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Finite element exterior calculus refers to the development of finite element methods for differential forms, generalizing several earlier finite element spaces of scalar fields and vector fields to arbitrary dimension n, arbitrary polynomial degree r, and arbitrary differential form degree k. The study of finite element exterior calculus began with the [Formula omitted] and [Formula omitted] families of finite element spaces on simplicial triangulations. In their development of these spaces, Arnold, Falk, and Winther rely on a duality relationship between [Formula omitted] and [Formula omitted] and between [Formula omitted] and [Formula omitted]. In this article, we show that this duality relationship is, in essence, Hodge duality of differential forms on the standard n-sphere, disguised by a change of coordinates. We remove the disguise, giving explicit correspondences between the [Formula omitted], [Formula omitted], [Formula omitted] and [Formula omitted] spaces and spaces of differential forms on the sphere. As a direct corollary, we obtain new pointwise duality isomorphisms between [Formula omitted] and [Formula omitted] and between [Formula omitted] and [Formula omitted]. These isomorphisms can be implemented via a simple computation, which we illustrate with examples., Author(s): Yakov Berchenko-Kogan [sup.1] Author Affiliations: (1) Honolulu, USA Introduction The finite element method is a tool for solving partial differential equations numerically that approximates solutions to the PDE by [...]

The flow problem of an incompressible axisymmetrical quasisteady translation and steady rotation of a porous spheroid in a concentric spheroidal container are studied analytically. The same small departure from a sphere is considered for each spheroidal surface. In the limit of small Reynolds number, the Brinkman equation for the flow inside the porous region and the Stokes equation for the outside region in their stream functions formulations and velocity components, which are proportional to the translational and angular velocities, respectively, are used. Explicit expressions are obtained for both inside and outside flow fields to the first order in a small parameter characterizing the deformation of the spheroidal surface from the spherical shape. The hydrodynamic drag force and couple exerted on the porous spheroid are obtained for the special cases of prolate and oblate spheroids in closed forms. The dependence of the normalized wall-corrected translational and rotational mobilities on permeability for a porous spheroid in an unbounded medium and for a solid spheroid in a cell on the particle volume fraction is discussed numerically and graphically for various values of the deformation parameter. In the limiting cases, the analytical solutions describing the drag force and torque or mobilities for a porous spheroid in the spheroidal vessel reduce to those for a solid sphere and for a porous sphere in a spherical cell. PACS Nos: 47.15.G-, 47.56.+r Nous etudions analytiquement le probleme de la translation incompressible, axysimeetrique et en rotation stationnaire d'un spheroide poreux dans un contenant spheroidal concentrique. Le meme petit ecart d'une sphere est etudie pour chaque surface spheroidale. Dans la limite des petits nombres de Reynolds, nous utilisons l'eequation de Brinkman pour l'ecoulement a l'inteerieur d'une region poreuse et l'eequation de Stokes pour la region exterieure dans leurs formulations de la fonction de courant et des composantes de la vitesse, qui sont proportionnelles aux vitesses de translation et de rotation respectivement, tel qu'utilisee. Nous obtenons des expressions explicites pour les ecoulements a la fois interieur et exterieur, au premier ordre du parametre definissant la non sphericite des surfaces spheroidales. Nous derivons analytiquement la force de trainee hydrodynamique et le couple exerces sur le spheroide poreux pour les cas speeciaux de spheroides allonges et aplatis. Nous discutons numeriquement et graphiquement pour un spheroide poreux la deependance des mobilites translationnelle et rotationnelle, normaliseies pour la correction due au mur, sur la permeiabilite; dans un milieu non borne; et pour un spheroide solide dans une cellule, la dependance sur la fraction de volume de la particule. Dans les cas limites, les solutions analytiques decrivant la force de trainee et le torque ou les mobilites pour un spheroide poreux dans un contenant spheroidal se reduisent a celles pour une sphere solide et pour une sphere poreuse dans une cellule spherique. [Traduit par la Redaction], 1. Introduction The problems of uniform flow past and through porous media have several important applications, notably in the flow of oil through porous rocks, extractions of energy from geothermal [...]

