The main purpose of this paper is to study biharmonic hypersurface in a quasi-paraSasakian manifold $mathbb{Q}^{2m+1}$. Biharmonic hypersurfaces are special cases of biharmonic maps and biharmonic maps are the critical points of the bienergy functional. The condition of biharmonicity for non-degenerate hypersurfaces in $mathbb{Q}^{2m+1}$ is investigated for both cases: either the characteristic vector field of $mathbb{Q}^{2m+1}$ is the unit normal vector field to the hypersurface or it belongs to the tangent space of the hypersurface. Some relevant examples are also illustrated.
{"title":"Characterization of Biharmonic Hypersurface","authors":"S. Srivastava, K. Sood, K. Srivastava","doi":"10.15421/242211","DOIUrl":"https://doi.org/10.15421/242211","url":null,"abstract":"The main purpose of this paper is to study biharmonic hypersurface in a quasi-paraSasakian manifold $mathbb{Q}^{2m+1}$. Biharmonic hypersurfaces are special cases of biharmonic maps and biharmonic maps are the critical points of the bienergy functional. The condition of biharmonicity for non-degenerate hypersurfaces in $mathbb{Q}^{2m+1}$ is investigated for both cases: either the characteristic vector field of $mathbb{Q}^{2m+1}$ is the unit normal vector field to the hypersurface or it belongs to the tangent space of the hypersurface. Some relevant examples are also illustrated.","PeriodicalId":52827,"journal":{"name":"Researches in Mathematics","volume":"4 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74412158","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 the present paper the spaces $Omega_n(m)$ are considered. The spaces $Omega_n(m)$, introduced in 2018 by A.M. Pasko and Y.O. Orekhova, are the generalization of the spaces $Omega_n$ (the space $Omega_n(2)$ coincides with $Omega_n$). The investigation of homotopy properties of the spaces $Omega_n$ has been started by V.I. Ruban in 1985 and followed by V.A. Koshcheev, A.M. Pasko. In particular V.A. Koshcheev has proved that the spaces $Omega_n$ are simply connected. We generalized this result proving that all the spaces $Omega_n(m)$ are simply connected. In order to prove the simply connectedness of the space $Omega_n(m)$ we consider the 1-skeleton of this space. Using 1-cells we form the closed ways that create the fundamental group of the space $Omega_n(m)$. Using 2-cells we show that all these closed ways are equivalent to the trivial way. So the fundamental group of the space $Omega_n(m)$ is trivial and the space $Omega_n(m)$ is simply connected.
{"title":"The fundamental group of the space $Omega_n(m)$","authors":"A. Paśko","doi":"10.15421/242207","DOIUrl":"https://doi.org/10.15421/242207","url":null,"abstract":"In the present paper the spaces $Omega_n(m)$ are considered. The spaces $Omega_n(m)$, introduced in 2018 by A.M. Pasko and Y.O. Orekhova, are the generalization of the spaces $Omega_n$ (the space $Omega_n(2)$ coincides with $Omega_n$). The investigation of homotopy properties of the spaces $Omega_n$ has been started by V.I. Ruban in 1985 and followed by V.A. Koshcheev, A.M. Pasko. In particular V.A. Koshcheev has proved that the spaces $Omega_n$ are simply connected. We generalized this result proving that all the spaces $Omega_n(m)$ are simply connected. In order to prove the simply connectedness of the space $Omega_n(m)$ we consider the 1-skeleton of this space. Using 1-cells we form the closed ways that create the fundamental group of the space $Omega_n(m)$. Using 2-cells we show that all these closed ways are equivalent to the trivial way. So the fundamental group of the space $Omega_n(m)$ is trivial and the space $Omega_n(m)$ is simply connected.","PeriodicalId":52827,"journal":{"name":"Researches in Mathematics","volume":"13 3","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72610068","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 article, we established new travelling wave solutions for the loaded Benjamin-Bona-Mahony and the loaded modified Benjamin-Bona-Mahony equation by the functional variable method. The performance of this method is reliable and effective and gives the exact solitary wave solutions and periodic wave solutions. All solutions of these equations have been examined and three dimensional graphics of the obtained solutions have been drawn by using the Matlab program. We get some traveling wave solutions, which are expressed by the hyperbolic functions and trigonometric functions. This method is effective to find exact solutions of many other similar equations.
