scholarly journals Application of slice regularity to functions of a dual-quaternionic variable

2021 ◽  
Vol 13 (2) ◽  
pp. 298-304
Author(s):  
Ji Eun Kim
Keyword(s):  
Power Series ◽  
Dual Quaternions ◽  
Algebraic Properties ◽  
Cauchy Riemann

In this paper, we present the algebraic properties of dual quaternions and define a slice regularity of a dual quaternionic function. Since the product of dual quaternions is non-commutative, slice regularity is derived in two ways. Thereafter, we propose the Cauchy-Riemann equations and a power series corresponding to dual quaternions.

scholarly journals Poisson superposition processes

2015 ◽  
Vol 52 (04) ◽  
pp. 1013-1027
Author(s):  
Harry Crane ◽  
Peter Mccullagh
Keyword(s):  
Power Series ◽  
Poisson Process ◽  
Explicit Expression ◽  
Negative Binomial ◽  
Poisson Processes ◽  
Domain Restriction ◽  
Point Configuration ◽  
Point Configurations ◽  
Algebraic Properties ◽  
Formal Power

Superposition is a mapping on point configurations that sends the n-tuple into the n-point configuration , counted with multiplicity. It is an additive set operation such that the superposition of a k-point configuration in is a kn-point configuration in . A Poisson superposition process is the superposition in of a Poisson process in the space of finite-length -valued sequences. From properties of Poisson processes as well as some algebraic properties of formal power series, we obtain an explicit expression for the Janossy measure of Poisson superposition processes, and we study their law under domain restriction. Examples of well-known Poisson superposition processes include compound Poisson, negative binomial, and permanental (boson) processes.

scholarly journals Some Algebraic Properties Of Asymptotic Power Series

1956 ◽  
Vol 8 ◽  
pp. 220-224
Author(s):  
T. E. Hull
Keyword(s):  
Asymptotic Expansion ◽  
Power Series ◽  
Asymptotic Power ◽  
Algebraic Properties ◽  
Image Position ◽  
The Right ◽  
The Given ◽  
Class Of Functions

1. Introduction. Let us consider all power series of the formIt was shown first by Borel (1) that to each such series there corresponds a non-empty class of functions such that each function in the class has the given series as its asymptotic expansion about z = 0, the expansion being valid in a sector of the right half z-plane with vertex at the origin.

scholarly journals The Regularity of Functions on Dual Split Quaternions in Clifford Analysis

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Ji Eun Kim ◽  
Kwang Ho Shon
Keyword(s):  
Power Series ◽  
Clifford Analysis ◽  
Differential Operators ◽  
Regular Function ◽  
Split Quaternion ◽  
Split Quaternions ◽  
Cauchy Riemann ◽  
Dual Split Quaternion

This paper shows some properties of dual split quaternion numbers and expressions of power series in dual split quaternions and provides differential operators in dual split quaternions and a dual split regular function onΩ⊂ℂ2×ℂ2that has a dual split Cauchy-Riemann system in dual split quaternions.

scholarly journals Generating Asymptotics for Factorially Divergent Sequences

10.37236/5999 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Michael Borinsky
Keyword(s):  
Asymptotic Expansion ◽  
Power Series ◽  
Formal Power Series ◽  
Asymptotic Expansions ◽  
Series Form ◽  
Chord Diagrams ◽  
Algebraic Properties ◽  
Chain Rules ◽  
Formal Power ◽  
And Inversion

The algebraic properties of formal power series, whose coefficients show factorial growth and admit a certain well-behaved asymptotic expansion, are discussed. It is shown that these series form a subring of $\mathbb{R}[[x]]$. This subring is also closed under composition and inversion of power series. An `asymptotic derivation' is defined which maps a power series to the asymptotic expansion of its coefficients. Product and chain rules for this derivation are deduced. With these rules asymptotic expansions of the coefficients of implicitly defined power series can be obtained. The full asymptotic expansions of the number of connected chord diagrams and the number of simple permutations are given as examples.

scholarly journals Poisson superposition processes

2015 ◽  
Vol 52 (4) ◽  
pp. 1013-1027
Author(s):  
Harry Crane ◽  
Peter Mccullagh
Keyword(s):  
Power Series ◽  
Poisson Process ◽  
Explicit Expression ◽  
Negative Binomial ◽  
Poisson Processes ◽  
Domain Restriction ◽  
Point Configuration ◽  
Point Configurations ◽  
Algebraic Properties ◽  
Formal Power