Linear consolidation analyses are usually treated either by means of Terzaghi-Rendulic uncoupled theory or Blot's consolidation theory. In this note, the problem of consolidation displacements around an axially loaded sphere was considered. It is demonstrated that both the uncoupled analysis and the coupled analysis give the same governing equation for pore fluid pressure dissipation with time. A simplified procedure for deriving transient strain components is illustrated. A general solution for time-dependent displacements is obtained using uncoupled consolidation analysis. Close agreement is evident between the new approximate uncoupled analysis solution and the existing coupled analysis solution with a maximum error of less than 0.5%. DOI: 10.1061/(ASCE)EM.1943-7889.0000149 CE Database subject headings: Elasticity; Poroelasticity; Spheres; Deformation; Coupling. Author keywords: Elastic analysis; Poroelasticity; Spheres; Deformation.

In this paper, the classic coupled thermoelasticity model of hollow and solid spheres under radial-symmetric loading condition (r, t) is considered. A full analytical method is used and an exact unique solution of the classic coupled equations is presented. The thermal and mechanical boundary conditions, the body force, and the heat source are considered in the most general forms, where no limiting assumption is used. This generality allows to simulate a variety of applicable problems. [DOI: 10.1115/1.4001198] Keywords: coupled thermoelasticity, hollow sphere, exact solution

Inverse light scattering analysis (ILSA) seeks to associate measured scattering properties with the most probable theoretical scattering distribution, making it a useful tool for assessing structure in biological materials. The accuracy of ILSA depends on the compatibility of tile light scattering geometry with the light scattering model. In this study, we compare the accuracy obtained when analyzing light scattering data from spheroids using a numerical implementation of Mie theory, and the T matrix, a numerical method of solving light scattering from spheroids. Our experimental data are acquired using novel optical phantoms containing spheroidal scatterers and angle-resolved low-coherence interferometry, a depth-and angle-resolved light scattering measurement modality The results show that Mie theory can accurately assess spheroidal structure despite the geometric incompatibility provided measurements are taken in multiple orientations of the sample relative to the incident polarization and the measured scattering angle. In comparison, analysis using tile T-matrix method is highly accurate and more reliable yet requires measurements from only a single orientation. OCIS codes: 290.1350, 290.4020, 290.3200, 170.6510.

Since their discovery by Milnor in 1956, exotic spheres have provided a fascinating object of study for geometers. In this article we survey what is known about the curvature of exotic spheres.

This paper introduces a novel approach for solving the problem of camera calibration from spheres. By exploiting the relationship between the dual images of spheres and the dual image of the absolute conic (IAC), it is shown that the common pole and polar with regard to the conic images of two spheres are also the pole and polar with regard to the IAC. This provides two constraints for estimating the IAC and, hence, allows a camera to be calibrated from an image of at least three spheres. Experimental results show the feasibility of the proposed approach. Index Terms--Calibration, sphere, silhouette, surface of revolution (SOR).

Spheroids are widely used in biology because they provide an in vitro 3-dimensional (3D) model to study proliferation, cell death, differentiation, and metabolism of cells in tumors and the response [...]

Byline: M. A. Stepanova (1) Abstract: Let f:S .sup.2 [right arrow]! R .sup.3 be a generic smooth immersion. The skeleton of f is the following triple (I, D, p): I a R .sup.3 is the 1-polyhedron of singular points of f, D = f.sup.-1(I) a S .sup.2 is also a 1-polyhedron, and p : D [right arrow] I, x [right arrow]| f(x), is the projection. For triples of the form (D,I,p), where I has at most four vertices, we give an iff-condition under which the triple is the skeleton of a smooth immersion f : S .sup.2 [right arrow]! R .sup.3. Bibliography: 4 titles. Author Affiliation: (1) Russian State Pedagogical University, St. Petersburg, Russia Article History: Registration Date: 05/10/2005 Received Date: 31/01/2003 Article note: Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 299, 2003, pp. 300--313.

