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Atomic systems with exponential-cosine-screened Coulomb (ECSC) or static screened Coulomb (SSC) potentials have drawn considerable attention recently due to the possible applications to atoms in plasma environments. In this work, we develop a computing scheme to deal with the electron–electron correlation terms with the screened Coulomb interactions instead of the conventional expansion method using the Gegenbauer's addition theorem. Based on this approach, we investigate the helium atom with the ECSC potentials. Bypassing the complex expansion functions for electron–electron interactions, the proposed approach simplifies the calculations greatly and provides an advantage on programming. The results are found to be in good agreement with the existing data. Bound-state energies, oscillator strengths, and multipole polarizabilities varying with the screening parameters are presented. Comparisons of the ECSC potentials are made with the SSC potentials. The influence of screening effect on the energies, oscillator strengths, and polarizabilities is discussed. © 2015 Wiley Periodicals, Inc.

The radial Slater-type orbitals (STO) can be simply obtained by repeated parametric differentiation of the Yukawa Potential with respect to α. A new compact two-range addition theorem (AdT) for the STO is herein derived by explicit parametric differentiation of the well-known Yukawa AdT. The resulting addition formula is combined with the single-range AdT for solid spherical harmonics to present a new AdT for three-dimensional spherical coordinate STOs. We advance the proposition that this formula is “canonical” in the same sense that the Laplace expansion of the Coulomb potential is considered canonical. We demonstrate how this procedure can be employed for all exponential-type orbitals. © 2012 Wiley Periodicals, Inc.

In this paper, we concentrate on the design of 1-resilient Boolean functions with desirable cryptographic properties. Firstly, we put forward a novel secondary construction to obtain 1-resilient functions. Next, we present the relationships between the properties of these constructed 1-resilient functions and that of the initial functions. Based on the construction and a class of bent functions on n variables, we can obtain a class of (n + 3)-variable 1-resilient non-separable cryptographic functions with a high algebraic immunity, whose nonlinearity is equal to the bent concatenation bound 2n + 2 − 2(n + 2)/2. Furthermore, we propose a set of 1-resilient non-separable functions on odd number of variables with an optimal algebraic degree, a high algebraic immunity, and a high nonlinearity. Copyright © 2012 John Wiley & Sons, Ltd.

In this study, we use the addition theorem and superposition technique to solve the scattering problem with multiple circular cylinders arising from point sound sources. Using the superposition technique, the problem can be decomposed into two individual parts. One is the free-space fundamental solution. The other is a typical boundary value problem (BVP) with specified boundary conditions derived from the addition theorem by translating the fundamental solution. Following the success of null-field boundary integral formulation to solve the typical BVP of the Helmholtz equation with Fourier densities, the second-part solution is easily obtained after collocating the observation point exactly on the real boundary and matching the boundary condition. The total solution is obtained by superimposing the two parts which are the fundamental solution and the semianalytical solution of the Helmholtz problem. An example was demonstrated to validate the present approach. The parameter study of size and spacing between cylinders are addressed. The results are well compared with the available theoretical solutions and experimental data. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2011

In this paper, the dual integral formulation is derived for the modified Helmholtz equation in the propagation of oblique incident wave passing a thin barrier (zero thickness) by employing the concept of fast multipole method (FMM) to accelerate the construction of an influence matrix. By adopting the addition theorem, the four kernels in the dual formulation are expanded into degenerate kernels that separate the field point and the source point. The source point matrices decomposed in the four influence matrices are similar to each other or only to some combinations. There are many zeros or the same influence coefficients in the field point matrices decomposed in the four influence matrices, which can avoid calculating the same terms repeatedly. The separable technique reduces the number of floating-point operations from O((N)2) to O(N loga(N)), where N is the number of elements and a is a small constant independent of N. Finally, the FMM is shown to reduce the CPU time and memory requirement, thus enabling us to apply boundary element method (BEM) to solve water scattering problems efficiently. Two-moment FMM formulation was found to be sufficient for convergence in the singular equation. The results are compared well with those of conventional BEM and analytical solutions and show the accuracy and efficiency of the FMM. Copyright © 2008 John Wiley & Sons, Ltd.

This paper considers a transient heat conduction problem for an infinite medium with multiple nonoverlapping spherical cavities. Suddenly applied, steady Dirichlet-, Neumannor Robin-type boundary conditions are assumed. The approach is based on the use of the general solution to the problem of a single cavity and superposition. Application of the Laplace transform and the so-called addition theorem results in a semi-analytical transformed solution for the case of multiple cavities. The solution in the time domain is obtained by performing a numerical inversion of the Laplace transform. A large-time asymptotic series for the temperature is obtained. The limiting case of infinitely large time results in the solution for the corresponding steady-state problem. Several numerical examples that demonstrate the accuracy and the efficiency of the method are presented. Copyright 2008 John Wiley & Sons, Ltd.

