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 Carpathian Mathematical PublicationsNumber of Followers: 1     Open Access journal ISSN (Print) 2075-9827 Published by Vasyl Stefanyk Precarpathian National University  [2 journals]
• Local nearrings on finite non-abelian $2$-generated $p$-groups

• Authors: I.Yu. Raievska; M.Yu. Raievska
Abstract: It is proved that for ${p>2}$ every finite non-metacyclic $2$-generated p-group of nilpotency class $2$ with cyclic commutator subgroup is the additive group of a local nearring and in particular of a nearring with identity. It is also shown that the subgroup of all non-invertible elements of this nearring is of index $p$ in its additive group.
PubDate: Mon, 29 Jun 2020 00:00:00 +000

• Certain results for a class of nonlinear functional spaces

• Authors: K. Soltanov; U. Sert
Abstract: In this article, we study properties of a class of functional spaces, so-called pn-spaces, which arise from investigation of nonlinear differential equations. We establish some integral inequalities to analyse the structures of the pn-spaces with the constant and variable exponent. We prove embedding theorems, which indicate the relation of these spaces with the well known classical Lebesgue and Sobolev spaces with the constant and variable exponents.
PubDate: Mon, 29 Jun 2020 00:00:00 +000

• On two long standing open problems on $L_p$-spaces

• Authors: M.M. Popov
Abstract: The present note was written during the preparation of the talk at the International Conference dedicated to 70-th anniversary of Professor O. Lopushansky, September 16-19, 2019, Ivano-Frankivsk (Ukraine). We focus on two long standing open problems. The first one, due to Lindenstrauss and Rosenthal (1969), asks of whether every complemented infinite dimensional subspace of $L_1$ is isomorphic to either $L_1$ or $\ell_1$. The second problem was posed by Enflo and Rosenthal in 1973: does there exist a nonseparable space $L_p(\mu)$ with finite atomless $\mu$ and $1<p<\infty$, $p \neq 2$, having an unconditional basis' We analyze partial results and discuss on some natural ideas to solve these problems.
PubDate: Mon, 29 Jun 2020 00:00:00 +000

• The generalized centrally extended Lie algebraic structures and related
integrable heavenly type equations

• Authors: O.Ye. Hentosh; A.A. Balinsky, A.K. Prykarpatski
Abstract: There are studied Lie-algebraic structures of a wide class of heavenly type non-linear integrable equations, related with coadjoint flows on the adjoint space to a loop vector field Lie algebra on the torus. These flows are generated by the loop Lie algebras of vector fields on a torus and their coadjoint orbits and give rise to the compatible Lax-Sato type vector field relationships. The related infinite hierarchy of conservations laws is analysed and its analytical structure, connected with the Casimir invariants, is discussed. We present the typical examples of such equations and demonstrate in details their integrability within the scheme developed. As examples, we found and described new multidimensional generalizations of the Mikhalev-Pavlov and Alonso-Shabat type integrable dispersionless equation, whose seed elements possess a special factorized structure, allowing to extend them to the multidimensional case of arbitrary dimension.
PubDate: Mon, 29 Jun 2020 00:00:00 +000

• The nonlocal boundary value problem with perturbations of mixed boundary
conditions for an elliptic equation with constant coefficients. II

• Authors: Ya.O. Baranetskij; P.I. Kalenyuk, M.I. Kopach, A.V. Solomko
Abstract: In this paper we continue to investigate the properties of the problem with nonlocal conditions, which are multipoint perturbations of mixed boundary conditions, started in the first part. In particular, we construct a generalized transform operator, which maps the solutions of the self-adjoint boundary-value problem with mixed boundary conditions to the solutions of the investigated multipoint problem. The system of root functions $V(L)$ of operator $L$ for multipoint problem is constructed. The conditions under which the system $V(L)$ is complete and minimal, and the conditions under which it is the Riesz basis are determined. In the case of an elliptic equation the conditions of existence and uniqueness of the solution for the problem are established.
PubDate: Sun, 28 Jun 2020 23:27:21 +000

• Bounded solutions of a difference equation with finite number of jumps of
operator coefficient

