Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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  • 2021-01-01

    Construction of two- or three-weight binary linear codes from Vasil'ev codes

    Jong Yoon Hyun, Jaeseon Kim

    Abstract : The set $D$ of column vectors of a generator matrix of a linear code is called a defining set of the linear code. In this paper we consider the problem of constructing few-weight (mainly two- or three-weight) linear codes from defining sets. It can be easily seen that we obtain an one-weight code when we take a defining set to be the nonzero codewords of a linear code. Therefore we have to choose a defining set from a non-linear code to obtain two- or three-weight codes, and we face the problem that the constructed code contains many weights. To overcome this difficulty, we employ the linear codes of the following form: Let $D$ be a subset of $\mathbb{F}_2^n$, and $W$ (resp.~$V$) be a subspace of $\mathbb{F}_2$ (resp.~$\mathbb{F}_2^n$). We define the linear code $\mathcal{C}_D(W; V)$ with defining set $D$ and restricted to $W, V$ by \[ \mathcal{C}_D(W; V) = \{(s+u\cdot x)_{x\in D^*} \,|\, s\in W, u\in V\}. \] We obtain two- or three-weight codes by taking $D$ to be a Vasil'ev code of length $n=2^m-1 (m \geq 3)$ and a suitable choices of $W$. We do the same job for $D$ being the complement of a Vasil'ev code. The constructed few-weight codes share some nice properties. Some of them are optimal in the sense that they attain either the Griesmer bound or the Grey-Rankin bound. Most of them are minimal codes which, in turn, have an application in secret sharing schemes. Finally we obtain an infinite family of minimal codes for which the sufficient condition of Ashikhmin and Barg does not hold.

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  • 2021-07-01

    A persistently singular map of $\mathbb{T}^n$ that is $C^2$ robustly transitive but is not $C^1$ robustly transitive

    Juan Carlos Morelli

    Abstract : Consider the high dimensional torus $\mathbb{T}^n$ and the set $\mathcal{E}$ of its endomorphisms. We construct a map in $\mathcal{E}$ that is robustly transitive if $\mathcal{E}$ is endowed with the $C^2$ topology but is not robustly transitive if $\mathcal{E}$ is endowed with the $C^1$ topology.

  • 2021-05-01

    Estimation algorithm for physical parameters in a shallow arch

    Semion Gutman, Junhong Ha, Sudeok Shon

    Abstract : Design and maintenance of large span roof structures require an analysis of their static and dynamic behavior depending on the physical parameters defining the structures. Therefore, it is highly desirable to estimate the parameters from observations of the system. In this paper we study the parameter estimation problem for damped shallow arches. We discuss both symmetric and non-symmetric shapes and loads, and provide theoretical and numerical studies of the model behavior. Our study of the behavior of such structures shows that it is greatly affected by the existence of critical parameters. A small change in such parameters causes a significant change in the model behavior. The presence of the critical parameters makes it challenging to obtain good estimation. We overcome this difficulty by presenting the Parameter Estimation Algorithm that identifies the unknown parameters sequentially. It is shown numerically that the algorithm achieves a successful parameter estimation for models defined by arbitrary parameters, including the critical ones.

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  • 2021-01-01

    Weighted projective lines with weight permutation

    Lina Han, Xintian Wang

    Abstract : Let $\mathbb X$ be a weighted projective line defined over the algebraic closure $k=\overline{\mathbb F}_q$ of the finite field $\bbf_q$ and $\sigma$ be a weight permutation of $\mathbb X$. By folding the category coh-$\mathbb{X}$ of coherent sheaves on $\mathbb X$ in terms of the Frobenius twist functor induced by $\sigma$, we obtain an $\bbf_q$-category, denoted by coh-$(\mathbb{X},\sigma;q)$. We then prove that $\coh(\mathbb{X},\sigma;q)$ is derived equivalent to the valued canonical algebra associated with $(\bbX,\sigma)$.

  • 2021-01-01

    Restriction of scalars and cubic twists of elliptic curves

    Dongho Byeon, Keunyoung Jeong, Nayoung Kim

    Abstract : Let $K$ be a number field and $L$ a finite abelian extension of $K$. Let $E$ be an elliptic curve defined over $K$. The restriction of scalars $\mathrm{Res}^{L}_{K}E$ decomposes (up to isogeny) into abelian varieties over $K$ $$ \mathrm{Res}^{L}_{K}E \sim \bigoplus_{F \in S}A_F, $$ where $S$ is the set of cyclic extensions of $K$ in $L$. It is known that if $L$ is a quadratic extension, then $A_L$ is the quadratic twist of $E$. In this paper, we consider the case that $K$ is a number field containing a primitive third root of unity, $L=K(\root 3\of D)$ is the cyclic cubic extension of $K$ for some $D\in K^{\times}/(K^{\times})^3$, $E=E_a: y^2=x^3+a$ is an elliptic curve with $j$-invariant $0$ defined over $K$, and $E_a^D: y^2=x^3+aD^2$ is the cubic twist of $E_a$. In this case, we prove $A_L$ is isogenous over $K$ to $E_a^D \times E_a^{D^2}$ and a property of the Selmer rank of $A_L$, which is a cubic analogue of a theorem of Mazur and Rubin on quadratic twists.

