Journal of the
Korean Mathematical Society
JKMS

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

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  • 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.

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  • 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.

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  • 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.

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  • 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 $m \geq 3$.

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  • 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.

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  • 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.

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July, 2022
Vol.59 No.4

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