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Inhomogeneous Cauchy-Riemann equation in octonion space

Applied Mathematics(A Journal of Chinese Universities) 2015年第4期30卷 447-452页

作者： GONG Ding-dong Department of Mathematical Science Zhejiang Sci-Tech University

The integral representation of differentiable functions in octonion space is obtained and the explicit solution of the inhomogeneous Cauchy-Riemann equation is given by integral representation. As an application, the ... 详细信息

关键词： octonion integral representation of differentiable functions inhoraogeneous Cauchy-RiemannEquation Cousin problem.

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TAYLOR SERIES AND ORTHOGONALITY OF THE octonion ANALYTIC FUNCTIONS

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Acta Mathematica Scientia 2001年第3期21卷 323-330页

作者：李兴民彭立中 Department of Mathematics Guangzhou Normal University Guangzhou 510405 China Department of Mathematics Beijing University Beijing 100871 China

The Taylor series of the octonion analytic function is given. And the orthogonal formula for the octonion analytic functions is also obtained.

关键词： octonion O-analytic function Taylor series orthogonality

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Three-line Theorem on the octonions

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Acta Mathematica Sinica,English Series 2004年第3期20卷 483-490页

作者： XingMinLI LiZhongPENG DepartmentofMathematics SouthChinaNormalUniversityGuangzhou510631P.R.China SchoolofMathematicalSciences PekingUniversityBeijing100871P.R.China

The three-line theorem on the octonions is obtained, which generalizes the result of J. Peetre and P. Sj?lin from the associative Clifford algebra to non-associative octonion algebra.

关键词： octonion O-analytic functions 3-line theorem

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Powers of octonions

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Applied Mathematics 2021年第2期12卷 75-84页

作者： W. E. Ahmed Mathematics and Statistics Department Faculty of Science Imam Mohammad Ibn Saud Islamic University Riyadh KSA Department of Basics and Engineering Sciences Faculty of Engineering University of Khartoum Khartoum Sudan

As it is known, Binomial expansion, De Moivre’s formula, and Euler’s formula are suitable methods for computing the powers of a complex number, but to compute the powers of an octonion number in easy way, we need to derive suitable formulas from these methods. In this paper, we present a novel way to compute the powers of an octonion number using formulas derived from the binomial expansion.

关键词： octonion Matrix Algebra

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Multidimensional Laplace Transforms over Quaternions, octonions and Cayley-Dickson Algebras, Their Applications to PDE

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Advances in Pure Mathematics 2012年第2期2卷 63-103页

作者： Sergey Victor Ludkovsky Department of Applied Mathematics Moscow State Technical University MIREA Moscow Russia

Multidimensional noncommutative Laplace transforms over octonions are studied. Theorems about direct and inverse transforms and other properties of the Laplace transforms over the Cayley-Dickson algebras are proved. Applications to partial differential equations including that of elliptic, parabolic and hyperbolic type are investigated. Moreover, partial differential equations of higher order with real and complex coefficients and with variable coefficients with or without boundary conditions are considered.

关键词： Laplace Transform Quaternion Skew Field octonion Algebra Cayley-Dickson Algebra Partial Differential Equation Non-Commutative Integration

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The Cyclic Universes Model Based on the Split Division Algebras

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Journal of Modern Physics 2018年第13期9卷 2257-2273页

作者： Ding-Yu Chung Utica Michigan USA

The proposed cyclic universes model based on the split division algebras accounts for the inflation, the Big Bang, gravity, dark energy, dark matter, the standard model, and the masses of all elementary particles. The split algebras (complex quaternion and complex octonion) as the Furey model generate the fixed spacetime dimension number for the observable universe with the fixed 4-dimensional spacetime (4D) standard model particles and the oscillating spacetime dimension number for the oscillating universes (hidden or dark energy) with the oscillation between 11D and 11D through 10D and between 10D and 10D through 4D. 11D has the lowest rest mass, the highest speed of light, and the highest vacuum energy, while 4D has the highest rest mass, the lowest (observed) speed of light, and zero vacuum energy. In the cyclic universes model, the universes start with the positive-energy and the negative-energy 11D membrane-antimembrane dual universes from the zero-energy inter-universal void, and are followed by the transformation of the 11D membrane-antimembrane dual universes into the 10D string-antistring dual universes and the external dual gravities as in the Randall-Sundrum model, resulting in the four equal and separate universes consisting of the positive-energy 10D universe, the positive-energy external gravity, the negative-energy 10D universe, and the negative-energy external gravity. Under the fixed spacetime dimension number, the positive-energy 10D universe is transformed into 4D standard model particles through the inflation and the Big Bang. Dark matter is the right-handed neutrino, exactly five times of baryonic matter in total mass in the universe. Under the oscillating spacetime dimension number, the other three universes oscillate between 10D and 10D through 4D, resulting in the hidden universes when D > 4 and dark energy (the maximum dark energy = 3/4 = 75%) when D = 4. Eventually, all four universes return to the 10D universes.

关键词： Cyclic Universes Model Division Algebras Furey Complex Quaternion Complex octonion Dark Energy Dark Matter Standard Model Gravity

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