Abstract:
We propose an analytical approach to the conformal mapping of (rectangular) polygons based on the theory of Riemann surfaces and theta functions.

Abstract:
Flat magnetic nano-elements are an essential component of current and future spintronic devices. By shaping an element it is possible to select and stabilize chosen metastable magnetic states, control its magnetization dynamics. Here, using a recent significant development in mathematics of conformal mapping, complex variable based approach to the description of magnetic states in planar nano-elements is extended to the case when elements are multiply-connected (that is, contain holes or magnetic anti-dots). We show that presence of holes implies a certain restriction on the set of magnetic states of nano-element.

Abstract:
In this note we present a universal formula in terms of theta functions for the Log- capacity of several segments on a line. The case of two segments was studied by N.I.Akhiezer (1930); three segments were considered by A.Sebbar and T.Falliero (2001).

Abstract:
Measurements of the suppression of the yield per nucleon and differential distributions of $J / \psi$ production for 920 GeV/c protons incident on heavy nuclear targets have been made with broad coverage in $p_T$ and negative coverage in $x_F$ of produced meson. Production ratios of $\psi(2S)$ to $J / \psi$ and $\chi_c$ to $J / \psi$ have been measured with a high accuracy. The $D^+$ and $D^0$ production cross sections as well as $D^+$ to $D^0$ ratio have been obtained on one of the highest statistics available in proton nucleus experiments.

Abstract:
Eigenvalue problem for Poincare-Steklov-3 integral equation is reduced to the solution of three transcendential equations for three unknown numbers, moduli of pants. The complete list of antisymmetric eigenfunctions of integral equation in terms of Kleinian membranes is given.

Abstract:
We give an explicit multi-parametric construction for Jenkins-Strebel differentials on real algebraic curves. Roughly speaking, the square of any real holomorphic abelian differential subjected to certain linear restrictions will be a JS quadratic differential.

Abstract:
J.Ritt has investigated the structure of complex polynomials with respect to superposition. In particular, he listed all the polynomials admitting different double decompositions into indecomposable polynomials. The analogues of Ritt theory for rational functions were constructed just for several particular classes of the said functions, say for Laurent polynomials (F.Pakovich). In this note we describe a certain class of double decompositions for rational functions. Essentially, described below rational functions were discovered by E.I.Zolotarev in 1877 as a solution of certain optimization problem. However, the double decomposition property for them was hidden until recently because of somewhat awkward representation. We give a (possibly new) symmetric representation of Zolotarev fractions resembling the parametric representation for Chebyshev polynomials, which are a special limit case of Zolotarev fraction.

Abstract:
For the evaluation and inversion of abelian integrals we show that the image of the Abel-Jacobi map of genus less than 5 hyperelliptic curve in its Jacobian is the intersection of shifted theta divisors with specified shifts. Therefore the image is a solution of a (slightly overdetermined) set of equations in the Jacobian.

Abstract:
Known properties of Chebyshev polynomials are the following: they have simple critical points with only two (finite) critical values. Those properties uniquely determine the named polynomials modulo affine transformations of dependent and independent variables. A similar property of Zolotarev fractions: simple critical points and only four critical values generates already many classes of rational functions modulo projective transformations of their dependent and independent variables. They are listed in this note.

Abstract:
More than a hundred years ago H.Poincare and V.A.Steklov considered a problem for the Laplace equation with spectral parameter in the boundary conditions. Today similar problems for two adjacent domains with the spectral parameter in the conditions on the common boundary of the domains arises in a variety of situations: in justification and optimization of domain decomposition method, simple 2D models of oil extraction, (thermo)conductivity of composite materials. Singular 1D integral Poincare-Steklov equation with spectral parameter naturally emerges after reducing this 2D problem to the common boundary of the domains. We present a constructive representation for the eigenvalues and eigenfunctions of this integral equation in terms of moduli of explicitly constructed pants, one of the simplest Riemann surfaces with boundary. Essentially the solution of integral equation is reduced to the solution of three transcendent equations with three unknown numbers, moduli of pants. The discreet spectrum of the equation is related to certain surgery procedure ('grafting') invented by B.Maskit (1969), D.Hejhal (1975) and D.Sullivan- W.Thurston (1983).