Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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  • 2022-05-31

    Weighted integral inequalities for modified integral Hardy operators

    Duranta Chutia, Rajib Haloi

    Abstract : In this article, we study the weak and extra-weak type integral inequalities for the modified integral Hardy operators. We provide suitable conditions on the weights $\omega, \rho, \phi$ and $\psi$ to hold the following weak type modular inequality \begin{align*} \mathcal{U}^{-1} \bigg ( \int_{ \{ | \mathcal{I}f | > \gamma\}} \mathcal{U} \Big(\gamma \omega \Big ) \rho \bigg ) & \leq \mathcal{V}^{-1} \bigg ( \int_{0}^{\infty} \mathcal{V} \Big ( C |f| \phi\Big) \psi \bigg ), \end{align*} where $\mathcal{I}$ is the modified integral Hardy operators. We also obtain a necesary and sufficient condition for the following extra-weak type integral inequality \begin{align*} \omega \bigg ( \Big\{ |\mathcal{I}f| > \gamma \Big \} \bigg) &\leq \mathcal{U}\circ \mathcal{V}^{-1} \bigg ( \int_{0}^{\infty} \mathcal{V} \bigg ( \dfrac{C |f| \phi}{\gamma} \bigg) \psi \bigg ). \end{align*} Further, we discuss the above two inequalities for the conjugate of the modified integral Hardy operators. It will extend the existing results for the Hardy operator and its integral version.

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  • 2022-03-31

    Characterizations of Jordan derivable mappings at the unit element

    Jiankui Li, Shan Li, Kaijia Luo

    Abstract : Let $\mathcal{A}$ be a unital Banach algebra, $\mathcal{M}$ a unital $\mathcal{A}$-bimodule, and $\delta$ a linear mapping from $\mathcal{A}$ into $\mathcal{M}$. We prove that if $\delta$ satisfies $\delta(A)A^{-1}+A^{-1}\delta(A)+A\delta(A^{-1})+\delta(A^{-1})A=0$ for every invertible element $A$ in $\mathcal{A}$, then $\delta$ is a Jordan derivation. Moreover, we show that $\delta$ is a Jordan derivable mapping at the unit element if and only if $\delta$ is a Jordan derivation. As an application, we answer the question posed in [4, Problem 2.6].

  • 2022-05-31

    The characterisation of $BMO$ via commutators in variable Lebesgue spaces on stratified groups

    Dongli Liu, Jian Tan, Jiman Zhao

    Abstract : Let $T$ be a bilinear Calder\'{o}n-Zygmund operator, $$b\in \cup_{q>1}L_{loc}^{q}(G).$$ We firstly obtain a constructive proof of the weak factorisation of Hardy spaces. Then we establish the characterization of $BMO$ spaces by the boundedness of the commutator $[b, T]_{j}$ in variable Lebesgue spaces.

  • 2022-03-31

    Some commutative rings defined by multiplication like-conditions

    Mohamed Chhiti, Soibri Moindze

    Abstract : In this article we investigate the transfer of multiplication-like properties to homomorphic images, direct products and amalgamated duplication of a ring along an ideal. Our aim is to provide examples of new classes of commutative rings satisfying the above-mentioned properties.

  • 2022-03-31

    Duadic codes over finite local rings

    Arezoo Soufi Karbaski, Karim Samei

    Abstract : In this paper, we introduce duadic codes over finite local rings and concentrate on quadratic residue codes. We study their properties and give the comprehensive method for the computing the unique idempotent generator of quadratic residue codes.

  • 2022-05-31

    On the sizes of dual groups

    Joungmin Song

    Abstract : We give a formula for the sizes of the dual groups. It is obtained by generalizing a size estimation of certain algebraic structure that lies in the heart of the proof of the celebrated primality test by Agrawal, Kayal and Saxena. In turn, by using our formula, we are able to give a streamlined survey of the AKS test.

  • 2023-07-31

    Semi-symmetric structure Jacobi operator for real hypersurfaces in the complex quadric

    Imsoon Jeong, Gyu Jong Kim, Changhwa Woo

    Abstract : In this paper, we introduce the notion of {\it semi-symmetric structure Jacobi operator } for Hopf real hypersufaces in the complex quad\-ric $Q^m = SO_{m+2}/SO_mSO_2$. Next we prove that there does not exist any Hopf real hypersurface in the complex quadric $Q^m = SO_{m+2}/SO_mSO_2$ with semi-symmetric structure Jacobi operator. As a corollary, we also get a non-existence property of Hopf real hypersurfaces in the complex quadric $Q^m$ with either symmetric (parallel), or recurrent structure Jacobi operator.

