We prove that the manifold of non-singular points of a stable real caustic germ of type and the manifolds of points of transversal intersection of its smooth branches consist only of contractible connected components. We also calculate the number of these components.

Jabbarov [1] obtained the exact value of the exponent of convergence of the singular integral in Tarry’s problem for homogeneous polynomials of degree . We extend this result to the case of polynomials of degree .

We study monotone path-connected sets and also strongly and weakly Menger-connected sets. We introduce the notion of -solarity and establish a connection with the notion of solarity. We prove that boundedly compact suns in are monotone path-connected.

We consider an elliptic boundary-value problem with a homogeneous Dirichlet boundary condition, a parameter and a discontinuous non-linearity. The positive parameter appears as a multiplicative term in the non-linearity, and the problem has a zero solution for any value of the parameter. The non-linearity has superlinear growth at infinity. We prove the existence of positive solutions by a topological

We construct an integrable function whose Fourier series possesses the following property. After an appropriate choice of signs of the coefficients of this series, the partial sums of the resulting series are dense in , .

We find sufficient conditions which are in a sense best possible that must be satisfied by the functions of an orthonormal system in order for the Fourier coefficients of functions of bounded variation to satisfy the hypotheses of the Men’shov–Rademacher theorem. We also prove a theorem saying that every system contains a subsystem with respect to which the Fourier coefficients of functions of bounded

We study a quite natural class of diffeomorphisms on , where is the infinite-dimensional torus (the direct product of countably many circles endowed with the topology of uniform coordinatewise convergence). The diffeomorphisms under consideration can be represented as the sums of a linear hyperbolic map and a periodic additional term. We find some constructive sufficient conditions, which imply that

In the paper, we obtain order estimates for the Kolmogorov widths of the intersection of weighted Sobolev classes with conditions on the first and zeroth derivatives; the weights have power-law form.

We consider the Cauchy problem for a model partial differential equation of order three with a non-linearity of the form . We prove that when the Cauchy problem in has no local-in-time weak solution for a large class of initial functions, while when there is a local weak solution.

We study the wedge of solutions of the inequality , where is a linear elliptic operator of order acting on functions of variables. We establish interior estimates of the form for the elements of this wedge, where is a compact subdomain of , is the Sobolev space, , is the Lebesgue space of integrable functions, and the constant is independent of .

This paper is devoted to investigating the weak solubility of the alpha-model for a fractional viscoelastic Voigt medium. The model involves the Voigt rheological relation with a left Riemann–Liouville fractional derivative, which accounts for the medium’s memory. The memory is considered along the trajectories of fluid particles determined by the velocity field. Since the velocity field is not smooth

We consider the Cauchy problem for a Schrdinger equation whose Hamiltonian is the difference of the operator of multiplication by the potential and the operator of taking the second derivative. Here the potential is a real differentiable function of a real variable such that this function and its derivative are bounded. This equation has been studied since the advent of quantum mechanics and is still

We prove that the Grothendieck standard conjecture of Lefschetz type holds for a smooth complex projective -dimensional variety fibred by Abelian varieties (possibly, with degeneracies) over a smooth projective curve if the endomorphism ring of the generic geometric fibre is not an order of an imaginary quadratic field. This condition holds automatically in the cases when the reduction of the generic

We make use of a finite support product of the Jensen minimal forcing to define a model of set theory in which the separation theorem fails for projective classes $\mathbf\Sigma^1_n$ and $\mathbf\Pi^1_n$, for a given $n\ge3$.

In this paper we consider nonlocal energies defined on probability measures in the plane, given by a convolution interaction term plus a quadratic confinement. The interaction kernel is $-\log|z|+\alpha\, x^2/|z|^2, \; z=x+iy,$ with $-1 < \alpha< 1.$ This kernel is anisotropic except for the Coulombic case $\alpha=0.$ We present a short compact proof of the known surprising fact that the unique minimiser

We give a geometric condition on a compact subset of a complex manifold which is necessary and sufficient for the existence of a smooth strictly plurisubharmonic function defined in a neighbourhood of this set.

We present a study of real Hurwitz numbers enumerating a special kind of real meromorphic functions, which we call simple framed purely real functions. We deduce partial differential equations of cut-and-join type for generating functions for these numbers. We also construct a topological field theory for them.

