Bulletin of the
Korean Mathematical Society
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ISSN(Print) 1015-8634
ISSN(Online) 2234-3016
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2024-01-31
List injective coloring of planar graphs with girth at least five
Hongyu ChenAbstract : A vertex coloring of a graph $G$ is called injective if any two vertices with a common neighbor receive distinct colors. A graph $G$ is injectively $k$-choosable if any list $L$ of admissible colors on $V(G)$ of size $k$ allows an injective coloring $\varphi$ such that $\varphi(v)\in L(v)$ whenever $v\in V(G)$. The least $k$ for which $G$ is injectively $k$-choosable is denoted by $\chi_{i}^{l}(G)$. For a planar graph $G$, Bu et al.~proved that $\chi_{i}^{l}(G)\leq\Delta+6$ if girth $g\geq5$ and maximum degree $\Delta(G)\geq8$. In this paper, we improve this result by showing that $\chi_{i}^{l}(G)\leq\Delta+6$ for $g\geq5$ and arbitrary $\Delta(G)$.
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2023-05-31
There are no numerical radius peak $n$-linear mappings on $c_0$
Sung Guen KimAbstract : 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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2023-03-31
On nuclearity of semigroup crossed products
Qing MengAbstract : In this paper, we study nuclearity of semigroup crossed products for quasi-lattice ordered groups. We show the relationships among nuclearity of the semigroup crossed product, amenability of the quasi-lattice ordered group and nuclearity of the underlying $C^*$-algebra.
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2024-03-31
Curvature estimates for a class of fully nonlinear elliptic equations with general right hand sides
Jundong ZhouAbstract : In this paper, we establish the curvature estimates for a class of curvature equations with general right hand sides depending on the gradient. We show an existence result by using the continuity method based on a priori estimates. We also derive interior curvature bounds for solutions of a class of curvature equations subject to affine Dirichlet data.
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2023-05-31
A new $q$-analogue of Van Hamme's $($G.2$)$ supercongruence for primes $p\equiv 3\pmod{4}$
Victor J. W. Guo
, Xiuguo LianAbstract : Van Hamme's (G.2) supercongruence modulo $p^4$ for primes $p\equiv 3 \pmod 4$ and $p>3$ was first established by Swisher. A $q$-analogue of this supercognruence was implicitly given by the first author and Schlosser. In this paper, we present a new $q$-analogue of Van Hamme's (G.2) supercongruence for $p\equiv 3\pmod{4}$.
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2024-01-31
Stability of total scalar curvature and the critical point equation
Seungsu Hwang, Gabjin YunAbstract : We consider the total scalar curvature functional, and show that if the second variation in the transverse traceless tensor direction is negative, then the metric is Einstein. We also find the relation between the second variation and the Lichnerowicz Laplacian.
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2024-03-31
Using rotationally symmetric planes to establish topological finiteness of manifolds
Eric ChoiAbstract : Let $(M, p)$ denote a noncompact manifold $M$ together with arbitrary basepoint $p$. In \cite{KonTan-II}, Kondo-Tanaka show that $(M, p)$ can be compared with a rotationally symmetric plane $M_m$ in such a way that if $M_m$ satisfies certain conditions, then $M$ is proved to be topologically finite. We substitute Kondo-Tanaka's condition of finite total curvature of $M_m$ with a weaker condition and show that the same conclusion can be drawn. We also use our results to show that when $M_m$ satisfies certain conditions, then $M$ is homeomorphic to $\mathbb{R}^n$.
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Bach almost solitons in paraSasakian geometry
Uday Chand De, Gopal Ghosh
Bull. Korean Math. Soc. 2023; 60(3): 763-774
https://doi.org/10.4134/BKMS.b220366 -
Brouwer degree for mean field equation on graph
Yang Liu
Bull. Korean Math. Soc. 2022; 59(5): 1305-1315
https://doi.org/10.4134/BKMS.b210756 -
Einstein-type manifolds with complete divergence of Weyl and Riemann tensor
Seungsu Hwang, Gabjin Yun
Bull. Korean Math. Soc. 2022; 59(5): 1167-1176
https://doi.org/10.4134/BKMS.b210671 -
The Matrix representation of a composition operator on the Hardy space
Young-Bok Chung
Bull. Korean Math. Soc. 2022; 59(5): 1119-1129
https://doi.org/10.4134/BKMS.b210622
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Regular $t$-balanced Cayley maps on split metacyclic $2$-groups
Haimiao Chen, Jingrui Zhang
Bull. Korean Math. Soc. 2024; 61(1): 13-27
https://doi.org/10.4134/BKMS.b220811 -
The classification of $\omega$-left-symmetric algebras in low dimensions
Zhiqi Chen, Yang Wu
Bull. Korean Math. Soc. 2023; 60(3): 747-762
https://doi.org/10.4134/BKMS.b220365 -
Some estimates for generalized commutators of multilinear Calder\'on-Zygmund operators
Honghai Liu, Zengyan Si, Ling Wang
Bull. Korean Math. Soc. 2023; 60(2): 541-560
https://doi.org/10.4134/BKMS.b220287 -
A note on Artinian local rings
Kui Hu, Hwankoo Kim, Dechuan Zhou
Bull. Korean Math. Soc. 2022; 59(5): 1317-1325
https://doi.org/10.4134/BKMS.b210759
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