Abstract : 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. We use covariant functors from the category of based topological spaces and based maps to the category of groups and homomorphisms to study the self-homotopy equivalences in the category of pairs. We introduce specific subgroups of groups of self-pair homotopy equivalences and put these groups together into certain sequences. We investigate properties of the sequence, in particular, the exactness. We apply the results to two special functors, homotopy and homology functors to determine the suggested several subgroups of groups of self-pair homotopy equivalences.
Abstract : The matrix completion problem is to predict missing entries of a data matrix using the low-rank approximation of the observed entries. Consequently, typical matrix completion problem approaches employ thresholding in 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 an accurate and robust estimation of the data matrix. Moreover, 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 show that the proposed algorithm yields a more accurate estimation with a faster execution than other matrix completion algorithms in image inpainting problems.
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 four special cases.
Abstract : 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}.
Abstract : 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 [13].
Abstract : Consider a noetherian domain R0 with quotient field K0. Let K be a finitely generated regular transcendental field extension of K0. We con- struct a noetherian domain R with Quot(R) = K that contains R0 and embed Spec(R0) 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 p for "almost all" p 2 Spec(R0).
Abstract : 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 is totally real (resp. commutative) with respect to the quaternionic Hermitian triple product if and only if it leaves a real (resp. complex) hyperbolic subspace invariant.
Abstract : 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.
Abstract : In this paper, we are devoted to focus on studying a forth order curve flow for a smooth closed curve in centro-affine geometry. Firstly, a new evolutionary equation about this curve flow is proposed. Then the related geometric quantities and some meaningful conclusions are obtained through the equation. Next, we obtain finite order differential inequalities for energy by applying interpolation inequalities, Cauchy Schwartz inequalities, etc. After using a completely new symbolic expression, the n-order differential inequality for energy is considered. Finally, by the means of energy estimation, we prove that the forth order curve flow has a smooth solution all the time for any smooth closed initial curve.
Abstract : We consider the following strongly damped wave equation on $\mathbb{R}^3$ with memory $$ u_{tt} - \alpha \Delta u_{t} - \beta \Delta u +\lambda u - \int_{0}^\infty \kappa'( s) \Delta u(t-s)ds+ f(x,u) +g(x,u_t)=h, $$ where a quite general memory kernel and the nonlinearity $f$ exhibit a critical growth. Existence, uniqueness and continuous dependence results are provided as well as the existence of regular global and exponential attractors of finite fractal dimension.
Cédric Bonnafé, Alessandra Sarti
J. Korean Math. Soc. 2023; 60(4): 695-743
https://doi.org/10.4134/JKMS.j220014
Mohamed Boucetta, Abdelmounaim Chakkar
J. Korean Math. Soc. 2022; 59(4): 651-684
https://doi.org/10.4134/JKMS.j210460
Jangwon Ju
J. Korean Math. Soc. 2023; 60(5): 931-957
https://doi.org/10.4134/JKMS.j220231
Sangtae Jeong
J. Korean Math. Soc. 2023; 60(1): 1-32
https://doi.org/10.4134/JKMS.j210494
Jangwon Ju
J. Korean Math. Soc. 2023; 60(5): 931-957
https://doi.org/10.4134/JKMS.j220231
Cédric Bonnafé, Alessandra Sarti
J. Korean Math. Soc. 2023; 60(4): 695-743
https://doi.org/10.4134/JKMS.j220014
Sangtae Jeong
J. Korean Math. Soc. 2023; 60(1): 1-32
https://doi.org/10.4134/JKMS.j210494
Mohamed Boucetta, Abdelmounaim Chakkar
J. Korean Math. Soc. 2022; 59(4): 651-684
https://doi.org/10.4134/JKMS.j210460
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