Abstract:
The aim of the current work is to synthesize the new heterocyclic pentacyclic condensed systems that combine benzothiophen and benzimidazole/triazole into one molecule. The dibenzothiophene was taken as an initial compound and by consistent “extension” was annihilated the imidazole and triazole nucleuses. As a result two new pentacyclic systems were produced: 3H-, 7H-diimidazole[4,5-b][5,4-g] dibenzothiophene-5,5-dioxide and 3H-, 7H-ditriazole[4,5-b][5,4-g] dibenzothiophene-5,5-dioxide with the promising antimi-crobial activity. Their spectral characteristics were studied.

Abstract:
A perfect cuboid (PC) is a rectangular parallelepiped with rational sides $a,b,c$ whose face diagonals $d_{ab}$, $d_{bc}$, $d_{ac}$ and space (body) diagonal $d_s$ are rationals. The existence or otherwise of PC is a problem known since at least the time of Leonhard Euler. This research establishes equivalent conditions of PC by nontrivial rational solutions $(X,Y)$} and $(Z,W)$} of congruent number equation $ y^2=x^3-N^2x$, where product $XZ$ is a square. By using such pair of solutions five parametrizations of nearly-perfect cuboid (NPC) (only one face diagonal is irrational) and five equivalent conditions for PC were found. Each parametrization gives all possible NPC. For example, by using one of them -- invariant parametrization for sides and diagonals of NPC are obtained: $a=2XZN$, $b=|YW|$, $c=|X-Z|\sqrt{XZ}\,N$,$d_{bc}=|XZ-N^2|\sqrt{XZ}$, $d_{ac}=|X+Z|\sqrt{XZ}\,N$, $d_s=(XZ+N^2)\sqrt{XZ}$; and condition of the existence of PC is the rationality of $d_{ab} = \sqrt{Y^2W^2+4N^2X^2Z^2}$. Because each parametrization is complete, inverse problem is discussed. For given NPC is found corresponding congruent number equation (i.e. congruent number) and its solutions.

Abstract:
We consider nearly-perfect cuboids (NPC), where the only irrational is one of the face diagonals. Obtained are three rational parametrizations for NPC with one parameter.

Abstract:
By using pairs of nontrivial rational solutions of congruent number equation $$ C_N:\;\;y^2=x^3-N^2x, $$ constructed are pairs of rational right (Pythagorean) triangles with one common side and the other sides equal to the sum and difference of the squares of the same rational numbers. The parametrizations are found for following Diophantine systems: \begin{align*} (p^2\pm q^2)^2-a^2 & =\square_{1,2}\,, \\[0.2cm] c^2-(p^2\pm q^2)^2 & =\square_{1,2}\,, \\[0.2cm] a^2+(p^2\pm q^2)^2 & =\square_{1,2}\,, \\[0.2cm] (p^2\pm q^2)^2-a^2 & =(r^2\pm s^2)^2. \end{align*}

Abstract:
It is proved that the third Mac Lane cohomology group of a ring R with coefficients in a bimodule B classifies categorical rings having R as the ring of isomorphism classes of objects and B as the bimodule of automorphisms of the neutral object.

Abstract:
Cohomology of a topological space with coefficients in stacks of abelian 2-groups is considered. A 2-categorical analog of the theorem of Grothendieck is proved, relating cohomology of the space with coefficients in a 2-stage spectrum and the Ext groups of appropriate stacks.

Abstract:
The article has been withdrawn - the proof of the final corollary turned out to rely on an unproven statement. If the authors will manage to repair the argument, resubmission will be made.

Abstract:
This paper provides some explicit expressions concerning the formal group laws of the Brown-Peterson cohomology, the cohomology theory obtained from Brown-Peterson theory by killing all but one Witt symbol, the Morava $K$-theory and the Abel cohomology.

Abstract:
We single out a large class of semisimple singularities with the property that all roots of the Poincar\'e polynomial of the Lie algebra of derivations of the corresponding suitably (not necessarily quasihomogeneously) graded moduli algebra lie on the unit circle; for a still larger class there might occur exactly four roots outside the unit circle.