This study treats the interaction of a plane compressional sound wave with a cluster of two fluid-saturated porous elastic cylinders submerged in an unbounded acoustic medium. The novel features of Biot dynamic model for description of poroelastic material behaviour along with the appropriate wave field expansions, the pertinent boundary conditions, and the translational addition theorem for cylindrical wave functions are employed to develop a closed-form solution in the form of infinite series. The theoretical model was extended to consider the case of sound diffraction by a pair of absorbing cylinders with locally reacting or extended reaction boundaries. The analytical results are illustrated with numerical examples in which two identical air-saturated glass wool cylinders are insonified by a plane sound wave at end-on/broadside incidence. The most fundamental acoustic field quantity, namely, the scattering form function amplitude is evaluated and discussed for representative values of the parameters characterizing the system. The effects of incident wave frequency, angle of incidence, and proximity of the two cylinders are examined. Particular attention has been focused on multiple scattering interactions as well as the surface reaction modelling effects. Numerical comparison is made with the case of acoustic diffraction by a pair of absorbing cylinders with locally reacting or extended reaction boundaries. Limiting cases involving two rigid cylinders and also a pair of penetrable (fluid) cylinders are considered and good agreements with well-known solutions are established.

Acoustic radiation of an infinite cylindrical surface vibrating with an arbitrary, time-harmonic surface velocity distribution, while positioned within an acoustic quarterspace with rigid boundary, is analyzed in an exact fashion using the classical method of separation of variables. The formulation utilizes the appropriate wave-field expansions and the method of images along with the translational addition theorem for cylindrical wave functions to develop a closed-form solution in form of infinite series. The analytical results are illustrated with numerical examples in which the cylindrical source, vibrating in the pulsating (n = 0) and the transversely oscillating (n = 1) modes, is positioned near the rigid boundary of an air-filled quarterspace. Subsequently, the basic acoustic field quantities such as the modal acoustic radiation impedance load and the sound radiation intensity distribution are evaluated for representative values of the parameters characterizing the system.

Addition theorems can be constructed by doing three-dimensional Taylor expansions according to $f (\mathbf{r} + \mathbf{r}') = \exp (\mathbf{r}' \cdot \mathbf{\nabla}) f (\mathbf{r})$. Since, however, one is normally interested in addition theorems of irreducible spherical tensors, the application of the translation operator in its Cartesian form $\exp (x' \partial /\partial x) \exp (y' \partial /\partial y) \exp (z' \partial /\partial z)$ would lead to enormous technical problems. A better alternative consists in using a series expansion for the translation operator $\exp (\mathbf{r}' \cdot \mathbf{\nabla})$ involving powers of the Laplacian $\mathbf{\nabla}^2$ and spherical tensor gradient operators $\mathcal{Y}_{\ell}^{m} (\nabla)$, which are irreducible spherical tensors of ranks zero and $\ell$, respectively [F.D.\ Santos, Nucl. Phys. A {\bf 212}, 341 (1973)]. In this way, it is indeed possible to derive addition theorems by doing three-dimensional Taylor expansions [E.J. Weniger, Int. J. Quantum Chem. {\bf 76}, 280 (2000)]. The application of the translation operator in its spherical form is particularly simple in the case of $B$ functions and leads to an addition theorem with a comparatively compact structure. Since other exponentially decaying functions like Slater-type functions, bound-state hydrogenic eigenfunctions, and other functions based on generalized Laguerre polynomials can be expressed by simple finite sums of $B$ functions, the addition theorems for these functions can be written down immediately., Comment: 17 pages, LaTeX2e, 0 figures. Submitted to the Per-Olof L\"owdin Honorary Volume, International Journal of Quantum Chemistry

of ON too. The basis of the FMM is the application of the addition theorem to the free-space Green's function 1 . Coifman, Rokhlin, and Wandzura 2 , Song and Chew 3 , and Sertel and Volakis 10 have discussed how to choose the multipole expansion orders for achieving high accuracy in the FMM. The accuracy of the multipole expansion of the scalar Green's function directly affects the FMM acceleration schemes. This thus shows the motivation of this paper, that is, to address the accuracy of addition theorems for the free- space scalar Green's function and the convergence of the FMM algorithms.