{"title":"Solitary and periodic wave solutions of the loaded modified Benjamin-Bona-Mahony equation via the functional variable method","authors":"B. Babajanov, F. Abdikarimov","doi":"10.15421/242202","DOIUrl":"https://doi.org/10.15421/242202","url":null,"abstract":"In this article, we established new travelling wave solutions for the loaded Benjamin-Bona-Mahony and the loaded modified Benjamin-Bona-Mahony equation by the functional variable method. The performance of this method is reliable and effective and gives the exact solitary wave solutions and periodic wave solutions. All solutions of these equations have been examined and three dimensional graphics of the obtained solutions have been drawn by using the Matlab program. We get some traveling wave solutions, which are expressed by the hyperbolic functions and trigonometric functions. This method is effective to find exact solutions of many other similar equations.","PeriodicalId":52827,"journal":{"name":"Researches in Mathematics","volume":"28 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87527218","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 obtain the strengthened Kolmogorov comparison theorem in asymmetric case.In particular, it gives us the opportunity to obtain the following strengthened Kolmogorov inequality in the asymmetric case:$$|x^{(k)}_{pm }|_{infty}le frac{|varphi _{r-k}( cdot ;;alpha ,beta )_pm |_{infty }}{E_0(varphi _r( cdot ;;alpha ,beta ))^{1-k/r}_{infty }}|||x|||^{1-k/r}_{infty}|alpha^{-1}x_+^{(r)}+beta^{-1}x_-^{(r)}|_infty^{k/r}$$for functions $x in L^r_{infty }(mathbb{R})$, where$$|||x|||_infty:=frac12 sup_{alpha ,beta}{ |x(beta)-x(alpha)|:x'(t)neq 0 ;;foralltin (alpha ,beta) }$$$k,r in mathbb{N}$, $k 0$, $varphi_r( cdot ;;alpha ,beta )_r$ is the asymmetric perfect spline of Euler of order $r$ and $E_0(x)_infty $ is the best uniform approximation of the function $x$ by constants.
{"title":"Strengthening the Comparison Theorem and Kolmogorov Inequality in the Asymmetric Case","authors":"V. Kofanov, K.D. Sydorovych","doi":"10.15421/242204","DOIUrl":"https://doi.org/10.15421/242204","url":null,"abstract":"We obtain the strengthened Kolmogorov comparison theorem in asymmetric case.In particular, it gives us the opportunity to obtain the following strengthened Kolmogorov inequality in the asymmetric case:$$|x^{(k)}_{pm }|_{infty}le frac{|varphi _{r-k}( cdot ;;alpha ,beta )_pm |_{infty }}{E_0(varphi _r( cdot ;;alpha ,beta ))^{1-k/r}_{infty }}|||x|||^{1-k/r}_{infty}|alpha^{-1}x_+^{(r)}+beta^{-1}x_-^{(r)}|_infty^{k/r}$$for functions $x in L^r_{infty }(mathbb{R})$, where$$|||x|||_infty:=frac12 sup_{alpha ,beta}{ |x(beta)-x(alpha)|:x'(t)neq 0 ;;foralltin (alpha ,beta) }$$$k,r in mathbb{N}$, $k 0$, $varphi_r( cdot ;;alpha ,beta )_r$ is the asymmetric perfect spline of Euler of order $r$ and $E_0(x)_infty $ is the best uniform approximation of the function $x$ by constants.","PeriodicalId":52827,"journal":{"name":"Researches in Mathematics","volume":"25 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74341683","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}
V. Babenko, V. Babenko, O. Kovalenko, N. Parfinovych
The goal of the article is to characterize continuous $(lambda,varphi)$-additive operators acting on measurable bounded functions with values in $L$-spaces. As an application, we prove a sharp Ostrowski type inequality for such operators.
{"title":"General form of $(lambda,varphi)$-additive operators on spaces of $L$-space-valued functions","authors":"V. Babenko, V. Babenko, O. Kovalenko, N. Parfinovych","doi":"10.15421/242201","DOIUrl":"https://doi.org/10.15421/242201","url":null,"abstract":"The goal of the article is to characterize continuous $(lambda,varphi)$-additive operators acting on measurable bounded functions with values in $L$-spaces. As an application, we prove a sharp Ostrowski type inequality for such operators.","PeriodicalId":52827,"journal":{"name":"Researches in Mathematics","volume":"17 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80884154","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}
Three- and four-term recurrence relations for hypergeometric functions of the second order (such as hypergeometric functions of Appell, Horn, etc.) are the starting point for constructing branched continued fraction expansions of the ratios of these functions. These relations are essential for obtaining the simplest structure of branched continued fractions (elements of which are simple polynomials) for approximating the solutions of the systems of partial differential equations, as well as some analytical functions of two variables. In this study, three- and four-term recurrence relations for Horn's hypergeometric function $H_4$ are derived. These relations can be used to construct branched continued fraction expansions for the ratios of this function and they are a generalization of the classical three-term recurrent relations for Gaussian hypergeometric function underlying Gauss' continued fraction.
{"title":"Three- and four-term recurrence relations for Horn's hypergeometric function $H_4$","authors":"R. Dmytryshyn, I.-A.V. Lutsiv","doi":"10.15421/242203","DOIUrl":"https://doi.org/10.15421/242203","url":null,"abstract":"Three- and four-term recurrence relations for hypergeometric functions of the second order (such as hypergeometric functions of Appell, Horn, etc.) are the starting point for constructing branched continued fraction expansions of the ratios of these functions. These relations are essential for obtaining the simplest structure of branched continued fractions (elements of which are simple polynomials) for approximating the solutions of the systems of partial differential equations, as well as some analytical functions of two variables. In this study, three- and four-term recurrence relations for Horn's hypergeometric function $H_4$ are derived. These relations can be used to construct branched continued fraction expansions for the ratios of this function and they are a generalization of the classical three-term recurrent relations for Gaussian hypergeometric function underlying Gauss' continued fraction.","PeriodicalId":52827,"journal":{"name":"Researches in Mathematics","volume":"24 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73442423","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 paper we obtained generalisations of the L. V. Taikov’s and N. Ainulloev’s sharp inequalities, which estimate a norm of function's first-order derivative (L. V. Taikov) and a norm of function's second-order derivative (N. Ainulloev) via the modulus of continuity or the modulus of smoothness of the function itself and the modulus of continuity or the modulus of smoothness of the function's second-order derivative. The generalisations are obtained on the power of unbounded self-adjoint operators which act in a Hilbert space. The moduli of continuity or smoothness are defined by a strongly continuous group of unitary operators.
