Journal of the
Korean Mathematical Society

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

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  • 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

    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

    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.

  • 2022-01-01

    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.

  • 2023-09-01

    Almost universal sums of triangular numbers with one exception

    Jangwon Ju

    Abstract : For an arbitrary integer $x$, an integer of the form $T(x)\!=\!\frac{x^2+x}{2}$ is called a triangular number. Let $\alpha_1,\dots,\alpha_k$ be positive integers. A sum $\Delta_{\alpha_1,\dots,\alpha_k}(x_1,\dots,x_k)=\alpha_1 T(x_1)+\cdots+\alpha_k T(x_k)$ of triangular numbers is said to be {\it almost universal with one exception} if the Diophantine equation $\Delta_{\alpha_1,\dots,\alpha_k}(x_1,\dots,x_k)=n$ has an integer solution $(x_1,\dots,x_k)\in\mathbb{Z}^k$ for any nonnegative integer $n$ except a single one. In this article, we classify all almost universal sums of triangular numbers with one exception. Furthermore, we provide an effective criterion on almost universality with one exception of an arbitrary sum of triangular numbers, which is a generalization of ``15-theorem" of Conway, Miller, and Schneeberger.

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  • 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

    The interior gradient estimate for a class of mixed Hessian curvature equations

    Jundong Zhou

    Abstract : In this paper, we are concerned with a class of mixed Hessian curvature equations with non-degeneration. By using the maximum principle and constructing an auxiliary function, we obtain the interior gradient estimate of $(k-1)$-admissible solutions.

  • 2023-07-01

    Complex reflection groups and K3 surfaces II. The groups ${\boldsymbol{G_{29}}}$, ${\boldsymbol{G_{30}}}$ and ${\boldsymbol{G_{31}}}$

    Cédric Bonnafé, Alessandra Sarti

    Abstract : We study some K3 surfaces obtained as minimal resolutions of quotients of subgroups of special reflection groups. Some of these were already studied in a previous paper by W.~Barth and the second author. We give here an easy proof that these are K3 surfaces, give equations in weighted projective space and describe their geometry.

  • 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.

  • 2023-01-01

    Minimal polynomial dynamics on the $p$-adic integers

    Sangtae Jeong

    Abstract : In this paper, we present a method of characterizing minimal polynomials on the ring ${\mathbf Z}_p$ of $p$-adic integers in terms of their coefficients for an arbitrary prime $p$. We first revisit and provide alternative proofs of the known minimality criteria given by Larin [11] for $p=2$ and Durand and Paccaut [6] for $p=3$, and then we show that for any prime $p\geq 5,$ the proposed method enables us to classify all possible minimal polynomials on ${\mathbf Z}_p$ in terms of their coefficients, provided that two prescribed prerequisites for minimality are satisfied.

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November, 2023
Vol.60 No.6

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