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.
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.
Abstract : We study the real-analytic continuation of local real-analytic solutions to the Cauchy problems of quasi-linear partial differential equations of first order for a scalar function. By making use of the first integrals of the characteristic vector field and the implicit function theorem we determine the maximal domain of the analytic extension of a local solution as a single-valued function. We present some examples including the scalar conservation laws that admit global first integrals so that our method is applicable.
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.
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 $f(u)$.
Abstract : In this paper, we introduce the notion of Gorenstein quasi-resolving subcategories (denoted by $\mathcal{GQR}_{\mathcal{X}}(\mathcal{A})$) in term of quasi-resolving subcategory $\mathcal{X}$. We define a resolution dimension relative to the Gorenstein quasi-resolving categories $\mathcal{GQR}_{\mathcal{X}}(\mathcal{A})$. In addition, we study the stability of $\mathcal{GQR}_{\mathcal{X}}(\mathcal{A})$ and apply the obtained properties to special subcategories and in particular to modules categories. Finally, we use the restricted flat dimension of right $B$-module $M$ to characterize the finitistic dimension of the endomorphism algebra $B$ of a $\mathcal{GQ}_{\mathcal{X}}$-projective $A$-module $M$.
Abstract : In this paper, for an $m$-dimensional ($m\geq5$) complete noncompact submanifold $M$ immersed in an $n$-dimensional ($n\geq6$) simply connected Riemannian manifold $N$ with negative sectional curvature, under suitable constraints on the squared norm of the second fundamental form of $M$, the norm of its weighted mean curvature vector $|\textbf{\emph{H}}_{f}|$ and the weighted real-valued function $f$, we can obtain:$\bullet$ several one-end theorems for $M$; $\bullet$ two Liouville theorems for harmonic maps from $M$ to complete Riemannian manifolds with nonpositive sectional curvature.
Abstract : Let $S$ and $R$ be rings and $_{S}C_{R}$ a semidualizing bimodule. We introduce the notion of $G_C$-$FP_n$-injective modules, which generalizes $G_C$-$FP$-injective modules and $G_C$-weak injective modules. The homological properties and the stability of $G_C$-$FP_n$-injective modules are investigated. When $S$ is a left $n$-coherent ring, several nice properties and new Foxby equivalences relative to $G_C$-$FP_n$-injective modules are given.
Abstract : In this paper, we investigate the nonnil-exact sequences and nonnil-commutative diagrams and show that they behave in a way similar to the classical ones in Abelian categories.
Abstract : We determine the $N\to\infty$ asymptotics of the expected value of entanglement entropy for pure states in $H_{1,N}\otimes H_{2,N}$, where $H_{1,N}$ and $H_{2,N}$ are the spaces of holomorphic sections of the $N$-th tensor powers of hermitian ample line bundles on compact complex manifolds.
Eon-Kyung Lee, Sang-Jin Lee
J. Korean Math. Soc. 2022; 59(4): 717-731
https://doi.org/10.4134/JKMS.j210578
Shaoyong He, Taotao Zheng
J. Korean Math. Soc. 2022; 59(3): 469-494
https://doi.org/10.4134/JKMS.j210115
Daewoong Cheong, Jinbeom Kim
J. Korean Math. Soc. 2023; 60(4): 799-822
https://doi.org/10.4134/JKMS.j220333
Nayandeep Deka Baruah, Hirakjyoti Das
J. Korean Math. Soc. 2022; 59(4): 685-697
https://doi.org/10.4134/JKMS.j210517
Zhiqiang Cheng, Guoqiang Zhao
J. Korean Math. Soc. 2024; 61(1): 29-40
https://doi.org/10.4134/JKMS.j220398
Jun Liu, Haonan Xia
J. Korean Math. Soc. 2023; 60(5): 1057-1072
https://doi.org/10.4134/JKMS.j220646
Ya Gao , Yanling Gao, Jing Mao , Zhiqi Xie
J. Korean Math. Soc. 2024; 61(1): 183-205
https://doi.org/10.4134/JKMS.j230283
Chong-Kyu Han, Taejung Kim
J. Korean Math. Soc. 2022; 59(6): 1171-1184
https://doi.org/10.4134/JKMS.j220043
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