A part of the theory of differential equations in the complex plane $mathbb C$ is the study of their solutions. To obtain them sometimes researchers can use local expand of solution in the integer degrees of an independent variable. In more difficult cases received local expand in fractional degrees of an independent variable, on so-called Newton - Poiseux series. A row of mathematicians for integration of linear differential equations applied a method of so-called generalized degree series, where meets irrational, in general real degree of an independent variable. One of the directions of the theory of differential equations in the complex plane $mathbb C$ is the construction a function $f$ according given sequence of zeros or poles, zeros of the derivative $f'$ and then find a differential equation for which this function be solution. Some authors studied sequences of zeros of solutions of the linear differential equation begin{equation*} f''+Af=0, end{equation*} where $A$ is entire or analytic function in a disk ${rm { z:|z| < 1} }$. In addition to the case when the above-mentioned differential equation has the non-trivial solution with given zero-sequences it is possible for consideration the case, when this equation has a solution with a given sequence of zeros (poles) and critical points. In this article we consider the question when the above-mentioned differential equation has the non-trivial solution $f$ such that $f^{1/alpha}$, $alpha in {mathbb R}backslash { 0;-1} $ is meromorphic function without zeros with poles in given sequence and the derivative of solution $f'$ has zeros in other given sequence, where $A$ is meromorphic function. Let's note, that representation of function by Weierstrass canonical product is the basic element for researches in the theory of the entire functions. Further we consider the question about construction of entire solution $f$ of the differential equation begin{equation*} f^{(n)} +Af^{m} =0, quad n,min {mathbb N}, end{equation*} where $A$ is meromorphic function such that $f$ has zeros in given sequence and the derivative of solution $f'$ has zeros in other given sequence.
{"title":"ON THE CONSTRUCTION OF SOLUTIONS OF DIFFERENTIAL EQUATIONS ACCORDING TO GIVEN SEQUENCES OF ZEROS AND CRITICAL POINTS","authors":"O. Shavala","doi":"10.31861/bmj2023.01.12","DOIUrl":"https://doi.org/10.31861/bmj2023.01.12","url":null,"abstract":"A part of the theory of differential equations in the complex plane $mathbb C$ is the study of their solutions. To obtain them sometimes researchers can use local expand of solution in the integer degrees of an independent variable. In more difficult cases received local expand in fractional degrees of an independent variable, on so-called Newton - Poiseux series. A row of mathematicians for integration of linear differential equations applied a method of so-called generalized degree series, where meets irrational, in general real degree of an independent variable. One of the directions of the theory of differential equations in the complex plane $mathbb C$ is the construction a function $f$ according given sequence of zeros or poles, zeros of the derivative $f'$ and then find a differential equation for which this function be solution. Some authors studied sequences of zeros of solutions of the linear differential equation begin{equation*} f''+Af=0, end{equation*} where $A$ is entire or analytic function in a disk ${rm { z:|z| < 1} }$. In addition to the case when the above-mentioned differential equation has the non-trivial solution with given zero-sequences it is possible for consideration the case, when this equation has a solution with a given sequence of zeros (poles) and critical points. In this article we consider the question when the above-mentioned differential equation has the non-trivial solution $f$ such that $f^{1/alpha}$, $alpha in {mathbb R}backslash { 0;-1} $ is meromorphic function without zeros with poles in given sequence and the derivative of solution $f'$ has zeros in other given sequence, where $A$ is meromorphic function. Let's note, that representation of function by Weierstrass canonical product is the basic element for researches in the theory of the entire functions. Further we consider the question about construction of entire solution $f$ of the differential equation begin{equation*} f^{(n)} +Af^{m} =0, quad n,min {mathbb N}, end{equation*} where $A$ is meromorphic function such that $f$ has zeros in given sequence and the derivative of solution $f'$ has zeros in other given sequence.","PeriodicalId":479563,"journal":{"name":"Bukovinsʹkij matematičnij žurnal","volume":"169 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135801174","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The theory of optimal control of systems, which is described by partial differential equations, is rich in results and is actively developing nowadays. The popularity of this kind of research is connected with its active use in solving problems of natural science, in particular hydro and gas dynamics, heat physics, diffusion, and the theory of biological populations. The problem of optimal control of the system described by the Dirichlet problem for the elliptic equation of the second order is studied. Cases of internal control are considered. The quality criterion is given by the volumetric integral. The coefficients of the equation admit power singularities of arbitrary order in any variables at some set of points. Solutions of auxiliary problems with smooth coefficients are studied to solve the given problem. Using a priori estimates, inequalities are established for solving problems and their derivatives in special Hölder spaces. Using the theorems of Archel and Riess, a convergent sequence is distinguished from a compact sequence of solutions to auxiliary problems, the limiting value of which will be the solution to the given problem. The necessary and sufficient conditions for the existence of the optimal solution of the system described by the Dirichlet problem for the elliptic equation with degeneracy have been established.
