Abstract:
In this paper we consider the original and different generalizations of Postnikov-Shapiro algebra, see~\cite{PSh}. Firstly, for a given graph $G$ and a positive integer $t$, we generalize the notion of Postnikov-Shapiro algebras counting forests in $G$ to an algebra counting $t$-labelled forests. We also prove that for large $t$ we can restore the Tutte polynomial of $G$ from the Hilbert series of such algebra. Secondly, we prove that the original Postnikov-Shapiro algebra counting forests depends only on the matroid of $G$. And conversely, we can reconstruct this matroid from the latter algebra. Similar facts hold for analogous algebras counting trees in connected graphs. Thirdly, we present a generalization of such algebras for hypergraphs. Namely, we construct a certain family of algebras for a given hypergraph, such that for almost algebras from this family, their Hilbert series is the same. Finally, we present the definition of a hypergraphical matroid, whose Tutte polynomial allows us to calculate this generic Hilbert series.

Abstract:
In this note by using elementary considerations, we settle Fr\"oberg's conjecture for a large number of cases, when all generators of ideals have the same degree.

Abstract:
A new shortest proof of Kotzig's Theorem about graphs with unique perfect matching is presented in this paper. It is well known that Kotzig's theorem is a consequence of Yeo's Theorem about edge-colored graph without alternating cycle. We present a proof of Yeo's Theorem based on the same ideas as our proof of Kotzig's theorem.

Abstract:
Given a plane curve $\gamma: S^1\to \mathbb R^2$, we consider the problem of determining the minimal number $I(\gamma)$ of inflections which curves $\mbox{diff}(\gamma)$ may have, where $\mbox{diff}$ runs over the group of diffeomorphisms of $\mathbb R^2$. We show that if $\gamma$ is an immersed curve with $D(\gamma)$ double points and no other singularities, then $I(\gamma)\leq 2D(\gamma)$. In fact, we prove the latter result for the so-called plane doodles which are finite collections of closed immersed plane curves whose only singularities are double points.

Abstract:
For a given graph $G$, we construct an associated commutative algebra, whose dimension is equal to the number of $t$-labeled forests of $G$. We show that the dimension of the $k$-th graded component of this algebra also has a combinatorial meaning and that its Hilbert polynomial can be expressed through the Tutte polynomial of $G$.

Abstract:
We analyze the spectral distribution of localisation in a 1D diagonally disordered chain of fragments each of which consist of m coupled two-level systems. The calculations performed by means of developed perturbation theory for joint statistics of advanced and retarded Green’s functions. We show that this distribution is rather inhomogeneous and reveals spectral regions of weakly localized states with sharp peaks of the localization degree in the centers of these regions.

Abstract:
The basic morphological aspects of auditory cortex organization in different orders of eutherian mammals are considered in the present review. The modern data describing a partitioning of mammalian auditory cortex into subfields are presented. A detailed observation of the structural organization of primary auditory cortex is given, as well as a review of recent morphological data about secondary auditory areas. Another section describes the system of auditory cortical projections. The data are considered from the perspective of possible homologies existing between the auditory cortices in different mammalian species.

Abstract:
For an oriented cohomology theory A and a relative cellular space X, we decompose the A-motive of X into a direct sum of twisted motives of the base spaces. We also obtain respective decompositions of the A-cohomology of X. Applying them, one can compute A(X), where X is an isotropic projective homogeneous variety and A means algebraic K-theory, motivic cohomology or algebraic cobordism MGL.

Abstract:
Let $D$ be a strongly connected digraphs on $n\ge 4$ vertices. A vertex $v$ of $D$ is noncritical, if the digraph $D-v$ is strongly connected. We prove, that if sum of the degrees of any two adjacent vertices of $D$ is at least $n+1$, then there exists a noncritical vertex in $D$, and if sum of the degrees of any two adjacent vertices of $D$ is at least $n+2$, then there exist two noncritical vertices in $D$. A series of examples confirm that these bounds are tight.