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The paper presents the results of the study of the correlation between the motivation for studying the disciplines of the mathematical cycle by students of engineering specialties of a higher educational institution and the level of their knowledge of these disciplines. Statistical analysis of the results of the survey shows that students' attitude to mathematics is related to the level of knowledge of the subject. Interest in school and university mathematics courses, sufficiency of applied effort and orientation to success in acquiring knowledge and skills in mathematical disciplines at university, as well as the usefulness of mathematical knowledge for future educational and career aspirations differ for students with different levels of knowledge, and namely, the higher the level of students' knowledge, the higher the level of the specified motivational factors. Difficulties in learning mathematical disciplines and anxiety during control measures do not have a significant statistical correlation with the level of knowledge. [ABSTRACT FROM AUTHOR]

Axiomatic Fredholm theory in unital C*-algebras was established in [D. Kečkić, Z. Lazović, Acta Sci. Math., 83(3-4):629--655, 2017]. Following the pure algebraic approach by Kečkić and Lazović, in the author's paper [S. Ivković, Banach J. Math. Anal., 17:51, 2023] we extended further this theory to axiomatic semi-Fredholm and semi-Weyl theory in unital C*-algebras. However, recently, in [S. Ivković, arXiv:2306.01133] we developed another approach to axiomatic Fredholm theory in unital C*-algebras which is based on the theory of Hilbert modules and is equivalent to the algebraic approach by Kečkić and Lazović. In this paper, we extend further that new Hilbert-module approach from Fredholm theory to semi-Fredholm and semi-Weyl theory in unital C*-algebras. Hence, we provide new proofs to the results in [S. Ivković, Banach J. Math. Anal., 17:51, 2023].

In this paper, we address the stabilization problem for nonlinear systems in a critical case. Namely, we study the class of canonical nonlinear systems. Canonical nonlinear systems or chain of power integrators is an important subject of research. Studying such systems is complicated by the fact that they cannot be mapped onto linear systems. Moreover, they have the uncontrollable first approximation. Previous results on smooth stabilization of such systems were obtained under the assumption that the powers in the right-hand side are strictly decreasing. In this work, we consider a case of non-increasing powers in the right-hand side for a three-dimensional system. A popular approach for studying such systems is the backstepping method, which is a method of step-wise stabilization. This method requires a sequential investigation of lower-dimensional subsystems. Backstepping enables the study of a wide range of nonlinear triangular systems but requires technically complex and cumbersome computations. Therefore, a natural question arises about constructing stabilizing controls of a simple form. Polynomial controls can serve as an example of such controls. In the paper, we demonstrate that linear controls can be considered as stabilizing controls. We derive sufficient conditions for the coefficients of the linear control that ensure the asymptotic stability of the zero equilibrium point of the corresponding closed-loop system. The asymptotic stability is proven using the Lyapunov function method, which is found as the sum of squares. The negative definiteness of the Lyapunov function derivative in a neighborhood of the origin guarantees asymptotic stability. In contrast to the case of strictly decreasing powers, additional conditions on the control coefficients, apart from their negativity, emerge. The obtained result extends to a broader class of nonlinear systems through stabilization by nonlinear approximation. This allows the consideration of systems with higher-order terms in the right-hand side. The effectiveness of the applied approach is illustrated by several model examples. The method used in this work to investigate the case of non-increasing powers can be applied to systems of higher dimensions.

In this paper, we consider polynomial composites with the coefficients from $K\subset L$. We already know many properties, but we do not know the answer to the question of whether there is a relationship between composites and field extensions. We present the characterization of some known field extensions in terms of polynomial composites. This paper contains the open problem of characterization of ideals in polynomial composites with respect to various field extensions. We also present the full possible characterization of certain field extensions. Moreover, in this paper we show that any finite group is a Galois group of some field extensions and present the inverse Galois problem solved.

This paper is an introduction to a sequence of nonlinear Baskakov-Durrmeyer operators $(NBD_{n})$ of the form \[ (NBD_{n})(f;x) =\int_{0}^\infty K_{n}(x,t,f(t))\,dt \] with $x\in [0,\infty)$ and $n\in\mathbb{N}$. While $K_{n}(x,t,u)$ provide convenient assumptions, these operators work on bounded functions, which are defined on all finite subintervals of $[0,\infty)$. This paper comprise some pointwise convergence results for these operators in certain functional spaces. As well as this study can be seen as a continuation of studies about nonlinear operators, it is the first study on nonlinear Baskakov-Durrmeyer or modified Baskakov operators, while there were more papers on linear part of the operators.

In this paper, we study the qualitative behavior of the rational recursive equation \begin{equation*} x_{n+1}=\frac{x_{n-4}}{\pm1\pm x_{n}x_{n-1}x_{n-2}x_{n-3}x_{n-4}}, \quad n \in \mathbb{N}_{0}:=\{0\}\cup\mathbb N, \end{equation*} where the initial conditions are arbitrary nonzero real numbers. The main goal of this paper, is to obtain the forms of the solutions of the nonlinear fifth-order difference equations, where the initial conditions are arbitrary positive real numbers. Moreover, we investigate stability, boundedness, oscillation and the periodic character of these solutions. The results presented in this paper improve and extend some corresponding results in the literature.

The topic of the paper is the investigation of the homology groups of the $(2n+1)$-dimensional CW-complex $\mathbb{C}\Omega_n$. The spaces $\mathbb{C}\Omega_n$ consist of complex-valued functions and are the analogue of the spaces $\Omega_n$, widely known in the approximation theory. The spaces $\mathbb{C}\Omega_n$ have been introduced in 2015 by A.M. Pasko who has built the CW-structure of the spaces $\mathbb{C}\Omega_n$ and using this CW-structure established that the spaces $\mathbb{C}\Omega_n$ are simply connected. Note that the mentioned CW-structure of the spaces $\mathbb{C}\Omega_n$ is the analogue of the CW-structure of the spaces $\Omega_n$ constructed by V.I. Ruban. Further A.M. Pasko found the homology groups of the space $\mathbb{C}\Omega_n$ in the dimensionalities $0, 1, \ldots, n, 2n-1, 2n, 2n+1$. The goal of the present paper is to find the homology group $H_{n+1}\left ( \mathbb{C}\Omega_n \right )$. It is proved that $H_{n+1} \left ( \mathbb{C}\Omega_n \right )=\mathbb{Z}^\frac{n+1}{2}$ if $n$ is odd and $H_{n+1} \left ( \mathbb{C}\Omega_n \right )=\mathbb{Z}^\frac{n+2}{2}$ if $n$ is even.

The present paper continues the study of quasi-geodesic mappings f:(Vn, gij, Fih) → (V'n,g'ij, Fih) of pseudo-Riemannian spaces Vn, V'n with a generalized-recurrent structure Fih of parabolic type. By a generalized recurrent structure of parabolic type on Vn we mean an almost Hermitian affinor structure of parabolic type for which the covariant derivative of the structural affinor Fih satisfies the condition F(i,j)h=q(i Fj)h. In the previous paper by the authors [Proc. Intern. Geom. Center, 13:3 (2020) 18-32] it was proved that the class of pseudo-Riemannian spaces with generalized-recurrent structure of parabolic type is closed with respect to the considered mappings and the generalized recurrence vectors in (Vn, gij,Fih) and (V'_n, g'ij, Fih) may be distinct. In this article, it is assumed that the mapping f preserves the generalized recurrence vector qi. We construct geometric objects that are invariant under the quasi-geodesic mapping of generalized-recurrent spaces of parabolic type and recurrent-parabolic spaces. A number of conditions are given on these objects, which lead to the fact that a generalized-recurrent space of parabolic type admits a parabolic K-structure, and a recurrent-parabolic space admits a Kählerian structure of parabolic type. We study special types of these mappings that preserve some tensors of an intrinsic nature.

