The problem of the equivalence of two systems with $n$ convolutional equalities arose in investigation of the conditions of similarity in spaces of sequences of operators which are left inverse to the $n$-th degree of the generalized integration operator. In this paper we solve this problem. Note that we first prove the equivalence of two corresponding systems with $n$ equalities in the spaces of analytic functions, and then, using this statement, the main result of paper is obtained. Let $X$ be a vector space of sequences of complex numbers with K$ddot{rm o}$the normal topology from a wide class of spaces, ${mathcal I}_{alpha}$ be a generalized integration operator on $X$, $ast$ be a nontrivial convolution for ${mathcal I}_{alpha}$ in $X$, and $(P_q)_{q=0}^{n-1}$ be a system of natural projectors with $displaystyle x = sumlimits_{q=0}^{n-1} P_q x$ for all $xin X$. We established that a set $(a^{(j)})_{j=0}^{n-1}$ with $$ maxlimits_{0le j le n-1}left{mathop{overline{lim}}limits_{mtoinfty} sqrt[m]{left|frac{a_{m}^{(j)}}{alpha_m}right|}right}
{"title":"ON THE EQUIVALENCE OF SOME CONVOLUTIONAL EQUALITIES IN SPACES OF SEQUENCES","authors":"M. Mytskan, T. Zvozdetskyi","doi":"10.31861/bmj2021.01.15","DOIUrl":"https://doi.org/10.31861/bmj2021.01.15","url":null,"abstract":"The problem of the equivalence of two systems with $n$ convolutional equalities arose in\u0000investigation of the conditions of similarity in spaces of sequences of operators which\u0000are left inverse to the $n$-th degree of the generalized integration operator. In this paper we solve this problem. Note that we first prove the equivalence of two corresponding systems with $n$ equalities in the spaces of analytic functions, and then, using this statement, the main result of paper is obtained.\u0000\u0000Let $X$ be a vector space of sequences of complex numbers with K$ddot{rm o}$the normal topology from a wide class of spaces,\u0000${mathcal I}_{alpha}$ be a generalized integration operator on $X$, $ast$ be a nontrivial convolution for ${mathcal I}_{alpha}$ in $X$,\u0000and $(P_q)_{q=0}^{n-1}$ be a system of natural projectors with $displaystyle x = sumlimits_{q=0}^{n-1} P_q x$ for all $xin X$.\u0000\u0000We established that a set $(a^{(j)})_{j=0}^{n-1}$ with\u0000$$\u0000maxlimits_{0le j le n-1}left{mathop{overline{lim}}limits_{mtoinfty} sqrt[m]{left|frac{a_{m}^{(j)}}{alpha_m}right|}right}<infty\u0000$$\u0000and a set $(b^{(j)})_{j=0}^{n-1}$ of elements of the space $X$ satisfy the system of equalities\u0000$$\u0000b^{(j)}=a^{(j)}+sumlimits_{k=0}^{n-1}({mathcal I}_{alpha}^{n-k-1} a^{(k)}) ast {(P_{k}b^{(j)})}, quad j = 0, 1, ... , , , n-1,\u0000$$\u0000if and only if they satisfy the system of equalities\u0000$$\u0000b^{(j)}=a^{(j)}+sumlimits_{k=0}^{n-1}({mathcal I}_{alpha}^{n-k-1} b^{(k)}) ast {(P_{k}a^{(j)})}, quad j = 0, 1, ... , , , n-1.\u0000$$\u0000\u0000Note that the assumption on the elements $(a^{(j)})_{j=0}^{n-1}$ of the space $X$ allows us to reduce the solution of this problem to the solution of an analogous problem in the space of functions analytic in a disc.","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":"69 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129242365","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Among other sequences of integers Fibonacci numbers and Lucas numbers are cituated in the central place. In spite of great amount of literature dedicated to Fibonacci and Lucas sequences, there are still a lot of intriguing questions and open problems in this direction, see, for instance, the ''The Fibonacci Quarterly'' journal or materials of the Biannual International Conference organized by Fibonacci Association.Among other sequences of integers Fibonacci numbers and Lucas numbers are cituated in the central place. In spite of great amount of literature dedicated to Fibonacci and Lucas sequences, there are still a lot of intriguing questions and open problems in this direction, see, for instance, the ''The Fibonacci Quarterly'' journal or materials of the Biannual International Conference organized by Fibonacci Association. We are motivated by the following simple observatoin. Consider the classical Fibonacci sequence defined by the rule $$ F_{n+2}=F_{n+1}+F_n, n=0,1,2,dots $$ with the initial values $F_0=0$, $F_1=1$: $$ 0,1,1,2,3,5, 8, 13, 21, 34, 55,dots $$ If we consider a little bit another sequence $$ G_{n+2}=G_{n+1}-G_n, n=0,1,2,dots, $$ then for $G_0=0$, $G_1=1$ the sequence $(G_n)_{n=0}^infty$ is of the form $$ 0,1,1,0,-1,-1,0,1,1,0,-1,-1,dots. $$ In other words, this sequence is periodic with period of the length $6$. Therefore, the next questions follow naturally from the previous observation:(i) under which conditions on its coefficients the reccurent sequence is periodic? (ii) How long may be a period of the reccurent sequence and how it depends on coefficients? (iii) Does the length of a period depends on initial values of the reccurent sequence? In the given paper we answer to these questions for the reccurent sequences of the second and the third order. We obtain necessary and sufficient conditions on coefficients $u_i$ for the periodicity of a recurrent sequence defined by the rule $a_{n+k}=u_{k-1}a_{n+k-1}+dots+u_0a_0$ for $n=0,1,dots$ and $u_iinmathbb R$, $i=0,dots,k-1$, in the case of $k=2,3$.
{"title":"ON PERIODICITY OF RECURRENT SEQUENCES OF THE SECOND AND THE THIRD ORDER","authors":"O. Karlova, K. Katyrynchuk, V. Protsenko","doi":"10.31861/bmj2022.02.08","DOIUrl":"https://doi.org/10.31861/bmj2022.02.08","url":null,"abstract":"Among other sequences of integers Fibonacci numbers and Lucas numbers are cituated in the central place. In spite of great amount of literature dedicated to Fibonacci and Lucas sequences, there are still a lot of intriguing questions and open problems in this direction, see, for instance, the ''The Fibonacci Quarterly'' journal or materials of the Biannual International Conference organized by Fibonacci Association.Among other sequences of integers Fibonacci numbers and Lucas numbers are cituated in the central place. In spite of great amount of literature dedicated to Fibonacci and Lucas sequences, there are still a lot of intriguing questions and open problems in this direction, see, for instance, the ''The Fibonacci Quarterly'' journal or materials of the Biannual International Conference organized by Fibonacci Association.\u0000\u0000We are motivated by the following simple observatoin. Consider the classical Fibonacci sequence defined by the rule\u0000\u0000$$ F_{n+2}=F_{n+1}+F_n, n=0,1,2,dots $$\u0000\u0000with the initial values\u0000\u0000$F_0=0$, $F_1=1$: $$ 0,1,1,2,3,5, 8, 13, 21, 34, 55,dots $$\u0000\u0000If we consider a little bit another sequence\u0000\u0000$$ G_{n+2}=G_{n+1}-G_n, n=0,1,2,dots, $$\u0000\u0000then for $G_0=0$, $G_1=1$ the sequence $(G_n)_{n=0}^infty$ is of the form\u0000\u0000$$ 0,1,1,0,-1,-1,0,1,1,0,-1,-1,dots. $$\u0000\u0000In other words, this sequence is periodic with period of the length $6$.\u0000\u0000Therefore, the next questions follow naturally from the previous observation:(i) under which conditions on its coefficients the reccurent sequence is periodic? (ii) How long may be a period of the reccurent sequence and how it depends on coefficients? (iii) Does the length of a period depends on initial values of the reccurent sequence? \u0000\u0000In the given paper we answer to these questions for the reccurent sequences of the second and the third order. We obtain necessary and sufficient conditions on coefficients $u_i$ for the periodicity of a recurrent sequence defined by the rule $a_{n+k}=u_{k-1}a_{n+k-1}+dots+u_0a_0$ for $n=0,1,dots$ and $u_iinmathbb R$, $i=0,dots,k-1$, in the case of $k=2,3$.","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122604992","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 this note, we focus on some aspects of the use of virtual ethics in the study of the scientific heritage of the outstanding Ukrainian mathematician Mykhailo Krawtchouk and its involvement in the invention of the first electronic computer by Atanasov and Berry. In particular, the biased and clearly propagandistic activity of the Canadian political scientist I. Kachanovsky is analyzed concerning the contrived contribution of Mykhailo Krawtchouk’s mathematical advice to an allegedly substantial solution of the designer G. Atanasov problems of implanting computational algorithms in his designed first electronic computing device. We also noted the ill-considered popularization of these false as well as harmful statements in scientific and popular science Ukrainian literature. Separately, we focused on the openly anti- Ukrainian propaganda activity of I. Kachanovsky, which concerns his clumsy efforts in investi- gating the activities of Ukrainian nationalists during World War II and the last events on the Maidan, and its aggressive dissemination in the press of insinuations, pseudo-historical and pseudo-scientific anti-Ukrainian insults.
