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We demonstrate that quantum Hamiltonian operator for a free transverse field within the framework of the second quantization reveals an alternative set of states satisfying the eigenstate functional equations. The construction is based upon extensions of the quadratic form of the transverse Laplace operator which are used as a source of spherical basis functions with singularity at the origin. This basis then naturally takes place of the one of plane or spherical waves in the process of Fourrier or spherical variable separation., Comment: 22 pages

We propose a method to generate an acoustical levitation trap at the same time as suppressing the sound in a multi-wavelength region of space. The method uses a spherical basis expansion of the sound field in the quiet zone, calculated by translating individual source expansions of elements in a transducer array. We show that it is possible to control the size of the quiet zone with the truncation order of the expansion, and explain the trade-off between field suppression in the quiet zone and stiffness loss of the levitation trap. Measurements of a generated sound field show the existence of a region of lower sound pressure. Simulations demonstrate a contrast up to 50 dB and sizes up to 60 mm for a 256 element array.

Using the solution of the problem of light scattering by a homogeneous spheroid, derived by separation of variables in spheroidal coordinates, we find the ‘spheroidal’ T -matrix. We suggest a transformation of this matrix into the ‘spherical’ T -matrix useful in applications. The procedure consists of two steps: first, transition from spheroidal to spherical basis, and then from spherical non-orthogonal to orthogonal basis.

Constrained spherical deconvolution (CSD) of diffusion‐weighted MRI (DW‐MRI) is a popular analysis method that extracts the full white matter (WM) fiber orientation density function (fODF) in the living human brain, noninvasively. It assumes that the DW‐MRI signal on the sphere can be represented as the spherical convolution of a single‐fiber response function (RF) and the fODF, and recovers the fODF through the inverse operation. CSD approaches typically require that the DW‐MRI data is sampled shell‐wise, and estimate the RF in a purely spherical manner using spherical basis functions, such as spherical harmonics (SH), disregarding any radial dependencies. This precludes analysis of data acquired with nonspherical sampling schemes, for example, Cartesian sampling. Additionally, nonspherical sampling can also arise due to technical issues, for example, gradient nonlinearities, resulting in a spatially dependent bias of the apparent tissue densities and connectivity information. Here, we adopt a compact model for the RFs that also describes their radial dependency. We demonstrate that the proposed model can accurately predict the tissue response for a wide range of b‐values. On shell‐wise data, our approach provides fODFs and tissue densities indistinguishable from those estimated using SH. On Cartesian data, fODF estimates and apparent tissue densities are on par with those obtained from shell‐wise data, significantly broadening the range of data sets that can be analyzed using CSD. In addition, gradient nonlinearities can be accounted for using the proposed model, resulting in much more accurate apparent tissue densities and connectivity metrics., Nonspherically sampled diffusion MRI data can arise either by choice of the sampling scheme (such as diffusion spectrum imaging) or due to gradient nonlinearities, precluding spherical deconvolution (SD) analysis. We adopt a compact response function model that accurately describes the b‐value dependency, enabling SD of nonspherically sampled data and the correction of gradient nonlinearities. With the proposed method, a broader range of data sets can be analyzed with SD approaches, as well as allowing more accurate downstream processing such as fiber tracking and connectomics.

Highly excited states in 156Gd, populated via the neutron pickup reaction 157Gd(3He,4He)156Gd, are investigated, and their spin–parity distribution P(Jπ,E) is examined. The cross section for one-neutron transfers to states above the neutron separation energy in 156Gd is calculated as coherent sum, using standard reaction codes that employ spherical basis states. Spectroscopic factors and form factors for the relevant states are obtained by expanding the deformed neutron wave functions in a spherical Sturmian basis. For the energy regime relevant to surrogate applications involving neutron absorption, 155Gd+n →156Gd⋆, the calculations show that the reaction 3He+157Gd →4He+156Gd⋆ induces a well-behaved formation probability P(Jπ,E) of approximately Gaussian shape. It is observed that the centroid and shape of the Gaussian distributions of the positive and negative parity states of the compound system can be significantly different from each other!

When considering light scattering from a sphere, the ratios between the expansion coefficients of the scattered and the incident field in a spherical basis are known as the Mie coefficients. Generally, Mie coefficients depend on many degrees of freedom, including the dimensions and electromagnetic properties of the spherical object. However, for fundamental research, it is important to have easy expressions for all possible values of Mie coefficients within the existing physical constraints and which depend on the least number of degrees of freedom. While such expressions are known for spheres made from non-absorbing materials, we present here, for the first time to our knowledge, corresponding expressions for spheres made from absorbing materials. To illustrate the usefulness of these expressions, we investigate the upper bound for the absorption cross section of a trimer made from electric dipolar spheres. Given the results, we have designed a dipolar ITO trimer that offers a maximal absorption cross section. Our approach is not limited to dipolar terms, but indeed, as demonstrated in the manuscript, can be applied to higher order terms as well. Using our model, one can scan the entire accessible parameter space of spheres for specific functionalities in systems made from spherical scatterers.