Byline: V. A. Zalgaller (1) Abstract: What is the form of the shortest curve C going outside the unit sphere S in [].sup.3 such that passing along C we can see all points of S from outside? How will the form of C change if we require that C has one (or both) of its endpoints on S? A solution to the latter problem also answers the following question. You are in a half-space at a unit distance from the boundary plane P, but you do not know where P is. What is the shortest space curve C such that going along C you will certainly come to P? Geometric arguments suggest that the required curves should be looked for in certain classes depending on several parameters. A computer-aided analysis yields the best curves in the classes. Some other questions are solved in a similar way. Bibliography: 4 titles. Author Affiliation: (1) St. Petersburg Department, Steklov Mathematical Institute, St. Petersburg, Russia Article History: Registration Date: 05/10/2005 Received Date: 25/12/2001 Article note: Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 299, 2003, pp. 87--108.

Clinical information on tumor growth is often limited to a few determinations of the size of the tumor burden taken at variable time. As a consequence, fitting of growth equations to clinical data is hampered by the small number of available data. On the other hand, characterising the tumor growth kinetics in terms of clinically relevant parameters, such as the doubling time of the tumors, is increasingly required to optimize and personalise treatments. A computational method is presented which can estimate the growth kinetics of tumors from as few as two determinations of its size taken at two successive time points, provided the size at which tumor growth saturates is known. The method is studied by using experimental data obtained in vitro with multicell tumor spheroids and in vivo with tumors grown in mice, and its outputs are compared to those obtained by tilting of experimental data with the Gompertz growth equation. Under certain assumptions and limitations the method provides comparable estimates of the doubling time of tumors with respect to the classical nonlinear fitting approach. The method is then tested against simulated tumor growth trajectories spanning the range of tumor sizes observed in the clinics. The simulations show that a relative classification of tumors on the basis of their growth kinetics can be obtained even if the size at which tumor growth saturates is not known. This result opens the possibility to classify patients bearing fast or slow growing tumors and, hence, to adapt therapeutic regimens under a more rationale basis. Index Terms--Gompertzian kinetics, in vivo data, mathematical computation, spheroids, tumor growth.

The modification of the Ornstein-Zernike (OZ) equation for mixtures is computed and numerically solved for binary mixtures of hard spheres (HS) or hard spheres and hard prolate spherocylinders (HPS) with length-to-breadth ratio gamma = 2.0. In the case of binary mixtures of hard spheres, the proposed theoretical method is found to be in agreement with Monte Carlo data.

This survey collects basic results concerning fractal and ergodic properties of Julia sets of rational functions of the Riemann sphere. Frequently these results are compared with their counterparts in the theory of Kleinian groups, and this enlarges the famous Sullivan dictionary. The topics concerning Hausdorff and packing measures and dimensions are given most attention. Then, conformal measures are constructed and their relations with Hausdorff and packing measures are discussed throughout the entire article. Also invariant measures absolutely continuous with respect to conformal measures are touched on. While the survey begins with facts concerning all rational functions, much time is devoted toward presenting the well-developed theory of hyperbolic and parabolic maps, and in Section 3 the class NCP is dealt with. This class consists of such rational functions f that all critical points of f which are contained in the Julia set of f are non-recurrent. The NCP class comprises ii1 particular hyperbolic, parabolic and subhyperbolic maps. Our last section collects some recent results about other subclasses of rational functions, e.g. Collet-Eckmann maps and Fibonacci maps. At the end of this article two appendices are included which are only loosely related to Sections 1-4. They contain a short description of tame mappings and the theory of equilibrium states and Perron-Frobenius operators associated with Holder continuous potentials.