{"title":"OPTIMAL CONTROL IN THE DIRICHLET PROBLEM FOR ELLIPTIC EQUATIONS WITH DEGENERATION","authors":"I. Pukalskyy, B. Yashan","doi":"10.31861/bmj2023.01.10","DOIUrl":"https://doi.org/10.31861/bmj2023.01.10","url":null,"abstract":"The theory of optimal control of systems, which is described by partial differential equations, is rich in results and is actively developing nowadays. The popularity of this kind of research is connected with its active use in solving problems of natural science, in particular hydro and gas dynamics, heat physics, diffusion, and the theory of biological populations. The problem of optimal control of the system described by the Dirichlet problem for the elliptic equation of the second order is studied. Cases of internal control are considered. The quality criterion is given by the volumetric integral. The coefficients of the equation admit power singularities of arbitrary order in any variables at some set of points. Solutions of auxiliary problems with smooth coefficients are studied to solve the given problem. Using a priori estimates, inequalities are established for solving problems and their derivatives in special Hölder spaces. Using the theorems of Archel and Riess, a convergent sequence is distinguished from a compact sequence of solutions to auxiliary problems, the limiting value of which will be the solution to the given problem. The necessary and sufficient conditions for the existence of the optimal solution of the system described by the Dirichlet problem for the elliptic equation with degeneracy have been established.","PeriodicalId":479563,"journal":{"name":"Bukovinsʹkij matematičnij žurnal","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135801173","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For infinite-symbol E-representation of numbers $x in (0, 1]$: [ x = sum_{n=1}^infty frac{1}{(2+g_1)ldots(2+g_1+g_2+ldots+g_n)} equiv Delta^E_{g_1g_2ldots g_nldots}, ] where $g_n in Z_0 = { 0, 1, 2, ldots }$, we consider a class of E-cylinders, i.e., sets defined by equality [ Delta^E_{c_1ldots c_m} = left{ x colon x = Delta^E_{c_1ldots c_mg_{m+1}ldots g_{m+k}ldots}, ; g_{m+k} in Z_0, ; k in N right}. ] We prove that, for determination (calculation) of fractal Hausdorff-Besicovitch dimension of any Borel set $B subset [0, 1]$, it is enough to use coverings of the set $B$ by connected unions of E-cylinders of the same rank that belong to the same cylinder of the previous rank.
