The article is dedicated to the issues of studying the approximation of functions by trigonometric polynomials with a spectrum from special sets. In this paper, these special sets are harmonic intervals. To study the approximation of functions over harmonic intervals, families of function classes have been created, designed as a subsidiary tool. These families of function classes are characterized through the best approximations of functions by trigonometric polynomials over such sets and are used in the research. For these families of function classes, their properties and the connection with classical Besov spaces are shown. The results of the study are presented in the form of theorems and lemmas. In carrying out the research presented in the article, the main apparatus for proving theorems are the fundamentals of approximation theory, the method of real interpolation of spaces, and the fundamentals of the theory of embedding classes of functions and functional spaces. The article is destined for mathematicians and can be used by researchers and specialists whose interests lie in the indicated areas of mathematics.
E.F. Abdyldaeva, A. Kerimbekov, T.K. Yuldashev
et al.
This article addresses the non-linear optimization problem of oscillatory processes governed by partial integro-differential equations involving a Fredholm integral operator. A distinctive feature of the problem is that both the objective functional and the functions describing external and boundary influences are non-linear with respect to the vector controls. The integro-differential equation describing the state of the oscillatory process includes Fredholm integral operator, which has a significant impact on the structure and properties of the solutions. The algorithm for constructing the complete solution to this problem, as well as the effect of the Fredholm integral operator on the solution of the corresponding boundary value problem, has been published in previous studies. This article is dedicated to the investigation of the convergence of approximate solutions to the exact solution of the considered non-linear optimization problem. The influence of the Fredholm integral operator on the convergence behavior of the approximations is examined. It is demonstrated that the presence of the integral operator necessitates the construction of three distinct types of approximations of the optimal process: “Resolvent” approximations, based on the resolvent of the kernel of the integral operator; Approximations by optimal controls, constructed through the approximation of control functions; Finite-dimensional approximations.
K.A. Бекмаганбетов, Г.А. Чечкин, В.В. Чепыжов
et al.
The Ginzburg-Landau equation with rapidly oscillating terms in the equation and boundary conditions in a perforated domain was considered. Proof was given that the trajectory attractors of this equation converge weakly to the trajectory attractors of the homogenized Ginzburg-Landau equation. To do this, we use the approach from the articles and monographs of V.V. Chepyzhov and M.I. Vishik about trajectory attractors of evolutionary equations, and we also use homogenization methods that appeared at the end of the 20th century. First, we use asymptotic methods to construct asymptotics formally, and then we justify the form of the main terms of the asymptotic series using functional analysis and integral estimates. By defining the corresponding auxiliary function spaces with weak topology, we derive a limit (homogenized) equation and prove the existence of a trajectory attractor for this equation. Then, we formulate the main theorems and prove them by using auxiliary lemmas. We prove that the trajectory attractors of this equation tend in a weak sense to the trajectory attractors of the homogenized Ginzburg-Landau equation in the subcritical case, and they disappear in the supercritical case.
The present paper establishes a formula of Reidemeister torsion for Schottky representations. The theoretical results are applied to 3−manifolds with boundary consisting orientable surfaces with genus at least 2.
Multiparametric difference schemes of the finite element method of a high order of accuracy for the Sobolevtype equation of the fourth-order in time are studied. In particular, the first boundary value problem for the equation of ion-acoustic waves in a magnetized plasma is considered. A high-order accuracy of the scheme is achieved due to the special discretization of time and space variables. The presence of parameters in the scheme makes it possible to regularize the accuracy of the schemes and optimize the implementation algorithm. An a priori estimate in a weak norm is obtained by the method of energy inequality. Based on this estimate and the Bramble-Hilbert lemma, the convergence of the constructed algorithms in classes of generalized solutions is proved. An algorithm for implementing the difference scheme is proposed.
The article studies a second-order integro-differential equation with difference kernels and power nonlinearity. A connection is established between this equation and an integral equation of the convolution type, which arises when describing the processes of liquid infiltration from a cylindrical reservoir into an isotropic homogeneous porous medium, the propagation of shock waves in pipes filled with gas and others. Since non-negative continuous solutions of this integral equation are of particular interest from an applied point of view, solutions of the corresponding integro-differential equation are sought in the cone of the space of continuously differentiable functions. Two-sided a priori estimates are obtained for any solution of the indicated integral equation, based on which the global theorem of existence and uniqueness of the solution is proved by the method of weighted metrics. It is shown that any solution of this integro-differential equation is simultaneously a solution of the integral equation and vice versa, under the additional condition on the kernel that any solution of this integral equation is a solution of this integro-differential equation. Using these results, a global theorem on the existence, uniqueness and method of finding a solution to an integrodifferential equation is proved. It is shown that this solution can be found by the method of successive approximations of the Picard type and an estimate for the rate of their convergence is established. Examples are given to illustrate the obtained results.
