Abstract : In 1963, Graham introduced a problem to find integer partitions such that the reciprocal sum of their parts is 1. Inspired by Graham's work and classical partition identities, we show that there is an integer partition of a sufficiently large integer $n$ such that the reciprocal sum of the parts is $1$, while the parts satisfy certain congruence conditions.
Abstract : Suppose that $T=\left(\begin{smallmatrix} A&0\\U&B \end{smallmatrix}\right)$ is a formal triangular matrix ring, where $A$ and $B$ are rings and $U$ is a $(B, A)$-bimodule. Let $\mathfrak{C}_{1}$ and $\mathfrak{C}_{2}$ be two classes of left $A$-modules, $\mathfrak{D}_{1}$ and $\mathfrak{D}_{2}$ be two classes of left $B$-modules. We prove that $(\mathfrak{C}_{1},\mathfrak{C}_{2})$ and $(\mathfrak{D}_{1},\mathfrak{D}_{2})$ are admissible balanced pairs if and only if $(\textbf{p}(\mathfrak{C}_{1}, \mathfrak{D}_{1}), \textbf{h}(\mathfrak{C}_{2}, \mathfrak{D}_{2}))$ is an admissible balanced pair in $T$-Mod. Furthermore, we describe when $(\mathfrak{P}^{\mathfrak{C}_{1}}_{\mathfrak{D}_{1}}, \mathfrak{I}^{\mathfrak{C}_{2}}_{\mathfrak{D}_{2}})$ is an admissible balanced pair in $T$-Mod. As a consequence, we characterize when $T$ is a left virtually Gorenstein ring.
Abstract : The main aim of the article is to introduce new generalizations of Fredholm and Browder classes of spectra when the underlying Hilbert space is not necessarily separable and study their properties. To achieve the goal the notions of $\alpha$-Browder operators, $\alpha$-B-Fredholm operators, $\alpha$-B-Browder operators and $\alpha$-Drazin invertibility have been introduced. The relation of these classes of operators with their corresponding weighted spectra has been investigated. An equivalence of $\alpha$-Drazin invertible operators with $\alpha$-Browder operators and $\alpha$-B-Browder operators has also been established. The weighted Browder spectrum of the sum of two bounded linear operators has been characterised in the case when the Hilbert space (not necessarily separable) is a direct sum of its closed invariant subspaces.
Abstract : The one-sided fattenings (called semi-ribbon graph in this paper) of the graph embedded in the real projective plane $\rrpp^{2}$ are completely classified up to topological equivalence. A planar graph (i.e., embedded in the plane), admitting the one-sided fattening, is known to be a cactus boundary. For the graphs embedded in $\rrpp^{2}$ admitting the one-sided fattening, unlike the planar graphs, a new building block appears: a bracelet along the M\"obius band, which is not a connected summand of the oriented surfaces.
Abstract : In this paper, we extend some previous works by Liu et al. on the existence of transcendental entire solutions of differential-difference equations of Fermat type. In addition, we also present a precise description of the associated entire solutions.
Abstract : In this article, the notion of $\ast$-conformal Ricci soliton is defined as a self similar solution of the $\ast$-conformal Ricci flow. A Sasakian 3-metric satisfying the $\ast$-conformal Ricci soliton is completely classified under certain conditions on the soliton vector field. We establish a relation with Fano manifolds and proves a homothety between the Sasakian 3-metric and the Berger Sphere. Also, the potential vector field $V$ is a harmonic infinitesimal automorphism of the contact metric structure.
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 : In this paper, we introduce duadic codes over finite local rings and concentrate on quadratic residue codes. We study their properties and give the comprehensive method for the computing the unique idempotent generator of quadratic residue codes.
