Bulletin of the
Korean Mathematical Society
BKMS

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

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  • 2023-01-31

    Lens spaces admitting minimal symplectic fillings with the second Betti number one

    Heesang Park, Dongsoo Shin

    Abstract : We classify lens spaces with the Milnor fillable contact structure that admit minimal symplectic fillings whose second Betti numbers are one.

  • 2023-07-31

    On the structure of certain subset of Farey sequence

    Xing-Wang Jiang, Ya-Li Li

    Abstract : Let $F_n$ be the Farey sequence of order $n$. For $S\subseteq F_n$, let $\mathcal{Q}(S)$ be the set of rational numbers $x/y$ with $x,y\in S,~x\leq y$ and $y\neq 0$. Recently, Wang found all subsets $S$ of $F_n$ with $|S|=n+1$ for which $\mathcal{Q}(S)\subseteq F_n$. Motivated by this work, we try to determine the structure of $S\subseteq F_n$ such that $|S|=n$ and $\mathcal{Q}(S)\subseteq F_n$. In this paper, we determine all sets $S\subseteq F_n$ satisfying these conditions for $n\in\{p,2p\}$, where $p$ is prime.

  • 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.

  • 2023-05-31

    Bach almost solitons in paraSasakian geometry

    Uday Chand De, Gopal Ghosh

    Abstract : If a paraSasakian manifold of dimension $(2n+1)$ represents Bach almost solitons, then the Bach tensor is a scalar multiple of the metric tensor and the manifold is of constant scalar curvature. Additionally it is shown that the Ricci operator of the metric $g$ has a constant norm. Next, we characterize 3-dimensional paraSasakian manifolds admitting Bach almost solitons and it is proven that if a 3-dimensional paraSasakian manifold admits Bach almost solitons, then the manifold is of constant scalar curvature. Moreover, in dimension 3 the Bach almost solitons are steady if $r=-6$; shrinking if $r>-6$; expanding if $r

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  • 2023-11-30

    On a generalization of unit regular rings

    Tahire Ozen

    Abstract : In this paper, we introduce a class of rings generalizing unit regular rings and being a subclass of semipotent rings, which is called idempotent unit regular. We call a ring $R$ an idempotent unit regular ring if for all $r\in R-J(R)$, there exist a non-zero idempotent $e$ and a unit element $u$ in $R$ such that $er=eu$, where this condition is left and right symmetric. Thus, we have also that there exist a non-zero idempotent $e$ and a unit $u$ such that $ere=eue$ for all $r\in R-J(R)$. Various basic characterizations and properties of this class of rings are proved and it is given the relationships between this class of rings and some well-known classes of rings such as semiperfect, clean, exchange and semipotent. Moreover, we obtain some results about when the endomorphism ring of a module in a class of left $R$-modules $X$ is idempotent unit regular.

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  • 2024-01-31

    On delay differential equations with meromorphic solutions of hyper-order less than one

    Risto Korhonen, Yan Liu

    Abstract : We consider the delay differential equations \begin{equation*} b(z)w(z+1)+c(z)w(z-1)+a(z)\frac{w'(z)}{w^k(z)}=\frac{P(z,w(z))}{Q(z,w(z))}, \end{equation*} where $k\in\{1,2\}$, $a(z)$, $b(z)\not\equiv 0$, $c(z)\not\equiv 0$ are rational functions, and $P(z,w(z))$ and $Q(z,w(z))$ are polynomials in $w(z)$ with rational coefficients satisfying certain natural conditions regarding their roots. It is shown that if this equation has a non-rational meromorphic solution $w$ with hyper-order $\rho_{2}(w)

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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 $\mathrm{Tor}_{s+t}^R(R/\mathfrak{a},X)\cong\mathrm{Tor}_{s}^R(R/\mathfrak{a},\mathrm{H}_{t}^\mathfrak{a}(X))$.

  • 2023-03-31

    Hankel determinants for starlike functions with respect to symmetrical points

    Nak Eun Cho, Young Jae Sim, Derek K. Thomas

    Abstract : We prove sharp bounds for Hankel determinants for starlike functions $f$ with respect to symmetrical points, i.e., $f$ given by $f(z)=z+\sum_{n=2}^{\infty}a_nz^n$ for $z\in \mathbb{D}$ satisfying $$ Re\dfrac{zf'(z)}{f(z)-f(-z)}>0, \quad z\in \mathbb{D}. $$ We also give sharp upper and lower bounds when the coefficients of $f$ are real.

  • 2023-07-31

    Hypersurfaces with prescribed mean curvature in measure metric space

    Zhengmao Chen

    Abstract : For any given function $f$, we focus on the so-called prescribed mean curvature problem for the measure $e^{-f(|x|^2)}dx$ provided that $e^{-f(|x|^2)}\in L^1(\mathbb{R}^{n+1})$. More precisely, we prove that there exists a smooth hypersurface $M$ whose metric is $ds^2=d\rho^2+\rho^2d\xi^2$ and whose mean curvature function is \begin{equation*} \frac{1}{n}\frac{u^p}{\rho^\beta}e^{f(\rho^2)}\psi(\xi) \end{equation*} for any given real constants $p$, $\beta$ and functions $f$ and $\psi$ where $u$ and $\rho$ are the support function and radial function of $M$, respectively. Equivalently, we get the existence of a smooth solution to the following quasilinear equation on the unit sphere $\mathbb{S}^{n}$, \begin{equation*} \sum\limits_{i,j}(\delta_{ij}-\frac{\rho_i\rho_j}{\rho^2+|\nabla\rho|^2})(-\rho_{ji} +\frac{2}{\rho}\rho_j\rho_i +\rho\delta_{ji})=\psi\frac{\rho^{2p+2-n-\beta} e^{f(\rho^2)}}{(\rho^2+|\nabla \rho|^2)^{\frac{p}{2}}} \end{equation*} under some conditions. Our proof is based on the powerful method of continuity. In particular, if we take $f(t)=\frac{t}{2}$, this may be prescribed mean curvature problem in Gauss measure space and it can be seen as an embedded result in Gauss measure space which will be needed in our forthcoming papers on the differential geometric analysis in Gauss measure space, such as Gauss-Bonnet-Chern theorem and its application on positive mass theorem and the Steiner-Weyl type formula, the Plateau problem and so on.

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  • 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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November, 2024
Vol.61 No.6

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