Journal of the
Korean Mathematical Society
JKMS
ISSN(Print) 0304-9914
ISSN(Online) 2234-3008
01 January 2025
Harmonic functions and end numbers on smooth metric measure spaces
Xuenan Fu; Jia-Yong Wu
J. Korean Math. Soc. 2025; 62: 1-31
In this paper, we study properties of functions on smooth metric measure space $(M,g,e^{-f}dv)$. We prove that any simply connected, negatively curved smooth metric measure space with a small bound of $|\nabla f|$ admits a unique $f$-harmonic function for a given boundary value at infinity. We also prove a sharp $L_f^2$-decay estimate for a Schr\"odinger equation under certain positive spectrum. As applications, we discuss the number of ends on smooth metric measure spaces. We show that the space with finite $f$-volume has a finite number of ends when the Bakry-\'Emery Ricci tensor and the bottom of Neumann spectrum satisfy some lower bounds. We also show that the number of ends with infinite $f$-volume is finite when the Bakry-\'Emery Ricci tensor is bounded below by certain positive spectrum. Finally we study the dimension of the first $L^2_f$-cohomology of the smooth metric measure space.
Keywords: Smooth metric measure space, Bakry-\'Emery Ricci tensor, harmonic function, Dirichlet problem, end, spectrum, Sobolev inequality, cohomology
01 January 2025
The constructions and deformations of Nijenhuis operators on 3-Hom-Lie algebras
Yizheng Li; Dingguo Wang
J. Korean Math. Soc. 2025; 62: 127-143
In this paper, we study Nijenhuis operators on 3-Hom-Lie algebras and provide some examples. Next, we give various constructions of Nijenhuis operators according to constructions of 3-Hom-Lie algebras. Furthermore, we define a cohomology of Nijenhuis operators on 3-Hom-Lie algebras with coefficients in a suitable representation. Finally, as an application, we study formal deformations of Nijenhuis operators that are generated by the above-defined cohomology.
Keywords: Nijenhuis operator, Rota-Baxter operator, cohomology, formal deformation
01 January 2025
Biconservative PNMCV surfaces in the arbitrary dimensional Minkowski space
Nurettin Cenk Turgay; Rüya Yeğin Şen
J. Korean Math. Soc. 2025; 62: 145-163
In this article, we study biconservative surfaces with parallel normalized mean curvature vector field in the arbitrary dimensional Minkowski space $\mathbb{E}^m_1$, where $m\geq 4$. Firstly, we obtain some geometric properties of these surfaces. In particular, we prove that if $M$ is a PNMCV biconservative surface in $\mathbb{E}^m_1$, then it must be contained in a 4-dimensional non-degenerated totally geodesic of $\mathbb{E}^m_1$ and all its shape operators are diagonalizable. Then, we give local classification theorems for biconservative PNMCV space-like and time-like surfaces in $\mathbb{E}^4_1$.
Keywords: Biconservative surfaces, parallel normalized mean curvature vectors, Minkowski space
01 January 2025
Local gradient estimate on Riemannian manifold with asymptotically non-negative Ricci curvature
Tian Chong; Hui Liu; Lingen Lu; Jingjing Zhang
J. Korean Math. Soc. 2025; 62: 165-178
In this paper, we study the asymptotic behavior of the energy densities of harmonic maps, exponentially harmonic functions and positive $p$-harmonic functions at infinity of a Riemannian manifold with asymptotically non-negative curvature. We prove that the energy densities of bounded harmonic maps, exponentially harmonic functions and positive $p$-harmonic functions all vanish at infinity.
Keywords: Asymptotically non-negative Ricci curvature, exponentially harmonic function, harmonic map, positive $p$-harmonic function, gradient estimate
01 January 2025
Weighted sums form of second main theorem on generalized $p$-parabolic manifolds for arbitrary closed subschemes
Tingbin Cao; Liang Wang; Chuanxi Zhu
J. Korean Math. Soc. 2025; 62: 179-195
Our purpose in this paper is to prove the weighted sums form of second main theorem for the case of meromorphic mappings from p-parabolic manifolds into projective spaces with closed subschemes without any general position condition, it generalizes previous results by Han [4], Chen-Thin [3], Quang [13], and Cao-Wang [1].
