Continuity of Multiplication in the Largest Compactification of a Locally Compact Group

G^LUC is the largest semigroup compactification of the locallycompact group G. When G is not compact, given q G* = G^LUC \G, there is p G* such that x qx is discontinuous at p (Theorem2). If G is -compact, there is one p which will serve for allq (Theorem 1).     

Separation of convex sets by extreme hyperplanes

The problem of separation of convex sets by extreme hyperplanes (functionals) in normed linear spaces is examined. The concept of a bar (a closed set of special form) is introduced; it is shown that a bar is characterized by the property that any point not lying in it can be separated from it by an extreme hyperplane. In two-dimensional spaces, in spaces with strictly convex dual, and in the space of continuous functions, any two bars are extremely separated. This property is shown to fail in the space of summable functions. A number of examples and generalizations are given.     

On Closed Ideals of βG

For every discrete group G, the Stone-Cech compactification βG has a natural structure of a right topological semigroup. Using the concept of Boolean group ideal on G, we propose a way of constructing closed ideals of the semigroup βG. In particular, we show that, for every infinite Abelian group G of cardinality κ, there are distinct closed ideals in βG.     

Deformation-induced structural transformations in Si and the initial stage of mechanical alloying of Si and Fe

The methods of X-ray diffraction, Mössbauer spectroscopy, IR spectroscopy, and laser diffractometry are employed to study the changes in the structure and phase transformations that accompany mechanical alloying of Si and ⁵⁷Fe used in an atomic ratio of 99: 1. It is established that the process comprises the development of a nanocrystalline state of Si with crystallite sizes smaller than 10 nm; the formation of an amorphous phase of Si at particle surfaces and in near-boundary distorted zones of interfaces in Si nanostructure; the penetration of ⁵⁷Fe atoms along grain boundaries; and the formation of Si-Fe clusters, the local environment of Fe atoms in which is typical of a deformed α-FeSi₂ phase, with these clusters being located in the interfaces.     

Chromium mask for plasma-chemical etching of Al x Ga1 − x N layers

The influence of the mask material used for formation of GaN mesa-structures using plasma-chemical etching in a BCl₃ : Ar : N₂ atmosphere is studied. It is shown that a bilayer SiO₂/Cr mask in which the thickness of the chromium layer is six to seven times smaller than a desired etch depth provides effective protection of the surface and allows the formation of structures with a flat surface for an etch depth of up to 2.5 μm. When the mask is produced by conventional lithography, the slope of the side walls is no more than 10° and decreases when liftoff lithography is applied.