• Authors: A. Chaikovs'kyi; O. Lagoda
Abstract: We study the problem of existence of a unique bounded solution of a difference equation with variable operator coefficient in a Banach space. There is well known theory of such equations with constant coefficient. In that case the problem is solved in terms of spectrum of the operator coefficient. For the case of variable operator coefficient correspondent conditions are known too. But it is too hard to check the conditions for particular equations. So, it is very important to give an answer for the problem for those particular cases of variable coefficient, when correspondent conditions are easy to check. One of such cases is the case of piecewise constant operator coefficient. There are well known sufficient conditions of existence and uniqueness of bounded solution for the case of one jump. In this work, we generalize these results for the case of finite number of jumps of operator coefficient. Moreover, under additional assumption we obtained necessary and sufficient conditions of existence and uniqueness of bounded solution.
PubDate: Sun, 28 Jun 2020 00:00:00 +000

• Nilpotent Lie algebras of derivations with the center of small corank

• Authors: Y.Y. Chapovskyi; L.Z. Mashchenko, A.P. Petravchuk
Abstract: Let $\mathbb K$ be a field of characteristic zero, $A$ be an integral domain over $\mathbb K$ with the field of fractions $R=Frac(A),$ and $Der_{\mathbb K}A$ be the Lie algebra of all $\mathbb K$-derivations on $A$. Let $W(A):=RDer_{\mathbb K} A$ and $L$ be a nilpotent subalgebra of rank $n$ over $R$ of the Lie algebra $W(A).$ We prove that if the center $Z=Z(L)$ is of rank $\geq n-2$ over $R$ and $F=F(L)$ is the field of constants for $L$ in $R,$ then the Lie algebra $FL$ is contained in a locally nilpotent subalgebra of $W(A)$ of rank $n$ over $R$ with a natural basis over the field $R.$ It is also proved that the Lie algebra $FL$ can be isomorphically embedded (as an abstract Lie algebra) into the triangular Lie algebra $u_n(F)$, which was studied early by other authors.
PubDate: Sun, 28 Jun 2020 00:00:00 +000

• Zero product preserving bilinear operators acting in sequence spaces

• Authors: E. Erdoğan
Abstract: Consider a couple of sequence spaces and a product function $-$ a canonical bilinear map associated to the pointwise product $-$ acting in it. We analyze the class of "zero product preserving" bilinear operators associated with this product, that are defined as the ones that are zero valued in the couples in which the product equals zero. The bilinear operators belonging to this class have been studied already in the context of Banach algebras, and allow a characterization in terms of factorizations through $\ell^r(\mathbb{N})$ spaces. Using this, we show the main properties of these maps such as compactness and summability.
PubDate: Fri, 19 Jun 2020 00:00:00 +000

• Symmetric functions on spaces $\ell_p(\mathbb{{R}}^n)$ and
$\ell_p(\mathbb{{C}}^n)$

• Authors: T.V. Vasylyshyn
Abstract: This work is devoted to the study of algebras of continuous symmetric polynomials, that is, invariant with respect to permutations of coordinates of its argument, and of $*$-polynomials on Banach spaces $\ell_p(\mathbb{R}^n)$ and $\ell_p(\mathbb{C}^n)$ of $p$-power summable sequences of $n$-dimensional vectors of real and complex numbers respectively, where $1\leq p < +\infty.$ We construct the subset of the algebra of all continuous symmetric polynomials on the space $\ell_p(\mathbb{R}^n)$ such that every continuous symmetric polynomial on the space $\ell_p(\mathbb{R}^n)$ can be uniquely represented as a linear combination of products of elements of this set. In other words, we construct an algebraic basis of the algebra of all continuous symmetric polynomials on the space $\ell_p(\mathbb{R}^n)$. Using this result, we construct an algebraic basis of the algebra of all continuous symmetric $*$-polynomials on the space $\ell_p(\mathbb{C}^n)$. Results of the paper can be used for investigations of algebras, generated by continuous symmetric polynomials on the space $\ell_p(\mathbb{R}^n)$, and algebras, generated by continuous symmetric $*$-polynomials on the space $\ell_p(\mathbb{C}^n)$.
PubDate: Fri, 12 Jun 2020 00:00:00 +000

• Analogues of the Newton formulas for the block-symmetric polynomials on
$\ell_p(\mathbb{C}^s)$