In this paper, we consider the stability of limit regimes for a general class of nonlinear distributed mathematical models named Reaction-Diffusion models. RD systems naturally arise in many applications. For instance, in the biological mathematical modeling and in the signal transmission theory the FitzHugh–Nagumo model, whose distributed variant is a particular case of general RD system, is widely used. We investigate the problem of stability of attracting sets for an infinite-dimensional RD system with respect to bounded external signals (disturbances). The interaction functions as well as nonlinear perturbations do not assume to be Lipschitz continuous. Therefore, we cannot expect the uniqueness of solution for the corresponding initial-value problem and we have to use a multi-valued semigroup approach. An undisturbed system is considered to have a global attractor, i.e., the minimal compact uniformly attracting set. The main purpose is to estimate the deviation of the trajectory of the disturbed system from the global attractor of the undisturbed one as a function of the magnitude of external signals. Such an estimate can be obtained in the framework of the theory of input-to-state stability (ISS). The paper proposes a new approach to obtaining estimates of robust stability of the attractor in the case of a multivalued evolutionary operator. In particular, it is proved that the multivalued semigroup generated by weak solutions of a nonlinear reaction-diffusion system has the property of local ISS with respect to the attractor of the undisturbed system.

The present paper continues the study of quasi-geodesic mappings f:(Vn, gij, Fih) → (V'n,g'ij, Fih) of pseudo-Riemannian spaces Vn, V'n with a generalized-recurrent structure Fih of parabolic type. By a generalized recurrent structure of parabolic type on Vn we mean an almost Hermitian affinor structure of parabolic type for which the covariant derivative of the structural affinor Fih satisfies the condition F(i,j)h=q(i Fj)h. In the previous paper by the authors [Proc. Intern. Geom. Center, 13:3 (2020) 18-32] it was proved that the class of pseudo-Riemannian spaces with generalized-recurrent structure of parabolic type is closed with respect to the considered mappings and the generalized recurrence vectors in (Vn, gij,Fih) and (V'_n, g'ij, Fih) may be distinct. In this article, it is assumed that the mapping f preserves the generalized recurrence vector qi. We construct geometric objects that are invariant under the quasi-geodesic mapping of generalized-recurrent spaces of parabolic type and recurrent-parabolic spaces. A number of conditions are given on these objects, which lead to the fact that a generalized-recurrent space of parabolic type admits a parabolic K-structure, and a recurrent-parabolic space admits a Kählerian structure of parabolic type. We study special types of these mappings that preserve some tensors of an intrinsic nature.

In the paper, we investigate an asymptotic behavior of the sharp upper bounds in the integral metric of deviations of the Abel-Poisson integrals from functions from the class $L^{\psi}_{\beta, 1}$. The Abel-Poisson integrals are solutions of the partial differential equations of elliptic type with corresponding boundary conditions, and they play an important role in applied problems. The approximative properties of the Abel-Poisson integrals on different classes of differentiable functions were studied in a number of papers. Nevertheless, a problem on the respective approximation on the classes $L^{\psi}_{\beta,1}$ in the metric of the space $L$ remained unsolved. We managed to obtain the estimates for the values of approximation of $(\psi, \beta)$-differentiable functions from the unit ball of the space $L$ by the Abel-Poisson integrals. In some cases, we also write down asymptotic equalities for these quantities, that is we solve the Kolmogorov-Nikol'skii problem for the the Abel-Poisson integrals on the classes $L^{\psi}_{\beta,1}$ in the integral metric.

The representations of partially ordered sets (abbreviated as posets), introduced by L. A. Nazarova and A. V. Roiter (in matrix form) in 1972, play an important role in the modern representation theory. In his first paper on this topic M. M. Kleiner proved that a poset S is of finite representation type (i.e. has a finite number, up to equivalence, of indecomposable representations) if and only if it does not contain subposets of the form K1= (1, 1, 1, 1), K2 = (2, 2, 2), K3 = (1, 3, 3), K4 = (1, 2, 5) and K5 = (N, 4). Specified posets are called critical posets relative to the finiteness of the type. They are also called the Kleiner's (critical) posets. In 1974 Yu. A. Drozd proved that a poset S has finite representation type if and only if its Tits quadratic form is weakly positive (i.e., positive on the set of non-negative vectors). Consequently, the Kleiner's posets are critical relative to weak positivity of the Tits quadratic form, and there are no (up to isomorphism) other such posets. In 2005 the authors proved that a poset is critical relative to the positivity of the Tits quadratic form if and only if it is minimax isomorphic to a Kleiner's poset. A similar situation takes place for posets of tame representation type. In 1975 L. A. Nazarova proved that a poset S is tame if and only if it does not contain subsets of the form N1 = (1, 1, 1, 1, 1), N2 = (1, 1, 1, 2), N3 = (2, 2, 3), N4 = (1, 3, 4), N5 = (1, 2, 6) and N6 = (N, 5). She called these posets supercritical; they are also critical relative to weak non-negativity of the Tits quadratic. In 2008 the authors proved that a poset is critical relative to non-negativity of the Tits quadratic form if and only if it is minimax isomorphic to a supercritical poset. In this paper we study the combinatorial properties of posets, minimax isomorphic to the supercritical poset of greatest height, i.e. (1, 2, 6). The importance of studying minimax isomorphic posets is determined by the fact that their Tits quadratic forms are ℤ-equivalent, and minimax isomorphism itself is a fairly general constructively defined ℤ- quivalence of the Tits quadratic forms for posets.

The paper is devoted to the investigation of the notion of a differentially prime ideal of a differential commutative semiring (i. e. a semiring equipped with a derivation), and its interrelation with the notions of a quasi-prime ideal and a primary ideal. The notion of a semiring derivation is traditionally defined as an additive map satisfying the Leibnitz rule, i. e. a map δ: R → R is called a derivation on R if δ (a + b)= δ (a) + δ (b) and δ (ab) = δ (a)b + aδ (b) for any a, b ∈ R. A differential ideal P of R is called a differentially prime ideal if for any a, b ∈ R, k ∈ ℕ0, ab(k) ∈ P follows a ∈ P or b ∈ P. It is proved that an ideal P of a semiring R is prime if and only if for any ideals I and J of R the inclusion IJ ⊆ P follows I ⊆ P or J ⊆ P. A quasi-prime ideal is a differential ideal of a semiring which is maximal among those ideals disjoint from some multiplicatively closed subset of a semiring. In this paper we investigate some properties of such differentially prime ideals, in particular in case of differential Noetherian semirings. The paper consists of two main parts. The first part of the paper is devoted to establishing some properties of differentially prime ideals and gives some examples of such ideals. In the second part, the author investigates the connection existing between quasi-prime ideals, primary ideals and differentially prime ideals in differential Noetherian semirings. It is established that in a differential Noetherian semiring R a differential ideal I of R is differentially prime if and only if I is a quasi-prime ideal.

