Bulletin of the
Korean Mathematical Society
BKMS

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

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  • 2022-01-31

    On weighted Browder spectrum

    Preeti Dharmarha, Sarita Kumari

    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.

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  • 2022-01-31

    On the existence of Graham partitions with congruence conditions

    Byungchan Kim, Ji Young Kim, Chong Gyu Lee, Sang June Lee, Poo-Sung Park

    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.

  • 2022-01-31

    Entire solutions of differential-difference equations of Fermat type

    Peichu Hu, Wenbo Wang, Linlin Wu

    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.

  • 2022-01-31

    Sasakian 3-Metric as a $\ast$-Conformal Ricci Soliton Represents a Berger Sphere

    Dibakar Dey

    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.

  • 2022-01-31

    Semiclassical asymptotics of infinitely many solutions for the infinite case of a nonlinear Schr\"odinger equation with critical frequency

    Ariel Aguas-Barreno, Jordy Cevallos-Ch\'avez, Juan Mayorga-Zambrano, Leonardo Medina-Espinosa

    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.

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  • 2022-01-31

    One-sided fattening of the graph in the real projective plane

    Jaeyoo Choy, Hahng-Yun Chu

    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.

  • 2022-03-31

    Duadic codes over finite local rings

    Arezoo Soufi Karbaski, Karim Samei

    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.

  • 2022-01-31

    Further results on biases in integer partitions

    Shane Chern

    Abstract : Let $p_{a,b,m}(n)$ be the number of integer partitions of $n$ with more parts congruent to $a$ modulo $m$ than parts congruent to $b$ modulo $m$. We prove that $p_{a,b,m}(n)\ge p_{b,a,m}(n)$ whenever $1\le a

  • 2022-01-31

    Bernstein-Walsh type inequalities for derivatives of algebraic polynomials

    Fahreddin G. Abdullayev, Cevahir D. G\"{u}n

    Abstract : In this work, we study Bernstein-Walsh-type estimations for the derivative of an arbitrary algebraic polynomial in regions with piecewise smooth boundary without cusps of the complex plane. Also, estimates are given on the whole complex plane.

  • 2022-01-31

    The images of locally finite $\mathcal{E}$-derivations of polynomial algebras

    Lintong Lv, Dan Yan

    Abstract : Let $K$ be a field of characteristic zero. We first show that images of the linear derivations and the linear $\mathcal{E}$-derivations of the polynomial algebra $K[x]=K[x_1,x_2,\ldots,x_n]$ are ideals if the products of any power of eigenvalues of the matrices according to the linear derivations and the linear $\mathcal{E}$-derivations are not unity. In addition, we prove that the images of $D$ and $\delta$ are Mathieu-Zhao spaces of the polynomial algebra $K[x]$ if $D=\sum_{i=1}^n(a_ix_i+b_i)\partial_i$ and $\delta=I-\phi$, $\phi(x_i)=\lambda_ix_i+\mu_i$ for $a_i,b_i,\lambda_i,\mu_i\in K$ for $1\leq i\leq n$. Finally, we prove that the image of an affine $\mathcal{E}$-derivation of the polynomial algebra $K[x_1,x_2]$ is a Mathieu-Zhao space of the polynomial algebra $K[x_1,x_2]$. Hence we give an affirmative answer to the LFED Conjecture for the affine $\mathcal{E}$-derivations of the polynomial algebra $K[x_1,x_2]$.

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March, 2022
Vol.59 No.2

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