• 대한수학회논문집
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• 제36권3호
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• pp.413-422
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• 2021

The Study determinant is known as one of replacements for the determinant of matrices with entries in a noncommutative ring. In this paper, we give a generalization of the Study determinant and show its relationship with the Cayley-Dickson process. We also give some properties of a non-associative ring obtained by the Cayley-Dickson process with a not necessarily commutative, but associative ring as the initial ring.

• Kyungpook Mathematical Journal
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• 제61권2호
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• pp.257-267
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• 2021

In this paper, we develop an iterative algorithm for obtaining common solutions to the Cayley inclusion problem and the set of fixed points of a non-expansive mapping in Hilbert spaces. A numerical example is given for the justification of our claim.

• 대한수학회지
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• 제45권3호
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• pp.781-794
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• 2008

Let $\mathbb{H}_g$ and $\mathbb{D}_g$ be the Siegel upper half plane and the generalized unit disk of degree g respectively. Let $\mathbb{C}^{(h,g)}$ be the Euclidean space of all $h{\times}g$ complex matrices. We present a partial Cayley transform of the Siegel-Jacobi disk $\mathbb{D}_g{\times}\mathbb{C}^{(h,g)}$ onto the Siegel-Jacobi space $\mathbb{H}_g{\times}\mathbb{C}^{(h,g)}$ which gives a partial bounded realization of $\mathbb{H}_g{\times}\mathbb{C}^{(h,g)}$ by $\mathbb{D}_g{\times}\mathbb{C}^{(h,g)}$. We prove that the natural actions of the Jacobi group on $\mathbb{D}_g{\times}\mathbb{C}^{(h,g)}$. and $\mathbb{H}_g{\times}\mathbb{C}^{(h,g)}$. are compatible via a partial Cayley transform. A partial Cayley transform plays an important role in computing differential operators on the Siegel Jacobi disk $\mathbb{D}_g{\times}\mathbb{C}^{(h,g)}$. invariant under the natural action of the Jacobi group $\mathbb{D}_g{\times}\mathbb{C}^{(h,g)}$ explicitly.

• 대한수학회보
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• 제59권4호
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• pp.929-949
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• 2022

Given an ACM set 𝕏 of points in a multiprojective space ℙ^m×ℙⁿ over a field of characteristic zero, we are interested in studying the Kähler different and the Cayley-Bacharach property for 𝕏. In ℙ¹×ℙ¹, the Cayley-Bacharach property agrees with the complete intersection property and it is characterized by using the Kähler different. However, this result fails to hold in ℙ^m×ℙⁿ for n > 1 or m > 1. In this paper we start an investigation of the Kähler different and its Hilbert function and then prove that 𝕏 is a complete intersection of type (d₁, …, d_m, d'₁, …, d'_n) if and only if it has the Cayley-Bacharach property and the Kähler different is non-zero at a certain degree. We characterize the Cayley-Bacharach property of 𝕏 under certain assumptions.

• Journal of applied mathematics & informatics
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• 제30권1_2호
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• pp.49-56
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• 2012

In this paper we introduce two new labelings called product antimagic labeling and total product antimagic labeling for directed graphs and show the existence of the same for Cayley digraphs of 2-generated 2-groups.

• 대한수학회보
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• 제35권1호
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• pp.141-148
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• 1998

In the present paper, some pinching theorems for the curvatures of the totally complex submanifolds of the Cayley projective plane $CaP^2$ are obtained.

• 한국수학사학회지
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• 제21권3호
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• pp.127-142
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• 2008

수학의 역사에서는 이미 발견되어 논증된 정리를 새로운 방법으로 공략함으로써 그 정리의 깊은 의미를 드러내는 작업의 기록을 쉽게 찾을 수 있다. 이 연구는 직관적으로 비교적 이해하기 쉬운 소재인 수형도를 대상으로 하여, 꼭지점의 집합이 결정되었을 때 수형도의 개수를 결정하여 주는 케일리 공식(Cayley formula)의 증명에 대한 서로 다른 네 가지 접근방법을 소개하는 것을 목적으로 한다. 네 가지 증명은 수형도의 성질로부터 유도된 재귀적 관계식을 이용한 케일리의 증명에서부터 특정한 수학적 대상과 수형도 사이의 일대일대응 관계에 주목하는 나머지 세 가지 증명으로 이루어진다. 특히, 마지막 증명은 순수한 수학적 작업이 다른 분야에 강력한 도구를 제공하는 전형적인 예를 보여준다.

• 대한수학회지
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• 제36권3호
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• pp.621-631
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• 1999

In this paper we briefly describe the geometry of the Cayley hyperbolic plane and we show that every uniform lattice in quaternionic space cannot be deformed in the Cayley hyperbolic 2-plane. We also describe the nongeometric bending deformation by developing the theory of the Cartan angular invariant for quaternionic hyperbolic space.

• 대한수학회보
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• 제53권4호
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• pp.1017-1031
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• 2016

Let R be a commutative ring with the non-zero identity and n be a natural number. ${\Gamma}^n_R$ is a simple graph with $R^n{\setminus}\{0\}$ as the vertex set and two distinct vertices X and Y in $R^n$ are adjacent if and only if there exists an $n{\times}n$ lower triangular matrix A over R whose entries on the main diagonal are non-zero such that $AX^t=Y^t$ or $AY^t=X^t$, where, for a matrix B, $B^t$ is the matrix transpose of B. ${\Gamma}^n_R$ is a generalization of Cayley graph. Let $T_n(R)$ denote the $n{\times}n$ upper triangular matrix ring over R. In this paper, for an arbitrary ring R, we investigate the properties of the graph ${\Gamma}^n_{T_n(R)}$.

• 대한수학회보
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• 제53권4호
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• pp.1197-1211
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• 2016

Let R be a finite commutative ring with nonzero identity. We define ${\Gamma}(R)$ to be the graph with vertex set R in which two distinct vertices x and y are adjacent if and only if there exists a unit element u of R such that x + uy is a unit of R. This graph provides a refinement of the unit and unitary Cayley graphs. In this paper, basic properties of ${\Gamma}(R)$ are obtained and the vertex connectivity and the edge connectivity of ${\Gamma}(R)$ are given. Finally, by a constructive way, we determine when the graph ${\Gamma}(R)$ is Hamiltonian. As a consequence, we show that ${\Gamma}(R)$ has a perfect matching if and only if ${\mid}R{\mid}$ is an even number.