Superposition is a mapping on point configurations that sends the n-tuple into the n-point configuration , counted with multiplicity. It is an additive set operation such that the superposition of a k-point configuration in is a kn-point configuration in . A Poisson superposition process is the superposition in of a Poisson process in the space of finite-length -valued sequences. From properties of Poisson processes as well as some algebraic properties of formal power series, we obtain an explicit expression for the Janossy measure of Poisson superposition processes, and we study their law under domain restriction. Examples of well-known Poisson superposition processes include compound Poisson, negative binomial, and permanental (boson) processes.

PROPERTIES OF REGULAR FUNCTIONS OF A QUATERNION VARIABLE MODIFIED WITH TRI-COMPLEX QUATERNION

2021 ◽  
Vol 10 (5) ◽  
pp. 2663-2673
Author(s):  
Ji Eun Kim
Keyword(s):  
Complex Number ◽  
Regular Function ◽  
Complex Numbers ◽  
Riemann Equation ◽  
Regular Functions ◽  
Complex Quaternion ◽  
Algebraic Properties ◽  
Quaternion Structure ◽  
Cauchy Riemann

In a quaternion structure composed of four real dimensions, we derive a form wherein three complex numbers are combined. Thereafter, we examined whether this form includes the algebraic properties of complex numbers and whether transformations were necessary for its application to the system. In addition, we defined a regular function in quaternions, expressed as a combination of complex numbers. Furthermore, we derived the Cauchy-Riemann equation to investigate the properties of the regular function in the quaternions coupled with the complex number.

Regular functions on dual split quaternions in Clifford analysis

Filomat ◽  
2017 ◽  
Vol 31 (1) ◽  
pp. 17-27
Author(s):  
Ji Kim ◽  
Kwang Shon
Keyword(s):  
Power Series ◽  
Clifford Analysis ◽  
Differential Operators ◽  
Split Quaternions ◽  
Regular Functions ◽  
Cauchy Riemann

This paper shows expressions of a power series for the form of dual split quaternions and provides differential operators in dual split quaternions. The paper also represents a power series of dual split regular functions by using a dual split Cauchy-Riemann system in dual split quaternions.

Algebraic properties of separated power series

2007 ◽  
Vol 259 (3) ◽  
pp. 681-695 ◽  
Author(s):  
Y. Fırat Çelikler
Keyword(s):  
Power Series ◽  
Algebraic Properties

Some algebraic properties of complex Segre quaternoins

2019 ◽  
Vol 33 ◽  
pp. 158-159
Author(s):  
Anatoliy Pogorui ◽  
Tamila Kolomiiets
Keyword(s):  
Matrix Representation ◽  
Complex Numbers ◽  
Polynomial Equations ◽  
Complex Field ◽  
Basic Properties ◽  
Algebraic Properties ◽  
Cauchy Riemann

This paper deals with the basic properties the algebra of Segre quaternions over the field of complex numbers. We study idempotents, ideals, matrix representation and the Peirce decomposition of this algebra. We also investigate the structure of zeros of a polynomial in Segre complex quaternions by reducing it to the system of four polynomial equations in the complex field. In addition, Cauchy-Riemann type conditions are obtained for the differentiability of a function on the complex Segre quaternionic algebra.

scholarly journals Vectorial Iterative Fractional Laplace Transform Method for the Analytic Solutions of Fractional Cauchy-Riemann Systems Partial Differential Equations

2020 ◽  
pp. 60-83
Author(s):  
Kebede Shigute Kenea
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Laplace Transform ◽  
Initial Conditions ◽  
Fractional Power ◽  
Power Series Solution ◽  
Laplace Transform Method ◽  
Transform Method ◽  
Cauchy Riemann ◽  
Fractional Power Series

The present study aims to obtain infinite fractional power series solution vectors of fractional Cauchy-Riemann systems equations with initial conditions by the use of vectorial iterative fractional Laplace transform method (VIFLTM). The basic idea of the VIFLTM was developed successfully and applied to four test examples to see its effectiveness and applicability. The infinite fractional power series form solutions were successfully obtained analytically. Thus, the results show that the VIFLTM works successfully in solving fractional Cauchy-Riemann system equations with initial conditions, and hence it can be extended to other fractional differential equations.

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