S.Zh. Igisinov, L.D. Zhumaliyeva, A.O. Suleimbekova
et al.
In this paper we study a class of degenerate elliptic equations with an arbitrary power degeneracy on the line. Based on the research carried out in the course of the work, the authors propose methods to overcome various difficulties associated with the behavior of functions from the definition domain for a differential operator with piecewise continuous coefficients in a bounded domain, which affect the spectral characteristics of boundary value problems for degenerate elliptic equations. It is shown the conditions imposed on the coefficients at the lowest terms of the equation, which ensure the existence and uniqueness of the solution. The existence, uniqueness, and smoothness of a solution are proved, and estimates are found for singular numbers (s-numbers) and eigenvalues of the semiperiodic Dirichlet problem for a class of degenerate elliptic equations with arbitrary power degeneration.
Geometrical frustration in thin sheets is ubiquitous across scales in biology and becomes increasingly relevant in technology. Previous research identified the origin of the frustration as the violation of Gauss’s Theorema Egregium. Such “Gauss frustration” exhibits rich phenomenology; it may lead to mechanical instabilities, anomalous mechanics, and shape-morphing abilities that can be harnessed in engineering systems. Here we report a new type of geometrical frustration, one that is as general as Gauss frustration. We show that its origin is the violation of Mainardi-Codazzi-Peterson compatibility equations and that it appears in Euclidean sheets. Combining experiments, simulations, and theory, we study the specific case of a Euclidean ribbon with radial and geodesic curvatures. Experiments, conducted using different materials and techniques, reveal shape transitions, symmetry breaking, and spontaneous stress focusing. These observations are quantitatively rationalized using analytic solutions and geometrical arguments. We expect this frustration to play a significant role in natural and engineering systems, specifically in slender 3D printed sheets.
We present a microfluidic method to measure the elastic properties of a population of microcapsules (liquid drops enclosed by a thin hyperelastic membrane). The method is based on the observation of flowing capsules in a cylindrical capillary tube and an automatic inverse analysis of the deformed profiles. The latter requires results from a full numerical model of the fluid–structure interaction accounting for nonlinear membrane elastic properties. For ease of use, we provide them under the form of databases, when the initially spherical capsule has a membrane governed by a neo-Hookean or a general Hooke's law with different surface Poisson ratios. Ultimately, the microfluidic method yields information on the type of elastic constitutive law that governs the capsule wall material together with the value of the elastic parameters. The method is applied to a population of ovalbumin microcapsules and is validated by means of independent experiments of the same capsules subjected to a different flow in a microrheological device. This is of great interest for quality control purposes, as small samples of capsule suspensions can be diverted to a measuring test section and mechanically tested with a 10 % precision using an automated process.
Combining a computationally efficient and affordable molecular dynamics approach, based on atom-centered density matrix propagation scheme, with the density functional tight binding semiempirical quantum mechanics, we study the vibrational dynamics of a single molecule at series of finite temperatures, spanning quite wide range. Data generated by molecular dynamics simulations are further analyzed and processed using time series analytic methods, based on correlation functions formalism, leading to both vibrational density of states spectra and infrared absorption spectra at finite temperatures. The temperature-induced dynamics in structural intramolecular parameters is correlated to the observed changes in the spectral regions relevant to molecular detection. In particular, we consider a case when an intramolecular X-H stretching vibrational states are notably dependent on the intramolecular torsional degree of freedom, the dynamics of which is, on the other hand, strongly temperature-dependent.
This article is devoted to the study of the solvability of a semi - periodic boundary value problem for an evolution equation of the pseudoparabolic type. Nonlocal problems for high order partial differential equations have been investigated by many authors [1-4]. A certain interest in the study of these problems is caused in connection with their applied values. These problems include highly porous media with a complex topology, and first of all, soil and ground. Such equations can also describe long waves in dispersed systems. To solve this problem, new functions are introduced in the work and the method of a parameterizations applied [5]. Then the boundary value problem for a third order differential equation is reduced to a periodic boundary value problem for a family of systems of ordinary differential equations [6-18]. New constructive algorithms for finding an approximate solution are proposed and in terms of the initial data, coefficient - like signs of the unique solvability of the problem under study are obtained.
