Abstract:
A method of constructing of Darboux coordinates on a space that is a symplectic reduction with respect to a diagonal action of $GL(m})$ on a Cartesian product of $N$ orbits of coadjoint representation of $GL(m)$ is presented. The method gives an explicit symplectic birational isomorphism between the reduced space on the one hand and a Cartesian product of $N-3$ coadjoint orbits of dimension $m(m-1)$ on an orbit of dimension $(m-1)(m-2)$ on the other hand. In a generic case of the diagonalizable matrices it gives just the isomorphism that is birational and symplectic between some open, in a Zariski topology, domain of the reduced space and the Cartesian product of the orbits in question. The method is based on a Gauss decomposition of a matrix on a product of upper-triangular, lower-triangular and diagonal matrices. It works uniformly for the orbits formed by diagonalizable or not-diagonalizable matrices. It is elaborated for the orbits of maximal dimension that is $m(m-1)$.

Abstract:
We simplify Hitchin's description of SU(2)-invariant self-dual Einstein metrics, making use of the tau-function of related four-pole Schlesinger system.

Abstract:
Several semianalytical approaches are now available for describing diffraction of a plane wave by the 2D (two dimensional) traction free isotropic elastic wedge. In this paper we follow Budaev and Bogy who reformulated the original diffraction problem as a singular integral one. This comprises two algebraic and two singular integral equations. Each integral equation involves two unknowns, a function and a constant. We discuss the underlying integral operators and develop a new semianalytical scheme for solving the integral equations. We investigate the properties of the obtained solution and argue that it is the solution of the original diffraction problem. We describe a comprehensive code verification and validation programme.

Abstract:
The geometric model of the pathway linking Schlesinger and Garnier--Painlev\'e~6 systems based on an original orthonormalization of a set of elements in ${sl(2,\mathbb C)}$ is constructed. The explicit polynomial map of the Cartesian products of $n-2$ quadrics (the Zariski-topology chart of the phase space of the Garnier--Painlev\'e~6 system) into the phase space of the Schlesinger system and the rational inverse to this map are presented.

Abstract:
A set of all linear transformations with a fixed Jordan structure $J$ is a symplectic manifold isomorphic to the coadjoint orbit $\mathcal O (J)$ of $GL(N,C)$. Any linear transformation may be projected along its eigenspace to (at least one) coordinate subspace of the complement dimension. The Jordan structure $\tilde J$ of the image is defined by the Jordan structure $J$ of the pre-image, consequently the projection $\mathcal O (J)\to \mathcal O (\tilde J)$ is the mapping of the symplectic manifolds. It is proved that the fiber $\mathcal E$ of the projection is a linear symplectic space and the map $\mathcal O(J) \to \mathcal E \times \mathcal O (\tilde J)$ is a birational symplectomorphysm. The iteration of the procedure gives the isomorphism between $\mathcal O (J)$ and the linear symplectic space, which is the direct product of all the fibers of the projections. The Darboux coordinates on $\mathcal O(J)$ are pull-backs of the canonical coordinates on the linear spaces in question.

Abstract:
We study the influence of energy levels broadening and electron subsystem overheating in island electrode (cluster) on current-voltage characteristics of three-electrode structure. A calculation scheme for broadening effect in one-dimensional case is suggested. Estimation of broadening is performed for electron levels in disc-like and spherical gold clusters. Within the two-temperature model of metallic cluster and by using a size dependence of the Debye frequency the effective electron temperature as a function of bias voltage is found approximately. We suggest that the effects of broadening and electron overheating are responsible for the strong smoothing of current-voltage curves, which is observed experimentally at low temperatures in structures based on clusters consisting of accountable number of atoms.

Abstract:
Pancreatic pseudocysts, abscesses, and walled-off pancreatic necrosis are types of pancreatic fluid collections that arise as a consequence of pancreatic injury. Pain, early satiety, biliary obstruction, and infection are all indications for drainage. Percutaneous-radiologic drainage, surgical drainage, and endoscopic drainage are the three traditional approaches to the drainage of pancreatic pseudocysts. The endoscopic approach to pancreatic pseudocysts has evolved over the past thirty years and endoscopists are often capable of draining these collections. In experienced centers endoscopic ultrasound-guided endoscopic drainage avoids complications related to percutaneous drainage and is less invasive than surgery.

Abstract:
We study the effect of temperature on the tunnel current in a structure based on gold clusters taking into consideration their discrete electronic spectra. We suggest that an overheating of electron subsystem leads to the disappearance of a current gap and gradual smoothing of current--voltage curves that is observed experimentally.

Abstract:
We suggest a method for the self-consistent calculations of characteristics of metal films in dielectric environment. Within a modified Kohn-Sham method and stabilized jellium model, the most interesting case of asymmetric metal-dielectric sandwiches is considered, for which dielectric media are different from the two sides of the film. As an example, we focus on Na, Al and Pb. We calculate the spectrum, electron work function, and surface energy of polycrystalline and crystalline films placed into passive isolators. We find that a dielectric environment generally leads to the decrease of both the electron work function and surface energy. It is revealed that the change of the work function is determined only by the average of dielectric constants from both sides of the film.