对于数字的无限符号e表示$x in (0, 1]$: [ x = sum_{n=1}^infty frac{1}{(2+g_1)ldots(2+g_1+g_2+ldots+g_n)} equiv Delta^E_{g_1g_2ldots g_nldots}, ]，其中$g_n in Z_0 = { 0, 1, 2, ldots }$，我们考虑一类e柱体，即由等式定义的集合[ Delta^E_{c_1ldots c_m} = left{ x colon x = Delta^E_{c_1ldots c_mg_{m+1}ldots g_{m+k}ldots}, ; g_{m+k} in Z_0, ; k in N right}. ]，我们证明，对于任意Borel集$B subset [0, 1]$的分形hausdoroff - besicovitch维的确定(计算)，通过属于前秩的同一柱体的相同秩的e柱体的连通并集来覆盖集合$B$就足够了。
{"title":"CYLINDRICAL SETS OF E-REPRESENTATION OF NUMBERS AND FRACTAL HAUSDORFF – BESICOVITCH DIMENSION","authors":"O. Baranovskyi, B. Hetman, M. Pratsiovytyi","doi":"10.31861/bmj2023.01.05","DOIUrl":"https://doi.org/10.31861/bmj2023.01.05","url":null,"abstract":"For infinite-symbol E-representation of numbers $x in (0, 1]$: [ x = sum_{n=1}^infty frac{1}{(2+g_1)ldots(2+g_1+g_2+ldots+g_n)} equiv Delta^E_{g_1g_2ldots g_nldots}, ] where $g_n in Z_0 = { 0, 1, 2, ldots }$, we consider a class of E-cylinders, i.e., sets defined by equality [ Delta^E_{c_1ldots c_m} = left{ x colon x = Delta^E_{c_1ldots c_mg_{m+1}ldots g_{m+k}ldots}, ; g_{m+k} in Z_0, ; k in N right}. ] We prove that, for determination (calculation) of fractal Hausdorff-Besicovitch dimension of any Borel set $B subset [0, 1]$, it is enough to use coverings of the set $B$ by connected unions of E-cylinders of the same rank that belong to the same cylinder of the previous rank.","PeriodicalId":479563,"journal":{"name":"Bukovinsʹkij matematičnij žurnal","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135801184","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
M. Pratsiovytyi, O. Bondarenko, N. Vasylenko, I. Lysenko
In the paper we justify existence and unity $B$-representation of numbers of segment $(0;1)$, which uses as a basis a positive number $a$ that satisfies the condition $00~forall nin Z$, $sumlimits_{n=-infty}^{+infty}Theta_n=1$, $b_{n+1}equivsumlimits_{i=-infty}^{n-1}=b_n+Theta_n$ $forall nin Z$. The geometry of $B$-representations of numbers is described (geometric content of numbers, properties of cylinder and tail sets, topological and metric properties of sets with restrictions on the use of numbers). The left and right shift operators of numbers are studied, a group of continuous transformations of the unit interval preserving the tails of the $B$-representation of numbers is described.
{"title":"INFINITE-SYMBOL B-REPRESENTATION OF REAL NUMBERS AND SOME OF ITS APPLICATIONS","authors":"M. Pratsiovytyi, O. Bondarenko, N. Vasylenko, I. Lysenko","doi":"10.31861/bmj2023.01.08","DOIUrl":"https://doi.org/10.31861/bmj2023.01.08","url":null,"abstract":"In the paper we justify existence and unity $B$-representation of numbers of segment $(0;1)$, which uses as a basis a positive number $a$ that satisfies the condition $0<a<frac{1}{3}$ in particular the positive root $tau$ of the equation $x^2+x-1=0$, bilateral sequence $(Theta_n)$: $Theta_0=frac{1-3a}{1-a}$, $Theta_{-n}=Theta_n=a^{|n|}$ and alphabet $Z={0,pm 1, pm 2, pm, dots },$ namely $$x=b_{alpha_1}+sumlimits_{k=2}^{m}b_{alpha_k}prodlimits_{i=1}^{k-1}Theta_{alpha_i}equiv Delta^{B}_{alpha_1alpha_2...alpha_m(emptyset)},$$ $$x=b_{alpha_1}+sumlimits_{k=2}^{infty}b_{alpha_k}prodlimits_{i=1}^{k-1}Theta_{alpha_i}equiv Delta^{B}_{alpha_1alpha_2...alpha_n...},$$ where $alpha_nin Z$, $Theta_n>0~forall nin Z$, $sumlimits_{n=-infty}^{+infty}Theta_n=1$, $b_{n+1}equivsumlimits_{i=-infty}^{n-1}=b_n+Theta_n$ $forall nin Z$. The geometry of $B$-representations of numbers is described (geometric content of numbers, properties of cylinder and tail sets, topological and metric properties of sets with restrictions on the use of numbers). The left and right shift operators of numbers are studied, a group of continuous transformations of the unit interval preserving the tails of the $B$-representation of numbers is described.","PeriodicalId":479563,"journal":{"name":"Bukovinsʹkij matematičnij žurnal","volume":"37 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135801177","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