  • 2022-05-31

    Existence of the continued fractions of $\sqrt{d}$ and its applications

    Jun Ho Lee

    Abstract : It is well known that the continued fraction expansion of $\sqrt{d}$ has the form $[a_0, \overline{a_1, \ldots, a_{l-1}, 2a_0}]$ and $a_1, \ldots, a_{l-1}$ is a palindromic sequence of positive integers. For a given positive integer $l$ and a palindromic sequence of positive integers $a_1, \ldots, a_{l-1}$, we define the set $S(l;a_1,$ $\ldots, a_{l-1}) :=\{d\in \mathbb{Z} \,| \, d>0, \sqrt{d}=[a_0, \overline{a_1, \ldots, a_{l-1}, 2a_0}], \, \textup{where} \, a_0=\lfloor \sqrt{d} \rfloor\}$. In this paper, we completely determine when $S(l;a_1, \ldots, a_{l-1})$ is not empty in the case that $l$ is $4$, $5$, $6$, or $7$. We also give similar results for $(1+\sqrt{d})/2$. For the case that $l$ is $4$, $5$, or $6$, we explicitly describe the fundamental units of the real quadratic field $\mathbb{Q}(\sqrt{d})$. Finally, we apply our results to the Mordell conjecture for the fundamental units of $\mathbb{Q}(\sqrt{d})$.

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  • 2022-05-31

    A generalization of $w$-linked extensions

    Xiaoying Wu

    Abstract : In this paper, the concepts of $w$-linked homomorphisms, the $w_{\phi}$-operation, and DW${}_{\phi}$ rings are introduced. Also the relationships between $w_{\phi}$-ideals and $w$-ideals over a $w$-linked homomorphism $\phi: R\ra T$ are discussed. More precisely, it is shown that every $w_{\phi}$-ideal of $T$ is a $w$-ideal of $T$. Besides, it is shown that if $T$ is not a DW${}_{\phi}$ ring, then $T$ must have an infinite number of maximal $w_{\phi}$-ideals. Finally we give an application of Cohen's Theorem over $w$-factor rings, namely it is shown that an integral domain $R$ is an SM-domain with $w$-$\dim(R)\leq 1$, if and only if for any nonzero $w$-ideal $I$ of $R$, $(R/I)_w$ is an Artinian ring, if and only if for any nonzero element $a\in R$, $(R/(a))_w$ is an Artinian ring, if and only if for any nonzero element $a\in R$, $R$ satisfies the descending chain condition on $w$-ideals of $R$ containing $a$.

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  • 2022-05-31

    Rings with a right duo factor ring by an ideal contained in the center

    Jeoung Soo Cheon, Tai Keun Kwak, Yang Lee, Zhelin Piao, Sang Jo Yun

    Abstract : This article concerns a ring property that arises from combining one-sided duo factor rings and centers. A ring $R$ is called {\it right CIFD} if $R/I$ is right duo by some proper ideal $I$ of $R$ such that $I$ is contained in the center of $R$. We first see that this property is seated between right duo and right $\pi$-duo\textbf{,} and not left-right symmetric. We prove, for a right CIFD ring $R$, that $W(R)$ coincides with the set of all nilpotent elements of $R$; that $R/P$ is a right duo domain for every minimal prime ideal $P$ of $R$; that $R/W(R)$ is strongly right bounded; and that every prime ideal of $R$ is maximal if and only if $R/W(R)$ is strongly regular, where $W(R)$ is the Wedderburn radical of $R$. It is also proved that a ring $R$ is commutative if and only if $D_3(R)$ is right CIFD, where $D_3(R)$ is the ring of $3$ by $3$ upper triangular matrices over $R$ whose diagonals are equal. Furthermore\textbf{,} we show that the right CIFD property does not pass to polynomial rings, and that the polynomial ring over a ring $R$ is right CIFD if and only if $R/I$ is commutative by a proper ideal $I$ of $R$ contained in the center of $R$.

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January, 2024
Vol.61 No.1

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