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Fuguo Ge

Inner Products, Group, Ring of Quaternion Numbers

In this article, we define the division of the quaternion numbers, we also give the definition of inner products, group, ring of the quaternion numbers, and we prove some of their properties.

MML identifier: QUATERN2, version: 7.8.10 4.100.1011

Open access

Xiquan Liang and Fuguo Ge

The Quaternion Numbers

In this article, we define the set H of quaternion numbers as the set of all ordered sequences q = <x,y,w,z> where x,y,w and z are real numbers. The addition, difference and multiplication of the quaternion numbers are also defined. We define the real and imaginary parts of q and denote this by x = ℜ(q), y = ℑ1(q), w = ℑ2(q), z = ℑ3(q). We define the addition, difference, multiplication again and denote this operation by real and three imaginary parts. We define the conjugate of q denoted by q *' and the absolute value of q denoted by |q|. We also give some properties of quaternion numbers.

Open access

Fuguo Ge and Bing Xie

Several Differentiation Formulas of Special Functions. Part VII

In this article, we prove a series of differentiation identities [2] involving the arctan and arccot functions and specific combinations of special functions including trigonometric and exponential functions.

MML identifier: FDIFF 11, version: 7.10.01 4.111.1036

Open access

Bing Xie, Xiquan Liang and Fuguo Ge

Inverse Trigonometric Functions Arcsec and Arccosec

This article describes definitions of inverse trigonometric functions arcsec and arccosec, as well as their main properties.

MML identifier: SINCOS10, version: 7.8.10 4.100.1011

Open access

Cuiying Peng, Fuguo Ge and Xiquan Liang

Several Integrability Formulas of Special Functions

In this article, we give several integrability formulas of special and composite functions including trigonometric function, inverse trigonometric function, hyperbolic function and logarithmic function.

Open access

Xiquan Liang, Fuguo Ge and Xiaopeng Yue

Some Special Matrices of Real Elements and Their Properties

This article describes definitions of positive matrix, negative matrix, nonpositive matrix, nonnegative matrix, nonzero matrix, module matrix of real elements and their main properties, and we also give the basic inequalities in matrices of real elements.

Open access

Yuzhong Ding, Fuguo Ge and Chenglong Wu

BCI-homomorphisms

In this article the notion of the power of an element of BCI-algebra and its period in the book [11], sections 1.4 to 1.5 are firstly given. Then the definition of BCI-homomorphism is defined and the fundamental theorem of homomorphism, the first isomorphism theorem and the second isomorphism theorem are proved following the book [9], section 1.6.

MML identifier: BCIALG 6, version: 7.9.03 4.108.1028