Abstract : Let $\mathcal{A}$ be a unital Banach algebra, $\mathcal{M}$ a unital $\mathcal{A}$-bimodule, and $\delta$ a linear mapping from $\mathcal{A}$ into $\mathcal{M}$. We prove that if $\delta$ satisfies $\delta(A)A^{-1}+A^{-1}\delta(A)+A\delta(A^{-1})+\delta(A^{-1})A=0$ for every invertible element $A$ in $\mathcal{A}$, then $\delta$ is a Jordan derivation. Moreover, we show that $\delta$ is a Jordan derivable mapping at the unit element if and only if $\delta$ is a Jordan derivation. As an application, we answer the question posed in \cite[Problem 2.6]{E}.
Abstract : We consider a nonlinear Schr\"odinger equation with critical frequency, $\displaystyle \left( \mathrm{P}_\varepsilon \right): \varepsilon^2 \Delta v(x) - V(x) v(x) + |v(x)|^{p-1} v(x) = 0$, $x\in \mathbb{R}^N$, and $v(x) \rightarrow 0$ as $|x|\rightarrow +\infty$, for the \emph{infinite case} as described by Byeon and Wang. \emph{Critical} means that $0\leq V\in \mathrm{C}(\mathbb{R}^N)$ verifies $\mathcal{Z} = \{V = 0 \} \neq \emptyset$. \emph{Infinite} means that $\mathcal{Z} = \{x_0\}$ and that, grossly speaking, the potential $V$ decays at an exponential rate as $x\rightarrow x_0$. For the semiclassical limit, $\varepsilon \rightarrow 0$, the infinite case has a characteristic limit problem, $\displaystyle \left( \mathrm{P}_{\mathrm{inf}} \right): \Delta u(x) - P(x) \, u(x) + |u(x)|^{p-1}\, u(x)=0$, $x\in \Omega$, with $u(x) = 0$ as $x\in \Omega$, where $\Omega\subseteq \mathbb{R}^N$ is a smooth bounded strictly star-shaped region related to the potential $V$. We prove the existence of an infinite number of solutions for both the original and the limit problem via a Ljusternik-Schnirelman scheme for even functionals. Fixed a topological level $k$ we show that $v_{k,\varepsilon}$, a solution of $(\mathrm{P}_\varepsilon)$, subconverges, up to a scaling, to a corresponding solution of $(\mathrm{P}_{\mathrm{inf}})$, and that $v_{k,\varepsilon}$ exponentially decays out of $\Omega$. Finally, uniform estimates on $\partial \Omega$ for scaled solutions of $(\mathrm{P}_\varepsilon)$ are obtained.
Byungchan Kim, Ji Young Kim, Chong Gyu Lee, Sang June Lee, Poo-Sung Park
Bull. Korean Math. Soc. 2022; 59(1): 15-25
https://doi.org/10.4134/BKMS.b200730
Lixin Mao
Bull. Korean Math. Soc. 2021; 58(6): 1387-1400
https://doi.org/10.4134/BKMS.b200924
Preeti Dharmarha, Sarita Kumari
Bull. Korean Math. Soc. 2022; 59(1): 1-13
https://doi.org/10.4134/BKMS.b200328
Jaeyoo Choy, Hahng-Yun Chu
Bull. Korean Math. Soc. 2022; 59(1): 27-43
https://doi.org/10.4134/BKMS.b201080
Preeti Dharmarha, Sarita Kumari
Bull. Korean Math. Soc. 2022; 59(1): 1-13
https://doi.org/10.4134/BKMS.b200328
Byungchan Kim, Ji Young Kim, Chong Gyu Lee, Sang June Lee, Poo-Sung Park
Bull. Korean Math. Soc. 2022; 59(1): 15-25
https://doi.org/10.4134/BKMS.b200730
Dibakar Dey
Bull. Korean Math. Soc. 2022; 59(1): 101-110
https://doi.org/10.4134/BKMS.b210125
Peichu Hu, Wenbo Wang, Linlin Wu
Bull. Korean Math. Soc. 2022; 59(1): 83-99
https://doi.org/10.4134/BKMS.b210099
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