Ideal on an arbitrary non-empty set $\Omega$ it's a non-empty family of subset $\mathfrak{I}$ of the set $\Omega$ which satisfies the following axioms: $\Omega \notin \mathfrak{I}$, if $A, B \in \mathfrak{I}$, then $A \cup B \in \mathfrak{I}$, if $A \in \mathfrak{I}$ and $D \subset A$, then $D \in \mathfrak{I}$. The ideal theory is a very popular branch of modern mathematical research. In our paper we study some classes of ideals on the set of all positive integers $\mathbb{N}$, namely the ideal of statistical convergence $\mathfrak{I}_s$ and the ideal $\mathfrak{I}_f$ generated by a modular function $f$. Statistical ideal it's a family of subsets of $\mathbb{N}$ whose natural density is equal to 0, i.e. $A \in \mathfrak{I}_s$ if and only if $\displaystyle\lim\limits_{n \rightarrow \infty}\frac{\#\{k \leq n: k \in A\}}{n} = 0$. A function $f:\mathbb{R}^+ \rightarrow \mathbb{R}^+$ is called a modular function, if $f(x) = 0$ only if $x = 0$, $f(x + y) \leq f(x) + f(y)$ for all $x, y \in\mathbb{R}^+$, $f(x) \le f(y)$ whenever $x \le y$, $f$ is continuous from the right 0, and finally $\lim\limits_{n \rightarrow \infty} f(n) = \infty$. Ideal, generated by the modular function $f$ it's a family of subsets of $\mathbb{N}$ with zero $f$-density, in other words, $A \in \mathfrak{I}_f$ if and only if $\displaystyle\lim\limits_{n \rightarrow \infty}\frac{f(\#\{k \leq n: k \in A\})}{f(n)} = 0$. It is known that for an arbitrary modular function $f$ the following is true: $\mathfrak{I}_f \subset \mathfrak{I}_s$. In our research we give the complete description of those modular functions $f$ for which $\mathfrak{I}_f = \mathfrak{I}_s$. Then we analyse obtained result, give some partial cases of it and prove one simple sufficient condition for the equality $\mathfrak{I}_f = \mathfrak{I}_s$. The last section of this article is devoted to examples of some modulus functions $f, g$ for which $\mathfrak{I}_f = \mathfrak{I}_s$ and $\mathfrak{I}_g \neq \mathfrak{I}_s$. Namely, if $f(x) = x^p$ where $p \in (0, 1]$ we have $\mathfrak{I}_f = \mathfrak{I}_s$; for $g(x) = \log(1 + x)$, we obtain $\mathfrak{I}_g \neq \mathfrak{I}_s$. Then we consider more complicated function $f$ which is given recursively to demonstrate that the conditions of the main theorem of our paper can't be reduced to the sufficient condition mentioned above.

The permutation code (or the code) is well known object of research starting from 1970s. The code and its properties is used in different algorithmic domains such as error-correction, computer search, etc. It can be defined as follows: the set of permutations with the minimum distance between every pair of them. The considered distance can be different. In general, there are studied codes with Hamming, Ulam, Levensteins, etc. distances. In the paper we considered permutations codes over 2-Sylow subgroups of symmetric groups with Hamming distance over them. For this approach representation of permutations by rooted labeled binary trees is used. This representation was introduced in the previous author's paper. We also study the property of the Hamming distance defined on permutations from Sylow 2-subgroup $Syl_2(S_{2^n})$ of symmetric group $S_{2^n}$ and describe an algorithm for finding the Hamming distance over elements from Sylow 2-subgroup of the symmetric group with complexity $O(2^n)$. The metric properties of the codes that are defined on permutations from Sylow 2-subgroup $Syl_2(S_{2^n})$ of symmetric group $S_{2^n}$ are studied. The capacity and number of codes for the maximum and the minimum non-trivial distance over codes are characterized.

A well known diameter search problem for finite groups with respect to its systems of generators is considered. The problem can be formulated as follows: find the diameter of a group over its system of generators. The diameter of a group over a specific system of generators is the diameter of the corresponding Cayley graph. It is considered alternating groups with classic irreducible system of generators consisting of cycles with length three of the form $(1,2,k)$. The main part of the paper concentrates on analysis how even permutations decompose with respect to this system of generators. The rules for moving generators from permutation's decomposition from left to right and from right to left are introduced. These rules give rise for transformations of decompositions, that do not increase their lengths. They are applied for removing fixed points of a permutation, that were included in its decomposition. Based on this rule the stability of system of generators is proved. The strict growing property of the system of generators is also proved, as the corollary of transformation rules and the stability property. It is considered homogeneous theory, that was introduced in the previous author's paper. For the series of alternating groups with systems of generators mentioned above it is shown that this series is uniform and homogeneous. It makes possible to apply the homogeneous down search algorithm to compute the diameter. This algorithm is applied and exact values of diameters for alternating groups of degree up to 43 are computed.

The paper considers machine learning algorithms. The focus is on semi-controlled learning, which seems to be the balance between teaching accuracy with a teacher and the cost of teaching methods without a teacher. Examples of careful processing of labelled data sets for which supervised learning can be very effective are considered. The semi-supervised and supervised approaches are compared, and the effectiveness of each is analyzed. The paper considers S3VM and TSVM approach. The work aimed to investigate whether semi-controlled methods can compete with controlled or even surpass them. Applying these approaches to the proposed dataset to determine a more accurate classification of data, namely at the reference limit, is described.

The notion of a semiring derivation is traditionally defined as an additive map satisfying the Leibnitz rule, i. e. a map δ: R →R is called a derivation on R if δ (a + b) = δ (a)+δ (b) and δ (ab) = δ (a) b + aδ (b) for any a,b ∈ R. The notion of a "quasi-prime ideal", for the first time, was introduced in differential commutative rings, i.e. commutative rings considered together with a derivation, as a differential ideal maximal among those disjoint from some multiplicatively closed subset of a ring. A subsemimodule P of a semimodule M is called prime if for any ideal I of R and any subsemimodule N of M the inclusion IN ⊆ P follows N ⊆ P or I ⊆ (P : M). A differential subsemimodule P of M is called a differentially prime subsemimodule if for any r ∈ R, m ∈ M, k ∈ N0, rm(k) ∈ P follows r ∈(P : M) or m ∈P. The present paper is devoted to investigating the notions of "differential subsemimodule", "differentially prime subsemimodule", and "quasi-prime subsemimodule" of a differential semimodule (which is defined as a semimodule together with a derivation on it related to the corresponding semiring derivation), not necessarily commutative. The objective of the article is to investigate some properties of such subsemimodules, and to show the interrelation between "quasi-prime ideals" and "differentially prime subsemimodule" in case of differential semimodules satisfying the ascending chain condition for differential subsemimodules. The paper consists of two main parts. In the first part, the author investigates some properties of differential subsemimodules and the corresponding differential ideals, and gives some examples of such subsemimodules. The second part of the paper is devoted to considering the connection existing between "quasi-prime subsemimodules" and "differentially prime subsemimodules". It is established that a differential subsemimodule N of M is differentially prime if and only if N is a quasi-prime subsemimodule for a differential semimodule M satisfying the ascending chain condition for differential subsemimodules.

The controllability function method, introduced by V. I. Korobov in late 1970s, is known to be an efficient tool for control systems design. It is developed for both linear/nonlinear and finite/infinite dimensional systems. This paper bridges the method with the homogeneity theory popular today. The standard homogeneity known since 18th century is a symmetry of function with respect to uniform scaling of its argument. Some generalizations of the standard homogeneity were introduced in 20th century. This paper shows that the so-called homogeneous norm is a controllability function of the linear autonomous control system and the corresponding closed-loop system is homogeneous in the generalized sense. This immediately yields many useful properties known for homogeneous systems such as robustness (Input-to-State Stability) with respect to a rather large class of perturbations, in particular, with respect to bounded additive measurement noises and bounded additive exogenous disturbances. The main theorem presented in this paper slightly refines the design of the controllability function for a multiply-input linear autonomous control systems. The design procedure consists in solving subsequently a linear algebraic equation and a system of linear matrix inequalities. The homogeneity itself and the use of the canonical homogeneous norm essentially simplify the design of a controllability function and the analysis of the closed-loop system. Theoretical results are supported with examples. The further study of homogeneity-based design of controllability functions seems to be a promising direction for future research.

