• Korean Journal of Mathematics
• /
• v.32 no.1
• /
• pp.83-95
• /
• 2024

The Quaternionic Dirac operator proves instrumental in tackling various challenges within spectral geometry processing and shape analysis. This work involves the introduction of the quaternionic Dirac operator on a symplectic submanifold of an exact symplectic product manifold. The self adjointness of the symplectic quaternionic Dirac operator is observed. This operator is verified for spin ${\frac{1}{2}}$ particles. It factorizes the Hodge Laplace operator on the symplectic submanifold of an exact symplectic product manifold. For achieving this a new complex structure and an almost quaternionic structure are formulated on this exact symplectic product manifold.

• Bulletin of the Korean Mathematical Society
• /
• v.59 no.5
• /
• pp.1215-1235
• /
• 2022

In this paper we study Szegö kernel projections for Hardy spaces in quaternionic Clifford analysis. At first we introduce the matrix Szegö projection operator for the Hardy space of quaternionic Hermitean monogenic functions by the characterization of the matrix Hilbert transform in the quaternionic Clifford analysis. Then we establish the Kerzman-Stein formula which closely connects the matrix Szegö projection operator with the Hardy projection operator onto the Hardy space, and we get the matrix Szegö projection operator in terms of the Hardy projection operator and its adjoint. At last, we construct the explicit matrix Szegö kernel function for the Hardy space on the sphere as an example, and get the solution to a Diriclet boundary value problem for matrix functions.

• Bulletin of the Korean Mathematical Society
• /
• v.54 no.2
• /
• pp.583-592
• /
• 2017

The paper gives the regularity of dual quaternionic functions and the dual Cauchy-Riemann system in dual quaternions. Also, the paper researches the polar representation and properties of a dual quaternionic function and their regular quaternionic functions.

• East Asian mathematical journal
• /
• v.32 no.3
• /
• pp.311-317
• /
• 2016

In this paper, we investigate several properties of quaternionic functions. We research some dierentials of quaternionic functions, and relations between the dierentials and the corresponding Cauchy Riemann system in Clifford analysis

• Honam Mathematical Journal
• /
• v.37 no.4
• /
• pp.569-575
• /
• 2015

In this paper, we research some properties of quaternionic regular functions in Clifford analysis. We investigate the corresponding Cauchy-Riemann system and find regularities of some hypercomplex valued functions.

• Nonlinear Functional Analysis and Applications
• /
• v.23 no.3
• /
• pp.551-558
• /
• 2018

This paper proposes a form and use of exponential of a matrix over dual quaternions. Due to the property of the product for dual quaternions, we give the way of computing the exponential of a matrix with the exponential map from their Lie algebras into the dual quaternionic matrices and a form of an eigenvalue of the dual quaternionic exponential of a matrix.

• The Pure and Applied Mathematics
• /
• v.14 no.2 s.36
• /
• pp.91-98
• /
• 2007

We give definitions of hyperholomorphic functions of quaternionic functions of two quaternionic variables. We investigate properties of hyperholomorphic functions on quaternion analysis, and obtain equivalence relations for domains of hyperholomorphy and hyper Stein domains in a domain of $C^2{\times}C^2$.

• East Asian mathematical journal
• /
• v.31 no.5
• /
• pp.653-658
• /
• 2015

In this paper, we consider split quaternionic functions defined on an open set of split quaternions and give the split quaternionic functions whose each inverse function is sp-hyperholomorphic almost everywhere on ${\Omega}$. Also, we describe the definitions and notions of pseudoholomorphic functions for split quaternions.

• Honam Mathematical Journal
• /
• v.37 no.4
• /
• pp.559-567
• /
• 2015

This paper gives the expression of dual quaternions and provides differential operators in dual quaternions. The paper also represents the derivative of dual quaternion-valued functions by using a corresponding Cauchy-Riemann system in dual quaternions.

• Nonlinear Functional Analysis and Applications
• /
• v.24 no.3
• /
• pp.543-553
• /
• 2019

In this paper, we develop and propose a modified quaternion, called the pseudo- dual-quaternion (PDQ). After identifying the differences between quaternions and PDQs, we define the PDQ and a function corresponding to the PDQ. Based on the operators provided in the PDQ, we give the form of power for the PDQ.