We describe a novel meshless Galerkin method for numerically solving semilinear parabolic equations on spheres. The new approximation method is based upon a discretization in space using spherical basis functions in a Galerkin approximation. As our spatial approximation spaces are built with spherical basis functions, they can be of arbitrary order and do not require the construction of an underlying mesh. We will establish convergence of the meshless method by adapting, to the sphere, a convergence result due to Thomee and Wahlbin. To do this requires proving new approximation results, including a novel inverse or Nikolskii inequality for spherical basis functions. We also discuss how the integrals in the Galerkin method can accurately and more efficiently be computed using a recently developed quadrature rule. These new quadrature formulas also apply to Galerkin approximations of elliptic partial differential equations on the sphere. Finally, we provide several numerical examples.

The differential scattering characteristics of an object having a uniaxial anisotropic inclusion, for incidence of an arbitrarily shaped beam, are theoretically investigated. The scattering problem is solved in a spherical basis by using the boundary conditions, method of moments technique and Schelkunoff’s equivalence theorem, which leads to a system of linear equations for the expansion coefficients of the scattered and internal fields when the description of the incident shaped beam is known. The theoretical formulation is reviewed, and the numerical problems are discussed. As examples, for a Gaussian beam, zero-order Bessel beam and Hertzian electric dipole radiation striking a spheroid with a uniaxial anisotropic spheroid inclusion and a circular cylinder with a uniaxial anisotropic circular cylinder inclusion, the normalized differential scattering cross sections are calculated, and the scattering properties are analyzed concisely.

The solution to the electromagnetic scattering of a sphere was published by Gustav Mie more than hundred years ago and is well-known as the Mie theory. In Mie theory, the ratio between the scattered and the incident field coefficients in the spherical basis is expressed in terms of the Mie coefficient. These Mie coefficients depend in an intricate manner on the spheres’ radius and the involved material properties of both the sphere and the ambient. To enable a systematic analysis of the accessible optical properties from spheres, these explicit expressions are not useful because they are too complicated. These Mie coefficients simply contain too many degrees of freedom. For fundamental research, it is of utmost importance to have easy expressions at hand that express all values of Mie coefficients that are accessible in general. The precise geometrical and material properties of a sphere that offer these coefficients can be identified in a secondary step. But if these accessible coefficients depend on the least number of degrees of freedom, they would allow for a systematic analysis of all observable effects using spherical scatterers. These desired simple expressions have been previously identified only for non-absorbing materials. However, a model for absorbing particles has never been reported. Here, while using the optical theorem we derive the generic equations to express any possible Mie coefficient of an absorbing sphere. Our model for absorbing particles can facilitate the study of absorbing systems such as perfect absorbers, optical torque calculations, cooling, thermal emitters etc. Using the proposed model, one can systematically analyse through all the possible space and search for specific functionality as it will be demonstrated at selected applications at the talk.

Semi-analytic expressions for the static limit of the T-matrix for electromagnetic scattering are derived for a circular torus, expressed in bases of both toroidal and spherical harmonics. The scattering problem for an arbitrary static excitation is solved using toroidal harmonics and the extended boundary condition method to obtain analytic expressions for auxiliary Q and P-matrices, from which the T-matrix is given by their division. By applying the basis transformations between toroidal and spherical harmonics, the quasi-static limit of the T-matrix block for electric multipole coupling is obtained. For the toroidal geometry there are two similar T-matrices on a spherical basis, for computing the scattered field both near the origin and in the far field. Static limits of the optical cross-sections are computed, and analytic expressions for the limit of a thin ring are derived., Same content as published version in JQSRT. Also contains content from a previous arxiv document "Relationships between spherical and toroidal harmonics"

Modeling a very broad spectrum of the Earth's gravity field needs observations from various measurement techniques with different spectral sensitivities. Typically, high-resolution regional gravity data are combined with low-resolution global observations. To exploit the gravitational information as optimally as possible, we set up a regional modeling approach using radial spherical basis functions, emphasizing the strengths of various data sets by the flexible combination of high- and middle-resolution terrestrial, airborne, shipborne, and altimetry measurements. The basis functions are defined and located in the region of interest in such a manner, which the highest measure of information of the input data is captured. Any functional of the Earth's gravity field can be derived, as, e.g., quasi-geoid heights or gravity anomalies. Here we present results of a study area in Northern Germany. A comprehensive cross validation to external observation data delivers standard deviations less than 5 cm. Differences to an existing regional quasi-geoid model count on average ±6 cm and proof the plausibility of our solution. The comparison with existing global models reaches higher standard deviations for the more sensitive gravity anomalies as for quasi-geoid heights, showing the additional value of our solution in the high frequency domain. Covering a broad frequency spectrum, our regional models can be used as basis for various applications, such as refinement of global models, national geoid determination, and detection of mass anomalies in the Earth's interior.