In this paper, we introduce the concept of orthogonal $m$-metric spaces and by using $F_m$-contraction in orthogonal $m$-metric spaces, we give the concept of orthogonal $F_m$-contraction (briefly, $\bot_{F_m}$-contraction) and investigate fixed point results for such mappings. Many existing results in the literature appear to be special case of results proved in this paper. An example to support our main results is also mentioned.

In the paper the homology groups of the $(2n+1)$-dimensional CW-complex $\mathbb{C}\Omega_n$ are investigated. The spaces $\mathbb{C}\Omega_n$ consist of complex-valued functions and generalize the widely known in the approximation theory spaces $\Omega_n$. The research of the homotopy properties of the spaces $\Omega_n$ has been started by V.I. Ruban who in 1985 found the n-dimensional homology group of the space $\Omega_n$ and in 1999 found all the cohomology groups of this space. The spaces $\mathbb{C}\Omega_n$ have been introduced by A.M. Pasko who in 2015 has built the structure of CW-complex on these spaces. This CW-structure is analogue of the CW-structure of the space $\Omega_n$ introduced by V.I. Ruban. In present paper in order to investigate the homology groups of the spaces $\mathbb{C}\Omega_n$ we calculate the relative homology groups $H_k(\mathbb{C}\Omega_n, \mathbb{C}\Omega_{n-1})$, it turned out that the groups $H_k \left (\mathbb{C}\Omega_n, \mathbb{C}\Omega_{n-1} \right )$ are trivial if $1\leq k < n$ and $H_k \left (\mathbb{C}\Omega_n, \mathbb{C}\Omega_{n-1} \right )=\mathbb{Z}^{C^{k-n}_{n+1}}$ if $n \leq k \leq 2n+1$, in particular $H_n \left (\mathbb{C}\Omega_n, \mathbb{C}\Omega_{n-1} \right )=\mathbb{Z}$. Further we consider the exact homology sequence of the pair $\left (\mathbb{C}\Omega_{n+1}, \mathbb{C}\Omega_n \right )$ and prove that its inclusion operator $i_*: H_k(\mathbb{C}\Omega_n) \rightarrow H_k(\mathbb{C}\Omega_{n+1})$ is zero. Taking into account that the relative homology groups $H_k \left (\mathbb{C}\Omega_{n+1}, \mathbb{C}\Omega_n \right )$ are zero if $1\leq k \leq n$ and the inclusion operator $i_*=0$ we have derived from the exact homology sequence of the pair $\left (\mathbb{C}\Omega_{n+1}, \mathbb{C}\Omega_n \right )$ that the homology groups $H_k \left ( \mathbb{C}\Omega_n \right ), 1\leq k

In this paper, based on generalized Herz-type function spaces $\dot{K}_{q}^{p}(\theta)$ were introduced by Y. Komori and K. Matsuoka in 2009, we define Herz-type Besov spaces $\dot{K}_{q}^{p}B_{\beta }^{s}(\theta)$ and Herz-type Triebel-Lizorkin spaces $\dot{K}_{q}^{p}F_{\beta }^{s}(\theta)$, which cover the Besov spaces and the Triebel-Lizorkin spaces in the homogeneous case, where $\theta=\left\{\theta(k)\right\} _{k\in\mathbb{Z}}$ is a sequence of non-negative numbers $\theta(k)$ such that \begin{equation*} C^{-1}2^{\delta (k-j)}\leq \frac{\theta(k)}{\theta(j)} \leq C2^{\alpha (k-j)},\quad k>j, \end{equation*} for some $C\geq 1$ ($\alpha$ and $\delta $ are numbers in $\mathbb{R}$). Further, under the condition mentioned above on ${\theta }$, we prove that $\dot{K}_{q}^{p}\left({\theta }\right)$ and $\dot{K}_{q}^{p}B_{\beta }^{s}\left({\theta }\right)$ are localizable in the $\ell _{q}$-norm for $p=q$, and $\dot{K}_{q}^{p}F_{\beta }^{s}\left({\theta }\right)$ is localizable in the $\ell _{q}$-norm, i.e. there exists $\varphi \in \mathcal{D}({\mathbb{R}}^{n})$ satisfying $\sum_{k\in \mathbb{Z}^{n}}\varphi \left( x-k\right) =1$, for any $x\in \mathbb{R}^{n}$, such that \begin{equation*} \left\Vert f|E\right\Vert \approx \Big(\underset{k\in \mathbb{Z}^{n}}{\sum }\left\Vert \varphi (\cdot-k)\cdot f|E\right\Vert ^{q}\Big)^{1/q}. \end{equation*} Results presented in this paper improve and generalize some known corresponding results in some function spaces.

The purpose of this paper is to obtain some sufficient conditions to determine the relation between a meromorphic function and an $L$-function when certain differential polynomial generated by them sharing a one degree polynomial. The main theorem of the paper extends and improves all the results due to W.J. Hao, J.F. Chen [Discrete Dyn. Nat. Soc. 2018, 2018, article ID 4673165], F. Liu, X.M. Li, H.X. Yi [Proc. Japan Acad. Ser. A Math. Sci. 2017, 93 (5), 41-46], P. Sahoo, S. Halder [Tbilisi Math. J. 2018, 11 (4), 67-78].

The paper treats geodesic mappings of quasi-Einstein spaces with gradient defining vector. Previously the authors defined three types of these spaces. In the present paper it is proved that there are no quasi-Einstein spaces of special type. It is demonstrated that quasi-Einstein spaces of main type are closed with respect to geodesic mappings. The spaces of particular type are proved to be geodesic $D$-symmetric spaces.

The problem of optimal recovery is considered for functions from their Fourier coefficients known with error. In a more general statement, this problem for the classes of smooth and atalytic functions defined on various compact manifolds can be found in the classical paper by G.G. Magaril-Il'yaev, K.Y. Osipenko. Namely, the paper is devoted to the recovery of continuous real-valued functions $y$ of one variable from the classes $W^{\psi}_{p}$, $1 \leq p< \infty$, that are defined in terms of generalized smoothness $\psi$ from their Fourier coefficients with respect to some complete orthonormal in the space $L_2$ system $\Phi = \{ \varphi_k \}_{k=1}^{\infty}$ of continuous functions, that are blurred by noise. Assume that for function $y$ we know the values $y_k^{\delta}$ of their noisy Fourier coefficients, besides $y_k^{\delta} = y_k + \delta \xi_k$, $k = 1,2, \dots$, where $y_k$ are the corresponding Fourier coefficients, $\delta \in (0,1)$, and $\xi = (\xi_k)_{k=1}^{\infty}$ is a noise. Additionally let the functions from the system $\Phi$ be continuous and satisfy the condition $\| \varphi_k \|_{C}\leq C_1 k^{\beta}$, $k=1,2,\dots$, where $C_1>0$, $\beta \geq 0$ are some constants, and $\| \cdot\|_{C}$ is the standart norm of the space $C$ of continuous on the segment $[0,1]$ functions. Under certain conditions on parameter $\psi$, we obtain order estimates of the approximation errors of functions from the classes $$ W^{\psi}_{p} = \left\{ y \in L_2\colon \| y \|^p_{W^{\psi}_{p}} = \sum\limits_{k=1}^{\infty} \psi^p(k) |y_k|^p \leq 1 \right\}, \quad 1 \leq p< \infty, $$ in metric of the space $C$ by the so-called $\Lambda$-method of series summation that is defined by the number triangular matrix $\Lambda = \{ \lambda_k^n \}_{k=1}^n$, $n=n(\delta) \in \mathbb{N}$, with some restrictions on its elements. Note, that we extend the known results [8, 7] to a more wide spectrum of the classes of functions and for a more general restrictions on the noise level. In our results a case is considered when the noise is stronger than those in the space $l_2$ of real number sequences, but not stochastic.