In this article are considered model - theoretical properties of chains of positive ( n1,n2) - Jonsson theories. Herewith considered theories is perfect in the sense of the existence of appropriate model companion. The main obtained results are as follows: introduced new concepts n 2 - elimination of quantifier for positive theory, ( n1,n2) - Jonsson theory, n1-Jonsson chain; indicated a feature of perfect ( n1,n2) - positive Jonsson theory
The paper considers a singularly perturbed control problem with a quadratic quality functional. Such problems in their standard formulation under known spectrum restrictions (the points of the spectrum of the optimal system are not purely imaginary and are located symmetrically with respect to the imaginary axis) were previously considered using the Vasilyeva - Butuzov method of boundary functions. If at least one of the points of the spectrum for some values of the independent variable falls on the imaginary axis, the boundary functions method does not work. It is precisely this situation with the assumption of purely imaginary points of the spectrum that is investigated in this paper. In this case, you have to develop a different approach based on the ideas of the regularization method S.A. Lomov. It should also be noted that in the control problems considered earlier, the cost functional either did not depend on a small parameter at all, or allowed a smooth dependence on the parameter...
In this paper we consider a telegraph equation. In the case of a rectangular domain for the Cauchy potential the lateral boundary conditions obtained. When considering the equation in the first quadrant a criterion for the existence of soliton solutions is obtained.
The artificial accelerated aging test of low density polyethylene (LDPE) was carried out for different time periods up to 64 days under UV environment to study the photo-oxidative degradation behavior and the rule. The influence on mechanical properties, chemical structure, thermal stability and melting property evolution of LDPE after photo-oxidation aging was studied by mechanics experiments, attenuated total reflection infrared spectroscopy (ATR-FTIR), thermogravimetry analysis (TGA) and differential scanning calorimetry (DSC). Analytic hierarchy procedure (AHP) was used to determine the weighing values for assessment index of LDPE, and the photo-oxidative degradation comprehensive evaluation model of LDPE was established on the base of improving traditional principal components analysis (PCA). The results show with increasing aging time, the tensile strength, bending strength and impact strength of LDPE are declined by 39.7%, 32.3% and 96.4%, respectively. The concentration of carbonyl and hydroxyl groups are increased, the rupture of molecular chain is intensified. The initial thermal decomposition temperature and melting temperature peak value are declined and the LDPE surface microstructure is seriously damaged and the aging is severe. The comprehensive evaluation parameters exhibit changing trend in three stages. The evaluation results of improved PCA are more reasonable and all the above-mentioned shows that improved PCA is appropriate for the comprehensive evaluation of photo-oxidative degradation behavior of LDPE.
Materials of engineering and construction. Mechanics of materials
Elena Beltrán-Heredia, Elena Beltrán-Heredia, Víctor G. Almendro-Vedia
et al.
Many cell division processes have been conserved throughout evolution and are being revealed by studies on model organisms such as bacteria, yeasts, and protozoa. Cellular membrane constriction is one of these processes, observed almost universally during cell division. It happens similarly in all organisms through a mechanical pathway synchronized with the sequence of cytokinetic events in the cell interior. Arguably, such a mechanical process is mastered by the coordinated action of a constriction machinery fueled by biochemical energy in conjunction with the passive mechanics of the cellular membrane. Independently of the details of the constriction engine, the membrane component responds against deformation by minimizing the elastic energy at every constriction state following a pathway still unknown. In this paper, we address a theoretical study of the mechanics of membrane constriction in a simplified model that describes a homogeneous membrane vesicle in the regime where mechanical work due to osmotic pressure, surface tension, and bending energy are comparable. We develop a general method to find approximate analytical expressions for the main descriptors of a symmetrically constricted vesicle. Analytical solutions are obtained by combining a perturbative expansion for small deformations with a variational approach that was previously demonstrated valid at the reference state of an initially spherical vesicle at isotonic conditions. The analytic approximate results are compared with the exact solution obtained from numerical computations, getting a good agreement for all the computed quantities (energy, area, volume, constriction force). We analyze the effects of the spontaneous curvature, the surface tension and the osmotic pressure in these quantities, focusing especially on the constriction force. The more favorable conditions for vesicle constriction are determined, obtaining that smaller constriction forces are required for positive spontaneous curvatures, low or negative membrane tension and hypertonic media. Conditions for spontaneous constriction at a given constriction force are also determined. The implications of these results for biological cell division are discussed. This work contributes to a better quantitative understanding of the mechanical pathway of cellular division, and could assist the design of artificial divisomes in vesicle-based self-actuated microsystems obtained from synthetic biology approaches.