Bulletin of the
Korean Mathematical Society
BKMS

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

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    January, 2024 | Volume 61, No. 1
  • 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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  • 2024-01-31

    Regular $t$-balanced Cayley maps on split metacyclic $2$-groups

    Haimiao Chen, Jingrui Zhang

    Abstract : A regular $t$-balanced Cayley map on a group $\Gamma$ is an embedding of a Cayley graph on $\Gamma$ into a surface with certain special symmetric properties. We completely classify regular $t$-balanced Cayley maps for a class of split metacyclic $2$-groups.

  • 2024-01-31

    Rotationally symmetric solutions of the prescribed higher mean curvature spacelike equations in Minkowski spacetime

    Man Xu

    Abstract : In this paper we consider the existence of rotationally symmetric entire solutions for the prescribed higher mean curvature spacelike equations in Minkowski spacetime. As a first step, we study the associated 0-Dirichlet problems on a ball, and then we prove that all possible solutions can be extended to $+\infty$. The proof of our main results are based upon the topological degree methods and the standard prolongability theorem of ordinary differential equations.

  • 2024-01-31

    The kernels of the linear maps of finite group algebras

    Dan Yan

    Abstract : Let $G$ be a finite group, $K$ a split field for $G$, and $L$ a linear map from $K[G]$ to $K$. In our paper, we first give sufficient and necessary conditions for $\operatorname{Ker}L$ and $\operatorname{Ker}L\cap Z(K[G])$, respectively, to be Mathieu-Zhao spaces for some linear maps $L$. Then we give equivalent conditions for $\operatorname{Ker}L$ to be Mathieu-Zhao spaces of $K[G]$ in term of the degrees of irreducible representations of $G$ over $K$ if $G$ is a finite Abelian group or $G$ has a normal Sylow $p$-subgroup $H$ and $L$ are class functions of $G/H$. In particular, we classify all Mathieu-Zhao spaces of the finite Abelian group algebras if $K$ is a split field for $G$.

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  • 2024-01-31

    Complex moments and the distribution of values of $L(1, \chi_{u})$ in even characteristic

    Sunghan Bae, Hwanyup Jung

    Abstract : In this paper, we announce that the strategy of comparing the complex moments of $L(1,\chi_{u})$ to that of a random Euler product $L(1, {\mb X})$ is also valid in even characteristic case. We give an asymptotic formulas for the complex moments of $L(1,\chi_{u})$ in a large uniform range. We also give $\Omega$-results for the extreme values of $L(1,\chi_{u})$.

  • 2024-01-31

    $S$-versions and $S$-generalizations of idempotents, pure ideals and Stone type theorems

    Bayram Ali Ersoy, Ünsal Tekir, Eda Yıldız

    Abstract : Let $R$ be a commutative ring with nonzero identity and $M$ be an $R$-module. In this paper, we first introduce the concept of $S$-idempotent element of $R$. Then we give a relation between $S$-idempotents of $R$ and clopen sets of $S$-Zariski topology. After that we define $S$-pure ideal which is a generalization of the notion of pure ideal. In fact, every pure ideal is $S$-pure but the converse may not be true. Afterwards, we show that there is a relation between $S$-pure ideals of $R$ and closed sets of $S$-Zariski topology that are stable under generalization.

  • 2024-01-31

    Unconditionally stable Gauge-Uzawa finite element methods for the Darcy-Brinkman equations driven by temperature and salt concentration

    Yangwei Liao, Demin Liu

    Abstract : In this paper, the Gauge-Uzawa methods for the Darcy-Brinkman equations driven by temperature and salt concentration \linebreak (DBTC) are proposed. The first order backward difference formula is adopted to approximate the time derivative term, and the linear term is treated implicitly, the nonlinear terms are treated semi-implicit. In each time step, the coupling elliptic problems of velocity, temperature and salt concentration are solved, and then the pressure is solved. The unconditional stability and error estimations of the first order semi-discrete scheme are derived, at the same time, the unconditional stability of the first order fully discrete scheme is obtained. Some numerical experiments verify the theoretical prediction and show the effectiveness of the proposed methods.

