Let \(\Omega = A^{N}\)  be a space of right-sided infinite sequences drawn from a finite alphabet \(A = \{0,1\}\),  \(N = \{1,2,\dots \}\),
\[\label{rho}
  \rho(\boldsymbol{x},\boldsymbol{y}) =
\sum_{k=1}^{\infty}|x_{k} - y_{k}|2^{-k}
\]
a metric on \(\Omega = A^{N}\),
and \(\mu\) is a probability measure on \(\Omega\). Let \(\boldsymbol{\xi_0}, \boldsymbol{\xi_1}, \dots, \boldsymbol{\xi_n}\) be independent identically distributed points on \(\Omega\). We study the estimator \(\eta_n^{(k)}(\gamma)\) of the reciprocal of the entropy \(1/h\) that are defined as

\[ \label{etan}
\eta_n^{(k)}(\gamma) = k \left(r_{n}^{(k)}(\gamma) - r_{n}^{(k+1)}(\gamma)\right),\]
where
\[\label{def_r}
r_n^{(k)}(\gamma) =
\frac{1}{n+1}\sum_{j=0}^{n} \gamma\left(\min_{i:i \neq j} {^{(k)}}
\rho(\boldsymbol{\xi_{i}}, \boldsymbol{\xi_{j}})\right),
\]
 \(\min ^{(k)}\{X_1,\dots,X_N\}=  X_k\), if  \(X_1\leq X_2\leq \dots\leq X_N\). The number \(k\) and the function \(\gamma(t)\) are auxiliary parameters.

The main result of this paper is

Theorem. Let \(\mu\) be the Bernoulli measure  with probabilities \(p_0,p_1>0\), \(p_0+p_1=1\), \(p_0=p_1^2\). There exists a function \(\gamma(t)\) such that
\[E\eta_n^{(k)}(\gamma) =  \frac1h.\]

 

The stability of the solutions of the linear equations arising in the theory of twodimensional digital filtration is studied. The different statements of the initial value problem are analysed. As the basic results, the corresponding stability criterion is obtained for each of them.

 

In the paper, a mathematical model of a neural network with an even number of ring synaptic interaction elements is considered. The model is a system of scalar nonlinear differentialdifference equations, the right parts of which depend on large parameters. The unknown functions included in the system characterize the membrane potentials of the neurons. The search of special impulse-refraction cycles within the system of equations is of interest. The functions with odd numbers of the impulse-refraction cycle have an asymptotically high pulses and the functions with even numbers are asymptotically small. Two changes allow to study a two-dimension nonlinear differential-difference system with two delays instead of the system. Further, a limit object that represents a relay system with two delays is defined by a large parameter tending to infinity. There exists the only periodic solution of the relay system with the initial function from a suitable function class. This is structurally proved, by using the step method. Next, the existence of relaxation periodic solutions of the two-dimension singularly perturbed system is proved by using the Poincare operator and the Schauder principle. The asymptotics of this solution is constructed, and it is proved that the solution is close to the decision of the relay system. Because of the exponential estimate of the Frechet derivative of the Poincare operator it implies the uniqueness and stability of solutions of the two-dimension differential-difference equation with two delays. Furthermore, with the help of reverse replacement the proved result is transferred to the original system.

 

We introduce the definition of locally convex curves and establish some properties of such curves. In the section 1, we consider the curve \(K\) allowing the parametric representation \(x = u(t),\, y = v(t), \, (a \leqslant t \leqslant b)\), where  \(u(t)\), \(v(t)\) are continuously differentiable on \([a,b]\) functions such that \(|u'(t)| + |v'(t)| > 0 \,\forall t \in [a,b]\). A continuous on  \([a,b]\)  function  \(\theta(t)\) is called  it  the angle function of the curve \(K\) if the following conditions hold: \(u'(t) = \sqrt{(u'(t))^2 + (v'(t))^2}\, \cos \theta(t), \quad v'(t) = \sqrt{(u'(t))^2 + (v'(t))^2}\, \sin \theta(t)\). The curve \(K\) is called it locally convex if its angle function \(\theta(t)\) is strictly monotonous on \([a,b]\). For a closed curve \(K\) the number \(deg K= \cfrac{\theta(b)- \theta(a)}{2 \pi}\) is whole. This number is equal to the number of rotations that the speed vector \((u'(t),v'(t))\) performs around the origin. The main result of the first section is the statement: if the curve \(K\) is locally convex, then for any straight line \(G\) the number \(N(K;G)\) of intersections of \(K\) and \(G\) is finite and the estimate \(N(K;G) \leqslant 2 |deg K|\) holds. We discuss versions of this estimate for closed and non-closed curves. In the sections 2 and 3, we consider curves arising in the investigation of a linear homogeneous differential equation of the form \(L(x) \equiv x^{(n)} + p_1(t) x^{(n-1)} + \cdots p_n(t) x = 0 \) with locally summable coefficients \(p_i(t)\, (i = 1, \cdots,n)\).
We demonstrate how conditions of disconjugacy of the differential operator \(L\) that were established in works of G.A. Bessmertnyh and A.Yu.Levin, can be applied.

