The Central Limit Problem for Random Vectors with Symmetries

Motivated by the central limit problem for convex bodies, we study normal approximation of linear functionals of high-dimensional random vectors with various types of symmetries. In particular, we obtain results for distributions which are coordinatewise symmetric, uniform in a regular simplex, or spherically symmetric. Our proofs are based on Stein’s method of exchangeable pairs; as far as we know, this approach has not previously been used in convex geometry. The spherically symmetric case is treated by a variation of Stein’s method which is adapted for continuous symmetries. This work was done while at Stanford University.     

Research problems

This section appears from time to time. Contributions are invited, and should be submitted to R. C. Thompson, Mathematics Department, University of California, Santa Barbara, CA 93106.     

On the bidirectional vortex and other similarity solutions in spherical coordinates

The bidirectional vortex refers to the bipolar, coaxial swirling motion that can be triggered, for example, in cyclone separators and some liquid rocket engines with tangential aft-end injectors. In this study, we present an exact solution to describe the corresponding bulk motion in spherical coordinates. To do so, we examine both linear and nonlinear solutions of the momentum and vorticity transport equations in spherical coordinates. The assumption will be that of steady, incompressible, inviscid, rotational, and axisymmetric flow. We further relate the vorticity to some power of the stream function. At the outset, three possible types of similarity solutions are shown to fulfill the momentum equation. While the first type is incapable of satisfying the conditions for the bidirectional vortex, it can be used to accommodate other physical settings such as Hill’s vortex. This case is illustrated in the context of inviscid flow over a sphere. The second leads to a closed-form analytical expression that satisfies the boundary conditions for the bidirectional vortex in a straight cylinder. The third type is more general and provides multiple solutions. The spherical bidirectional vortex is derived using separation of variables and the method of variation of parameters. The three-pronged analysis presented here increases our repertoire of general mean flow solutions that rarely appear in spherical geometry. It is hoped that these special forms will permit extending the current approach to other complex fluid motions that are easier to capture using spherical coordinates.     

Numerical study of simultaneous natural convection heat transfer from both surfaces of a uniformly heated thin plate with arbitrary inclination

 Numerical studies were conducted to investigate the natural convection heat transfer around a uniformly heated thin plate with arbitrary inclination in an infinite space. The numerical approach was based on the finite volume technique with a nonstaggered grid arrangement. For handling the pressure–velocity coupling the SIMPLE-algorithm was used. QUICK scheme and first order upwind scheme were employed for discretization of the momentum and energy convective terms respectively. Plate width and heating rate were used to vary the modified Rayleigh number over the range of 4.8×10⁶ to 1.87×10⁸. Local and average heat transfer characteristics were compared with regarding to the inclination angle. The empirical expressions for local and average Nusselt number were correlated. It has been found that for inclination angle less than 10^∘, the flow and heat transfer characteristics are complicated and the average Nusselt number can not be correlated by one equation while for inclination angle larger than 10^∘, the average Nusselt number can be correlated into an elegant correlation. Received on 18 April 2001 / Published online: 29 November 2001     

Studies on Rare-Earth/Iron and Actinide/Iron Multilayers

Hyperfine Interactions - Multilayers of Ce/Fe and U/Fe were fabricated by sputtering and studied by X-ray diffraction and reflectivity, Mössbauer spectroscopy and polarised neutron...     

Finite element method in the solution of the Euler and Navier-Stokes equations for internal flow

Finite element solutions of the Euler and Navier-Stokes equations are presented, using a simple dissipation model. The discretization is based on the weak-Galerkin weighted residual method and equal interpolation functions for all the unknowns are permitted. The nonlinearity is iterated upon using a Newton method and at each iteration the linear algebraic system is solved by a direct solver with all unknowns fully coupled. Results are presented for two-dimensional transonic inviscid flows and two- and three-dimensional incompressible viscous flows. Convergence of the algorithm is shown to be quadratic, reaching machine accuracy in very few iterations. The inviscid results demonstrate the existence of nonunique numerical solutions to the steady Euler equations.     

Controllability of Functional Integro-Differential Inclusions with an Unbounded Delay

In this paper, a sufficient condition is established for the controllability of neutral functional integro-differential inclusions with an unbounded delay in Banach spaces. The approach used is a fixed-point theorem for condensing maps due to Martelli and the theory of analytic semigroup of linear operators. Communicated by F. Zirilli Research supported by NNSF of China, by the Teaching and Research Award Program for Outstanding Young Teachers in Higher Education Institutions of the Ministry of Education of China, and by the Qing Lan Talent Engineering Fund QL-05-164 of Lanzhou Jiaotong University. The authors are grateful to Professor F. Zirilli and two anonymous referees for valuable suggestions improving this paper.     

Discovery of a new long-lived isomeric state in 125Ce

A new long-lived isomeric state in the near proton dripline nucleus ¹²⁵Ce has been identified with Schottky mass spectrometry at GSI. The excitation energy E ^* = 103(12)keV and the decay time of 193(1)s have been obtained from a single stored fully ionized ^125m Ce⁵⁸⁺ ion. The data implies an E3 transition and a 1/2⁺ assignment for the spin of the isomer.