Journal of the
Korean Mathematical Society
JKMS

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

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  • 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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  • 2024-01-01

    Combinatorial supersymmetry: Supergroups, superquasigroups, and their multiplication groups

    Bokhee Im, Jonathan D.H. Smith

    Abstract : The Clifford algebra of a direct sum of real quadratic spaces appears as the superalgebra tensor product of the Clifford algebras of the summands. The purpose of the current paper is to present a purely set-theoretical version of the superalgebra tensor product which will be applicable equally to groups or to their non-associative analogues --- quasigroups and loops. Our work is part of a project to make supersymmetry an effective tool for the study of combinatorial structures. Starting from group and quasigroup structures on four-element supersets, our superproduct unifies the construction of the eight-element quaternion and dihedral groups, further leading to a loop structure which hybridizes the two groups. All three of these loops share the same character table.

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  • 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

    An intrinsic proof of Numata's theorem on Landsberg spaces

    Salah Gomaa Elgendi, Amr Soleiman

    Abstract : In this paper, we study the unicorn's Landsberg problem from an intrinsic point of view. Precisely, we investigate a coordinate-free proof of Numata's theorem on Landsberg spaces of scalar curvature. In other words, following the pullback approach to Finsler geometry, we prove that all Landsberg spaces of dimension $n\geq 3$ of non-zero scalar curvature are Riemannian spaces of constant curvature.

  • 2022-05-01

    Boundedness of Calder\'{o}n-Zygmund operators on inhomogeneous product Lipschitz spaces

    Shaoyong He, Taotao Zheng

    Abstract : In this paper, we study the boundedness of a class of inhomogeneous Journ\'{e}'s product singular integral operators on the inhomogeneous product Lipschitz spaces. The consideration of such inhomogeneous Journ\'{e}'s product singular integral operators is motivated by the study of the multi-parameter pseudo-differential operators. The key idea used here is to develop the Littlewood-Paley theory for the inhomogeneous product spaces which includes the characterization of a special inhomogeneous product Besov space and a density argument for the inhomogeneous product Lipschitz spaces in the weak sense.

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  • 2022-07-01

    Regularity relative to a hereditary torsion theory for modules over a commutative ring

    Lei Qiao, Kai Zuo

    Abstract : In this paper, we introduce and study regular rings relative to the hereditary torsion theory $w$ (a special case of a well-centered torsion theory over a commutative ring), called $w$-regular rings. We focus mainly on the $w$-regularity for $w$-coherent rings and $w$-Noetherian rings. In particular, it is shown that the $w$-coherent $w$-regular domains are exactly the Pr\"ufer $v$-multiplication domains and that an integral domain is $w$-Noetherian and $w$-regular if and only if it is a Krull domain. We also prove the $w$-analogue of the global version of the Serre--Auslander-Buchsbaum Theorem. Among other things, we show that every $w$-Noetherian $w$-regular ring is the direct sum of a finite number of Krull domains. Finally, we obtain that the global weak $w$-projective dimension of a $w$-Noetherian ring is 0, 1, or $\infty$.

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  • 2022-09-01

    Spectral decomposition for homeomorphisms on non-metrizable totally disconnected spaces

    Jumi Oh

    Abstract : We introduce the notions of symbolic expansivity and symbolic shadowing for homeomorphisms on non-metrizable compact spaces which are generalizations of expansivity and shadowing, respectively, for metric spaces. The main result is to generalize the Smale's spectral decomposition theorem to symbolically expansive homeomorphisms with symbolic shadowing on non-metrizable compact Hausdorff totally disconnected spaces.

  • 2022-09-01

    Multiple solutions of a perturbed Yamabe-type equation on graph

    Yang Liu

    Abstract : Let $u$ be a function on a locally finite graph $G=(V, E)$ and $\Omega$ be a bounded subset of $V$. Let $\varepsilon>0$, $p>2$ and $0\leq\lambda<\lambda_1(\Omega)$ be constants, where $\lambda_1(\Omega)$ is the first eigenvalue of the discrete Laplacian, and $h: V\rightarrow\mathbb{R}$ be a function satisfying $h\geq 0$ and $h\not\equiv 0$. We consider a perturbed Yamabe equation, say\begin{equation*}\left\{\begin{array}{lll} -\Delta u-\lambda u=|u|^{p-2}u+\varepsilon h, &{\rm in}& \Omega,\\ u=0,&{\rm on}&\partial\Omega,\end{array}\ri.\end{equation*}where $\Omega$ and $\partial\Omega$ denote the interior and the boundary of $\Omega$, respectively. Using variational methods,we prove thatthere exists some positive constant $\varepsilon_0>0$ such that for all $\varepsilon\in(0,\varepsilon_0)$, the above equationhas two distinct solutions. Moreover, we consider a more general nonlinear equation\begin{equation*}\left\{\begin{array}{lll} -\Delta u=f(u)+\varepsilon h, &{\rm in}& \Omega,\\ u=0, &{\rm on}&\partial\Omega,\end{array}\ri.\end{equation*}and  prove similar result for certain nonlinear term $f(u)$.

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  • 2023-05-01

    Symmetry of the twisted Gromov-Witten classes of projective line

    Hyenho Lho

    Abstract : We study the rationality and symmetry of the Gromov-Witten invariants of the projective line twisted by certain line bundles.

  • 2022-05-01

    Zeros of new Bergman kernels

    Noureddine Ghiloufi , Safa Snoun

    Abstract : In this paper we determine explicitly the kernels $\mathbb K_{\alpha,\beta}$ associated with new Bergman spaces $\mathcal A_{\alpha,\beta}^2(\mathbb D)$ considered recently by the first author and M. Zaway. Then we study the distribution of the zeros of these kernels essentially when $\alpha\in\mathbb N$ where the zeros are given by the zeros of a real polynomial $Q_{\alpha,\beta}$. Some numerical results are given throughout the paper.

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March, 2024
Vol.61 No.2

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