Let  \(n\in{\mathbb N}\),  \(Q_n=[0,1]^n.\) For a nondegenerate simplex  \(S\subset {\mathbb R}^n\), by  \(\sigma S\) we denote the homothetic image of  \(S\) with the center of homothety in the center of gravity of  \(S\) and ratio of homothety  \(\sigma\). By  \(d_i(S)\)  we mean the  \(i\)-th axial diameter of  \(S\), i.e. the maximum length of a line segment in  \(S\) parallel to the  \(i\)th coordinate axis. Let  \(\xi(S)=\min \{\sigma\geq 1: Q_n\subset \sigma S\},\)  \(\xi_n=\min \{ \xi(S): \, S\subset Q_n \}.\) By  \(\alpha(S)\) we denote the minimal  \(\sigma>0\) such that  \(Q_n\) is contained in a translate of simplex  \(\sigma S\). Consider  \((n+1)\times(n+1)\)-matrix  \({\bf A}\) with the rows containing coordinates of vertices of  \(S\); the last column of  \({\bf A}\) consists of 1's. Put  \({\bf A}^{-1}\)  \(=(l_{ij})\). Denote by  \(\lambda_j\) a linear function on  \({\mathbb R}^n\) with coefficients from the  \(j\)-th column of  \({\bf A}^{-1}\), i.\,e.  \(\lambda_j(x)= l_{1j}x_1+\ldots+ l_{nj}x_n+l_{n+1,j}.\) Earlier, the first author proved the equalities  \( \frac{1}{d_i(S)}=\frac{1}{2}\sum_{j=1}^{n+1} \left|l_{ij}\right|, \ \alpha(S) =\sum_{i=1}^n\frac{1}{d_i(S)}.\) In the present paper, we consider the case  \(S\subset Q_n\). Then all the  \(d_i(S)\leq 1\), therefore,  \(n\leq \alpha(S)\leq \xi(S).\) If for some simplex  \(S^\prime\subset Q_n\) holds  \(\xi(S^\prime)=n,\) then  \(\xi_n=n\),   \(\xi(S^\prime)=\alpha(S^\prime)\), and  \(d_i(S^\prime)=1\). However, such simplices  \(S^\prime\) do not exist for all the dimensions  \(n\). The first value of  \(n\) with such a property is equal to  \(2\). For each 2-dimensional simplex,  \(\xi(S)\geq \xi_2=1+\frac{3\sqrt{5}}{5}=2.34 \ldots>2\). We have an estimate  \(n\leq \xi_n<n+1\). The equality  \(\xi_n=n\) takes place if there exists an Hadamard matrix of order  \(n+1\). Further study showed that  \(\xi_n=n\) also for some other  \(n\). In particular, simplices with the condition  \(S\subset Q_n\subset nS\) were built for any odd  \(n\)  in the interval  \(1\leq n\leq 11\). In the first part of the paper, we present some new results concerning simplices with such a condition. If  \(S\subset Q_n\subset nS\), the center of gravity of  \(S\) coincide, with the center of  \(Q_n\). We prove that  \(\sum_{j=1}^{n+1} |l_{ij}|=2 \quad (1\leq i\leq n), \ \sum_{i=1}^{n} |l_{ij}|=\frac{2n}{n+1} \  (1\leq j\leq n+1).\) Also we give some corollaries. In the second part of the paper, we consider the following conjecture. { Let for simplex  \(S\subset Q_n\) an equality  \(\xi(S)=\xi_n\) holds. Then  \((n-1)\)-dimensional hyperplanes containing the faces of  \(S\) cut from the cube  \(Q_n\) the equal-sized parts. Though it is true for  \(n=2\) and  \(n=3\), in the general case this conjecture is not valid.

We investigate the problem of constructing the asymptotics for weak solutions of certain class of linear differential equations in the Banach space as the independent variable tends to infinity. The studied class of equations is the perturbation of linear autonomous equation, generally speaking, with an unbounded operator. The perturbation takes the form of the family of the bounded operators that, in a sense, decreases oscillatory at infinity. The unperturbed equation satisfies the standard requirements of the center manifold theory. The essence of the proposed asymptotic integration method is to prove the existence for the initial equation of the center-like manifold (critical manifold). This manifold is positively invariant with respect to the initial equation and attracts all the trajectories of the weak solutions. The dynamics of the initial equation on the critical manifold is described by the finite-dimensional ordinary differential system. The asymptotics for the fundamental matrix of this system may be constructed by using the method proposed by the author for asymptotic integration of the systems with oscillatory decreasing coefficients. We illustrate the suggested technique by constructing the asymptotic formulas for solutions of the perturbed heat equation.

