Journal of the
Korean Mathematical Society

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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  • 2022-01-01

    Synchronized components of a subshift

    Manouchehr Shahamat

    Abstract : 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.

  • 2022-01-01

    Regularity of the generalized Poisson operator

    Pengtao Li, Zhiyong Wang, Kai Zhao

    Abstract : 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 $P^{L}_{t,\sigma}$.

  • 2022-01-01

    Construction for self-orthogonal codes over a certain non-chain Frobenius ring

    Boran Kim

    Abstract : 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.

  • 2022-01-01

    Parabolic quaternionic Monge-Amp\`{e}re equation on compact manifolds with a flat hyperK\"ahler metric

    Jiaogen Zhang

    Abstract : 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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  • 2022-01-01

    Katok-Hasselblatt-kinematic expansive flows

    Hien Minh Huynh

    Abstract : In this paper we introduce a new notion of expansive flows, which is the combination of expansivity in the sense of Katok and Hasselblatt and kinematic expansivity, named KH-kinematic expansivity. We present new properties of several variations of expansivity. A new hierarchy of expansive flows is given.

  • 2022-01-01

    Global nonexistence for the wave equation with boundary variable exponent nonlinearities

    Tae Gab Ha, Sun-Hye Park

    Abstract : 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.

  • 2022-01-01

    On cylindrical smooth rational Fano fourfolds

    Nguyen Thi Anh Hang, Michael Hoff, Hoang Le Truong

    Abstract : 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.

  • 2022-01-01

    Practical FHE parameters against lattice attacks

    Jung Hee Cheon, Yongha Son, Donggeon Yhee

    Abstract : 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.

  • 2022-03-01

    Hardy type estimates for Riesz transforms associated with Schr\"{o}dinger operators on the Heisenberg group

    Chunfang Gao

    Abstract : 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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  • 2022-03-01

    Margin-based generalization for classifications with input noise

    Hi Jun Choe, Hayeong Koh, Jimin Lee

    Abstract : 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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July, 2022
Vol.59 No.4

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