Journal of the
Korean Mathematical Society
JKMS
ISSN(Print) 0304-9914
ISSN(Online) 2234-3008
01 May 2024
Self-pair homotopy equivalences related to co-variant functors
Ho Won Choi, Kee Young Lee, Hye Seon Shin
J. Korean Math. Soc. 2024; 61: 409-425
The category of pairs is the category whose objects are maps between two based spaces and morphisms are pair-maps from one object to another object. To study the self-homotopy equivalences in the category of pairs, we use covariant functors from the category of pairs to the group category whose objects are groups and morphisms are group homomorphisms. We introduce specific subgroups of groups of self-pair homotopy equivalences and put these groups together into certain sequences. We investigate properties of these sequences, in particular, the exactness and split. We apply the results to two special functors, homotopy and homology functors and determine the suggested several subgroups of groups of self-pair homotopy equivalences.
Keywords: Category of pairs, self homotopy equivalence, co-variant functor
01 May 2024
Smooth singular value thresholding algorithm for low-rank matrix completion problem
Geunseop Lee
J. Korean Math. Soc. 2024; 61: 427-444
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.
Keywords: Matrix completion, singular value thresholding, Stein unbiased risk estimator
01 May 2024
Properties of positive solutions for the fractional Laplacian systems with positive-negative mixed powers
Zhongxue Lü, Mengjia Niu, Yuanyuan Shen et al.
J. Korean Math. Soc. 2024; 61: 445-459
In this paper, by establishing the direct method of moving planes for the fractional Laplacian system with positive-negative mixed powers, we obtain the radial symmetry and monotonicity of the positive solutions for the fractional Laplacian systems with positive-negative mixed powers in the whole space. We also give two special cases.
Keywords: The fractional Laplacian, positive-negative mixed powers, method of moving planes, radial symmetry, monotonicity
01 May 2024
Continuity of the maximal commutators in Sobolev spaces
Xixi Jiang, Feng Liu
J. Korean Math. Soc. 2024; 61: 461-494
We prove the Sobolev continuity of maximal commutator and its fractional variant with Lipschitz symbols, both in the global and local cases. The main result in global case answers a question originally posed by Liu and Wang in \cite{LW}.
Keywords: Maximal commutator, fractional maximal commutator, Sobolev space, continuity
01 May 2024
Singular hyperbolicity of $C^1$ generic three dimensional vector fields
Manseob Lee
J. Korean Math. Soc. 2024; 61: 495-513
In the paper, we show that for a generic $C^1$ vector field $X$ on a closed three dimensional manifold $M$, any isolated transitive set of $X$ is singular hyperbolic. It is a partial answer of the conjecture in \cite{MP}.
Keywords: Transitive set, generic, local star, Lyapunove stable, singular hyperbolic
01 May 2024
Reduction of abelian varieties and curves
Moshe Jarden, Aharon Razon
J. Korean Math. Soc. 2024; 61: 515-545
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 $\Quot(R)=K$ that contains $R_0$ and embed $\Spec(R_0)$ into $\Spec(R)$. Then, we prove that key properties of abelian varieties and smooth geometrically integral projective curves over $K$ are preserved under reduction modulo $\frp$ for ``almost all'' $\frp\in\Spec(R_0)$.
Keywords: Reduction, isotriviality, abelian variety, moduli space
01 May 2024
Totally real and complex subspaces of a right quaternionic vector space with a Hermitian form of signature $(n,1)$
Sungwoon Kim
J. Korean Math. Soc. 2024; 61: 547-564
We study totally real and complex subsets of a right quaternionic vector space of dimension $n+1$ with a Hermitian form of signature $(n,1)$ and extend these notions to right quaternionic projective space. Then we give a necessary and sufficient condition for a subset of a right quaternionic projective space to be totally real or complex in terms of the quaternionic Hermitian triple product. As an application, we show that the limit set of a non-elementary quaternionic Kleinian group $\Gamma$ is totally real (resp.~commutative) with respect to the quaternionic Hermitian triple product if and only if $\Gamma$ leaves a real (resp.~complex) hyperbolic subspace invariant.
Keywords: Quaternionic vector space, quaternionic hyperbolic space, quaternionic Hermitian triple product
01 May 2024
Collective behaviors of second-order nonlinear consensus models with a bonding force
Hyunjin Ahn, Junhyeok Byeon, Seung-Yeal Ha et al.
J. Korean Math. Soc. 2024; 61: 565-602
We study the collective behaviors of two second-order nonlinear consensus models with a bonding force, namely the Kuramoto model and the Cucker-Smale model with inter-particle bonding force. The proposed models contain feedback control terms which induce collision avoidance and emergent consensus dynamics in a suitable framework. Through the cooperative interplays between feedback controls, initial state configuration tends to an ordered configuration asymptotically under suitable frameworks which are formulated in terms of system parameters and initial configurations. For a two-particle system on the real line, we show that the relative state tends to the preassigned value asymptotically, and we also provide several numerical examples to analyze the possible nonlinear dynamics of the proposed models, and compare them with analytical results.