Byline: A. N. Glushakov (1), S. E. Kozlov (1) Abstract: A canonical bijection between the set of extreme points of the comass-unit sphere S.sub.2,2k.sup.* [subset] [LAMBDA].sup.2 (R.sup.2k) and the manifold of orthogonal complex structures in R.sup.2k is described, under which unitary bases correspond to decompositions of the forms realizing the conjugate norm of the mass. This correspondence is used for obtaining a classification of faces of the sphere S.sub.2,n.sup.* and the known classification of faces of the set polar to S.sub.2,n.sup.* . Bibliography: 10 titles. Author Affiliation: (1) St.Petersburg State University, Russia Article History: Registration Date: 22/10/2004

Abstract: By using generalized functions, we obtain the set of thermoelasticity equations for a hollow thermosensitive, transversally isotropic sphere with a thin foreign inclusion. By approximating the thermoelastic characteristics of the basic material and the material of the inclusion by piecewise-constant functions, we deduce a differential equation with coefficients of the type of impulse functions to determine a radial displacement. We carry out the numerical analysis of the temperature and level of thermal stresses in a system made of graphites. Article History: Registration Date: 09/10/2004

Byline: J. Krawczyk (1) Abstract: We consider a thin linear elastic isotropic shell of constant thickness. We assume that it undergoes deformations of the Kirchhoff--Love type and derive some explicit formulas for certain infinitesimal isometric deformations of a pseudospherical shell by considering a linearized version of this problem. Author Affiliation: (1) Institute of Mathematics, Opole University, Opole, Poland Article History: Registration Date: 09/10/2004

Byline: E. S. Dubtsov (1) Abstract: Let K be a compact space, let X be a closed subspace of C(K), and let [mu] be a positive measure on K. The triple ( X,K,[mu] ) is said to be regular if, for any positive function [phi] [member of] C( K ) and for any [epsilon] >0 , there exists a function f [member of] X such that | f | [phi] on K and [mu] \ t [member of] K:| f( t ) |[phi] ( t ) \ < [epsilon] . The case where K is the unit sphere in C.sup.n and the subspace X is invariant with respect to the unitary group is investigated. Sufficient spectral conditions and a necessary condition for the regularity of a triple are obtained. Connections with compactness of certain Hankel operators and applications to interpolation problems are presented. Bibliography: 16 titles. Author Affiliation: (1) St.Petersburg State University, Russia Article History: Registration Date: 11/10/2004

Abstract: We construct models of Gaussian isotropic random fields on a sphere in an n-dimensional space. The models approximate the fields with a given reliability and accuracy in the space L .sub.p, p aY= 2. Article History: Registration Date: 23/10/2004

Mathematical issues concerning four mutually tangent spheres are examined. Topics include the six points of tangencies, and the resulting three opposing pairs of tangenciens, which define triangle centers.

Longitudinal and transverse polarizability of conducting spheres of arbitrary radii and arbitrary angles of intersection was investigated.

This paper presents a new approach to using virtual reality (VR) to design spherical mechanisms. VR provides a three-dimensional (3-D) design space where a designer can input design positions using a combination of hand gestures and motions and view the resultant mechanism in stereo using natural head movement to change the viewpoint. Because of the three-dimensional nature of the design and verification of spherical mechanisms, VR is examined as a new design interface in this research. In addition to providing a VR environment for design, the research presented in this paper has focused on developing a 'design in context' approach to spherical mechanism design. Previous design methods have involved placing coordinate frames along the surface of a constraint sphere. The new 'design in context' approach allows a designer to freely place geometric models of movable objects inside an environment consisting of fixed objects. The fixed objects could either act as a base for a mechanism or be potential sources of interference with the motion of the mechanism. This approach allows a designer to perform kinematic synthesis of a mechanism while giving consideration to the interaction of that mechanism with its application environment.

Spherical 4R mechanisms are studied for which the crank is relatively smaller than the remaining links. The theory of small-crank mechanisms is applied to obtain approximate descriptions for the follower angular displacement in terms of the input crank angle. The follower angle is presumed to comprise a mean and a perturbational motion. This results in an approximate expression in which the follower displacement is given as a linear combination of simple harmonic functions of the first and second harmonics of the crank angle. The approximate equations are utilized for synthesis of spherical 4R mechanisms for function generation. In contrast to the conventional design procedures, the use of the approximate equations allows the synthesis of spherical mechanisms in which a prescribed function is satisfied for the entire motion of the mechanism. In addition to design examples, sample error charts are provided to assist the designer in ascertaining feasible ranges for design and corresponding orders of error.