• Authors: V.V. Kravtsiv
Abstract: The classical Newton formulas gives recurrent relations between algebraic bases of symmetric polynomials. They are true, of course, for symmetric polynomials on infinite-dimensional Banach sequence spaces. In this paper, we consider block-symmetric polynomials (or MacMahon symmetric polynomials) on Banach spaces $\ell_p(\mathbb{C}^s),$ $1\le p\le \infty.$ We prove an analogue of the Newton formula for the block-symmetric polynomials for the case $p=1.$ In the case $1< p$ we have no classical elementary block-symmetric polynomials. However, we extend the obtained Newton type formula for $\ell_1(\mathbb{C}^s)$ to the case of $\ell_p(\mathbb{C}^s),$ $1< p\le \infty$, and in this way we found a natural definition of elementary block-symmetric polynomials on $\ell_p(\mathbb{C}^s).$
PubDate: Fri, 12 Jun 2020 00:00:00 +000

• Nonlocal inverse boundary-value problem for a 2D parabolic equation with
integral overdetermination condition

• Authors: E.I. Azizbayov; Y.T. Mehraliyev
Abstract: This article studies a nonlocal inverse boundary-value problem for a two-dimensional second-order parabolic equation in a rectangular domain. The purpose of the article is to determine the unknown coefficient and the solution of the considered problem. To investigate the solvability of the inverse problem, we transform the original problem into some auxiliary problem with trivial boundary conditions. Using the contraction mappings principle, existence and uniqueness of the solution of an equivalent problem are proved. Further, using the equivalency, the existence and uniqueness theorem of the classical solution of the original problem is obtained.
PubDate: Fri, 12 Jun 2020 00:00:00 +000

• Mersenne-Horadam identities using generating functions

• Authors: R. Frontczak; T.P. Goy
Abstract: The main object of the present paper is to reveal connections between Mersenne numbers $M_n=2^n-1$ and generalized Fibonacci (i.e., Horadam) numbers $w_n$ defined by a second order linear recurrence $w_n=pw_{n-1}+qw_{n-2}$, $n\geq 2$, with $w_0=a$ and $w_1=b$, where $a$, $b$, $p>0$ and $q\ne0$ are integers. This is achieved by relating the respective (ordinary and exponential) generating functions to each other. Several explicit examples involving Fibonacci, Lucas, Pell, Jacobsthal and balancing numbers are stated to highlight the results.
PubDate: Fri, 12 Jun 2020 00:00:00 +000

• Sufficient conditions for the improved regular growth of entire functions
in terms of their averaging

• Authors: R.V. Khats'
Abstract: Let $f$ be an entire function of order $\rho\in (0,+\infty)$ with zeros on a finite system of rays $\{z: \arg z=\psi_{j}\}$, $j\in\{1,\ldots,m\}$, $0\le\psi_1<\psi_2<\ldots<\psi_m<2\pi$ and $h(\varphi)$ be its indicator. In 2011, the author of the article has been proved that if $f$ is of improved regular growth (an entire function $f$ is called a function of improved regular growth if for some $\rho\in (0,+\infty)$ and $\rho_1\in (0,\rho)$, and a $2\pi$-periodic $\rho$-trigonometrically convex function $h(\varphi)\not\equiv -\infty$ there exists a set $U\subset\mathbb C$ contained in the union of disks with finite sum of radii and such that $\log {f(z)} = z ^\rho h(\varphi)+o( z ^{\rho_1})$, $U\not\ni z=re^{i\varphi}\to\infty$), then for some $\rho_3\in (0,\rho)$ the relation \begin{equation*} \int_1^r {\frac{\log {f(te^{i\varphi})} }{t}}\, dt=\frac{r^\rho}{\rho}h(\varphi)+o(r^{\rho_3}),\quad r\to +\infty, \end{equation*} holds uniformly in $\varphi\in [0,2\pi]$. In the present paper, using the Fourier coefficients method, we establish the converse statement, that is, if for some $\rho_3\in (0,\rho)$ the last asymptotic relation holds uniformly in $\varphi\in [0,2\pi]$, then $f$ is a function of improved regular growth. It complements similar results on functions of completely regular growth due to B. Levin, A. Grishin, A. Kondratyuk, Ya. Vasyl'kiv and Yu. Lapenko.
PubDate: Fri, 12 Jun 2020 00:00:00 +000