Keywords: $p$-parabolic manifolds, Nevanlinna theory, Seshadri constants, second main theorem, closed subscheme
01 January 2025
On index divisors of certain number fields defined by $x^{11}+ax^2+b$
Omar Kchit
J. Korean Math. Soc. 2025; 62: 197-216
In this paper, for any number field $K$ generated by a root $\alpha$ of a monic irreducible trinomial $F(x)=x^{11}+ax^2+b \in \mathbb{Z}[x]$ and for every rational prime $p$, we characterize when $p$ divides the index of $K$. We also describe the prime power decomposition of the index $i(K)$. In such a way we give a partial answer of Problem $22$ of Narkiewicz (\cite{Nar}) for this family of number fields. As an application of our results, if $i(K)\neq1$, then $K$ is not monogenic. We illustrate our results by some computational examples.
Keywords: Theorem of Dedekind, theorem of Ore, prime ideal factorization, Newton polygon, index of a number field, power integral basis, monogenic
01 January 2025
Harmonic curvature in dimension four
Andrzej Derdzinski
J. Korean Math. Soc. 2025; 62: 217-252
We provide a step towards classifying Riemannian four-man\-i\-folds in which the curvature tensor has zero divergence, or -- equivalently -- the Ricci tensor Ric satisfies the Codazzi equation. Every known compact manifold of this type belongs to one of five otherwise-familiar classes of examples. The main result consists in showing that, if such a manifold (not necessarily compact or even complete) lies outside of the five classes -- a non-vacuous assumption -- then, at all points of a dense open subset, Ric has four distinct eigenvalues, while suitable local coordinates simultaneously diagonalize Ric, the metric and, in a natural sense, also the curvature tensor. Furthermore, in a local orthonormal frame formed by Ricci eigenvectors, the connection form (or, curvature tensor) has just twelve (or, respectively, six) possibly-nonzero components, which together satisfy a specific system, not depending on the point, of homogeneous polynomial equations. A part of the classification problem is thus reduced to a question in real algebraic geometry.
Keywords: Harmonic curvature, Co\-daz\-zi tensor
01 January 2025
Finding tensor decompositions with sparse optimization
Jeong-Hoon Ju; Taehyeong Kim; Yeongrak Kim
J. Korean Math. Soc. 2025; 62: 33-49
In this paper, we suggest a new method for a given tensor to find CP decompositions using a less number of rank $1$ tensors. The main ingredient is the Least Absolute Shrinkage and Selection Operator (LASSO) by considering the decomposition problem as a sparse optimization problem. As applications, we design experiments to find some CP decompositions of the matrix multiplication and determinant tensors. In particular, we find a new formula for the $4 \times 4$ determinant tensor as a sum of $12$ rank $1$ tensors.
Keywords: CP decomposition, tensor rank, LASSO, determinant
01 January 2025
Classification of homogeneous structures on 4-dimensional nilpotent Lie groups
Wafaa Batat; Rabea Taleb
J. Korean Math. Soc. 2025; 62: 51-75
We determine, for all left-invariant Lorentzian metrics, the set of homogeneous structures on the four-dimensional 3-step nilpotent Lie group $G_{4}$. Combined with the results of \cite{Rabea}, this provides a complete classification of homogeneous structures on four-dimensional nilpotent Lie groups. As an application, we explore the distinct characteristics of each structure and demonstrate the existence of homogeneous structures that are not canonical. We then identify scenarios in which the metrics exhibit natural reductiveness, proving that a naturally reductive homogeneous structure can exist for left-invariant Lorentzian metrics admitting a parallel null vector on $G_{4}$. This highlights a significant distinction between Riemannian and pseudo-Riemannian geometries, as Gordon's result \cite{Gordon} does not apply in the Lorentzian context, where the Lie group is not restricted to being 2-step nilpotent.