In earlier papers we introduced a representation of isotopy classes of compact surfaces embedded in the three-sphere S3 by so called rectangular diagrams. The formalism proved useful for comparing Legendrian knots. The aim of this paper is to prove a Reidemeister type theorem for rectangular diagrams of surfaces.

The paper is devoted to the study of properties of pseudo-Riemannian spaces admitting nontrivial geodesic mappings. Necessary and sufficient conditions are found for A-threesymmetric spaces to admit nontrivial geodesic mappings. The research is carried out locally, in tensor form without restrictions to the sign of the metric tensor and the signature of the space.

This article is devoted to the study of one generalization of the Riemann integral. Namely, in the paper, it was observed that the classical definition of the Riemann integral over a finite segment as a limit of integral sums, when the diameter of the division of the segment tends to zero, can be replaced by a limit of integral sums over a filter of sets, which can be described in a certain "good way". This idea was continued, and in the work we propose a new concept - the integral of a function over a filter on the set of all tagged partitions of a segment. Using of filters is a very good method in questions related to convergence or some of its analogues in general topological vector spaces. Namely, if the space is non-metrizable, then the concept of convergence is introduced precisely with the help of filters. Also, using filters, you can formulate the concept of completeness and its analogues. The completeness of spaces is one of the central concepts of the theory of topological vector spaces, since Banach spaces are complete. That is, using a generalization of the completeness of spaces constructed using filters, we can explore various generalizations of Banach spaces. We study standard issues related to integration. For example, does the integrability of the filter function imply its boundedness? The answer to this question is affirmative. Namely: the concept of filter boundedness of a function is introduced, and it is shown that if a function is integrable over filter, then its integral sums are bounded over the filter, and this function itself is bounded in the classical sense. Next, we showed that the filter integral satisfies the linearity property, namely, the integral over filter of the sum of two functions is the sum of the filter integrals of these functions. In addition, we can to subtract the constant factor from the sign of the integral over filter. We introduce the concept of an exactly tagged filter, and with the help of such filters we study the filter integrability of unbounded functions on a segment. We give an example of a specific unbounded function and a specific filter under which this function is integrable. Next, we prove a theorem that describes unbounded filter-integrable functions on a segment. The last section of the article is devoted to the integration of functions relative to the filter on a subsegment of this segment.

Matroid is defined as a pair $(X,\mathcal{I})$, where $X$ is a nonempty finite set, and $\mathcal{I}$ is a nonempty set of subsets of $X$ that satisfies the Hereditary Axiom and the Augmentation Axiom. The paper investigates for which semigroups (primarily finite) $S$, the pair $(\widehat{S}, \mathcal{I})$ will be a matroid.

The paper considers the problem of establishing the convergence criteria of the branched continued fraction expansion of the ratio of Horn's hypergeometric functions $H_4$. To solve it, the technique of expanding the domain of convergence of the branched continued fraction from the known small domain of convergence to a wider domain of convergence is used. For the real and complex parameters of the Horn hypergeometric function $H_4$, a number of convergence criteria of the branched continued fraction expansion under certain conditions to its coefficients in various unbounded domains of the space have been established.

Let $k$ be a field and let $N$ be a nilpotent minimax torsion-free group acted by a solvable group of operators $G$ of finite rank. In the presented paper we study properties of some types of $G$-invariant ideals of the group ring $kN$.

For given unit vectors $x_1, \cdots, x_n$ of a real Banach space $E,$ we define $$NA({\mathcal L}(^nE))(x_1, \cdots, x_n)=\{T\in {\mathcal L}(^nE): |T(x_1, \cdots, x_n)|=\|T\|=1\},$$ where ${\mathcal L}(^nE)$ denotes the Banach space of all continuous $n$-linear forms on $E$ endowed with the norm $\|T\|=\sup_{\|x_k\|=1, 1\leq k\leq n}{|T(x_1, \ldots, x_n)|}$. In this paper, we classify $NA({\mathcal L}(^2 d_{*}(1, w)^2))(Z_1, Z_2)$ for unit vectors $Z_1, Z_2\in d_{*}(1, w)^2,$ where $d_{*}(1, w)^2=\mathbb{R}^2$ with the norm of weight $0

Regression analysis is a huge part of mathematical and applied statistics. Nonlinear regression analysis is a significant extension and complication of classical linear regression analysis, due to the use of nonlinear or partially nonlinear in parameters models that describe more adequately than linear model phenomena requiring statistical analysis. A large number of applied problems in the numerical scientific, technical, and humanitarian fields of knowledge give impetus to the development of nonlinear regression analysis. The task of estimation the vector signal parameter in the «signal + noise» observation models is a well-known problem of statistics of stochastic processes, and in the case of a nonlinear signal parameter is the problem of nonlinear regression analysis. Among the variety of nonlinear regression analysis problems the problem of estimating amplitudes and angular frequencies of the sum of harmonic oscillations that are observed against the background of a random noise, takes significant place due to its numerous applications. Statistical model of such a type is said to be trigonometric regression, and the problem of statistical estimation is called the problem of detecting hidden periodicities. The paper is devoted to the study of time continuous trigonometric regression model where the random noise is a linear L´evy driven stationary of the fourth order stochastic process with zero mean, integrable and square integrable impulse transmission function. This assumption leads to the integrability of the noise covariance function and cumulant function of the fourth order. To estimate unknown amplitudes and angular frequencies of such a trigonometric model we use the least squares estimators in the Walker sense, that is special parametric set are considered to distinguish properly different angular frequencies in the sum of harmonic oscillations. Theorem on strong consistency of the least squares estimators is proved in the paper under the assumption on the random noise described above. To obtain such a result a very important lemma was proved on the uniform tending to zero almost surely of the average value of L´evy-driven linear stochastic process Fourier transform. This Lemma is the main tool of the strong consistency Theorem proof. To prove the Theorem we, firstly, find some expressions for the least squares estimates of amplitudes via corresponding estimates of angular frequencies. Secondly, we substitute these formulas into the functional of the least squares method. The last step of the proof consists of the L2- norm transformation of the difference between empirical trigonometric regression function and true regression function such that this norm tends to zero almost surely if and on if the estimators are strongly consistent.

The notion of a \verb"quasi-prime ideal", for the first time, was introduced in differential commutative rings, i.e. commutative rings considered together with a derivation, as differential ideals maximal among those not meeting some multiplicatively closed subset of a ring. The notion of a semiring derivation is traditionally defined as an additive map satisfying the Leibnitz rule. Due to rapid development of semiring theory in recent years, the need of considering ideals in semirings defined by similar conditions arose. The present paper is devoted to investigating the notion of a \verb"quasi-prime ideal" of differential semiring (which is defined as a semiring together with a derivation on it), not necessarily commutative. It aims to show, how \verb"quasi-prime ideals" are related to prime differential ideals, primary ideals, maximal ideals and other types of ideals of semirings. The paper consists of two main parts. In the first part, the author investigates some properties of quasi-prime differential ideals, and gives some examples of such semiring ideals, such as prime differential, maximal differential ideals, or ideal obtained by derivation operator acting on a prime ideal of a semiring. It contains a theorem, which gives equivalent conditions for a quasi-prime semiring ideal to be prime. The second part of the paper is devoted to considering chains of \verb"quasi-prime ideals". In this part, the interrelation between \verb"quasi-prime ideals" and other types of differential ideals of semirings is established. It contains a theorem, which gives a characterization of such ideals in case of a commutative semiring. This characterization involves the notion of the radical of an ideal of a semiring and a derivation operator for semirings. The paper ends with a theorem, which states that every chain of quasi-prime ideals of a semiring has the least upper bound and the greatest lower bound. It is also proven that every \verb"quasi-prime ideal" containing some differential ideal contains a \verb"quasi-prime ideal" minimal among all the quasi-prime ideals of the given semiring, which contain the above mentioned differential ideal.

