Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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  • 2023-11-01

    On the $\eta$-parallelism in almost Kenmotsu $3$-manifolds

    Jun-ichi Inoguchi, Ji-Eun Lee

    Abstract : In this paper, we study the $\eta$-parallelism of the Ricci operator of almost Kenmotsu $3$-manifolds. First, we prove that an almost Kenmotsu $3$-manifold $M$ satisfying $\nabla_{\xi}h=-2\alpha h \varphi$ for some constant $\alpha$ has dominantly $\eta$-parallel Ricci operator if and only if it is locally symmetric. Next, we show that if $M$ is an $H$-almost Kenmotsu $3$-manifold satisfying $\nabla_{\xi}h=-2\alpha h \varphi$ for a constant $\alpha$, then $M$ is a Kenmotsu $3$-manifold or it is locally isomorphic to certain non-unimodular Lie group equipped with a left invariant almost Kenmotsu structure. The dominantly $\eta$-parallelism of the Ricci operator is equivalent to the local symmetry on homogeneous almost Kenmotsu $3$-manifolds.

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  • 2024-01-01

    Stable automorphic forms for the general linear group

    Jae-Hyun Yang

    Abstract : In this paper, we introduce the notion of the stability of automorphic forms for the general linear group and relate the stability of automorphic forms to the moduli space of real tori and the Jacobian real locus.

  • 2023-01-01

    Simple zeros of $L$-functions and the Weyl-type subconvexity

    Peter Jaehyun Cho , Gyeongwon Oh

    Abstract : Let $f$ be a self-dual primitive Maass or modular forms for level $4$. For such a form $f$, we define \begin{align*} N_f^s(T)\!:=\!|\{\rho \in \mathbb{C} : |\Im(\rho)| \leq T, \text{ $\rho$ is a non-trivial simple zero of $L_f(s)$} \}|. \end{align*} We establish an omega result for $N_f^s(T)$, which is $N_f^s(T)=\Omega \big( T^{\frac{1}{6}-\epsilon} \big)$ for any $\epsilon>0$. For this purpose, we need to establish the Weyl-type subconvexity for $L$-functions attached to primitive Maass forms by following a recent work of Aggarwal, Holowinsky, Lin, and Qi.

  • 2022-09-01

    Fourier transform of anisotropic mixed-norm Hardy spaces with applications to Hardy--Littlewood inequalities

    Jun Liu, Yaqian Lu, Mingdong Zhang

    Abstract : Let $\vec{p}\in(0,1]^n$ be an $n$-dimensional vector and $A$ a dilation. Let $H_A^{\vec{p}}(\mathbb{R}^n)$ denote the anisotropic mixed-norm Hardy space defined via the radial maximal function. Using the known atomic characterization of $H_{A}^{\vec{p}}(\mathbb{R}^n)$ and establishing a uniform estimate for corresponding atoms, the authors prove that the Fourier transform of $f\in H_A^{\vec{p}}(\mathbb{R}^n)$ coincides with a continuous function $F$ on $\mathbb{R}^n$ in the sense of tempered distributions. Moreover, the function $F$ can be controlled pointwisely by the product of the Hardy space norm of $f$ and a step function with respect to the transpose matrix of $A$. As applications, the authors obtain a higher order of convergence for the function $F$ at the origin, and an analogue of Hardy--Littlewood inequalities in the present setting of $H_A^{\vec{p}}(\mathbb{R}^n)$.

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  • 2022-09-01

    Dynamic behavior of cracked beams and shallow arches

    Semion Gutman, Junhong Ha, Sudeok Shon

    Abstract : We develop a rigorous mathematical framework for studying dynamic behavior of cracked beams and shallow arches. The governing equations are derived from the first principles, and stated in terms of the subdifferentials of the bending and the axial potential energies. The existence and the uniqueness of the solutions is established under various conditions. The corresponding mathematical tools dealing with vector-valued functions are comprehensively developed. The motion of beams and arches is studied under the assumptions of the weak and strong damping. The presence of cracks forces weaker regularity results for the arch motion, as compared to the beam case.

