Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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  • 2023-01-31

    Homology and Serre class in $\mathrm{D}(R)$

    Zhicheng Wang

    Abstract : Let $\mathcal{S}$ be a Serre class in the category of modules and $\mathfrak{a}$ an ideal of a commutative Noetherian ring $R$. We study the containment of Tor modules, Koszul homology and local homology in $\mathcal{S}$ from below. With these results at our disposal, by specializing the Serre class to be Noetherian or zero, a handful of conclusions on Noetherianness and vanishing of the foregoing homology theories are obtained. We also determine when $\mathrm{Tor}_{s+t}^R(R/\mathfrak{a},X)\cong\mathrm{Tor}_{s}^R(R/\mathfrak{a},\mathrm{H}_{t}^\mathfrak{a}(X))$.

  • 2022-11-30

    Estimate for bilinear Calder\'{o}n-Zygmund operator and its commutator on product of variable exponent spaces

    Guanghui Lu, Shuangping Tao

    Abstract : The goal of this paper is to establish the boundedness of bilinear Calder\'{o}n-Zygmund operator $BT$ and its commutator $[b_{1},b_{2},BT]$ which is generated by $b_{1}, b_{2}\in\mathrm{BMO}(\mathbb{R}^{n})$ (or $\dot{\Lambda}_{\alpha}(\mathbb{R}^{n})$) and the $BT$ on generalized variable exponent Morrey spaces $\mathcal{L}^{p(\cdot),\varphi}(\mathbb{R}^{n})$. Under assumption that the functions $\varphi_{1}$ and $\varphi_{2}$ satisfy certain conditions, the authors proved that the $BT$ is bounded from product of spaces $\mathcal{L}^{p_{1}(\cdot),\varphi_{1}}(\mathbb{R}^{n}) \times\mathcal{L}^{p_{2}(\cdot),\varphi_{2}}(\mathbb{R}^{n})$ into space $\mathcal{L}^{p(\cdot),\varphi}(\mathbb{R}^{n})$. Furthermore, the boundedness of commutator $[b_{1},b_{2},BT]$ on spaces $L^{p(\cdot)}(\mathbb{R}^{n})$ and on spaces $\mathcal{L}^{p(\cdot),\varphi}(\mathbb{R}^{n})$ is also established.

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  • 2023-09-30

    Complex symmetric weighted composition-differentiation operators on $H^2$

    Lian Hu, Songxiao Li, Rong Yang

    Abstract : In this paper, we study the complex symmetric weighted composition-differentiation operator $D_{\psi,\phi}$ with respect to the conjugation $ JW_{\xi, \tau}$ on the Hardy space $H^2$. As an application, we characterize the necessary and sufficient conditions for such an operator to be normal under some mild conditions. Finally, the spectrum of $D_{\psi,\phi}$ is also investigated.

  • 2023-01-31

    S-curvature and geodesic orbit property of invariant $(\alpha_{1},\alpha_{2})$-metrics on spheres

    Huihui An, Zaili Yan, Shaoxiang Zhang

    Abstract : Geodesic orbit spaces are homogeneous Finsler spaces whose geodesics are all orbits of one-parameter subgroups of isometries. Such Finsler spaces have vanishing S-curvature and hold the Bishop-Gromov volume comparison theorem. In this paper, we obtain a complete description of invariant $(\alpha_{1},\alpha_{2})$-metrics on spheres with vanishing S-curvature. Also, we give a description of invariant geodesic orbit $(\alpha_{1},\alpha_{2})$-metrics on spheres. We mainly show that a ${\mathrm S}{\mathrm p}(n+1)$-invariant $(\alpha_{1},\alpha_{2})$-metric on $\mathrm{S}^{4n+3}={\mathrm S}{\mathrm p}(n+1)/{\mathrm S}{\mathrm p}(n)$ is geodesic orbit with respect to ${\mathrm S}{\mathrm p}(n+1)$ if and only if it is ${\mathrm S}{\mathrm p}(n+1){\mathrm S}{\mathrm p}(1)$-invariant. As an interesting consequence, we find infinitely many Finsler spheres with vanishing S-curvature which are not geodesic orbit spaces.

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  • 2022-07-31

    The K\"{a}hler Different of a Set of Points in~$\mathbb{P}^{m}\!\times\mathbb{P}^{n}$

    Nguyen T. Hoa, Tran N. K. Linh, Le N. Long, Phan T. T. Nhan, Nguyen T. P. Nhi

    Abstract : Given an ACM set $\mathbb{X}$ of points in a multiprojective space $\mathbb{P}^{m}\!\times\mathbb{P}^{n}$ over a field of characteristic zero, we are interested in studying the K\"ahler different and the Cayley-Bacharach property for $\mathbb{X}$. In $\mathbb{P}^1\times \mathbb{P}^1$, the Cayley-Bacharach property agrees with the complete intersection property and it is characterized by using the K\"ahler different. However, this result fails to hold in $\mathbb{P}^{m}\!\times\mathbb{P}^{n}$ for $n>1$ or $m>1$. In this paper we start an investigation of the K\"ahler different and its Hilbert function and then prove that $\mathbb{X}$ is a complete intersection of type $(d_1,\ldots,d_m,d'_1,\ldots,d'_n)$ if and only if it has the Cayley-Bacharach property and the K\"ahler different is non-zero at a certain degree. We characterize the Cayley-Bacharach property of $\mathbb{X}$ under certain assumptions.

