Pub Date : 2023-01-16DOI: 10.30970/ms.58.2.133-141
A. Z. Ansari, N. Rehman
Consider a ring $R$, which is semiprime and also having $k$-torsion freeness. If $F, d : Rto R$ are two additive maps fulfilling the algebraic identity $$F(x^{n+m})=F(x^m) x^n+ x^m d(x^n)$$ for each $x$ in $R.$ Then $F$ will be a generalized derivation having $d$ as an associated derivation on $R$. On the other hand, in this article, it is also derived that $f$ is a generalized left derivation having a linked left derivation $delta$ on $R$ if they satisfy the algebraic identity $$f(x^{n+m})=x^n f(x^m)+ x^m delta(x^n)$$ for each $x$ in $R$ and $kin {2, m, n, (n+m-1)!}$ and at last an application on Banach algebra is presented.
考虑一个环$R$，它是半素数并且具有$k$ -扭自由度。如果$F, d : Rto R$是两个相加的映射，满足$R.$中的每个$x$的代数恒等式$$F(x^{n+m})=F(x^m) x^n+ x^m d(x^n)$$，那么$F$将是一个广义的派生，在$R$上有一个关联的派生$d$。另一方面，本文还推导出$f$是一个广义左导，在$R$上有一个链接左导$delta$，如果它们满足$R$和$kin {2, m, n, (n+m-1)!}$中每个$x$的代数恒等式$$f(x^{n+m})=x^n f(x^m)+ x^m delta(x^n)$$，最后给出了在Banach代数上的应用。
{"title":"Identities on additive mappings in semiprime rings","authors":"A. Z. Ansari, N. Rehman","doi":"10.30970/ms.58.2.133-141","DOIUrl":"https://doi.org/10.30970/ms.58.2.133-141","url":null,"abstract":"Consider a ring $R$, which is semiprime and also having $k$-torsion freeness. If $F, d : Rto R$ are two additive maps fulfilling the algebraic identity $$F(x^{n+m})=F(x^m) x^n+ x^m d(x^n)$$ for each $x$ in $R.$ Then $F$ will be a generalized derivation having $d$ as an associated derivation on $R$. On the other hand, in this article, it is also derived that $f$ is a generalized left derivation having a linked left derivation $delta$ on $R$ if they satisfy the algebraic identity $$f(x^{n+m})=x^n f(x^m)+ x^m delta(x^n)$$ for each $x$ in $R$ and $kin {2, m, n, (n+m-1)!}$ and at last an application on Banach algebra is presented.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135695195","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}
Pub Date : 2023-01-16DOI: 10.30970/ms.58.2.115-132
L. P. Bedratyuk, A. I. Bedratyuk
The aim of this paper is to clear up the problem of the connection between the 3D geometric moments invariants and the invariant theory, considering a problem of describing of the 3D geometric moments invariants as a problem of the classical invariant theory. Using the remarkable fact that the groups $SO(3)$ and $SL(2)$ are locally isomorphic, we reduced the problem of deriving 3D geometric moments invariants to the well-known problem of the classical invariant theory. We give a precise statement of the 3D geometric invariant moments computation, introducing the notions of the algebras of simultaneous 3D geometric moment invariants, and prove that they are isomorphic to the algebras of joint $SL(2)$-invariants of several binary forms. To simplify the calculating of the invariants we proceed from an action of Lie group $SO(3)$ to an action of its Lie algebra $mathfrak{sl}_2$. The author hopes that the results will be useful to the researchers in the fields of image analysis and pattern recognition.
