# Journal of theKorean Mathematical SocietyJKMS

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

• ### 2021-09-01

#### The stability of weak solutions to an anisotropic polytropic infiltration equation

Huashui Zhan

Abstract : This paper considers an anisotropic polytropic infiltration equation with a source term $${u_t}= \sum_{i=1}^N\frac{\partial }{\partial x_i}\left(a_i(x)|u|^{\alpha_i}{\left| {u_{x_i}} \right|^{p_i-2}}u_{x_i}\right)+f(x,t,u),$$ where $p_i>1$, $\alpha_i >0$, $a_i(x)\geq 0$. The existence of weak solution is proved by parabolically regularized method. Based on local integrability $u_{x_i}\in W^{1,p_i}_{loc}(\Omega)$, the stability of weak solutions is proved without boundary value condition by the weak characteristic function method. One of the essential characteristics of an anisotropic equation different from an isotropic equation is found originally.

• ### 2021-01-01

#### Dagger-sharp Tits octagons

Bernhard M\"uhlherr, Richard M. Weiss

Abstract : The spherical buildings associated with absolutely simple algebraic groups of relative rank~$2$ are all Moufang polygons. Tits polygons are a more general class of geometric structures that includes Moufang polygons as a special case. Dagger-sharp Tits $n$-gons exist only for $n=3$, $4$, $6$ and~$8$. Moufang octagons were classified by Tits. We show here that there are no dagger-sharp Tits octagons that are not Moufang. As part of the proof it is shown that the same conclusion holds for a certain class of dagger-sharp Tits quadrangles.

• ### 2021-01-01

#### Singular minimal translation graphs in Euclidean spaces

Muh{$\dot{\textsc i}$}tt{$\dot{\textsc i}$}n Evren Aydin, Ayla Erdur, Mahmut Erg\"ut

Abstract : In this paper, we consider the problem of finding the hypersurface $M^{n}$ in the Euclidean $\left( n+1\right)$-space $\mathbb{R}^{n+1}$ that satisfies an equation of mean curvature type, called singular minimal hypersurface equation. Such an equation physically characterizes the surfaces in the upper halfspace $\mathbb{R}_{+}^{3}\left( \mathbf{u} \right)$ with lowest gravity center, for a fixed unit vector $\mathbf{u}\in \mathbb{R}^{3}$. We first state that a singular minimal cylinder $M^{n}$ in $\mathbb{R}^{n+1}$ is either a hyperplane or a $\alpha$-catenary cylinder. It is also shown that this result remains true when $M^{n}$ is a translation hypersurface and $\mathbf{u}$ is a horizantal vector. As a further application, we prove that a singular minimal translation graph in $\mathbb{R }^{3}$ of the form $z=f(x)+g(y+cx),$ $c\in \mathbb{R-\{}0\},$ with respect to a certain horizantal vector $\mathbf{u}$ is either a plane or a $\alpha$- catenary cylinder.

• ### 2021-01-01

#### Weighted $L^p$-boundedness of singular integrals with rough kernel associated to surfaces

Ronghui Liu, Huoxiong Wu

Abstract : In this paper, we prove weighted norm inequalities for rough singular integrals along surfaces with radial kernels $h$ and sphere kernels $\Omega$ by assuming $h\in{\triangle}_{\gamma}(\mathbb{R}_+)$ and $\Omega\in\mathcal{WG}_\beta({\rm S}^{n-1})$ for some $\gamma>1$ and $\beta>1$. Here $\Omega\in\mathcal{WG}_\beta({\rm S}^{n-1})$ denotes the variant of Grafakos-Stefanov type size conditions on the unit sphere. Our results essentially improve and extend the previous weighted results for the rough singular integrals and the corresponding maximal truncated operators.

• ### 2020-07-01

#### Properties of operator matrices

Il Ju An, Eungil Ko, Ji Eun Lee

Abstract : Let ${\mathcal S}$ be the collection of the operator matrices $\left(\begin{smallmatrix} A & C \cr Z & B\end{smallmatrix}\right)$ where the range of $C$ is closed. In this paper, we study the properties of operator matrices in the class ${\mathcal S}$. We first explore various local spectral relations, that is, the property $(\beta)$, decomposable, and the property $(C)$ between the operator matrices in the class $\mathcal{S}$ and their component operators. Moreover, we investigate Weyl and Browder type spectra of operator matrices in the class $\mathcal S$, and as some applications, we provide the conditions for such operator matrices to satisfy $a$-Weyl's theorem and $a$-Browder's theorem, respectively.

• ### 2021-09-01

#### A sufficient condition for a toric weak Fano 4-fold to be deformed to a Fano manifold

Hiroshi Sato

Abstract : In this paper, we introduce the notion of toric special weak Fano manifolds, which have only special primitive crepant contractions. We study their structure, and in particular completely classify smooth toric special weak Fano $4$-folds. As a result, we can confirm that almost every smooth toric special weak Fano $4$-fold is a weakened Fano manifold, that is, a weak Fano manifold which can be deformed to a Fano manifold.

• ### 2021-07-01

#### Prime knots with arc index 12 up to 16 crossings

Gyo Taek Jin, Hyuntae Kim, Seungwoo Lee, Hun Joo Myung

Abstract : We obtain the list of prime knots with arc index 12 up to 16 crossings and their minimal grid diagrams. This is a continuation of the works \cite{Jin2006} and \cite{Jin2011} in which Cromwell matrices were generated to obtain minimal grid diagrams of all prime knots up to arc index 11. We provide minimal grid diagrams of the prime alternating knots with arc index 12. They are the 10 crossing prime alternating knots. The full list of 19,513 prime knots of arc index 12 up to 16 crossings and their minimal grid diagrams can be found in the arXiv~\cite{Jin2020}.

