Bulletin of the
Korean Mathematical Society
BKMS
ISSN(Print) 1015-8634
ISSN(Online) 2234-3016
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2023-11-30
Homogeneous geodesics in homogeneous sub-Finsler manifolds
Zaili Yan, Tao ZhouAbstract : In this paper, we mainly study the problem of the existence of homogeneous geodesics in sub-Finsler manifolds. Firstly, we obtain a characterization of a homogeneous curve to be a geodesic. Then we show that every compact connected homogeneous sub-Finsler manifold and Carnot group admits at least one homogeneous geodesic through each point. Finally, we study a special class of $\ell^{p}$-type bi-invariant metrics on compact semi-simple Lie groups. We show that every homogeneous curve in such a metric space is a geodesic. Moreover, we prove that the Alexandrov curvature of the metric space is neither non-positive nor non-negative.
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2023-11-30
Singularity formation for a nonlinear variational sine-Gordon equation in a multidimensional space
Fengmei Qin, Kyungwoo Song, Qin WangAbstract : We study a multidimensional nonlinear variational sine-Gor\-don equation, which can be used to describe long waves on a dipole chain in the continuum limit. By using the method of characteristics, we show that a solution of a nonlinear variational sine-Gordon equation with certain initial data in a multidimensional space has a singularity in finite time.
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2023-11-30
Weak factorizations of $H^1(\mathbb{R}^n)$ in terms of multilinear fractional integral operator on variable Lebesgue spaces
Zongguang Liu, Huan ZhaoAbstract : This paper provides a constructive proof of the weak factorizations of the classical Hardy space $H^1(\mathbb{R}^n)$ in terms of multilinear fractional integral operator on the variable Lebesgue spaces, which the result is new even in the linear case. As a direct application, we obtain a new proof of the characterization of $\mathrm{BMO}(\mathbb{R}^n)$ via the boundedness of commutators of the multilinear fractional integral operator on the variable Lebesgue spaces.
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2023-11-30
Time analyticity for the heat equation under Bakry-\'Emery Ricci curvature condition
Ling WuAbstract : Inspired by Hongjie Dong and Qi S. Zhang's article [3], we find that the analyticity in time for a smooth solution of the heat equation with exponential quadratic growth in the space variable can be extended to any complete noncompact Riemannian manifolds with Bakry-\'Emery Ricci curvature bounded below and the potential function being of at most quadratic growth. Therefore, our result holds on all gradient Ricci solitons. As a corollary, we give a necessary and sufficient condition on the solvability of the backward heat equation in a class of functions with the similar growth condition. In addition, we also consider the solution in certain $L^p$ spaces with $p\in[2,+\infty)$ and prove its analyticity with respect to time.
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2023-11-30
Two-weight norm estimates for square functions associated to fractional Schr{\"o}dinger operators with Hardy potential
Tongxin Kang, Yang ZouAbstract : Let $d\in\mathbb{N}$ and ${\alpha}\in(0,\min\{2,d\})$. For any $a\in[a^\ast,\infty)$, the fractional Schr\"odinger operator $\mathcal{L}_a$ is defined by \begin{equation*} \mathcal{L}_a:=(-\Delta)^{{\alpha}/2}+a{|x|}^{-{\alpha}}, \end{equation*} where $a^*:=-{\frac{2^{\alpha}{\Gamma}((d+{\alpha})/4)^2}{{\Gamma}((d-{\alpha})/4)^2}}$. In this paper, we study two-weight Sobolev inequalities associated with $\mathcal{L}_a$ and two-weight norm estimates for several square functions associated with $\mathcal{L}_a$.
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2024-01-31
On delay differential equations with meromorphic solutions of hyper-order less than one
Risto Korhonen, Yan LiuAbstract : We consider the delay differential equations \begin{equation*} b(z)w(z+1)+c(z)w(z-1)+a(z)\frac{w'(z)}{w^k(z)}=\frac{P(z,w(z))}{Q(z,w(z))}, \end{equation*} where $k\in\{1,2\}$, $a(z)$, $b(z)\not\equiv 0$, $c(z)\not\equiv 0$ are rational functions, and $P(z,w(z))$ and $Q(z,w(z))$ are polynomials in $w(z)$ with rational coefficients satisfying certain natural conditions regarding their roots. It is shown that if this equation has a non-rational meromorphic solution $w$ with hyper-order $\rho_{2}(w)
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2023-09-30
Commutators of the maximal functions on Banach function spaces
Müjdat Ağcayazı, Pu ZhangAbstract : Let $M$ and $M^{\#}$ be Hardy-Littlewood maximal operator and sharp maximal operator, respectively. In this article, we present necessary and sufficient conditions for the boundedness properties for commutator operators $[M,b]$ and $[M^{\#},b]$ in a general context of Banach function spaces when $b$ belongs to $\operatorname{BMO}(\mathbb{R}^{n})$ spaces. Some applications of the results on weighted Lebesgue spaces, variable Lebesgue spaces, Orlicz spaces and Musielak--Orlicz spaces are also given.
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2023-05-31
An altered group ring construction of the $[24,12,8]$ and $[48,24,12]$ Type II linear block code
Shefali Gupta, Dinesh UdarAbstract : In this paper, we present a new construction for self-dual codes that uses the concept of double bordered construction, group rings, and reverse circulant matrices. Using groups of orders $2,3,4,$ and $5$, and by applying the construction over the binary field and the ring $F_{2}+uF_{2}$, we obtain extremal binary self-dual codes of various lengths: $12, 16, 20, 24, 32, 40,$ and $48$. In particular, we show the significance of this new construction by constructing the unique Extended Binary Golay Code $[24,12,8]$ and the unique Extended Quadratic Residue $[48,24,12]$ Type II linear block code. Moreover, we strengthen the existing relationship between units and non-units with the self-dual codes presented in [10] by limiting the conditions given in the corollary. Additionally, we establish a relationship between idempotent and self-dual codes, which is done for the first time in the literature.
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2023-03-31
Pricing American lookback options under a stochastic volatility model
Donghyun Kim, Junhui Woo, Ji-Hun YoonAbstract : In this study, we deal with American lookback option prices on dividend-paying assets under a stochastic volatility (SV) model. By using the asymptotic analysis introduced by Fouque et al. [17] and the Laplace-Carson transform (LCT), we derive the explicit formula for the option prices and the free boundary values with a finite expiration whose volatility is driven by a fast mean-reverting Ornstein-Uhlenbeck process. In addition, we examine the numerical implications of the SV on the American lookback option with respect to the model parameters and verify that the obtained explicit analytical option price has been obtained accurately and efficiently in comparison with the price obtained from the Monte-Carlo simulation.
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2023-01-31
Bisectors in the Heisenberg group I
Gaoshun Gou, Yueping Jiang, Ioannis D. PlatisAbstract : We show that metric bisectors with respect to the Kor\'anyi metric in the Heisenberg group are spinal spheres and vice versa. We also calculate explicitly their horizontal mean curvature.
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Rings and modules which are stable under nilpotents of their injective hulls
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On the geometry of complex metallic Norden manifolds
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Colocalization of generalized local homology modules
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Bull. Korean Math. Soc. 2022; 59(4): 917-928
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Existence of the continued fractions of $\sqrt{d}$ and its applications
Jun Ho Lee
Bull. Korean Math. Soc. 2022; 59(3): 697-707
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Sharp inequalities involving the Chen-Ricci inequality for slant Riemannian submersions
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Fixed-width partitions according to the parity of the even parts
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