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  • 2024-01-31

    On the Semigroup of partition-preserving transformations whose characters are bijective

    Mosarof Sarkar, Shubh Narayan Singh

    Abstract : Let $\mathcal{P}=\{X_i\colon i\in I\}$ be a partition of a set $X$. We say that a transformation $f\colon X \to X$ preserves $\mathcal{P}$ if for every $X_i \in \mathcal{P}$, there exists $X_j \in \mathcal{P}$ such that $X_if \subseteq X_j$. Consider the semigroup $\mathcal{B}(X,\mathcal{P})$ of all transformations $f$ of $X$ such that $f$ preserves $\mathcal{P}$ and the character (map) $\chi^{(f)}\colon I \to I$ defined by $i\chi^{(f)}=j$ whenever $X_if\subseteq X_j$ is bijective. We describe Green's relations on $\mathcal{B}(X,\mathcal{P})$, and prove that $\mathcal{D} = \mathcal{J}$ on $\mathcal{B}(X,\mathcal{P})$ if $\mathcal{P}$ is finite. We give a necessary and sufficient condition for $\mathcal{D} = \mathcal{J}$ on $\mathcal{B}(X,\mathcal{P})$. We characterize unit-regular elements in $\mathcal{B}(X,\mathcal{P})$, and determine when $\mathcal{B}(X,\mathcal{P})$ is a unit-regular semigroup. We alternatively prove that $\mathcal{B}(X,\mathcal{P})$ is a regular semigroup. We end the paper with a conjecture.

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  • 2024-01-31

    A function-field analogue of the Goldbach counting function and the associated Dirichlet series

    Shigeki Egami, Kohji Matsumoto

    Abstract : We consider a function-field analogue of Dirichlet series associated with the Goldbach counting function, and prove that it can, or cannot, be continued meromorphically to the whole plane. When it cannot, we further prove the existence of the natural boundary of it.

  • 2024-01-31

    Depth and Stanley depth of two special classes of monomial ideals

    Xiaoqi Wei

    Abstract : In this paper, we define two new classes of monomial ideals $I_{l,d}$ and $J_{k,d}$. When $d\geq 2k+1$ and $l\leq d-k-1$, we give the exact formulas to compute the depth and Stanley depth of quotient rings $S/I_{l,d}^{t}$ for all $t\geq 1$. When $d=2k=2l$, we compute the depth and Stanley depth of quotient ring $S/I_{l,d}$. When $d\geq 2k$, we also compute the depth and Stanley depth of quotient ring $S/J_{k,d}$.

  • 2024-01-31

    Well-Posedness and asymptotic behavior of partly dissipative reaction diffusion systems with memory

    Vu Trong Luong, Nguyen Duong Toan

    Abstract : In this paper, we consider the asymptotic behavior of solutions for the partly dissipative reaction diffusion systems of the FitzHugh-Nagumo type with hereditary memory and a very large class of nonlinearities, which have no restriction on the upper growth of the nonlinearity. We first prove the existence and uniqueness of weak solutions to the initial boundary value problem for the above-mentioned model. Next, we investigate the existence of a uniform attractor of this problem, where the time-dependent forcing term $h \in L^2_b(\mathbb{R}; H^{-1}(\mathbb{R}^N))$ is the only translation bounded instead of translation compact. Finally, we prove the regularity of the uniform attractor $\mathcal{A}$, i.e., $\mathcal{A}$ is a bounded subset of $ H^2(\mathbb{R}^N)\times H^1(\mathbb{R}^N)\times L^2_\mu(\mathbb{R}^+, H^2(\mathbb{R}^N))$. The results in this paper will extend and improve some previously obtained results, which have not been studied before in the case of non-autonomous, exponential growth nonlinearity and contain memory kernels.

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  • 2024-01-31

    Periodic shadowable points

    Namjip Koo, Hyunhee Lee, Nyamdavaa Tsegmid

    Abstract : In this paper, we consider the set of periodic shadowable points for homeomorphisms of a compact metric space, and we prove that this set satisfies some properties such as invariance and being a $G_{\delta}$ set. Then we investigate implication relations related to sets consisting of shadowable points, periodic shadowable points and uniformly expansive points, respectively. Assume that the set of periodic points and the set of periodic shadowable points of a homeomorphism on a compact metric space are dense in $X$. Then we show that a homeomorphism has the periodic shadowing property if and only if so is the restricted map to the set of periodic shadowable points. We also give some examples related to our results.

