In the present article, we study almost continuous operators on the set of left computably enumerable (left-c.e.) real numbers (called L -operators). In constructive analysis developed in Markov’s school, almost continuity is one of effective modifications of the classical continuity. We suggest a criterion for almost continuity of an L -operator F that increases on a segment [a,b] ; namely, for every positive real ε and every number α in the image of [a,b] under F , there exists a number β in the same image such that α <β <α +ε . We compare this criterion with the well-known criterion from classical calculus for an increasing function f on a segment [a,b] to be continuous; namely, every real in the segment [f(a),f(b)] belongs to the range of f . We show that there exists an increasing almost continuous L -operator whose range is the set of all left-c.e. reals that do not belong to a prescribed interval [c,d] with rational endpoints and c⩽ d . We construct an increasing L -operator that is not pseudocontinuous at 0 . We also construct an increasing almost continuous L -operator F such that F admits no inverse L -operator on [0,1] and is not continuous on this segment but each left-c.e. real in [F(0),F(1)] belongs to the range of F .
We consider the composition (or chain) operation on mappings of two-step Carnot groups and explicitly derive properties of mappings that guarantee that their composition is a Lipschitz continuous mapping. We present examples and consider an important case in which one of mappings is a projection.
This research proposes a statistical test that accounts for changes in binary outcomes over time—an exact repeated measures test for binary observations. The test is particularly well suited to settings in which the objective is to assess the temporal effect of a factor, such as a treatment, advertising, or a political campaign. The proposed test is not time-symmetric: interchanging the vectors of initial and final values may lead to different test results. The test follows the Barnard test approach in computing the p -values.
We consider a class of systems of nonautonomous nonlinear differential equations of neutral type with concentrated and distributed delays. We use a Lyapunov–Krasovskii functional and establish estimates that allow us to make a conclusion on stability of solutions. In the case of exponential stability, we find estimates for an attraction set and a stabilization rate of the solutions at infinity.
We consider a delay differential equation together with a sequence of high-dimensional systems of ordinary differential equations. We establish a connection between the regions of asymptotic stability of their zero solutions and study limit properties of the sequence of Lyapunov’s integrals.
We consider approximation by piecewise approximatively compact Chebyshev sets in linear normed spaces. We show that, in locally uniformly convex Banach spaces, such sets are suns and are B -path-connected.
We study properties of weighted Radon transforms of symmetric 2 -tensor fields in ℝ^3 . We establish connections between weighted Radon transforms of 2 -tensor fields and the Radon transform of their potentials. These connections describe the kernel and image of tomographic integral operators acting on tensor fields. We consider formulations of problems on reconstruction of symmetric 2 -tensor fields from the values of the normal, longitudinal, and weighted Radon transforms and obtain inversion formulas.
We consider the question on diagram preserving extension of D -types. Let D denote the diagram of a λ -homogeneous model and let S be a set of D -types whose domains are subsets of cardinality less than λ of a suitable D -set B . We prove that a model V_0 of the set theory ZFC admits a generic extension V such that each type in S extends to a D -type over B ; moreover, if a cardinal of V_0 does not exceed λ and λ =κ ^+ then it is preserved in V . We also find conditions for preservation of all cardinals of V_0 in V .
In this work, we investigate the conditions under which a Volterra quadratic stochastic operator with variable coefficients can exhibit infinitely many periodic points. Our approach combines analytical methods from mathematical analysis with computational techniques. We begin by identifying conditions under which the operator admits a 3-periodic orbit, assuming the variation rule of its parameter is unknown but itself possesses a 3-periodic behavior. Subsequently, we analyze the set of periodic points when the parameter variation is modeled by a piecewise-linear function. By applying Sharkovsky’s theorem to the parameter’s dynamics, we demonstrate that the operator can indeed exhibit an infinite number of periodic points.
