Bulletin of the
Korean Mathematical Society
BKMS
ISSN(Print) 1015-8634
ISSN(Online) 2234-3016
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2022-09-30
On transcendental meromorphic solutions of certain types of differential equations
Abhijit Banerjee, Tania Biswas, Sayantan MaityAbstract : In this paper, for a transcendental meromorphic function $f$ and $a\in \mathbb{C}$, we have exhaustively studied the nature and form of solutions of a new type of non-linear differential equation of the following form which has never been investigated earlier: \[f^n+af^{n-2}f'+ P_d(z,f) = \sum_{i=1}^{k}p_i(z)e^{\alpha_i(z)}, \] where $P_d(z,f)$ is a differential polynomial of $f$, $p_i$'s and $\alpha_{i}$'s are non-vanishing rational functions and non-constant polynomials, respectively. When $a=0$, we have pointed out a major lacuna in a recent result of Xue [17] and rectifying the result, presented the corrected form of the same equation at a large extent. In addition, our main result is also an improvement of a recent result of Chen-Lian [2] by rectifying a gap in the proof of the theorem of the same paper. The case $a\neq 0$ has also been manipulated to determine the form of the solutions. We also illustrate a handful number of examples for showing the accuracy of our results.
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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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2023-07-31
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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2022-09-30
Enumeration of relaxed complete partitions and double-complete partitions
Suhyung An, Hyunsoo Cho
Abstract : A partition of $n$ is complete if every positive integer from $1$ to $n$ can be represented by the sum of its parts. The concept of complete partitions has been extended in several ways. In this paper, we consider the number of $k$-relaxed $r$-complete partitions of $n$ and the number of double-complete partitions of $n$.
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2023-11-30
Weak factorizations of $H^1(\mathbb{R}^n)$ in terms of multilinear fractional integral operator on variable Lebesgue spaces
Zongguang Liu, Huan ZhaoAbstract : This paper provides a constructive proof of the weak factorizations of the classical Hardy space $H^1(\mathbb{R}^n)$ in terms of multilinear fractional integral operator on the variable Lebesgue spaces, which the result is new even in the linear case. As a direct application, we obtain a new proof of the characterization of $\mathrm{BMO}(\mathbb{R}^n)$ via the boundedness of commutators of the multilinear fractional integral operator on the variable Lebesgue spaces.
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2023-09-30
Commutators of the maximal functions on Banach function spaces
Müjdat Ağcayazı, Pu ZhangAbstract : Let $M$ and $M^{\#}$ be Hardy-Littlewood maximal operator and sharp maximal operator, respectively. In this article, we present necessary and sufficient conditions for the boundedness properties for commutator operators $[M,b]$ and $[M^{\#},b]$ in a general context of Banach function spaces when $b$ belongs to $\operatorname{BMO}(\mathbb{R}^{n})$ spaces. Some applications of the results on weighted Lebesgue spaces, variable Lebesgue spaces, Orlicz spaces and Musielak--Orlicz spaces are also given.
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2023-09-30
Spin structures on complex projective spaces and circle actions
Donghoon JangAbstract : It is known that the complex projective space $\mathbb{CP}^n$ admits a spin structure if and only if $n$ is odd. In this paper, we provide another proof that $\mathbb{CP}^{2m}$ does not admit a spin structure, by using a circle action.
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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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2023-03-31
$\Delta$-transitivity for semigroup actions
Tiaoying ZengAbstract : In this paper, we study $\Delta$-transitivity, $\Delta$-weak mixing and $\Delta$-mixing for semigroup actions and give several characterizations of them, which generalize related results in the literature.
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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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The nilpotency of the prime radical of a Goldie module
John A. Beachy, Mauricio Medina-B\'arcenas
Bull. Korean Math. Soc. 2023; 60(1): 185-201
https://doi.org/10.4134/BKMS.b220053 -
Characterizing $S$-flat modules and $S$-von Neumann regular rings by uniformity
Xiaolei Zhang
Bull. Korean Math. Soc. 2022; 59(3): 643-657
https://doi.org/10.4134/BKMS.b210291 -
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 -
On the Pocklington-Peralta square root algorithm in finite fields
Chang Heon Kim, Namhun Koo
, Soonhak KwonBull. Korean Math. Soc. 2022; 59(6): 1523-1537
https://doi.org/10.4134/BKMS.b210870
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Vanishing theorems for weighted harmonic $1$-forms on smooth metric measure spaces
Xiaoli Chao, Weili Wang
Bull. Korean Math. Soc. 2023; 60(3): 623-636
https://doi.org/10.4134/BKMS.b210927 -
The $u$-$S$-global dimensions of commutative rings
Wei Qi, Xiaolei Zhang
Bull. Korean Math. Soc. 2023; 60(6): 1523-1537
https://doi.org/10.4134/BKMS.b220677 -
Bredon homology of wallpaper groups
Ramon Flores
Bull. Korean Math. Soc. 2023; 60(6): 1497-1522
https://doi.org/10.4134/BKMS.b220669 -
On the construction of optimal linear codes of dimension four
ATSUYA KATO, TATSUYA MARUTA, KEITA NOMURA
Bull. Korean Math. Soc. 2023; 60(5): 1237-1252
https://doi.org/10.4134/BKMS.b220613
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