Abstract : Let $j$ be a nonnegative integer. We define the Toeplitz-type operators $T_{a}^{(j)}$ with symbol $a\in L^{\infty}(C)$, which are variants of the traditional Toeplitz operators obtained for $j=0$. In this paper, we study the boundedness of these operators and characterize their compactness in terms of its Berezin transform.
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.
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
Abstract : A ring $R$ is called a UN ring if every non unit of it can be written as product of a unit and a nilpotent element. We obtain results about lifting of conjugate idempotents and unit regular elements modulo an ideal $I$ of a UN ring $R$. Matrix rings over UN rings are discussed and it is obtained that for a commutative ring $R$, a matrix ring $M_n(R)$ is UN if and only if $R$ is UN. Lastly, UN group rings are investigated and we obtain the conditions on a group $G$ and a field $K$ for the group algebra $KG$ to be UN. Then we extend the results obtained for $KG$ to the group ring $RG$ over a ring $R$ (which may not necessarily be a field).
Abstract : For any prime number $p$, let $J(p)$ be the set of positive integers $n$ such that the numerator of the $n^{th}$ harmonic number in the lowest terms is divisible by this prime number $p$. We consider an extension of this set to the generalized harmonic numbers, which are a natural extension of the harmonic numbers. Then, we present an upper bound for the number of elements in this set. Moreover, we state an explicit condition to show the finiteness of our set, together with relations to Bernoulli and Euler numbers.
Abstract : Main objective of the present paper is to establish Chen inequalities for slant Riemannian submersions in contact geometry. In this manner, we give some examples for slant Riemannian submersions and also investigate some curvature relations between the total space, the base space and fibers. Moreover, we establish Chen-Ricci inequalities on the vertical and the horizontal distributions for slant Riemannian submersions from Sasakian space forms.
Abstract : Let $ p $ be an odd prime. Let $ E $ be an elliptic curve defined over a quadratic imaginary field, where $ p $ splits completely. Suppose $ E $ has supersingular reduction at primes above $ p $. Under appropriate hypotheses, we extend the results of \cite{SujFil} to $ \mathbb{Z}_p^{2} $-extensions. We define and study the fine double-signed residual Selmer groups in these settings. We prove that for two residually isomorphic elliptic curves, the vanishing of the signed $ \mu $-invariants of one elliptic curve implies the vanishing of the signed $ \mu $-invariants of the other. Finally, we show that the Pontryagin dual of the Selmer group and the double-signed Selmer groups have no non-trivial pseudo-null submodules for these extensions.
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.
Abstract : The main motivation of this paper is to introduce and study the notions of strong Dedekind rings and semi-regular injective modules. Specifically, a ring $R$ is called strong Dedekind if every semi-regular ideal is $Q_0$-invertible, and an $R$-module $E$ is called a semi-regular injective module provided ${\rm Ext}^1_R(T,E)=0$ for every $\mathcal{Q}$-torsion module $T$. In this paper, we first characterize rings over which all semi-regular injective modules are injective, and then study the semi-regular injective envelopes of $R$-modules. Moreover, we introduce and study the semi-regular global dimensions $sr$-gl.dim$(R)$ of commutative rings $R$. Finally, we obtain that a ring $R$ is a ${\rm DQ}$-ring if and only if $sr$-gl.dim$(R)=0$, and a ring $R$ is a strong Dedekind ring if and only if $sr$-gl.dim$(R)\leq 1$, if and only if any semi-regular ideal is projective. Besides, we show that the semi-regular dimensions of strong Dedekind rings are at most one.
Abstract : In this paper, we show that a complete translating soliton $\Sigma^m$ in $\mathbb R^n$ for the mean curvature flow is stable with respect to weighted volume functional if $\Sigma$ satisfies that the $L^m$ norm of the second fundamental form is smaller than an explicit constant that depends only on the dimension of $\Sigma$ and the Sobolev constant provided in Michael and Simon [12]. Under the same assumption, we also prove that under this upper bound, there is no non-trivial $f$-harmonic $1$-form of $L^2_f$ on $\Sigma$. With the additional assumption that $\Sigma$ is contained in an upper half-space with respect to the translating direction then it has only one end.
Ali Benhissi, Abdelamir Dabbabi
Bull. Korean Math. Soc. 2022; 59(6): 1349-1357
https://doi.org/10.4134/BKMS.b210425
Yuxia Liang, Zhi-Yuan Xu, Ze-Hua Zhou
Bull. Korean Math. Soc. 2023; 60(2): 293-305
https://doi.org/10.4134/BKMS.b210802
Donghyun Kim, Junhui Woo, Ji-Hun Yoon
Bull. Korean Math. Soc. 2023; 60(2): 361-388
https://doi.org/10.4134/BKMS.b220134
Jorge Ferreira, Nazl\i \, Irk\i l, Erhan Pi\c{s}kin, Carlos Raposo , Mohammad Shahrouzi
Bull. Korean Math. Soc. 2022; 59(6): 1495-1510
https://doi.org/10.4134/BKMS.b210853
Yu Wang
Bull. Korean Math. Soc. 2023; 60(4): 1025-1034
https://doi.org/10.4134/BKMS.b220460
Xing-Wang Jiang, Ya-Li Li
Bull. Korean Math. Soc. 2023; 60(4): 915-931
https://doi.org/10.4134/BKMS.b220396
Enkhbayar Azjargal, Zorigt Choinkhor, Nyamdavaa Tsegmid
Bull. Korean Math. Soc. 2023; 60(4): 1131-1139
https://doi.org/10.4134/BKMS.b220595
Weiguo Lu, Ce Xu, Jianing Zhou
Bull. Korean Math. Soc. 2023; 60(4): 985-1001
https://doi.org/10.4134/BKMS.b220447
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