We consider a discrete-continuous predator-prey model and a derived discrete model of isolated population. Unlike the well-known Lotka–Volterra model [32], we assume that new individuals are generated at fixed moments. Thus, we obtain a mathematical model represented by a system of ordinary differential equations with impulses. From this system we obtain a model of isolated population in the form of a nonlinear difference second-order equation and study its dynamic regimes and phase changes in both conservative and nonconservative cases. The model is relevant because it agrees with experimental data from freely distributed databases on populations.
The consistency of new universal kernel estimators for conditional variance function in a heteroscedastic nonparametric regression model has been proven. The new estimators are insensitive to the nature of the design dependence. For design that can be either fixed or random, only the following condition is used: the design points densely fill the domain of regression function. As a consequence, we consider the problem of constructing a confidence region for a regression function under the above-mentioned very general conditions on the design in terms of dense data.
We consider an inverse problem on finding unknown functions σ _0 and σ _1 that occur in the formula σ (x,u^2)=σ _0+σ _1 u^2 for the absorption coefficient in electrodynamics equation. We obtain an a priori estimate for a solution to the direct problem and prove existence and uniqueness theorems for solutions to the direct and inverse problems.
In the present article, we consider a model of Hopfield neural network described by a system of neutral type differential equations with several delays. We use the method of Lyapunov–Krasovskiĭ functionals and find conditions on parameters of this model that guarantee exponential stability of the stationary solution to the system. Under these conditions, we obtain estimates characterizing stabilization rate of solutions at the infinity.
Let X_1, X_2, … , X_n be independent random variables, and let 𝐄^(l) for l=1, … , n be symmetric deterministic matrices. We study matrices of the type 𝐖 = ∑ _l=1^n X_l 𝐄^(l) . The convergence of empirical spectral distribution of 𝐖 to the normal law is established under certain conditions on matrices 𝐄^(l) . Applications include estimating the convergence rate for the spectral distribution functions of palindromic and circulant random matrices. The analysis employs a method introduced by the author in his 1980 work “On the Rate of Convergence in the Central Limit Theorem for Weakly Dependent Variables”.
In the present article, we consider systems of linear difference equations with periodic coefficients such that the spectrum of the monodromy matrix and the unit circle are disjoint. We find bounded solutions to such systems. For these solutions, we prove uniqueness theorems, obtain formulas, and estimate the norm. In certain partial cases, the estimates coincide with Kreĭn’s inequalities.
We describe an algorithm for constructing self-similar dendrites such that the intersection of their copies is a polygonal tree.
We construct a solution of a plane two-period problem on loading an infinite elastic isotropic plane with a grid of square inclusions. The plane is under one of two loads. It is either stretched at some angle to the X -axis or has a pure shear at the infinity. Each regular square periodicity cell contains a single square inclusion whose sides are perpendicular to the sides of the cell. The size of each inclusion is considerably greater than the thickness of the plate. The stresses are located near a stress concentrator at the frontier between the inclusion and matrix. Solution of the problem is reduced to finding complex-valued functions on the basis of boundary conditions obtained from the equalities of the normal forces and displacements of the matrix and inclusions. We use conformal mappings and integrate by the Muskhelishvili method. The effect of noncentral inclusions is expressed by using the small parameter method. As a result, we obtain a system of linear algebraic equations for solving the two-period problem under consideration and find its solutions for several partial cases. We compare our result with the numerical solution obtained with the use of the Abaqus software (which is based on the finite element method). Solution of such a problem is actual because it models loading of a fiber composite. There are comparatively few articles on fiber composites in mechanics. The majority of them is devoted to analysis of either experiments or numerical solutions. Therefore, the presented analytic solution is of significant scientific value.
We consider an initial-boundary value problem in a quarter-space for a pseudohyperbolic equation that is not resolved with respect to the highest time derivative. We find sufficient conditions on the right-hand side of this equation that guarantee existence of solutions of the problem in Sobolev spaces with exponential weight.
