Bulletin of the
Korean Mathematical Society
BKMS
ISSN(Print) 1015-8634
ISSN(Online) 2234-3016
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2023-11-30
Automorphisms of K3 surfaces with Picard number two
Kwangwoo LeeAbstract : It is known that the automorphism group of a K3 surface with Picard number two is either an infinite cyclic group or an infinite dihedral group when it is infinite. In this paper, we study the generators of such automorphism groups. We use the eigenvector corresponding to the spectral radius of an automorphism of infinite order to determine the generators.
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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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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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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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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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2023-07-31
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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2023-07-31
Classification of twisted product lightlike submanifolds
Sangeet Kumar, Megha PruthiAbstract : In this paper, we introduce the idea of twisted product lightlike submanifolds of semi-Riemannian manifolds and provide non-trivial examples of such lightlike submanifolds. Then, we prove the non-existence of proper isotropic or totally lightlike twisted product submanifolds of a semi-Riemannian manifold. We also show that for a twisted product lightlike submanifold of a semi-Riemannian manifold, the induced connection $\nabla$ is not a metric connection. Further, we prove that a totally umbilical $SCR$-lightlike submanifold of an indefinite Kaehler manifold $\tilde{M}$ does not admit any twisted product $SCR$-lightlike submanifold of the type $M_{\perp}\times_{\phi}M_{T}$, where $M_{\perp}$ is a totally real submanifold and $M_{T}$ is a holomorphic submanifold of $\tilde{M}$. Consequently, we obtain a geometric inequality for the second fundamental form of twisted product $SCR$-lightlike submanifolds of the type $M_{T}\times_{\phi}M_{\perp}$ of an indefinite Kaehler manifold $\tilde{M}$, in terms of the gradient of $\ln \phi$, where $\phi$ stands for the twisting function. Subsequently, the equality case of this inequality is discussed. Finally, we construct a non-trivial example of a twisted product $SCR$-lightlike submanifold in an indefinite Kaehler manifold.
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2023-05-31
When all permutations are combinatorial similarities
Viktoriia Bilet, Oleksiy DovgosheyAbstract : Let \((X, d)\) be a semimetric space. A permutation \(\Phi\) of the set \(X\) is a combinatorial self similarity of \((X, d)\) if there is a bijective function \(f \colon d(X \times X) \to d(X \times X)\) such that \[ d(x, y) = f(d(\Phi(x), \Phi(y))) \] for all \(x\), \(y \in X\). We describe the set of all semimetrics \(\rho\) on an arbitrary nonempty set \(Y\) for which every permutation of \(Y\) is a combinatorial self similarity of \((Y, \rho)\).
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Integrability of an almost complex structure on $S^4\times_f V^2$
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The third Hermitian-Toeplitz and Hankel determinants for parabolic starlike functions
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$P$-extremal functions and Bernstein-Markov properties associated to compact sets in $\mathbb R^d$
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Some results on 2-strongly Gorenstein projective modules and related rings
Dong Chen, Kui Hu
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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
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Hypersurfaces with prescribed mean curvature in measure metric space
Zhengmao Chen
Bull. Korean Math. Soc. 2023; 60(4): 1085-1100
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A model of retirement and consumption-portfolio choice
Junkee Jeon, Hyeng Keun Koo
Bull. Korean Math. Soc. 2023; 60(4): 1101-1129
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