# Bulletin of theKorean Mathematical SocietyBKMS

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

## Current Issue

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March, 2022 | Volume 59, No. 2
• ### 2022-03-31

#### Duadic codes over finite local rings

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.

• ### 2022-03-31

#### Characterizations of Jordan derivable mappings at the unit element

Jiankui Li, Shan Li, Kaijia Luo

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}.

• ### 2022-03-31

#### Random sampling and reconstruction of signals with finite rate of innovation

Yingchun Jiang, Junjian Zhao

Abstract : In this paper, we mainly study the random sampling and reconstruction of signals living in the subspace $V^p(\Phi,\Lambda)$ of $L^p(\mathbb{R}^d)$, which is generated by a family of molecules $\Phi$ located on a relatively separated subset $\Lambda\subset \mathbb{R}^d$. The space $V^p(\Phi,\Lambda)$ is used to model signals with finite rate of innovation, such as stream of pulses in GPS applications, cellular radio and ultra wide-band communication. The sampling set is independently and randomly drawn from a general probability distribution over $\mathbb{R}^d$. Under some proper conditions for the generators $\Phi=\{\phi_\lambda:\lambda\in \Lambda\}$ and the probability density function $\rho$, we first approximate $V^{p}(\Phi,\Lambda)$ by a finite dimensional subspace $V^{p}_N(\Phi,\Lambda)$ on any bounded domains. Then, we prove that the random sampling stability holds with high probability for all signals in $V^{p}(\Phi,\Lambda)$ whose energy concentrate on a cube when the sampling size is large enough. Finally, a reconstruction algorithm based on random samples is given for signals in $V^{p}_N(\Phi,\Lambda)$.

• ### 2022-03-31

#### Variations in writhes of virtual knots under a local move

Abstract : $n$-writhes denoted by $J_n(K)$ are virtual knot invariants for $n\neq 0$ and are closely associated with coefficients of some polynomial invariants of virtual knots. In this work, we investigate the variations of $J_n(K)$ under arc shift move and conclude that $n$-writhes $J_n(K)$ vary randomly in the sense that it may change by any random integer value under one arc shift move. Also, for each $n\neq 0$ we provide an infinite family of virtual knots which can be distinguished by $n$-writhes $J_n(K)$, whereas odd writhe $J(K)$ fails to do so.

• ### 2022-03-31

#### Positive expansivity, chain transitivity, rigidity, and specification on general topological spaces

Thiyam Thadoi Devi, Khundrakpam Binod Mangang

Abstract : We discuss the notions of positive expansivity, chain transitivity, uniform rigidity, chain mixing, weak specification, and pseudo orbital specification in terms of finite open covers for Hausdorff topological spaces and entourages for uniform spaces. We show that the two definitions for each notion are equivalent in compact Hausdorff spaces and further they are equivalent to their standard definitions in compact metric spaces. We show that a homeomorphism on a Hausdorff uniform space has uniform $h$-shadowing if and only if it has uniform shadowing and its inverse is uniformly equicontinuous. We also show that a Hausdorff positively expansive system with a Hausdorff shadowing property has Hausdorff $h$-shadowing.

• ### 2022-03-31

#### Packing trees into complete k-partite graph

Yanling Peng, Hong Wang

Abstract : In this work, we confirm a weak version of a conjecture proposed by Hong Wang. The ideal of the work comes from the tree packing conjecture made by Gy\'arf\'as and Lehel. Bollob\'as confirms the tree packing conjecture for many small tree, who showed that one can pack $T_1,T_2,\ldots,T_{n/\sqrt{2}}$ into $K_n$ and that a better bound would follow from a famous conjecture of Erd\H{o}s. In a similar direction, Hobbs, Bourgeois and Kasiraj made the following conjecture: Any sequence of trees $T_1,T_2,\ldots,T_n$, with $T_i$ having order $i$, can be packed into $K_{n-1,\lceil n/2\rceil}$. Further Hobbs, Bourgeois and Kasiraj \cite{3} proved that any two trees can be packed into a complete bipartite graph $K_{n-1,\lceil n/2\rceil}$. Motivated by the result, Hong Wang propose the conjecture: For each $k$-partite tree $T(\mathbb{X})$ of order $n$, there is a restrained packing of two copies of $T(\mathbb{X})$ into a complete $k$-partite graph $B_{n+m}(\mathbb{Y})$, where $m=\lfloor\frac{k}{2}\rfloor$. Hong Wong \cite{4} confirmed this conjecture for $k=2$. In this paper, we prove a weak version of this conjecture.

• ### 2022-03-31

#### Rigidity of a rank $1$ cusp of punctured-surface groups in hyperbolic $4$-space

Youngju Kim

Abstract : We prove that a punctured-torus group of hyperbolic $4$-space which keeps an embedded hyperbolic $2$-plane invariant has a strictly parabolic commutator. More generally, this rigidity persists for a punctured-surface group.

• ### 2022-03-31

#### Some irrational quartic threefolds

Kyusik Hong, Joonyeong Won

Abstract : We study the factoriality of a nodal quartic hypersurface $V_4$ in $\mathbb{P}^4$ when there is a hyperplane in $\mathbb{P}^4$ containing all the nodes of $V_4$. As an application, we obtain new examples of irrational quartic $3$-folds.