• ### 2021-07-01

#### A new classification of real hypersurfaces with Reeb parallel structure Jacobi operator in the complex quadric

Hyunjin Lee, Young Jin Suh

Abstract : In this paper, first we introduce the full expression of the Riemannian curvature tensor of a real hypersurface $M$ in the complex quadric~$Q^{m}$ from the equation of Gauss and some important formulas for the structure Jacobi operator ~$R_{\xi}$ and its derivatives $\nabla R_{\xi}$ under the Levi-Civita connection $\nabla$ of $M$. Next we give a complete classification of Hopf real hypersurfaces with Reeb parallel structure Jacobi operator, $\nabla_{\xi}R_{\xi}=0$, in the complex quadric $Q^{m}$ for $m \geq 3$. In addition, we also consider a new notion of $\mathcal C$-parallel structure Jacobi operator of $M$ and give a nonexistence theorem for Hopf real hypersurfaces with $\mathcal C$-parallel structure Jacobi operator in $Q^{m}$, for $m \geq 3$.

• ### 2021-07-01

#### Asymptotics for an extended inverse Markovian Hawkes process

Youngsoo Seol

Abstract : Hawkes process is a self-exciting simple point process with clustering effect whose jump rate depends on its entire past history and has been widely applied in insurance, finance, queueing theory, statistics, and many other fields. Seol~\cite{Seol5} proposed the inverse Markovian Hawkes processes and studied some asymptotic behaviors. In this paper, we consider an extended inverse Markovian Hawkes process which combines a Markovian Hawkes process and inverse Markovian Hawkes process with features of several existing models of self-exciting processes. We study the limit theorems for an extended inverse Markovian Hawkes process. In particular, we obtain a law of large number and central limit theorems.

• ### 2021-07-01

#### Normal complex symmetric weighted composition operators on the Hardy space

Hang Zhou, Ze-Hua Zhou

Abstract : In this paper, we investigate the normal and complex symmetric weighted composition operators $W_{\psi,\varphi}$ on the Hardy space $H^2(\mathbb{D})$. Firstly, we give the explicit conditions of weighted composition operators to be normal and complex symmetric with respect to conjugations $\mathcal{C}_1$ and $\mathcal{C}_2$ on $H^2(\mathbb{D})$, respectively. Moreover, we particularly investigate the weighted composition operators $W_{\psi,\varphi}$ on $H^2(\mathbb{D})$ which are normal and complex symmetric with respect to conjugations $\mathcal{J}$, $\mathcal{C}_1$ and $\mathcal{C}_2$, respectively, when $\varphi$ has an interior fixed point, $\varphi$ is of hyperbolic type or parabolic type.

## Current Issue

• ### Erratum to Static and related critical spaces with harmonic curvature and three Ricci eigenvalues'' [J. Korean Math. Soc. 57 (2020), No. 6, pp. 1435--1449]

Jongsu Kim

J. Korean Math. Soc. 2022; 59(3): 649-650
https://doi.org/10.4134/JKMS.j210761

• ### Radius constants for functions associated with a limacon domain

Nak Eun Cho, Anbhu Swaminathan, Lateef Ahmad Wani

J. Korean Math. Soc. 2022; 59(2): 353-365
https://doi.org/10.4134/JKMS.j210246

• ### Inclusion relations and radius problems for a subclass of starlike functions

Prachi Gupta, Sumit Nagpal, Vaithiyanathan Ravichandran

J. Korean Math. Soc. 2021; 58(5): 1147-1180
https://doi.org/10.4134/JKMS.j200465

• ### On the scaled inverse of $(x^i-x^j)$ modulo cyclotomic polynomial of the form $\Phi_{p^s}(x)$ or $\Phi_{p^s q^t}(x)$

Jung Hee Cheon, Dongwoo Kim, Duhyeong Kim, Keewoo Lee

J. Korean Math. Soc. 2022; 59(3): 621-634
https://doi.org/10.4134/JKMS.j210446

• ### Erratum to Static and related critical spaces with harmonic curvature and three Ricci eigenvalues'' [J. Korean Math. Soc. 57 (2020), No. 6, pp. 1435--1449]

Jongsu Kim

J. Korean Math. Soc. 2022; 59(3): 649-650
https://doi.org/10.4134/JKMS.j210761

• ### On the scaled inverse of $(x^i-x^j)$ modulo cyclotomic polynomial of the form $\Phi_{p^s}(x)$ or $\Phi_{p^s q^t}(x)$

Jung Hee Cheon, Dongwoo Kim, Duhyeong Kim, Keewoo Lee

J. Korean Math. Soc. 2022; 59(3): 621-634
https://doi.org/10.4134/JKMS.j210446

• ### Constructions of Segal algebras in $L^1(G)$ of LCA groups $G$ in which a generalized Poisson summation formula holds

Jyunji Inoue, Sin-Ei Takahasi

J. Korean Math. Soc. 2022; 59(2): 367-377
https://doi.org/10.4134/JKMS.j210290

• ### Every abelian group is the class group of a ring of Krull type

Gyu Whan Chang

J. Korean Math. Soc. 2021; 58(1): 149-171
https://doi.org/10.4134/JKMS.j200010