# Bulletin of theKorean Mathematical SocietyBKMS

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

## Most Read

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• ### 2023-05-31

#### There are no numerical radius peak $n$-linear mappings on $c_0$

Sung Guen Kim

Abstract : For $n\geq 2$ and a real Banach space $E$, ${\mathcal L}(^n E:E)$ denotes the space of all continuous $n$-linear mappings from $E$ to itself. Let $$\Pi(E)=\{[x^*, (x_1, \ldots, x_n)]: x^{*}(x_j)=\|x^{*}\|=\|x_j\|=1~\mbox{for}~{j=1, \ldots, n}~\}.$$ An element $[x^*, (x_1, \ldots, x_n)]\in \Pi(E)$ is called a {\em numerical radius point} of $T\in {\mathcal L}(^n E:E)$ if $|x^{*}(T(x_1, \ldots, x_n))|=v(T)$, where the numerical radius $v(T)=\sup_{[y^*, y_1, \ldots, y_n]\in \Pi(E)}\Big|y^{*}\Big(T(y_1, \ldots,y_n)\Big)\Big|$. For $T\in {\mathcal L}(^n E:E)$, we define \begin{align*} {Nradius}({T})=&\ \{[x^*, (x_1, \ldots, x_n)]\in \Pi(E): [x^*, (x_1, \ldots, x_n)]\\ &\quad \mbox{is a numerical radius point of}~T\}. \end{align*} $T$ is called a {\em numerical radius peak $n$-linear mapping} if there is a unique $[x^{*}, (x_1, \ldots, x_n)]\in \Pi(E)$ such that ${Nradius}({T})=\{\pm [x^{*}, (x_1, \ldots, x_n)]\}$. In this paper we present explicit formulae for the numerical radius of $T$ for every $T\in {\mathcal L}(^n E:E)$ for $E=c_0$ or $l_{\infty}$. Using these formulae we show that there are no numerical radius peak mappings of ${\mathcal L}(^n c_0:c_0)$.

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• ### 2022-09-30

#### Brouwer degree for mean field equation on graph

Yang Liu

Abstract : Let $u$ be a function on a connected finite graph $G=(V, E)$. We consider the mean field equation $$\label{5} -\Delta u=\rho\bigg{(}\frac{he^u}{\int_V he^ud\mu}-\frac{1}{|V|}\bigg{)},$$ where $\Delta$ is $\mu$-Laplacian on the graph, $\rho\in \mathbb{R}\backslash\{0\}$, $h: V\ra\mathbb{R^+}$ is a function satisfying $\min_{x\in V}h(x)>0$. Following Sun and Wang \cite{S-w}, we use the method of Brouwer degree to prove the existence of solutions to the mean field equation $(\ref{5})$. Firstly, we prove the compactness result and conclude that every solution to the equation $(\ref{5})$ is uniformly bounded. Then the Brouwer degree can be well defined. Secondly, we calculate the Brouwer degree for the equation $(\ref{5})$, say \begin{equation*}d_{\rho,h}=\left\{\begin{array}{lll} -1,\quad \rho>0,\\ \ 1,\quad \ \rho

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• ### 2023-03-31

#### Hyperbolic and spherical power of a circle

Young Wook Kim, Sung-Eun Koh, Hyung Yong Lee, Heayong Shin, Seong-Deog Yang

Abstract : Suppose that a line passing through a given point $P$ intersects a given circle $\mathcal{C}$ at $Q$ and $R$ in the Euclidean plane. It is well known that $|PQ||PR|$ is independent of the choice of the line as long as the line meets the circle at two points. It is also known that similar properties hold in the 2-sphere and in the hyperbolic plane. New proofs for the similar properties in the 2-sphere and in the hyperbolic plane are given.

• ### 2024-01-31

#### A cotorsion pair induced by the class of Gorenstein $(m,n)$-flat modules

Qiang Yang

Abstract : In this paper, we introduce the notion of Gorenstein $(m,n)$-flat modules as an extension of $(m,n)$-flat left $R$-modules over a ring $R$, where $m$ and $n$ are two fixed positive integers. We demonstrate that the class of all Gorenstein $(m,n)$-flat modules forms a Kaplansky class and establish that ($\mathcal{GF}_{m,n}(R)$,$\mathcal{GC}_{m,n}(R)$) constitutes a hereditary perfect cotorsion pair (where $\mathcal{GF}_{m,n}(R)$ denotes the class of Gorenstein $(m,n)$-flat modules and $\mathcal{GC}_{m,n}(R)$ refers to the class of Gorenstein $(m,n)$-cotorsion modules) over slightly $(m,n)$-coherent rings.

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• ### 2023-09-30

#### Commutators of the maximal functions on Banach function spaces

Müjdat Ağcayazı, Pu Zhang

Abstract : Let $M$ and $M^{\#}$ be Hardy-Littlewood maximal operator and sharp maximal operator, respectively. In this article, we present necessary and sufficient conditions for the boundedness properties for commutator operators $[M,b]$ and $[M^{\#},b]$ in a general context of Banach function spaces when $b$ belongs to $\operatorname{BMO}(\mathbb{R}^{n})$ spaces. Some applications of the results on weighted Lebesgue spaces, variable Lebesgue spaces, Orlicz spaces and Musielak--Orlicz spaces are also given.

