Free boundary problem for a fully nonlinear and degenerate parabolic equation in an angular domain

This paper concerns with the problem of how to running an insurance company to maximize his total discounted expected dividends. In our model, the dividend rate is limited in $[0,M]$ and the company is allowed to transfer any proportion of risk by reinsuring. So there are two strategies which we call dividend strategy and reinsurance strategy. The objective function and the corresponding optimal two strategies are the solution and the two free boundaries of the following Barenblatt parabolic equation

$v_t?\underset{0\le a\le 1}{\max }?(\frac{1}{2}{a}^2{\sigma }^2{v}_{xx}+a\mu v_x)+cv?\underset{0\le l\le M}{\max }?[(1?v_x)l]=0$

under certain boundary conditions on an angular domain

$Q_T=\{(x,t)|0<x<Mt,0<t\le T\}.$

The main effort is to analyze the properties of the solution and the free boundaries to show the optimal decision for the insurance company.     

Boundary value problems for a quasilinear parabolic equation with an unknown coefficient

The work is connected with the mathematical modeling of physical–chemical processes in which inner characteristics of materials are subjected to changes. The considered nonlinear parabolic models consist of a boundary value problem for a quasilinear parabolic equation with an unknown coefficient multiplying the derivative with respect to time and, moreover, involve an additional relationship for a time dependence of this coefficient. For such a system, conditions of unique solvability in a class of smooth functions are studied on the basis of the Rothe method. The proposed approach involves the proof of a priori estimates in the difference-continuous Hölder spaces for the corresponding differential-difference nonlinear system that approximates the original system by the Rothe method. These estimates allow one to establish the existence of the smooth solutions and to obtain the error estimates of the approximate solutions.As examples of applications of the considered nonlinear boundary value problems, the models of destruction of heat-protective composite under the influence of high temperature heating are discussed.     

On a nonlocal degenerate parabolic problem

Conditions for the existence and uniqueness of weak solutions for a class of nonlinear nonlocal degenerate parabolic equations are established. The asymptotic behaviour of the solutions as time tends to infinity are also studied. In particular, the finite time extinction and polynomial decay properties are proved.     

平均值不等式的证明及其应用

n n n设 a1，a2，…，an为正数，若∏i＝1 ai ＝1或∑i＝1 ai ＝1，借助数学归纳法可相应地证明∑ai ≥ n或i＝1 n nn∏ai ≤1．这两个不等式可用于证明平均值不等式，并由此得出三者相互等价．实例说明平均值不等式在求数列极限方面的应用． i＝1     

Modern information technologies in construction of kinetic models for reactions of metal complex catalysis

For detailed study of complex chemical reactions mechanisms experiment is conducted for selected private reactions. This causes a problem of kinetic parameters getting—the same set of rate constants must describe both public and private reaction stages, and also a general mechanism. In this paper, solution of this problem for a reaction of olefins hydroalumination is proposed. To optimize the computational process a methodology of parallelization is elaborated. On the base of parallel computations, a kinetic model for the reaction assigned is constructed, and on its base, the physical and chemical conclusions about reaction mechanism are done.     

Quantification of 1‐(propan‐2‐ylamino)‐4‐propoxy‐9H‐thioxanthen‐9‐one (TX5), a newly synthetized P‐glycoprotein inducer/activator,in biological samples: method development and validation

A simple, rapid and economical method was developed and validated for the analysis and quantification of 1‐(propan‐2‐ylamino)‐4‐propoxy‐9H ‐thioxanthen‐9‐one (TX5), a P‐glycoprotein inducer/activator, in biological samples, using reverse‐phase high‐performance liquid chromatography (HPLC). A C₁₈ column and a mobile phase composed of methanol–water (90/10, v /v) with 1% (v/v) triethylamine, at a flow rate of 1 mL/min, were used for chromatographic separation. TX5 standards (0.5–150 μm ) were prepared in human serum. Methanol was used for TX5 extraction and serum protein precipitation. After filtration, samples were injected into the HPLC apparatus and TX5 was quantified by a conventional UV detector at 255 nm. The TX5 retention time was 13 min in this isocratic system. The method was validated according to ICH guidelines for specificity/selectivity, linearity, accuracy, precision, limits of detection and quantification (LOD and LOQ) and recovery. The method was proved to be selective, as there were no interferences of endogenous compounds with the same retention time of TX5. Also, the developed method was linear (r ² ≥ 0.99) for TX5 concentrations between 0.5 and 150 μm and the LOD and LOQ were 0.08 and 0.23 μm , respectively. The results indicated that the reported method could meet the requirements for TX5 analysis in the trace amounts expected to be present in biological samples.     

STEREOSELECTIVE REACTIONS OF CHIRAL AMINES WITH RACEMIC CHLOROPHOSPHINES

Racemic chlorophosphines react stereoselectively with chiral l-phenylethylamines or amino acid esters to give diastereomerically enriched aminophosphines 3, which were isolated as diastereomerically pure crystalline borane complexes. Oxidation, thionation, the reaction with methyl iodide provide optically active derivatives of aminophosphines. (R,S)- and (S,S)-stereomers of phosphinic acid amides were separated by crystallization and a flash-chromatography. The stereochemical properties of phosphorus acid amides were investigated. The mechanism of asymmetric induction at the trivalent phosphorus atom was rationalized.