In this paper we describe the group of symmetries of a two-dimensional continued fraction. As a multidimensional generalization of continued fractions we consider Klein polyhedra. We distinguish two types of symmetries: the Dirichlet-type ones, that correspond to the multiplication by units of the corresponding number field, and so called palindromic ones. The main result of the paper is a criterion

Approximation in measure is employed to solve an asymptotic Dirichlet problem on arbitrary open sets and to show that many functions, including the Riemann zeta-function, are universal in measure. Connections with the Riemann Hypothesis are suggested.

Boileau and Rudolph called a link $L$ in the $3$-sphere a $\bf C$-boundary if it can be realized as the intersection of an algebraic curve $A$ in $\bf C^2$ with the boundary of a smooth embedded $4$-ball $B$. They showed that some links are not $\bf C$-boundaries. We say that $L$ is a strong $\bf C$-boundary if $A\setminus B$ is connected. In particular, all quasipositive links are strong $\bf C$-boundaries

We prove that if a countable group is elementarily equivalent to a non-abelian free group and all of its finitely generated abelian subgroups are cyclic, then the group is a union of a chain of regular NTQ groups (i.e., hyperbolic towers).

We study proper holomorphic mappings between strictly pseudoconvex domains with low boundary regularity.

In this paper we show that the space of holomorphic immersions from any given open Riemann surface, $M$, into the Riemann sphere $\mathbb{CP}^1$ is weakly homotopy equivalent to the space of continuous maps from $M$ to the complement of the zero section in the tangent bundle of $\mathbb{CP}^1$. We show in particular that this space has $2^k$ path components, where $H_1(M,\mathbb Z)={\mathbb Z}^k$.

For every positive integer and every metric space we consider the class of all parametric families , where , , , of linear differential systems whose coefficients are piecewise continuous and, generally speaking, unbounded on the time semi-axis for every fixed value of the parameter such that if a sequence converges to in the space of parameters, then the sequence converges uniformly on the semi-axis

Let sequences , satisfy the relations , , , as , and let and . We redefine the function as on the interval by polygonal arcs in such a way that the function remains continuous and vanishes on a neighbourhood of the ends of the interval. Also let the function and the pair of sequences , be connected by the equiconvergence condition. Then for the classical Lagrange–Jacobi interpolation processes to approximate

We consider the problem of minimizing an integral functional on the solutions of a controlled system described by a non-linear differential equation in a separable Banach space and a variational inequality. The variational inequality determines a hysteresis operator whose input is a trajectory of the controlled system and whose output occurs in the right-hand side of the differential equation, in the

We describe and classify the thresholds of the continuous spectrum and the resulting resonances for general formally self-adjoint elliptic systems of second-order differential equations with Dirichlet or Neumann boundary conditions in domains with cylindrical and periodic outlets to infinity (in waveguides). These resonances arise because there are “almost standing” waves, that is, non-trivial solutions

Suppose that and are real numbers with . Let be a non-empty open subset of , . We consider the inequality where is the gradient operator, and are certain functions and for almost all and all . We obtain estimates for solutions of this inequality using the geometry of . In particular, these estimates yield regularity conditions for boundary points.

Consider a tree $\mathbb T$, all whose vertices have countable valence; its boundary is the Baire space $\mathbb{B} \simeq\mathbb{N}^{\mathbb N}$; continued fractions expansions identify the set of irrational numbers $\mathbb{R}\setminus \mathbb{Q}$ with $\mathbb B$. Removing $k$ edges from $\mathbb T$ we get a forest consisting of copies of $\mathbb T$. A spheromorphism (or hierarchomorphism) of $\mathbb

We study homotopes of alternative algebras over an algebraically closed field of characteristic different from ##IMG## [http://ej.iop.org/images/1064-5632/84/5/1002/toc_IZV_84_5_1002ieqn1.gif] {$3$} . We prove an analogue of Albert’s theorem on isotopes of associative algebras: in the class of finite-dimensional unital alternative algebras every isotopy is an isomorphism. We also prove that every ##IMG##

We obtain a class of explicit formulae each of which gives an expression for the remainder term in the asymptotic equation for the Chebyshev function in terms of the spectrum of the Laplace operator on the fundamental domain of the modular group.