Research was conducted to provide a solution to the problem of electrostatics of the conducting double spheres of arbitrary radii placed in an arbitrary potential field. The approach was based on Kelvin's transformation combined with reflection and translation characteristics for harmonic functions. The proposed theorem admits interpretation of the image system for several imposed potentials in the presence of the two-sphere assembly.

The axial pressure distribution in the entrance area of a randomly packed bed of spheres at Reynolds numbers 1036-2749 was measured for expansion angles 0, 30, 60 and 90 degrees. For the 0 and 30 inlet geometries, a necessary constant pressure gradient was achieved within about one and one half sphere diameters. Thus, it is determined that the entrance length is of the same measurement for these cases. For the 60 and 90 geometry angles, non-constant pressure gradients were obtained.

As an extension of the knife-edge test, a noninterferometric method for inspecting circularly symmetric aspheres is proposed in which the test surface is illuminated by a spherical wave. When a small circular stop is placed around the curvature center of the best-fitting sphere, only rays characteristic for the deviation from a perfect sphere remain unobstructed. An image of the tested surface is thereby obtained, showing light and dark regions whose boundaries are correlated to the surface profile, the stop size, and the stop position along the symmetry axis. The experiment has been carried out with a paraboloid.

This paper investigates the tensile stress distribution within a solid circular cylinder of diameter D and length 2L subjected to double diametral indentors or diametral point load strength test (PLST). The contact problem between the indentors and the cylinder is solved analytically, and exact solution for stress distribution is obtained through the use of displacement functions and double Fourier expansion in [Theta]- and z-coordinates. Numerical results show that the tensile stress at the center of the cylinders, in general, decreases with Poisson's ratio v and increases with 2L/D for 2L/D < 1, but remains roughly constant for 2L/D > 1. This prediction agrees exactly with the shape effect observed experimentally for the PLST for marbles, granite, and tuff found in Hong Kong and with previous published results for other rocks. The magnitude of tensile stress at the center is about three times the prediction obtained by using Wijk's formula, but seems more comparable with experimental results. The size effect of the point load strength index observed experimentally is also predicted by our theory.

A new approach suitable for solving inverse problems in multiangle light scattering is presented. The method takes advantage of multidimensional function approximation capability of radial basis function neural networks. An algorithm for training the networks is described in detail. It is shown that the radius and refractive index of homogeneous spheres can be recovered accurately and quickly, with maximum relative errors of the order of [10.sup.-3] and mean errors as low as [10.sup.-5]. The influence of the angular range of available scattering data on the loss of information and inversion accuracy is investigated, and it is shown that more than two thirds of input data can be removed before substantial degradation of accuracy occurs.

A previously developed theoretical procedure for determination of electromagnetic fields associated with the interaction of a higher-order Gaussian beam with a homogeneous spherical particle is used to investigate the effects of incident beam type on far-field scattering. Far-field scattering patterns are calculated for (0,0), (0,1), and (1,1) mode Hermite-Gaussian beams and for the helix doughnut mode beam. The effects of incident beam type on the angular distribution of far-field scattering, for both on-sphere-center and off-sphere-center focusing, are examined.

Oscillations that are superimposed on the smooth, exponential decay of light from microcavity modes are observed, and they are characteristic of the temporal beating of adjacent, degeneracy-split azimuthal modes in distorted spheres. The perturbation theory result for the frequency splitting of azimuthal modes in distorted spheres is used to determine the distortion amplitude from the temporal oscillations associated with mode beating. The beating period can be used to determine whether the spectrally measured resonance linewidth [Delta][[Omega].sub.l,n], is broadened by closely spaced, degeneracy-split azimuthal modes of slightly nonspherical droplets.