We have examined an analog to the extended boundary conditions method (EBCM) with the standard spherical basis, which is popular in light scattering theory, with respect to its applicability to the solution of an electrostatic problem that arises for multilayer scatterers the sizes of which are smaller compared to the wavelength of the incident radiation. It has been found that, in the case of two or more layers, to determine the polarizability and other optical characteristics of particles in the far-field zone, the parameters of the surfaces of layers should obey the condition max{σ1 (j)} < min{σ2 (j)}. In this case, appearing infinite systems of linear equations for expansion coefficients of unknown fields have a unique solution, which can be found by the reduction method. For nonspheroidal particles, this condition is related to the convergence radii of expansions of regular and irregular fields outside and inside of the particle, including its shells—R 1 (j) = σ1 (j) and R 2 (j) = σ2 (j). In other words, a spherical shell should exist in which expansions of all regular and irregular fields converge simultaneously. This condition is a natural generalization of the result for homogeneous particles, for which such a condition is imposed only on expansions of the “scattered” and internal fields—R 1 < R 2. For spheroidal multilayer particles, which should be singled out into a separate class, the EBCM applicability condition is written as max{σ1 (1), σ1 (2), …, σ1 (J−1), σ1 (J)} < min{σ2 (1), σ2 (2), …, σ2 (J−1)} and parameters σ2 (j) of the surfaces of shells are not related to corresponding convergence radii R 2 j of irregular fields. Numerical calculations for two-layer spheroids and pseudospheroids have confirmed completely theoretical inferences. Apart from the EBCM algorithm, an approximate formula has been proposed for the calculation of the polarizability of two-layer particles, in which the polarizability of a two-layer particle is interpreted as a linear combination of the polarizabilities of homogeneous particles that consist of the materials of the shell and core proportionally to their volumes. The range of applicability of this formula is wider than that for the EBCM, and the calculation error is smaller than 1%.

Several quaternion Fourier transforms have received considerable attention in the last years. In this paper, based on the symplectic decomposition of quaternions, we prove that the Hermite functions in the spherical basis are the eigenfunctions of the QFTs. The relationship with Radon transform, the real and complex Paley-Wiener theorem, as well as the sampling formula for the 2-sided QFT are established in an easy way. Moreover, a two parameter fractional Fourier transform is introduced based on a new direct sum decomposition of the L2 space. The explicit formula and bound of the kernel are obtained. Compared with the Clifford-Fourier transform defined using eigenfunctions, the main adventure of this transform is that the integral kernel has a uniform bound for all dimensions.

The Rayleigh approximation and solution of the electrostatic problem are considered for pseudospheroids that are obtained as a result of inversion of usual spheroidal particles: $$r\left( \theta \right) = ab/{r_{{\text{spheroid}}}}\left( \theta \right) = a\sqrt {1 - { \in ^2}{{\cos }^2}\theta } $$ , where e2 = 1 - (b/a)2. The approach is based on using the extended boundary condition method (EBCM) for the calculation of polarizability. The approach is studied analytically. The study has been shown that, in the near zone with respect to the particle, expansions of potentials of the “scattered” and internal fields in terms of the spherical basis converge up to the surface only under the conditions $$a/b \prec \sqrt 5 $$ and $$a/b \prec \sqrt 2 $$ (the external and internal Rayleigh hypotheses, respectively). In the far zone, the ЕВСМ can be justifiably applied for the ratio of pseudospheroid parameters $$a/b \prec \sqrt 2 + 1$$ . In this case, there exists a nonempty intersection of analytical continuations of expansions in terms of the spherical basis of the “scattered” and internal fields. The radii of convergence R 1, R 2 of these expansions have been found; they are determined by the presence of singular points. The relation between singular points for pseudospheroids and corresponding spheroids is considered. Numerical analysis of the problem under consideration has completely verified results of the analytical study, in particular, the presence of corresponding singular points. To obtain the necessary accuracy in calculations of elements of linear algebraic systems, the corresponding integrals are transformed into series containing gamma functions. Numerical calculations involving the EBCM were successfully verified using the condition of symmetry of the T matrix. For the applicability area of the ЕВСМ, calculation results for the polarizability of pseudospheroids are discussed according to the exact algorithm and within the approximation of the homogeneous internal field.

The transition (T) matrix of Waterman has been very useful for computing fast, accurate acoustic scattering predictions for axisymmetric elastic objects, but this technique is usually limited to fairly smooth objects that are not too aspherical unless complex basis functions or stabilization schemes are used. To ease this limitation, a spherical-basis formulation adapted from approaches proposed recently by Waterman [J. Acoust. Soc. Am. 125(1), 42-51 (2009)] and Doicu, Eremin, and Wriedt [Acoustic and Electromagnetic Scattering Analysis Using Discrete Sources (Academic, London, 2000)] is suggested. This is implemented by simply transforming the high-order outgoing spherical basis functions within standard T-matrix formulations to low-order functions distributed along the object's symmetry axis. A free-field T matrix is produced in a nonstandard form, but computations with it become much more stable for elongated aspherical elastic shapes. Some advantages of this approach over the approaches of Waterman and Doicu, Eremin, and Wriedt are noted, and sample calculations for a 10:1 Al prolate spheroid and a 10:1 Al superspheroid of order 10 are given to demonstrate the enhanced stability.