We consider an abstract Cauchy problem with non-linear operator coefficients and prove the existence of a unique non-extendable classical solution. Under certain sufficient close-to-necessary conditions, we obtain finite-time blow-up conditions and upper and lower bounds for the blow-up time. Moreover, under certain sufficient close-to-necessary conditions, we obtain a result on the existence of a

We consider orientation-preserving ##IMG## [http://ej.iop.org/images/1064-5632/84/5/862/IZV_84_5_862ieqn1.gif] {$A$} -diffeomorphisms of orientable surfaces of genus greater than one with a one-dimensional spaciously situated perfect attractor. We show that the topological classification of restrictions of diffeomorphisms to such basic sets can be reduced to that of pseudo-Anosov homeomorphisms with

We study the covariant bundles (Mostow bundles) for simply connected homogeneous manifolds, establish their relation to homogeneous bundles and consider classes of homogeneous manifolds for which the Mostow bundle is trivial (resp. non-trivial). We also give a classification of non-compact simply connected homogeneous manifolds of dimension not exceeding seven.

The paper contains two main results. First we describe one-dimensional Franklin series converging everywhere except possibly on a finite set to an everywhere-finite integrable function. Second we establish a class of subsets of ##IMG## [http://ej.iop.org/images/1064-5632/84/5/829/IZV_84_5_829ieqn1.gif] {$[0, 1]^2$} with the following property. If a double Franklin series converges everywhere except

We prove that the Grothendieck standard conjecture of Lefschetz type holds for a complex projective 3-dimensional variety fibred by curves (possibly with degeneracies) over a smooth projective surface provided that the endomorphism ring of the Jacobian variety of some smooth fibre coincides with the ring of integers and the corresponding Kodaira–Spencer map has rank ##IMG## [http://ej.iop.org/imag

We classify uniruled compact Kahler threefolds whose groups of bimeromorphic selfmaps do not have Jordan property.

In this paper subdivision schemes, which are used for functions approximation and curves generation, are considered. In classical case, for the functions defined on the real line, the theory of subdivision schemes is widely known due to multiple applications in constructive approximation theory, signal processing as well as for generating fractal curves and surfaces. Subdivision schemes on a dyadic

We study the existence and uniqueness as well as the asymptotic behaviour of solutions of a certain boundary-value problem for a convolution integral equation on the whole line with monotone non-linearity. In some special cases, there are concrete applications to ##IMG## [http://ej.iop.org/images/1064-5632/84/4/807/IZV_84_4_807ieqn1.gif] {$p$} -adic string theory, the mathematical theory of the geographical

We study deformations of a long (narrow after rescaling) Kirchhoff plate with periodic (rapidly oscillating) boundary. We deduce a limiting system of two ordinary differential equations of orders 4 and 2 which describe the deflection and torsion of a two-dimensional plate in the leading order. We also consider point supports (Sobolev conditions) whose configuration influences the result of homogenizing

We study problems of the absolute convergence of Fourier series of differentiable functions with respect to general orthonormal systems (ONS). We obtain results which show that under certain conditions on functions in the ONS, the Fourier series of an arbitrary function in a certain differentiability class is absolutely convergent. We finally note that our results cannot be improved.

Suppose that ##IMG## [http://ej.iop.org/images/1064-5632/84/4/627/IZV_84_4_627ieqn1.gif] {$0\le k<\infty$} . We prove that there is a dense open subset of the Grassmann space ##IMG## [http://ej.iop.org/images/1064-5632/84/4/627/IZV_84_4_627ieqn2.gif] {$\operatorname{Gr}(2k+1,m)$} such that the orthogonal projection of the standard Nöbeling space ##IMG## [http://ej.iop.org/images/1064-5632/84/4/627/IZV_84_4_627ieqn3

We study measures on a real separable Hilbert space ##IMG## [http://ej.iop.org/images/1064-5632/84/4/694/IZV_84_4_694ieqn1.gif] {$E$} that are invariant under translations by arbitrary vectors in ##IMG## [http://ej.iop.org/images/1064-5632/84/4/694/IZV_84_4_694ieqn1.gif] {$E$} . We define the Hilbert space ##IMG## [http://ej.iop.org/images/1064-5632/84/4/694/IZV_84_4_694ieqn2.gif] {$\mathcal H$} of

