Spherical halloysite aggregates have been identified for the first time in mineral matter isolated from bituminous coals. The spherules, found in Permian coals of the Sydney basin, New South Wales, range from 0.4 to 0.6 µm in diameter and have a delicate ring-like structure that helps to confirm the halloysite identification. They appear from their location to be related to influxes of pyroclastic debris, either directly or from nearby soils, into the original peat accumulation. Analytical electron microscopy indicates higher proportions of Si and Fe than coexisting particles of hexagonal platy kaolinite, and electron diffraction reveals a typical disordered halloysite structure. The aggregates are larger than those normally reported in soils, and comparison to growth rates in soils suggests development over a significantly longer time than that expected for accumulation of the host coal seams. The buckled structure in the ring-like pattern and the related crude polyhedral outlines probably reflect shrinkage with dehydration during the coalification process, but it may also be due to the different sample preparation techniques.

In this paper, we prove a rigidity result for the product metric on the product of spheres $S^1\times S^{n-1}$.

In this paper, we completely solve the $L^{2}\to L^{r}$ extension conjecture for the zero radius sphere over finite fields. We also obtain the sharp $L^{p}\to L^{4}$ extension estimate for non-zero radii spheres over finite fields, which improves the previous result of the first and second authors significantly.

We prove a topological rigidity theorem for closed hypersurfaces of the Euclidean sphere and of an elliptic space form. It asserts that, under a lower bound hypothesis on the absolute value of the principal curvatures, the hypersurface is diffeomorphic to a sphere or to a quotient of a sphere by a group action. We also prove another topological rigidity result for hypersurfaces of the sphere that involves the spherical image of its usual Gauss map.

The purpose of the present study was to do a psychometric evaluation of the somatic and psychological health report (SPHERE) among Chinese adolescents. Our participants were 116 twins (50 females). Psychometric evaluation indicated that the reliability and validity of this scale were good. The internal consistencies and split-half reliabilities of all subscales were above 0.80. Furthermore, the item-total correlations were acceptable for all the subscales (all the values were higher than 0.20). The present findings suggest that the SPHERE can be well used to measure Chinese adolescents’ somatic and psychological health.

A finite measure supported by the unit sphere 𝕊n−1 in ℝn and absolutely continuous with respect to the natural measure on 𝕊n−1 is entirely determined by the restriction of its Fourier transform to a sphere of radius r if and only 2πr is not a zero of any Bessel function Jd+(n−2)/2 with d a nonnegative integer.

Spherical radial basis functions are used to define approximate solutions to pseudodifferential equations of negative order on the unit sphere. These equations arise from geodesy. The approximate solutions are found by the collocation method. A salient feature of our approach in this paper is a simple error analysis for the collocation method using the same argument as that for the Galerkin method.

In this paper we derive local error estimates for radial basis function interpolation on the unit sphere . More precisely, we consider radial basis function interpolation based on data on a (global or local) point set for functions in the Sobolev space with norm , where s>1. The zonal positive definite continuous kernel ϕ, which defines the radial basis function, is chosen such that its native space can be identified with . Under these assumptions we derive a local estimate for the uniform error on a spherical cap S(z;r): the radial basis function interpolant ΛXf of satisfies , where h=hX,S(z;r) is the local mesh norm of the point set X with respect to the spherical cap S(z;r). Our proof is intrinsic to the sphere, and makes use of the Videnskii inequality. A numerical test illustrates the theoretical result.

Given a group, it is a basic problem to determine which manifolds can occur as a fixed-point set of a smooth action of this group on a sphere. The current article answers this problem for a family of finite groups including perfect groups and nilpotent Oliver groups. We obtain the answer as an application of a new deleting and inserting theorem which is formulated to delete (or insert) fixed-point sets from (or to) disks with smooth actions of finite groups. One of the keys to the proof is an equivariant interpretation of the surgery theory of S. E. Cappell and J. L. Shaneson, for obtaining homology equivalences.

This paper studies topological and metric rigidity theorems for hypersurfaces in a Euclidean sphere. We first show that an $n({\geq}\,2)$-dimensional complete connected oriented closed hypersurface with non-vanishing Gauss–Kronecker curvature immersed in a Euclidean open hemisphere is diffeomorphic to a Euclidean $n$-sphere. We also show that an $n({\geq}\,2)$-dimensional complete connected orientable hypersurface immersed in a unit sphere $S^{n+1}$ whose Gauss image is contained in a closed geodesic ball of radius less than $\pi/2$ in $S^{n+1}$ is diffeomorphic to a sphere. Finally, we prove that an $n({\geq}\,2)$-dimensional connected closed orientable hypersurface in $S^{n+1}$ with constant scalar curvature greater than $n(n-1)$ and Gauss image contained in an open hemisphere is totally umbilic.

Based on the theory of spherical harmonics for measures invariant under a finite reflection group developed by Dunkl recently, we study orthogonal polynomials with respect to the weight functions |x1|α1 . . . |xd|αd on the unit sphere Sd-1 in ℝd. The results include explicit formulae for orthonormal polynomials, reproducing and Poisson kernel, as well as intertwining operator.

Let be a real-valued, homogeneous, and isotropic random field indexed in . When restricted to those indices with , the Euclidean length of , equal to r (a positive constant), then the random field resides on the surface of a sphere of radius r. Using a modified stratified spherical sampling plan (Brown (1993a)) on the sphere, define to be a realization of the random process and to be the cardinality of . A bootstrap algorithm is presented and conditions for strong uniform consistency of the bootstrap cumulative distribution function of the standardized sample mean, , are given. We illustrate the bootstrap algorithm with global land-area data.