• Legendrian normally flat submanifols of $\mathcal{S}$-space forms

• Authors: F. Mahi; M. Belkhelfa
Abstract: In the present study, we consider a Legendrian normally flat submanifold $M$ of $(2n+s)$-dimensional $\mathcal{S}$-space form $\widetilde{M}^{2n+s}(c)$ of constant $\varphi$-sectional curvature $c$. We have shown that if $M$ is pseudo-parallel then $M$ is semi-parallel or totally geodesic. We also prove that if $M$ is Ricci generalized pseudo-parallel, then either it is minimal or $L=\frac{1}{n-1}$, when $c\neq -3s$.
PubDate: Fri, 12 Jun 2020 00:00:00 +000

• Construction of the fundamental solution of a class of degenerate
parabolic equations of high order

• Authors: H.P. Malytska; I.V. Burtnyak
Abstract: In the article, using the modified Levy method, a Green's function for a class of ultraparabolic equations of high order with an arbitrary number of parabolic degeneration groups is constructed. The modified Levy method is developed for high-order Kolmogorov equations with coefficients depending on all variables, while the frozen point, which is a parametrix, is chosen so that an exponential estimate of the fundamental solution and its derivatives is conveniently used.
PubDate: Fri, 12 Jun 2020 00:00:00 +000

• Isomorphic spectrum and isomorphic length of a Banach space

• Authors: O. Fotiy; M. Ostrovskii, M. Popov
Abstract: We prove that, given any ordinal $\delta < \omega_2$, there exists a transfinite $\delta$-sequence of separable Banach spaces $(X_\alpha)_{\alpha < \delta}$ such that $X_\alpha$ embeds isomorphically into $X_\beta$ and contains no subspace isomorphic to $X_\beta$ for all $\alpha < \beta < \delta$. All these spaces are subspaces of the Banach space $E_p = \bigl( \bigoplus_{n=1}^\infty \ell_p \bigr)_2$, where $1 \leq p < 2$. Moreover, assuming Martin's axiom, we prove the same for all ordinals $\delta$ of continuum cardinality.
PubDate: Fri, 12 Jun 2020 00:00:00 +000

• Related fixed point results via $\mathit{C_{*}}$-class functions on
$C^{*}$-algebra-valued $G_{b}$-metric spaces

• Authors: B. Moeini; H. Işik, H. Aydi
Abstract: We initiate the concept of $C^{*}$-algebra-valued $G_{b}$-metric spaces. We study some basic properties of such spaces and then prove some fixed point theorems for Banach and Kannan types via $\mathit{C_{*}}$-class functions. Also, some nontrivial examples are presented to ensure the effectiveness and applicability of the obtained results.
PubDate: Fri, 12 Jun 2020 00:00:00 +000

• A note on approximation of continuous functions on normed spaces

• Authors: M.A. Mytrofanov; A.V. Ravsky
Abstract: Let $X$ be a real separable normed space $X$ admitting a separating polynomial. We prove that each continuous function from a subset $A$ of $X$ to a real Banach space can be uniformly approximated by restrictions to $A$ of functions, which are analytic on open subsets of $X$. Also we prove that each continuous function to a complex Banach space from a complex separable normed space, admitting a separating $*$-polynomial, can be uniformly approximated by $*$-analytic functions.
PubDate: Fri, 12 Jun 2020 00:00:00 +000

• On the estimation of functions belonging to Lipschitz class by block pulse
functions and hybrid Legendre polynomials

• Authors: S. Lal; V.K. Sharma
Abstract: In this paper, block pulse functions and hybrid Legendre polynomials are introduced. The estimators of a function $f$ having first and second derivative belonging to $Lip_\alpha[a,b]$ class, $0 < \alpha \leq 1$, and $a$, $b$ are finite real numbers, by block pulse functions and hybrid Legendre polynomials have been calculated. These calculated estimators are new, sharp and best possible in wavelet analysis. An example has been given to explain the validity of approximation of functions by using the hybrid Legendre polynomials approximation method. A real-world problem of radioactive decay is solved using this hybrid Legendre polynomials approximation method. Moreover, the Hermite differential equation of order zero is solved by using hybrid Legendre polynomials approximation method to explain the importance and the application of the technique of this method.
PubDate: Fri, 12 Jun 2020 00:00:00 +000

