Multipole expansion of integral powers of cosine theta

Legendre polynomials form the basis for multipole expansion of spatially varying functions. The technique allows for decomposition of the function into two separate parts with one depending on the radial coordinates only and the other depending on the angular variables. In this work, the angular function $$\cos ^k \theta$$ cos k θ is expanded in the Legendre polynomial basis and the algorithm for determining the corresponding coefficients of the Legendre polynomials is generated. This expansion together with the algorithm can be generalized to any case in which a dot product of any two vectors appears. Two alternative multipole expansions for the electron–electron Coulomb repulsion term are obtained. It is shown that the conventional multipole expansion of the Coulomb repulsion term is a special case for one of the expansions generated in this work.

ANALYSIS OF ORDER OF STRESS SINGULARITY AT A VERTEX IN 3D TRANSVERSELY ISOTROPIC PIEZOELECTRIC DISSIMILAR BONDED JOINTS

The order of singularity near the vertex of bonded joints is one of the main factors responsible fordebonding under mechanical or thermal loading. The distribution of stress singularity field near the vertex ofbonded joints is very important to maintain the reliability of intelligent materials. In this paper, order of stresssingularity at vertex in 3D transversely isotropic piezoelectric dissimilar bonded joints is analyzed. Eigenanalysis based on FEM is used for stress singularity field analysis of piezoelectric bonded joints. The eigenequation is used for calculating the order of stress singularity, and the angular function. The numerical resultshows that the angular functions have large value near the interface edge than the inner portion of the joint.Therefore, there is a possibility to debond and delamination may occur at the interface edge of the piezoelectricbonded joints due to the higher stress and electric displacement concentration at the free edge.DOI: http://dx.doi.org/10.3329/jme.v44i1.19490

Expansions of fields by angular functions

The notion of an angular function has been introduced by Zilber as one possible way of connecting non-commutative geometry with two ‘counterexamples’ from model theory: the non-classical Zariski curves of Hrushovski and Zilber, and Poizat's field with green points. This article discusses some questions of Zilber relating to existentially closed structures in the class of algebraically closed fields with an angular function.

Two-Dimensional Tilt Illusions Induced by Orthogonal Plaid Patterns: Effects of Plaid Motion, Orientation, Spatial Separation, and Spatial Frequency

Tilt illusions occur when a drifting vertical test grating is surrounded by a drifting plaid pattern composed of orthogonal moving gratings. The angular function of this illusion was measured as the plaid orientation (and therefore its drift direction) varied over a 180° range, This was done when the test and inducing stimuli abutted and had the same spatial frequency, and when the test and inducing stimuli either differed in frequency by an octave, or were spatially separated by a 2 deg blank annulus, or both differed in frequency and were also separated by the annulus (experiments 1–4). The obtained angular function was virtually identical to that obtained previously with the rod and frame effect and other cases involving orthogonal inducing components, with evidence for illusions induced both by real-line components and by virtual axes of symmetry. Although the magnitude of the illusion was very similar in all four experiments, there was evidence to suggest that largest real-line effects occurred in the abutting same-frequency condition, with a pattern of results similar to that obtained previously with the simple one-dimensional tilt illusion. On the other hand, virtual-axis effects were more prominent with gaps between test and inducing stimuli. A fifth, repeated-measures, experiment confirmed this pattern of results. It is suggested that this pattern-induced tilt effect reflects both striate and extrastriate mechanisms and that the apparent influence of spatially distal virtual axes of symmetry upon perceived orientation implies the existence of AND-gate mechanisms, or conjunction detectors, in the orientation domain.