This paper presents a fast multipole boundary element method (FMBEM) for the 3-D elastodynamic boundary integral equation in the `low frequency' regime. New compact recursion relations for the second-order Cartesian partial derivatives of the spherical basis functions are derived for the expansion of the elastodynamic fundamental solutions. Numerical solution is achieved via a novel combination of a nested outer---inner generalized minimum residual (GMRES) solver and a sparse approximate inverse preconditioner. Additionally translation stencils are newly applied to the elastodynamic FMBEM and an implementation of the 8, 4 and 2-box stencils is presented, which is shown to reduce the number of translations per octree level by up to $$60\,\%$$60%. This combination of strategies converges 2---2.5 times faster than the standard GMRES solution of the FMBEM. Numerical examples demonstrate the algorithmic and memory complexities of the model, which are shown to be in good agreement with the theoretical predictions.

Summary The method of regularized Stokeslets (MRS) is a numerical approach using regularized fundamental solutions to compute the flow due to an object in a viscous fluid where inertial effects can be neglected. The elastic object is represented as a Lagrangian structure, exerting point forces on the fluid. The forces on the structure are often determined by a bending or tension model, previously calculated using finite difference approximations. In this paper, we study spherical basis function (SBF), radial basis function (RBF), and Lagrange–Chebyshev parametric models to represent and calculate forces on elastic structures that can be represented by an open curve, motivated by the study of cilia and flagella. The evaluation error for static open curves for the different interpolants, as well as errors for calculating normals and second derivatives using different types of clustered parametric nodes, is given for the case of an open planar curve. We determine that SBF and RBF interpolants built on clustered nodes are competitive with Lagrange–Chebyshev interpolants for modeling twice-differentiable open planar curves. We propose using SBF and RBF parametric models within the MRS for evaluating and updating the elastic structure. Results for open and closed elastic structures immersed in a 2D fluid are presented, showing the efficacy of the RBF–Stokeslets method. Copyright © 2015 John Wiley & Sons, Ltd.

The extended boundary condition method (EBCM) in solving the electrostatic problem for axisymmetric Chebyshev particles the surface of which is described by the equation r(θ) = a(1 + ɛcosnθ) is studied. The main attention is paid to the case n = 1. The problem is reduced to solving infinite systems of linear algebraic equations (ISLAEs) for expansion coefficients of internal and “scattered” fields in terms of a spherical basis where matrix elements of the fields are integrals of products of Legendre functions and power functions. Radii of convergence R2 and R1 of these expansions, respectively, have been found analytically. For the considered particles, depending on perturbation parameter ɛ, conditions of the applicability of the EBCM have been obtained, i.e., conditions of correct construction of the T matrix (R1 maxr(θ)) Rayleigh hypotheses under which expansions of the “scattered” and internal fields in terms of the spherical basis converge up to the surface of the particle beyond and inside it, respectively. In the particular case n = 1, numerical calculations have been performed; in this process, integral ISLAE elements have been represented as finite sums the summands of which depend on gamma functions. Calculations of matrix elements by explicit formulas have made it possible to considerably increase the dimension of the solved reduced ISLAE with preservation of the necessary accuracy. Analysis of results of numerical calculations verified their agreement with theoretical conclusions.

Author(s): Macchiavelli, AO; Crawford, HL; Campbell, CM; Clark, RM; Cromaz, M; Fallon, P; Jones, MD; Lee, IY; Salathe, M | Abstract: Spectroscopic factors in Be10, Be11, and Be12, extracted from (d,p), one-neutron knockout, and (p,d) reactions, are interpreted within the rotational model. Assuming that the ground state and first excited state of Be11 can be associated with the 12[220] and 12[101] Nilsson levels, the strong-coupling limit gives simple expressions that relate the amplitudes of these wave functions (in the spherical basis) with the measured cross sections and derived spectroscopic factors. We obtain good agreement with both the measured magnetic moment of the ground state in Be11 and the reaction data.

Circular polarization of the cosmic microwave background (CMB) arises in the standard cosmological model from Faraday conversion of the linear polarization generated at the surface of last scatter by various sources of birefringence along the line of sight. If the sources of birefringence are generated at linear order in primordial density perturbations the principal axes of the index-of-refraction tensor are determined by gradients of the primordial density field. Since linear polarization at the surface of last scatter is generated at linear order in density perturbations, the circular polarization thus arises at second order in primordial perturbations. Here, we re-visit the calculation of the circular polarization using the total-angular-momentum formalism, which allows for some simplifications in the calculation of the angular power spectrum of the circular polarization---especially for the dominant photon-photon scattering contribution---and also provides some new intuition., Comment: 6 pages