{"title":"3D geometric moment invariants from the point of view of the classical invariant theory","authors":"L. P. Bedratyuk, A. I. Bedratyuk","doi":"10.30970/ms.58.2.115-132","DOIUrl":"https://doi.org/10.30970/ms.58.2.115-132","url":null,"abstract":"The aim of this paper is to clear up the problem of the connection between the 3D geometric moments invariants and the invariant theory, considering a problem of describing of the 3D geometric moments invariants as a problem of the classical invariant theory. Using the remarkable fact that the groups $SO(3)$ and $SL(2)$ are locally isomorphic, we reduced the problem of deriving 3D geometric moments invariants to the well-known problem of the classical invariant theory. We give a precise statement of the 3D geometric invariant moments computation, introducing the notions of the algebras of simultaneous 3D geometric moment invariants, and prove that they are isomorphic to the algebras of joint $SL(2)$-invariants of several binary forms. To simplify the calculating of the invariants we proceed from an action of Lie group $SO(3)$ to an action of its Lie algebra $mathfrak{sl}_2$. The author hopes that the results will be useful to the researchers in the fields of image analysis and pattern recognition.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135594156","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}
Pub Date : 2023-01-16DOI: 10.30970/ms.58.2.174-181
P. Filevych, O. B. Hrybel
Consider an entire (absolutely convergent in $mathbb{C}$) Dirichlet series $F$ with the exponents $lambda_n$, i.e., of the form $F(s)=sum_{n=0}^infty a_ne^{slambda_n}$, and, for all $sigmainmathbb{R}$, put $mu(sigma,F)=max{|a_n|e^{sigmalambda_n}:nge0}$ and $M(sigma,F)=sup{|F(s)|:operatorname{Re}s=sigma}$. Previously, the first of the authors and M.M.~Sheremeta proved that if $omega(lambda)1$. In the present article we prove that the exponents of entire Dirichlet series with the same property can form an arbitrary sequence $lambda=(lambda_n)_{n=0}^infty$ not satisfying $omega(lambda)
{"title":"On regular variation of entire Dirichlet series","authors":"P. Filevych, O. B. Hrybel","doi":"10.30970/ms.58.2.174-181","DOIUrl":"https://doi.org/10.30970/ms.58.2.174-181","url":null,"abstract":"Consider an entire (absolutely convergent in $mathbb{C}$) Dirichlet series $F$ with the exponents $lambda_n$, i.e., of the form $F(s)=sum_{n=0}^infty a_ne^{slambda_n}$, and, for all $sigmainmathbb{R}$, put $mu(sigma,F)=max{|a_n|e^{sigmalambda_n}:nge0}$ and $M(sigma,F)=sup{|F(s)|:operatorname{Re}s=sigma}$. Previously, the first of the authors and M.M.~Sheremeta proved that if $omega(lambda)<C(rho)$, then the regular variation of the function $lnmu(sigma,F)$ with index $rho$ implies the regular variation of the function $ln M(sigma,F)$ with index $rho$, and constructed examples of entire Dirichlet series $F$, for which $lnmu(sigma,F)$ is a regularly varying function with index $rho$, and $ln M(sigma,F)$ is not a regularly varying function with index $rho$. For the exponents of the constructed series we have $lambda_n=lnln n$ for all $nge n_0$ in the case $rho=1$, and $lambda_nsim(ln n)^{(rho-1)/rho}$ as $ntoinfty$ in the case $rho>1$. In the present article we prove that the exponents of entire Dirichlet series with the same property can form an arbitrary sequence $lambda=(lambda_n)_{n=0}^infty$ not satisfying $omega(lambda)<C(rho)$. More precisely, if $omega(lambda)ge C(rho)$, then there exists a regularly varying function $Phi(sigma)$ with index $rho$ such that, for an arbitrary positive function $l(sigma)$ on $[a,+infty)$, there exists an entire Dirichlet series $F$ with the exponents $lambda_n$, for which $ln mu(sigma, F)simPhi(sigma)$ as $sigmato+infty$ and $M(sigma,F)ge l(sigma)$ for all $sigmagesigma_0$.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46614713","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}
Let $boldsymbol{B}_{omega}^{mathscr{F}}$ be the bicyclic semigroup extension for the family $mathscr{F}$ of ${omega}$-closed subsets of $omega$ which is introduced in cite{Gutik-Mykhalenych=2020}.We study topologizations of the semigroup $boldsymbol{B}_{omega}^{mathscr{F}}$ for the family $mathscr{F}$ of inductive ${omega}$-closed subsets of $omega$. We generalize Eberhart-Selden and Bertman-West results about topologizations of the bicyclic semigroup cite{Bertman-West-1976, Eberhart-Selden=1969} and show that every Hausdorff shift-continuous topology on the semigroup $boldsymbol{B}_{omega}^{mathscr{F}}$ is discrete and if a Hausdorff semitopological semigroup $S$ contains $boldsymbol{B}_{omega}^{mathscr{F}}$ as a proper dense subsemigroup then $Ssetminusboldsymbol{B}_{omega}^{mathscr{F}}$ is an ideal of $S$. Also, we prove the following dichotomy: every Hausdorff locally compact shift-continuous topology on $boldsymbol{B}_{omega}^{mathscr{F}}$ with an adjoined zero is either compact or discrete. As a consequence of the last result we obtain that every Hausdorff locally compact semigroup topology on $boldsymbol{B}_{omega}^{mathscr{F}}$ with an adjoined zero is discrete and every Hausdorff locally compact shift-continuous topology on the semigroup $boldsymbol{B}_{omega}^{mathscr{F}}sqcup I$ with an adjoined compact ideal $I$ is either compact or the ideal $I$ is open, which extent many results about locally compact topologizations of some classes of semigroups onto extensions of the semigroup $boldsymbol{B}_{omega}^{mathscr{F}}$.
