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A semi-analytical method is presented for the assessment of the induced electromagnetic field inside a multilayer spherical head model exposed to radiation of an arbitrary source antenna. First, the isolated source antenna is simulated in a full-wave software to evaluate its radiation characteristics. By sampling the radiated fields, their spherical vector wave function (SVWF) amplitudes are evaluated. Next, the well-known translation addition theorem for SVWFs is implemented to translate the SVWFs of the radiated fields to the local coordinate system of the head model. It’s assumed that the presence of the head model does not affect the primary radiated fields by the antenna due to the adequate distance between them. By applying the boundary conditions on the head model, the unknown SVWF amplitudes of the induced fields inside each layer as well as those of the scattered field outside the model are evaluated. The verification of the proposed method is shown through some numerical examples. In comparison with a fullwave numerical method, the proposed method provides an efficient repeatable simulation approach due to the independent analysis of the source and head model, provided that the reaction of the head model to the antenna is negligible.

A new approach to overcome the ill-conditioning of the Method of Fundamental Solutions (MFS) combining Singular Value Decomposition (SVD) and an adequate change of basis was introduced in [1] as MFS-SVD. The original formulation considered polar coordinates and harmonic polynomials as basis functions and is restricted to the Laplace equation in 2D. In this work, we start by adapting the approach to the Helmholtz equation in 2D and later extending it to elliptic coordinates. As in the Laplace case, the approach in polar coordinates has very good numerical results both in terms of conditioning and accuracy for domains close to a disk but does not perform so well for other domains, such as an eccentric ellipse. We therefore consider the MFS-SVD approach in elliptic coordinates with Mathieu functions as basis functions for the latter. We illustrate the feasibility of the approach by numerical examples in both cases. [ABSTRACT FROM AUTHOR]

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Based on the image method and addition theorem of spherical vector wave functions, an analytical solution of the multiple scattering by a buried medium-coated conducting sphere is proposed in this paper. An iterative process to obtain the scattered electromagnetic field is discussed on the basis of the continuous boundary condition in the plane boundary, the medium inner and the outer surface of a coated conducting sphere, respectively. Applying an image method and the addition theorem of spherical vector wave functions, the scattering electromagnetic fields by the plane in a local coordinate system can be transformed into the globe coordinate, and they can be regarded as the next incident electromagnetic fields to the buried medium-coated conducting sphere. This process does not end until the scattering electrical field on the plane boundary is accurate enough. Numerical results are given and compared with commercial software FEKO, they coincide enough; the calculation times of the present method are very short compared to those of the software FEKO, and some discussion is given at the end of this paper. [ABSTRACT FROM AUTHOR]

This work establishes the existence of addition theorems and double-angle formulas for C k real scalar functions. Moreover, we determine necessary and sufficient conditions for a bivariate function to be an addition formula for a C k real function. The double-angle formulas allow us to generate a duplication algorithm, which can be used as an alternative to the classical numerical methods to obtain an approximation for the solution of an ordinary differential equation. We demonstrate that this algorithm converges uniformly in any compact domain contained in the maximal domain of that solution. Finally, we carry out some numerical simulations showing a good performance of the duplication algorithm when compared with standard numerical methods. [ABSTRACT FROM AUTHOR]

Electromagnetic response of clustered charged particles is the foundation of electromagnetic wave interaction with various natural phenomena, such as sandstorm, cloud, and volcano eruption. Previous studies usually employed assumption of independent charged particles, without considering the coupling between them. Here, we build up a general numerical model considering the multiple scattering effect, and test it with a charged two- and four-particle system. The numerical results show that independence assumption fails, while the number density of clustered charged particles is getting larger. This work may pave the way for deeper understanding on the electromagnetic interaction of clustered charged particles.