Introduction. In the theoretical description of the behavior of a hydrogen-like atom in external electric and magnetic fields, it is convenient to use Coulomb spheroidal wave functions. Generally, they are defined in two limiting cases of large and small distances R between the foci of the spheroidal coordinate system. The Coulomb spheroidal wave functions are presented in the form of a linear combination of the Coulomb parabolic functions in the first case and the spherical functions in the second case. In order to find the expansions, which connect these bases, the standard perturbation theory as well as the additional integrals of motion were used.Purpose. Determine the spheroidal corrections to the spherical and parabolic bases of the hydrogen atom at large and small intercenter distances R.Results. The asymptotical expressions for the eigenvalues and eigenfunctions (the Coulomb spheroidal functions) of the hydrogen atom system in the form of series in R for small (R<>1) internuclear distances were obtained. For this purpose, the additional integrals of motion and the standard Raeleigh-Schrödinger perturbation theory scheme within the terms of 3rd order were used. It is shown that in each order of perturbation theory the corrections to the Coulomb spheroidal functions are expressed in a finite number of the basic functions of the corresponding representation.Conclusion. In this paper, a brief analysis of the fundamental bases of the hydrogen atom, which are the eigenfunctions of its Hamiltonian and of the one of the generators of the hidden symmetry group SO (4) is carried out. The expansion of the one of the fundamental bases with respect to another one was analyzed in terms of additional integrals of motion. The information about additional integrals of motion allowed to calculate the spheroidal corrections to the spherical and parabolic bases of the hydrogen atom at small and large intercenter distances R using a purely algebraic scheme of the Rayleigh-Schrödinger perturbation theory., Введены сферический, параболический и сфероидальный базисы атома водорода, а также приведены дополнительные интегралы движения. Вычислены сфероидальные поправки к сферическому и параболическому базисам атома водорода методами теории возмущений при больших и малых расстояниях R между фокусами сфероидальной системы координат. Показано, что в каждом порядке теории возмущений поправки к кулоновским сфероидальным волновым функциям выражается через конечное число базисных функций представления., Введено сферичний, параболічний та сфероїдальний базиси атома водню, а також приведено додаткові інтеграли руху. Обчислено сфероїдальні поправки до сферичного і параболічного базисів атома водню методами теорії збурень при великих та малих відстанях R між фокусами сфероїдальної системи координат. Показано, що в кожному порядку теорії збурень поправки до кулонівських сфероїдальних хвильових функцій виражаються через скінчене число базисних функцій зображення.

Any acoustic sensor disturbs the spatial acoustic field to certain extent, and a recorded field is different from a field that would have existed if a sensor were absent. Recovery of the original (incident) field is a fundamental task in spatial audio. For some sensor geometries, the disturbance of the field by the sensor can be characterized analytically and its influence can be undone; however, for arbitrary-shaped sensor numerical methods have to be employed. In the current work, the sensor influence on the field is characterized using numerical (specifically, boundary-element) methods, and a framework to recover the incident field, either in the plane-wave or in the spherical wave function basis, is developed. Field recovery in terms of the spherical basis allows the generation of a higher-order Ambisonics representation of the spatial audio scene. Experimental results using a complex-shaped scatterer are presented.

Most spatial audio reproduction systems have the constraint that all loudspeakers must be equidistant from the listener, a property which is difficult to achieve in real rooms. In traditional Ambisonics this arises because the spherical harmonic functions, which are used to encode the spatial sound-field, are orthonormal over a sphere and because loudspeaker proximity is not fully addressed. Recently, significant progress to lift this restriction has been made through the theory of sound field synthesis, which formalizes various spatial audio systems in a mathematical framework based on the single layer potential. This approach has shown many benefits but the theory, which treats audio rendering as a sound-soft scattering problem, can appear one step removed from the physical reality and also possesses frequencies where the solution is non-unique. In the time-domain Boundary Element Method approaches to address such non-uniqueness amount to statements which test the flow of acoustic energy rather than considering pressure alone. This paper applies that notion to spatial audio rendering by re-examining the Kirchhoff-Helmholtz integral equation as a wave-matching metric, and suggests a physical interpretation of its kernel in terms of common acoustic power flux density between waves. It is shown that the spherical basis functions (spherical harmonics multiplied by spherical Bessel or Hankel functions) are orthogonal over any arbitrary surface with respect to this metric. Finally other applications are discussed, including design of high-order microphone arrays and the coupling of virtual acoustic models to auralization hardware.

High resolution transformations between regular geophysical data and harmonic model coefficients can be most efficiently computed by Fast Fourier Transform (FFT). However, a prerequisite is that the data grids are given in the appropriate geometrical domain. For example, if the data are situated on the ellipsoid at equi-angular reduced latitudes, spherical harmonic analysis can be employed and the coefficients subsequently converted by Jekeli’s transformation. This results in the spherical harmonic spectrum in the domain of geocentric latitudes. However, the data are most likely given at geodetic (ellipsoidal) latitudes which means that the FFT base needs to be shifted by latitude dependent phase lags in order to obtain the correct spherical harmonic spectrum. This requires appropriate sample rate conversion about the shifted latitudes by means of Fourier summation and cannot be treated efficiently by an FFT algorithm. In this article another solution is discussed instead. Since the variable heights between the spherical and ellipsoidal surfaces can be accurately approximated by a series of Tschebyshev polynomials, they can be convolved into the spherical basis. It will be shown how this new type of transformation to and from the ellipsoid in combination with Jekeli’s conversion of the spectra between the two surfaces allows eventually the sample rate conversion to shifted latitudes. This avoids the inexpedient Fourier summation mentioned previously. In this paper three applications for FFT in the domain of spherical and ellipsoidal surfaces, and using geocentric, reduced and geodetic latitudes are discussed. The Earth gravitational model EGM2008 of 5 arcminutes resolution has been used to demonstrate numerical results and computational advantages.