{"title":"On a semitopological semigroup $boldsymbol{B}_{omega}^{mathscr{F}}$ when a family $mathscr{F}$ consists of inductive non-empty subsets of $omega$","authors":"O. Gutik, M. Mykhalenych","doi":"10.30970/ms.59.1.20-28","DOIUrl":"https://doi.org/10.30970/ms.59.1.20-28","url":null,"abstract":"Let $boldsymbol{B}_{omega}^{mathscr{F}}$ be the bicyclic semigroup extension for the family $mathscr{F}$ of ${omega}$-closed subsets of $omega$ which is introduced in cite{Gutik-Mykhalenych=2020}.We study topologizations of the semigroup $boldsymbol{B}_{omega}^{mathscr{F}}$ for the family $mathscr{F}$ of inductive ${omega}$-closed subsets of $omega$. We generalize Eberhart-Selden and Bertman-West results about topologizations of the bicyclic semigroup cite{Bertman-West-1976, Eberhart-Selden=1969} and show that every Hausdorff shift-continuous topology on the semigroup $boldsymbol{B}_{omega}^{mathscr{F}}$ is discrete and if a Hausdorff semitopological semigroup $S$ contains $boldsymbol{B}_{omega}^{mathscr{F}}$ as a proper dense subsemigroup then $Ssetminusboldsymbol{B}_{omega}^{mathscr{F}}$ is an ideal of $S$. Also, we prove the following dichotomy: every Hausdorff locally compact shift-continuous topology on $boldsymbol{B}_{omega}^{mathscr{F}}$ with an adjoined zero is either compact or discrete. As a consequence of the last result we obtain that every Hausdorff locally compact semigroup topology on $boldsymbol{B}_{omega}^{mathscr{F}}$ with an adjoined zero is discrete and every Hausdorff locally compact shift-continuous topology on the semigroup $boldsymbol{B}_{omega}^{mathscr{F}}sqcup I$ with an adjoined compact ideal $I$ is either compact or the ideal $I$ is open, which extent many results about locally compact topologizations of some classes of semigroups onto extensions of the semigroup $boldsymbol{B}_{omega}^{mathscr{F}}$.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46389928","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 paper we examined the some effects of derivation, trace of permuting tri-derivation and endomorphism on each other in prime and semiprime ring.Let $R$ be a $2,3$-torsion free prime ring and $F:Rtimes Rtimes Rrightarrow R$ be a permuting tri-derivation with trace $f$, $ d:Rrightarrow R$ be a derivation. In particular, the following assertions have been proved:1) if $[d(r),r]=f(r)$ for all $rin R$, then $R$ is commutative or $d=0$ (Theorem 1); 2) if $g:Rrightarrow R$ is an endomorphism such that $F(d(r),r,r)=g(r)$ for all $rin R$, then $F=0$ or $d=0$ (Theorem 2); 3) if $F(d(r),r,r)=f(r)$ for all $rin R$, then $(i)$ $F=0$ or $d=0$, $(ii)$ $d(r)circ f(r)=0$ for all $rin R$ (Theorem 3). In the other hand, if there exist permuting tri-derivations $F_{1},F_{2}:Rtimes Rtimes Rrightarrow R$ such that $F_{1}(f_{2}(r),r,r)=f_{1}(r)$ for all $rin R$, where $f_{1}$ and $%f_{2}$ are traces of $F_{1}$ and $F_{2}$, respectively, then $(i)$ $F_{1}=0$ or $F_{2}=0$, $(ii)$ $f_{1}(r)circ f_{2}(r)=0$ for all $rin R$ (Theorem 4).
