On the properties of an operator of stochastic differentiation constructed on a group

We construct a differential operator by an admissible group in the space L ₂ (?^m,P)and study its properties.     

Properties of the Flows Generated by Stochastic Equations with Reflection

We consider the properties of a random set ϕ _t (ℝ ₊ ^d ), where ϕ _t (x) is a solution of a stochastic differential equation in ℝ ₊ ^d with normal reflection from the boundary that starts from a point x. We characterize inner and boundary points of the set ϕ _t (ℝ ₊ ^d ) and prove that the Hausdorff dimension of the boundary ∂ϕ _t (ℝ ₊ ^d ) does not exceed d − 1. __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 8, pp. 1069 – 1078, August, 2005.     

Effect of magnesium cations on the activity and stability of β-galactosidases

It was shown that the presence of magnesium cations in the reaction mixture increases, approximately twofold, the activity of bacterial Escherichia coli and yeast Kluyveromyces lactis β-galactosidases but does not affect the activity of bovine liver and fungous Penicillium canescens β-galactosidases. The catalytic constants for E. coli and yeast K. lactis β-galactosidases in the presence of 0.01 M and in the absence of Mg²⁺ cations were determined (490 and 220 s^?1 and 59.8 and 37.4 s^?1, respectively). It was shown that the Michaelis constants for these two enzymes are higher in the presence of Mg²⁺ cations, that the thermal stability of E. coli and K. Lactis β-galactosidases is higher in the presence of 0.01 M Mg²⁺, and that the effective rate constants of thermal inactivation of the enzymes are two-to eightfold lower, depending on conditions, in the presence of Mg²⁺ cations. The maximum stabilizing effect of magnesium cations was observed at weak alkaline pH values (7.5–8.5).     

Dissociation and catalytic activity of oligomer forms of β-galactosidases

The dissociation of oligomer forms of bacterial Escherichia coli, yeast Kluyveromices fragilis, and bovine liver β-galactosidases was studied. The catalytic constants for the dimers and tetramers of the bacterial enzyme, dimers and monomers of the animal enzyme, and dimers of the yeast enzyme in the reaction of hydrolysis of 2-nitrophenyl-β-D-galactopyranoside were determined. At 25°C, these values were found to be 180 and 400 s^?1 for the bacterial enzyme, 0.01 and 0.08 s^?1 for the bovine liver enzyme, and 45.4 s^?1 for the yeast enzyme, respectively. The other oligomer forms of the β-galactosidases were inactive under conditions of these experiments.     

Crack propagation as a free boundary problem

A sharp interface model of crack propagation as a phase transition process is discussed. We develop a multipole expansion technique to solve this free boundary problem numerically. We obtain steady state solutions with a self-consistently selected propagation velocity and shape of the crack, provided that elastodynamic effects are taken into account. Also, we find a saturation of the steady state crack velocity below the Rayleigh speed, tip blunting with increasing driving force, and a tip splitting instability above a critical driving force.