Journal of the
Korean Mathematical Society JKMS

Erratum ``The BFK-gluing formula for zeta-determinants and the value of relative zeta functions at zero" [J. Korean Math. Soc. 45 (2008), No. 5, pp. 1255--1274]

Properties of operator matrices

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.

Infinitely many solutions for fractional Schr\"odinger equation with superquadratic conditions or combined nonlinearities

Abstract : We obtain infinitely many solutions for a class of fractional Schr\"odinger equation, where the nonlinearity is superquadratic or involves a combination of superquadratic and subquadratic terms at infinity. By using some weaker conditions, our results extend and improve some existing results in the literature.

On the ratio of biomass to total carrying capacity in high dimensions

Abstract : This paper is concerned with a reaction-diffusion logistic model. In \cite{L06}, Lou observed that a heterogeneous environment with diffusion makes the total biomass greater than the total carrying capacity. Regarding the ratio of biomass to carrying capacity, Ni \cite{HN16} raised a conjecture that the ratio has a upper bound depending only on the spatial dimension. For the one-dimensional case, Bai, He, and Li \cite{BHL16} proved that the optimal upper bound is $3$. Recently, Inoue and Kuto \cite{IK20} showed that the supremum of the ratio is infinity when the domain is a multi-dimensional ball. In this paper, we generalized the result of \cite{IK20} to an arbitrary smooth bounded domain in $\mathbb{R}^n, n \geq 2$. We use the sub-solution and super-solution method. The idea of the proof is essentially the same as the proof of \cite{IK20} but we have improved the construction of sub-solutions. This is the complete answer to the conjecture of Ni.

The stability of weak solutions to an anisotropic polytropic infiltration equation

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.

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

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.

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

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 $m \geq 3$.

Numerical solutions for one and two dimensional nonlinear problems related to dispersion managed solitons

Abstract : We study behavior of numerical solutions for a nonlinear eigenvalue problem on $\R^n$ that is reduced from a dispersion managed nonlinear Schr\"{o}dinger equation. The solution operator of the free Schr\"{o}dinger equation in the eigenvalue problem is implemented via the finite difference scheme, and the primary nonlinear eigenvalue problem is numerically solved via Picard iteration. Through numerical simulations, the results known only theoretically, for example the number of eigenpairs for one dimensional problem, are verified. Furthermore several new characteristics of the eigenpairs, including the existence of eigenpairs inherent in zero average dispersion two dimensional problem, are observed and analyzed.

Asymptotics for an extended inverse Markovian Hawkes process

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.

Examples of simply reducible groups

Abstract : Simply reducible groups are important in physics and chemistry, which contain some of the important groups in condensed matter physics and crystal symmetry. By studying the group structures and irreducible representations, we find some new examples of simply reducible groups, namely, dihedral groups, some point groups, some dicyclic groups, generalized quaternion groups, Heisenberg groups over prime field of characteristic $2$, some Clifford groups, and some Coxeter groups. We give the precise decompositions of product of irreducible characters of dihedral groups, Heisenberg groups, and some Coxeter groups, giving the Clebsch-Gordan coefficients for these groups. To verify some of our results, we use the computer algebra systems GAP and SAGE to construct and get the character tables of some examples.