Bulletin of the
Korean Mathematical Society
BKMS
ISSN(Print) 1015-8634
ISSN(Online) 2234-3016
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2022-05-31
Existence of the continued fractions of $\sqrt{d}$ and its applications
Jun Ho LeeAbstract : It is well known that the continued fraction expansion of $\sqrt{d}$ has the form $[a_0, \overline{a_1, \ldots, a_{l-1}, 2a_0}]$ and $a_1, \ldots, a_{l-1}$ is a palindromic sequence of positive integers. For a given positive integer $l$ and a palindromic sequence of positive integers $a_1, \ldots, a_{l-1}$, we define the set $S(l;a_1,$ $\ldots, a_{l-1}) :=\{d\in \mathbb{Z} \,| \, d>0, \sqrt{d}=[a_0, \overline{a_1, \ldots, a_{l-1}, 2a_0}], \, \textup{where} \, a_0=\lfloor \sqrt{d} \rfloor\}$. In this paper, we completely determine when $S(l;a_1, \ldots, a_{l-1})$ is not empty in the case that $l$ is $4$, $5$, $6$, or $7$. We also give similar results for $(1+\sqrt{d})/2$. For the case that $l$ is $4$, $5$, or $6$, we explicitly describe the fundamental units of the real quadratic field $\mathbb{Q}(\sqrt{d})$. Finally, we apply our results to the Mordell conjecture for the fundamental units of $\mathbb{Q}(\sqrt{d})$.
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2023-09-30
Complex symmetric weighted composition-differentiation operators on $H^2$
Lian Hu, Songxiao Li, Rong YangAbstract : In this paper, we study the complex symmetric weighted composition-differentiation operator $D_{\psi,\phi}$ with respect to the conjugation $ JW_{\xi, \tau}$ on the Hardy space $H^2$. As an application, we characterize the necessary and sufficient conditions for such an operator to be normal under some mild conditions. Finally, the spectrum of $D_{\psi,\phi}$ is also investigated.
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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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2023-01-31
Stability and topology of translating solitons for the mean curvature flow with the small $L^m$ norm of the second fundamental form
Eungmo Nam, Juncheol PyoAbstract : In this paper, we show that a complete translating soliton $\Sigma^m$ in $\mathbb R^n$ for the mean curvature flow is stable with respect to weighted volume functional if $\Sigma$ satisfies that the $L^m$ norm of the second fundamental form is smaller than an explicit constant that depends only on the dimension of $\Sigma$ and the Sobolev constant provided in Michael and Simon [12]. Under the same assumption, we also prove that under this upper bound, there is no non-trivial $f$-harmonic $1$-form of $L^2_f$ on $\Sigma$. With the additional assumption that $\Sigma$ is contained in an upper half-space with respect to the translating direction then it has only one end.
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2023-09-30
Sharp inequalities involving the Chen-Ricci inequality for slant Riemannian submersions
Mehmet Akif Akyol, Nergiz (Önen) PoyrazAbstract : Main objective of the present paper is to establish Chen inequalities for slant Riemannian submersions in contact geometry. In this manner, we give some examples for slant Riemannian submersions and also investigate some curvature relations between the total space, the base space and fibers. Moreover, we establish Chen-Ricci inequalities on the vertical and the horizontal distributions for slant Riemannian submersions from Sasakian space forms.
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2023-05-31
$p$-Biharmonic hypersurfaces in Einstein space and conformally flat space
Ahmed Mohammed Cherif, Khadidja MouffokiAbstract : In this paper, we present some new properties for $p$-biharmon\-ic hypersurfaces in a Riemannian manifold. We also characterize the $p$-biharmonic submanifolds in an Einstein space. We construct a new example of proper $p$-biharmonic hypersurfaces. We present some open problems.
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2022-07-31
A new characterization of type $(A)$ and ruled real hypersurfaces in nonflat complex space forms
Yaning WangAbstract : In this paper, we obtain an inequality involving the squared norm of the covariant differentiation of the shape operator for a real hypersurface in nonflat complex space forms. It is proved that the equality holds for non-Hopf case if and only if the hypersurface is ruled and the equality holds for Hopf case if and only if the hypersurface is of type $(A)$.