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  • 2023-09-01

    Erratum to ``Pseudo-Riemannian Sasaki solvmanifolds'' [J. Korean Math. Soc. 60 (2023), No. 1, pp. 115--141]

    Diego Conti, Federico A. Rossi, Romeo Segnan Dalmasso

    Abstract : In this erratum, we offer a correction to [J. Korean Math. Soc. 60 (2023), No. 1, pp. 115--141]. We rectify Theorem 5.7 and Table 1 of the original paper.

  • 2023-11-01

    Some abelian McCoy rings

    Rasul Mohammadi, Ahmad Moussavi, masoome zahiri

    Abstract : We introduce two subclasses of abelian McCoy rings, so-called $\pi$-\textit{CN}-rings and $\pi$-duo rings, and systematically study their fundamental characteristic properties accomplished with relationships among certain classical sorts of rings such as $2$-primal rings, bounded rings etc. It is shown that a ring $R$ is $\pi$-\textit{CN} whenever every nilpotent element of index $2$ in $R$ is central. These rings naturally generalize the long-known class of \textit{CN}-rings, introduced by Drazin \cite{drz}. It is proved that $\pi$-\textit{CN}-rings are abelian, McCoy and $2$-primal. We also show that, $\pi$-duo rings are strongly McCoy and abelian and also they are strongly right $AB$. If $R$ is $\pi$-duo, then $R[x]$ has property ($A$). If $R$ is $\pi$-duo and it is either right weakly continuous or every prime ideal of $R$ is maximal, then $R$ has property ($A$). A $\pi$-duo ring $R$ is left perfect if and only if $R$ contains no infinite set of orthogonal idempotents and every left $R$-module has a maximal submodule. Our achieved results substantially improve many existing results.

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  • 2023-01-01

    Rational homology disk smoothings and Lefschetz fibrations

    Hakho Choi

    Abstract : In this article, we generalize the results discussed in \cite{MR2783383} by introducing a genus to generic fibers of Lefschetz fibrations. That is, we give families of relations in the mapping class groups of genus-1 surfaces with boundaries that represent rational homology disk smoothings of weighted homogeneous surface singularities whose resolution graphs are $3$-legged with a bad central vertex.

  • 2022-07-01

    C*-algebraic Schur product theorem, P\'{O}lya-Szeg\H{O}-Rudin question and Novak's conjecture

    Krishnanagara Mahesh Krishna

    Abstract : Striking result of Vyb\'{\i}ral \cite{VYBIRAL} says that Schur product of positive matrices is bounded below by the size of the matrix and the row sums of Schur product. Vyb\'{\i}ral used this result to prove the Novak's conjecture. In this paper, we define Schur product of matrices over arbitrary C*-algebras and derive the results of Schur and Vyb\'{\i}ral. As an application, we state C*-algebraic version of Novak's conjecture and solve it for commutative unital C*-algebras. We formulate P\'{o}lya-Szeg\H{o}-Rudin question for the C*-algebraic Schur product of positive matrices.

  • 2024-01-01

    Existence of solutions to a generalized self-dual Chern-Simons equation on finite graphs

    Yuanyang Hu

    Abstract : Let $G=(V,E)$ be a connected finite graph. We study the existence of solutions for the following generalized Chern-Simons equation on $G$ \begin{equation*} \Delta u=\lambda \mathrm{e}^{u}\left(\mathrm{e}^{u}-1\right)^{5}+4 \pi \sum_{s=1}^{N} \delta_{p_{s}}, \end{equation*} where $\lambda>0$, $\delta_{p_{s}}$ is the Dirac mass at the vertex $p_s$, and $p_1, p_2,\dots, p_N$ are arbitrarily chosen distinct vertices on the graph. We show that there exists a critical value $\hat{\lambda}$ such that when $\lambda > \hat{\lambda}$, the generalized Chern-Simons equation has at least two solutions, when $\lambda = \hat{\lambda}$, the generalized Chern-Simons equation has a solution, and when $\lambda < \hat\lambda$, the generalized Chern-Simons equation has no solution.

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March, 2024
Vol.61 No.2

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