Objective: This study extends the scope of a previous analysis on the time-averaged acoustic radiation force on a rigid (sound impenetrable) cylinder near a flat boundary [F.G. Mitri, J. Phys. Commun. 2 (2018) 045019] to the case of a viscous compressible fluid (sound penetrable) particle, and determine the time-averaged acoustic radiation torque as well. Motivation and novelty: Previous analytical formalisms did not consider the case of a sound penetrable cylindrical particle insonified at an arbitrary angle of incidence (in the polar plane) near a reflecting boundary. This work fills this gap, and provides exact expressions and computations for the acoustic radiation force and torque components. Method: The partial-wave series expansion method, in conjunction with the method of images and the translational addition theorem of cylindrical wave functions are used to derive the analytical expressions for the longitudinal and transverse acoustic radiation force components. Moreover, the emergence of a radiation torque that causes the particle to rotate around its center of mass is computed using an exact partial-wave series expression. Results, key conclusion and some perspectives: Attractive (pulling), repulsive (pushing) and neutral (zero) forces arise depending on the particle-boundary distance, the cylinder size parameter as well as the angle of incidence (in the polar plane) of the insonifying waves. Emphasis is also given on the emergence of an acoustic radiation torque (that vanishes for a rigid or non-viscous circular cylinder). Computations for the axial radiation torque efficiency anticipate the generation of positive radiation torque, its reversal, in addition to a zero efficiency, leading, respectively, to counter-clockwise, clockwise or lack of particle rotation as the angle of the incident waves deviates from normal incidence with respect to the boundary surface. The extension to the case of an elliptical/oval cylinder near a boundary is mentioned, and replies to some misleading and obtuse comments on the paper [F.G. Mitri, Phys. Fluids 28 (2016) 077104] are provided.

In this paper, the systems with solutions in the form of degenerate hypergeometric Humbert functions of two variables reduced to Bessel functions of two variables are established and studied. The connections between the Humbert and Bessel functions of two variables are revealed, their differential properties are investigated. The addition and multiplication theorems are proved. In future, these proven properties allow us to establish recurrent relations between degenerate hypergeometric functions of two variables, similarly to extend these properties to the case of many variables. The connection between type systems of Bessel and Whittaker is shown. Using the Frobenius - Latysheva method, the singularities of constructing normalregular solutions of the newly established Bessel - type system are studied.

Iterative procedure is implemented to derive rigorous solution to the problem of plane electromagnetic wave scattering by couple of perfect electromagnetic conducting (PEMC) elliptic cylinders due co and cross polarized scattered fields among cylinders. The translation addition theorem for Mathieu functions is enforced to compute the higher order scattered fields by single PEMC elliptic cylinder in terms of the other elliptic cylinder coordination system to impose the boundary conditions. The kth co and cross polarized scattered field coefficient expressions are extracted by iteration procedure without using matrix inversion.

Electronic states of a single electron in doubly eccentric cylindrical quantum wire are theoretically investigated in this paper. The motion of electron in quantum wire is free along axial direction in a cylindrical quantum wire and restricted in annular regions by three different parallel finite cylindrical barriers as soft wall confinement. The effective mass Schrödinger equation with effective mass boundary conditions is used to find energy eigenvalues and corresponding wavefunctions. Addition theorem for cylindrical Bessel functions is used to shift the origin for applying boundary conditions at different circular boundaries. Fourier expansion is applied after addition theorem to get wavefunctions in analytical form. A determinant equation is obtained as a result of applications of effective mass boundary conditions which roots gives energy of various electronic states. The lowest root gives ground state energy. The variation in ground state energy with eccentricity is obtained numerically and presented graphically. Electronic states in massive wall confinement and hard wall confinement is further obtained as limiting behavior of the states obtained in soft wall confinement. The knowledge of electronic states in such cylindrical hetrostructures semiconductor material can lead to improve the efficiency of many quantum devices. [ABSTRACT FROM AUTHOR]

The paper provides a way to model axially symmetric random fields defined over the two-dimensional unit sphere embedded in the three-dimensional Euclidean space. Specifically, our strategy is to integrate an isotropic random field on the sphere over longitudinal arcs with a given central angle. The resulting random field is shown to be axially symmetric and to have the arc central angle as a tuning parameter that allows for isotropy as well as for longitudinal independence as limit cases. We then consider multivariate longitudinally integrated random fields, having the same properties of axial symmetry and a tuning parameter (arc central angle) proper to each random field component. This construction allows for a unified framework for vector-valued random fields that can be geodesically isotropic, axially symmetric, or longitudinally independent. Additionally, all the components of the vector random field are allowed to be cross-correlated. We finally show how to simulate the proposed axially symmetric scalar and vector random fields through a computationally efficient algorithm that exactly reproduces the desired covariance structure and provides approximately Gaussian finite-dimensional distributions. [ABSTRACT FROM AUTHOR]