本文研究了素数环和半素数环上的导数、置换三导数和自同态的相互影响。设$R$是一个2,3$无扭素环，$F:R乘以R右列R$是一个有迹$F $的置换三重导数，$ d:R右列R$是一个导数。特别地，证明了下列断言:1)如果$[d(r)，r]=f(r)$对于r $中的所有$r是交换的或$d=0$(定理1);2)如果$g: r右列r $是自同态使得$ f(d(r)，r,r)=g(r)$对于r $中的所有$r是自同态，则$ f =0$或$d=0$(定理2);3)如果$ F (d (r), r, r) = F (r) $ r r美元,然后(i) $ F = 0美元或美元d = 0美元,美元(ii) $ $ d (r) 保监会F (r) = 0中所有$ r r美元(定理3)。在另一方面,如果存在交换tri-derivations $ F {1}, F{2}: 乘以r r rightarrow r F {1} $, $ (F {2} (r), r, r) = F {1} (r)为所有r r美元,美元$ F{1} $和$ % F {2} $ $ F{1} $的痕迹和F{2},美元,那么美元(i) $ $ F {1} = 0 F{2} = 0美元或美元,美元(ii) $ $ F {1} (r) F{2} 保监会(r) = 0中所有$ r r美元(定理4)。
{"title":"On the trace of permuting tri-derivations on rings","authors":"D. Yılmaz, H. Yazarli","doi":"10.30970/ms.58.1.20-25","DOIUrl":"https://doi.org/10.30970/ms.58.1.20-25","url":null,"abstract":"In the paper we examined the some effects of derivation, trace of permuting tri-derivation and endomorphism on each other in prime and semiprime ring.Let $R$ be a $2,3$-torsion free prime ring and $F:Rtimes Rtimes Rrightarrow R$ be a permuting tri-derivation with trace $f$, $ d:Rrightarrow R$ be a derivation. In particular, the following assertions have been proved:1) if $[d(r),r]=f(r)$ for all $rin R$, then $R$ is commutative or $d=0$ (Theorem 1); \u00002) if $g:Rrightarrow R$ is an endomorphism such that $F(d(r),r,r)=g(r)$ for all $rin R$, then $F=0$ or $d=0$ (Theorem 2); \u00003) if $F(d(r),r,r)=f(r)$ for all $rin R$, then $(i)$ $F=0$ or $d=0$, $(ii)$ $d(r)circ f(r)=0$ for all $rin R$ (Theorem 3). \u0000In the other hand, if there exist permuting tri-derivations $F_{1},F_{2}:Rtimes Rtimes Rrightarrow R$ such that $F_{1}(f_{2}(r),r,r)=f_{1}(r)$ for all $rin R$, where $f_{1}$ and $%f_{2}$ are traces of $F_{1}$ and $F_{2}$, respectively, then $(i)$ $F_{1}=0$ or $F_{2}=0$, $(ii)$ $f_{1}(r)circ f_{2}(r)=0$ for all $rin R$ (Theorem 4).","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46873557","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. Baksa, Andriy Ivanovych Bandura, T. Salo, O. Skaskiv
We study a composition of two functions belonging to a class of slice holomorphic functions in the whole $n$-dimensional complex space. The slice holomorphy in the space means that for some fixed direction $mathbf{b}inmathbb{C}^nsetminus{mathbf{0}}$ and for every point $z^0inmathbb{C}^n$ the function is holomorphic on its restriction on the slice ${z^0+tmathbf{b}: tinmathbb{C}}.$ An additional assumption on joint continuity for these functions allows to construct an analog of theory of entire functions having bounded index. The analog is applicable to study properties of slice holomorphic solutions of directional differential equations, describe local behavior and value distribution.In particular, we found conditions providing boundedness of $L$-index in the direction $mathbf{b}$ for a function $f(underbrace{Phi(z),ldots,Phi(z)}_{mtext{ times}}),$where $f: mathbb{C}^ntomathbb{C}$ is a slice entire function, $Phi: mathbb{C}^ntomathbb{C}$ is a slice entire function,${L}: mathbb{C}^ntomathbb{R}_+$ is a continuous function.The obtained results are also new in one-dimensional case, i.e. for $n=1,$ $m=1.$ They are deduced using new approach in this area analog of logarithmic criterion.For a class of nonvanishing outer functions in the composition the sufficient conditions obtained by logarithmic criterion are weaker than the conditions by the Hayman theorem.
