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:
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:
Non analytic behaviour of Hanle effect in InGaAs quantum dots is described in terms of a simple 4-level model. Despite simplicity the model makes it possible to explain the observed fracture of Hanle curve at zero magnetic field and obtain quantitative agreement with the experiment.

Abstract:
For a generic (polynomial) one-parameter deformation of a complete intersection, there is defined its monodromy zeta-function. We provide explicit formulae for this zeta-function in terms of the corresponding Newton polyhedra in the case the deformation is non-degenerate with respect to its Newton polyhedra. Using this result we obtain the formula for the monodromy zeta-function at the origin of a polynomial on a complete intersection, which is an analog of the Libgober--Sperber theorem.

Abstract:
We construct a monomorphism from the differential algebra $k\{x\} / [x^m]$ to a Grassmann algebra endowed with a structure of differential algebra. Using this monomorphism we prove primality of $k\{x\} / [x^m]$ and its algebra of differential polynomials, solve one of Ritt's problems and give a new proof of integrality of the ideal $[x^m]$.

Abstract:
In this paper the local singularities of integrable Hamiltonian systems with two degrees of freedom are studied. The topological obstruction to the existence of focus-focus singularity with given complexity was found. It has been showed that only simple focus-focus singularities can appear in a typical mechanical system. The model examples of mechanical systems with complex focus-focus singularity are given.

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:
We study the cluster monomials and cluster complex in $\mathbb C[GL_n/N]$. For we consider the {\em tableau basis} in $\mathbb C[GL_n/N]$. Namely, an element $\Delta_T$ of the tableau basis labeled by a semistandard Young tableau $T$ is the product of the flag minors corresponding to columns of $T$. Our main results state: (i) cluster monomials in $\mathbb C[GL_n/N]$ can be labeled by semistandard Young tableaux such that any cluster monomial has the form $\Delta_T+$ lexicographically smaller terms; (ii) such labeling distinguish the cluster monomial; (iii) for any seed of the cluster algebra on $\mathbb C[GL_n/N]$, we define a cone in $\mathbf D(n)$ generated by tableaux which label the cluster variables of the seed, then these cones form a simlicial fan in $\mathbf D(n)$ ($\mathbf D(n)$ is linear isomorphic to the Gelfand Tseitlin cone).

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:
We present and compare various approaches to a classical selection problem on Graphics Processing Units (GPUs). The selection problem consists in selecting the $k$-th smallest element from an array of size $n$, called $k$-th order statistic. We focus on calculating the median of a sample, the $n/2$-th order statistic. We introduce a new method based on minimization of a convex function, and show its numerical superiority when calculating the order statistics of very large arrays on GPUs. We outline an application of this approach to efficient estimation of model parameters in high breakdown robust regression.