The starting point of the present paper is a result by E.A. Boltarovych (1989) on convergence regions, dealing with branched continued fraction \[\sum_{i_1=1}^N\frac{a_{i(1)}}{1}{\atop+}\sum_{i_2=1}^N\frac{a_{i(2)}}{1}{\atop+}\ldots{\atop+}\sum_{i_n=1}^N\frac{a_{i(n)}}{1}{\atop+}\ldots,\] where $|a_{i(2n-1)}|\le\alpha/N,$ $i_p=\overline{1,N},$ $p=\overline{1,2n-1},$ $n\ge1,$ and for each multiindex $i(2n-1)$ there is a single index $j_{2n},$ $1\le j_{2n}\le N,$ such that $|a_{i(2n-1),j_{2n}}|\ge R,$ $i_p=\overline{1,N},$ $p=\overline{1,2n-1},$ $n\ge1,$ and $|a_{i(2n)}|\le r/(N-1),$ $i_{2n}\ne j_{2n},$ $i_p=\overline{1,N},$ $p=\overline{1,2n},$ $n\ge1,$ where $N>1$ and $\alpha,$ $r$ and $R$ are real numbers that satisfying certain conditions. In the present paper conditions for these regions are replaced by $\sum_{i_1=1}^N|a_{i(1)}|\le\alpha(1-\varepsilon),$ $\sum_{i_{2n+1}=1}^N|a_{i(2n+1)}|\le\alpha(1-\varepsilon),$ $i_p=\overline{1,N},$ $p=\overline{1,2n},$ $n\ge1,$ and for each multiindex $i(2n-1)$ there is a single index $j_{2n},$ $1\le j_{2n}\le N,$ such that $|a_{i(2n-1),j_{2n}}|\ge R$ and $\sum_{i_{2n}\in\{1,2,\ldots,N\}\backslash\{j_{2n}\}}|a_{i(2n)}|\le r,$ $i_p=\overline{1,N},$ $p=\overline{1,2n-1},$ $n\ge1,$ where $\varepsilon,$ $\alpha,$ $r$ and $R$ are real numbers that satisfying certain conditions, and better convergence speed estimates are obtained.

In practice, process values are usually observed at certain points in time. And based on this data, we need to draw conclusions about the behavior of the process that is being monitored. That is why the primary purpose of the paper is to evaluate the covariance function of such a process. For this purpose, in this paper, we consider a Gaussian stationary random process $X$ with unknown mean, when its values are known in a finite set of points and the task is to estimate the covariance function of such a random process. One feature of estimating the correlation function of a random process with an unknown mean is that the use of correlograms as an estimator is not possible, since the correlogram in this case is a biased estimation of the correlation function. Therefore, to prove the theorems, it was necessary to construct a statistics that would be an unbiased estimate of the covariance function of a Gaussian stationary random process. In addition, as shown in some of our previous papers and in this work, we are dealing with quadratic-Gaussian processes when estimating the deviations of the correlation function of a Gaussian stationary random process from a correlogram in the $L_p$-metric. Therefore, to prove this estimate was used the theory of quadratic-Gaussian random processes. Using this theory, we obtain estimates of the deviations of the correlation function of a Gaussian stationary random process with an unknown mean, when its values in the finite set of points of this process from its estimate in $L_p$-metric are known. The paper also builds a criterion for testing the hypothesis of the appearance of the correlation function of such a random process. This criterion was formulated using the obtained estimates.

The subject of this work is at the intersection of two branches of mathematics: mathematical physics and stochastic processes. The physical formulation of problems of mathematical physics with random factors was studied by Kampe de Feriet. In the works by E. Beisenbaev, Yu.V. Kozachenko and V.V. Buldygin a new approach studying the solutions of partial differential equations with random initial conditions was proposed. The authors investigate the convergence in probability of the sequence of function spaces of partial sums approximating the solution of a problem. The mentioned approach was used in the works by E. Barrasa de La Krus, Endzhyrgly, Ya.A. Kovalchuk. In the paper by V.V. Buldygin and Yu.V. Kozachenko the application of the Fourier method for the homogeneous hyperbolic equation with Gaussian initial conditions is justified and existence conditions in terms of correlation functions are studied. Homogeneous hyperbolic equation with random initial conditions from the space Sub ϕ (Ω) are considered in works by B. V. Dovgay, G.I. Slyvka-Tylyshchak. The model of a solution of a hyperbolic type equation with random initial conditions was investigated in the papers by G.I. Slyvka-Tylyshchak. There is studied a boundary value problem of mathematical physics for the inhomogeneous hyperbolic equation with ϕ-subgaussian in right part in works by B. V. Dovgay. The parabolic type equations of Mathematical Physics with random factors of Orlicz spaces have been studied in the papers by Yu.V. Kozachenko and K.J. Veresh. Properties of the classical solution of the heat equation on a line with a random right part are considered in works by Yu.V. Kozachenko and G.I. Slyvka-Tylyshchak. We consider a Cauchy problem for the heat equations with a random right part. We study the inhomogeneous heat equation on a line with a random right part. We consider the right part as a random function of the space L p (Ω). The conditions of existence with probability one generalized solution of the problem are investigated.

In the paper, we solve one extremal problem of the theory of approximation of functional classes by linear methods. Namely, questions are investigated concerning the approximation of classes of differentiable functions by $\lambda$-methods of summation for their Fourier series, that are defined by the set $\Lambda =\{{{\lambda }_{\delta }}(\cdot )\}$ of continuous on $\left[ 0,\infty \right)$ functions depending on a real parameter $\delta$. The Kolmogorov-Nikol'skii problem is considered, that is one of the special problems among the extremal problems of the theory of approximation. That is, the problem of finding of asymptotic equalities for the quantity $$\mathcal{E}{{\left( \mathfrak{N};{{U}_{\delta}} \right)}_{X}}=\underset{f\in \mathfrak{N}}{\mathop{\sup }}\,{{\left\| f\left( \cdot \right)-{{U}_{\delta }}\left( f;\cdot;\Lambda \right) \right\|}_{X}},$$ where $X$ is a normalized space, $\mathfrak{N}\subseteq X$ is a given function class, ${{U}_{\delta }}\left( f;x;\Lambda \right)$ is a specific method of summation of the Fourier series. In particular, in the paper we investigate approximative properties of the three-harmonic Poisson integrals on the Weyl-Nagy classes. The asymptotic formulas are obtained for the upper bounds of deviations of the three-harmonic Poisson integrals from functions from the classes $W^{r}_{\beta,\infty}$. These formulas provide a solution of the corresponding Kolmogorov-Nikol'skii problem. Methods of investigation for such extremal problems of the theory of approximation arised and got their development owing to the papers of A.N. Kolmogorov, S.M. Nikol'skii, S.B. Stechkin, N.P. Korneichuk, V.K. Dzyadyk, A.I. Stepanets and others. But these methods are used for the approximations by linear methods defined by triangular matrices. In this paper we modified the mentioned above methods in order to use them while dealing with the summation methods defined by a set of functions of a natural argument.