{"title":"MYKHAILO KRAWTCHOUK AND COMPUTING DEVICES. ON ETHIC OF INVESTIGATIONS IN HISTORY OF EXACT SCIENCES","authors":"A. Prykarpatsky, A. Plichko","doi":"10.31861/bmj2021.01.22","DOIUrl":"https://doi.org/10.31861/bmj2021.01.22","url":null,"abstract":"In this note, we focus on some aspects of the use of virtual ethics in the study of the\u0000scientific heritage of the outstanding Ukrainian mathematician Mykhailo Krawtchouk and its involvement in the invention of the first electronic computer by Atanasov and Berry. In particular, the biased and clearly propagandistic activity of the Canadian political scientist I. Kachanovsky is analyzed concerning the contrived contribution of Mykhailo Krawtchouk’s mathematical advice to an allegedly substantial solution of the designer G. Atanasov problems of implanting computational algorithms in his designed first electronic computing device. We also noted the ill-considered popularization of these false as well as harmful statements in scientific and popular science Ukrainian literature. Separately, we focused on the openly anti- Ukrainian propaganda activity of I. Kachanovsky, which concerns his clumsy efforts in investi- gating the activities of Ukrainian nationalists during World War II and the last events on the Maidan, and its aggressive dissemination in the press of insinuations, pseudo-historical and pseudo-scientific anti-Ukrainian insults.","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":"79 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114580975","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 work examines integro-differential equations Fredholm with a degenerate kernel with Hilbert control spaces. The need to study these equations is related to numerous ones applications of integro- differential equations in mathematics, physics, technology, economy and other fields. Complexity the study of integro-differential equations is connected with the fact that the integral-differential operator is not solvable everywhere. There are different approaches to the solution of not everywhere solvable linear operator equations: weak perturbation of the right-hand side of this equation with further application of the Vishyk-Lyusternyk method, introduction to system of impulse action, control, etc. The problem of obtaining coefficient conditions of solvability and analytical presentation of general solutions of integro-differential equations is a rather difficult problem, so frequent solutions will suffice are obtained by numerical methods. In this connection, Fredholm’s integro-differential equations with degenerate kernel and control in Hilbert spaces no were investigated. Therefore, the task of establishing conditions is urgent controllability, construction of general solutions in an analytical form and corresponding general controls of integro-differential equations with a degenerate kernel in abstract Hilbert spaces. As an intermediate result in the work using the results of pseudoinversion of integral operators in Hilbert spaces the solvability criterion and the form of general solutions are established integro-differential equations without control in the abstract Hilbert spaces. To establish the controllability criterion is not solvable everywhere integro-differential equations with Hilbert control spaces, the general theory of research is not applied everywhere solvable operator equations. At the same time, they are used significantly orthoprojectors, pseudo-inverse operators to normally solvable ones operators in Hilbert spaces. With the use of orthoprojectors, pseudo-inverse operators and pseudoinversion of integraloperators, a criterion is obtained solutions and the general form of solutions of integro-differential equations with a degenerate kernel with control y Hilbert spaces. An image of the general appearance is obtained control under which these solutions exist.
