Journal of the
Korean Mathematical Society
JKMS
ISSN(Print) 0304-9914
ISSN(Online) 2234-3008
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2023-09-01
Forbidden theta graph, bounded spectral radius and size of non-bipartite graphs
Shuchao Li, Wanting Sun, Wei WeiAbstract : Zhai and Lin recently proved that if $G$ is an $n$-vertex connected $\theta(1, 2, r+1)$-free graph, then for odd $r$ and $n \geqslant 10r$, or for even $r$ and $n \geqslant 7r$, one has $\rho(G) \le \sqrt{\lfloor\frac{n^2}{4}\rfloor}$, and equality holds if and only if $G$ is $K_{\lceil\frac{n}{2}\rceil, \lfloor\frac{n}{2}\rfloor}$. In this paper, for large enough $n$, we prove a sharp upper bound for the spectral radius in an $n$-vertex $H$-free non-bipartite graph, where $H$ is $\theta(1, 2, 3)$ or $\theta(1, 2, 4)$, and we characterize all the extremal graphs. Furthermore, for $n \geqslant 137$, we determine the maximum number of edges in an $n$-vertex $\theta(1, 2, 4)$-free non-bipartite graph and characterize the unique extremal graph.
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2023-09-01
A conjecture of Gross and Zagier: case $E(\mathbb{Q})_{\rm{tor}} \cong \mathbb{Z}/2\mathbb{Z}$ or $\mathbb{Z}/4\mathbb{Z}$
Dongho Byeon, Taekyung Kim, Donggeon YheeAbstract : Let $E$ be an elliptic curve defined over $\mathbb{Q}$ of conductor $N$, $c$ the Manin constant of $E$, and $m$ the product of Tamagawa numbers of $E$ at prime divisors of $N$. Let $K$ be an imaginary quadratic field where all prime divisors of $N$ split in $K$, $P_K$ the Heegner point in $E(K)$, and ${\rm III}(E/K)$ the Shafarevich-Tate group of $E$ over $K$. Let $2u_K$ be the number of roots of unity contained in $K$. Gross and Zagier conjectured that if $P_K$ has infinite order in $E(K)$, then the integer $ c \cdot m \cdot u_K \cdot |{\rm III}(E/K)|^{\frac{1}{2}}$ is divisible by $|E(\mathbb{Q})_{\rm{tor}} |$. In this paper, we prove that this conjecture is true if $E(\mathbb{Q})_{\rm{tor}} \cong \mathbb{Z}/2\mathbb{Z}$ or $\mathbb{Z}/4\mathbb{Z}$ except for two explicit families of curves. Further, we show these exceptions can be removed under Stein--Watkins conjecture.
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2024-03-01
Multiple weighted estimates for multilinear commutators of multilinear singular integrals with generalized kernels
Liwen Gao, Yan Lin, Shuhui YangAbstract : In this paper, the weighted $L^{p}$ boundedness of multilinear commutators and multilinear iterated commutators generated by the multilinear singular integral operators with generalized kernels and BMO functions is established, where the weight is multiple weight. Our results are generalizations of the corresponding results for multilinear singular integral operators with standard kernels and Dini kernels under certain conditions.
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2024-05-01
Reduction of abelian varieties and curves
Moshe Jarden, Aharon RazonAbstract : Consider a Noetherian domain $R_0$ with quotient field $K_0$. Let $K$ be a finitely generated regular transcendental field extension of $K_0$. We construct a Noetherian domain $R$ with $\mathrm{Quot}(R)=K$ that contains $R_0$ and embed $\mathrm{Spec}(R_0)$ into $\mathrm{Spec}$. Then, we prove that key properties of abelian varieties and smooth geometrically integral projective curves over $K$ are preserved under reduction modulo $\mathfrak{p}$ for ``almost all'' $\mathfrak{p}\in\mathrm{Spec}(R_0)$.
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2024-05-01
Smooth singular value thresholding algorithm for low-rank matrix completion problem
Geunseop LeeAbstract : The matrix completion problem is to predict missing entries of a data matrix using the low-rank approximation of the observed entries. Typical approaches to matrix completion problem often rely on thresholding the singular values of the data matrix. However, these approaches have some limitations. In particular, a discontinuity is present near the thresholding value, and the thresholding value must be manually selected. To overcome these difficulties, we propose a shrinkage and thresholding function that smoothly thresholds the singular values to obtain more accurate and robust estimation of the data matrix. Furthermore, the proposed function is differentiable so that the thresholding values can be adaptively calculated during the iterations using Stein unbiased risk estimate. The experimental results demonstrate that the proposed algorithm yields a more accurate estimation with a faster execution than other matrix completion algorithms in image inpainting problems.