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  • 2024-01-31

    A note on variation continuity for multilinear maximal operators

    Xiao Zhang

    Abstract : This note is devoted to establishing the variation continuity of the one-dimensional discrete uncentered multilinear maximal operator. The above result is based on some refine variation estimates of the above maximal functions on monotone intervals. The main result essentially improves some known ones.

  • 2024-01-31

    $C^*$-algebra of local conjugacy equivalence relation on strongly irreducible subshift of finite type

    Chengjun Hou, Xiangqi Qiang

    Abstract : Let $G$ be an infinite countable group and $A$ be a finite set. If $ \Sigma \subseteq A^{G}$ is a strongly irreducible subshift of finite type and $\mathcal{G}$ is the local conjugacy equivalence relation on $ \Sigma$. We construct a decreasing sequence $\mathcal{R}$ of unital $C^*$-subalgebras of $C(\Sigma)$ and a sequence of faithful conditional expectations $\mathcal{E}$ defined on $C(\Sigma)$, and obtain a Toeplitz algebra $\mathcal{T}(\mathcal{R},\mathcal{E})$ and a $C^*$-algebra $C^*(\mathcal{R},\mathcal{E})$ for the pair $(\mathcal{R},\mathcal{E})$. We show that $C^*(\mathcal{R},\mathcal{E})$ is $\ast$-isomorphic to the reduced groupoid $C^*$-algebra $C_r^*(\mathcal{G})$.

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  • 2024-01-31

    On delay differential equations with meromorphic solutions of hyper-order less than one

    Risto Korhonen, Yan Liu

    Abstract : We consider the delay differential equations \begin{equation*} b(z)w(z+1)+c(z)w(z-1)+a(z)\frac{w'(z)}{w^k(z)}=\frac{P(z,w(z))}{Q(z,w(z))}, \end{equation*} where $k\in\{1,2\}$, $a(z)$, $b(z)\not\equiv 0$, $c(z)\not\equiv 0$ are rational functions, and $P(z,w(z))$ and $Q(z,w(z))$ are polynomials in $w(z)$ with rational coefficients satisfying certain natural conditions regarding their roots. It is shown that if this equation has a non-rational meromorphic solution $w$ with hyper-order $\rho_{2}(w)

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  • 2024-01-31

    Topological sensitivity and its stronger forms on semiflows

    Ruchi Das, Devender Kumar, Mohammad Salman

    Abstract : In this paper we introduce and study the notions of topological sensitivity and its stronger forms on semiflows and on product semiflows. We give a relationship between multi-topological sensitivity and thick topological sensitivity on semiflows. We prove that for a Urysohn space $X$, a syndetically transitive semiflow $(T,X,\pi)$ having a point of proper compact orbit is syndetic topologically sensitive. Moreover, it is proved that for a $T_3$ space $X$, a transitive, nonminimal semiflow $(T,X,\pi)$ having a dense set of almost periodic points is syndetic topologically sensitive. Also, wherever necessary examples/counterexamples are given.

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  • 2024-01-31

    List injective coloring of planar graphs with girth at least five

    Hongyu Chen

    Abstract : 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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  • 2024-01-31

    An Abelian category of weakly cofinite modules

    Gholamreza Pirmohammadi

    Abstract : Let $I$ be an ideal of a commutative Noetherian semi-local ring $R$ and $M$ be an $R$-module. It is shown that if $\dim M\leq 2$ and $\Supp_R M\subseteq V(I)$, then $M$ is $I$-weakly cofinite if (and only if) the $R$-modules $\Hom_R(R/I,M)$ and $\Ext^1_R(R/I,M)$ are weakly Laskerian. As a consequence of this result, it is shown that the category of all $I$-weakly cofinite modules $X$ with $\dim X\leq 2$, forms an Abelian subcategory of the category of all $R$-modules. Finally, it is shown that if $\dim R/I\leq 2$, then for each pair of finitely generated $R$-modules $M$ and $N$ and each pair of the integers $i,j\geq 0$, the $R$-modules $\Tor_i^R(N,H^j_I(M))$ and $\Ext^i_R(N,H^j_I(M))$ are $I$-weakly cofinite.

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  • 2024-01-31

    Stability of total scalar curvature and the critical point equation

    Seungsu Hwang, Gabjin Yun

    Abstract : 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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January, 2024
Vol.61 No.1

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