A detailed study of rainbow thermometry and its application to droplets in reactive systems is presented. To this end the light-scattering history of a vaporizing droplet under unsteady conditions is discussed. Unlike in previous papers, the reduction of the droplet's diameter is also taken into account in addition to the variation of the refractive-index profile. A finely stratified sphere model with thousands of layers (i.e., 20,000) is used to compute the scattering patterns of a radially inhomogeneous evaporating droplet at different heating times and therefore with different diameters and refractive-index profiles. In the studied case the temperatures inferred from rainbow thermometry do not represent the actual temperatures inside the droplet. They do not represent an average internal temperature or even the surface or the core temperature. For droplets with a temperature that increases from the core to the surface, the inferred values are always lower than the minimum temperature inside the droplet. Therefore the rainbow technique should be applied with caution in all cases in which droplet inhomogeneities are suspected. In addition, a careful analysis of the scattering in the rainbow region is presented. Because of the physical structure of the rainbow, a marked uncertainty in the inferred temperatures also has to be considered in the case of homogenous droplets. For inhomogeneous spheres this intrinsic uncertainty has to be added to the effects caused by the internal profiles of the refractive index.

A path is defined as the vector's sum of the translation and rotation component of the length unit belonging to the mass entity in motion on the sphere. The fine structure constant is an irrational number being a mirror of the path complexity as well as the sphere curvature where the path is made. The inverse value in the Euclidean plane yields [[alpha].sup.-1] = [square root of [[pi].sup.2] + [137.sup.2]]. The inverse fine structure constant on the elliptic sphere is smaller and on the hyperbolic sphere is greater. The electron in the Hydrogen atom should move on the elliptic sphere of the radius of 3679 Compton wavelengths of the electron according to the CODATA 2012 recommended empirical value [[alpha].sup.-1] = 137.035999074. Such a small sphere radius implies the heterogeneous curvature ofthe present universe., 1 Theoretical background In motion is an entity having some mass. Respecting Compton the length unit is attributed to that mass: [lambda] = h/mc = 1. (1) The infinite mass [...]

We describe a method for integrating analytically, over a circular aperture, the emission from an oscillating dipole inside a dielectric sphere. The model is useful for investigating fluorescence, Raman, or other emission from molecules inside of spherical particles or droplets. The analysis is performed for two cases: (a) the dipole emits from a fixed orientation, and (b) the dipole emits from all orientations and the collected energy is summed. This second case models the collection of emission from a molecule that is excited repeatedly; after each excitation it rotates to a random orientation before emitting. These results are applicable to single-molecule detection techniques employing microdroplets and to other techniques for characterizing microparticles with luminescence or inelastic scattering. Key words: Fluorescence, electromagnetic waves, emission, microspheres, scattering, single-molecule detection.

We present laboratory microwave scattering measurements for complex amplitude scattering matrices of three aggregates of 2, 8, and 27 identical spheres and compare them with theoretical predictions. Electromagnetic multiparticle-scattering calculations involve the determination of a large number of vector translation coefficients introduced by the addition theorems for vector spherical harmonics. For one of the two classes of vector translation coefficients there is an overall-sign discrepancy between two groups of formulations that exist in the literature. We compare our experimental data with the theoretical results from scattering calculations using the two different sets of formulas for computation of the translation coefficients. This comparison of experiment with theory reveals that Cruzan's original research on the vector addition theorems [Q. Appl. Math. 20, 33-40 (1962)] is correct, although many authors believe that Cruzan's formulation contains an overall-sign error. Key words: Multiparticle light scattering, vector addition theorems, vector translation coefficients, particles, aggregates of spheres.