This paper is devoted to a partial differential equation approach to the problem of construction of Feller semigroups associated with one-dimensional diffusion processes with boundary conditions in theory of stochastic processes. In this paper we investigate the boundary-value problem for a one-dimensional linear parabolic equation of the second order (backward Kolmogorov equation) in curvilinear bounded domain with one of the variants of nonlocal Feller-Wentzell boundary condition. We restrict our attention to the case when the boundary condition has only one term and it is of the integral type. The classical solution of the last problem is obtained by the boundary integral equation method with the use of the fundamental solution of backward Kolmogorov equation and the associated parabolic potentials. This solution is used to construct the Feller semigroup corresponding to such a diffusion phenomenon that a Markovian particle leaves the boundary of the domain by jumps.

In this paper, we show that harmonic convex functions $f$ is strongly $(p, h)$-harmonic convex functions if and only if it can be decomposed as $g(x) = f(x) - c (\frac{1}{x^p})^2,$ where $g(x)$ is $(p, h)$-harmonic convex function. We obtain some new estimates class of strongly $(p, h)$-harmonic convex functions involving hypergeometric and beta functions. As applications of our results, several important special cases are discussed. We also introduce a new class of harmonic convex functions, which is called strongly $(p, h)$-harmonic $\log$-convex functions. Some new Hermite-Hadamard type inequalities for strongly $(p, h)$-harmonic $log$-convex functions are obtained. These results can be viewed as important refinement and significant improvements of the new and previous known results. The ideas and techniques of this paper may stimulate further research.

In this paper we study the following hypergeometric polynomials: $$ \mathcal{P}_n(x) = \mathcal{P}_n(x;\alpha,\beta,\delta_1, \dots,\delta_\rho,\kappa_1,\dots,\kappa_\rho) = $$ $$ = {}_{\rho+2} F_{\rho+1} (-n,n+\alpha+\beta+1,\delta_1+1, \dots,\delta_\rho+1;\alpha+1,\kappa_1+\delta_1+1, \dots,\kappa_\rho+\delta_\rho+1;x), $$ and $$ \mathcal{L}_n(x) = \mathcal{L}_n(x;\alpha,\delta_1,\dots, \delta_\rho,\kappa_1,\dots,\kappa_\rho) = $$ $$ = {}_{\rho+1} F_{\rho+1} (-n,\delta_1+1,\dots,\delta_\rho+1; \alpha+1,\kappa_1+\delta_1+1,\dots,\kappa_\rho+\delta_\rho+1;x), \qquad n\in\mathbb{Z}_+, $$ where $\alpha,\beta,\delta_1,\dots,\delta_\rho\in(-1,+\infty)$, and $\kappa_1,\dots,\kappa_\rho\in\mathbb{Z}_+$, are some parameters. The natural number $\rho$ of the continuous parameters $\delta_1,\dots,\delta_\rho$ can be chosen arbitrarily large. It is seen that the special case $\kappa_1=\dots=\kappa_\rho=0$ leads to Jacobi and Laguerre orthogonal polynomials. Of course, such polynomials and more general ones appeared in the literature earlier. Our aim here is to show that polynomials $\mathcal{P}_n(x)$ and $\mathcal{L}_n(x)$ are Sobolev orthogonal polynomials on the real line with some explicit matrices of measures. The importance of the orthogonality property was our main reason to concentrate our attention on polynomials $\mathcal{P}_n(x)$ and $\mathcal{L}_n(x)$. Here we shall use some our tools developed earlier. In particular, it was shown recently that Sobolev orthogonal polynomials are related by a differential equation with orthogonal systems $\mathcal{A}$ of functions acting in the direct sums of usual $L^2_\mu$ spaces of square-summable (classes of the equivalence of) functions with respect to a positive measure $\mu$. The case of a unique $L^2_\mu$ is of a special interest, since it allows to use OPRL to obtain explicit systems of Sobolev orthogonal polynomials. The main problem here is \textit{to choose a suitable linear differential operator in order to get explicit representations for Sobolev orthogonal polynomials}. The proof of the orthogonality relations is then a verification of such a choice and it goes in another direction: we start from the already known polynomials to their properties. We also study briefly such properties of the above polynomials: integral representations, differential equations and location of zeros. A system of such polynomials with a kind of the bispectrality property is constructed.

The article presents the results of the authors’ research on the cosmological views of Johannes Keppler in his Latin treatise Harmonices Mundi. The authors discuss the background of composing the treatise concerning the philosophical, religious, and cultural contexts of that period, and give a brief overview of the structure of the treatise, Kepler’s theory and his third law of harmony. The paper demonstrates the connection and reception of philosophical ideas on the harmony of the world in Kepler’s treatise and ancient philosophers, such as Pythagoras, Plato, and Proclus. This shows that Kepler’s philosophical and scientific views largely depended on his theological background. The authors discuss the perception of ancient cosmological ideas in Kepler’s Harmonices Mundi relating to Pythagorean tetractys, five Platonic solids or regular polyhedral, and cosmic harmony as well as Proclus’ theoretical mathematics.

Switched systems is a special case of hybrid dynamical systems with discrete and continuous dynamics. They are widely applied when a real system cannot be described by one single model. In theoretical works on switched systems, switching signals and times can be random or given by some law. Stability depends both on vector fields and on the switching law. In the present paper, a different formulation of the problem is considered, that is the case, when switching signal is under our control. Namely, a switched system is called controllable if for any two points there exists a switching signal that allows to get from the first point to the second one. In the paper the controllability of linear switched systems of a special type is studied. More specifically, we consider a switch, that is carried out between two 2x2 matrices with purely imaginary eigenvalues of both matrices. In the first section we discuss the physical meaning of switched systems of this type. Namely, the problem of oscillation of a spring pendulum with a switchable stiffness coefficient is considered with the series and parallel connection of an additional spring to the system with one given spring. We prove that such a system is controllable, and propose the method of finding the controlling switching signal. In the second section we present the main result of the work. We formulate an algorithm that allows finding a set of switching signals for getting from any given initial point to any given end point. We present an example of such controlling switching signals, simulated in MATLAB. In the last section we propose a generalization of the obtained result and formulate the theorem that states the controllability of the special type switched system with a block-diagonal matrix of high dimension. The method presented in the paper can be generalized to study of controllability of linear switched systems of more general form.

We study the time-optimal control problem for an unmanned aerial vehicle (drone) moving in the plane of a constant altitude. A kinematic model is considered where the angular velocity is a control. Such a system is described by Markov-Dubins equations; a large number of works are devoted to solving different optimal and admissible control and stabilization problems for such models. In the papers [T. Maillot, U. Boscain, J.-P. Gauthier, U. Serres, Lyapunov and minimum-time path planning for drones, J. Dyn. Control Syst., V. 21 (2015)] and [M.A.~Lagache, U. Serres, V. Andrieu, Minimal time synthesis for a kinematic drone model, Mathematical Control and Related Fields, V. 7 (2017)] the time optimal control problem is solved where the drone must reach a given unit circle in the minimal possible time and stay on this circle rotating counterclockwise. In particular, in the mentioned works it is shown that is this case the problem is simplified; namely, the problem becomes two-dimensional. In the present paper we consider a natural generalization of the formulation mentioned above: in our problem, the drone must reach a given unit circle in the minimal possible time and stay on this circle, however, both rotating directions are admissible. That is, the drone can rotate clockwise or counterclockwise; the direction is chosen for reasons of minimizing the time of movement. Such a reformulation leads to the time-optimal control problem with two final points. In the paper, we obtain a complete solution of this time-optimal control problem. In particular, we show that the optimal control takes the values $\pm1$ or $0$ and has no more than two switchings. If the optimal control is singular, i.e., contains a piece $u=0$, then this piece is unique and the duration of the last piece equals $\pi/3$; moreover, in this case the optimal control ins non-unique and the final point can be $(0,1)$ as well as $(0,-1)$. If the optimal control is non-singular, i.e., takes the values $\pm1$, then it is unique (except the case when the duration of the last piece equals $\pi/3$) and the optimal trajectory entirely lies in the upper or lower semi-plane. Also, we give a solution of the optimal synthesis problem.