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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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2024-05-01
\boldmath$(\mathcal{V},\mathcal{W},\mathcal{Y},\mathcal{X})$-Gorenstein complexes
yanjie Li, Renyu ZhaoAbstract : Let $\mathcal{V}$, $\mathcal{W}$, $\mathcal{Y}$, $\mathcal{X}$ be four classes of left $R$-modules. The notion of $(\mathcal{V, W, Y, X})$-Gorenstein $R$-complexes is introduced, and it is shown that under certain mild technical assumptions on $\mathcal{V}$, $\mathcal{W}$, $\mathcal{Y}$, $\mathcal{X}$, an $R$-complex ${M}$ is $(\mathcal{V, W, Y, X})$-Gorenstein if and only if the module in each degree of ${M}$ is $(\mathcal{V, W, Y, X})$-Gorenstein and the total Hom complexs Hom$_R({Y},{M})$, Hom$_R({M},{X})$ are exact for any ${Y}\in\widetilde{\mathcal{Y}}$ and any ${X}\in\widetilde{\mathcal{X}}$. Many known results are recovered, and some new cases are also naturally generated.
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2023-05-01
Gradient type estimates for linear elliptic systems from composite materials
Youchan Kim, Pilsoo ShinAbstract : In this paper, we consider linear elliptic systems from composite materials where the coefficients depend on the shape and might have the discontinuity between the subregions. We derive a function which is related to the gradient of the weak solutions and which is not only locally piecewise H\"{o}lder continuous but locally H\"{o}lder continuous. The gradient of the weak solutions can be estimated by this derived function and we also prove the local piecewise gradient H\"{o}lder continuity which was obtained by the previous results.
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2024-03-01
Positive solutions to discrete harmonic functions in unbounded cylinders
Fengwen Han, Lidan WangAbstract : In this paper, we study the positive solutions to a discrete harmonic function for a random walk satisfying finite range and ellipticity conditions, killed at the boundary of an unbounded cylinder in $\mathbb{Z}^d$. We first prove the existence and uniqueness of positive solutions, and then establish that all the positive solutions are generated by two special solutions, which are exponential growth at one end and exponential decay at the other. Our method is based on maximum principle and a Harnack type inequality.
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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.
Most Read
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Symmetry of the twisted Gromov-Witten classes of projective line
Hyenho Lho
J. Korean Math. Soc. 2023; 60(3): 479-501
https://doi.org/10.4134/JKMS.j210448 -
Generalized hexagons embedded in metasymplectic spaces
Sebastian Petit, Hendrik Van Maldeghem
J. Korean Math. Soc. 2023; 60(4): 907-929
https://doi.org/10.4134/JKMS.j220528 -
Li-Yau gradient estimates on closed manifolds under Bakry-\'Emery Ricci curvature conditions
Xing Yu Song, Ling Wu
J. Korean Math. Soc. 2023; 60(5): 1023-1041
https://doi.org/10.4134/JKMS.j220589 -
Time periodic solution for the compressible magneto-micropolar fluids with external forces in $\mathbb{R}^3$
Qingfang Shi, Xinli Zhang
J. Korean Math. Soc. 2023; 60(3): 587-618
https://doi.org/10.4134/JKMS.j220285
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A note on unicity of meromorphic functions in several variables
yezhou Li, heqing sun
J. Korean Math. Soc. 2023; 60(4): 859-876
https://doi.org/10.4134/JKMS.j220473 -
The automorphism groups of Artin groups of edge-separated CLTTF graphs
Byung Hee An, Youngjin Cho
J. Korean Math. Soc. 2023; 60(6): 1171-1213
https://doi.org/10.4134/JKMS.j220343 -
Gorenstein $FP_{n}$-injective modules with respect to a semidualizing bimodule
Zhiqiang Cheng, Guoqiang Zhao
J. Korean Math. Soc. 2024; 61(1): 29-40
https://doi.org/10.4134/JKMS.j220398 -
Multiple weighted estimates for multilinear commutators of multilinear singular integrals with generalized kernels
Liwen Gao, Yan Lin, Shuhui Yang
J. Korean Math. Soc. 2024; 61(2): 207-226
https://doi.org/10.4134/JKMS.j210768
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