Abstract : In this paper, we are devoted to studying the mixed radial-angular integrabilities for Hardy type operators. As an application, the upper and lower bounds are obtained for the fractional Hardy operator. In addition, we also establish the sharp weak-type estimate for the fractional Hardy operator.
Abstract : For solving complex symmetric positive definite linear systems, we propose a single step real-valued (SSR) iterative method, which does not involve the complex arithmetic. The upper bound on the spectral radius of the iteration matrix of the SSR method is given and its convergence properties are analyzed. In addition, the quasi-optimal parameter which minimizes the upper bound for the spectral radius of the proposed method is computed. Finally, numerical experiments are given to demonstrate the effectiveness and robustness of the propose methods.
Abstract : In this paper, we give an improvement for the second main theorems of algebraically non-degenerate meromorphic maps from generalized $ p $-parabolic manifolds into projective varieties intersecting hypersurfaces in subgeneral position with some index, which extends the results of Han [6] and Chen-Thin [3].
Abstract : We show the asymptotics of the volume density function in the class of central harmonic manifolds can be specified arbitrarily and do not determine the geometry.
Abstract : In this article, we study the weak and extra-weak type integral inequalities for the modified integral Hardy operators. We provide suitable conditions on the weights $\omega, \rho, \phi$ and $\psi$ to hold the following weak type modular inequality \begin{align*} \mathcal{U}^{-1} \bigg ( \int_{ \{ | \mathcal{I}f | > \gamma\}} \mathcal{U} \Big(\gamma \omega \Big ) \rho \bigg ) & \leq \mathcal{V}^{-1} \bigg ( \int_{0}^{\infty} \mathcal{V} \Big ( C |f| \phi\Big) \psi \bigg ), \end{align*} where $\mathcal{I}$ is the modified integral Hardy operators. We also obtain a necesary and sufficient condition for the following extra-weak type integral inequality \begin{align*} \omega \bigg ( \Big\{ |\mathcal{I}f| > \gamma \Big \} \bigg) &\leq \mathcal{U}\circ \mathcal{V}^{-1} \bigg ( \int_{0}^{\infty} \mathcal{V} \bigg ( \dfrac{C |f| \phi}{\gamma} \bigg) \psi \bigg ). \end{align*} Further, we discuss the above two inequalities for the conjugate of the modified integral Hardy operators. It will extend the existing results for the Hardy operator and its integral version.
Abstract : We explicitly construct the smooth toric Fano variety which is isomorphic to the blow-up of the projective space at torus invariant points in codimension one by anti-flips.
Abstract : In 2010, Li-Ye [13, Theorem 0.1] proved that \begin{equation}\nonumber P\left(\zeta(z),\zeta'(z),\ldots,\zeta^{(m)}(z),\Gamma(z),\Gamma'(z),\Gamma^{''}(z)\right) \not\equiv 0\quad\text{in }\ \mathbb{C}, \end{equation} where $m$ is a non-negative integer, and $P(u_{0},u_{1}, \ldots, u_{m},v_{0},v_{1},v_{2})$ is any non-trivial polynomial in its arguments with coefficients in the field $\mathbb{C}$. Later on, Li-Ye [15, Theorem 1] proved that \begin{equation}\nonumber P\left(z,\Gamma(z),\Gamma'(z),\ldots,\Gamma^{(n)}(z), \zeta(z)\right)\not\equiv 0 \end{equation} in $z\in \mathbb{C}$ for any non-trivial distinguished polynomial $P(z,u_0, u_1,\ldots$, $u_n, v)$ with coefficients in a set $L_\delta$ of the zero function and a class of non-zero functions $f$ from $\mathbb{C}$ to $\mathbb{C}\cup\{\infty\}$ (cf. [15, Definition 1]). In this paper, we prove that $P\left(z,\zeta(z),\zeta'(z),\ldots,\zeta^{(m)}(z),\Gamma(z),\Gamma'(z),\ldots,\Gamma^{(n)}(z)\right)\not\equiv 0$ in $z\in\mathbb{C}$, where $m$ and $n$ are two non-negative integers, and $$P(z, u_0,u_1,\ldots,u_m,v_0,v_1,\ldots,v_n)$$ is any non-trivial polynomial in the $m+n+2$ variables $$u_0,u_1,\ldots,u_m,v_0,v_1,\ldots,v_n$$ with coefficients being meromorphic functions of order less than one, and the polynomial $P(z, u_0,u_1,\ldots,u_m,v_0,v_1,\ldots,v_n)$ is a distinguished polynomial in the $n+1$ variables $v_0,v_1,\ldots, v_n$. The question studied in this paper is concerning the conjecture of Markus from [16]. The main results obtained in this paper also extend the corresponding results from Li-Ye [12] and improve the corresponding results from Chen-Wang [5] and Wang-Li-Liu-Li [23], respectively.