The paper tackles the problem of simulating isotropic vector-valued Gaussian random fields defined over the unit two-dimensional sphere embedded in the three-dimensional Euclidean space. Such random fields are used in different disciplines of the natural sciences to model observations located on the Earth or in the sky, or direction-dependent subsoil properties measured along borehole core samples. The simulation is obtained through a weighted sum of finitely many spherical harmonics with random degrees and orders, which allows accurately reproducing the desired multivariate covariance structure, a construction that can actually be generalized to the simulation of isotropic vector random fields on the d-dimensional sphere. The proposed algorithm is illustrated with the simulation of bivariate random fields whose covariances belong to the F , spectral Matérn and negative binomial classes of covariance functions on the two-dimensional sphere. [ABSTRACT FROM AUTHOR]

We introduce a new class of locally compact groups, namely the strongly compactly covered groups, which are the Hausdorff topological groups G such that every element of G is contained in a compact open normal subgroup of G. For continuous endomorphisms ϕ : G → G of these groups we compute the algebraic entropy and study its properties. Also an Addition Theorem is available under suitable conditions. [ABSTRACT FROM AUTHOR]

In this paper we strengthen the result of Fomin and Zelevinsky (2002) on the Laurent phenomenon for Somos-4 and Somos-5 sequences. [ABSTRACT FROM AUTHOR]

Three-dimensional time-harmonic acoustic multiple scattering problems are considered for a finite number of prolate and oblate spheroidal objects adjacent to flat surfaces. Wave propagation by spheroids is modelled by the method of separation of variables equipped with the addition theorems in the spheroidal coordinates. The effect of flat surfaces is accounted for by using the method of images; hence, the flat surfaces are of (semi-)infinite extent and perfectly reflecting: either rigid or pressure release. Wedge-shaped acoustic domains are constructed including half-space and right-angled corners with the wedge angle of |$\pi /n$| rad with positive integer |$n$|⁠. First, Euler angles are implemented to rotate image spheroids to realize the mirror reflection. Then, the 'image conditions' are developed to reduce the number of unknowns by expressing the unknown expansion coefficients of image-scattered fields in terms of real counterparts. Use of image conditions to 2D wedges, therefore, leads to the |$4n^2$| -fold reduction in the size of a matrix for direct solvers and |$2n$| -times faster computation than the approach without using them; for 3D wedges, the savings are |$16n^2$| -fold and |$4n$| -times, respectively. Multiple scattering models (MSMs) are also formulated for fluid, rigid and pressure-release spheroids under either plane- or spherical-wave incidence; novel addition theorems are also derived for spheroidal wavefunctions by using two rotations of spherical wavefunctions and a |$z$| -axis translation in-between, which is shown numerically more efficient than other addition theorems based on an arbitrary-direction translation and a single rotation. Finally, MSMs using image conditions are numerically validated by the boundary element method for a configuration populated with both prolate and oblate spheroids. [ABSTRACT FROM AUTHOR]

Abstract In recent years, several fast solvers for the solution of the Lippmann–Schwinger integral equation that mathematically models the scattering of time-harmonic acoustic waves by penetrable inhomogeneous obstacles, have been proposed. While many of these fast methodologies exhibit rapid convergence for smoothly varying scattering configurations, the rate for most of them reduce to either linear or quadratic when material properties are allowed to jump across the interface. A notable exception to this is a recently introduced Nyström scheme (Anand et al., 2016 [22]) that utilizes a specialized quadrature in the boundary region for a high-order treatment of the material interface. In this text, we present a solution framework that relies on the specialized boundary integrator to enhance the convergence rate of other fast, low order methodologies without adding to their computational complexity of O (N log ⁡ N) for an N -point discretization. In particular, to demonstrate the efficacy of the proposed framework, we explain its implementation to enhance the order to convergence of two schemes, one introduced by Duan and Rokhlin (2009) [13] that is based on a pre-corrected trapezoidal rule while the other by Bruno and Hyde (2004) [12] which relies on a suitable decomposition of the Green's function via Addition theorem. In addition to a detailed description of these methodologies, we also present a comparative performance study of the improved versions of these two and the Nyström solver in Anand et al. (2016) [22] through a wide range of numerical experiments. [ABSTRACT FROM AUTHOR]

