The combinatorial derivation and its inverse mapping

Let G be a group and P _G be the Boolean algebra of all subsets of G. A mapping Δ: P _G → P _G defined by Δ(A) = {g ∈ G: gA ∩ A is infinite} is called the combinatorial derivation. The mapping Δ can be considered as an analogue of the topological derivation d: P _X → P _X , A ? A ^d , where X is a topological space and A ^d is the set of all limit points of A. We study the behaviour of subsets of G under action of Δ and its inverse mapping ?. For example, we show that if G is infinite and I is an ideal in P _G such that Δ(A) ∈ I and ?(A) ? I for each A ∈ I then I = P _G .     

When do several linear operators share an invariant cone?

We establish a criterion for a finite family of matrices to possess a common invariant cone. The criterion reduces the problem of existence of an invariant cone to equality of two special numbers that depend on the family. In spite of theoretical simplicity, the practical use of the criterion may be difficult. We show that the problem of existence of a common invariant cone for four matrices with integral entries is algorithmically undecidable. Corollaries of the criterion, which give sufficient and necessary conditions, are derived. Finally, we introduce a “co-directional number” of several matrices. We prove that this parameter is close to zero iff there is a small perturbation of matrices, after which they get an invariant cone. An algorithm for its computation is presented.     

On linear selections of convex set-valued maps

We study continuous subadditive set-valued maps taking points of a linear space X to convex compact subsets of a linear space Y. The subadditivity means that φ(x ₁ + x ₂) ⊂ φ(x ₁) + φ(x ₂). We characterize all pairs of locally convex spaces (X, Y) for which any such map has a linear selection, i.e., there exists a linear operator A: X → Y such that Ax ∈ φ(x), x ∈ X. The existence of linear selections for a class of subadditive maps generated by differences of a continuous function is proved. This result is applied to the Lipschitz stability problem for linear operators in Banach spaces.     

Balleans of topological groups

A subset S of a topological group G is called bounded if, for every neighborhood U of the identity of G, there exists a finite subset F such that S ⊆ FU, S ⊆ UF. The family of all bounded subsets of G determines two structures on G, namely the left and right balleans B _l (G) and B _r (G), which are counterparts of the left and right uniformities of G. We study the relationships between the uniform and ballean structures on G, describe all topological groups admitting a metric compatible both with uniform and ballean structures, and construct a group analogue of Higson’s compactification of a proper metric space.     

Purification of human recombinant anti-mullerian hormone and its derivatives

Anti-mullerian hormone (AMH) is a cytokine of transforming growth factor β (TGF-β) superfamily able to induce apoptosis in cells bearing specific AMH type II receptors (AMHRII). AMHRII is overexpressed in some malignant cells, so at present recombinant AMH (rAMH) is considered as a new candidate antineoplastic drug. The use of rAMH may be especially effective in case of such severe diseases as ovarian, prostate and breast cancer. However, the development of a new drug is hampered by the laboriousness of obtaining highly purified rAMH and by the lack of data about the pharmacological characteristics of rAMH derivatives. In this work, we obtained preparations of prohormone, half-cleaved rAMH and a C-terminal fragment of rAMH, which was confirmed by qualitative and quantitative analyses. To obtain rAMH and its derivatives we used a previously developed highly effective producer strain containing the optimized human AMH gene. The production process has been divided into several stages: (a) rAMH biosynthesis in the bioreactor; (b) culture media preparation; (c) purification of rAMH and its derivatives using immunoaffinity chromatography and reversed-phase HPLC; (d) identification of the purified proteins by immunoblotting and analytical reversed-phase HPLC; and (e) evaluation of the hormone forms activity. The obtained proteins may be used in preclinical trials and in vitro study of rAMH derivatives properties.     

Small subsets of groups

Given an infinite group G and an infinite cardinal κ|G|, we say that a subset A of G is κ-large (κ-small) if there exists F[G]^<κ such that G=FA (GFA is κ-large for each F[G]^<κ). The subject of the paper is the family of all κ-small subsets. We describe the left ideal of the right topological semigroup βG determined by . We study interrelations between κ-small and other (P_κ-small and κ-thin) subsets of groups, and prove that G can be generated by some 2-thin subsets. We partition G in countable many subsets which are κ-small for each κω. We show that [G]^<κ is dual to provided that either κ is regular and κ=|G|, or G is Abelian and κ is a limit cardinal, or G is a divisible Abelian group.     

Use of short-wavelength synchrotron radiation to measure luminescence quantum yield and the luminescence excitation spectrum of polymers

We describe an optical diagnostics module and the instrumental and methodological features of ultrahigh vacuum experiments investigating the optical characteristics of condensed media in the short-wavelength (hv ~ 3.5–25 eV) range of the spectrum of probing synchrotron radiation. We give a brief presentation of the results of an experimental determination of the spectral dependence of the luminescence quantum yield and the luminescence excitation spectrum of ablatable polymer dielectrics on the Kurchatov synchrotron radiation source at values of the probing radiation power density (I ₀ ~ 10¹² photons/cm²∙sec) that are below threshold for extended surface vaporization and a surface temperature of the condensed targets equal to 77–300 K.     

On the Suslin Number of Totally-Bounded Left-Topological Groups

For every infinite cardinal , we construct a zero-dimensional totally-bounded left-topological group with Suslin number .