• ### 2023-03-31

#### Some estimates for generalized commutators of multilinear Calder\'on-Zygmund operators

Honghai Liu, Zengyan Si, Ling Wang

Abstract : Let $T$ be an $m$-linear Calder\'on-Zygmund operator. $T_{\vec{b},S}$ is the generalized commutator of $T$ with a class of measurable functions $\{b_{i}\}_{i=1}^\infty$. In this paper, we will give some new estimates for $T_{\vec{b},S}$ when $\{b_{i}\}_{i=1}^\infty$ belongs to Orlicz-type space and Lipschitz space, respectively.

• ### 2023-03-31

#### Some integral inequalities for the Laplacian with density on weighted manifolds with boundary

Fanqi Zeng

Abstract : In this paper, we derive a Reilly-type inequality for the Laplacian with density on weighted manifolds with boundary. As its applications, we obtain some new Poincar\'{e}-type inequalities not only on weighted manifolds, but more interestingly, also on their boundary. Furthermore, some mean-curvature type inequalities on the boundary are also given.

• ### 2023-01-31

#### Representations by quaternary quadratic forms with coefficients $1$, $2$, $11$ and $22$

B\"{u}lent K\"{o}kl\"{u}ce

Abstract : In this article, we find bases for the spaces of modular forms $M_{2}(\Gamma _{0}(88),\big( \frac{d}{\cdot }\big) )$ for $d=1,8,44\text{ and }88$. We then derive formulas for the number of representations of a positive integer by the diagonal quaternary quadratic forms with coefficients $1,2,11$ and $22$.

• ### 2024-03-31

#### Some factorization properties of idealization in commutative rings with zero divisors

Sina Eftekhari, Sayyed Heidar Jafari, Mahdi Reza Khorsandi

Abstract : We study some factorization properties of the idealization $R$(+)$M$ of a module $M$ in a commutative ring $R$ which is not necessarily a domain. We show that $R$(+)$M$ is ACCP if and only if $R$ is ACCP and $M$ satisfies ACC on its cyclic submodules. We give an example to show that the BF property is not necessarily preserved in idealization, and give some conditions under which $R$(+)$M$ is a BFR. We also characterize the idealization rings which are UFRs.

• ### 2023-05-31

#### The critical pods of planar quadratic polynomial maps of topological degree 2

Misong Chang, Sunyang Ko, Chong Gyu Lee, Sang-Min Lee

Abstract : {Let $K$ be an algebraically closed field of characteristic 0 and let $f$ be a non-fibered planar quadratic polynomial map of topological degree 2 defined over $K$. We assume further that the meromorphic extension of $f$ on the projective plane has the unique indeterminacy point.} We define \emph{the critical pod of $f$} where $f$ sends a critical point to another critical point. By observing the behavior of $f$ at the critical pod, we can determine a good conjugate of $f$ which shows its statue in GIT sense.

## Most Read

• ### Biharmonic hypersurfaces with recurrent operators in the Euclidean space

Esmaiel Abedi, Najma Mosadegh

Bull. Korean Math. Soc. 2022; 59(6): 1595-1603
https://doi.org/10.4134/BKMS.b210910

• ### On weakly $S$-prime submodules

Hani A. Khashan, Ece Yetkin~Celikel

Bull. Korean Math. Soc. 2022; 59(6): 1387-1408
https://doi.org/10.4134/BKMS.b210784

• ### UN rings and group rings

Kanchan Jangra, Dinesh Udar

Bull. Korean Math. Soc. 2023; 60(1): 83-91
https://doi.org/10.4134/BKMS.b210917

• ### Lens spaces admitting minimal symplectic fillings with the second Betti number one

Heesang Park, Dongsoo Shin

Bull. Korean Math. Soc. 2023; 60(1): 113-122
https://doi.org/10.4134/BKMS.b210923

## Most Downloaded

• ### On a generalization of unit regular rings

Tahire Ozen

Bull. Korean Math. Soc. 2023; 60(6): 1463-1475
https://doi.org/10.4134/BKMS.b220573

• ### Automorphisms of K3 surfaces with Picard number two

Kwangwoo Lee

Bull. Korean Math. Soc. 2023; 60(6): 1427-1437
https://doi.org/10.4134/BKMS.b220371

• ### $p$-Biharmonic hypersurfaces in Einstein space and conformally flat space

Ahmed Mohammed Cherif, Khadidja Mouffoki

Bull. Korean Math. Soc. 2023; 60(3): 705-715
https://doi.org/10.4134/BKMS.b220347

• ### Structure of idempotents in polynomial rings and matrix rings

Juan Huang, Tai Keun Kwak, Yang Lee, Zhelin Piao

Bull. Korean Math. Soc. 2023; 60(5): 1321-1334
https://doi.org/10.4134/BKMS.b220692