Journal of the
Korean Mathematical Society
JKMS
ISSN(Print) 0304-9914
ISSN(Online) 2234-3008
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2022-01-01
Synchronized components of a subshift
Manouchehr ShahamatAbstract : We introduce the notion of a minimal synchronizing word; that is a synchronizing word whose proper subwords are not synchronized. This has been used to give a new shorter proof for a theorem in [6]. Also, the common synchronized components of a subshift and its derived set have been characterized.
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2022-01-01
Parabolic quaternionic Monge-Amp\`{e}re equation on compact manifolds with a flat hyperK\"ahler metric
Jiaogen ZhangAbstract : The quaternionic Calabi conjecture was introduced by Alesker-Verbitsky, analogous to the K\"ahler case which was raised by Calabi. On a compact connected hypercomplex manifold, when there exists a flat hyperK\"ahler metric which is compatible with the underlying hypercomplex structure, we will consider the parabolic quaternionic Monge-Amp\`{e}re equation. Our goal is to prove the long time existence and $C^{\infty}$ convergence for normalized solutions as $t\rightarrow\infty$. As a consequence, we show that the limit function is exactly the solution of quaternionic Monge-Amp\`{e}re equation, this gives a parabolic proof for the quaternionic Calabi conjecture in this special setting.
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2022-03-01
Margin-based generalization for classifications with input noise
Hi Jun Choe, Hayeong Koh, Jimin LeeAbstract : Although machine learning shows state-of-the-art perfor\-man\-ce in a variety of fields, it is short a theoretical understanding of how machine learning works. Recently, theoretical approaches are actively being studied, and there are results for one of them, margin and its distribution. In this paper, especially we focused on the role of margin in the perturbations of inputs and parameters. We show a generalization bound for two cases, a linear model for binary classification and neural networks for multi-classification, when the inputs have normal distributed random noises. The additional generalization term caused by random noises is related to margin and exponentially inversely proportional to the noise level for binary classification. And in neural networks, the additional generalization term depends on (input dimension) $\times$ (norms of input and weights). For these results, we used the PAC-Bayesian framework. This paper is considering random noises and margin together, and it will be helpful to a better understanding of model sensitivity and the construction of robust generalization.
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2022-03-01
Hardy type estimates for Riesz transforms associated with Schr\"{o}dinger operators on the Heisenberg group
Chunfang GaoAbstract : Let $\mathbb{H}^{n}$ be the Heisenberg group and $Q=2n+2$ be its homogeneous dimension. Let $\mathcal{L}=-\Delta_{\mathbb{H}^{n}}+V$ be the Schr\"{o}dinger operator on $\mathbb{H}^{n}$, where $\Delta_{\mathbb{H}^{n}}$ is the sub-Laplacian and the nonnegative potential $V$ belongs to the reverse H\"{o}lder class $B_{q_{1}}$ for $q_{1}\geq Q/2$. Let ${H_{\mathcal{L}}^{p}(\mathbb{H}^{n})}$ be the Hardy space associated with the Schr\"{o}dinger operator $\mathcal{L}$ for $Q/(Q+\delta_{0})
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2022-01-01
Regularity of the generalized Poisson operator
Pengtao Li, Zhiyong Wang, Kai ZhaoAbstract : Let $L=-\Delta+V$ be a Schr\"{o}dinger operator, where the potential $V$ belongs to the reverse H\"{o}lder class. In this paper, by the subordinative formula, we investigate the generalized Poisson operator $P^{L}_{t,\sigma}$, $0<\sigma<1$, associated with $L$. We estimate the gradient and the time-fractional derivatives of the kernel of $P^{L}_{t,\sigma}$, respectively. As an application, we establish a Carleson measure characterization of the Campanato type space $\mathcal{C}^{\gamma}_{L}(\mathbb{R}^{n})$ via $P^{L}_{t,\sigma}$.
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2022-01-01
Practical FHE parameters against lattice attacks
Jung Hee Cheon, Yongha Son, Donggeon YheeAbstract : We give secure parameter suggestions to use sparse secret vectors in $\mathsf{LWE}$ based encryption schemes. This should replace existing security parameters, because homomorphic encryption (HE) schemes use quite different variables from the existing parameters. In particular, HE schemes using sparse secrets should be supported by experimental analysis, here we summarize existing attacks to be considered and security levels for each attacks. Based on the analysis and experiments, we compute optimal scaling factors for CKKS.