This paper details the results of upgrading an effective numerical technique (derived for multiple-scattering simulations in photon correlation=cross-correlation with a plane-wave light source) for the modeling of multiple scattering in a laser beam. The off-axis shape coefficients of an arbitrary beam are computed starting from the set of known beam-shape coefficients for an on-axis location by using the addition theorem for the spherical vector wavefunctions of the first kind. The discussed technique is verified by comparison with a localized approximation for a focused Gaussian beam and with Barton’s spheres-arbitrary beam interaction theory. An additional advantage of the proposed technique (selftesting of the computation accuracy by comparison of the off-axis beam-shape coefficients evaluated from two different on-axis origins) is demonstrated.

The scattering of a TM-polarized plane wave by some bianisotropic eccentric cylinders is considered. Such composite structures include a one-layer bianisotropic eccentric cylindrical shell, a two-layer bianisotropic eccentric cylinder, and an impedance cylinder embedded in an eccentric bianisotropic cylinder, respectively. An algorithm based on the boundary value technique combined with the addition theorem for cylindrical functions is constructed, and is also suitable for the obliquely incident case. Numerical results are presented to show the different effects of various geometrical and constitutive parameters on the copolarized and cross-polarized bistatic echo widths of bianisotropic eccentric cylinders. © 1995 John Wiley & Sons, Inc.

A closed-form solution of two-dimensional scattering of plane SH waves by a cylindrical alluvial valley of circular-arc cross-section in a half-space is presented using the wave functions expansion. The solution is reduced to solving a set of infinite linear algebraic equations using the exterior region form of Graf's addition theorem. Numerical solutions are obtained by truncation of the infinite equations and their accuracies are demonstrated by convergence of the numerical results to the exact boundary condition with the increasing of the truncation order. The present solution is compared with the existing one presented by Todorovska and Lee for the same problem and their differences are analysed. Complicated effects of the depth-to-width ratio of the alluvial valley on surface ground motion are finally illustrated.

A new efficient algorithm based on the decomposition of strong and weak interactions among scatterers is proposed. The weak interactions, which account for the majority of the required CPU time and memory, are calculated using a canonical grid with a translation addition theorem. This facilitates the use of FFT and results in an N log N-type efficiency for CPU and O(N) for memory.

The one-range addition theorem [J. Fernandez Rico, R. Lopez, and G. Ramirez, Int. J. Quantum Chem. 34, 121 (1988)] for the Yukawa potential is rederived in a very simple manner using weakly convergent expansions [E. J. Weniger, J. Math. Phys. 26, 276 (1985)] of plane waves in the Fourier representation. It is discussed in which sense the addition theorem converges. The relation to Sturmian expansions of Coulomb Green functions is pointed out. © 1992 John Wiley & Sons, Inc.

An alternative derivation of the vector addition theorem is presented using the completeness of vector wave functions and integration by parts. The advantage of this derivation is that it leads directly to the simplified results of Bruning and Lo and of Stein. Moreover, the dichotomous results of the addition theorem when a spherical Hankel function is involved can be derived by contour integration.

A completely analytic method for evaluating three-center nuclear attraction integrals for STOS is presented. The method exploits a separation of the STO into an ‘evenly loaded’ solid harmonic and a OS STO. The harmonics are translated to the molecular center of mass in closed finite terms. The OS STO is translated using the Gegenbauer addition theorem; 1s STOS are translated using a single parametric differentiation of the OS formula. Explicit formulas for the integrals are presented for arbitrarily located atoms. A numerical example is given to illustrate the method.

The electrostatic potential in systems of spherical polyelectrolytes is determined by superposition of spherically symmetric solutions of the linearized Poisson-Boltzmann equation. The case of constant surface potential is investigated in detail. Higher-order approximations are given by use of a generalization of the addition theorem for modified Bessel functions.

Starting with the partial wave method generalized to non-spherically symmetric potentials by Demkov and Rudakov a purely algebraic scheme is proposed how to calculate the generalized phase shifts for a cluster of non-overlapping muffin-tin potentials. These cluster phase shifts and the corresponding partial wave amplitudes are determined by a system of homogeneous linear equations, into which the phase shifts of each muffin-tin potential and the structure enter separately, completely similar to the KKR method for the band structure of a crystal. By the proposed scheme bound states can also be calculated, the number of which is limited by the properties of cluster phase shifts. As a mathematical remedy an addition theorem about the spherical Neumann functions is derived. [Russian Text Ignored].

Publisher Summary This chapter reviews an effective lower bound on the diophantine approximation of algebraic functions by rational functions. In an article in the Proceedings in 1959, Professor Ellis Kolchin showed an analogue of Liouville's theorem, which dealt with the approximation of solutions of algebraic differential equations. The most concrete application of Kolchin's theorem is to the approximation by rational functions of formal power series solutions of a nontrivial algebraic differential equation. Kolchin's theorem says that there exists a natural number n and a real number c such that for every r, s ≠ 0: ord(y1 - r/s)