{"title":"Note on boundedness of the $L$-index in the direction of the composition of slice entire functions","authors":"V. Baksa, Andriy Ivanovych Bandura, T. Salo, O. Skaskiv","doi":"10.30970/ms.58.1.58-68","DOIUrl":"https://doi.org/10.30970/ms.58.1.58-68","url":null,"abstract":"We study a composition of two functions belonging to a class of slice holomorphic functions in the whole $n$-dimensional complex space. The slice holomorphy in the space means that for some fixed direction $mathbf{b}inmathbb{C}^nsetminus{mathbf{0}}$ and for every point $z^0inmathbb{C}^n$ the function is holomorphic on its restriction on the slice ${z^0+tmathbf{b}: tinmathbb{C}}.$ An additional assumption on joint continuity for these functions allows to construct an analog of theory of entire functions having bounded index. The analog is applicable to study properties of slice holomorphic solutions of directional differential equations, describe local behavior and value distribution.In particular, we found conditions providing boundedness of $L$-index in the direction $mathbf{b}$ for a function $f(underbrace{Phi(z),ldots,Phi(z)}_{mtext{ times}}),$where $f: mathbb{C}^ntomathbb{C}$ is a slice entire function, $Phi: mathbb{C}^ntomathbb{C}$ is a slice entire function,${L}: mathbb{C}^ntomathbb{R}_+$ is a continuous function.The obtained results are also new in one-dimensional case, i.e. for $n=1,$ $m=1.$ They are deduced using new approach in this area analog of logarithmic criterion.For a class of nonvanishing outer functions in the composition the sufficient conditions obtained by logarithmic criterion are weaker than the conditions by the Hayman theorem.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47041447","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}
Let $h$ be a positive continuous increasing to $+infty$ function on $mathbb{R}$. It is proved that for an arbitrary complex sequence $(zeta_n)$ such that $0<|zeta_1|le|zeta_2|ledots$ and $zeta_ntoinfty$ as $ntoinfty$, there exists an entire function $f$ whose zeros are the $zeta_n$, with multiplicities taken into account, for which$$ln m_2(r,f)=o(N(r)),quad rnotin E, rto+infty.$$with a set $E$ satisfying $int_{Ecap(1,+infty)}h(r)dr<+infty$, if and only if $ln h(r)=O(ln r)$ as $rto+infty$.Here $N(r)$ is the integrated counting function of the sequence $(zeta_n)$ and$$m_2(r,f)=left(frac{1}{2pi}int_0^{2pi}|ln|f(re^{itheta})||^2dthetaright)^{1/2}.$$
{"title":"Minimal growth of entire functions with prescribed zeros outside exceptional sets","authors":"I. Andrusyak, P. Filevych, O. Oryshchyn","doi":"10.30970/ms.58.1.51-57","DOIUrl":"https://doi.org/10.30970/ms.58.1.51-57","url":null,"abstract":"Let $h$ be a positive continuous increasing to $+infty$ function on $mathbb{R}$. It is proved that for an arbitrary complex sequence $(zeta_n)$ such that $0<|zeta_1|le|zeta_2|ledots$ and $zeta_ntoinfty$ as $ntoinfty$, there exists an entire function $f$ whose zeros are the $zeta_n$, with multiplicities taken into account, for which$$ln m_2(r,f)=o(N(r)),quad rnotin E, rto+infty.$$with a set $E$ satisfying $int_{Ecap(1,+infty)}h(r)dr<+infty$, if and only if $ln h(r)=O(ln r)$ as $rto+infty$.Here $N(r)$ is the integrated counting function of the sequence $(zeta_n)$ and$$m_2(r,f)=left(frac{1}{2pi}int_0^{2pi}|ln|f(re^{itheta})||^2dthetaright)^{1/2}.$$","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41834068","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}
B. Ráth, D. V. Krishna, K. S. Kumar, G. K. S. Viswanadh
We study the sharp bound for the third Hankel determinant for the inverse function $f$, when it belongs to of the class of starlike functions with respect to symmetric points.Let $mathcal{S}^{ast}_{s}$ be the class of starlike functions with respect to symmetric points. In the article proves the following statements (Theorem): If $fin mathcal{S}^{ast}_{s}$ thenbegin{equation*}big|H_{3,1}(f^{-1})big|leq1,end{equation*}and the result is sharp for $f(z)=z/(1-z^2).$
