The Clifford algebra is a unification and generalization of real number, complex number, quaternion, and vector algebra, which accurately and faithfully characterizes the intrinsic properties of space-time, providing ...The Clifford algebra is a unification and generalization of real number, complex number, quaternion, and vector algebra, which accurately and faithfully characterizes the intrinsic properties of space-time, providing a unified, standard, elegant, and open language and tools for numerous complicated mathematical and physical theories. So it is worth popularizing in the teaching of undergraduate physics and mathematics. Clifford algebras can be directly generalized to 2<sup>n</sup>-ary associative algebras. In this generalization, the matrix representation of the orthonormal basis of space-time plays an important role. The matrix representation carries more information than the abstract definition, such as determinant and the definition of inverse elements. Without this matrix representation, the discussion of hypercomplex numbers will be difficult. The zero norm set of hypercomplex numbers is a closed set of special geometric meanings, like the light-cone in the realistic space-time, which has no substantial effect on the algebraic calculus. The physical equations expressed in Clifford algebra have a simple formalism, symmetrical structure, standard derivation, complete content. Therefore, we can hope that this magical algebra can complete a new large synthesis of modern science.展开更多
In this paper, it is considered for some two-dimensional singular integral equations of the hypercomplex functions in the Douglis sense. In some special cases, the Fredholm' conditions and index formula of such eq...In this paper, it is considered for some two-dimensional singular integral equations of the hypercomplex functions in the Douglis sense. In some special cases, the Fredholm' conditions and index formula of such equations are obtained.展开更多
In order to calculate the cross-correlation of two color images treated as vector in a holistic manner,a rapid vertical/parallel decomposition algorithm for quaternion is presented.The calculation for decomposition is...In order to calculate the cross-correlation of two color images treated as vector in a holistic manner,a rapid vertical/parallel decomposition algorithm for quaternion is presented.The calculation for decomposition is reduced from 21 times to 4 times real number multiplications with the same results.An algorithm for cross-correlation of color images based on decomposition in time domain is put forward,in which some properties pointed out in this paper can be utilized to reduce the computational complexity.Simulation results show the effectiveness and superiority of the proposed method.展开更多
文摘The Clifford algebra is a unification and generalization of real number, complex number, quaternion, and vector algebra, which accurately and faithfully characterizes the intrinsic properties of space-time, providing a unified, standard, elegant, and open language and tools for numerous complicated mathematical and physical theories. So it is worth popularizing in the teaching of undergraduate physics and mathematics. Clifford algebras can be directly generalized to 2<sup>n</sup>-ary associative algebras. In this generalization, the matrix representation of the orthonormal basis of space-time plays an important role. The matrix representation carries more information than the abstract definition, such as determinant and the definition of inverse elements. Without this matrix representation, the discussion of hypercomplex numbers will be difficult. The zero norm set of hypercomplex numbers is a closed set of special geometric meanings, like the light-cone in the realistic space-time, which has no substantial effect on the algebraic calculus. The physical equations expressed in Clifford algebra have a simple formalism, symmetrical structure, standard derivation, complete content. Therefore, we can hope that this magical algebra can complete a new large synthesis of modern science.
文摘In this paper, it is considered for some two-dimensional singular integral equations of the hypercomplex functions in the Douglis sense. In some special cases, the Fredholm' conditions and index formula of such equations are obtained.
基金supported by the National Natural Science Foundation of China (6060402160874054)
文摘In order to calculate the cross-correlation of two color images treated as vector in a holistic manner,a rapid vertical/parallel decomposition algorithm for quaternion is presented.The calculation for decomposition is reduced from 21 times to 4 times real number multiplications with the same results.An algorithm for cross-correlation of color images based on decomposition in time domain is put forward,in which some properties pointed out in this paper can be utilized to reduce the computational complexity.Simulation results show the effectiveness and superiority of the proposed method.