• Some properties of generalized hypergeometric Appell polynomials

• Authors: L. Bedratyuk; N. Luno
Abstract: Let $x^{(n)}$ denotes the Pochhammer symbol (rising factorial) defined by the formulas $x^{(0)}=1$ and $x^{(n)}=x(x+1)(x+2)\cdots (x+n-1)$ for $n\geq 1$. In this paper, we present a new real-valued Appell-type polynomial family $A_n^{(k)}(m,x)$, $n, m \in {\mathbb{N}}_0$, $k \in {\mathbb{N}},$ every member of which is expressed by mean of the generalized hypergeometric function ${}_{p} F_q \begin{bmatrix} \begin{matrix} a_1, a_2, \ldots, a_p \:\\ b_1, b_2, \ldots, b_q \end{matrix} \: \Bigg \:z \end{bmatrix}= \sum\limits_{k=0}^{\infty} \frac{a_1^{(k)} a_2^{(k)} \ldots a_p^{(k)}}{b_1^{(k)} b_2^{(k)} \ldots b_q^{(k)}} \frac{z^k}{k!}$ as follows $$A_n^{(k)}(m,x)= x^n{}_{k+p} F_q \begin{bmatrix} \begin{matrix} {a_1}, {a_2}, {\ldots}, {a_p}, {\displaystyle -\frac{n}{k}}, {\displaystyle -\frac{n-1}{k}}, {\ldots}, {\displaystyle-\frac{n-k+1}{k}}\:\\ {b_1}, {b_2}, {\ldots}, {b_q} \end{matrix} \: \Bigg \: \displaystyle \frac{m}{x^k} \end{bmatrix}$$ and, at the same time, the polynomials from this family are Appell-type polynomials. The generating exponential function of this type of polynomials is firstly discovered and the proof that they are of Appell-type ones is given. We present the differential operator formal power series representation as well as an explicit formula over the standard basis, and establish a new identity for the generalized hypergeometric function. Besides, we derive the addition, the multiplication and some other formulas for this polynomial family.
PubDate: Fri, 12 Jun 2020 00:00:00 +000

• Asymptotics of approximation of functions by conjugate Poisson integrals

• Authors: I.V. Kal'chuk; Yu.I. Kharkevych, K.V. Pozharska
Abstract: Among the actual problems of the theory of approximation of functions one should highlight a wide range of extremal problems, in particular, studying the approximation of functional classes by various linear methods of summation of the Fourier series. In this paper, we consider the well-known Lipschitz class $\textrm{Lip}_1\alpha$, i.e. the class of continuous $2\pi$-periodic functions satisfying the Lipschitz condition of order $\alpha$, $0<\alpha\le 1$, and the conjugate Poisson integral acts as the approximating operator. One of the relevant tasks at present is the possibility of finding constants for asymptotic terms of the indicated degree of smallness (the so-called Kolmogorov-Nikol'skii constants) in asymptotic distributions of approximations by the conjugate Poisson integrals of functions from the Lipschitz class in the uniform metric. In this paper, complete asymptotic expansions are obtained for the exact upper bounds of deviations of the conjugate Poisson integrals from functions from the class $\textrm{Lip}_1\alpha$. These expansions make it possible to write down the Kolmogorov-Nikol'skii constants of the arbitrary order of smallness.
PubDate: Fri, 12 Jun 2020 00:00:00 +000

• Approximation of the Nikol'skii-Besov functional classes by entire
functions of a special form

• Authors: S.Ya. Yanchenko
Abstract: We establish the exact-order estimates for the approximation of functions from the Nikol'skii-Besov classes $S^{\boldsymbol{r}}_{1,\theta}B(\mathbb{R}^d)$, $d\geq 1$, by entire functions of exponential type with some restrictions for their spectrum. The error of the approximation is estimated in the metric of the Lebesgue space $L_{\infty}(\mathbb{R}^d)$.
PubDate: Fri, 12 Jun 2020 00:00:00 +000

• On convergence criteria for branched continued fraction

• Authors: T.M. Antonova
Abstract: 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.
PubDate: Fri, 12 Jun 2020 00:00:00 +000

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