Abstract : In this paper, Glift codes, generalized lifted polynomials, matrices are introduced. The advantage of Glift code is ``distance preserving" over the ring $\mathcal{R}$. Then optimal codes can be obtained over the rings by using Glift codes and lifted polynomials. Zero divisors are classified to satisfy ``distance preserving" for codes over non-chain rings. Moreover, Glift codes apply on MDS codes and MDS codes are obtained over the ring $\mathcal{R}$ and the non-chain ring $\mathcal{R}_{e,s}$.
Abstract : In view of Nevanlinna theory, we investigate the meromorphic solutions of q-difference differential equations and our results give the estimates about counting function and proximity function of meromorphic solutions to these equations. In addition, some interesting results are obtained for two general equations and a class of system of q-difference differential equations.
Abstract : For $n\geq 2$ and a real Banach space $E$, ${\mathcal L}(^n E:E)$ denotes the space of all continuous $n$-linear mappings from $E$ to itself. Let $$\Pi(E)=\{[x^*, (x_1, \ldots, x_n)]: x^{*}(x_j)=\|x^{*}\|=\|x_j\|=1~\mbox{for}~{j=1, \ldots, n}~\}.$$ An element $[x^*, (x_1, \ldots, x_n)]\in \Pi(E)$ is called a {\em numerical radius point} of $T\in {\mathcal L}(^n E:E)$ if $|x^{*}(T(x_1, \ldots, x_n))|=v(T)$, where the numerical radius $v(T)=\sup_{[y^*, y_1, \ldots, y_n]\in \Pi(E)}\Big|y^{*}\Big(T(y_1, \ldots,y_n)\Big)\Big|$. For $T\in {\mathcal L}(^n E:E)$, we define \begin{align*} {Nradius}({T})=&\ \{[x^*, (x_1, \ldots, x_n)]\in \Pi(E): [x^*, (x_1, \ldots, x_n)]\\ &\quad \mbox{is a numerical radius point of}~T\}. \end{align*} $T$ is called a {\em numerical radius peak $n$-linear mapping} if there is a unique $[x^{*}, (x_1, \ldots, x_n)]\in \Pi(E)$ such that ${Nradius}({T})=\{\pm [x^{*}, (x_1, \ldots, x_n)]\}$. In this paper we present explicit formulae for the numerical radius of $T$ for every $T\in {\mathcal L}(^n E:E)$ for $E=c_0$ or $l_{\infty}$. Using these formulae we show that there are no numerical radius peak mappings of ${\mathcal L}(^n c_0:c_0)$.
Abstract : In this paper, we derive a Reilly-type inequality for the Laplacian with density on weighted manifolds with boundary. As its applications, we obtain some new Poincar\'{e}-type inequalities not only on weighted manifolds, but more interestingly, also on their boundary. Furthermore, some mean-curvature type inequalities on the boundary are also given.
Abstract : Let $T$ be an $m$-linear Calder\'on-Zygmund operator. $T_{\vec{b},S}$ is the generalized commutator of $T$ with a class of measurable functions $\{b_{i}\}_{i=1}^\infty$. In this paper, we will give some new estimates for $T_{\vec{b},S}$ when $\{b_{i}\}_{i=1}^\infty$ belongs to Orlicz-type space and Lipschitz space, respectively.
Abstract : In this article, we find bases for the spaces of modular forms $M_{2}(\Gamma _{0}(88),\big( \frac{d}{\cdot }\big) )$ for $d=1,8,44\text{ and }88$. We then derive formulas for the number of representations of a positive integer by the diagonal quaternary quadratic forms with coefficients $1,2,11$ and $ 22 $.
Abstract : Suppose that a line passing through a given point $P$ intersects a given circle $\mathcal{C}$ at $Q$ and $R$ in the Euclidean plane. It is well known that $|PQ||PR|$ is independent of the choice of the line as long as the line meets the circle at two points. It is also known that similar properties hold in the 2-sphere and in the hyperbolic plane. New proofs for the similar properties in the 2-sphere and in the hyperbolic plane are given.
Abstract : In this paper, we establish the boundedness and continuity for variation operators for $\theta$-type Calder\'{o}n--Zygmund singular integrals and their commutators on the Triebel--Lizorkin spaces. As applications, we obtain the corresponding results for the Hilbert transform, the Hermit Riesz transform, Riesz transforms and rough singular integrals as well as their commutators.
Abstract : In this paper, we introduce the notion of Gorenstein $(m,n)$-flat modules as an extension of $(m,n)$-flat left $R$-modules over a ring $R$, where $m$ and $n$ are two fixed positive integers. We demonstrate that the class of all Gorenstein $(m,n)$-flat modules forms a Kaplansky class and establish that ($\mathcal{GF}_{m,n}(R)$,$\mathcal{GC}_{m,n}(R)$) constitutes a hereditary perfect cotorsion pair (where $\mathcal{GF}_{m,n}(R)$ denotes the class of Gorenstein $(m,n)$-flat modules and $\mathcal{GC}_{m,n}(R)$ refers to the class of Gorenstein $(m,n)$-cotorsion modules) over slightly $(m,n)$-coherent rings.
Abstract : In this note, we study a comparison principle for elliptic obstacle problems of $p$-Laplacian type with $L^1$-data. As a consequence, we improve some known regularity results for obstacle problems with zero Dirichlet boundary conditions.
Nguyen Van Duc
Bull. Korean Math. Soc. 2022; 59(3): 709-723
https://doi.org/10.4134/BKMS.b210426
Gaoshun Gou, Yueping Jiang, Ioannis D. Platis
Bull. Korean Math. Soc. 2023; 60(1): 225-235
https://doi.org/10.4134/BKMS.b220059
Florin Pop
Bull. Korean Math. Soc. 2022; 59(3): 659-669
https://doi.org/10.4134/BKMS.b210392
Henrique F. de~Lima , F\'{a}bio R. dos~Santos, Lucas S. Rocha
Bull. Korean Math. Soc. 2022; 59(3): 789-799
https://doi.org/10.4134/BKMS.b210479
Dong-Soo Kim, Young Ho Kim
Bull. Korean Math. Soc. 2023; 60(4): 905-913
https://doi.org/10.4134/BKMS.b220393
Jun Ho Lee
Bull. Korean Math. Soc. 2023; 60(2): 315-323
https://doi.org/10.4134/BKMS.b220094
Liufeng Cao
Bull. Korean Math. Soc. 2023; 60(6): 1687-1695
https://doi.org/10.4134/BKMS.b220845
Jiale Chen
Bull. Korean Math. Soc. 2023; 60(5): 1201-1219
https://doi.org/10.4134/BKMS.b220578
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