Bulletin of the
Korean Mathematical Society
BKMS
ISSN(Print) 1015-8634
ISSN(Online) 2234-3016
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2023-07-31
$A_{\alpha}$-spectral extrema of graphs with given size and matching number
Xingyu Lei, Shuchao Li, Jianfeng WangAbstract : In 2017, Nikiforov proposed the $A_{\alpha}$-matrix of a graph $G$. This novel matrix is defined as $$A_{\alpha}(G)=\alpha D(G)+(1- \alpha )A(G),~\alpha \in [0,1],$$ where $D(G)$ and $A(G)$ are the degree diagonal matrix and adjacency matrix of $G$, respectively. Recently, Zhai, Xue and Liu [39] considered the Brualdi-Hoffman-type problem for $Q$-spectra of graphs with given matching number. As a continuance of it, in this contribution we consider the Brualdi-Hoffman-type problem for $A_{\alpha}$-spectra of graphs with given matching number. We identify the graphs with given size and matching number having the largest $A_{\alpha}$-spectral radius for $\alpha \in [\frac{1}{2},1)$.
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2023-11-30
Some one-dimensional Noetherian domains and G-projective modules
Kui Hu, Hwankoo Kim, Dechuan ZhouAbstract : Let $R$ be a one-dimensional Noetherian domain with quotient field $K$ and $T$ be the integral closure of $R$ in $K$. In this note we prove that if the conductor ideal $(R:_KT)$ is a nonzero prime ideal, then every finitely generated reflexive (and hence finitely generated $G$-projective) $R$-module is isomorphic to a direct sum of some ideals.
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2023-05-31
Characterization of weakly cofinite local cohomology modules
Moharram Aghapournahr, Marziye HatamkhaniAbstract : Let $R$ be a commutative Noetherian ring, $\mathfrak{a}$ an ideal of $R$,$M$ an arbitrary $R$-module and $X$ a finite $R$-module. We prove a characterization for ${H} \DeclareMathOperator{\End}{End}_{\mathfrak{a}}^{i}(M)$ and ${H} \DeclareMathOperator{\End}{End}_{\mathfrak{a}}^{i}(X,M)$ to be $\mathfrak{a}$-weakly cofinite for all $i$, whenever one of the following cases holds:(a) ${ara} (\mathfrak{a})\leq 1$, (b) $\dim R/\mathfrak{a} \leq 1$ or (c) $\dim R\leq 2$. We alsoprove that, if $M$ is a weakly Laskerian $R$-module, then ${H} \DeclareMathOperator{\End}{End}_{\mathfrak{a}}^{i}(X,M)$ is $\mathfrak{a}$-weakly cofinite for all $i$, whenever $\dim X\leq 2$ or $\dim M\leq 2$ (resp.$(R,\mathfrak{m})$ a local ring and $\dim X\leq 3$ or $\dim M\leq 3$). Let $d=\dim M<\infty$, we prove an equivalent condition for top local cohomology module ${H} \DeclareMathOperator{\End}{End}_{\mathfrak{a}}^{d}(M)$ to be weakly Artinian.
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2023-09-30
Relationship between the structure of a factor ring $R/P$ and derivations of $R$
Karim Bouchannafa, Moulay Abdallah Idrissi, Lahcen OukhtiteAbstract : The purpose of this paper is to study the relationship between the structure of a factor ring $R/P$ and the behavior of some derivations of $R$. More precisely, we establish a connection between the commutativity of $R/P$ and derivations of $R$ satisfying specific identities involving the prime ideal $P$. Moreover, we provide an example to show that our results cannot be extended to semi-prime ideals.
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2023-07-31
Some results on 2-strongly Gorenstein projective modules and related rings
Dong Chen, Kui HuAbstract : In this paper, we give some results on 2-strongly Gorenstein projective modules and related rings. We first investigate the relationship between strongly Gorenstein projective modules and periodic modules and then give the structure of modules over strongly Gorenstein semisimple rings. Furthermore, we prove that a ring $R$ is 2-strongly Gorenstein hereditary if and only if every ideal of $R$ is Gorenstein projective and the class of 2-strongly Gorenstein projective modules is closed under extensions. Finally, we study the relationship between 2-Gorenstein projective hereditary and 2-Gorenstein projective semisimple rings, and we also give an example to show the quotient ring of a 2-Gorenstein projective hereditary ring is not necessarily 2-Gorenstein projective semisimple.
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2023-05-31
On the hybrid mean value of generalized Dedekind sums, generalized Hardy sums and Kloosterman sums
Qing Tian, Yan WangAbstract : The main purpose of this paper is to study the hybrid mean value problem involving generalized Dedekind sums, generalized Hardy sums and Kloosterman sums. Some exact computational formulas are given by using the properties of Gauss sums and the mean value theorem of the Dirichlet L-function. A result of W. Peng and T. P. Zhang [12] is extended. The new results avoid the restriction that $q$ is a prime.