Keywords: Nilpotent Lie groups, homogeneous structures, naturally reductive, left-invariant Lorentzian metrics, curvature
01 January 2025
Mixed Finite Volume Method for Two-dimensional Maxwell's Equations
Kwang-Yeon Kim; Do Young Kwak
J. Korean Math. Soc. 2025; 62: 77-96
We propose and analyze a mixed finite volume method for the two-dimensional time-harmonic Maxwell's equations which simultaneously approximates the vector field $\boldsymbol{u}$ and the scalar function $\xi = \mu^{-1}\operatorname{curl}\boldsymbol{u}$. The method chooses the lowest-order N\'{e}d\'{e}lec edge element for $\boldsymbol{u}$ and the $P1$ Crouzeix--Raviart nonconforming element for $\xi$ on triangular meshes. It is shown that the method is reduced to a modified $P1$ nonconforming FEM for $\xi$ or a modified edge element method for $\boldsymbol{u}$ by eliminating the discrete variable of $\boldsymbol{u}$ or $\xi$. After solving the reduced method, the eliminated discrete variable can be recovered from the other one via a simple local formula. Using this feature, we also derive optimal a priori error estimates under weak regularity assumptions and show that the approximation to $\xi$ has a higher-order of convergence in the $L^2$ norm than the one obtained by direct differentiation of the approximation to $\boldsymbol{u}$ when the exact solution is sufficiently smooth.
Keywords: Maxwell's equations, mixed finite volume method, N\'{e}d\'{e}lec edge element, $P1$ nonconforming finite element
01 January 2025
Identities for the Rogers-Ramanujan continued fraction
Nayandeep Deka Baruah; Pranjal Talukdar
J. Korean Math. Soc. 2025; 62: 97-126
We prove some new modular identities for the Rogers-\linebreak Ramanujan continued fraction. For example, if $R(q)$ denotes the Rogers-Ramanujan continued fraction, then \begin{align*}&R(q)R(q^4)=\dfrac{R(q^5)+R(q^{20})-R(q^5)R(q^{20})}{1+R(q^{5})+R(q^{20})},\\ &\dfrac{1}{R(q^{2})R(q^{3})}+R(q^{2})R(q^{3})= 1+\dfrac{R(q)}{R(q^{6})}+\dfrac{R(q^{6})}{R(q)}, \end{align*} and \begin{align*}&R(q^2)\\ =&\ \dfrac{R(q)R(q^3)}{R(q^6)}\cdot\dfrac{R(q) R^2(q^3) R(q^6)+2 R(q^6) R(q^{12})+ R(q) R(q^3) R^2(q^{12})}{R(q^3) R(q^6)+2 R(q) R^2(q^3) R(q^{12})+ R^2(q^{12})}.\end{align*} In the process, we also find some new relations for the Rogers-Ramanujan functions by using dissections of theta functions and the quintuple product identity.
Keywords: Rogers-Ramanujan functions, Rogers-Ramanujan continued fraction, Ramanujan's notebooks, Ramanujan's lost notebook, modular identities, theta functions
January 1, 2025Current Issue Vol. 62 No. 1
-
2025-01-01
Harmonic functions and end numbers on smooth metric measure spaces
Xuenan Fu
; Jia-Yong Wu Abstract : In this paper, we study properties of functions on smooth metric measure space $(M,g,e^{-f}dv)$. We prove that any simply connected, negatively curved smooth metric measure space with a small bound of $|\nabla f|$ admits a unique $f$-harmonic function for a given boundary value at infinity. We also prove a sharp $L_f^2$-decay estimate for a Schr\"odinger equation under certain positive spectrum. As applications, we discuss the number of ends on smooth metric measure spaces. We show that the space with finite $f$-volume has a finite number of ends when the Bakry-\'Emery Ricci tensor and the bottom of Neumann spectrum satisfy some lower bounds. We also show that the number of ends with infinite $f$-volume is finite when the Bakry-\'Emery Ricci tensor is bounded below by certain positive spectrum. Finally we study the dimension of the first $L^2_f$-cohomology of the smooth metric measure space.
