Synthesis of novel 2-perfluoroacylcyclohexane-1,3-diones

A one-pot synthesis of 2-perfluoroalkanoylcyclohexane-1,3-diones via C-acylation of cyclohexane-1,3-diones with N-perfluoroacylimidazole as an acylating agent is reported. A reaction was examined with isolated N-trifluoroacetylimidazole and with N-perfluoroacylimidazoles generated in situ from perfluorocarboxylic acid anhydrides or perfluorocarboxylic acids.     

The generation of three-dimensional internal waves and attendant boundary layers in a viscous continuously stratified fluid. Construction of an analytical solution

An analytical solution of a linearized problem of the emission of periodic internal waves by part of a plane which oscillates with a small amplitude in an arbitrary direction in a viscous exponentially stratified fluid is constructed. Solutions of the dispersion equation are given for all positions of the emitting surface (arbitrary, vertical, horizontal, and critical when one of the beam propagation directions is collinear with the emitting surface). The possibility of transition to the case of a uniform fluid, which is important for applications, is analyzed.     

Investigation of multicomponent fluoride optical materials in the UV spectral region: I. Single crystals of Ca1−x R xF2+x (R = Sc, Y, La, Yb, Lu) solid solutions

Crystalline materials that are transparent in the vacuum UV spectral region and currently used have been reviewed. Transmission of crystals of solid solutions with the fluorite structure Ca_1?x R _xF_2+x (R = Sc, Y, La, Yb, Lu) in the UV and vacuum UV spectral regions has been investigated. It is shown that application of different methods of purification of fluorides from some impurities can significantly improve the optical quality of fluoride multicomponent crystals in the short-wavelength spectral region.     

Coset extensions of <Emphasis Type=”Italic”>S</Emphasis>-sets

Summary Three weak variants of compactness which lie strictly between compactness and quasicompactness, are introduced. Their basic properties are studied. The interplay with mapping and their direct and inverse preservation under mappings are investigated. In the process three decompositions of compactness are observed.     

Admissible slopes for monotone and convex interpolation

Summary In many applications, interpolation of experimental data exhibiting some geometric property such as nonnegativity, monotonicity or convexity is unacceptable unless the interpolant reflects these characteristics. This paper identifies admissible slopes at data points of variousC ¹ interpolants which ensure a desirable shape. We discuss this question, in turn for the following function classes commonly used for shape preserving interpolations: monotone polynomials,C ¹ monotone piecewise polynomials, convex polynomials, parametric cubic curves and rational functions.     

INTERSECTION OF PRIME SUBMODULES AND DIMENSION OF MODULES

The aim of this paper is to study the conditions by which a P-prime submodule can be expressed as a finite intersection or union of P-prime submodules. Also corresponding to dimension and rank of modules, some equivalent conditions for a ring to be a Dedekind domain are given.     

Inverse Problem for a Mathematical Model of Adsorption Dynamics

A mixed problem is considered for a system of partial differential equations modeling the process of adsorption dynamics. An existence and uniqueness theorem is proved for this problem, and the solution properties are investigated. The inverse problem is posed, involving the determination of the system coefficient given additional information about the solution. A uniqueness theorem is proved for the solution of the inverse problem.__________Translated from Prikladnaya Matematika i Informatika, No. 16, pp. 5 – 14, 2004.     

Generalized lattice model of liquid state and its relation to Ginzburg–Landau theory

Basic principles of the generalized lattice model of multicomponent condensed systems are formulated. Short-range parts of interatomic interactions are taken into account by means of the geometric constraints method. Long-range parts of the interactions are taken into account in mean field approximation. The expression for Helmholtz free energy is obtained. A system of integral equations for the equilibrium distributions of components is derived. The asymptotic properties of its solutions are investigated. Moment expansion of interatomic interactions and localization of integral terms in free energy is obtained. A Ginzburg–Landau-like functional of free energy is derived.     

Simulation of viscous water column collapse using adapting hierarchical grids

An adaptive hierarchical grid‐based method for predicting complex free surface flows is used to simulate collapse of a water column. Adapting quadtree grids are combined with a high‐resolution interface‐capturing approach and pressure‐based coupling of the Navier–Stokes equations. The Navier–Stokes flow solution scheme is verified for simulation of flow in a lid‐driven cavity at Re=1000. Two approaches to the coupling of the Navier–Stokes equations are investigated as are alternative face velocity and hanging node interpolations. Collapse of a water column as well as collapse of a water column and its subsequent interaction with an obstacle are simulated. The calculations are made on uniform and adapting quadtree grids, and the accuracy of the quadtree calculations is shown to be the same as those made on the equivalent uniform grids. Results are in excellent agreement with experimental and other numerical data. A sharp interface is maintained at the free surface. The new adapting quadtree‐based method achieves a considerable saving in the size of the computational grid and CPU time in comparison with calculations made on equivalent uniform grids. Copyright © 2005 John Wiley & Sons, Ltd.