An article consists of two parts. In the first part the sufficient and necessary conditions for an integral representation of hyperbolically convex (h.c.) functions $k(x)$ $left(xin mathbb{R}^{infty}= mathbb{R}^1timesmathbb{R}^1times dotsright)$ are proved. For this purpose in $mathbb{R}^{infty}$ we introduce measures $omega_1(x)$, $omega_{frac{1}{2}}(x)$. The positive definiteness of a function will be understood on the integral sense with respect to the measure $omega_1(x)$. Then we proved that the measure $rho(lambda)$ in the integral representation is concentrated on $l_2^+=bigg{lambda in mathbb{R}_+^{infty}= mathbb{R}_+^1timesmathbb{R}_+^1times dotsBig|sumlimits_{n=1}^{infty}lambda_n^2
{"title":"INTEGRAL REPRESENTATION OF HYPERBOLICALLY CONVEX FUNCTIONS","authors":"O. Lopotko","doi":"10.31861/bmj2023.01.02","DOIUrl":"https://doi.org/10.31861/bmj2023.01.02","url":null,"abstract":"An article consists of two parts. In the first part the sufficient and necessary conditions for an integral representation of hyperbolically convex (h.c.) functions $k(x)$ $left(xin mathbb{R}^{infty}= mathbb{R}^1timesmathbb{R}^1times dotsright)$ are proved. For this purpose in $mathbb{R}^{infty}$ we introduce measures $omega_1(x)$, $omega_{frac{1}{2}}(x)$. The positive definiteness of a function will be understood on the integral sense with respect to the measure $omega_1(x)$. Then we proved that the measure $rho(lambda)$ in the integral representation is concentrated on $l_2^+=bigg{lambda in mathbb{R}_+^{infty}= mathbb{R}_+^1timesmathbb{R}_+^1times dotsBig|sumlimits_{n=1}^{infty}lambda_n^2<inftybigg}$. The equality for $k(x)$ $left(xinmathbb{R}^{infty} right)$ is regarded as an equality for almost all $xinmathbb{R}^{infty}$ with respect to measure $omega_{frac{1}{2}}(x)$. In the second part we proved the sufficient and necessary conditions for integral representation of h.c. functions $k(x)$ $big(xin mathbb{R}_0^{infty}$ $mathrm{~is~a~nuclear~space}big)$. The positive definiteness of a function $k(x)$ will be understood on the pointwise sense. For this purpose we shall construct a rigging (chain) $mathbb{R}_0^{infty}subset l_2 subset mathbb{R}^{infty}$. Then, given that the projection and inductive topologies are coinciding, we shall obtaine the integral representation for $k(x)$ $left(xin mathbb{R}_0^{infty}right)$","PeriodicalId":479563,"journal":{"name":"Bukovinsʹkij matematičnij žurnal","volume":"37 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135801182","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For regularly converging in ${Bbb C}$ series $A_j(z)=sumlimits_{n=1}^{infty}a_{n,j}f(lambda_nz)$, $1le jle p$, where $f$ is an entire transcendental function, the asymptotic behavior of a Hadamard composition $A(z)=break=(A_1*...