A novel domain decomposition procedure is presented to analyze the electromagnetic wave multiple scattering among separate PEC objects. The method is based on the equivalence principle algorithm (EPA) that transfers the primary unknowns on the objects to the unknowns on the equivalence surfaces. Each equivalence surface is considered to be a spherical surface encompassing a PEC object. Inside the equivalence spheres, the analysis is performed based on the Rao–Wilton–Glisson (RWG) basis functions where the electromagnetic interactions outside them are considered by means of the addition theorem for spherical wave harmonics (SWHs). The translation procedures are presented to translate the RWG basis functions to SWHs and vice versa. The reduction of unknowns in the presented method versus the conventional EPA (which uses translation operators among equivalence surfaces) shows its efficiency. Some numerical examples are presented to evaluate the efficacy of the proposed approach. The acceptable consistency among the results of the proposed method and the conventional EPA as well as the direct method of moments validates the proposed method. [ABSTRACT FROM AUTHOR]

Let g be a linear combination with quasipolynomial coefficients of shifts of the Jacobi theta function and its derivatives in the argument. All entire functions f: ℂ → ℂ satisfying f(x+y)g(x−y) = α1(x)β1(y)+· · ·+αr(x)βr(y) for some r ∈ ℕ and αj, βj: ℂ → ℂ are described. [ABSTRACT FROM AUTHOR]

The purpose of the present study is the investigation of the hydrodynamic interactions induced by arrays of elliptical cylinders subjected to regular waves. The solution methodology employs linear potential theory and is based on pure analytic considerations. The interaction phenomena are approached using the Mathieu functions addition theorem which converts elliptical harmonics from one elliptic coordinate system to a remote elliptic coordinate system. The followed approach resembles the ‘direct’ solution methodology which is used for the analytical solution of the diffraction problem by arrays of circular cylinders. The ‘direct’ approach allows, among others, the construction of a linear matrix equation for the calculation of the expansion coefficients of the diffraction component(s) which accordingly is used to trace the wavenumber(s) under which trapped modes may be stimulated to induce wave trapping in the array, and reduction of the energy radiated to the far-field. The numerous computations which are performed, specifically verify that any array of elliptical cylinders, in accord with arrays of circular cylinders, is potentially a wave trapping configuration, which is connected mainly i) with sharp amplifications in all modes of hydrodynamic loading (both forces and moments) and ii) strong free-surface elevations in the liquid regions between the cylinders and on the cylinders' surfaces accordingly. [ABSTRACT FROM AUTHOR]

Abstract: Coding representation of metamaterials builds up a bridge between the physical world and the digital world, making it possible to manipulate electromagnetic (EM) waves by digital coding sequences and reach field‐programmable metamaterials. Here, the coding space is extended to complex domain and proposed complex digital codes to provide closer essence of EM‐wave propagation. Based on the analytic geometry and complex variable functions, an addition theorem on complex coding is established, which reveals inherent connections among digital codes with different bits and enables all higher‐bit digital codes to be represented by the 1‐bit complex codes. According to the complex coding and addition theorem, multifunctional metamaterials can be directly designed and realized without considering mutual coupling. When two different coding patterns with different functions are added together via the addition theorem in complex form, the combined coding pattern will directly generate the two functions simultaneously without any perturbations. A series of realistic coding metasurfaces is presented to demonstrate the powerful and flexible performance of the complex coding and addition theorem for independent controls of EM waves to reach multiple functions. Good agreements between numerical simulations and experimental results prove the feasibility of the proposed concept and theorem in practical applications. [ABSTRACT FROM AUTHOR]

Mutual coupling between two identical radiators is computed using vector spherical harmonics and reaction theorem for small element spacing cases. For closely spaced antennas, the minimum spheres that enclose the antennas overlap, and the vector spherical harmonic expansions do not converge on the surface that separates the two antennas. The proposed method mainly uses addition theorems to translate the vector spherical harmonic expansions to a different center. These operations on spherical waves enable to redefine the minimum spheres and convergence regions. The proposed method has been validated by analyzing the coupling between two half-wave dipoles in a side-by-side configuration. The results clearly show excellent agreement with that obtained from induced EMF method. The theory is also tested experimentally. A printed dipole antenna is designed, and its spherical harmonic coefficients are obtained by spherical near field measurement. Mutual impedance between two such dipoles is computed using the proposed method. The computed results for different element spacings in a side-by-side configuration are compared to those obtained from s -parameter measurements. [ABSTRACT FROM AUTHOR]