We develop an approach which we used to deduce conditions of a new type for the existence of periodic solutions of ordinary differential equations and functional differential equations of point type. These conditions are based on the use of asymptotic properties of solutions of differential equations which can be observed on shifts of solutions and stated in terms of averages over the period on a distinguished

In this paper we propose a new effective approach to the problem of finding and constructing non-trivial ##IMG## [http://ej.iop.org/images/1064-5632/84/2/392/toc_IZV_84_2_392ieqn1.gif] {$S$} -units of a hyperelliptic field ##IMG## [http://ej.iop.org/images/1064-5632/84/2/392/IZV_84_2_392ieqn2.gif] {$L$} for a set ##IMG## [http://ej.iop.org/images/1064-5632/84/2/392/IZV_84_2_392ieqn3.gif] {$S=S_h$}

We construct orthogonal trigonometric polynomials satisfying a new spectral condition and such that their ##IMG## [http://ej.iop.org/images/1064-5632/84/2/361/IZV_84_2_361ieqn1.gif] {$L^{1}$} -norms are bounded below and the uniform norm of their partial sums has extremely small order of growth. We obtain new results that relate the uniform norm and ##IMG## [http://ej.iop.org/images/1064-5632/84/2

In this paper, we describe the set of solutions of the monomial equation ##IMG## [http://ej.iop.org/images/1064-5632/84/2/348/IZV_84_2_348ieqn2.gif] {$x^k=a$} over ##IMG## [http://ej.iop.org/images/1064-5632/84/2/348/IZV_84_2_348ieqn3.gif] {$\mathbb Q_p$} . Moreover, as an application, we study some perturbations of the equation under consideration over the ##IMG## [http://ej.iop.org/images/1064-5

We prove that weakly ##IMG## [http://ej.iop.org/images/1064-5632/84/2/324/toc_IZV_84_2_324ieqn1.gif] {$o$} -minimal theories of finite convexity rank having less than ##IMG## [http://ej.iop.org/images/1064-5632/84/2/324/IZV_84_2_324ieqn2.gif] {$2^{\omega}$} countable models are binary. Our main result is the confirmation of Vaught’s conjecture for weakly ##IMG## [http://ej.iop.org/images/1064-5632

We describe acts over rectangular bands that have modular, distributive or linearly ordered congruence lattice. It turns out that such acts have at most 11 elements, and their congruence lattice has at most 300 elements. Furthermore, certain facts are established about the structure of acts with modular congruence lattice over an arbitrary semigroup and about the structure of the congruence lattice

We consider a boundary-value problem for a system of two second-order ODE with distinct powers of a small parameter at the second derivative in the first and second equations. When one of the two equations of the degenerate system has a double root, the asymptotic behaviour of the boundary-layer solution of the boundary-value problem turns out to be qualitatively different from the known asymptotic

We define various algorithms for greedy approximations by elements of an arbitrary set ##IMG## [http://ej.iop.org/images/1064-5632/84/2/246/IZV_84_2_246ieqn1.gif] {$M$} in a Banach space. We study the convergence of these algorithms in a Hilbert space under various geometric conditions on ##IMG## [http://ej.iop.org/images/1064-5632/84/2/246/IZV_84_2_246ieqn1.gif] {$M$} . As a consequence, we obtain

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We consider two classifications of real Kummer quartics. They use the Heisenberg invariance of Kummer quartics. The first divides the whole variety of real Kummer quartics into four classes according to the Heisenberg-invariance type and then subdivides each class into subclasses to obtain a deformation classification. This subdivision into subclasses is performed by means of the topological classification

We establish an area formula for graph mappings on two-step sub-Lorentzian structures with an arbitrary number of spatial and temporal directions. In a particular case, we consider an alternative approach that requires no additional smoothness of the mapping from which the graph is constructed.

In this paper we prove an existence theorem concerning linear forms of a given Diophantine type and apply it to study the structure of the spectrum of lattice exponents.

We consider one-parameter families (general unfoldings) of two-dimensional reversible diffeomorphisms that contain a diffeomorphism with a symmetric non-transversal heteroclinic cycle. We show that in such families there exist Newhouse intervals of parameters such that the values corresponding to the co-existence of infinitely many stable, completely unstable, saddle and symmetric elliptic periodic

We consider entire functions of finite order (greater than or equal to 1) which take rational integer values at the points of a rather general discrete set. We show that under certain conditions all such functions can only be exponential polynomials of a special form.

We study effective divisors ##IMG## [http://ej.iop.org/images/1064-5632/84/1/146/IZV_84_1_146ieqn1.gif] {$D$} on surfaces with ##IMG## [http://ej.iop.org/images/1064-5632/84/1/146/IZV_84_1_146ieqn2.gif] {$H^0(\mathcal{O}_D)=\Bbbk$} and ##IMG## [http://ej.iop.org/images/1064-5632/84/1/146/IZV_84_1_146ieqn3.gif] {$H^1(\mathcal{O}_D)=H^0(\mathcal{O}_D(D))=0$} . We give a numerical criterion for such divisors

In this paper we introduce a direct family of simple polytopes ##IMG## [http://ej.iop.org/images/1064-5632/83/6/1081/IZV_83_6_1081ieqn1.gif] {$P^{0}\,{\subset}\, P^{1}\,{\subset}{\kern1pt}{\cdots}$} such that for any ##IMG## [http://ej.iop.org/images/1064-5632/83/6/1081/IZV_83_6_1081ieqn2.gif] {$2\,{\leq}\,k\,{\leq}\,n$} there are non-trivial strictly defined Massey products of order ##IMG## [http://ej

We consider the Cauchy problem for a model third-order partial differential equation with non-linearity of the form ##IMG## [http://ej.iop.org/images/1064-5632/83/6/1174/IZV_83_6_1174ieqn1.gif] {$|\nabla u|^q$} . We prove that for ##IMG## [http://ej.iop.org/images/1064-5632/83/6/1174/IZV_83_6_1174ieqn2.gif] {$q\in(1,2]$} the Cauchy problem in ##IMG## [http://ej.iop.org/images/1064-5632/83/6/1174/IZV_83_6_1174ieqn3

The authors have recently introduced the class of topological billiards. Topological billiards are glued from elementary planar billiard sheets (bounded by arcs of confocal quadrics) along intervals of their boundaries. It turns out that the integrability of the elementary billiards implies that of the topological billiards. We show that all classical linearly and quadratically integrable geodesic