-
2025-01-01
Finding tensor decompositions with sparse optimization
Jeong-Hoon Ju; Taehyeong Kim ; Yeongrak KimAbstract : In this paper, we suggest a new method for a given tensor to find CP decompositions using a less number of rank $1$ tensors. The main ingredient is the Least Absolute Shrinkage and Selection Operator (LASSO) by considering the decomposition problem as a sparse optimization problem. As applications, we design experiments to find some CP decompositions of the matrix multiplication and determinant tensors. In particular, we find a new formula for the $4 \times 4$ determinant tensor as a sum of $12$ rank $1$ tensors.
-
2025-01-01
Classification of homogeneous structures on 4-dimensional nilpotent Lie groups
Wafaa Batat; Rabea TalebAbstract : We determine, for all left-invariant Lorentzian metrics, the set of homogeneous structures on the four-dimensional 3-step nilpotent Lie group $G_{4}$. Combined with the results of \cite{Rabea}, this provides a complete classification of homogeneous structures on four-dimensional nilpotent Lie groups. As an application, we explore the distinct characteristics of each structure and demonstrate the existence of homogeneous structures that are not canonical. We then identify scenarios in which the metrics exhibit natural reductiveness, proving that a naturally reductive homogeneous structure can exist for left-invariant Lorentzian metrics admitting a parallel null vector on $G_{4}$. This highlights a significant distinction between Riemannian and pseudo-Riemannian geometries, as Gordon's result \cite{Gordon} does not apply in the Lorentzian context, where the Lie group is not restricted to being 2-step nilpotent.
-
2025-01-01
Mixed Finite Volume Method for Two-dimensional Maxwell's Equations
Kwang-Yeon Kim; Do Young KwakAbstract : We propose and analyze a mixed finite volume method for the two-dimensional time-harmonic Maxwell's equations which simultaneously approximates the vector field $\boldsymbol{u}$ and the scalar function $\xi = \mu^{-1}\operatorname{curl}\boldsymbol{u}$. The method chooses the lowest-order N\'{e}d\'{e}lec edge element for $\boldsymbol{u}$ and the $P1$ Crouzeix--Raviart nonconforming element for $\xi$ on triangular meshes. It is shown that the method is reduced to a modified $P1$ nonconforming FEM for $\xi$ or a modified edge element method for $\boldsymbol{u}$ by eliminating the discrete variable of $\boldsymbol{u}$ or $\xi$. After solving the reduced method, the eliminated discrete variable can be recovered from the other one via a simple local formula. Using this feature, we also derive optimal a priori error estimates under weak regularity assumptions and show that the approximation to $\xi$ has a higher-order of convergence in the $L^2$ norm than the one obtained by direct differentiation of the approximation to $\boldsymbol{u}$ when the exact solution is sufficiently smooth.
-
2025-01-01
Identities for the Rogers-Ramanujan continued fraction
Nayandeep Deka Baruah; Pranjal TalukdarAbstract : We prove some new modular identities for the Rogers-\linebreak Ramanujan continued fraction. For example, if $R(q)$ denotes the Rogers-Ramanujan continued fraction, then \begin{align*}&R(q)R(q^4)=\dfrac{R(q^5)+R(q^{20})-R(q^5)R(q^{20})}{1+R(q^{5})+R(q^{20})},\\ &\dfrac{1}{R(q^{2})R(q^{3})}+R(q^{2})R(q^{3})= 1+\dfrac{R(q)}{R(q^{6})}+\dfrac{R(q^{6})}{R(q)}, \end{align*} and \begin{align*}&R(q^2)\\ =&\ \dfrac{R(q)R(q^3)}{R(q^6)}\cdot\dfrac{R(q) R^2(q^3) R(q^6)+2 R(q^6) R(q^{12})+ R(q) R(q^3) R^2(q^{12})}{R(q^3) R(q^6)+2 R(q) R^2(q^3) R(q^{12})+ R^2(q^{12})}.\end{align*} In the process, we also find some new relations for the Rogers-Ramanujan functions by using dissections of theta functions and the quintuple product identity.
