Table of contents

We construct polynomial models for germs of real submanifolds in complex space. It was shown earlier that the properties of models of degree 3 (for appropriate values of the codimension) are similar to well-known properties of tangent quadrics. In this paper we construct models of arbitrarily high degree. They have all these properties with one exception: from degree 5 onwards, they are not completely universal.

We show that the differential field generated by Siegel modular forms and the differential field generated by exponentials of polynomials are linearly disjoint over . Combined with our previous work [3], this provides a complete multidimensional extension of Mahler's theorem on the transcendence degree of the field generated by the exponential function and the derivatives of a modular function. We give two proofs of our result, one purely algebraic, the other analytic, but both based on arguments from differential algebra and on the stability under the action of the symplectic group of the differential field generated by rational and modular functions.

In this paper we investigate the convergence set of a regular C-fraction with limit-periodic coefficients. This investigation is based on a general assertion concerning the convergence of composites of linear-fractional transformations whose coefficients are limit-periodic functions depending holomorphically on a parameter. We show that the singularity set of such a C-fraction possesses an extremal property stated in terms of the transfinite diameter (capacity) of sets.

The real analogues of many results about complex monodromies of singularities can be formulated and proved in terms of partial orderings on A'Campo-Gusein-Zade diagrams, the real versions of Coxeter-Dynkin diagrams of singularities. In this paper it is proved that the only diagrams among the A'Campo-Gusein-Zade diagrams of singularities that determine partially ordered sets of finite type (in the sense of representations of a quiver) are the diagrams of simple singularities. To encode the real decompositions of a singularity the analogue of Vasilev invariants turn out to be surjections of a partially ordered set onto a chain. Formulae are proved for Arnold (mod 2)-invariants of plane curves in terms of the corresponding A'Campo-Gusein-Zade diagrams. We define, in the context of higher Bruhat orders, higher partially ordered sets and we describe their connection with the higher M-Morsifications A_n. We also consider certain previously known results about real singularities from the point of view of partially ordered sets.

We consider two classes of second-order parabolic matrix-vector systems (with solutions , ) that can be reduced to a single second-order parabolic equation for a scalar function , where is a fixed stochastic constant vector. We consider the first boundary-value problem for a scalar second-order parabolic equation (with unbounded coefficients) in a domain unbounded with respect to  under the assumption of strong absorption at infinity. We obtain an a priori estimate for solutions of the first boundary-value problem in the generalized Tikhonov-Täcklind classes. (The problem under investigation has at most one solution in these classes.)

We study the behaviour of fractional parts of functions , where  is a real algebraic number of degree and  is an arbitrary positive number less than one.

Although we consider only the concrete problem indicated in the title (the proofs of general theorems would be bulky), our arguments can be adapted for a wide class of singularly perturbed systems of reaction-diffusion type with time-delay in the ordinary part.

We find the eigenvalues and eigenfunctions for the spectral Tricomi problem for an equation of mixed type with two lines of degeneracy and solve the planar and spatial Tricomi problems for equations with the Lavrent'ev-Bitsadze operator.

This paper is the third in a series in which we complete the description of the finite vertex stabilizers for connected graphs with projective suborbits and, as a corollary, of the vertex stabilizers for finite connected graphs in groups of automorphisms that act transitively on 2-arcs. In this part we complete the treatment of the collineation case under the assumption that the projective dimension of the suborbit is equal to 4.

We present a new idea of quantization of classical mechanical systems, which uses the constructions of [2], [7] and [1]. As a first step, we verify the correspondence between the Poisson brackets on the initial symplectic manifold and on the moduli space of half-weighted Bohr-Sommerfeld Lagrangian cycles of a fixed volume.

We establish conditions under which there is a dual representation of a superlinear functional on a projective limit of vector lattices. These results will enable us, in the second part of this paper, to pose new dual problems for weighted spaces of holomorphic functions of one or several variables defined on a domain in , namely, the problems of non-triviality of a given space, description of null sets, description of sets of (non-)uniqueness, existence of holomorphic functions of certain classes that play the role of multipliers "suppressing" the growth of a given holomorphic function, and representation of meromorphic functions by quotients of holomorphic functions from a given space.