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  • 2022-07-31

    Knots in homology lens spaces determined by their complements

    Kazuhiro Ichihara , Toshio Saito

    Abstract : In this paper, we consider the knot complement problem for not null-homologous knots in homology lens spaces. Let $M$ be a homology lens space with $H_1(M; \mathbb{Z}) \cong \mathbb{Z}_p$ and $K$ a not null-homologous knot in $M$. We show that, $K$ is determined by its complement if $M$ is non-hyperbolic, $K$ is hyperbolic, and $p$ is a prime greater than 7, or, if $M$ is actually a lens space $L(p,q)$ and $K$ represents a generator of $H_1(L(p,q))$.

  • 2022-09-30

    Szeg\"{o} projections for Hardy spaces in quaternionic Clifford analysis

    Fuli He , Song Huang, Min Ku

    Abstract : In this paper we study Szeg\"{o} kernel projections for Hardy spaces in quaternionic Clifford analysis. At first we introduce the matrix Szeg\"{o} projection operator for the Hardy space of quaternionic Hermitean monogenic functions by the characterization of the matrix Hilbert transform in the quaternionic Clifford analysis. Then we establish the Kerzman-Stein formula which closely connects the matrix Szeg\"{o} projection operator with the Hardy projection operator onto the Hardy space, and we get the matrix Szeg\"{o} projection operator in terms of the Hardy projection operator and its adjoint. At last, we construct the explicit matrix Szeg\"o kernel function for the Hardy space on the sphere as an example, and get the solution to a Diriclet boundary value problem for matrix functions.

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  • 2022-11-30

    Modified cyclotomic polynomials

    Ae-Kyoung Cha, Miyeon Kwon, Ki-Suk Lee, Seong-Mo Yang

    Abstract : Let $H$ be a subgroup of $\mathbb{Z}_n^\ast$ (the multiplicative group of integers modulo $n$) and $h_1,h_2,\ldots,h_l$ distinct representatives of the cosets of $H$ in $\mathbb{Z}_n^\ast$. We now define a polynomial $J_{n,H}(x)$ to be \begin{align*} \begin{split} J_{n,H}(x)=\prod\limits_{j=1}^{l} \bigg( x-\sum\limits_{h \in H}\zeta_n^{h_jh} \bigg), \end{split} \end{align*} where $\zeta_n=e^{\frac{2\pi i}{n}}$ is the $n$th primitive root of unity. Polynomials of such form generalize the $n$th cyclotomic polynomial $\Phi_n(x)=\prod_{k \in \mathbb{Z}_n^\ast}(x-\zeta_n^k)$ as $J_{n,\{1\}}(x)=\Phi_n(x)$. While the $n$th cyclotomic polynomial $\Phi_n(x)$ is irreducible over $\mathbb{Q}$, $J_{n,H}(x)$ is not necessarily irreducible. In this paper, we determine the subgroups $H$ for which $J_{n,H}(x)$ is irreducible over $\mathbb{Q}$.

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  • 2022-03-31

    Various shadowing properties for time varying maps

    Javad Nazarian Sarkooh

    Abstract : This paper is concerned with the study of various notions of shadowing of dynamical systems induced by a sequence of maps, so-called time varying maps, on a metric space. We define and study the shadowing, h-shadowing, limit shadowing, s-limit shadowing and exponential limit shadowing properties of these dynamical systems. We show that h-shadowing, limit shadowing and s-limit shadowing properties are conjugacy invariant. Also, we investigate the relationships between these notions of shadowing for time varying maps and examine the role that expansivity plays in shadowing properties of such dynamical systems. Specially, we prove some results linking s-limit shadowing property to limit shadowing property, and h-shadowing property to s-limit shadowing and limit shadowing properties. Moreover, under the assumption of expansivity, we show that the shadowing property implies the h-shadowing, s-limit shadowing and limit shadowing properties. Finally, it is proved that the uniformly contracting and uniformly expanding time varying maps exhibit the shadowing, limit shadowing, s-limit shadowing and exponential limit shadowing properties.

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  • 2023-01-31

    Zero sums of dual Toeplitz products on the orthogonal complement of the Dirichlet space

    Young Joo Lee

    Abstract : We consider dual Toeplitz operators acting on the orthogonal complement of the Dirichlet space on the unit disk. We give a characterization of when a finite sum of products of two dual Toeplitz operators is equal to $0$. Our result extends several known results by using a unified way.

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January, 2024
Vol.61 No.1

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