Deformed single particle energies obtained by averaging a particle–core Hamiltonian with a projected spherical basis depend on a deformation parameter and an arbitrary constant defining the canonical transformation relating the collective quadrupole coordinates and momenta with the boson operators. When the mentioned basis describes the single particle motion of either protons or neutrons the parameters involved are isospin dependent. An algorithm for fixing these parameters is formulated and then applied for 194 isotopes covering a good part of the nuclide chart. Relation with the Nilsson deformed basis is pointed out in terms of deformation dependence of the corresponding single particle energies as well as of the nucleon densities and their symmetries. The proposed projected spherical basis provides an efficient tool for the description of spherical and deformed nuclei in a unified fashion.

Researchers have found functional-based methods to improve data storage, transfer, and interpretation. These functional expansions have been developed and utilized across a range of geometries, from cartesian to cylindrical to the surfaces of spheres. However, a volumetric spherical basis set had not yet been computationally implemented. This paper presents such a basis set and demonstrates its numerical features and benefits.

Computing the matrix elements of the linear operator, which transforms the spherical basis of SO(3, 1)-representation space into the hyperbolic basis, we present an integral formula involving the product of two Legendre functions of the first kind expressed in terms of 4F3-hypergeometric function. Also, using the general Mehler–Fock transform, we obtain another integral formula for the Legendre function of the first kind. A relevant connection of one of the results presented here with a known integral formula is also pointed out.

The applicability of an analog of the extended boundary condition method, which is popular in light-scattering theory, is studied in combination with the standard spherical basis for the solution of an electrostatic problem appearing for spheroidal layered scatterers the sizes of which are small as compared to the incident radiation wavelength. In the case of two or more layers, polarizability and other optical characteristics of particles in the far zone are shown to be undeterminable if the condition under which the appearing systems of linear equations for expansion coefficients of unknown fields are Fredholm systems solvable by the reduction method is broken. For two-layer spheroids with confocal boundaries, this condition is transformed into a simple restriction on the ratio of particle semiaxes a/b< \(\sqrt 2 \) + 1. In the case of homogeneous particles, the solvability condition is that the radius of convergence of the internal-field expansion must exceed that of the expansion of an analog of the scattering field. Since homogeneous spheroids (ellipsoids) are unique particles inside which the electrostatic field is homogeneous, it is shown that the solution can be always found in this case. The obtained results make it possible to match in principle the results of theoretical and numerical determinations of the domain of applicability for the extended boundary condition method with a spherical basis for spheroidal scatterers.

This paper discusses local uniform error estimates for spherical basis functions (SBFs) interpolation, where error bounds for target functions are restricted on spherical cap. The discussion is first carried out in the native space associated with the smooth SBFs, which is generated by a strictly positive definite zonal kernel. Then, the smooth SBFs are embedded in a larger space that is generated by a less smooth kernel, and for the target functions outside the original native space, the local uniform error estimates are established. Finally, some numerical experiments are given to illustrate the theoretical results. Copyright © 2013 John Wiley & Sons, Ltd.

Using the spherical basis of the spin-ν operator, together with an appropriate normalized complex (2ν +1)-spinor on S 3 we obtain spin-ν representation of the U(1) Hopf fibration S 3 → S 2 as well as its associated fuzzy version. Also, to realize the first Hopf map via the spherical basis of the spin-1 operator with even winding numbers, we present an appropriate normalized complex three-spinor. We put the winding numbers in one-to-one correspondence with the monopole charges corresponding to different associated complex vector bundles.

The low-lying high-spin yrast band structure of neutron-rich 113 , 118 – 121 Ag has been established for the first time using prompt γ-ray spectroscopy of isotopically identified fission fragments produced in the 9Be(238U, fγ) fusion- and transfer-induced fission processes. The newly obtained level energies follow the systematics of the neighboring isotopes. The sequences of levels exhibit an energy inheritance from states in the corresponding Cd core. A striking constancy of a large signature splitting in odd-A Ag throughout the long chain of isotopes with 50 N 82 and a signature inversion in even-A Ag isotopes, which are indications of triaxiality, were evidenced. These observed features were reproduced by large-scale shell-model calculations with a spherical basis for the first time in the Ag isotopic chain, revealing microscopically their complex nature with severely broken seniority ordering. The essential features of the observed signature splitting were further examined in the light of simplified, two-orbital shell-model calculations including only two intruder orbitals π g 9 / 2 and ν h 11 / 2 from two consecutive shells above Z = 50 and N = 82 for protons and neutrons respectively, resulting in the π g 9 / 2 − 3 × ν h 11 / 2 m configurations. The newly established bands were understood as fairly pure, built mainly on unique-parity intruder configurations and coupled to the basic states of the Cd core.

We developed a deformed quasi-particle random phase approximation (DQRPA) to describe the Gamow–Teller (GT) transitions on even–even neutron-rich nuclei. To describe deformed nuclei, we exploited the deformed axially symmetric Woods–Saxon potential, the deformed BCS, and the deformed QRPA with realistic two-body interaction calculated by Brueckner G-matrix based on Bonn CD potential. The deformed single particle states are expanded in terms of the spherical harmonic oscillator basis in order to take the realistic G-matrix stored in the spherical basis. We calculated GT strength distributions, B(GT), of two nuclei 12,14Be for many different deformation parameter β 2 values as a function of the excitation energy E ex w.r.t. the ground state of a parent nucleus. Our results for 12Be predict to prefer a prolate shape and B(GT) results of 14Be turn out to be independent of the β 2 values.

