Journal of the
Korean Mathematical Society

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

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  • 2021-03-01

    Killing structure Jacobi operator of a real hypersurface in a complex projective space

    Juan de Dios P\'{e}rez

    Abstract : We prove non-existence of real hypersurfaces with Killing structure Jacobi operator in complex projective spaces. We also classify real hypersurfaces in complex projective spaces whose structure Jacobi operator is Killing with respect to the $k$-th generalized Tanaka-Webster connection.

  • 2020-11-01

    Asymptotic behavior of the inverse of tails of Hurwitz zeta function

    Ho-Hyeong Lee, Jong-Do Park

    Abstract : This paper deals with the inverse of tails of Hurwitz zeta function. More precisely, for any positive integer $s\geq2$ and {$0\leq a

  • 2020-07-01

    Equidistribution of higher dimensional generalized Dedekind sums and exponential sums

    Hi-joon Chae, Byungheup Jun, Jungyun Lee

    Abstract : We consider generalized Dedekind sums in dimension $n$, defined as sum of products of values of periodic Bernoulli functions. For the generalized Dedekind sums, we associate a Laurent polynomial. Using this, we associate an exponential sum of a Laurent polynomial to the generalized Dedekind sums and show that this exponential sum has a nontrivial bound that is sufficient to fulfill the equidistribution criterion of Weyl and thus the fractional part of the generalized Dedekind sums are equidistributed in $\FR/\FZ$.

  • 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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  • 2021-03-01

    A natural topological manifold structure of phase tropical hypersurfaces

    Young Rock Kim, Mounir Nisse

    Abstract : First, we define phase tropical hypersurfaces in terms of a degeneration data of smooth complex algebraic hypersurfaces in $(\mathbb{C}^*)^n$. Next, we prove that complex hyperplanes are homeomorphic to their degeneration called phase tropical hyperplanes. More generally, using Mikhalkin's decomposition into pairs-of-pants of smooth algebraic hypersurfaces, we show that a phase tropical hypersurface with smooth tropicalization is naturally a topological manifold. Moreover, we prove that a phase tropical hypersurface is naturally homeomorphic to a symplectic manifold.

  • 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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  • 2021-07-01

    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.

  • 2021-01-01

    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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  • 2021-11-01

    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

    Weighted estimates for certain rough operators with applications to vector valued inequalities

    Feng Liu, Qingying Xue

    Abstract : Under certain rather weak size conditions assumed on the kernels, some weighted norm inequalities for singular integral operators, related maximal operators, maximal truncated singular integral operators and Marcinkiewicz integral operators in nonisotropic setting will be shown. These weighted norm inequalities will enable us to obtain some vector valued inequalities for the above operators.

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July, 2022
Vol.59 No.4

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