In this article, we define the set H of quaternion numbers as the set of all ordered sequences q = 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...
In this paper we defined the reduced residue system and proved its fundamental properties. Then we proved the basic properties of the order function. Finally, we defined the primitive root and proved its fundamental properties. Our work is based on [12], [8], and [11].
This article describes definitions of inverse trigonometric functions arctan, arccot and their main properties, as well as several differentiation formulas of arctan and arccot.MML identifier: SIN COS9, version: 7.8.10 4.100.1011
This article describes some properties of p-groups and some properties of commutative p-groups.
This article describes a rough subgroup with respect to a normal subgroup of a group, and some properties of the lower and the upper approximations in a group.
In this article, we give some important theorems of forward difference, backward difference, central difference and difference quotient and forward difference, backward difference, central difference and difference quotient formulas of some special functions.
This article introduces definitions of circulant matrices, line-and column-circulant matrices as well as anti-circular matrices and describes their main properties.MML identifier: MATRIX16, version: 7.9.03 4.108.1028
This article describes definitions of subsymmetric matrix, anti-subsymmetric matrix, central symmetric matrix, symmetry circulant matrix and their basic properties.
This article provides definitions of idempotent, nilpotent, involutory, self-reversible, similar, and congruent matrices, the trace of a matrix and their main properties.
In this paper, we defined the congruence relation and proved its fundamental properties on the base of some useful theorems. Then we proved the existence of solution and the number of incongruent solution to a linear congruence and the linear congruent equation class, in particular, we proved the Chinese Remainder Theorem. Finally, we defined the complete residue system and proved its fundamental properties.
In this article, we give several differentiation formulas of special and composite functions including trigonometric function, inverse trigonometric function, polynomial function and logarithmic function.
In this article, we give several integrability formulas of special and composite functions including trigonometric function, inverse trigonometric function, hyperbolic function and logarithmic function.
In this article, we give the definitions of forward difference, backward difference, central difference and difference quotient, and some of their important properties.
In this article the general theory of Commutative BCK-algebras and BCI-algebras and several classes of BCK-algebras are given according to [2].
This article describes definitions of elementary operations about matrix and their main 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.
In this article, we give some important theorems of forward difference, backward difference, central difference and difference quotient and forward difference, backward difference, central difference and difference quotient formulas of some special functions.
In this article we will first investigate the elementary properties of BCI-algebras with condition (S), see [8]. And then we will discuss the three classes of algebras: commutative, positive-implicative and implicative BCK-algebras with condition (S).MML identifier: BCIALG 4, version: 7.8.09 4.97.1001
In this article, we shall extend the result of [19] to discuss partial differentiation of real ternary functions (refer to [8] and [16] for partial differentiation).
In this paper, we defined the quadratic residue and proved its fundamental properties on the base of some useful theorems. Then we defined the Legendre symbol and proved its useful theorems [14], [12]. Finally, Gauss Lemma and Law of Quadratic Reciprocity are proven.MML identifier: INT 5, version: 7.8.05 4.89.993
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