An experimental investigation has been conducted to compare and contrast the dynamic load-transfer process in single-chain assemblies of two- and three-dimensional particles. In particular, single-chain assemblies of disks and spheres were subjected to explosive loadings and strain gauges were used to collect the resulting strain pulse information. The data were analyzed to compare the pulse velocity, contact load attenuation, dispersion, and pulse breakup in disks and spheres. The comparison shows that the average pulse velocity in disks is higher than in spheres and the pulse attenuation is lower in the disks. The results also show that there is a characteristic pulse that will propagate through a single chain of disks without significant dispersion. However, the characteristic pulse propagating in assemblies of spheres shows dispersion. For both disks and spheres, when the input pulse is sufficiently long, it undergoes a ringing process, which breaks the long pulse up into smaller signals of size approximately equal to the characteristic length. The larger the input pulse is with respect to the characteristic pulse length, the longer the breakup takes. This breakup is also accompanied by a redistribution of energy, which is easily seen in the frequency domain.

A study developed a simple theory of macroscopic elasticity of granular systems based on several models of the grain-grain contact forces which are related to the fundamental Hertz theory. The study considered only the special cases of perfect friction or no tangential friction. The result showed that the energy is well defined for all values of the macroscopic strain when the tangential contact stiffness is independent of the normal force.

A study was made on the computation of elastothermodynamic damping in a matrix reinforced with hollow spheres. The numerical results are presented for the case of alumina micro-balloons subjected to a uniform radial stress at the outer boundary. Observations showed that the total damping gradually disappears when the cavity takes up more than 75% volume fraction of the inclusion.

A theoretical procedure has been developed for the determination of internal and external electromagnetic fields that result from the interaction of a higher-order Gaussian beam with a homogeneous sphere. Specific calculations are performed for (1, 0), (0, 1), and (1, 1) mode Hermite-Gaussian beams and for doughnut mode beams of four different polarizations (radial, angular, arced, and helix). The effects of incident beam type on resonance excitation, and on the spatial distribution of the internal and near-surface electromagnetic fields, are examined. Key words: Higher-order Gaussian beams, laser-particle interactions, electromagnetic fields, spherical particle.

The flux-form advection scheme of Bott is modified for the spherical coordinates, combined with the expanded-polar-zone (EPZ) technique to improve the overall performance of the advection calculations. With the EPZ technique, this Eulerian scheme has comparable efficiency as semi-Lagrangian methods for advection of nonreactive tracers on a sphere but with somewhat better overall numerical accuracy. The conservation of global tracer mass and the positive definiteness of the algorithm are achieved to machine precision. For the test problem of solid body rotations on a sphere, this scheme shows small numerical diffusion, almost undetectable phase errors, and very little artificial deformation of the test shape even far cross-polar transport. In comparison with some semi-Lagrangian schemes and other high-order Eulerian methods, it shows very competitive performance. Numerical tests also indicate that, without any modifications, it performs just as well on slightly nonuniform Gaussian grid as on uniform grid. For the vertical advection, a fourth-order and two second-order versions of this scheme formulated on a nonuniform grid system have also been derived. The performance of these versions is tested with a nonuniform sigma grid system by using ideal one-dimensional test problems. This accurate numerical scheme is recommended for models where resolving the sharp vertical gradients of atmospheric trace species such as water vapor is important.

An elastic sphere solution is derived for the sphere's normal effect against a rigid plane. Equilibrium conditions must be assumed for the solution's validity. However, the solution can also account for the dynamic problem given specifications on impact duration and the elastic wave. Variation is observed on the relationship between initial contact and detachment time for the new solution and for Hertz's frictionless contact theory.

Edge functions T1 and T2, which describe the polarization-dependent edge contribution to forward scattering by spheres, are derived from the exact Mie solution. All the relative refractive indices and the 64 < x < 2048 size parameter range are considered. The edge functions significantly improve the approximation methods that can be used to calculate forward-scattering patterns. For m close to 1, an asymptotic approximation is used. Otherwise, the familiar geometrical optics approximation and the similar physical optics approximation for glow rays are used. Both geometrical and physical optics equations can be deduced from the above-mentioned asymptotic approximation. Key words: Forward light scattering, sphere, Mie, geometrical optics, physical optics, anomalous diffraction, edge function.