Let $P$ be a probability on a finite group $G$, $U(g)=\frac{1}{|G|}$ the uniform (trivial) probability on the group $G$, $P^{(n)}=P *\ldots*P$ an $n$-fold convolution of $P$. A lot of estimates of the rate of the convergence $P^{(n)}\rightarrow U$ are found in different norms. It is well known conditions under which $P^{(n)}\rightarrow U$ if $n\rightarrow\infty$. Many papers are devoted to estimating the rate of this convergence for different norms. We consider finite groups that have a double transitive representation by substitutions and the probability that naturally arises in this image. This probability on each element of the group is proportional to the number of fixed (or stationary) points of this element, which is considered as a substitution. In other words, this probability is a character of the substitution representation of the group. A probability is called class if it takes the same values on each class of conjugate elements of a group, that is, it is a function of the class. The considered probability is class because any character of a group takes on the same values on conjugate elements. Any probability (and, in general, functions with values in an arbitrary ring) on a group can be associated with an element of the group algebra of this group over this ring. The class probability corresponds to an element of the center of this group algebra; that is why the class probability is also called central. On an abelian group, any probability is class (central). In the paper convergence with respect to the norm $\|F\|=\sum\limits_{g\in G} |F(g)|$, where $F(g)$ is a function on group $G$, is considered. For the norm an exact formula not estimate only, as usual for rate of convergence of convolution $P^{(n)}\rightarrow U$ is given. It turns out that the norm of the difference $\|P^{(n)}-U\|$ is determined by the order of the group, degree the group as a substitution group, and the number of regular substitutions in the group. A substitution is called regular if it has no fixed points. Special cases are considered the symmetric group, the alternating group, the Zassenhaus group, and the Frobenius group of order $p(p-1)$ with the Frobenius core of order $p$ ($p$ is a prime number). A Zassenhaus group is a double transitive substitution group of a finite set in which only a trivial substitution leaves more than two elements of this set fixed.

This paper deals with a study on classes of non linear operators. Let $SL(X,Y)$ be the set of all sublinear operators between two Riesz spaces $X$ and $Y$. It is a convex cone of the space $H(X,Y)$ of all positively homogeneous operators. In this paper we study some spaces generated by this cone, therefore we study several properties, which are well known in the theory of Riesz spaces, like order continuity, order boundedness etc. Finally, we try to generalise the concept of adjoint operator. First, by using the analytic form of Hahn-Banach theorem, we adapt the notion of adjoint operator to the category of positively homogeneous operators. Then we apply it to the class of operators generated by the sublinear operators.

Many objects of the Gaussian white noise analysis (spaces of test and generalized functions, stochastic integrals and derivatives, etc.) can be constructed and studied in terms of so-called chaotic decompositions, based on a chaotic representation property (CRP): roughly speaking, any square integrable with respect to the Gaussian measure random variable can be decomposed in a series of Ito's stochastic integrals from nonrandom functions. In the Levy analysis there is no the CRP (except the Gaussian and Poissonian particular cases). Nevertheless, there are different generalizations of this property. Using these generalizations, one can construct different spaces of test and generalized functions. And in any case it is necessary to introduce a natural product on spaces of generalized functions, and to study related topics. This product is called a Wick product, as in the Gaussian analysis. The construction of the Wick product in the Levy analysis depends, in particular, on the selected generalization of the CRP. In this paper we deal with Lytvynov's generalization of the CRP and with the corresponding spaces of regular generalized functions. The goal of the paper is to introduce and to study the Wick product on these spaces, and to consider some related topics (Wick versions of holomorphic functions, interconnection of the Wick calculus with operators of stochastic differentiation). Main results of the paper consist in study of properties of the Wick product and of the Wick versions of holomorphic functions. In particular, we proved that an operator of stochastic differentiation is a differentiation (satisfies the Leibniz rule) with respect to the Wick multiplication.

Development of a theory of test and generalized functions depending on infinitely many variables is an important and actual problem, which is stipulated by requirements of physics and mathematics. One of successful approaches to building of such a theory consists in introduction of spaces of the above-mentioned functions in such a way that the dual pairing between test and generalized functions is generated by integration with respect to some probability measure. First it was the Gaussian measure, then it were realized numerous generalizations. In particular, important results can be obtained if one uses the Levy white noise measure, the corresponding theory is called the Levy white noise analysis. In the Gaussian case one can construct spaces of test and generalized functions and introduce some important operators (e.g., stochastic integrals and derivatives) on these spaces by means of a so-called chaotic representation property (CRP): roughly speaking, any square integrable random variable can be decomposed in a series of repeated Itos stochastic integrals from nonrandom functions. In the Levy analysis there is no the CRP, but there are different generalizations of this property. In this paper we deal with one of the most useful and challenging generalizations of the CRP in the Levy analysis, which is proposed by E.W. Lytvynov, and with corresponding spaces of nonregular generalized functions. The goal of the paper is to introduce a natural product (a Wick product) on these spaces, and to study some related topics. Main results are theorems about properties of the Wick product and of Wick versions of holomorphic functions. In particular, we prove that an operator of stochastic differentiation satisfies the Leibniz rule with respect to the Wick multiplication. In addition we show that the Wick products and the Wick versions of holomorphic functions, defined on the spaces of regular and nonregular generalized functions, constructed by means of Lytvynov's generalization of the CRP, coincide on intersections of these spaces. Our research is a contribution in a further development of the Levy white noise analysis.

The paper deals with research of convergence for one of the generalizations of continued fractions -- branched continued fractions of the special form with two branches. Such branched continued fractions, similarly as the two-dimensional continued fractions and the branched continued fractions with two independent variables are connected with the problem of the correspondence between a formal double power series and a sequence of the rational approximants of a function of two variables. Unlike continued fractions, approximants of which are constructed unambiguously, there are many ways to construct approximants of branched continued fractions of the general and the special form. The paper examines the ordinary approximants and one of the structures of figured approximants of the studied branched continued fractions, which is connected with the problem of correspondence. We consider some properties of approximants of such fractions, whose partial numerators are positive and alternating-sign and partial denominators are equal to one. Some necessary and sufficient conditions for figured convergence are established. It is proved that under these conditions from the convergence of the sequence of figured approximants it follows the convergence of the sequence of ordinary approximants to the same limit.

The result presented in this paper is the second part of the paper {BoldyrevaZhmykhova2016}, where was constructed an equation which allows calculating the probability of non-ruin for the classical model of risk when an insurance company has promotional costs. There is considered the case of investment in the financial (B,S)-market, precisely in riskless part of it, in the bank deposit, in this article. Some numerical results for the probability of non-ruin by MHull of 10 Top Insurance companies of Ukraine for the period 01/01/2015 to 06/30/2015 based on the developed formulae are provided. Some conclusions about the influence of advertising company and investing in the bank deposit on the probability of non-ruin are carried out.