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Korean Mathematical Society
BKMS
ISSN(Print) 1015-8634
ISSN(Online) 2234-3016
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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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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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2023-01-31
The nilpotency of the prime radical of a Goldie module
John A. Beachy, Mauricio Medina-B\'arcenasAbstract : 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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2024-01-31
$C^*$-algebra of local conjugacy equivalence relation on strongly irreducible subshift of finite type
Chengjun Hou, Xiangqi QiangAbstract : Let $G$ be an infinite countable group and $A$ be a finite set. If $ \Sigma \subseteq A^{G}$ is a strongly irreducible subshift of finite type and $\mathcal{G}$ is the local conjugacy equivalence relation on $ \Sigma$. We construct a decreasing sequence $\mathcal{R}$ of unital $C^*$-subalgebras of $C(\Sigma)$ and a sequence of faithful conditional expectations $\mathcal{E}$ defined on $C(\Sigma)$, and obtain a Toeplitz algebra $\mathcal{T}(\mathcal{R},\mathcal{E})$ and a $C^*$-algebra $C^*(\mathcal{R},\mathcal{E})$ for the pair $(\mathcal{R},\mathcal{E})$. We show that $C^*(\mathcal{R},\mathcal{E})$ is $\ast$-isomorphic to the reduced groupoid $C^*$-algebra $C_r^*(\mathcal{G})$.
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2023-05-31
Compact composition operators on Besov spaces on the unit ball
Chao ZhangAbstract : In this paper, we give new necessary and sufficient conditions for the compactness of composition operator on the Besov space and the Bloch space of the unit ball, which, to a certain extent, generalizes the results given by M. Tjani in [10].
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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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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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2022-07-31
Magnetic geodesics on the space of K\"{a}hler potentials
Sibel \c{S}ahinAbstract : In this work, magnetic geodesics over the space of K\"{a}hler potentials are studied through a variational method for a generalized Landau-Hall functional. The magnetic geodesic equation is calculated in this setting and its relation to a perturbed complex Monge-Amp\`{e}re equation is given. Lastly, the magnetic geodesic equation is considered over the special case of toric K\"{a}hler potentials over toric K\"{a}hler manifolds.
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2022-11-30
Glift codes over chain ring and non-chain ring $R_{e,s}$
Elif Segah OztasAbstract : In this paper, Glift codes, generalized lifted polynomials, matrices are introduced. The advantage of Glift code is ``distance preserving" over the ring $\mathcal{R}$. Then optimal codes can be obtained over the rings by using Glift codes and lifted polynomials. Zero divisors are classified to satisfy ``distance preserving" for codes over non-chain rings. Moreover, Glift codes apply on MDS codes and MDS codes are obtained over the ring $\mathcal{R}$ and the non-chain ring $\mathcal{R}_{e,s}$.
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Some factorization properties of idealization in commutative rings with zero divisors
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