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1.

Babich, M. V.

et al.

St. Petersburg Department of Steklov Mathematical Institute .

Bordag, Ljudmila A.

Halmstad University, School of Information Science, Computer and Electrical Engineering (IDE), Halmstad Embedded and Intelligent Systems Research (EIS), Applied Mathematics and Physics (CAMP).

We consider a two-dimensional steady motion of an inviscid incompressible fluid described by the equation &UDelta; u(x,y) = F(u(x,y)), where u(x,y) is the streamfunction, &UDelta; is the Laplace operator, and F((.)) an arbitrary function measuring the flow vorticity. Apparently, until now, the only way to treat an equation of the above type with nontrivial function F analytically is to use the algebro-geometric method for integrable equations. In particular, we investigate the Cosh-Laplace equation (ChL) &UDelta; u(x,y) = &PLUSMN; 4cosh(u(x,y)) by means of the special technique of finite-gap integration, which allows us to obtain real solutions of the ChL equation by using a Riemann surface with appropriate symmetry. We study the first nontrivial case corresponding to a Riemann surface of genus g = 3. The hydrodynamical interpretation of finite-gap solutions is meaningful, and we try to understand the fluid processes described by these solutions. To this end, we take a Riemann surface with additional symmetry properties. We present four five-parameter families of exact solutions. These solutions are given in terms of Jacobi elliptic functions, which enables us to directly investigate the relevant properties. We also find explicit formulas for the lines of singularity. It is of interest from the point of view of algebraic geometry that the structure of the theta divisor can be described.

2.

Chechkin, G.A.

et al.

Department of Mechanics and Mathematics, Moscow State University.

Koroleva, Yulia O.

Meidell, Annette

Narvik University College, 8505 Narvik, Norway.

Persson, Lars-Erik

Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.

This paper is devoted to the asymptotic analysis of functions depending on a small parameter characterizing the microinhomogeneous structure of the domain on which the functions are defined. We derive the Friedrichs inequality for these functions and prove the convergence of solutions to corresponding problems posed in a domain perforated aperiodically along the boundary. Moreover, we use numerical simulation to illustrate the results.

We study the two-dimensional problem of propagation of linear water waves in deep water in the presence of a submerged body. Under some geometrical requirements, we derive an explicit bound for the solution depending on the domain and the functions on the right-hand side.

The paper is the first in a series of papers on the use of measures and generalized measures in quantum theory. In particular, a survey of the proofs of equivalence of various definitions of the density operator is presented. The exposition is of algebraic nature, and analytic assumptions are usually omitted.

The traction boundary value problem for anisotropic elasticity is considered. For polyhedral domains in R-3, it is proved that the displacements are Holder continuous. In the n-dimensional case, n > 3, the Holder continuity is proved for domains with conic points on the boundary. The proof is based on the study of spectrum of operator pencils associated with singularities of the boundary, which is of independent interest.

6.

Oleschko, K.

et al.

UNAM, Mexico.

Khrennikov, Andrei

Linnaeus University, Faculty of Technology, Department of Mathematics.

This paper is about a novel mathematical framework to model transport (of, e.g., fluid or gas) through networks of capillaries. This framework takes into account the tree structure of the networks of capillaries. (Roughly speaking, we use the tree-like system of coordinates.) As is well known, tree-geometry can be topologically described as the geometry of an ultrametric space, i.e., a metric space in which the metric satisfies the strong triangle inequality: in each triangle, the third side is less than or equal to the maximum of two other sides. Thus transport (e.g., of oil or emulsion of oil and water in porous media, or blood and air in biological organisms) through networks of capillaries can be mathematically modelled as ultrametric diffusion. Such modelling was performed in a series of recently published papers of the authors. However, the process of transport through capillaries can be only approximately described by the linear diffusion, because the concentration of, e.g., oil droplets, in a capillary can essentially modify the dynamics. Therefore nonlinear dynamical equations provide a more adequate model of transport in a network of capillaries. We consider a nonlinear ultrametric diffusion equation with quadratic nonlinearity - to model transport in such a network. Here, as in the linear case, we apply the theory of ultrametric wavelets. The paper also contains a simple introduction to theory of ultrametric spaces and analysis on them.