Rabi Bhattacharya

A statistical approach to estimate the 3D size distribution of spheres from 2D size distributions

Size distribution of rigidly embedded spheres in a groundmass is usually determined from measurements of the radii of the two-dimensional (2D) circular cross sections of the spheres in random flat planes of a sample, such as in thin sections or polished slabs. Several methods have been devised to find a simple factor to convert the mean of such 2D size distributions to the actual 3D mean size of the spheres without a consensus. We derive an entirely theoretical solution based on well-established probability laws and not constrained by limitations of absolute size, which indicates that the ratio of the means of measured 2D and estimated 3D grain size distribution should be π/4 (≈.785). Actual 2D size distribution of the radii of submicron sized, pure Fe 0 globules in lunar agglutinitic glass, determined from backscattered electron images, is tested to fit the gamma size distribution model better than the log-normal model. Numerical analysis of 2D size distributions of Fe 0 globules in 9 lunar soils shows that the average mean of 2D/3D ratio is 0.84, which is very close to the theoretical value. These results converge with the ratio 0.8 that Hughes (1978) determined for millimeter-sized chondrules from empirical measurements. We recommend that a factor of 1.273 (reciprocal of 0.785) be used to convert the determined 2D mean size (radius or diameter) of a population of spheres to estimate their actual 3D size.

On geometric ergodicity of nonlinear autoregressive models

A criterion is derived for the geometric Harris ergodicity of general nonlinear autoregressive models, which imposes a condition on the forcing function only at infinity and does not require that the function be continuous.

Nonparametic estimation of location and dispersion on Riemannian manifolds

A central limit theorem for intrinsic means on a complete flat manifold and some asymptotic properties of the intrinsic total sample variance on an arbitrary complete manifold are given. A studentized pivotal statistic and its bootstrap analogue which yield confidence regions for the intrinsic mean on a complete flat manifold are also derived.

Majorizing kernels and stochastic cascades with applications to incompressible Navier-Stokes equations

A general method is developed to obtain conditions on initial data and forcing terms for the global existence of unique regular solutions to incompressible 3d Navier-Stokes equations. The basic idea generalizes a probabilistic approach introduced by LeJan and Sznitman (1997) to obtain weak solutions whose Fourier transform may be represented by an expected value of a stochastic cascade. A functional analytic framework is also developed which partially connects stochastic iterations and certain Picard iterates. Some local existence and uniqueness results are also obtained by contractive mapping conditions on the Picard iteration.

On a Class of Stable Random Dynamical Systems: Theory and Applications

Abstract We consider a random dynamical system in which the state space is an interval, and possible laws of motion are monotone functions. It is shown that if the Markov process generated by this system satisfies a splitting condition, it converges to a unique invariant distribution exponentially fast in the Kolmogorov distance. A central limit theorem on the time-averages of observed values of the states is also proved. As an application we consider a system that captures an interaction of growth and cyclical forces: of two possible laws, one is monotone, but the other is unimodal with two periodic points. Journal of Economic Literature Classification Numbers: C6, D9.

Controlled semi-Markov models, the discounted case

Abstract Let the state space S be a Borel subset of a complete separable metric space, the action space A compact metric. Existence of stationary optimal policies is proved for general semi-Markov models with possibly unbounded rewards. The corresponding dynamic programming equations are also derived. The paper presents a synthesis and extensions of earlier results.

Asymptotics of solute dispersion in periodic porous media

The concentration

Ergodicity of nonlinear first order autoregressive models

C( {\bf x},t )

Random Iterations of Two Quadratic Maps

of a solute in a saturated porous medium is governed by a second-order parabolic equation

Random Dynamical Systems

\partial C / \partial t = U_0 {\bf b} \cdot \nabla C + \tfrac{1}{2} \sum D_{ij} \partial ^2 C /\partial x_i \partial x_j