We describe a novel discretisation method for numerically solving (systems of) semilinear parabolic equations on Euclidean spheres. The new approximation method is based upon a discretisation in space using spherical basis functions and can be of arbitrary order. This, together with the fact that the solutions of semilinear parabolic problems are known to be infinitely smooth, at least locally in time, allows us to prove stability and convergence of the discretisation in a straight-forward way.

Energy evaluation using fast Fourier transforms enables sampling billions of putative complex structures and hence revolutionized rigid protein-protein docking. However, in current methods efficient acceleration is achieved only in either the translational or the rotational subspace. Developing an efficient and accurate docking method that expands FFT based sampling to 5 rotational coordinates is an extensively studied but still unsolved problem. The algorithm presented here retains the accuracy of earlier methods but yields at least tenfold speedup. The improvement is due to two innovations. First, the search space is treated as the product manifold $\mathbf{SO(3)x(SO(3)\setminus S^1)}$, where $\mathbf{SO(3)}$ is the rotation group representing the space of the rotating ligand, and $\mathbf{(SO(3)\setminus S^1)}$ is the space spanned by the two Euler angles that define the orientation of the vector from the center of the fixed receptor toward the center of the ligand. This representation enables the use of efficient FFT methods developed for $\mathbf{SO(3)}$. Second, we select the centers of highly populated clusters of docked structures, rather than the lowest energy conformations, as predictions of the complex, and hence there is no need for very high accuracy in energy evaluation. Therefore it is sufficient to use a limited number of spherical basis functions in the Fourier space, which increases the efficiency of sampling while retaining the accuracy of docking results. A major advantage of the method is that, in contrast to classical approaches, increasing the number of correlation function terms is computationally inexpensive, which enables using complex energy functions for scoring.

Probe Correction (PC) in Spherical Near Field (SNF) measurements is typically performed involving the so called first order PC algorithm [1]–[3] which assumes probes with limited |μ|=1 spherical wave spectrum. Lot of effort is typically put in the developments of probes with such requirement [4]–[5] but their design is in many case very challenging especially for wide-band applications. In order to have less restrictions in the selection of the probe, different full probe correction techniques have been recently proposed [6]–[9]. In this paper the accuracy and limitation of the first and higher order PC algorithms are investigated considering the SNF measurement of a wideband horn measured with a first order probe and with a standard gain antenna with higher modal content. The full PC algorithm here considered is based on the modification of the spherical basis functions that are properly elaborated taking into account the effect of the probe.

We present a simple approach to the voxelwise classification of brain tissue acquired with diffusion weighted MRI (DWI). The approach leverages the power of spherical harmonics to summarise the diffusion information, sampled at many points over a sphere, using only a handful of coefficients. We use simple features that are invariant to the rotation of the highly orientational diffusion data. This provides a way to directly classify voxels whose diffusion characteristics are similar yet whose primary diffusion orientations differ. Subsequent application of machine-learning to the spherical harmonic coefficients therefore may permit classification of DWI voxels according to their inferred underlying fibre properties, whilst ignoring the specifics of orientation. After smoothing apparent diffusion coefficients volumes, we apply a spherical harmonic transform, which models the multi-directional diffusion data as a collection of spherical basis functions. We use the derived coefficients as voxelwise feature vectors for classification. Using a simple Gaussian mixture model, we examined the classification performance for a range of sub-classes (3-20). The results were compared against existing alternatives for tissue classification e.g. fractional anisotropy (FA) or the standard model used by Camino.1 The approach was implemented on both two publicly-available datasets: an ex-vivo pig brain and in-vivo human brain from the Human Connectome Project (HCP). We have demonstrated how a robust classification of DWI data can be performed without the need for a model reconstruction step. This avoids the potential confounds and uncertainty that such models may impose, and has the benefit of being computable directly from the DWI volumes. As such, the method could prove useful in subsequent pre-processing stages, such as model fitting, where it could inform about individual voxel complexities and improve model parameter choice.

Some mathematical models in geophysics and graphic processing need to compute integrals with scattered data on the sphere. Thus cubature formula plays an important role in computing these spherical integrals. This paper is devoted to establishing an exact positive cubature formula for spherical basis function networks. The authors give an existence proof of the exact positive cubature formula for spherical basis function networks, and prove that the cubature points needed in the cubature formula are not larger than the number of the scattered data.

The separation of variables method (SVM), which uses a spheroidal basis, is proposed. According to this method, fields are presented in the form of expansion in terms of spheroidal functions. The previously conducted analysis of various methods using a spherical basis showed that the SVM is applicable in a broader area for numerical calculations, while the proposed approach using a spheroidal basis yields reliable results in the case of spheroids with a high degree of asphericity where other methods and approaches cannot be used. Importantly, the method includes an SVM that uses a spherical basis as the limiting case. Thus, the proposed method has all chances of being highly efficient for calculation of optical characteristics of various nonspherical particles in a wide range of parameters of the formulated problem.