*A_p)_m(z)=sumlimits_{n=1}^{infty} left(sumlimits_{k_1+dots+k_p=m}c_{k_1...k_p}a_{n,1}^{k_1}cdot...cdot a_{n,p}^{k_p}right)f(lambda_nz)$ of genus m is investigated. The function $A_1$ is called dominant, if $|c_{m0...0}||a_{n,1}|^m not=0$ and $|a_{n,j}|=o(|a_{n,1}|)$ as $ntoinfty$ for $2le jle p$. The generalized order of a function $A_j$ is called the quantity $varrho_{alpha,beta}[A_j]=break=varlimsuplimits_{rto+infty}dfrac{alpha(ln,mathfrak{M}(r,A_j))}{beta(ln,r)}$, where $mathfrak{M}(r,A_j)=sumlimits_{n=1}^{infty} |a_{n,j}|M_f(rlambda_n)$, $ M_f(r)=max{|f(z)|:,|z|=r}$ and the functions $alpha$ and $beta$ are positive, continuous and increasing to $+infty$. Under certain conditions on $alpha$, $beta$, $M_f(r)$ and $(lambda_n)$, it is proved that if among the functions $A_j$ there exists a dominant one, then $varrho_{alpha,beta}[A]=max{varrho_{alpha,beta}[A_j]:,1le jle p}$. In terms of generalized orders, a connection is established between the growth of the maximal terms of power expansions of the functions $(A^{(k)}_1*...*A^{(k)}_p)_m$ and $((A_1*...*A_p)_m)^{(k)}$. Unresolved problems are formulated
{"title":"HADAMARD COMPOSITION OF SERIES IN SYSTEMS OF FUNCTIONS","authors":"M. Sheremeta","doi":"10.31861/bmj2023.01.03","DOIUrl":"https://doi.org/10.31861/bmj2023.01.03","url":null,"abstract":"For regularly converging in ${Bbb C}$ series $A_j(z)=sumlimits_{n=1}^{infty}a_{n,j}f(lambda_nz)$, $1le jle p$, where $f$ is an entire transcendental function, the asymptotic behavior of a Hadamard composition $A(z)=break=(A_1*...*A_p)_m(z)=sumlimits_{n=1}^{infty} left(sumlimits_{k_1+dots+k_p=m}c_{k_1...k_p}a_{n,1}^{k_1}cdot...cdot a_{n,p}^{k_p}right)f(lambda_nz)$ of genus m is investigated. The function $A_1$ is called dominant, if $|c_{m0...0}||a_{n,1}|^m not=0$ and $|a_{n,j}|=o(|a_{n,1}|)$ as $ntoinfty$ for $2le jle p$. The generalized order of a function $A_j$ is called the quantity $varrho_{alpha,beta}[A_j]=break=varlimsuplimits_{rto+infty}dfrac{alpha(ln,mathfrak{M}(r,A_j))}{beta(ln,r)}$, where $mathfrak{M}(r,A_j)=sumlimits_{n=1}^{infty} |a_{n,j}|M_f(rlambda_n)$, $ M_f(r)=max{|f(z)|:,|z|=r}$ and the functions $alpha$ and $beta$ are positive, continuous and increasing to $+infty$. Under certain conditions on $alpha$, $beta$, $M_f(r)$ and $(lambda_n)$, it is proved that if among the functions $A_j$ there exists a dominant one, then $varrho_{alpha,beta}[A]=max{varrho_{alpha,beta}[A_j]:,1le jle p}$. In terms of generalized orders, a connection is established between the growth of the maximal terms of power expansions of the functions $(A^{(k)}_1*...*A^{(k)}_p)_m$ and $((A_1*...*A_p)_m)^{(k)}$. Unresolved problems are formulated","PeriodicalId":479563,"journal":{"name":"Bukovinsʹkij matematičnij žurnal","volume":"121 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135801181","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider finite class of functions defined by parameters $e_0,e_1,e_2$ belonging to the set $A={0,1}$. The digits of the continued fraction $A_2$-representation of the argument $$x=frac{1}{alpha_1+frac{1}{alpha_2+_{ddots}}}equiv Delta^A_{a_1...a_n...},$$ where $alpha_nin {frac{1}{2};1}$, $a_n=2alpha_n-1$, $nin N$, and the values of the function are in a recursive dependence, namely: $$f(x=Delta^A_{a_1...a_{2n}...})=Delta^A_{b_1b_2...b_n...},$$ begin{equation*} b_1=begin{cases} e_0 &mbox{ if } (a_1,a_2)=(e_1,e_2), 1-e_0 &mbox{ if } (a_1,a_2)neq(e_1,e_2), end{cases} end{equation*} begin{equation*} b_{k+1}=begin{cases} b_k &mbox{ if } (a_{2k+1},a_{2k+2})neq(a_{2k-1},a_{2k}), 1-b_k &mbox{ if } (a_{2k+1},a_{2k+2})=(a_{2k-1},a_{2k}). end{cases} end{equation*} In the article, we justify the well-defined of the function, continuous and nowhere monotonic function. The variational properties of the function were studied and the unbounded variation was proved.