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  • 2020-11-01

    Lower order eigenvalues for the bi-drifting Laplacian on the Gaussian shrinking soliton

    Lingzhong Zeng

    Abstract : It may very well be difficult to prove an eigenvalue inequality of Payne-P\'{o}lya-Weinberger type for the bi-drifting Laplacian on the bounded domain of the general complete metric measure spaces. Even though we suppose that the differential operator is bi-harmonic on the standard Euclidean sphere, this problem still remains open. However, under certain condition, a general inequality for the eigenvalues of bi-drifting Laplacian is established in this paper, which enables us to prove an eigenvalue inequality of Ashbaugh-Cheng-Ichikawa-Mametsuka type (which is also called an eigenvalue inequality of Payne-P\'{o}lya-Weinberger type) for the eigenvalues with lower order of bi-drifting Laplacian on the Gaussian shrinking soliton.

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  • 2020-11-01

    Bifurcation problem for a class of quasilinear fractional Schr\"{o}dinger equations

    Imed Abid

    Abstract : We study bifurcation for the following fractional Schr\"{o}dinger equation \begin{eqnarray*}\left\{ \begin{array}{rlll} (-\Delta)^{s}u+V(x)u& = \lambda\,f(u)& \hbox{in}\,\Omega \\ u&>0& \hbox{in}\,\Omega\\ u &=0 &\hbox{in}\,\R^n\setminus\Omega \\ \end{array} \right. \end{eqnarray*} where $0

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  • 2021-11-01

    Preservation of expansivity in hyperspace dynamical systems

    Namjip Koo, Hyunhee Lee

    Abstract : In this paper we study the preservation of various notions of expansivity in discrete dynamical systems and the induced map for $n$-fold symmetric products and hyperspaces. Then we give a characterization of a compact metric space admitting hyper $N$-expansive homeomorphisms via the topological dimension. More precisely, we show that $C^0$-generically, any homeomorphism on a compact manifold is not hyper $N$-expansive for any $N\in \mathbb{N}$. Also we give some examples to illustrate our results.

  • 2021-07-01

    Symmetry and monotonicity of solutions to fractional elliptic and parabolic equations

    Fanqi Zeng

    Abstract : In this paper, we first apply parabolic inequalities and a maximum principle to give a new proof for symmetry and monotonicity of solutions to fractional elliptic equations with gradient term by the method of moving planes. Under the condition of suitable initial value, by maximum principles for the fractional parabolic equations, we obtain symmetry and monotonicity of positive solutions for each finite time to nonlinear fractional parabolic equations in a bounded domain and the whole space. More generally, if bounded domain is a ball, then we show that the solution is radially symmetric and monotone decreasing about the origin for each finite time. We firmly believe that parabolic inequalities and a maximum principle introduced here can be conveniently applied to study a variety of nonlocal elliptic and parabolic problems with more general operators and more general nonlinearities.

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  • 2021-05-01

    Homotopy properties of $\text{map}(\Sigma^n \mathbb C P^2,S^m)$

    Jin-ho Lee

    Abstract : For given spaces $X$ and $Y$, let $map(X,Y)$ and $map_\ast(X,Y)$ be the unbased and based mapping spaces from $X$ to $Y$, equipped with compact-open topology respectively. Then let $map(X,Y;f)$ and $map_\ast(X,$ $Y;g)$ be the path component of $map(X,Y)$ containing $f$ and $map_\ast(X,Y)$ containing $g$, respectively. In this paper, we compute cohomotopy groups of suspended complex plane $\pi^{n+m}(\Sigma^n \C P^2)$ for $m=6,7$. Using these results, we classify path components of the spaces $map(\Sigma^n \C P^2,S^m)$ up to homotopy equivalence. We also determine the generalized Gottlieb groups $G_n(\C P^2,S^m)$. Finally, we compute homotopy groups of mapping spaces $map(\Sigma^n \mathbb{C}P^2,S^m;f)$ for all generators $[f]$ of $[\Sigma^n \C P^2,S^m]$, and Gottlieb groups of mapping components containing constant map $map(\Sigma^n \C P^2,S^m;\ast)$.

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May, 2022
Vol.59 No.3

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