{"title":"CONTROLLABILITY OF FREDHOLM’S INTEGRO-DIFFERENTIAL EQUATIONS WITH BY A DEGENERATE KERNEL IN HILBERT SPACES","authors":"V. Zhuravlov, N. Gongalo, I. Slusarenko","doi":"10.31861/bmj2022.01.05","DOIUrl":"https://doi.org/10.31861/bmj2022.01.05","url":null,"abstract":"The work examines integro-differential equations Fredholm with a degenerate kernel with Hilbert control spaces. \u0000\u0000The need to study these equations is related to numerous ones applications of integro-\u0000differential equations in mathematics, physics, technology, economy and other fields. Complexity the study of integro-differential equations is connected with the fact that the integral-differential operator is not solvable everywhere.\u0000\u0000There are different approaches to the solution of not everywhere solvable linear operator\u0000equations: weak perturbation of the right-hand side of this equation with further application of the Vishyk-Lyusternyk method, introduction to system of impulse action, control, etc.\u0000The problem of obtaining coefficient conditions of solvability and analytical presentation\u0000of general solutions of integro-differential equations is a rather difficult problem, so frequent solutions will suffice are obtained by numerical methods.\u0000In this connection, Fredholm’s integro-differential equations with degenerate kernel and\u0000control in Hilbert spaces no were investigated. Therefore, the task of establishing conditions is urgent controllability, construction of general solutions in an analytical form and corresponding general controls of integro-differential equations with a degenerate kernel in abstract Hilbert spaces.\u0000As an intermediate result in the work using the results of pseudoinversion of integral\u0000operators in Hilbert spaces the solvability criterion and the form of general solutions are established integro-differential equations without control in the abstract Hilbert spaces.\u0000To establish the controllability criterion is not solvable everywhere integro-differential equations with Hilbert control spaces, the general theory of research is not applied everywhere solvable operator equations. At the same time, they are used significantly orthoprojectors, pseudo-inverse operators to normally solvable ones operators in Hilbert spaces.\u0000With the use of orthoprojectors, pseudo-inverse operators and pseudoinversion of integraloperators, a criterion is obtained solutions and the general form of solutions of integro-differential equations with a degenerate kernel with control y Hilbert spaces. An image of the general appearance is obtained control under which these solutions exist.","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":"78 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121117101","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}
This paper investigates the properties of the solutions of the equation of heat conduction with dissipation, which is associated with a harmonic oscillator - the operator $-d^2/dx^2 + x^2$, $xin mathbb{R}$ (non-negative and self-adjoint in $L_2(mathbb{R})$). An explicit form of the function is given, which is analogous to the fundamental solution of the Cauchy problem for the heat conduction equation. A formula that describes all infinitely differentiable (with respect to the variable $x$) solutions of such an equation was found, well-posedness of the Cauchy problem for the heat conduction equation with dissipation with the initial function, which is an element of the space of generalized functions $(S_{1/2}^{1/2})'$, is established. It is established that $(S_{1/2}^{1/2})'$ is the "maximum" space of initial data of the Cauchy problem, for which the solutions are infinite functions differentiable by spatial variable. The main means of research are formal Hermite series, which are identified with linear continuous functionals defined on $S_{1/2}^{1/2}$.