Several relevant properties of the Na clusters were studied by using a projected spherical single particle states.The proposed model is able to describe in an unified fashion the spherical and deformed clusters. Photoabsorbtion cross section is realistically explained within an RPA approach and a Shiff dipole moment as a transition operator, 11 pages, 7 figures

The separation of variables method for coordinate system, the extended boundary condition method, and the point-matching method that are used to solve the problem of light scattering by nonspherical particles are considered from a unified viewpoint. It is shown that, if the mathematical correctness condition (the Rayleigh hypothesis) holds, these methods are interrelated and are equivalent. The applicability ranges of the methods in the near- and far-field zones are analyzed, discussed, and compared on both analytical (based on analytical investigations) and practical (based on numerical calculations) grounds.

The purpose of this paper is to establish L p error estimates, a Bernstein inequality, and inverse theorems for approximation by a space comprising spherical basis functions located at scattered sites on the unit n-sphere. In particular, the Bernstein inequality estimates L p Bessel-potential Sobolev norms of functions in this space in terms of the minimal separation and the L p norm of the function itself. An important step in its proof involves measuring the L p stability of functions in the approximating space in terms of the l p norm of the coefficients involved. As an application of the Bernstein inequality, we derive inverse theorems for SBF approximation in the L P norm. Finally, we give a new characterization of Besov spaces on the n-sphere in terms of spaces of SBFs.

The main purpose of the present paper is to employ spherical basis functions (SBFs) to study uniform distribution of points on spheres. We extend Weyl's criterion for uniform distribution of points on spheres to include a characterization in terms of an SBF. We show that every set of minimal energy points associated with an SBF is uniformly distributed on the spheres. We give an error estimate for numerical integration based on the minimal energy points. We also estimate the separation of the minimal energy points.

A comparative analysis of the widely known methods for solving the problem of light scattering by nonspherical particles—of the method of separation of variables (MSV), of the extended boundary condition method (EBCM), and the point-matching method (PMM), which use the spherical wave functions as a basis for the expansions of the fields—is carried out. In the scientific literature, these methods have been analyzed independently of one another in spite of their evident similarity: The same expansion coefficients are determined from similar set of equations and all optical characteristics are calculated with the same formulas. The ranges of applicability of the methods for dielectric spheroids and Chebyshev particles are studied in the same manner. It was found that, when considering the far-field zone, theoretical conditions of mathematical correctness of the EBCM and the MSV, apparently, differ fundamentally, although, as was shown, the methods themselves are extremely closely related. The performed numerical calculations suggest that the EBCM is preferable for spheroids, the MSV is preferable for Chebyshev particles, and the PMM, which is the most time-consuming method, gives satisfactory results in many cases when two other methods are inapplicable. Since the methods supplement one another well and their programs differ only in several tens of operators, we propose combining these methods within the framework of one universal program.

In this paper, we propose a natural way to extend a bivariate Powell-Sabin (PS) B-spline basis on a planar polygonal domain to a PS B-spline basis defined on a subset of the unit sphere in R 3 . The spherical basis inherits many properties of the bivariate basis such as local support, the partition of unity property and stability. This allows us to construct a C1 continuous hierarchical basis on the sphere that is suitable for preconditioning fourth-order elliptic problems on the sphere. We show that the stiffness matrix relative to this hierarchical basis has a logarithmically growing condition number, which is a suboptimal result compared to standard multigrid methods. Nevertheless, this is a huge improvement over solving the discretized system without preconditioning, and its extreme simplicity contributes to its attractiveness. Furthermore, we briefly describe a way to stabilize the hierarchical basis with the aid of the lifting scheme. This yields a wavelet basis on the sphere for which we find a uniformly well-conditioned and (quasi-) sparse stiffness matrix.

The purpose of this paper is to get error estimates for spherical basis function (SBF) interpolation and approximation for target functions in Sobolev spaces less smooth than the SBFs, and to show that the rates achieved are, in a sense, best possible. In addition, we establish a Bernstein-type theorem, where the smallest separation between data sites plays the role of a Nyquist frequency. We then use these Berstein-type estimates to derive inverse estimates for interpolation via SBFs.

In this paper we investigate the approximation of a class of parabolic partial differential equations on the unit spheres S n⊂Rn+1 using spherical basis functions. Error estimates in the Sobolev norm are derived.

A recently proposed computationally tractable version of the Bohr collective model is developed to the extent that we are now justified in describing it as an algebraic collective model. The model has an SU ( 1 , 1 ) × SO ( 5 ) algebraic structure and a continuous set of exactly solvable limits. Moreover, it provides bases for mixed symmetry collective model calculations. However, unlike the standard realization of SU ( 1 , 1 ) , used for computing beta wave functions and their matrix elements in a spherical basis, the algebraic collective model makes use of an SU ( 1 , 1 ) algebra that generates wave functions appropriate for deformed nuclei with intrinsic quadrupole moments ranging from zero to any large value. A previous paper focused on the SO ( 5 ) wave functions, as SO ( 5 ) (hyper-)spherical harmonics, and computation of their matrix elements. This paper gives analytical expressions for the beta matrix elements needed in applications of the model and illustrative results to show the remarkable gain in efficiency that is achieved by using such a basis in collective model calculations for deformed nuclei.