Brownian motion reflected on Brownian motion

 We study Brownian motion reflected on an ``independent' Brownian path. We prove results on the joint distribution of both processes and the support of the parabolic measure in the space-time domain bounded by a Brownian path. We show that there exist two different natural local times for a Brownian path reflected on a Brownian path. Received: 25 October 2000 / Revised version: 30 March 2001 / Published online: 20 December 2002     

Organosulphur compounds-XXXVIII Axial preference of a 2-dimethoxyphosphoryl substituent in 1,3,5-trithiane

¹H-NMR studies and X-ray analysis indicate that 2-dimethoxyphosphoryl-1,3,5-trithiane exists both in solution and in the crystal in a chair conformation with the dimethoxyphosphoryl group at C-2 being axial.     

Crystal structure of two oximes incis-pinane series

The structures of racemic 2-hydroxy-4-hydroxyiminopinan-3-one (1) and 3-hydroxyimino-cis-pinan-4-one monohydrate (2) have been determined by X-ray diffraction. The crystals of1 are monoclinic:P2₁/c, a=7.838(2),b=10.776(3),c=12.025(3) Å, =105.09(1)°,Z=4, room temp.; the crystals of2 are orthorhombic;P2₁2₁2₁,a=8.181(3),b=9.250(3),c=14.085(6)Å,Z=4, T=80K. Both structures were solved by direct methods, and refined toR=0.037 for 1920 reflections (1) and 0.032 for 1776 reflections (2). The absolute configuration of2 was not established. The hydroxyimine groups in both crystals and water molecules in2 participate in the network O–H...O of hydrogen bonds.     

The Boundary Harnack Principle for Non-Divergence form Elliptic Operators

If L is a uniformly elliptic operator in non-divergence form,the boundary Harnack principle for the ratio of positive L-harmonicfunctions holds in Hölder domains of order if > . A counterexample shows that is sharp. For Hölder domainsof order with (0,1], the boundary Harnack principle holds providedthe domain also satisfies a strong uniform regularity condition.     

An ‘odd’ formula for the volume of three-dimensional lattice polyhedra

Let be the tiling of R ³ with unit cubes whose vertices belong to the fundamental lattice L ₁ of points with integer coordinates. Denote by L _n the lattice consisting of all points x in R ³ such that nx belongs to L ₁. When the vertices of a polyhedron P in R ³ are restricted to lie in L ₁ then there is a formula which relates the volume of P to the numbers of all points of two lattices L ₁ and L _n lying in the interior and on the boundary of P. In the simplest case of the lattices L ₁ and L ₂ there are 27 points in each cube from whose relationships to the polyhedron P must be examined. In this note we present a new formula for the volume of lattice polyhedra in R ³ which involves only nine points in each cube of : one from L ₂ and eight belonging to L ₄. Another virtue of our formula is that it does not employ any additional parameters, such as the Euler characteristic.     

Transplanting maximal inequalities between Laguerre and Hankel multipliers

We supplement our previous paper [9] by adding a theorem that transplantsL ^p -norm maximal inequalities for Laguerre multipliers. As an immediate consequence we obtain negative results concerningL ^p -estimates of partial sum maximal operators for Laguerre expansions.Research supported in part by KBN grant No. 2 PO3A 030 09.