{"title":"PROPERTIES OF THE EQUATION OF HEAT CONDUCTION WITH DISSIPATION SOLUTIONS","authors":"V. Horodets’kyi, O. Martynyuk","doi":"10.31861/bmj2022.02.06","DOIUrl":"https://doi.org/10.31861/bmj2022.02.06","url":null,"abstract":"This paper investigates the properties of the solutions of the equation of heat conduction with dissipation, which is associated with a harmonic oscillator - the operator $-d^2/dx^2 + x^2$, $xin mathbb{R}$ (non-negative and self-adjoint in $L_2(mathbb{R})$). An explicit form of the function is given, which is analogous to the fundamental solution of the Cauchy problem for the heat conduction equation. A formula that describes all infinitely differentiable (with respect to the variable $x$) solutions of such an equation was found, well-posedness of the Cauchy problem for the heat conduction equation with dissipation with the initial function, which is an element of the space of generalized functions $(S_{1/2}^{1/2})'$, is established. It is established that $(S_{1/2}^{1/2})'$ is the \"maximum\" space of initial data of the Cauchy problem, for which the solutions are infinite functions differentiable by spatial variable. The main means of research are formal Hermite series, which are identified with linear continuous functionals defined on $S_{1/2}^{1/2}$.","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":"71 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122157548","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}
{"title":"WEAKLY NONLINEAR BOUNDARY-VALUE PROBLEMS FOR SYSTEMS OF INTEGRODIFFERENTIAL EQUATIONS. CRITICAL CASE OF THE SECOND ORDER,","authors":"I. Bondar","doi":"10.31861/bmj2019.02.014","DOIUrl":"https://doi.org/10.31861/bmj2019.02.014","url":null,"abstract":"","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125519717","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 [4] by the Fourier coefficients method there were obtained some necessary and sufficient conditions for the sequence of zeros $(lambda_{nu})$ of holomorphic in the unit disk ${z:|z|<1}$ functions $f$ from the class that determined by the majorant $eta :[0;+infty)to [0;+infty )$ that is an increasing function of arbitrary growth. Using that result in present paper it is proved that if $(lambda_{nu})$ is a sequence of zeros and $(mu_ {j})$ is a sequence of poles of the meromorphic function $f$ in the unit disk, such that for some $A>0, B>0$ and for all $rin(0;1): T(r;f)leqslant Aetaleft(frac B{1-|z|}right)$, where $T(r;f):=m(r;f)+N(r;f); m(r;f)=frac{1}{2pi }intlimits_0^{2pi } ln ^{+}|f(re^{ivarphi})|dvarphi$, then for some positive constants $A_1, A’_1, B_1, B’_1, A_2, B_2$ and for all $k inmathbb{N}$, $r, r_1$ from $(0;1)$, $r_2in(r_1;1)$ and $sigmain(1;1/r_2)$ the next conditions hold $N (r,1/f) leq A_1 etaleft(frac{B_1}{1-r}right)$, $N(r,f)leq A'_1eta left( frac{B'_1}{1-r}right) $, $$frac1{2k}left|sumlimits_{r_1 <|lambda_{nu}|leqslant r_{2}} frac1{lambda_{nu}^k} -sumlimits_{r_1 < |mu_j|leqslant r_2} frac 1{mu_j^{k}} right| leq frac{A_{2}}{r_{1}^{k}}etaleft(frac{B_{2}}{1 -r_1}right ) +frac{A_{2}}{r_{2}^{k}}maxleft{ 1;frac 1{kln sigma}right}etaleft(frac{B_{2}}{1 -sigma r_{2}}right)$$ It is also shown that if sequence $(lambda_{nu})$ satisfies the condition $N (r,1/f) leq A_1 etaleft(frac{B_1}{1-r}right)$ and $$frac1{2k}left|sumlimits_{r_1 <|lambda_{nu}|leqslant r_{2}} frac1{lambda_{nu}^k} right| leq frac{A_{2}}{r_{1}^{k}}etaleft(frac{B_{2}}{1-r_{1}}right) +frac{A_{2}}{r_{2}^{k}}maxleft{ 1;frac 1{kln sigma}right}etaleft(frac{B_{2}}{1 -sigma r_{2}}right)$$ there is possible to construct a meromorphic function from the class $T(r;f)leqslant frac{A}{sqrt{1-r}}etaleft(frac B{1-r}right)$, for which the given sequence is a sequence of zeros or poles.
