Journal of the
Korean Mathematical Society
JKMS
ISSN(Print) 0304-9914
ISSN(Online) 2234-3008
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2024-01-01
Combinatorial supersymmetry: Supergroups, superquasigroups, and their multiplication groups
Bokhee Im, Jonathan D.H. SmithAbstract : 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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2022-11-01
Maximal domains of solutions for analytic quasilinear differential equations of first order
Chong-Kyu Han, Taejung KimAbstract : 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.
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2024-01-01
Complete noncompact submanifolds of manifolds with negative curvature
Ya Gao
, Yanling Gao, Jing Mao , Zhiqi XieAbstract : 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.
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2023-09-01
Infinite families of congruences modulo $2$ for $2$-core and $13$-core partitions
Ankita Jindal, Nabin Kumar MeherAbstract : A partition of $n$ is called a $t$-core partition if none of its hook number is divisible by $t$. In 2019, Hirschhorn and Sellers [5] obtained a parity result for $3$-core partition function $a_3(n)$. Motivated by this result, both the authors [8] recently proved that for a non-negative integer $\alpha$, $a_{3^{\alpha} m}(n)$ is almost always divisible by an arbitrary power of $2$ and $3$ and $a_{t}(n)$ is almost always divisible by an arbitrary power of $p_i^j$, where $j$ is a fixed positive integer and $t= p_1^{a_1}p_2^{a_2}\cdots p_m^{a_m}$ with primes $p_i \geq 5.$ In this article, by using Hecke eigenform theory, we obtain infinite families of congruences and multiplicative identities for $a_2(n)$ and $a_{13}(n)$ modulo $2$ which generalizes some results of Das [2].
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2022-09-01
Multiple solutions of a perturbed Yamabe-type equation on graph
Yang LiuAbstract : 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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2024-01-01
Average entropy and asymptotics
Tatyana Barron, Manimugdha SaikiaAbstract : 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.
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2024-01-01
Gorenstein $FP_{n}$-injective modules with respect to a semidualizing bimodule
Zhiqiang Cheng, Guoqiang ZhaoAbstract : 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.
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2022-11-01
$(\mathcal{F}, \mathcal{A})$-Gorenstein flat homological dimensions
V\'ictor BecerrilAbstract : In this paper we develop the homological properties of the Gorenstein $(\mathcal{L}, \mathcal{A})$-flat $R$-modules $\mathcal{GF}_{(\mathcal{F} (R), \mathcal{A})}$ proposed by Gillespie, \linebreak where the class $\mathcal{A} \subseteq \mathrm{Mod} (R^{\mathrm{op}})$ sometimes corresponds to a duality pair $(\mathcal{L}, \mathcal{A})$. We study the weak global and finitistic dimensions that come with the class $\mathcal{GF}_{(\mathcal{F} (R), \mathcal{A})}$ and show that over a $(\mathcal{L}, \mathcal{A})$-Gorenstein ring, the functor $-\otimes _R-$ is left balanced over $\mathrm{Mod} (R^{\mathrm{op}}) \times \mathrm{Mod} (R)$ by the classes $\mathcal{GF}_{(\mathcal{F} (R^{\mathrm{op}}), \mathcal{A})} \times \mathcal{GF}_{(\mathcal{F} (R), \mathcal{A})}$. When the duality pair is $(\mathcal{F} (R), \mathcal{FP}Inj (R^{\mathrm{op}}))$ we recover the G. Yang's result over a Ding-Chen ring, and we see that is new for $(\mathrm{Lev} (R), \mathrm{AC} (R^{\mathrm{op}}))$ among others.
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2024-01-01
An intrinsic proof of Numata's theorem on Landsberg spaces
Salah Gomaa Elgendi, Amr SoleimanAbstract : 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.
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2022-07-01
Gorenstein quasi-resolving subcategories
Weiqing Cao, Jiaqun WeiAbstract : 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 $M$.
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Minimal surface system in Euclidean four-space
Hojoo Lee
J. Korean Math. Soc. 2023; 60(1): 71-90
https://doi.org/10.4134/JKMS.j220095 -
Uniqueness of quasi-roots in right-angled Artin groups
Eon-Kyung Lee, Sang-Jin Lee
J. Korean Math. Soc. 2022; 59(4): 717-731
https://doi.org/10.4134/JKMS.j210578 -
Weakly equivariant classification of small covers over a product of simplicies
Aslı Güçlükan İlhan, Sabri Kaan Gürbüzer
J. Korean Math. Soc. 2022; 59(5): 963-986
https://doi.org/10.4134/JKMS.j220104 -
On weighted compactness of commutators of bilinear fractional maximal operator
Qianjun He, Juan Zhang
J. Korean Math. Soc. 2022; 59(3): 495-517
https://doi.org/10.4134/JKMS.j210188
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$(\mathcal{F}, \mathcal{A})$-Gorenstein flat homological dimensions
V\'ictor Becerril
J. Korean Math. Soc. 2022; 59(6): 1203-1227
https://doi.org/10.4134/JKMS.j220153 -
Complete noncompact submanifolds of manifolds with negative curvature
Ya Gao
, Yanling Gao, Jing Mao , Zhiqi XieJ. Korean Math. Soc. 2024; 61(1): 183-205
https://doi.org/10.4134/JKMS.j230283 -
Periodic surface homeomorphisms and contact structures
Dheeraj Kulkarni, Kashyap Rajeevsarathy, Kuldeep Saha
J. Korean Math. Soc. 2024; 61(1): 1-28
https://doi.org/10.4134/JKMS.j220244 -
Summability in Musielak--Orlicz Hardy spaces
Jun Liu, Haonan Xia
J. Korean Math. Soc. 2023; 60(5): 1057-1072
https://doi.org/10.4134/JKMS.j220646
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