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ISSN(Online) 2234-3016
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2022-07-31
Colocalization of generalized local homology modules
Marziyeh HatamkhaniAbstract : Let $R$ be a commutative Noetherian ring and $I$ an ideal of $R$. In this paper, we study colocalization of generalized local homology modules. We intend to establish a dual case of local-global principle for the finiteness of generalized local cohomology modules. Let $M$ be a finitely generated $R$-module and $N$ a representable $R$-module. We introduce the notions of the representation dimension $r^I (M, N)$ and artinianness dimension $a^I (M, N)$ of $M,N$ with respect to $I$ by $r^I (M, N)= \inf\{i\in \mathbb{N}_0 : H^I_i(M,N) \text{ is not representable}\}$ and $a^I (M, N)= \inf\{i\in \mathbb{N}_0 : H^I_i(M,N)\text{ is not artinian}\}$ and we show that $a^I (M, N)=r^I (M, N)$ $=\inf\{r^{IR_{\mathfrak{p}}}(M_{\mathfrak{p}},_{\mathfrak{p}}N) : \mathfrak{p}\in \operatorname{Spec}(R)\}\geq \inf\{a^{IR_{\mathfrak{p}}}(M_{\mathfrak{p}},_{\mathfrak{p}}N) : \mathfrak{p}\in \operatorname{Spec}(R)\}$. Also, in the case where $R$ is semi-local and $N$ a semi discrete linearly compact $R$-module such that $N/\bigcap_{t>0} I^tN$ is artinian we prove that $\inf\{i: H^I_i(M,N) \text{ is not minimax}\}\!=\!\inf\{r^{IR_{\mathfrak{p}}}(M_{\mathfrak{p}},_{\mathfrak{p}}N) : \mathfrak{p}\in \operatorname{Spec}(R)\setminus\operatorname{Max}(R)\}.$
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2022-09-30
Estimates for the Riesz transforms associated with Schr\"odinger type operators on the Heisenberg group
Yanhui WangAbstract : We consider the Schr\"odinger type operator \(\mathcal{L}=(-\Delta_{\mathbb{H}^n})^2+V^2 \) on the Heisenberg group $\mathbb{H}^n,$ where $\Delta_{\mathbb{H}^n}$ is the sub-Laplacian and the non-negative potential \(V\) belongs to the reverse H\"older class \(RH_s$ for $ s\geq Q/2$ and $Q\geq 6.\) We shall establish the $(L^p,L^q)$ estimates for the Riesz transforms $ T_{\alpha,\beta,j} =V^{2\alpha}\nabla_{\mathbb{H}^n}^j \mathcal{L}^{-\beta},~j=0,1,2,3,$ where $\nabla_{\mathbb{H}^n}$ is the gradient operator on $\mathbb{H}^n,~ 0
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2022-05-31
Finiteness and vanishing results on hypersurfaces with finite index in $\mathbb{R}^{n+1}$: a revision
Nguyen Van DucAbstract : In this note, we revise some vanishing and finiteness results on hypersurfaces with finite index in $\mathbb{R}^{n+1}$. When the hypersurface is stable minimal, we show that there is no nontrivial $L^{2p}$ harmonic $1$-form for some $p$. The our range of $p$ is better than those in [7]. With the same range of $p$, we also give finiteness results on minimal hypersurfaces with finite index.
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2023-01-31
The nilpotency of the prime radical of a Goldie module
John A. Beachy, Mauricio Medina-B\'arcenas
Abstract : With the notion of prime submodule defined by F. Raggi et al. we prove that the intersection of all prime submodules of a Goldie module $M$ is a nilpotent submodule provided that $M$ is retractable and $M^{(\Lambda)}$-projective for every index set $\Lambda$. This extends the well known fact that in a left Goldie ring the prime radical is nilpotent.
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2023-01-31
Inductive limit in the category of $C^{\ast}$-ternary rings
Arpit Kansal, Ajay Kumar, Vandana RajpalAbstract : We show the existence of inductive limit in the category of $C^{\ast}$-ternary rings. It is proved that the inductive limit of $C^{\ast}$-ternary rings commutes with the functor $\mathcal{A}$ in the sense that if $(M_n, \phi_n)$ is an inductive system of $C^{\ast}$-ternary rings, then $\varinjlim \mathcal{A}(M_n)=\mathcal{A}(\varinjlim M_n)$. Some local properties (such as nuclearity, exactness and simplicity) of inductive limit of $C^{\ast}$-ternary rings have been investigated. Finally we obtain $\varinjlim M_n^{\ast\ast}=(\varinjlim M_n)^{\ast\ast}$.