The smooth closed curve sigma's immersion on the unit sphere S2 is considered, where sigma's normalized velocity vector tau, or the tantrix of sigma, produces a tangent indicatrix of sigma. Sigma's immersion of the circle in the unit sphere also produces an immersion in tau. A tantrix is formed if and only if the curve's velocity in S2 never vanishes and has no subarc with total geodesic curvature pi.

We calculate the moments of inertia of D-dimensional spheres and spherical shells, where D is a complex number. We also examine the moments of inertia of fractional-dimensional geometrical objects such as the Cantor set and the Sierpinski carpet and their D-dimensional analogs.

Differential geometry and multivariable calculus can be used to prove that the rotational motion of the Foucault pendulum is determined by the earth's rotation and the latitude on which the pendulum is located. A solution involves understanding parallel vectors on a sphere, which leads to the development of a theorem about the angle of rotation.

We have extended the work of others in developing a model for the motion of a ball on a rotating turntable. The numerical model does not impose the constraint of rolling without slipping, and includes both rolling and sliding friction. Computer output from the model is compared with experimental trajectories obtained from time exposures, and found to be in excellent agreement.

Image theory, previously developed for an electric low-frequency dipole in front of a dielectric sphere, is formulated for the magnetically permeable sphere. The low-frequency image source corresponding to an electric dipole outside the sphere arises only in the case when the orientation of the dipole is perpendicular to the radial direction. Two alternative image representations are developed. One is a combination of a magnetic strip current and a bifilar magnetic line current, both flowing between the center of the sphere and the Kelvin inversion point on the axis connecting the sphere and the dipole. The other is a combination of a magnetic strip current and an electric strip current flowing in orthogonal directions between the center and the inversion point of the sphere. A simple approximation for the image source is also derived.

In this work we have obtained the fields for external points created by uniformly magnetized ellipsoids and spheroids. The problem is solved directly in rectangular coordinates, obtaining the potential created by a uniformly charged ellipsoid everywhere by a variational method, in contrast to the usual method due to Dirichlet. Using the preceding result, we calculate the potential created by a uniformly magnetized ellipsoid in any external and internal points, as a superposition of the potentials created by two uniformly charged ellipsoids, which have opposite charge densities and slightly shifted centers. Finally, we find the solutions for the potential and field in any point external to a uniformly magnetized spheroid in rectangular coordinates. The results are checked with the classical ones obtained in spheroidal coordinates, finding a complete agreement except in one case. A discussion of this discrepancy is given. Finally, we determine how the magnetic field behaves with distance from the ellipsoid boundaries, which is very useful for corrections of the stray field flux and search coil design and also for models of particulate recording media with uniaxial anisotropy.

The study of two penny-shaped crazes occurring at the ends of the pole in a spherical inhomogeneity reveals that the reaction between the inhomogeneity and craze is dependent on the elastic properties of the matrix. The inhomogeneity is enclosed by an infinitely large elastic body subjected to uniaxial tension. Different types of elastic moduli are exhibited by the matrix and the inhomogeneity. The Eshelby's equivalent inclusion technique and the superposition principle of elasticity are used in the study of the crazes.

In some textbooks, the case of two charged conducting spheres connected by a wire is used to explain the effect of a sharp point on a conductor. Because the wire is a part of the system, there must be charge on it. In this paper, the wire is modeled by a chain of small conducting spheres, making it possible to calculate the charge on the wire.

The formalism that was previously devised (J. Opt. Soc. Am. A 9, 1327 (1992)) to deal with the optical properties of homogeneous spheres containing an eccentric spherical inclusion is extended to the case of several inclusions. The extinction efficiency of dielectric spheres containing two identical metallic inclusions is calculated for a few significant geometries. Extinction by a low-density dispersion of the anisotropic scatterers mentioned above is also evaluated. Our results show that the subdivision of the included material has quite visible effects that strongly depend on both the polarization of the incident light and the geometric arrangement of the inclusions.