On the martingale property of certain local martingales

The stochastic exponential ${Z_t= {\rm exp}\{M_t-M_0-(1/2)\langle M,M\rangle_t\}}$ of a continuous local martingale M is itself a continuous local martingale. We give a necessary and sufficient condition for the process Z to be a true martingale in the case where ${M_t=\int_0^t b(Y_u)\,dW_u}$ and Y is a one-dimensional diffusion driven by a Brownian motion W. Furthermore, we provide a necessary and sufficient condition for Z to be a uniformly integrable martingale in the same setting. These conditions are deterministic and expressed only in terms of the function b and the drift and diffusion coefficients of Y. As an application we provide a deterministic criterion for the absence of bubbles in a one-dimensional setting.     

New Two-Dimensional Ice Models

This paper presents a new approach for enumerating all hydrogen bond arrangements of ice-like systems with periodic boundary conditions. It is founded on a topological procedure for the dimensional reduction and a new variant of the transfer matrix method based on small conditional transfer matrices. We consider a couple of new two-dimensional ice models on very unusual lattices. One of them is the twisted square ice model with crossing H-bonds. The other is the digonal-hexagonal model with double H-bonds. In spite of their uncommonness, these models are quite realistic, because from the standpoint of combinatorics and topology they are equivalent to the layers of usual hexagonal ice Ih under periodic boundary conditions in one of the directions. The exact proton configuration statistics for a number of 2D-expanded unit cells of hexagonal ice Ih and the residual entropy of the new ice models in the large system limit are presented.     

Excited State Intramolecular Proton Transfer in O-Tosylaminobenzaldehyde

Excited state intramolecular proton transfer (ESIPT) in o-tosylaminobenzaldehyde has been investigated. According to quantum-chemical calculations ESIPT in o-tosylaminobenzaldehyde is barrierless. Product of ESIPT undergoes efficient nonradiative deactivation caused by internal rotation of C(H)OH-group. The solvent orientational relaxation in anionic form of o-tosylaminobenzaldehyde was detected. The mechanism of anionic form fluorescence quenching at the addition of the base in a protic solvent is proposed. It consists in the intermolecular proton transfer from the protonated base to oxygen atom of aldehyde group followed by the internal rotation of C(H)OH-group.     

Combined natural convection heat and mass transfer in an enclosure having finite thickness walls

Unsteady three-dimensional conjugate heat and mass transfer in an enclosure having finite thickness heat-conducting walls has been analyzed numerically. The governing unsteady, three-dimensional flow, energy and contaminant transport equations for the gas cavity and unsteady heat conduction equation for solid walls, written in dimensionless terms of the vector potential functions, the vorticity vector, the temperature and the concentration, have been solved using an iterative implicit finite-difference method. Main attention was paid to the effects of the Rayleigh number, buoyancy ratio and the dimensionless time on the flow structure and heat and mass transfer regimes. It should be noted that the dominant cause of the oscillations in the dimensionless time dependences of the average Nusselt number on the heat source surface and the average Sherwood number on the contaminant source surface at Ra>5?10⁵ is the mutual influence of the analyzed object geometry and the thermo-diffusivity impact on the flow. The change in the buoyancy ratio can lead to the essential modifications of the flow, temperature and concentration fields owing to the significant influence of the concentration gradient.     

The Role of Noise in Finite Ensembles of Nanomagnetic Particles

The dynamics of finitely many nanomagnetic particles are described by the stochastic Landau–Lifshitz–Gilbert equation. We show that the system relaxes exponentially quickly to the unique invariant measure which is described by a Boltzmann distribution. We present two approaches to verify this result. The first uses the general theory of (Meyn and Tweedy, Adv Appl Prob 24:542–574, 1992; Meyn and Tweedy, Adv Appl Prob 25:487–517 1993; Meyn and Tweedy, Adv Appl Prob 25:518–548, 1993) for Markov chains, which involves the concepts of a Lyapunov structure, and irreducibility of transition probabilities; we show exponential convergence in a supremum topology, but lack explicit rates. The second approach shows exponential ergodicity in a weaker L ² topology, with an explicit rate of convergence of the Arrhenius type law. Then, we discuss two implicit discretizations to approximate transition functions at both finite and infinite times: the first scheme is shown to inherit the geometric ‘unit-length’ property of single spins, as well as the Lyapunov structure, and is shown to be geometrically ergodic; moreover, iterates converge strongly with a rate for finite times. The second scheme is computationally more efficient since it is linear; it is shown to converge weakly at an optimal rate for all finite times. We use a general result of Shardlow and Stuart (Siam J Numer Anal 37(4):1120–1137 2000) to conclude convergence to the invariant measure of the limiting problem for both discretizations.     

Application of a high-speed laser-induced fluorescence technique for studying the three-dimensional structure of annular gas–liquid flow

The wavy structure of liquid film in annular gas?Cliquid flow was studied using a high-speed modification of the laser-induced fluorescence (LIF) technique, which was adapted for three-dimensional measurements. The three-dimensional structure of different types of waves in regimes with and without liquid entrainment was investigated. A comparison of the circumferential size of different types of waves was performed. Disturbance waves at high liquid Reynolds numbers were shown to be circumferentially non-uniform, and it was shown that this non-uniformity affects the generation of ripples.     

Synthesis of the new adducts of imines and enamines with PH acids and their derivatives

Nucleophilic or radical addition of esters of trivalent organophosphorus acids with PH fragments to various imines and enamines is proposed as convenient methods for the synthesis of new substituted aminomethyl organophosphorus compounds with three‐, four‐, and five‐coordinated phosphorus. Also the new functionalized derivatives of these compounds with acyl and methanesulfonyl moieties are synthesized, and some properties of the obtained compounds are presented. © 2009 Wiley Periodicals, Inc. Heteroatom Chem 20:70–80, 2009; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/hc.20513