Let ##IMG## [http://ej.iop.org/images/1064-5632/84/4/780/IZV_84_4_780ieqn1.gif] {$R$} be a regular local ring containing a field. Let ##IMG## [http://ej.iop.org/images/1064-5632/84/4/780/IZV_84_4_780ieqn2.gif] {$\mathbf{G}$} be a reductive group scheme over ##IMG## [http://ej.iop.org/images/1064-5632/84/4/780/IZV_84_4_780ieqn1.gif] {$R$} . We prove that a principal ##IMG## [http://ej.iop.org/image

We study expansions with integer coefficients of elements in the multidimensional spaces ##IMG## [http://ej.iop.org/images/1064-5632/84/4/796/IZV_84_4_796ieqn1.gif] {$L_p\{(0,1]^m\}$} , ##IMG## [http://ej.iop.org/images/1064-5632/84/4/796/IZV_84_4_796ieqn2.gif] {$1\leq p<\infty$} , in systems of translates and dilates of a single function. We describe models useful in applications, including those

We consider functions $f(T,R)$ of pairs of noncommuting contractions on Hilbert space and study the problem for which functions $f$ we have Lipschitz type estimates in Schatten--von Neumann norms. We prove that if $f$ belongs to the Besov class $(B_{\infty,1}^1)_+({\Bbb D}^2)$ of analytic functions in the bidisk, then we have a Lipschitz type estimate for functions $f(T,R)$ of pairs of not necessarily

There is a standard method to calculate the cohomology of torus-invariant sheaves $L$ on a toric variety via the simplicial cohomology of associated subsets $V(L)$ of the space $N_{\mathbb R}$ of 1-parameter subgroups of the torus. For a line bundle $L$ represented by a formal difference $\Delta^+-\Delta^-$ of polyhedra in the character space $M_{\mathbb R}$, [ABKW18] contains a simpler formula for

We study an elliptic boundary-value problem in a bounded domain with inhomogeneous Dirichlet condition, discontinuous non-linearity and a positive parameter occurring as a factor in the non-linearity. The non-linearity is in the right-hand side of the equation. It is non-positive (resp. equal to zero) for negative (resp, non-negative) values of the phase variable. Let ##IMG## [http://ej.iop.org/im

Let ##IMG## [http://ej.iop.org/images/1064-5632/84/3/545/IZV_84_3_545ieqn1.gif] {$k$} be a number field and ##IMG## [http://ej.iop.org/images/1064-5632/84/3/545/IZV_84_3_545ieqn2.gif] {$S$} , ##IMG## [http://ej.iop.org/images/1064-5632/84/3/545/IZV_84_3_545ieqn3.gif] {$T$} sets of places of ##IMG## [http://ej.iop.org/images/1064-5632/84/3/545/IZV_84_3_545ieqn1.gif] {$k$} . For each prime ##IMG## [http://ej

We study uniform approximation in the open unit disc ##IMG## [http://ej.iop.org/images/1064-5632/84/3/437/IZV_84_3_437ieqn2.gif] {$D=\{z\colon |z|<1\}$} by logarithmic derivatives of ##IMG## [http://ej.iop.org/images/1064-5632/84/3/437/IZV_84_3_437ieqn3.gif] {$C$} -polynomials, that is, polynomials whose zeros lie on the unit circle ##IMG## [http://ej.iop.org/images/1064-5632/84/3/437/IZV_84_3_437ieqn4

We establish an asymptotic formula for the rate of approximation of Fourier series of individual periodic functions by linear averages with an error ##IMG## [http://ej.iop.org/images/1064-5632/84/3/608/IZV_84_3_608ieqn1.gif] {$\omega_{2m}(f;{1}/{n})$} , ##IMG## [http://ej.iop.org/images/1064-5632/84/3/608/IZV_84_3_608ieqn2.gif] {$m\in\mathbb{N}$} . This formula is applicable to the means of Riesz,

We prove some properties of real Segre cubics. In particular, we find the topological types of the real parts of Segre cubics as well as the topological types of the real parts of the complements of the Segre planes. We prove some differential-geometric properties of the real parts of real Segre cubics and Kummer quartics. We study the automorphism groups of real Segre cubics and, in particular, their

We consider two model non-linear equations describing electric oscillations in systems with distributed parameters on the basis of diodes with non-linear characteristics. We obtain equivalent integral equations for classical solutions of the Cauchy problem and the first and second initial-boundary value problems for the original equations in the half-space ##IMG## [http://ej.iop.org/images/1064-56

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

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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