formulae for some invariantmanifolds in the Fréchet and Weibull cases are also presented. Finally, the similaritybetween renormalization flows for extreme value statistics and the central limitproblem is stressed, whence follows the equivalence of the formulae for Weibulldistributions and the moment generating function of symmetric Lévy stable distributions.

A Discrete Analogue of the Dirac Operator and the Discrete Modified Novikov–Veselov Hierarchy

We construct a discrete analogue of the integrable two-dimensional Dirac operator and describe the spectral properties of its eigenfunctions. We construct an integrable discrete analogue of the modified Novikov-Veselov hierarchy. We derive the first two equations of the hierarchy and give explicit formulas for the eigenfunctions in terms of the theta functions of the associated spectral curve.

Integrable Structure in the Theory of Stress Concentration in Shear-Strained Nonlinear Elastic Materials

It is shown that the Neuber equations governing the stress and deformation in a shear-strained prismatical nonlinear elastic body may be completely characterized in terms of eigenfunctions of the integrable mNVN hierarchy. A Backlund transformation and nonlinear superposition principle are derived, whereby privileged constitutive laws may be generated iteratively.

Spin response of a normal Fermi liquid with noncentral interactions

We consider the spin response of a normal Fermi liquid with noncentral interactions under conditions intermediate between the collisionless and hydrodynamic regimes. This problem is of importance for calculations of neutrino properties in dense matter. By expressing the deviation of the quasiparticle distribution function from equilibrium in terms of eigenfunctions of the transport equation under the combined influence of collisions and an external field, we derive a closed expression for the spin-density--spin-density response function and compare its predictions with that of a relaxation time approximation. Our results indicate that the relaxation time approximation is reliable for neutrino properties under astrophysically relevant conditions.

Rotational Invariance Based on Fourier Analysis in Polar and Spherical Coordinates

In this paper, polar and spherical Fourier analysis are defined as the decomposition of a function in terms of eigenfunctions of the Laplacian with the eigenfunctions being separable in the corresponding coordinates. The proposed transforms provide effective decompositions of an image into basic patterns with simple radial and angular structures. The theory is compactly presented with an emphasis on the analogy to the normal Fourier transform. The relation between the polar or spherical Fourier transform and the normal Fourier transform is explored. As examples of applications, rotation-invariant descriptors based on polar and spherical Fourier coefficients are tested on pattern classification problems.

A unitary group formulation of open-shell electron propagator theory

By replacing the usual annihilation and creation operators of second quantization by appropriately normalized fundamental Wigner operators of the unitary group U(n) and by representing the many‐electron spin eigenfunctions in terms of the Gelfand–Tsetlin basis for the appropriate irreducible representation, we have succeeded in developing an attractive electron propagator formalism which incorporates closed‐shell, open‐shell, or multiconfigurational reference states. Matrix element evaluation for the fundamental U(n) Wigner operators is treated, and illustrative three‐orbital examples involving reference states of doublet and triplet spin symmetry are presented.

Squeezing a wave packet with an angular-dependent mass

We present a new effect of position-dependent mass (PDM) systems: the possibility of creating squeezed wave packets at the partial revival times. We solve exactly the PDM Schrödinger equation for the two-dimensional quantum rotor with two effective masses μ(θ), both free and interacting with a uniform electric field, and present their energy eigenvalues and eigenfunctions in terms of Mathieu functions. For the first one, in order to squeeze the wave packet it is necessary to apply an electric field; for the second one such an effect can be achieved without the field.

Structural components in functional data

Analyzing functional data often leads to finding common factors, for which functional principal component analysis proves to be a useful tool to summarize and characterize the random variation in a function space. The representation in terms of eigenfunctions is optimal in the sense of L 2 approximation. However, the eigenfunctions are not always directed towards an interesting and interpretable direction in the context of functional data and thus could obscure the underlying structure. To overcome such difficulty, an alternative to functional principal component analysis is proposed that produces directed components which may be more informative and easier to interpret. These structural components are similar to principal components, but are adapted to situations in which the domain of the function may be decomposed into disjoint intervals such that there is effectively independence between intervals and positive correlation within intervals. The approach is demonstrated with synthetic examples as well as real data. Properties for special cases are also studied.

Bethe Ansatz and Geometry of the Classical Configuration Space

We demonstrate that the seminal one-dimensional model of the Heisenberg magnet, consisting of N spins 1/2 with the nearest-neighbour isotropic interaction, solved exactly by Bethe ansatz, admits an interpretation of a system of r = N/2 − M pseudoparticles (spin deviations) which are indistinguishable, have hard cores and move on the chain by local hoppings. Such an approach allows us to construct a manifold with some boundaries, which is genericly r-dimensional, and whose F -dimensional regions, 0 < F < r, point out all l-strings. The latter classify, in terms of rigged string configurations of Kerov, Kirillov and Reshetikhin, all exact Bethe eigenfunctions. In this way, we interpret these eigenfunctions in terms of the classical configuration space, in particular on the structure of islands of adjacent spin deviations, in a way independent of the size N .

Radiation of Hertzian Dipole in Cylindrical Cavity with Narrow Slots

Radiation of a Hertzian dipole placed within a cylindrical cavity with narrow slots is investigated. Narrow axial and transverse slots are considered. Scattered fields are expanded in terms of eigenfunctions and boundary conditions are enforced to obtain a set of simultaneous equations. Computations are performed to check the validity of the formulation.

Cylindrical Cavity-Backed Antenna

A cylindrical cavity-backed antenna excited by Hertzian dipoles is investigated. A Green's function approach is used to account for arbitrarily oriented dipoles. Scattered fields are expanded in terms of eigenfunctions, and boundary conditions are enforced to obtain a set of simultaneous equations. A circularly polarized antenna is fabricated, and its radiation characteristics are compared to the theory.

Horseshoes in multidimensional scaling and local kernel methods

Classical multidimensional scaling (MDS) is a method for visualizing high-dimensional point clouds by mapping to low-dimensional Euclidean space. This mapping is defined in terms of eigenfunctions of a matrix of interpoint dissimilarities. In this paper we analyze in detail multidimensional scaling applied to a specific dataset: the 2005 United States House of Representatives roll call votes. Certain MDS and kernel projections output ``horseshoes'' that are characteristic of dimensionality reduction techniques. We show that, in general, a latent ordering of the data gives rise to these patterns when one only has local information. That is, when only the interpoint distances for nearby points are known accurately. Our results provide a rigorous set of results and insight into manifold learning in the special case where the manifold is a curve.