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2022-09-30
Szeg\"{o} projections for Hardy spaces in quaternionic Clifford analysis
Fuli He, Song Huang, Min KuAbstract : In this paper we study Szeg\"{o} kernel projections for Hardy spaces in quaternionic Clifford analysis. At first we introduce the matrix Szeg\"{o} projection operator for the Hardy space of quaternionic Hermitean monogenic functions by the characterization of the matrix Hilbert transform in the quaternionic Clifford analysis. Then we establish the Kerzman-Stein formula which closely connects the matrix Szeg\"{o} projection operator with the Hardy projection operator onto the Hardy space, and we get the matrix Szeg\"{o} projection operator in terms of the Hardy projection operator and its adjoint. At last, we construct the explicit matrix Szeg\"o kernel function for the Hardy space on the sphere as an example, and get the solution to a Diriclet boundary value problem for matrix functions.
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2022-11-30
Tamed exhaustion functions and Schwarz type lemmas for almost Hermitian manifolds
Weike YuAbstract : In this paper, we study a special exhaustion function on almost Hermitian manifolds and establish the existence result by using the Hessian comparison theorem. From the viewpoint of the exhaustion function, we establish a related Schwarz type lemma for almost holomorphic maps between two almost Hermitian manifolds. As corollaries, we deduce several versions of Schwarz and Liouville type theorems for almost holomorphic maps.
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2022-11-30
Irreducibility of the moduli space for the quotient singularity $\frac{1}{2k+1}(k+1,1,2k)$
Seung-Jo JungAbstract : A 3-fold quotient terminal singularity is of the type $\frac{1}{r}(b,1$, $-1)$ with $\gcd(r,b)=1$. In \cite{Terminal}, it is proved that the economic resolution of a 3-fold terminal quotient singularity is isomorphic to a distinguished component of a moduli space $\mathcal M_{\theta}$ of $\theta$-stable $G$-constellations for a suitable $\theta$. This paper proves that each connected component of the moduli space $\\mathcal M_{\theta}$ has a torus fixed point and classifies all torus fixed points on $\mathcal M_{\theta}$. By product, we show that for $\frac{1}{2k+1}(k+1,1,-1)$ case the moduli space $\mathcal M_{\theta}$ is irreducible.
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2023-05-31
Some results on meromorphic solutions of q-difference differential equations
Lingyun Gao, Zhenguang Gao, Manli LiuAbstract : In view of Nevanlinna theory, we investigate the meromorphic solutions of q-difference differential equations and our results give the estimates about counting function and proximity function of meromorphic solutions to these equations. In addition, some interesting results are obtained for two general equations and a class of system of q-difference differential equations.
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2022-11-30
Modified cyclotomic polynomials
Ae-Kyoung Cha, Miyeon Kwon, Ki-Suk Lee, Seong-Mo YangAbstract : Let $H$ be a subgroup of $\mathbb{Z}_n^\ast$ (the multiplicative group of integers modulo $n$) and $h_1,h_2,\ldots,h_l$ distinct representatives of the cosets of $H$ in $\mathbb{Z}_n^\ast$. We now define a polynomial $J_{n,H}(x)$ to be \begin{align*} \begin{split} J_{n,H}(x)=\prod\limits_{j=1}^{l} \bigg( x-\sum\limits_{h \in H}\zeta_n^{h_jh} \bigg), \end{split} \end{align*} where $\zeta_n=e^{\frac{2\pi i}{n}}$ is the $n$th primitive root of unity. Polynomials of such form generalize the $n$th cyclotomic polynomial $\Phi_n(x)=\prod_{k \in \mathbb{Z}_n^\ast}(x-\zeta_n^k)$ as $J_{n,\{1\}}(x)=\Phi_n(x)$. While the $n$th cyclotomic polynomial $\Phi_n(x)$ is irreducible over $\mathbb{Q}$, $J_{n,H}(x)$ is not necessarily irreducible. In this paper, we determine the subgroups $H$ for which $J_{n,H}(x)$ is irreducible over $\mathbb{Q}$.
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Knots in homology lens spaces determined by their complements
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Starlike functions associated with a petal shaped domain
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Modified cyclotomic polynomials
Ae-Kyoung Cha, Miyeon Kwon, Ki-Suk Lee, Seong-Mo Yang
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Zero sums of dual Toeplitz products on the orthogonal complement of the Dirichlet space
Young Joo Lee
Bull. Korean Math. Soc. 2023; 60(1): 161-170
https://doi.org/10.4134/BKMS.b220042
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On delay differential equations with meromorphic solutions of hyper-order less than one
Risto Korhonen, Yan Liu
Bull. Korean Math. Soc. 2024; 61(1): 229-246
https://doi.org/10.4134/BKMS.b230089 -
Time analyticity for the heat equation under Bakry-\'Emery Ricci curvature condition
Ling Wu
Bull. Korean Math. Soc. 2023; 60(6): 1673-1685
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Commutators of the maximal functions on Banach function spaces
Müjdat Ağcayazı, Pu Zhang
Bull. Korean Math. Soc. 2023; 60(5): 1391-1408
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Pricing American lookback options under a stochastic volatility model
Donghyun Kim, Junhui Woo, Ji-Hun Yoon
Bull. Korean Math. Soc. 2023; 60(2): 361-388
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