Abstract : Let $F$ be a periodic singular fiber of genus $g$ with dual fiber $F^*$, and let $T$ (resp.~$T^*$) be the set of the components of $F$ (resp.~$F^*$) by removing one component with multiplicity one. We give a formula to compute the determinant $|\det T\,|$ of the intersect form of $T$. As a consequence, we prove that $|\det T\,|=|\det T^*\,|$. As an application, we compute the Mordell-Weil group of a fibration $f:S\to \mathbb P^1$ of genus $2$ with two singular fibers.
Abstract : This article concerns a ring property that arises from combining one-sided duo factor rings and centers. A ring $R$ is called {\it right CIFD} if $R/I$ is right duo by some proper ideal $I$ of $R$ such that $I$ is contained in the center of $R$. We first see that this property is seated between right duo and right $\pi$-duo\textbf{,} and not left-right symmetric. We prove, for a right CIFD ring $R$, that $W(R)$ coincides with the set of all nilpotent elements of $R$; that $R/P$ is a right duo domain for every minimal prime ideal $P$ of $R$; that $R/W(R)$ is strongly right bounded; and that every prime ideal of $R$ is maximal if and only if $R/W(R)$ is strongly regular, where $W(R)$ is the Wedderburn radical of $R$. It is also proved that a ring $R$ is commutative if and only if $D_3(R)$ is right CIFD, where $D_3(R)$ is the ring of $3$ by $3$ upper triangular matrices over $R$ whose diagonals are equal. Furthermore\textbf{,} we show that the right CIFD property does not pass to polynomial rings, and that the polynomial ring over a ring $R$ is right CIFD if and only if $R/I$ is commutative by a proper ideal $I$ of $R$ contained in the center of $R$.
Abstract : A Hilbert space operator $A\in{\mathcal B(H)}$ is a generalised \linebreak $n$-projection, denoted $A\in (G-n-P)$, if ${A^*}^n=A$. $(G-n-P)$-operators $A$ are normal operators with finitely countable spectra $\sigma(A)$, subsets of the set $\{0\}\cup\{\sqrt[n+1]{1}\}$. The Aluthge transform $\tilde{A}$ of $A\in{\mathcal B(H)}$ may be $(G-n-P)$ without $A$ being $(G-n-P)$. For doubly commuting operators $A, B\in{\mathcal B(H)}$ such that $\sigma(AB)=\sigma(A)\sigma(B)$ and $\|A\|\|B\|\leq \left\|\widetilde{AB}\right\|$, $\widetilde{AB}\in (G-n-P)$ if and only if $A=\left\|\tilde{A}\right\|(A_{00}\oplus(A_{0}\oplus A_u))$ and $B=\left\|\tilde{B}\right\|(B_0\oplus B_u)$, where $A_{00}$ and $B_0$, and $A_0\oplus A_u$ and $B_u$, doubly commute, $A_{00}B_0$ and $A_0$ are 2 nilpotent, $A_u$ and $B_u$ are unitaries, $A^{*n}_u=A_u$ and $B^{*n}_u=B_u$. Furthermore, a necessary and sufficient condition for the operators $\alpha A$, $\beta B$, $\alpha \tilde{A}$ and $\beta \tilde{B}$, $\alpha=\frac{1}{\left\|\tilde{A}\right\|}$ and $\beta=\frac{1}{\left\|\tilde{B}\right\|}$, to be $(G-n-P)$ is that $A$ and $B$ are spectrally normaloid at $0$.
Donghyun Kim, Junhui Woo, Ji-Hun Yoon
Bull. Korean Math. Soc. 2023; 60(2): 361-388
https://doi.org/10.4134/BKMS.b220134
Bull. Korean Math. Soc. 2022; 59(6): 1327-1337
https://doi.org/10.4134/BKMS.b210183
Shahram Rezaei, Behrouz Sadeghi
Bull. Korean Math. Soc. 2023; 60(1): 149-160
https://doi.org/10.4134/BKMS.b220003
Nguyen Thi Thu Ha
Bull. Korean Math. Soc. 2023; 60(2): 339-348
https://doi.org/10.4134/BKMS.b220123
Kwangwoo Lee
Bull. Korean Math. Soc. 2023; 60(6): 1427-1437
https://doi.org/10.4134/BKMS.b220371
Ahmed Mohammed Cherif, Khadidja Mouffoki
Bull. Korean Math. Soc. 2023; 60(3): 705-715
https://doi.org/10.4134/BKMS.b220347
Eungmo Nam, Juncheol Pyo
Bull. Korean Math. Soc. 2023; 60(1): 171-184
https://doi.org/10.4134/BKMS.b220049
Yaning Wang
Bull. Korean Math. Soc. 2022; 59(4): 897-904
https://doi.org/10.4134/BKMS.b210514
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