{"title":"The sharp bound of the third Hankel determinants for inverse of starlike functions with respect to symmetric points","authors":"B. Ráth, D. V. Krishna, K. S. Kumar, G. K. S. Viswanadh","doi":"10.30970/ms.58.1.45-50","DOIUrl":"https://doi.org/10.30970/ms.58.1.45-50","url":null,"abstract":"We study the sharp bound for the third Hankel determinant for the inverse function $f$, when it belongs to of the class of starlike functions with respect to symmetric points.Let $mathcal{S}^{ast}_{s}$ be the class of starlike functions with respect to symmetric points. In the article proves the following statements (Theorem): If $fin mathcal{S}^{ast}_{s}$ thenbegin{equation*}big|H_{3,1}(f^{-1})big|leq1,end{equation*}and the result is sharp for $f(z)=z/(1-z^2).$","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43898841","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}
Pub Date : 2022-10-31DOI: 10.30970/ms.58.1.103-112
I. Argyros, S. Shakhno, H. Yarmola
We study a local and semi-local convergence of Kurchatov's method and its two-step modification for solving nonlinear equations under the classical Lipschitz conditions for the first-order divided differences. To develop a convergence analysis we use the approach of restricted convergence regions in a combination to our technique of recurrent functions. The semi-local convergence is based on the majorizing scalar sequences. Also, the results of the numerical experiment are given.
{"title":"On the convergence of Kurchatov-type methods using recurrent functions for solving equations","authors":"I. Argyros, S. Shakhno, H. Yarmola","doi":"10.30970/ms.58.1.103-112","DOIUrl":"https://doi.org/10.30970/ms.58.1.103-112","url":null,"abstract":"We study a local and semi-local convergence of Kurchatov's method and its two-step modification for solving nonlinear equations under the classical Lipschitz conditions for the first-order divided differences. To develop a convergence analysis we use the approach of restricted convergence regions in a combination to our technique of recurrent functions. The semi-local convergence is based on the majorizing scalar sequences. Also, the results of the numerical experiment are given.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48578956","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}
Let $G$ be a group with identity $e$. Let $R$ be a $G$-graded commutative ring and $M$ a graded $R$-module. In this paper, we introduce the concept of graded $WAG2$-absorbing submodule. A number of results concerning of these classes of graded submodules and their homogeneous components are given. Let $N=bigoplus _{hin G}N_{h}$ be a graded submodule of $M$ and $hin G.$ We say that $N_{h}$ is a $h$-$WAG2$-absorbing submodule of the $R_{e}$-module $M_{h}$ if $N_{h}neq M_{h}$; and whenever $r_{e},s_{e}in R_{e}$ and $m_{h}in M_{h}$ with $0neq r_{e}s_{e}m_{h}in N_{h}$, then either $%r_{e}^{i}m_{h}in N_{h}$ or $s_{e}^{j}m_{h}in N_{h}$ or $%(r_{e}s_{e})^{k}in (N_{h}:_{R_{e}}M_{h})$ for some $i,$ $j,$ $k$ $inmathbb{N}.$ We say that $N$ is {a graded }$WAG2${-absorbing submodule of }$M$ if $Nneq M$; and whenever $r_{g},s_{h}in h(R)$ and $%m_{lambda }in h(M)$ with $0neq r_{g}s_{h}m_{lambda }in N$, then either $r_{g}^{i}m_{lambda }in N$ or $s_{h}^{j}m_{lambda }in N$ or $%(r_{g}s_{h})^{k}in (N:_{R}M)$ for some $i,$ $j,$ $k$ $in mathbb{N}.$ In particular, the following assertions have been proved: Let $R$ be a $G$-graded ring, $M$ a graded cyclic $R$-module with $%Gr((0:_{R}M))=0$ and $N$ a graded submodule of $M.$ If $N$ is a graded $WAG2$% {-absorbing submodule of }$M,$ thenlinebreak $Gr((N:_{R}M))$ is a graded $WAG2$% -absorbing ideal of $R$ (Theorem 4).Let $R_{1}$ and $R_{2}$ be a $G$-graded rings. Let $R=R_{1}bigoplus R_{2}$ be a $G$-graded ring and $M=M_{1}bigoplus M_{2}$ a graded $R$-module. Let $N_{1},$ $N_{2}$ be a proper graded submodule of $M_{1}$, $M_{2}$ respectively. If $N=N_{1}bigoplus N_{2}$ is a graded $WAG2$-absorbing submodule of $M,$ then $N_{1}$ and $N_{2}$ are graded weakly primary submodule of $R_{1}$-module $M_{1},$ $R_{2}$-module $M_{2},$ respectively. Moreover, If $N_{2}neq 0$ $(N_{1}neq 0),$ then $N_{1}$ is a graded weak primary submodule of $R_{1}$-module $M_{1}$ $(N_{2}$ is a graded weak primary submodule of $R_{2}$-module $M_{2})$ (Theorem 7).
