Bulletin of the
Korean Mathematical Society
BKMS
ISSN(Print) 1015-8634
ISSN(Online) 2234-3016
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2024-01-31
An Abelian category of weakly cofinite modules
Gholamreza PirmohammadiAbstract : 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
Periodic shadowable points
Namjip Koo, Hyunhee Lee, Nyamdavaa TsegmidAbstract : 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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2022-05-31
Application of Rothe's method to a nonlinear wave equation on graphs
Yong Lin, Yuanyuan XieAbstract : We study a nonlinear wave equation on finite connected weig\-hted graphs. Using Rothe's and energy methods, we prove the existence and uniqueness of solution under certain assumption. For linear wave equation on graphs, Lin and Xie [10] obtained the existence and uniqueness of solution. The main novelty of this paper is that the wave equation we considered has the nonlinear damping term $|u_t|^{p-1}\cdot u_t$ ($p>1$).
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2022-09-30
A note on Artinian local rings
Kui Hu, Hwankoo Kim, Dechuan ZhouAbstract : In this note, we prove that an Artinian local ring is G-semisimple (resp., SG-semisimple, $2$-SG-semisimple) if and only if its maximal ideal is G-projective (resp., SG-projective, $2$-SG-projective). As a corollary, we obtain the global statement of the above. We also give some examples of local G-semisimple rings whose maximal ideals are $n$-generated for some positive integer $n$.
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2024-01-31
Well-Posedness and asymptotic behavior of partly dissipative reaction diffusion systems with memory
Vu Trong Luong, Nguyen Duong ToanAbstract : 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
The kernels of the linear maps of finite group algebras
Dan YanAbstract : 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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2023-05-31
The classification of $\omega$-left-symmetric algebras in low dimensions
Zhiqi Chen, Yang WuAbstract : $\omega$-left-symmetric algebras contain left-symmetric algebras as a subclass and the commutator defines an $\omega$-Lie algebra. In this paper, we classify $\omega$-left-symmetric algebras in dimension 3 up to an isomorphism based on the classification of $\omega$-Lie algebras and the technique of Lie algebras.
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2022-05-31
On the sizes of dual groups
Joungmin SongAbstract : 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.
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2022-07-31
Integral operators on Ces\`{a}ro function spaces
Kwok-Pun Ho
Abstract : This paper studies the boundedness of integral operators on the Ces\`{a}ro function spaces. As applications of the main result, we obtain the Hilbert inequalities, the boundedness of the Erd\'{e}lyi-Kober fractional integrals and the Mellin fractional integrals on the Ces\`{a}ro function spaces.
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2022-05-31
Constructions of regular sparse anti-magic squares
Guangzhou Chen, Wen Li, Bangying Xin, Ming ZhongAbstract : For positive integers $n$ and $d$ with $d
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A new characterization of type $(A)$ and ruled real hypersurfaces in nonflat complex space forms
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Integral operators on Ces\`{a}ro function spaces
Kwok-Pun Ho
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B\'ezout rings and weakly B\'ezout rings
Haitham El~Alaoui
Bull. Korean Math. Soc. 2022; 59(4): 843-852
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Finiteness and vanishing results on hypersurfaces with finite index in $\mathbb{R}^{n+1}$: a revision
Nguyen Van Duc
Bull. Korean Math. Soc. 2022; 59(3): 709-723
https://doi.org/10.4134/BKMS.b210426
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When all permutations are combinatorial similarities
Viktoriia Bilet, Oleksiy Dovgoshey
Bull. Korean Math. Soc. 2023; 60(3): 733-746
https://doi.org/10.4134/BKMS.b220355 -
Copure projective modules over FGV-domains and Gorenstein Pr\"{u}fer domains
Shiqi Xing
Bull. Korean Math. Soc. 2023; 60(4): 971-983
https://doi.org/10.4134/BKMS.b220431 -
Certain properties of the class of univalent functions with real coefficients
Milutin Obradovic, Nikola Tuneski
Bull. Korean Math. Soc. 2023; 60(5): 1253-1263
https://doi.org/10.4134/BKMS.b220643 -
An extension of Schneider's characterization theorem for ellipsoids
Dong-Soo Kim, Young Ho Kim
Bull. Korean Math. Soc. 2023; 60(4): 905-913
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