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2022-01-01
Global nonexistence for the wave equation with boundary variable exponent nonlinearities
Tae Gab Ha, Sun-Hye ParkAbstract : This paper deals with a nonlinear wave equation with boundary damping and source terms of variable exponent nonlinearities. This work is devoted to prove a global nonexistence of solutions for a nonlinear wave equation with nonnegative initial energy as well as negative initial energy.
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2022-01-01
Construction for self-orthogonal codes over a certain non-chain Frobenius ring
Boran KimAbstract : We present construction methods for free self-orthogonal (self-dual or Type II) codes over $\mathbb Z_4[v]/\langle v^2+2v \rangle$ which is one of the finite commutative local non-chain Frobenius rings of order $16$. By considering their Gray images on $\mathbb Z_4$, we give a construct method for a code over $\mathbb Z_4$. We have some new and optimal codes over $\mathbb Z_4$ with respect to the minimum Lee weight or minimum Euclidean weight.
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2022-01-01
Gorenstein sequences of high socle degrees
Jung Pil Park, Yong-Su ShinAbstract : In [4], the authors showed that if an $h$-vector $(h_0,h_1,\dots,h_e)$ with $h_1=4e-4$ and $h_i\le h_1$ is a Gorenstein sequence, then $h_1=h_i$ for every $1\le i\le e-1$ and $e\ge 6$. In this paper, we show that if an $h$-vector $(h_0,h_1,\dots,h_e)$ with $h_1=4e-4$, $h_2=4e-3$, and $h_i\le h_2$ is a Gorenstein sequence, then $h_2=h_i$ for every $2\le i\le e-2$ and $e\ge 7$. We also propose an open question that if an $h$-vector $(h_0,h_1,\dots,h_e)$ with $h_1=4e-4$, $4e-3<h_2\le {(h_1)_{(1)}}| ^{+1}_{+1}$, and $h_2\le h_i$ is a Gorenstein sequence, then $h_2=h_i$ for every $2\le i\le e-2$ and $e\ge 6$.
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2022-01-01
On cylindrical smooth rational Fano fourfolds
Nguyen Thi Anh Hang, Michael Hoff, Hoang Le TruongAbstract : We construct new families of smooth Fano fourfolds with Picard rank $1$ which contain open $\mathbb A^1$-cylinders, that is, Zariski open subsets of the form $Z \times \mathbb A^1$, where $Z$ is a quasiprojective variety. In particular, we show that every Mukai fourfold of genus $8$ is cylindrical and there exists a family of cylindrical Gushel-Mukai fourfolds.
Most Read
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Synchronized components of a subshift
Manouchehr Shahamat
J. Korean Math. Soc. 2022; 59(1): 1-12
https://doi.org/10.4134/JKMS.j200112 -
Regularity of the generalized Poisson operator
Pengtao Li, Zhiyong Wang, Kai Zhao
J. Korean Math. Soc. 2022; 59(1): 129-150
https://doi.org/10.4134/JKMS.j210224 -
Construction for self-orthogonal codes over a certain non-chain Frobenius ring
Boran Kim
J. Korean Math. Soc. 2022; 59(1): 193-204
https://doi.org/10.4134/JKMS.j210357 -
Parabolic quaternionic Monge-Amp\`{e}re equation on compact manifolds with a flat hyperK\"ahler metric
Jiaogen Zhang
J. Korean Math. Soc. 2022; 59(1): 13-33
https://doi.org/10.4134/JKMS.j200626
Most Downloaded
-
Synchronized components of a subshift
Manouchehr Shahamat
J. Korean Math. Soc. 2022; 59(1): 1-12
https://doi.org/10.4134/JKMS.j200112 -
Parabolic quaternionic Monge-Amp\`{e}re equation on compact manifolds with a flat hyperK\"ahler metric
Jiaogen Zhang
J. Korean Math. Soc. 2022; 59(1): 13-33
https://doi.org/10.4134/JKMS.j200626 -
Margin-based generalization for classifications with input noise
Hi Jun Choe, Hayeong Koh, Jimin Lee
J. Korean Math. Soc. 2022; 59(2): 217-233
https://doi.org/10.4134/JKMS.j200406 -
Hardy type estimates for Riesz transforms associated with Schr\"{o}dinger operators on the Heisenberg group
Chunfang Gao
J. Korean Math. Soc. 2022; 59(2): 235-254
https://doi.org/10.4134/JKMS.j200484
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