Journal of the
Korean Mathematical Society
JKMS

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

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  • 2024-03-01

    Shadowing property for ADMM flows

    Yoon Mo Jung , Bomi Shin , Sangwoon Yun

    Abstract : There have been numerous studies on the characteristics of the solutions of ordinary differential equations for optimization methods, including gradient descent methods and alternating direction methods of multipliers. To investigate computer simulation of ODE solutions, we need to trace pseudo-orbits by real orbits and it is called shadowing property in dynamics. In this paper, we demonstrate that the flow induced by the alternating direction methods of multipliers (ADMM) for a $C^2$ strongly convex objective function has the eventual shadowing property. For the converse, we partially answer that convexity with the eventual shadowing property guarantees a unique minimizer. In contrast, we show that the flow generated by a second-order ODE, which is related to the accelerated version of ADMM, does not have the eventual shadowing property.

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  • 2024-03-01

    On the linear independence measures of logarithms of rational numbers. II

    Abderraouf Bouchelaghem, Yuxin He, Yuanhang Li, Qiang Wu

    Abstract : In this paper, we give a general method to compute the linear independence measure of $1, \log(1-1/r),\log(1+1/s)$ for infinitely many integers $r$ and $s$. We also give improvements for the special cases when $r=s$, for example, $\nu(1, \log 3/4, \log 5/4) \leq 9.197$.

  • 2024-03-01

    Positive solutions to discrete harmonic functions in unbounded cylinders

    Fengwen Han, Lidan Wang

    Abstract : In this paper, we study the positive solutions to a discrete harmonic function for a random walk satisfying finite range and ellipticity conditions, killed at the boundary of an unbounded cylinder in $\mathbb{Z}^d$. We first prove the existence and uniqueness of positive solutions, and then establish that all the positive solutions are generated by two special solutions, which are exponential growth at one end and exponential decay at the other. Our method is based on maximum principle and a Harnack type inequality.

  • 2024-03-01

    Remarks on Ulrich bundles of small ranks over quartic fourfolds

    Yeongrak Kim

    Abstract : In this paper, we investigate a few strategies to construct Ulrich bundles of small ranks over smooth fourfolds in $\mathbb{P}^5$, with a focus on the case of special quartic fourfolds. First, we give a necessary condition for Ulrich bundles over a very general quartic fourfold in terms of the rank and the Chern classes. Second, we give an equivalent condition for Pfaffian fourfolds in every degree in terms of arithmetically Gorenstein surfaces therein. Finally, we design a computer-based experiment to look for Ulrich bundles of small rank over special quartic fourfolds via deformation theory. This experiment gives a construction of numerically Ulrich sheaf of rank $4$ over a random quartic fourfold containing a del Pezzo surface of degree $5$.

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  • 2024-03-01

    A note on Sobolev type trace inequalities for $s$-harmonic extensions

    Yongrui Tang, Shujuan Zhou

    Abstract : In this paper, apply the regularities of the fractional Poisson kernels, we establish the Sobolev type trace inequalities of homogeneous Besov spaces, which are invariant under the conformal transforms. Also, by the aid of the Carleson measure characterizations of Q type spaces, the local version of Sobolev trace inequalities are obtained.

  • 2024-03-01

    Conics in quintic del Pezzo varieties

    Kiryong Chung, Sanghyeon Lee

    Abstract : The smooth quintic del Pezzo variety $Y$ is well-known to be obtained as a linear sections of the Grassmannian variety $\mathrm{Gr}(2,5)$ under the Pl\"ucker embedding into $\mathbb{P}^{9}$. Through a local computation, we show the Hilbert scheme of conics in $Y$ for $\text{dim} Y \ge 3$ can be obtained from a certain Grassmannian bundle by a single blowing-up/down transformation.

  • 2024-03-01

    Real hypersurfaces in the complex hyperbolic quadric with cyclic parallel structure Jacobi operator

    Jin Hong Kim, Hyunjin Lee, Young Jin Suh

    Abstract : Let $M$ be a real hypersurface in the complex hyperbolic quadric~${Q^{m}}^{*}$, $m \geq 3$. The Riemannian curvature tensor field~$R$ of~$M$ allows us to define a symmetric Jacobi operator with respect to the Reeb vector field~$\xi$, which is called the structure Jacobi operator~$R_{\xi} = R(\, \cdot \, , \xi) \xi \in \text{End}(TM)$. On the other hand, in~\cite{Semm03}, Semmelmann showed that the cyclic parallelism is equivalent to the Killing property regarding any symmetric tensor. Motivated by his result above, in this paper we consider the cyclic parallelism of the structure Jacobi operator~$R_{\xi}$ for a real hypersurface~$M$ in the complex hyperbolic quadric~${Q^{m}}^{*}$. Furthermore, we give a complete classification of Hopf real hypersurfaces in ${Q^{m}}^{*}$ with such a property.

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March, 2024
Vol.61 No.2

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