Multi-vector Spherical Monogenics, Spherical Means and Distributions in Clifford Analysis

New higher-dimensional distributions have been introduced in the framework of Clifford analysis in previous papers by Brackx, Delanghe and Sommen. Those distributions were defined using spherical co-ordinates, the ＂finite part＂ distribution Fp x＋^μ on the real line and the generalized spherical means involving vector-valued spherical monogenics. In this paper, we make a second generalization, leading to new families of distributions, based on the generalized spherical means involving a multivector-valued spherical monogenic. At the same time, as a result of our attempt at keeping the paper self-contained, it offers an overview of the results found so far.     

Optimization of the parameters of the ideal 180° spherical analyzer

Resolution and intensity behaviour of the spherical analyzer are mainly determined by the entrance parameters (aperture angle 2α _m , relative width of the entrance slita/r ₀) and by the relative width of the exit slitb/r ₀. The mutual relation of these parameteres can be so optimized that independent of the required resolution the quantities étendue or luminosity of the spherical analyzer attain a maximum.     

Fast algorithms for spherical harmonic expansions,III

We accelerate the computation of spherical harmonic transforms, using what is known as the butterfly scheme. This provides a convenient alternative to the approach taken in the second paper from this series on “Fast algorithms for spherical harmonic expansions”. The requisite precomputations become manageable when organized as a “depth-first traversal” of the program’s control-flow graph, rather than as the perhaps more natural “breadth-first traversal” that processes one-by-one each level of the multilevel procedure. We illustrate the results via several numerical examples.     

A comparison of high-order time integrators for thermal convection in rotating spherical shells

A numerical study of several time integration methods for solving the three-dimensional Boussinesq thermal convection equations in rotating spherical shells is presented. Implicit and semi-implicit time integration techniques based on backward differentiation and extrapolation formulae are considered. The use of Krylov techniques allows the implicit treatment of the Coriolis term with low storage requirements. The codes are validated with a known benchmark, and their efficiency is studied. The results show that the use of high-order methods, especially those with time step and order control, increase the efficiency of the time integration, and allows to obtain more accurate solutions.     

Control over parameters of photonic nanojets of dielectric microspheres

We report on the results of theoretical investigation of the parameters of photonic nanojets formed near the shadow surface of micron-sized dielectric spheres irradiated by a laser radiation. The longitudinal and transversal dimensions of the photonic nanojet and its peak intensity also are calculated as the functions of particle size, index of absorption, and optical contrast of the particulate material. The photonic nano-flux was simulated numerically in composite particles made of a core and a shell with different refractive indices and variable shell thickness. Some practical conclusions are drawn concerning the possible ways to gain the control over the nanojet parameters in micron-sized spherical particles.     

Gegenbauer expansion to model the incident wave-field of a high-order Bessel vortex beam in spherical coordinates

The aim of this short communication is to report that Gegenbauer’s (partial-wave) expansion, that may be used (under some specific conditions) to represent the incident field of an acoustical (or optical) high-order Bessel beam (HOBB) in spherical coordinates, anticipates earlier expressions for undistorted waves. The incident wave-field is written in terms of the spherical Bessel function of the first kind, the gamma function as well as the Gegenbauer or ultraspherical functions given in terms of the associated Legendre functions when the order m of the HOBB is an integer number. Expressions for high-order and zero-order Bessel beams as well as for plane progressive waves reported in prior works can be deduced from Gegenbauer’s partial-wave expansion by appropriate choice of the beams’ parameters. Hence the value of this note becomes historical. In addition, Gegenbauer’s expansion in spherical coordinates may be used to advantage to model the wave-field of a fractional HOBB at the origin (i.e. z = 0).     

A new version of Bishop frame and an application to spherical images

In this work, we introduce a new version of Bishop frame using a common vector field as binormal vector field of a regular curve and call this frame as “Type-2 Bishop Frame”. Thereafter, by translating type-2 Bishop frame vectors to the center of unit sphere of three-dimensional Euclidean space, we introduce new spherical images and call them as type-2 Bishop spherical images. Frenet-Serret apparatus of these new spherical images are obtained in terms of base curve's type-2 Bishop invariants. Additionally, we express some interesting relations and illustrate two examples of our main results.     

Limit analysis and numerical modeling of spherically porous solids with Coulomb and Drucker-Prager matrices

The first goal of this work was to develop efficient limit analysis (la) tools to investigate the macroscopic criterion of a porous material on the basis of the hollow sphere model used by Gurson, here with a Coulomb matrix. Another goal was to give the resulting rigorous lower and upper bounds to the macroscopic criterion to enable comparisons and validations with further analytical or numerical studies on this micro-macro problem. In both static and kinematic approaches of la, a quadratic formulation was used to represent the stress and displacement velocity fields, in triangular finite elements. A significant improvement of the quality of the results was obtained by superimposing, on the fem fields, analytical fields which are the solutions to the problem under isotropic loadings.The final problems result in conic optimization, or linear programming after linearization of the criterion, so as to determine the “Porous Coulomb” criterion. A fine iterative post-analysis strictly restores the admissibility of the static and kinematic solutions. After presenting the results for various values of the porosity and internal friction angle, a comparison with a heuristic Cam-Clay-like criterion shows that this criterion cannot be considered a precise general approximation. Then a comparison with the “Porous Drucker-Prager” criterion treated by specific 3D codes is presented. With the same numerical tools, a final analysis of recent results in the literature is detailed, and tables of selected numerical data are presented in the appendices.     

On a generalization of distance sets

A subset X in the d-dimensional Euclidean space is called a k-distance set if there are exactly k distinct distances between two distinct points in X and a subset X is called a locally k-distance set if for any point x in X, there are at most k distinct distances between x and other points in X.Delsarte, Goethals, and Seidel gave the Fisher type upper bound for the cardinalities of k-distance sets on a sphere in 1977. In the same way, we are able to give the same bound for locally k-distance sets on a sphere. In the first part of this paper, we prove that if X is a locally k-distance set attaining the Fisher type upper bound, then determining a weight function w, (X,w) is a tight weighted spherical 2k-design. This result implies that locally k-distance sets attaining the Fisher type upper bound are k-distance sets. In the second part, we give a new absolute bound for the cardinalities of k-distance sets on a sphere. This upper bound is useful for k-distance sets for which the linear programming bound is not applicable. In the third part, we discuss about locally two-distance sets in Euclidean spaces. We give an upper bound for the cardinalities of locally two-distance sets in Euclidean spaces. Moreover, we prove that the existence of a spherical two-distance set in (d−1)-space which attains the Fisher type upper bound is equivalent to the existence of a locally two-distance set but not a two-distance set in d-space with more than d(d+1)/2 points. We also classify optimal (largest possible) locally two-distance sets for dimensions less than eight. In addition, we determine the maximum cardinalities of locally two-distance sets on a sphere for dimensions less than forty.