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2023-11-30
On a generalization of unit regular rings
Tahire OzenAbstract : In this paper, we introduce a class of rings generalizing unit regular rings and being a subclass of semipotent rings, which is called idempotent unit regular. We call a ring $R$ an idempotent unit regular ring if for all $r\in R-J(R)$, there exist a non-zero idempotent $e$ and a unit element $u$ in $R$ such that $er=eu$, where this condition is left and right symmetric. Thus, we have also that there exist a non-zero idempotent $e$ and a unit $u$ such that $ere=eue$ for all $r\in R-J(R)$. Various basic characterizations and properties of this class of rings are proved and it is given the relationships between this class of rings and some well-known classes of rings such as semiperfect, clean, exchange and semipotent. Moreover, we obtain some results about when the endomorphism ring of a module in a class of left $R$-modules $X$ is idempotent unit regular.
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2023-09-30
Structure of idempotents in polynomial rings and matrix rings
Juan Huang, Tai Keun Kwak, Yang Lee, Zhelin PiaoAbstract : An idempotent $e$ of a ring $R$ is called {\it right} (resp., {\it left}) {\it semicentral} if $er=ere$ (resp., $re =ere$) for any $r\in R$, and an idempotent $e$ of $R\backslash \{0,1\}$ will be called {\it right} (resp., {\it left}) {\it quasicentral} provided that for any $r\in R$, there exists an idempotent $f=f(e,r)\in R\backslash \{0,1\}$ such that $er=erf$ (resp., $re=fre$). We show the whole shapes of idempotents and right (left) semicentral idempotents of upper triangular matrix rings and polynomial rings. We next prove that every nontrivial idempotent of the $n$ by $n$ full matrix ring over a principal ideal domain is right and left quasicentral and, applying this result, we can find many right (left) quasicentral idempotents but not right (left) semicentral.
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2023-03-31
Uniqueness results on meromorphic functions and their difference operators sharing targets with weight
Thu Thuy Hoang, Hong Nhat Nguyen, Duc Thoan PhamAbstract : Let $f$ be a nonconstant meromorphic function of hyper-order strictly less than 1, and let $c\in\mathbb C\setminus\{0\}$ such that $f(z + c) \not\equiv f(z)$. We prove that if $f$ and its exact difference $\Delta_cf(z) = f(z + c) - f(z)$ share partially $0, \infty$ CM and share 1 IM, then $\Delta_cf = f$, where all 1-points with multiplicities more than 2 do not need to be counted. Some similar uniqueness results for such meromorphic functions partially sharing targets with weight and their shifts are also given. Our results generalize and improve the recent important results.
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The third Hermitian-Toeplitz and Hankel determinants for parabolic starlike functions
Rosihan M. Ali, Sushil Kumar, Vaithiyanathan Ravichandran
Bull. Korean Math. Soc. 2023; 60(2): 281-291
https://doi.org/10.4134/BKMS.b210368 -
Integrability of an almost complex structure on $S^4\times_f V^2$
Jong Taek Cho, Sun Hyang Chun, Yunhee Euh
Bull. Korean Math. Soc. 2022; 59(4): 801-810
https://doi.org/10.4134/BKMS.b200606 -
An associated sequence of ideals of an increasing sequence of rings
Ali Benhissi, Abdelamir Dabbabi
Bull. Korean Math. Soc. 2022; 59(6): 1349-1357
https://doi.org/10.4134/BKMS.b210425 -
$P$-extremal functions and Bernstein-Markov properties associated to compact sets in $\mathbb R^d$
Hoang Thieu Anh, Kieu Phuong Chi, Nguyen Quang Dieu, Tang Van Long
Bull. Korean Math. Soc. 2022; 59(4): 811-825
https://doi.org/10.4134/BKMS.b210341
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On the structure of certain subset of Farey sequence
Xing-Wang Jiang, Ya-Li Li
Bull. Korean Math. Soc. 2023; 60(4): 915-931
https://doi.org/10.4134/BKMS.b220396 -
Some results on 2-strongly Gorenstein projective modules and related rings
Dong Chen, Kui Hu
Bull. Korean Math. Soc. 2023; 60(4): 895-903
https://doi.org/10.4134/BKMS.b220392 -
Hypersurfaces with prescribed mean curvature in measure metric space
Zhengmao Chen
Bull. Korean Math. Soc. 2023; 60(4): 1085-1100
https://doi.org/10.4134/BKMS.b220531 -
UN rings and group rings
Kanchan Jangra, Dinesh Udar
Bull. Korean Math. Soc. 2023; 60(1): 83-91
https://doi.org/10.4134/BKMS.b210917
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