Complex Hyperspherical Equations, Nodal-Partitioning, and First-Excited-State Theorems in ℝ n

Three problems related to the spherical quantum billiard in are considered. In the first, a compact form of the hyperspherical equations leads to their complex contracted representation. Employing these contracted equations, a proof is given of Courant's nodal-symmetry intersection theorem for diagonal eigenstates of spherical-like quantum billiards in . The second topic addresses the first-excited-state theorem for the spherical quantum billiard in . Wavefunctions for this system are given by the product form, ( )Z _q+()Y ⁽ⁿ⁾ , where is dimensionless displacement, is angular-momentum number, qis an integer function of dimension, Z() is either a spherical Bessel function (nodd) or a Bessel function of the first kind (neven) and represents (n– 1) independent angular components. Generalized spherical harmonics are written . It is found that the first excited state (i.e., the second eigenstate of the Laplacian) for the spherical quantum billiard in is n-fold degenerate and a first excited state for this quantum billiard exists which contains a nodal bisecting hypersurface of mirror symmetry. These findings establish the first-excited-state theorem for the spherical quantum billiard in . In a third study, an expression is derived for the dimension of the th irreducible representation (irrep) of the rotation group O(n) in by enumerating independent degenerate product eigenstates of the Laplacian.     

Two-dimensional asymptotic expansions for large deviations of spherically distributed random vectors if the dominating point degenerates asymptotically

A geometric approach to asymptotic expansions for large-deviation probabilities, developed for the Gaussian law by Breitung and Richter [J. Multivariate Anal.,58, 1–20 (1996)], will be extended in the present paper to the class of spherical measures by utilizing their common geometric properties. This approach consists of rewriting the probabilities under consideration as large parameter values of the Laplace transform of a suitably defined function, expanding this function in a power series, and then applying Watson’s lemma. A geometric representation of the Laplace transform allows one to combine the global and local properties of both the underlying measure and the large-deviation domain. A special new type of difficulty is to be dealt with because the so-called dominating points of the large-deviation domain degenerate asymptotically. As is shown in Richter and Schumacher (in print), the typical statistical applications of large-deviation theory lead to such situations. In the present paper, consideration is restricted to a certain two-dimensional domain of large-deviations having asymptotically degenerating dominating points. The key assumption is a parametrized expansion for the inverse $\bar g^{ - 1} $ of the negative logarithm of the density-generating function of the two-dimensional spherical law under consideration.     

On pointwise convergence of Fourier series of radial functions in several variables

We prove the pointwise convergence of the Fourier series for radial functions in several variables, which in the case is the Dirichlet-Jordan theorem itself. In our proof the method for the case of the indicator function of the ball is very useful.

     

Shrinkage estimators of the location parameter for certain spherically symmetric distributions

We consider estimation of a location vector for particular subclasses of spherically symmetric distributions in the presence of a known or unknown scale parameter. Specifically, for these spherically symmetric distributions we obtain slightly more general conditions and larger classes of estimators than Brandwein and Strawderman (1991,Ann. Statist.,19, 1639–1650) under which estimators of the formX +ag(X) dominateX for quadratic loss, concave functions of quadratic loss and general quadratic loss.Research supported by NSF grant DMS-88-22622     

简易弧高计的误差分析

本文介绍了使用简易弧高计测量球面弧高时的误差公式，并用实例说明了此类弧高计在一般光学元件件加工测量中的可靠性。     

Spherical functions and invariant differential operators on complex Grassmann manifolds

Proofs are given of two theorems of Berezin and Karpelevič, which as far as we know never have been proved correctly. By using eigenfunctions of the Laplace-Beltrami operator it is shown that the spherical functions on a complex Grassmann manifold are given by a determinant of certain hypergeometric functions. By application of this result, it is proved that a certain system of operators, fow which explicit expressions are given, generates the algebra of radial parts of invariant differential operators.     

Calculation of the Slip Velocity of a Rarefied Gas on a Solid Spherical Surface with Allowance for Accommodation Coefficients

The slip velocity of a rarefied gas with inhomogeneous temperature and mass velocity on a solid spherical surface is calculated with the use of a twomoment boundary condition in the linear approximation in terms of the Knudsen number. The dependence of the slip velocity on accommodation coefficients of the two first moments of the distribution function is studied.     

New type of spherical pore Al alloy foam with low porosity and high strength

The Ultra Light Metal Structure with variouskinds of pores (open, close) realizes the lightness andmultifunction of structural material. The functionsinclude lightness (ρ < 1), high specific strength, highenergy absorption, sound insulation, heat insulationand electromagnetism shield. It is one of the hotspotsin material research nowadays[1— . The requirements 6]of high speed movement and high technology make Alalloy foam, which has higher specific st…     

High Refractive-Index Microspheres as Optical Cavity Structure

Microspheres of refractive index of n_D > 2.0 have been investigated. The organic-inorganic hybrid microspheres of n_D = 1.72 were prepared by the vibrating orifice technique using titanium-tetra-n-butoxide (TTBu) and diphenyldimethoxysilane (DPhDMS). For lasing demonstration, Eu³⁺-doped microspheres were prepared using europium (III) thenoyltrifluoroacetonate [Eu(TTFA)₃] as the dopant. The particles have good spherical shape, smooth surface and high optical transparency. The diameters of the particles could be controlled to within 0.1 m. Subsequent heating of the microspheres at 550°C under oxygen atmosphere resulted in an increase in the refractive index up to n_D = 2.6 with retention of the spherical shape. Resonant emission was confirmed from Eu^{3 +}-doped microspheres after heating at 400–550°C, by pumping with the 514.5 nm line of a CW- Ar⁺laser.     

Continuous Restricted Radial Motion of a Gas under the Action of a Piston

The possibility of continuous conjugation of the straightline radial motion of a gas sphere toward the center and away from the center with the motion where the gas in the entire sphere stops simultaneously is shown. The motion is described by an invariant submodel of rank 1. Time reflections allow one to construct a solution that describes a periodic continuous restricted motion of the gas sphere under the action of a piston.