{"title":"SOME NOTICES ON ZEROS AND POLES OF MEROMORPHIC FUNCTIONS IN A UNIT DISK FROM THE CLASSES DEFINED BY THE ARBITRARY GROWTH MAJORANT","authors":"I. Sheparovych","doi":"10.31861/bmj2021.02.10","DOIUrl":"https://doi.org/10.31861/bmj2021.02.10","url":null,"abstract":"In [4] by the Fourier coefficients method there were obtained some necessary and sufficient conditions for the sequence of zeros $(lambda_{nu})$ of holomorphic in the unit disk ${z:|z|<1}$ functions $f$ from the class that determined by the majorant $eta :[0;+infty)to [0;+infty )$ that is an increasing function of arbitrary growth.\u0000Using that result in present paper it is proved that if $(lambda_{nu})$ is a sequence of zeros and $(mu_ {j})$ is a sequence of poles of the meromorphic function $f$ in the unit disk, such that for some $A>0, B>0$ and for all $rin(0;1): T(r;f)leqslant Aetaleft(frac B{1-|z|}right)$, where $T(r;f):=m(r;f)+N(r;f); m(r;f)=frac{1}{2pi }intlimits_0^{2pi } ln ^{+}|f(re^{ivarphi})|dvarphi$, then for some positive constants $A_1, A’_1, B_1, B’_1, A_2, B_2$ and for all $k inmathbb{N}$, $r, r_1$ from $(0;1)$, $r_2in(r_1;1)$ and $sigmain(1;1/r_2)$ the next conditions hold\u0000$N (r,1/f) leq A_1 etaleft(frac{B_1}{1-r}right)$, $N(r,f)leq A'_1eta left( frac{B'_1}{1-r}right) $,\u0000$$frac1{2k}left|sumlimits_{r_1 <|lambda_{nu}|leqslant r_{2}} frac1{lambda_{nu}^k} -sumlimits_{r_1 < |mu_j|leqslant r_2} frac 1{mu_j^{k}} right| leq frac{A_{2}}{r_{1}^{k}}etaleft(frac{B_{2}}{1 -r_1}right ) +frac{A_{2}}{r_{2}^{k}}maxleft{ 1;frac 1{kln sigma}right}etaleft(frac{B_{2}}{1 -sigma r_{2}}right)$$\u0000It is also shown that if sequence $(lambda_{nu})$ satisfies the condition $N (r,1/f) leq A_1 etaleft(frac{B_1}{1-r}right)$ and\u0000$$frac1{2k}left|sumlimits_{r_1 <|lambda_{nu}|leqslant r_{2}} frac1{lambda_{nu}^k} right| leq frac{A_{2}}{r_{1}^{k}}etaleft(frac{B_{2}}{1-r_{1}}right) +frac{A_{2}}{r_{2}^{k}}maxleft{ 1;frac 1{kln sigma}right}etaleft(frac{B_{2}}{1 -sigma r_{2}}right)$$\u0000there is possible to construct a meromorphic function from the class $T(r;f)leqslant frac{A}{sqrt{1-r}}etaleft(frac B{1-r}right)$, for which the given sequence is a sequence of zeros or poles.","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133863069","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For the generating function $$ G_n(mathbi{x},mathbi{t})=sum_{lambda} mathbi{s}_{lambda}(x_1,x_2,ldots, x_n) t_1^{lambda_1 } t_2^{lambda_2 } cdots t_n^{lambda_n}, $$ where the Sсhur polynomials $mathbi{s}_{lambda}(x_1,x_2,ldots, x_n) $ are indexed by partitions $ lambda $ of length no more than $ n $ the explicit form for $ n = 2,3 $ is calculated and a recurrent relation for an arbitrary $ n $ is found. It is proved that $ G_n (mathbi {x}, mathbi {t}) $ is a rational function $$G_n(boldsymbol{x}, boldsymbol{t})=frac{P(boldsymbol{x}, boldsymbol{t})}{Q(boldsymbol{x}, boldsymbol{t})},$$ the numerator and denominator of which belong to the kernel of the differential operator $$ mathcal{D}_n=sum_{i=1}^n x_i frac{partial}{partial x_i}- sum_{i=1}^n t_i frac{partial}{partial t_i}. $$ For the numerator $ P (boldsymbol {x}, boldsymbol {t}) $ we find its specialization at $ t_1 = t_2 = cdots = t_n = 1. $
