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.     

Metric marginal problems for set-valued or non-measurable variables

Summary In a separable metric space, if two Borel probability measures (laws) are nearby in a suitable metric, then there exist random variables with those laws which are nearby in probability. Specifically, by a well-known theorem of Strassen, the Prohorov distance between two laws is the infimum of Ky Fan distances of random variables with those laws. The present paper considers possible extensions of Strassen's theorem to two random elements one of which may be (compact) set-valued and/or non-measurable. There are positive results in finite-dimensional spaces, but with factors depending on the dimension. Examples show that such factors cannot entirely be avoided, so that the extension of Strassen's theorem to the present situation fails in infinite dimensions.This research was partially supported by a Guggenheim Fellowship, by National Science Foundation grant DMS 8505550 at MSRI-Berkeley, and other NSF grants     

On order-preservation and positive correlations for multidimensional diffusion processes

Summary As a continuation of the study by Herbst and Pitt (1991), this note presents two criteria. The first one is on the order-preservation for two (may be different) multidimensional diffusion processes. The second one is on the preservation of positive correlations for a diffusion process.Research supported in part by the Ying-Tung Fok Educational Foundation and the National Natural Science Foundation of China     

Echo planar imaging of perfluorocarbons

Emulsions of perfluorotributylamine (FTBA) and perflubron were evaluated for their utility in ¹⁹F echo planar imaging. Fluorine images of the emulsions were obtained in a phantom and two mice that had been predosed. Both agents, but particularly perflubron, show potential for fluorine echo planar studies because of the long spin-spin relaxation times of the CF₃ resonances. High resolution thin slice images obtained in as little as 26.6 ms are presented.     

Riccati-based preconditioner for computing invariant subspaces of large matrices

Summary. This paper introduces and analyzes the convergence properties of a method that computes an approximation to the invariant subspace associated with a group of eigenvalues of a large not necessarily diagonalizable matrix. The method belongs to the family of projection type methods. At each step, it refines the approximate invariant subspace using a linearized Riccati's equation which turns out to be the block analogue of the correction used in the Jacobi-Davidson method. The analysis conducted in this paper shows that the method converges at a rate quasi-quadratic provided that the approximate invariant subspace is close to the exact one. The implementation of the method based on multigrid techniques is also discussed and numerical experiments are reported. Received June 15, 2000 / Revised version received January 22, 2001 / Published online October 17, 2001     

Coupling for Markov renewal processes and the rate of convergence in ergodic theorems for processes with semi-Markov switchings

Coupling procedures for Markov renewal processes are described. Applications to ergodic theorems for processes with semi-Markov switchings are considered.This paper was partly prepared with the support of NFR Grant F-UP 10257-300.     

Nonstationary curves in Hilbert spaces and their correlation functions II

Partially supportted by Grant MM-22/1991 of MESC     

Asymptotics in the random assignment problem

Summary We show that, in the usual probabilistic model for the random assignment problem, the optimal cost tends to a limit constant in probability and in expectation. The method involves construction of an infinite limit structure, in terms of which the limit constant is defined. But we cannot improve on the known numerical bounds for the limit.Research supported by NSF Grant MCS90-01710     

Minimizing the number of transitions with respect to observation equivalence

Labeled transition systems (lts) provide an operational semantics for many specification languages. In order to abstract unrelevant details of lts's, manybehavioural equivalences have been defined; here observation equivalence is considered. We are interested in the following problem:Given a finite lts, which is the minimal observation equivalent lts corresponding to it? It is well known that the number of states of an lts can be minimized by applying arelational coarsest partition algorithm. However, the obtained lts is not unique (up to the renaming of the states): for an lts there may exist several observation equivalent lts's which have the minimal number of states but varying number of transitions. In this paper we show how the number of transitions can be minimized, obtaining a unique lts.