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Bull. Korean Math. Soc. 2023; 60(4): 849~1139
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2023-07-31
Semi-symmetric structure Jacobi operator for real hypersurfaces in the complex quadric
Imsoon Jeong, Gyu Jong Kim, Changhwa WooAbstract : In this paper, we introduce the notion of {\it semi-symmetric structure Jacobi operator } for Hopf real hypersufaces in the complex quad\-ric $Q^m = SO_{m+2}/SO_mSO_2$. Next we prove that there does not exist any Hopf real hypersurface in the complex quadric $Q^m = SO_{m+2}/SO_mSO_2$ with semi-symmetric structure Jacobi operator. As a corollary, we also get a non-existence property of Hopf real hypersurfaces in the complex quadric $Q^m$ with either symmetric (parallel), or recurrent structure Jacobi operator.
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2023-07-31
On the greatest common divisor of binomial coefficients
Sunben Chiu, Pingzhi Yuan, Tao ZhouAbstract : Let $n\geqslant 2$ be an integer, we denote the smallest integer $b$ such that $\gcd\qty{\binom nk: b<k<n-b}>1$ as $b(n)$. For any prime $p$, we denote the highest exponent $\alpha$ such that $p^\alpha\mid n$ as $v_p(n)$. In this paper, we partially answer a question asked by Hong in 2016. For a composite number $n$ and a prime number $p$ with $p\mid n$, let $n=a_mp^m+r$, $0\leqslant r<p^m$, $0<a_m<p$. Then we have\\ \resizebox{\linewidth}{4.5mm}{ $\displaystyle v_p\qty(\gcd\qty{\binom nk: b(n)<k<n-b(n),\ (n,k)>1})= \begin{cases} 1,&a_m=1\text{ and }r=b(n), \\ 0,&\text{otherwise}. \end{cases} $}
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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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2023-07-31
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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2023-07-31
An extension of Schneider's characterization theorem for ellipsoids
Dong-Soo Kim, Young Ho KimAbstract : Suppose that $M$ is a strictly convex hypersurface in the $(n+1)$-dimensional Euclidean space ${\mathbb E}^{n+1}$ with the origin $o$ in its convex side and with the outward unit normal $N$. For a fixed point $p \in M$ and a positive constant $t$, we put $\Phi_t$ the hyperplane parallel to the tangent hyperplane $\Phi$ at $p$ and passing through the point $q=p-tN(p)$. We consider the region cut from $M$ by the parallel hyperplane $\Phi_t$, and denote by $I_p(t)$ the $(n+1)$-dimensional volume of the convex hull of the region and the origin $o$. Then Schneider's characterization theorem for ellipsoids states that among centrally symmetric, strictly convex and closed surfaces in the 3-dimensional Euclidean space ${\mathbb E}^{3}$, the ellipsoids are the only ones satisfying $I_p(t)=\phi(p)t$, where $\phi$ is a function defined on $M$. Recently, the characterization theorem was extended to centrally symmetric, strictly convex and closed hypersurfaces in ${\mathbb E}^{n+1}$ satisfying for a constant $\beta$, $I_p(t)=\phi(p)t^{\beta}$. In this paper, we study the volume $I_p(t)$ of a strictly convex and complete hypersurface in ${\mathbb E}^{n+1}$ with the origin $o$ in its convex side. As a result, first of all we extend the characterization theorem to strictly convex and closed (not necessarily centrally symmetric) hypersurfaces in ${\mathbb E}^{n+1}$ satisfying $I_p(t)=\phi(p)t^{\beta}$. After that we generalize the characterization theorem to strictly convex and complete (not necessarily closed) hypersurfaces in ${\mathbb E}^{n+1}$ satisfying $I_p(t)=\phi(p)t^{\beta}$.
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2023-07-31
On the structure of certain subset of Farey sequence
Xing-Wang Jiang, Ya-Li LiAbstract : Let $F_n$ be the Farey sequence of order $n$. For $S\subseteq F_n$, let $\mathcal{Q}(S)$ be the set of rational numbers $x/y$ with $x,y\in S,~x\leq y$ and $y\neq 0$. Recently, Wang found all subsets $S$ of $F_n$ with $|S|=n+1$ for which $\mathcal{Q}(S)\subseteq F_n$. Motivated by this work, we try to determine the structure of $S\subseteq F_n$ such that $|S|=n$ and $\mathcal{Q}(S)\subseteq F_n$. In this paper, we determine all sets $S\subseteq F_n$ satisfying these conditions for $n\in\{p,2p\}$, where $p$ is prime.
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2023-07-31
On the $p$-adic valuation of generalized harmonic numbers
Çağatay AltuntaşAbstract : For any prime number $p$, let $J(p)$ be the set of positive integers $n$ such that the numerator of the $n^{th}$ harmonic number in the lowest terms is divisible by this prime number $p$. We consider an extension of this set to the generalized harmonic numbers, which are a natural extension of the harmonic numbers. Then, we present an upper bound for the number of elements in this set. Moreover, we state an explicit condition to show the finiteness of our set, together with relations to Bernoulli and Euler numbers.
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2023-07-31
Toeplitz-type operators on the Fock space $F_{\alpha}^{2}$
Chunxu Xu, Tao YuAbstract : Let $j$ be a nonnegative integer. We define the Toeplitz-type operators $T_{a}^{(j)}$ with symbol $a\in L^{\infty}(C)$, which are variants of the traditional Toeplitz operators obtained for $j=0$. In this paper, we study the boundedness of these operators and characterize their compactness in terms of its Berezin transform.
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2023-07-31
Copure projective modules over FGV-domains and Gorenstein Pr\"{u}fer domains
Shiqi XingAbstract : In this paper, we prove that a domain $R$ is an FGV-domain if every finitely generated torsion-free $R$-module is strongly copure projective, and a coherent domain is an FGV-domain if and only if every finitely generated torsion-free $R$-module is strongly copure projective. To do this, we characterize G-Pr\"{u}fer domains by G-flat modules, and we prove that a domain is G-Pr\"{u}fer if and only if every submodule of a projective module is G-flat. Also, we study the $D+M$ construction of G-Pr\"{u}fer domains. It is seen that there exists a non-integrally closed G-Pr\"{u}fer domain that is neither Noetherian nor divisorial.
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2023-07-31
Some relations on parametric linear Euler sums
Weiguo Lu, Ce Xu, Jianing ZhouAbstract : Recently, Alzer and Choi [2] introduced and studied a set of the four linear Euler sums with parameters. These sums are parametric extensions of Flajolet and Salvy's four kinds of linear Euler sums [9]. In this paper, by using the method of residue computations, we will establish two explicit combined formulas involving two parametric linear Euler sums $S_{p,q}^{++}(a,b)$ and $S_{p,q}^{+-}(a,b)$ defined by Alzer and Choi, which can be expressed in terms of a linear combinations of products of trigonometric functions, digamma functions and Hurwitz zeta functions.
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