Keywords: Barbalat's lemma, bonding control, complete synchronization, Cucker-Smale model, flocking, Kuramoto model
01 May 2024
\boldmath$(\mathcal{V},\mathcal{W},\mathcal{Y},\mathcal{X})$-Gorenstein complexes
yanjie Li, Renyu Zhao
J. Korean Math. Soc. 2024; 61: 603-620
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 $\bm{M}$ is $(\mathcal{V, W, Y, X})$-Gorenstein if and only if the module in each degree of $\bm{M}$ is $(\mathcal{V, W, Y, X})$-Gorenstein and the total Hom complexs Hom$_R(\bm{Y},\bm{M})$, Hom$_R(\bm{M},\bm{X})$ are exact for any $\bm{Y}\in\widetilde{\mathcal{Y}}$ and any $\bm{X}\in\widetilde{\mathcal{X}}$. Many known results are recovered, and some new cases are also naturally generated.
Keywords: Gorenstein module, Gorenstein complex
May 1, 2024Current Issue Vol. 61 No. 3
-
2024-05-01
Self-pair homotopy equivalences related to co-variant functors
Ho Won Choi, Kee Young Lee, Hye Seon ShinAbstract : The category of pairs is the category whose objects are maps between two based spaces and morphisms are pair-maps from one object to another object. To study the self-homotopy equivalences in the category of pairs, we use covariant functors from the category of pairs to the group category whose objects are groups and morphisms are group homomorphisms. We introduce specific subgroups of groups of self-pair homotopy equivalences and put these groups together into certain sequences. We investigate properties of these sequences, in particular, the exactness and split. We apply the results to two special functors, homotopy and homology functors and determine the suggested several subgroups of groups of self-pair homotopy equivalences.
-
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.
-
2024-05-01
Properties of positive solutions for the fractional Laplacian systems with positive-negative mixed powers
Zhongxue Lü, Mengjia Niu, Yuanyuan Shen et al.Abstract : In this paper, by establishing the direct method of moving planes for the fractional Laplacian system with positive-negative mixed powers, we obtain the radial symmetry and monotonicity of the positive solutions for the fractional Laplacian systems with positive-negative mixed powers in the whole space. We also give two special cases.
-
2024-05-01
Continuity of the maximal commutators in Sobolev spaces
Xixi Jiang, Feng LiuAbstract : We prove the Sobolev continuity of maximal commutator and its fractional variant with Lipschitz symbols, both in the global and local cases. The main result in global case answers a question originally posed by Liu and Wang in \cite{LW}.
-
2024-05-01
Singular hyperbolicity of $C^1$ generic three dimensional vector fields
Manseob LeeAbstract : In the paper, we show that for a generic $C^1$ vector field $X$ on a closed three dimensional manifold $M$, any isolated transitive set of $X$ is singular hyperbolic. It is a partial answer of the conjecture in \cite{MP}.
-
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 $\Quot(R)=K$ that contains $R_0$ and embed $\Spec(R_0)$ into $\Spec(R)$. Then, we prove that key properties of abelian varieties and smooth geometrically integral projective curves over $K$ are preserved under reduction modulo $\frp$ for ``almost all'' $\frp\in\Spec(R_0)$.
-
2024-05-01
Totally real and complex subspaces of a right quaternionic vector space with a Hermitian form of signature $(n,1)$
Sungwoon KimAbstract : We study totally real and complex subsets of a right quaternionic vector space of dimension $n+1$ with a Hermitian form of signature $(n,1)$ and extend these notions to right quaternionic projective space. Then we give a necessary and sufficient condition for a subset of a right quaternionic projective space to be totally real or complex in terms of the quaternionic Hermitian triple product. As an application, we show that the limit set of a non-elementary quaternionic Kleinian group $\Gamma$ is totally real (resp.~commutative) with respect to the quaternionic Hermitian triple product if and only if $\Gamma$ leaves a real (resp.~complex) hyperbolic subspace invariant.
-
2024-05-01
Collective behaviors of second-order nonlinear consensus models with a bonding force
Hyunjin Ahn
, Junhyeok Byeon, Seung-Yeal Ha et al.Abstract : We study the collective behaviors of two second-order nonlinear consensus models with a bonding force, namely the Kuramoto model and the Cucker-Smale model with inter-particle bonding force. The proposed models contain feedback control terms which induce collision avoidance and emergent consensus dynamics in a suitable framework. Through the cooperative interplays between feedback controls, initial state configuration tends to an ordered configuration asymptotically under suitable frameworks which are formulated in terms of system parameters and initial configurations. For a two-particle system on the real line, we show that the relative state tends to the preassigned value asymptotically, and we also provide several numerical examples to analyze the possible nonlinear dynamics of the proposed models, and compare them with analytical results.
-
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 $\bm{M}$ is $(\mathcal{V, W, Y, X})$-Gorenstein if and only if the module in each degree of $\bm{M}$ is $(\mathcal{V, W, Y, X})$-Gorenstein and the total Hom complexs Hom$_R(\bm{Y},\bm{M})$, Hom$_R(\bm{M},\bm{X})$ are exact for any $\bm{Y}\in\widetilde{\mathcal{Y}}$ and any $\bm{X}\in\widetilde{\mathcal{X}}$. Many known results are recovered, and some new cases are also naturally generated.
May, 2024 | Volume 61, No. 3
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