考虑由参数$e_0,e_1,e_2$定义的有限类函数，该类函数属于集合$A={0,1}$。连分式的数字$A_2$ -参数$$x=frac{1}{alpha_1+frac{1}{alpha_2+_{ddots}}}equiv Delta^A_{a_1...a_n...},$$的表示形式，其中$alpha_nin {frac{1}{2};1}$、$a_n=2alpha_n-1$、$nin N$和函数的值是递归依赖的，即:$$f(x=Delta^A_{a_1...a_{2n}...})=Delta^A_{b_1b_2...b_n...},$$begin{equation*} b_1=begin{cases} e_0 &mbox{ if } (a_1,a_2)=(e_1,e_2), 1-e_0 &mbox{ if } (a_1,a_2)neq(e_1,e_2), end{cases} end{equation*}begin{equation*} b_{k+1}=begin{cases} b_k &mbox{ if } (a_{2k+1},a_{2k+2})neq(a_{2k-1},a_{2k}), 1-b_k &mbox{ if } (a_{2k+1},a_{2k+2})=(a_{2k-1},a_{2k}). end{cases} end{equation*}在本文中，我们证明了函数的定义，连续函数和无处单调函数。研究了该函数的变分性质，证明了该函数的无界变分性。
{"title":"CONTINUOUS NOWHERE MONOTONIC FUNCTION DEFINED IT TERM CONTINUED A_2-FRACTIONS REPRESENTATION OF NUMBERS","authors":"S. Ratushniak","doi":"10.31861/bmj2023.01.11","DOIUrl":"https://doi.org/10.31861/bmj2023.01.11","url":null,"abstract":"We consider finite class of functions defined by parameters $e_0,e_1,e_2$ belonging to the set $A={0,1}$. The digits of the continued fraction $A_2$-representation of the argument $$x=frac{1}{alpha_1+frac{1}{alpha_2+_{ddots}}}equiv Delta^A_{a_1...a_n...},$$ where $alpha_nin {frac{1}{2};1}$, $a_n=2alpha_n-1$, $nin N$, and the values of the function are in a recursive dependence, namely: $$f(x=Delta^A_{a_1...a_{2n}...})=Delta^A_{b_1b_2...b_n...},$$ begin{equation*} b_1=begin{cases} e_0 &mbox{ if } (a_1,a_2)=(e_1,e_2), 1-e_0 &mbox{ if } (a_1,a_2)neq(e_1,e_2), end{cases} end{equation*} begin{equation*} b_{k+1}=begin{cases} b_k &mbox{ if } (a_{2k+1},a_{2k+2})neq(a_{2k-1},a_{2k}), 1-b_k &mbox{ if } (a_{2k+1},a_{2k+2})=(a_{2k-1},a_{2k}). end{cases} end{equation*} In the article, we justify the well-defined of the function, continuous and nowhere monotonic function. The variational properties of the function were studied and the unbounded variation was proved.","PeriodicalId":479563,"journal":{"name":"Bukovinsʹkij matematičnij žurnal","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135801175","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The technique of decomposition for functions into the sum or product of two functions is often used to facilitate the study of properties of functions. Some decomposition problems in the weighted Hardy space, Paley-Wiener space, and Bergman space are well known. Usually, in these spaces, functions are represented as the sum of two functions, each of them is "big" only in the first or only in the second quarter. The problem of decomposition of functions has practical applications, particularly in information theory. In these applications, it is often necessary to find those solutions of the decomposition problem whose growth on the negative real semi-axis is "small". In this article we consider the decomposition problem for an entire function of any small exponential type in ${z:Re z<0}$. We obtain conditions for the existence of solutions of the above problem.
