scholarly journals Characteristics of Regular Functions Defined on a Semicommutative Subalgebra of 4-Dimensional Complex Matrix Algebra

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Ji Eun Kim
Keyword(s):  
Matrix Algebra ◽  
Matrix Form ◽  
Subject Classification ◽  
Complex Matrix ◽  
Regular Functions ◽  
Cauchy Riemann ◽  
Complex Components

In this paper, we give an extended quaternion as a matrix form involving complex components. We introduce a semicommutative subalgebra ℂ ℂ 2 of the complex matrix algebra M 4 , ℂ . We exhibit regular functions defined on a domain in ℂ 4 but taking values in ℂ ℂ 2 . By using the characteristics of these regular functions, we propose the corresponding Cauchy–Riemann equations. In addition, we demonstrate several properties of these regular functions using these novel Cauchy–Riemann equations. Mathematical Subject Classification is 32G35, 32W50, 32A99, and 11E88.

scholarly journals A generalization of the matrix form of the Brunn-Minkowski inequality

2007 ◽  
Vol 83 (1) ◽  
pp. 125-134 ◽  
Author(s):  
Jun Yuan ◽  
Gangsong Lenga
Keyword(s):  
Minkowski Inequality ◽  
Matrix Form ◽  
Subject Classification ◽  
Primary 52A40 ◽  
The Matrix

AbstractIn this paper, we establish an extension of the matrix form of the Brunn-Minkowski inequality. As applications, we give generalizations on the metric addition inequality of Alexander.2000 Mathematics subject classification: primary 52A40.

scholarly journals Sebuah Telaah Elips dan Lingkaran Melalui Sebuah Pendekatan Aljabar Matriks

CAUCHY ◽  
2010 ◽  
Vol 1 (2) ◽  
pp. 85 ◽  
Author(s):  
Rahmat Sagara
Keyword(s):  
Fixed Point ◽  
Matrix Algebra ◽  
Positive Definite Matrix ◽  
Positive Definite ◽  
Point Of View ◽  
Matrix Form ◽  
Algebra Approach ◽  
The Difference ◽  
Definition Of ◽  
Set Of Points

In this article, ellipse and circle will be learnt in depth via matrix algebra approach. The discussion of the both is started from their classic definition continued by surveying ellipse in matrix form. During the survey, some properties about ellipse will be explained and also, the procedure in drawing the figures can be obtained geometrically using some aspect in geometry: rotation and translation. At the end of the discussion, the new definition of the figures is deduced. Both of them are defined as” a set of points in a plane that are the same distance from a fixed point” but in different point of view about the ‘distance’. The ‘distance’ in the definition is derived from different norm definition. The difference lies on the positive definite matrix used in the norm definition. Base on the new definition, we’ll have the conclusion that circle is a special type of ellipse.

scholarly journals C*-ALGEBRAS THAT ARE ISOMORPHIC AFTER TENSORING AND FULL PROJECTIONS

2004 ◽  
Vol 47 (3) ◽  
pp. 659-668
Author(s):  
Kazunori Kodaka
Keyword(s):  
Matrix Algebra ◽  
Subject Classification ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Primary 46L05 ◽  
C Algebra ◽  
C Algebras ◽  
Necessary And Sufficient

AbstractLet $A$ be a unital $C^*$-algebra and for each $n\in\mathbb{N}$ let $M_n$ be the $n\times n$ matrix algebra over $\mathbb{C}$. In this paper we shall give a necessary and sufficient condition that there is a unital $C^*$-algebra $B$ satisfying $A\not\cong B$ but for which $A\otimes M_n\cong B\otimes M_n$ for some $n\in\mathbb{N}\setminus\{1\}$. Also, we shall give some examples of unital $C^*$-algebras satisfying the above property.AMS 2000 Mathematics subject classification: Primary 46L05

scholarly journals PROPAGATION OF SMALLNESS FOR SOLUTIONS OF GENERALIZED CAUCHY–RIEMANN SYSTEMS

2004 ◽  
Vol 47 (1) ◽  
pp. 191-204 ◽  
Author(s):  
E. Malinnikova
Keyword(s):  
Unit Ball ◽  
Lebesgue Measure ◽  
Compact Subset ◽  
Subject Classification ◽  
Positive Lebesgue Measure ◽  
Secondary 35B35 ◽  
Closed Set ◽  
Cauchy Riemann

AbstractLet $u$ be a solution of a generalized Cauchy–Riemann system in $\mathbb{R}^n$. Suppose that $|u|\le1$ in the unit ball and $|u|\le\varepsilon$ on some closed set $E$. Classical results say that if $E$ is a set of positive Lebesgue measure, then $|u|\le C\varepsilon^\alpha$ on any compact subset of the unit ball. In the present work the same estimate is proved provided that $E$ is a subset of a hyperplane and the (capacitary) dimension of $E$ is greater than $n-2$. The proof gives control of constants $C$ and $\alpha$.AMS 2000 Mathematics subject classification: Primary 31B35. Secondary 35B35; 35J45

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.

scholarly journals Regularity of Functions on the Reduced Quaternion Field in Clifford Analysis

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Ji Eun Kim ◽  
Su Jin Lim ◽  
Kwang Ho Shon
Keyword(s):  
Exponential Function ◽  
Clifford Analysis ◽  
Regular Function ◽  
Regular Functions ◽  
Quaternion Field ◽  
Hypercomplex Structure ◽  
Cauchy Riemann

We define a new hypercomplex structure ofℝ3and a regular function with values in that structure. From the properties of regular functions, we research the exponential function on the reduced quaternion field and represent the corresponding Cauchy-Riemann equations in hypercomplex structures ofℝ3.

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