Short-time isoprene polymerization under the action of a titanium–magnesium catalyst

The short-time polymerization of isoprene under the action of a TiCl₄/MgCl2?i-Bu₃Al heterogeneous catalyst has been investigated. Pulse mixing of the catalyst and monomer in a cylindrical tubular reactor with a certain length followed by ethanol injection has made it possible to carry out polymerization for 0.1?0.7 s. In the first 0.3 s, when there is a considerable rise in the activity of the catalyst, living polymerization of isoprene takes place. In this period, polyisoprene has up to 95% trans-1,4 units. Extending the polymerization time to 0.7 s diminishes the average molar mass of polyisoprene, broadens its molar mass distribution, and decreases the concentration of trans-1,4 units to 83%. The data of this study have been analyzed on the basis of the kinetic continuity of the polymer chain initiation and growth.     

General branching processes: Convergence to irzhina processes

Translated from Problemy Ustoichivosti Stokhasticheskikh Modelei, Trudy Seminara, pp. 74–82, 1990.     

Analyticity with respect to a parameter of a solution of a general linear boundary-value problem for a functional-differential equation

Translated fromIssledovaniya po Prikladnoi Matematike, No. 19, 1992, pp. 120–125.     

Extinction times for a birth–death process with weak competition

We consider a birth–death process with birth rates iλ and death rates iμ+i(i?1)θ, where i is the current state of the process. A positive competition rate θ is assumed to be small. In the supercritical case where λ > μ, this process can be viewed as a demographic model for a population with high carrying capacity around (λ?μ)/θ. The article reports in a self-contained manner on the asymptotic properties of the time to extinction for this logistic branching process as θ → 0. All three reproduction regimes λ > μ, λ < μ, and λ = μ are studied.     

Limiting behavior of general branching processes

Mathematical Notes -     

Structural and kinetic continuum in the butadiene polymerization with titanium catalysts

The short-time polymerization of butadiene induced by heterogeneous titanium catalysts was studied for the first time. In the 0.1?0.3 s time interval, polymerization is characterized by a considerable decrease in the chain propagation rate constant and 1,4-stereospecificity of the catalysts. A structure kinetic continuum model was proposed in which the initiation step and the first propagation steps are kinetically continuous. The violation of this continuity changes the catalyst stereospecificity and creates conditions for chain transfer reactions.     

Evolution of branching processes in a random environment

This review paper presents the known results on the asymptotics of the survival probability and limit theorems conditioned on survival of critical and subcritical branching processes in independent and identically distributed random environments. This is a natural generalization of the time-inhomogeneous branching processes. The key assumptions of the family of population models in question are nonoverlapping generations and discrete time. The reader should be aware of the fact that there are many very interesting papers covering other issues in the theory of branching processes in random environments which are not mentioned here.     

Correctness of the Dirichlet problem for a class of multidimensional hyperbolic-parabolic equations

It is shown that the Dirichlet problem in a cylindrical domain for a class of multidimensional hyperbolic-parabolic equations is uniquely solvable. The criterion of uniqueness of a regular solution is obtained.