本文得到了L. V. Taikov和N. Ainulloev尖锐不等式的推广，它们通过函数本身的连续模或平滑模和函数二阶导数的连续模或平滑模来估计函数一阶导数的范数(L. V. Taikov)和函数二阶导数的范数(N. Ainulloev)。在Hilbert空间中的无界自伴随算子的幂上得到了这些推广。连续或平滑的模由一组强连续的酉算子来定义。
{"title":"Two sharp inequalities for operators in a Hilbert space","authors":"N. Kriachko","doi":"10.15421/242206","DOIUrl":"https://doi.org/10.15421/242206","url":null,"abstract":"In this paper we obtained generalisations of the L. V. Taikov’s and N. Ainulloev’s sharp inequalities, which estimate a norm of function's first-order derivative (L. V. Taikov) and a norm of function's second-order derivative (N. Ainulloev) via the modulus of continuity or the modulus of smoothness of the function itself and the modulus of continuity or the modulus of smoothness of the function's second-order derivative. The generalisations are obtained on the power of unbounded self-adjoint operators which act in a Hilbert space. The moduli of continuity or smoothness are defined by a strongly continuous group of unitary operators.","PeriodicalId":52827,"journal":{"name":"Researches in Mathematics","volume":"34 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78158207","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 paper, we introduce a parametric type of Bernoulli polynomials with higher level and study their characteristic and combinatorial properties. We also give determinant expressions of a parametric type of Bernoulli polynomials with higher level. The results are generalizations of those with level 2 by Masjed-Jamei, Beyki and Koepf and with level 3 by the author.
{"title":"A parametric type of Bernoulli polynomials with higher level","authors":"T. Komatsu","doi":"10.15421/242205","DOIUrl":"https://doi.org/10.15421/242205","url":null,"abstract":"In this paper, we introduce a parametric type of Bernoulli polynomials with higher level and study their characteristic and combinatorial properties. We also give determinant expressions of a parametric type of Bernoulli polynomials with higher level. The results are generalizations of those with level 2 by Masjed-Jamei, Beyki and Koepf and with level 3 by the author.","PeriodicalId":52827,"journal":{"name":"Researches in Mathematics","volume":"73 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90424734","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 continue our study on relationships between Fibonacci (Lucas) numbers and Bernoulli numbers and polynomials. The derivations of our results are based on functional equations for the respective generating functions, which in our case are combinations of hyperbolic functions. Special cases and some corollaries will highlight interesting aspects of our findings.
{"title":"Additional Fibonacci-Bernoulli relations","authors":"K. Adegoke, R. Frontczak, T. Goy","doi":"10.15421/242208","DOIUrl":"https://doi.org/10.15421/242208","url":null,"abstract":"We continue our study on relationships between Fibonacci (Lucas) numbers and Bernoulli numbers and polynomials. The derivations of our results are based on functional equations for the respective generating functions, which in our case are combinations of hyperbolic functions. Special cases and some corollaries will highlight interesting aspects of our findings.","PeriodicalId":52827,"journal":{"name":"Researches in Mathematics","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88264436","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 paper, our main aim is to obtain two different discrete chaotic dynamical systems on the Box fractal ($B$). For this goal, we first give two composition functions (which generate Box fractal and filled-square respectively via escape time algorithm) of expanding, folding and translation mappings. In order to examine the properties of these dynamical systems more easily, we use the intrinsic metric which is defined by the code representation of the points on $B$ and express these dynamical systems on the code sets of this fractal. We then obtain that they are chaotic in the sense of Devaney and give an algorithm to compute periodic points.
{"title":"On the Construction of Chaotic Dynamical Systems on the Box Fractal","authors":"N. Aslan, M. Saltan","doi":"10.15421/242105","DOIUrl":"https://doi.org/10.15421/242105","url":null,"abstract":"In this paper, our main aim is to obtain two different discrete chaotic dynamical systems on the Box fractal ($B$). For this goal, we first give two composition functions (which generate Box fractal and filled-square respectively via escape time algorithm) of expanding, folding and translation mappings. In order to examine the properties of these dynamical systems more easily, we use the intrinsic metric which is defined by the code representation of the points on $B$ and express these dynamical systems on the code sets of this fractal. We then obtain that they are chaotic in the sense of Devaney and give an algorithm to compute periodic points.","PeriodicalId":52827,"journal":{"name":"Researches in Mathematics","volume":"43 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86012215","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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