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2023-11-30
Results on the algebraic differential independence of the Riemann zeta function and the Euler gamma function
Xiao-Min Li, Yi-Xuan LiAbstract : In 2010, Li-Ye [13, Theorem 0.1] proved that \begin{equation}\nonumber P\left(\zeta(z),\zeta'(z),\ldots,\zeta^{(m)}(z),\Gamma(z),\Gamma'(z),\Gamma^{''}(z)\right) \not\equiv 0\quad\text{in }\ \mathbb{C}, \end{equation} where $m$ is a non-negative integer, and $P(u_{0},u_{1}, \ldots, u_{m},v_{0},v_{1},v_{2})$ is any non-trivial polynomial in its arguments with coefficients in the field $\mathbb{C}$. Later on, Li-Ye [15, Theorem 1] proved that \begin{equation}\nonumber P\left(z,\Gamma(z),\Gamma'(z),\ldots,\Gamma^{(n)}(z), \zeta(z)\right)\not\equiv 0 \end{equation} in $z\in \mathbb{C}$ for any non-trivial distinguished polynomial $P(z,u_0, u_1,\ldots$, $u_n, v)$ with coefficients in a set $L_\delta$ of the zero function and a class of non-zero functions $f$ from $\mathbb{C}$ to $\mathbb{C}\cup\{\infty\}$ (cf. [15, Definition 1]). In this paper, we prove that $P\left(z,\zeta(z),\zeta'(z),\ldots,\zeta^{(m)}(z),\Gamma(z),\Gamma'(z),\ldots,\Gamma^{(n)}(z)\right)\not\equiv 0$ in $z\in\mathbb{C}$, where $m$ and $n$ are two non-negative integers, and $$P(z, u_0,u_1,\ldots,u_m,v_0,v_1,\ldots,v_n)$$ is any non-trivial polynomial in the $m+n+2$ variables $$u_0,u_1,\ldots,u_m,v_0,v_1,\ldots,v_n$$ with coefficients being meromorphic functions of order less than one, and the polynomial $P(z, u_0,u_1,\ldots,u_m,v_0,v_1,\ldots,v_n)$ is a distinguished polynomial in the $n+1$ variables $v_0,v_1,\ldots, v_n$. The question studied in this paper is concerning the conjecture of Markus from [16]. The main results obtained in this paper also extend the corresponding results from Li-Ye [12] and improve the corresponding results from Chen-Wang [5] and Wang-Li-Liu-Li [23], respectively.
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2023-07-31
Fixed-width partitions according to the parity of the even parts
John Maxwell CampbellAbstract : A celebrated result in the study of integer partitions is the identity due to Lehmer whereby the number of partitions of $n$ with an even number of even parts minus the number of partitions of $n$ with an odd number of even parts equals the number of partitions of $n$ into distinct odd parts. Inspired by Lehmer's identity, we prove explicit formulas for evaluating generating functions for sequences that enumerate integer partitions of fixed width with an even/odd number of even parts. We introduce a technique for decomposing the even entries of a partition in such a way so as to evaluate, using a finite sum over $q$-binomial coefficients, the generating function for the sequence of partitions with an even number of even parts of fixed, odd width, and similarly for the other families of fixed-width partitions that we introduce.
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2023-11-30
Bredon homology of wallpaper groups
Ramon FloresAbstract : In this paper we compute the Bredon homology of wallpaper groups with respect to the family of finite groups and with coefficients in the complex representation ring. We provide explicit bases of the homology groups in terms of irreducible characters of the stabilizers.
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2023-05-31
Erratum/Addendum to ``Biisometric operators and biorthogonal sequences'' [Bull. Korean Math. Soc. {\bf 56} (2019), No. 3, pp. 585--596]
Carlos Kubrusly, Nhan LevanAbstract : Erratum/Addendum to the paper ``Biisometric operators and biorthogonal sequences" [Bull. Korean Math. Soc. {\bf 56} (2019), No. 3, pp. 585--596].
Most Read
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Szeg\"{o} projections for Hardy spaces in quaternionic Clifford analysis
Fuli He
, Song Huang, Min KuBull. Korean Math. Soc. 2022; 59(5): 1215-1235
https://doi.org/10.4134/BKMS.b210688 -
Knots in homology lens spaces determined by their complements
Kazuhiro Ichihara
, Toshio SaitoBull. Korean Math. Soc. 2022; 59(4): 869-877
https://doi.org/10.4134/BKMS.b210503 -
S-curvature and geodesic orbit property of invariant $(\alpha_{1},\alpha_{2})$-metrics on spheres
Huihui An, Zaili Yan, Shaoxiang Zhang
Bull. Korean Math. Soc. 2023; 60(1): 33-46
https://doi.org/10.4134/BKMS.b210835 -
The K\"{a}hler Different of a Set of Points in~$\mathbb{P}^{m}\!\times\mathbb{P}^{n}$
Nguyen T. Hoa, Tran N. K. Linh, Le N. Long, Phan T. T. Nhan, Nguyen T. P. Nhi
Bull. Korean Math. Soc. 2022; 59(4): 929-949
https://doi.org/10.4134/BKMS.b210544
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Homogeneous geodesics in homogeneous sub-Finsler manifolds
Zaili Yan, Tao Zhou
Bull. Korean Math. Soc. 2023; 60(6): 1607-1620
https://doi.org/10.4134/BKMS.b220758 -
Singularity formation for a nonlinear variational sine-Gordon equation in a multidimensional space
Fengmei Qin, Kyungwoo Song, Qin Wang
Bull. Korean Math. Soc. 2023; 60(6): 1697-1704
https://doi.org/10.4134/BKMS.b220859 -
Two-weight norm estimates for square functions associated to fractional Schr{\"o}dinger operators with Hardy potential
Tongxin Kang, Yang Zou
Bull. Korean Math. Soc. 2023; 60(6): 1567-1605
https://doi.org/10.4134/BKMS.b220752 -
Weak factorizations of $H^1(\mathbb{R}^n)$ in terms of multilinear fractional integral operator on variable Lebesgue spaces
Zongguang Liu, Huan Zhao
Bull. Korean Math. Soc. 2023; 60(6): 1439-1451
https://doi.org/10.4134/BKMS.b220496
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