{"title":"GENERATING FUNCTION FOR SCHUR POLYNOMIALS","authors":"L. Bedratyuk","doi":"10.31861/bmj2022.01.04","DOIUrl":"https://doi.org/10.31861/bmj2022.01.04","url":null,"abstract":"For the generating function $$\u0000G_n(mathbi{x},mathbi{t})=sum_{lambda} mathbi{s}_{lambda}(x_1,x_2,ldots, x_n) t_1^{lambda_1 } t_2^{lambda_2 } cdots t_n^{lambda_n},\u0000$$ where the Sсhur polynomials $mathbi{s}_{lambda}(x_1,x_2,ldots, x_n) $ are indexed by partitions $ lambda $ of length no more than $ n $ the explicit form for $ n = 2,3 $ is calculated and a recurrent relation for an arbitrary $ n $ is found. It is proved that $ G_n (mathbi {x}, mathbi {t}) $ is a rational function\u0000$$G_n(boldsymbol{x}, boldsymbol{t})=frac{P(boldsymbol{x}, boldsymbol{t})}{Q(boldsymbol{x}, boldsymbol{t})},$$\u0000the numerator and denominator of which belong to the kernel of the differential operator\u0000$$\u0000mathcal{D}_n=sum_{i=1}^n x_i frac{partial}{partial x_i}- sum_{i=1}^n t_i frac{partial}{partial t_i}.\u0000$$\u0000For the numerator $ P (boldsymbol {x}, boldsymbol {t}) $ we find its specialization at $ t_1 = t_2 = cdots = t_n = 1. $","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":"39 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133896137","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}
{"title":"Cardinality of the set of continuous functions preserving digit 1 of Q3-representation of a number","authors":"Y. Maslova, M. Pratsiovytyi, N. Vasylenko","doi":"10.31861/bmj2019.01.069","DOIUrl":"https://doi.org/10.31861/bmj2019.01.069","url":null,"abstract":"","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132780032","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 article examines the activities of the special Academic Board K 76.051.02 at the Yuriy Fedkovich Chernivtsi National University in 1990-2021. It mentions the list of the members and heads of the Board for all periods of its tenure. During the work of the Board, 124 theses were defended on the specialties of differential equations, mathematical analysis, mathematical physics, mathematical modeling and computational methods. The article provides data on the applicants who defended their candidate theses, the thesis supervisors, the opponents, and the thesis topics. It also presents the geography of the applicants, the thesis supervisors' statistics and the participation in the defense of opponents.
{"title":"ON SPECIAL ACADEMIC BOARD K 76.051.02 AT THE YURIY FEDKOVICH CHERNIVTSI NATIONAL UNIVERSITY","authors":"Ya. I. Bihun, R. Petryshyn","doi":"10.31861/bmj2022.01.01","DOIUrl":"https://doi.org/10.31861/bmj2022.01.01","url":null,"abstract":"The article examines the activities of the special Academic Board K 76.051.02 at the Yuriy\u0000Fedkovich Chernivtsi National University in 1990-2021. It mentions the list of the members\u0000and heads of the Board for all periods of its tenure. During the work of the Board, 124 theses were defended on the specialties of differential equations, mathematical analysis, mathematical physics, mathematical modeling and computational methods. The article provides data on the applicants who defended their candidate theses, the thesis supervisors, the opponents, and the thesis topics. It also presents the geography of the applicants, the thesis supervisors' statistics and the participation in the defense of opponents.","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":"63 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130122634","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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