 

In this paper, a differential partial equation with an unknown function of three variables time and two spatial variables – is considered. The given equation is commonly called the generalized Kuramoto–Sivashinsky (gKS) equation. This equation represents a model of the formation of a nanorelief on a surface by ion bombardment. In the work, this equation is considered with the homogeneous Neumann boundary conditions. Local bifurcations of spatially inhomogeneous equilibrium states is studied in the case of their stability changes. It is shown that the inhomogeneous surface relief can occur when the stability of the homogeneous states of equilibrium changes. The conditions were obtained for coefficients when the stability changes. In the cases close to critical cases the local bifurcation problems are considered. It was shown that a question about the formation of inhomogeneous surface relief from a mathematical point of view is reduced to the study of auxiliary ordinary differential equations which are called a Poincare–Dulac normal form. The stability analysis of spatially homogeneous equilibrium states is given, as well as local bifurcations are studied in the case of their stability changes. The method of invariant manifolds coupled with the normal form theory were used to solve this problem. For the bifurcating solutions the asymptotic formulas are given.

 

The new versions of the collocations and least residuals (CLR) method of high-order accuracy are proposed and implemented for the numerical solution of the boundary value problems for PDE in the convex quadrangular domains. Their implementation and numerical experiments are performed by the examples of solving the biharmonic and Poisson equations. The solution of the biharmonic equation is used for simulation of the stress-strain state of an isotropic plate under the action of the transverse load. Differential problems are projected into the space of fourth-degree polynomials by the CLR method. The boundary conditions for the approximate solution are put down exactly on the boundary of the computational domain. The versions of the CLR method are implemented on the grids, which are constructed by two different ways. In the first version, a “quasiregular” grid is constructed in the domain, the extreme lines of this grid coincide with the boundaries of the domain. In the second version, the domain is initially covered by a regular grid with rectangular cells. Herewith, the collocation and matching points that are situated outside the domain are used for approximation of the differential equations in the boundary cells that had been crossed by the boundary. In addition the “small” irregular triangular cells that had been cut off by the domain boundary from rectangular cells of the initial regular grid are joined to adjacent quadrangular cells. This technique allowed to essentially reduce the conditionality of the system of linear algebraic equations of the approximate problem in comparison with the case when small irregular cells together with other cells were used as independent ones for constructing an approximate solution of the problem. It is shown that the approximate solution of problems converges with high order and matches with high accuracy with the analytical solution of the test problems in the case of the known solution in numerical experiments on the convergence of the solution of various problems on a sequence of grids.

 

In this paper, we consider the nonlocal dynamics of the model of two coupled oscillators with delayed feedback. This model has the form of a system of two differential equations with delay. The feedback function is non-linear, finite and smooth. The main assumption in the problem is that the coupling between the generators is sufficiently small. With the help of asymptotic methods we investigate the existence of relaxation periodic solutions of a given system. For this purpose, a special set is constructed in the phase space of the original system. Then we build an asymptotics of the solutions of the given system with initial conditions from this set. Using this asymptotics, a special mapping is constructed. Dynamics of this map describes the dynamics of the original problem in general. It is proved that all solutions of this mapping are non-rough cycles of period two. As a result, we formulate conditions for the coupling parameter such that the initial system has a two-parameter family of nonrough inhomogeneous relaxation periodic asymptotic (with respect to the residual) solutions.

 

We built first order elliptic systems with any possible number of unknown functions and the maximum possible number of unknowns, i.e, in general. These systems provide the basis for studying the properties of any first order elliptic systems. The study of the Cauchy-Riemann system and its generalizations led to the identification of a class of elliptic systems of first-order of a special structure. An integral representation of solutions is of great importance in the study of these systems. Only by means of a constructive method of integral representations we can solve a number of problems in the theory of elliptic systems related mainly to the boundary properties of solutions. The obtained integral representation could be applied to solve a number of problems that are hard to solve, if you rely only on the non-constructive methods. Some analogues of the theorems of Liouville, Weierstrass, Cauchy, Gauss, Morera, an analogue of Green’s formula are established, as well as an analogue of the maximum principle. The used matrix operators allow the new structural arrangement of the maximum number of linearly independent vector fields on spheres of any possible dimension. Also the built operators allow to obtain a constructive solution of the extended problem ”of the sum of squares” known in algebra.