• ### 2022-03-31

#### Harmonicity of almost Norden submersions between almost Norden manifolds

Garima Gupta, Rakesh Kumar, Rachna Rani, Rashmi Sachdeva

Abstract : We define an almost Norden submersion (holomorphic and semi-Riemannian submersion) between almost Norden manifolds and \linebreak show that, in most of the cases, the base manifold has the similar kind of structure as that of total manifold. We obtain necessary and sufficient conditions for almost Norden submersion to be a totally geodesic map. We also derive decomposition theorems for the total manifold of such submersions. Moreover, we study the harmonicity of almost Norden submersions between almost Norden manifolds and between Kaehler-Norden manifolds. Finally, we derive conditions for an almost Norden submersion to be a harmonic morphism.

• ### 2022-03-31

#### Some commutative rings defined by multiplication like-conditions

Mohamed Chhiti, Soibri Moindze

Abstract : In this article we investigate the transfer of multiplication-like properties to homomorphic images, direct products and amalgamated duplication of a ring along an ideal. Our aim is to provide examples of new classes of commutative rings satisfying the above-mentioned properties.

• ### 2022-03-31

#### Area properties associated with strictly convex curves

Shin-Ok Bang, Dong-Soo Kim, Incheon Kim

Abstract : Archimedes proved that for a point $P$ on a parabola $X$ and a chord $AB$ of $X$ parallel to the tangent of $X$ at $P$, the area of the region bounded by the parabola $X$ and the chord $AB$ is four thirds of the area of the triangle $\bigtriangleup ABP$. This property was proved to be a characteristic of parabolas, so called the Archimedean characterization of parabolas. In this article, we study strictly convex curves in the plane ${\mathbb R}^{2}$. As a result, first using a functional equation we establish a characterization theorem for quadrics. With the help of this characterization we give another proof of the Archimedean characterization of parabolas. Finally, we present two related conditions which are necessary and sufficient for a strictly convex curve in the plane to be an open arc of a parabola.

• ### 2022-03-31

#### Trace Properties and Integral Domains, III

Thomas G. Lucas, Abdeslam Mimouni

Abstract : An integral domain $R$ is an RTP domain (or has the radical trace property) (resp.~an $LTP$ domain) if $I(R:I)$ is a radical ideal for each nonzero noninvertible ideal $I$ (resp.~$I(R:I)R_P=PR_P$ for each minimal prime $P$ of $I(R:I)$). Clearly each RTP domain is an LTP domain, but whether the two are equivalent is open except in certain special cases. In this paper, we study the descent of these notions from particular overrings of $R$ to $R$ itself.

• ### 2022-03-31

#### Mean values of derivatives of $L$-functions in even characteristic

Sunghan Bae, Hwanyup Jung

Abstract : For any positive integer $\mu$, we compute the mean value of the $\mu$-th derivative of quadratic Dirichlet $L$-functions over the rational function field $\mathbb{F}_{q}(t)$, where $q$ is a power of $2$.

• ### 2022-03-31

#### Goldie extending property on the class of $z$-closed submodules

Adnan Tercan, Ramazan Ya\c{s}ar, Canan Celep Y\"ucel

Abstract : In this article, we define a module $M$ to be $G^{\, z}$-extending if and only if for each $z$-closed submodule $X$ of $M$ there exists a direct summand $D$ of $M$ such that $X\cap D$ is essential in both $X$ and $D$. We investigate structural properties of $G^{\, z}$-extending modules and locate the implications between the other extending properties. We deal with decomposition theory as well as ring and module extensions for $G^{\, z}$-extending modules. We obtain that if a ring is right $G^{\, z}$-extending, then so is its essential overring. Also it is shown that the $G^{\, z}$-extending property is inherited by its rational hull. Furthermore it is provided some applications including matrix rings over a right $G^{\, z}$-extending ring.

• ### 2022-03-31

#### On vanishing theorems for locally conformally flat Riemannian manifolds

Dang Tuyen Nguyen, Duc Thoan Pham

Abstract : In this paper, we obtain some vanishing theorems for $p$-harmonic 1-forms on locally conformally flat Riemannian manifolds which admit an integral pinching condition on the curvature operators.

• ### 2022-03-31

#### Time stepwise local volatility

Hyeong-Ohk Bae, Hyuncheul Lim

Abstract : We propose a path integral method to construct a time stepwise local volatility for the stock index market under Dupire's model. Our method is focused on the pricing with the Monte Carlo Method (MCM). We solve the problem of randomness of MCM by applying numerical integration. We reconstruct this task as a matrix equation. Our method provides the analytic Jacobian and Hessian required by the nonlinear optimization solver, resulting in stable and fast calculations.

## Current Issue

• ### On the existence of Graham partitions with congruence conditions

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

• ### On weighted Browder spectrum

Preeti Dharmarha, Sarita Kumari

Bull. Korean Math. Soc. 2022; 59(1): 1-13
https://doi.org/10.4134/BKMS.b200328

• ### Admissible balanced pairs over formal triangular matrix rings

Lixin Mao

Bull. Korean Math. Soc. 2021; 58(6): 1387-1400
https://doi.org/10.4134/BKMS.b200924

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

Jaeyoo Choy, Hahng-Yun Chu

Bull. Korean Math. Soc. 2022; 59(1): 27-43
https://doi.org/10.4134/BKMS.b201080

• ### On weighted Browder spectrum

Preeti Dharmarha, Sarita Kumari

Bull. Korean Math. Soc. 2022; 59(1): 1-13
https://doi.org/10.4134/BKMS.b200328

• ### On the existence of Graham partitions with congruence conditions

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

• ### Entire solutions of differential-difference equations of Fermat type

Peichu Hu, Wenbo Wang, Linlin Wu

Bull. Korean Math. Soc. 2022; 59(1): 83-99
https://doi.org/10.4134/BKMS.b210099

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

Dibakar Dey

Bull. Korean Math. Soc. 2022; 59(1): 101-110
https://doi.org/10.4134/BKMS.b210125