{"title":"On graded WAG2-absorbing submodule","authors":"K. Al-Zoubi, Mariam Al-Azaizeh","doi":"10.30970/ms.58.1.13-19","DOIUrl":"https://doi.org/10.30970/ms.58.1.13-19","url":null,"abstract":"Let $G$ be a group with identity $e$. Let $R$ be a $G$-graded commutative ring and $M$ a graded $R$-module. In this paper, we introduce the concept of graded $WAG2$-absorbing submodule. A number of results concerning of these classes of graded submodules and their homogeneous components are given. \u0000Let $N=bigoplus _{hin G}N_{h}$ be a graded submodule of $M$ and $hin G.$ We say that $N_{h}$ is a $h$-$WAG2$-absorbing submodule of the $R_{e}$-module $M_{h}$ if $N_{h}neq M_{h}$; and whenever $r_{e},s_{e}in R_{e}$ and $m_{h}in M_{h}$ with $0neq r_{e}s_{e}m_{h}in N_{h}$, then either $%r_{e}^{i}m_{h}in N_{h}$ or $s_{e}^{j}m_{h}in N_{h}$ or $%(r_{e}s_{e})^{k}in (N_{h}:_{R_{e}}M_{h})$ for some $i,$ $j,$ $k$ $inmathbb{N}.$ We say that $N$ is {a graded }$WAG2${-absorbing submodule of }$M$ if $Nneq M$; and whenever $r_{g},s_{h}in h(R)$ and $%m_{lambda }in h(M)$ with $0neq r_{g}s_{h}m_{lambda }in N$, then either $r_{g}^{i}m_{lambda }in N$ or $s_{h}^{j}m_{lambda }in N$ or $%(r_{g}s_{h})^{k}in (N:_{R}M)$ for some $i,$ $j,$ $k$ $in mathbb{N}.$ In particular, the following assertions have been proved: \u0000Let $R$ be a $G$-graded ring, $M$ a graded cyclic $R$-module with $%Gr((0:_{R}M))=0$ and $N$ a graded submodule of $M.$ If $N$ is a graded $WAG2$% {-absorbing submodule of }$M,$ thenlinebreak $Gr((N:_{R}M))$ is a graded $WAG2$% -absorbing ideal of $R$ (Theorem 4).Let $R_{1}$ and $R_{2}$ be a $G$-graded rings. Let $R=R_{1}bigoplus R_{2}$ be a $G$-graded ring and $M=M_{1}bigoplus M_{2}$ a graded $R$-module. Let $N_{1},$ $N_{2}$ be a proper graded submodule of $M_{1}$, $M_{2}$ respectively. If $N=N_{1}bigoplus N_{2}$ is a graded $WAG2$-absorbing submodule of $M,$ then $N_{1}$ and $N_{2}$ are graded weakly primary submodule of $R_{1}$-module $M_{1},$ $R_{2}$-module $M_{2},$ respectively. Moreover, If $N_{2}neq 0$ $(N_{1}neq 0),$ then $N_{1}$ is a graded weak primary submodule of $R_{1}$-module $M_{1}$ $(N_{2}$ is a graded weak primary submodule of $R_{2}$-module $M_{2})$ (Theorem 7).","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43887036","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：