-
2025-01-01
The constructions and deformations of Nijenhuis operators on 3-Hom-Lie algebras
Yizheng Li; Dingguo WangAbstract : In this paper, we study Nijenhuis operators on 3-Hom-Lie algebras and provide some examples. Next, we give various constructions of Nijenhuis operators according to constructions of 3-Hom-Lie algebras. Furthermore, we define a cohomology of Nijenhuis operators on 3-Hom-Lie algebras with coefficients in a suitable representation. Finally, as an application, we study formal deformations of Nijenhuis operators that are generated by the above-defined cohomology.
-
2025-01-01
Biconservative PNMCV surfaces in the arbitrary dimensional Minkowski space
Nurettin Cenk Turgay; Rüya Yeğin Şen Abstract : In this article, we study biconservative surfaces with parallel normalized mean curvature vector field in the arbitrary dimensional Minkowski space $\mathbb{E}^m_1$, where $m\geq 4$. Firstly, we obtain some geometric properties of these surfaces. In particular, we prove that if $M$ is a PNMCV biconservative surface in $\mathbb{E}^m_1$, then it must be contained in a 4-dimensional non-degenerated totally geodesic of $\mathbb{E}^m_1$ and all its shape operators are diagonalizable. Then, we give local classification theorems for biconservative PNMCV space-like and time-like surfaces in $\mathbb{E}^4_1$.
-
2025-01-01
Local gradient estimate on Riemannian manifold with asymptotically non-negative Ricci curvature
Tian Chong; Hui Liu; Lingen Lu; Jingjing ZhangAbstract : In this paper, we study the asymptotic behavior of the energy densities of harmonic maps, exponentially harmonic functions and positive $p$-harmonic functions at infinity of a Riemannian manifold with asymptotically non-negative curvature. We prove that the energy densities of bounded harmonic maps, exponentially harmonic functions and positive $p$-harmonic functions all vanish at infinity.
-
2025-01-01
Weighted sums form of second main theorem on generalized $p$-parabolic manifolds for arbitrary closed subschemes
Tingbin Cao; Liang Wang; Chuanxi ZhuAbstract : Our purpose in this paper is to prove the weighted sums form of second main theorem for the case of meromorphic mappings from p-parabolic manifolds into projective spaces with closed subschemes without any general position condition, it generalizes previous results by Han [4], Chen-Thin [3], Quang [13], and Cao-Wang [1].
-
2025-01-01
On index divisors of certain number fields defined by $x^{11}+ax^2+b$
Omar KchitAbstract : In this paper, for any number field $K$ generated by a root $\alpha$ of a monic irreducible trinomial $F(x)=x^{11}+ax^2+b \in \mathbb{Z}[x]$ and for every rational prime $p$, we characterize when $p$ divides the index of $K$. We also describe the prime power decomposition of the index $i(K)$. In such a way we give a partial answer of Problem $22$ of Narkiewicz (\cite{Nar}) for this family of number fields. As an application of our results, if $i(K)\neq1$, then $K$ is not monogenic. We illustrate our results by some computational examples.
-
2025-01-01
Harmonic curvature in dimension four
Andrzej DerdzinskiAbstract : We provide a step towards classifying Riemannian four-man\-i\-folds in which the curvature tensor has zero divergence, or -- equivalently -- the Ricci tensor Ric satisfies the Codazzi equation. Every known compact manifold of this type belongs to one of five otherwise-familiar classes of examples. The main result consists in showing that, if such a manifold (not necessarily compact or even complete) lies outside of the five classes -- a non-vacuous assumption -- then, at all points of a dense open subset, Ric has four distinct eigenvalues, while suitable local coordinates simultaneously diagonalize Ric, the metric and, in a natural sense, also the curvature tensor. Furthermore, in a local orthonormal frame formed by Ricci eigenvectors, the connection form (or, curvature tensor) has just twelve (or, respectively, six) possibly-nonzero components, which together satisfy a specific system, not depending on the point, of homogeneous polynomial equations. A part of the classification problem is thus reduced to a question in real algebraic geometry.
January, 2025 | Volume 62, No. 1
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