On integral conditions in the mapping theory

Interconnections between various integral conditions that play an important role in the theory of space mappings and in the theory of degenerate Beltrami equations in the plane are established.     

Mathematical Models of Incompressible Fluids as Singular Limits of Complete Fluid Systems

A rigorous justification of several well-known mathematical models of incompressible fluid flows can be given in terms of singular limits of the scaled Navier-Stokes-Fourier system, where some of the characteristic numbers become small or large enough. We discuss the problem in the framework of global-in-time solutions for both the primitive and the target system.     

An addendum to the multiplication and factorization theorems of brodskiǐ—livšic—potapov

In this paper the classical Brodskiǐ—Liv?ic— operator colligation is generalized to certain pairs of bounded linear operators and the corresponding characteristic operator—valued (transfer) function is introduced. The fundamental results due to M.S. Brodskiǐ, M.S. Liv?ic, and V.P. Potapov are then extended to such colligations. These new types of colligations can be used to obtain, for instance, realization results for general Herglotz—Nevanlinna functions.     

Productive Syntheses of Privileged Scaffolds Inspired by the Recognition of a Diels–Alder Pattern Common to Three Classes of Natural Products

Identification of a common Diels–Alder pattern in three classes of bioactive natural products led us to study the synthesis and cycloaddition of a new class of cyclic dienes readily available from β,γ-unsaturated lactams. A practical and readily scalable route to the parent p-methoxybenzyl-protected 6- and 7-membered β,γ-unsaturated lactams was developed. These were readily transformed into the corresponding O-silylated dienes, which were reacted with dimethyl and diethyl fumarate to yield stereoselectively highly functionalized bicyclic adducts. These exhibited unexpected and versatile transformations upon acid hydrolysis depending on the nature of the dienophile substituents and the acid catalyst. All reactions have been performed on multigram quantities. These transformations provide a convenient, economical, and easily scalable pathway for the rapid construction of functionally and stereochemically dense privileged scaffolds for the construction of libraries of natural products-inspired molecules of pharmacological relevance.