• Communications of the Korean Mathematical Society
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• v.35 no.2
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• pp.487-497
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• 2020

In this paper, we establish a complex matrix representation of the Clifford algebra Cℓ_p,q. The size of our representation is significantly smaller than the size of the elements in L_p,q(ℝ). Additionally, we give detailed information about the spectrum of the constructed matrix representation.

• The Pure and Applied Mathematics
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• v.20 no.2
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• pp.129-136
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• 2013

We research properties of ternary numbers with values in ${\Lambda}(2)$. Also, we represent dual ternary numbers in the sense of Clifford algebras of real six dimensional spaces. We give generation theorems in dual ternary number systems in view of Clifford analysis, and obtain Cauchy theorems with respect to dual ternary numbers.

• Journal of the Korean Mathematical Society
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• v.61 no.1
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• pp.109-132
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• 2024

The Clifford algebra of a direct sum of real quadratic spaces appears as the superalgebra tensor product of the Clifford algebras of the summands. The purpose of the current paper is to present a purely settheoretical version of the superalgebra tensor product which will be applicable equally to groups or to their non-associative analogues - quasigroups and loops. Our work is part of a project to make supersymmetry an effective tool for the study of combinatorial structures. Starting from group and quasigroup structures on four-element supersets, our superproduct unifies the construction of the eight-element quaternion and dihedral groups, further leading to a loop structure which hybridizes the two groups. All three of these loops share the same character table.

• Bulletin of the Korean Mathematical Society
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• v.56 no.6
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• pp.1551-1567
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• 2019

In this paper, Seiberg-Witten-like equations are written down on 3-manifolds. Then, it has been proved that the $L^2$-solutions of these equations are trivial on ${\mathbb{R}}^3$. Finally, a global solution is obtained on the strictly pseudoconvex CR-3 manifolds for a given constant negative scalar curvature.

• Bulletin of the Korean Mathematical Society
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• v.22 no.1
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• pp.11-15
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• 1985

Manddelberg [9] has shown that a Clifford algebra of a free quadratic space over an arbitrary semi-local ring R in Brawer-Wall group BW(R) is determined by its rank, determinant, and Hasse invariant. In this paper, we prove a corresponding result when R is a full ring.Throughout this paper, unless otherwise specified, we assume that R is a commutative ring having 2 a unit. A quadratic space (V, B, .phi.) over R is a finitely generated projective R-module V with a symmetric bilinear mapping B: V*V.rarw.R which is non-degenerate (i.e., the natural mapping V.rarw.Ho $m_{R}$(V,R) induced by B is an isomorphism), and with a quadratic mapping .phi.: V.rarw.R such that B(x,y)=1/2(.phi.(x+y)-.phi.(x)-.phi.(y)) and .phi.(rx) = $r^{2}$.phi.(x) for all x, y in V and r in R. We denote the group of multiplicative units of R by U9R). If (V, B, .phi.) is a free rank n quadratic space over R with an orthogonal basis { $x_{1}$,.., $x_{n}$}, we will write < $a_{1}$,.., $a_{n}$> for (V, B, .phi.) where the $a_{i}$=.phi.( $x_{i}$) are in U(R), and denote the space by the table [ $a_{ij}$ ] where $a_{ij}$ =B( $x_{i}$, $x_{j}$). In the case n=2 and B( $x_{1}$, $x_{2}$)=1/2 we reserve the notation [a $a_{11}$, $a_{22}$] for the space. A commutative ring R having 2 a unit is called full [10] if for every triple $a_{1}$, $a_{2}$, $a_{3}$ of elements in R with ( $a_{1}$, $a_{2}$, $a_{3}$)=R, there is an element w in R such that $a_{1}$+ $a_{2}$w+ $a_{3}$ $w^{2}$=unit.TEX>=unit.t.t.t.