{"title":"ON THE DECOMPOSITION PROBLEM FOR FUNCTIONS OF SMALL EXPONENTIAL TYPE","authors":"Kh. Voitovych","doi":"10.31861/bmj2023.01.04","DOIUrl":"https://doi.org/10.31861/bmj2023.01.04","url":null,"abstract":"The technique of decomposition for functions into the sum or product of two functions is often used to facilitate the study of properties of functions. Some decomposition problems in the weighted Hardy space, Paley-Wiener space, and Bergman space are well known. Usually, in these spaces, functions are represented as the sum of two functions, each of them is \"big\" only in the first or only in the second quarter. The problem of decomposition of functions has practical applications, particularly in information theory. In these applications, it is often necessary to find those solutions of the decomposition problem whose growth on the negative real semi-axis is \"small\". In this article we consider the decomposition problem for an entire function of any small exponential type in ${z:Re z<0}$. We obtain conditions for the existence of solutions of the above problem.","PeriodicalId":479563,"journal":{"name":"Bukovinsʹkij matematičnij žurnal","volume":"37 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135801180","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Differential-difference and differential-functional equations are mathematical models of ma-ny applied problems in automatic control and management systems, chemical, biological, technical, economic and other processes whose evolution depends on prehistory. In the study of the problems of stability, oscillation, bifurcation, control, and stabilization of solutions of linear differential-difference equations, the location of the roots of the corresponding characteristic equations is very important. Note that there are currently no effective algorithms for finding the zeros of quasipolynomials. When studying the approximation of a system of linear differential-difference equations, it was found that the approximation of nonsymptotic roots of their quasi-polynomials can be found with the help of characteristic polynomials of the corresponding approximating systems of ordinary differential equations . This paper investigates the application of approximation schemes for differential-difference equations to construct algorithms for the approximate finding of nonsymptotic roots of quasipolynomials and their application to study the stability of solutions of systems of linear differential equations with many delays. The equivalence of the exponential stability of systems with delay and of the proposed system of ordinary differential equations is established. This allowed us to build an algorithm for studying the location of non-asymptotic roots of quasi-polynomials, which are implemented on a computer. Computational experiments on special test examples showed the high efficiency of the proposed algorithms for studying the stability of linear differential-difference equations.
{"title":"MODELING STABILITY OF DIFFERENTIAL-DIFFERENCE EQUATIONS WITH DELAY","authors":"I. Vizinska","doi":"10.31861/bmj2023.01.06","DOIUrl":"https://doi.org/10.31861/bmj2023.01.06","url":null,"abstract":"Differential-difference and differential-functional equations are mathematical models of ma-ny applied problems in automatic control and management systems, chemical, biological, technical, economic and other processes whose evolution depends on prehistory. In the study of the problems of stability, oscillation, bifurcation, control, and stabilization of solutions of linear differential-difference equations, the location of the roots of the corresponding characteristic equations is very important. Note that there are currently no effective algorithms for finding the zeros of quasipolynomials. When studying the approximation of a system of linear differential-difference equations, it was found that the approximation of nonsymptotic roots of their quasi-polynomials can be found with the help of characteristic polynomials of the corresponding approximating systems of ordinary differential equations . This paper investigates the application of approximation schemes for differential-difference equations to construct algorithms for the approximate finding of nonsymptotic roots of quasipolynomials and their application to study the stability of solutions of systems of linear differential equations with many delays. The equivalence of the exponential stability of systems with delay and of the proposed system of ordinary differential equations is established. This allowed us to build an algorithm for studying the location of non-asymptotic roots of quasi-polynomials, which are implemented on a computer. Computational experiments on special test examples showed the high efficiency of the proposed algorithms for studying the stability of linear differential-difference equations.","PeriodicalId":479563,"journal":{"name":"Bukovinsʹkij matematičnij žurnal","volume":"91 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135801178","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In paper the topological structure of generalized spaces of $ S $ type and the basic operations in such spaces was investigated. The question of quasi-analyticity (non-quasi-analyticity) of generalized spaces of $ S $ type was studied. Some classes of pseudodifferential operators, properties of Fourier transformation of generalized functions from spaces of type $S'$, convolutions, convoluters and multipliers was investigated.
本文研究了$ S $型广义空间的拓扑结构及其基本运算。研究了$ S $型广义空间的拟解析性(非拟解析性)问题。研究了一类伪微分算子，S'$空间中广义函数的傅里叶变换性质，卷积，卷积和乘子。
{"title":"GENERALIZED SPACES OF S AND S′ TYPES","authors":"V. Gorodetskiy, R. Kolisnyk, N. Shevchuk","doi":"10.31861/bmj2023.01.01","DOIUrl":"https://doi.org/10.31861/bmj2023.01.01","url":null,"abstract":"In paper the topological structure of generalized spaces of $ S $ type and the basic operations in such spaces was investigated. The question of quasi-analyticity (non-quasi-analyticity) of generalized spaces of $ S $ type was studied. Some classes of pseudodifferential operators, properties of Fourier transformation of generalized functions from spaces of type $S'$, convolutions, convoluters and multipliers was investigated.","PeriodicalId":479563,"journal":{"name":"Bukovinsʹkij matematičnij žurnal","volume":"71 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135800904","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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