# Journal of theKorean Mathematical SocietyJKMS

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

• ### 2022-11-01

#### On limit behaviours for Feller's unfair-fair-game and its related model

Jun An

Abstract : Feller introduced an unfair-fair-game in his famous book \cite{Feller-1968}. In this game, at each trial, player will win $2^k$ yuan with probability $p_k=1/2^kk(k+1)$, $k\in \mathbb{N}$, and zero yuan with probability $p_0=1-\sum_{k=1}^{\infty}p_k$. Because the expected gain is 1, player must pay one yuan as the entrance fee for each trial. Although this game seemed fair", Feller \cite{Feller-1945} proved that when the total trial number $n$ is large enough, player will loss $n$ yuan with its probability approximate 1. So it's an unfair" game. In this paper, we study in depth its convergence in probability, almost sure convergence and convergence in distribution. Furthermore, we try to take $2^k=m$ to reduce the values of random variables and their corresponding probabilities at the same time, thus a new probability model is introduced, which is called as the related model of Feller's unfair-fair-game. We find out that this new model follows a long-tailed distribution. We obtain its weak law of large numbers, strong law of large numbers and central limit theorem. These results show that their probability limit behaviours of these two models are quite different.

• ### 2024-05-01

#### Self-pair homotopy equivalences related to co-variant functors

Ho Won Choi, Kee Young Lee, Hye Seon Shin

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. To study the self-homotopy equivalences in the category of pairs, we use covariant functors from the category of pairs to the group category whose objects are groups and morphisms are group homomorphisms. We introduce specific subgroups of groups of self-pair homotopy equivalences and put these groups together into certain sequences. We investigate properties of these sequences, in particular, the exactness and split. We apply the results to two special functors, homotopy and homology functors and determine the suggested several subgroups of groups of self-pair homotopy equivalences.

• ### 2024-05-01

#### Properties of positive solutions for the fractional Laplacian systems with positive-negative mixed powers

Zhongxue Lü, Mengjia Niu, Yuanyuan Shen, Anjie Yuan

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 two special cases.

• ### 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 $5$.

• ### 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

#### 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.

• ### 2024-05-01

#### Smooth singular value thresholding algorithm for low-rank matrix completion problem

Geunseop Lee

Abstract : The matrix completion problem is to predict missing entries of a data matrix using the low-rank approximation of the observed entries. Typical approaches to matrix completion problem often rely on thresholding 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 more accurate and robust estimation of the data matrix. Furthermore, 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 demonstrate that the proposed algorithm yields a more accurate estimation with a faster execution than other matrix completion algorithms in image inpainting problems.

• ### 2024-05-01

#### \boldmath$(\mathcal{V},\mathcal{W},\mathcal{Y},\mathcal{X})$-Gorenstein complexes

yanjie Li, Renyu Zhao

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.

• ### 2024-05-01

#### Totally real and complex subspaces of a right quaternionic vector space with a Hermitian form of signature $(n,1)$

Sungwoon Kim

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 $\Gamma$ is totally real (resp.~commutative) with respect to the quaternionic Hermitian triple product if and only if $\Gamma$ leaves a real (resp.~complex) hyperbolic subspace invariant.

• ### 2023-03-01

#### The gradient flow equation of Rabinowitz action functional in a symplectization

Urs Frauenfelder

Abstract : Rabinowitz action functional is the Lagrange multiplier functional of the negative area functional to a constraint given by the mean value of a Hamiltonian. In this note we show that on a symplectization there is a one-to-one correspondence between gradient flow lines of Rabinowitz action functional and gradient flow lines of the restriction of the negative area functional to the constraint. In the appendix we explain the motivation behind this result. Namely that the restricted functional satisfies Chas-Sullivan additivity for concatenation of loops which the Rabinowitz action functional does in general not do.

## Current Issue

• ### New congruences for $\ell$-regular overpartitions

Ankita Jindal, Nabin K. Meher

J. Korean Math. Soc. 2022; 59(5): 945-962
https://doi.org/10.4134/JKMS.j220011

• ### Preresolving subcategories in extriangulated categories

Songsong Liu, Jiaqun Wei

J. Korean Math. Soc. 2023; 60(4): 877-905
https://doi.org/10.4134/JKMS.j220485

• ### Two-weighted conditions and characterizations for a class of multilinear fractional new maximal operators

Rui Li, Shuangping Tao

J. Korean Math. Soc. 2023; 60(1): 195-212
https://doi.org/10.4134/JKMS.j220250

• ### Generalized hexagons embedded in metasymplectic spaces

Sebastian Petit, Hendrik Van Maldeghem

J. Korean Math. Soc. 2023; 60(4): 907-929
https://doi.org/10.4134/JKMS.j220528

• ### Existence of solutions to a generalized self-dual Chern-Simons equation on finite graphs

Yuanyang Hu

J. Korean Math. Soc. 2024; 61(1): 133-147
https://doi.org/10.4134/JKMS.j230254

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

Abderraouf Bouchelaghem, Yuxin He, Yuanhang Li, Qiang Wu

J. Korean Math. Soc. 2024; 61(2): 293-307
https://doi.org/10.4134/JKMS.j230133

• ### Remarks on Ulrich bundles of small ranks over quartic fourfolds

Yeongrak Kim

J. Korean Math. Soc. 2024; 61(2): 279-291
https://doi.org/10.4134/JKMS.j230104

• ### Rigidity and nonexistence of Riemannian immersions in semi-Riemannian warped products via parabolicity

Railane Antonia, Henrique de Lima, Márcio Santos

J. Korean Math. Soc. 2024; 61(1): 41-63
https://doi.org/10.4134/JKMS.j220523