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The formal structure of the early Einstein’s Special Relativity follows the axiomatic deductive method of Euclidean geometry. In this paper we show the deep-rooted relation between Euclidean and space-time geometries that are both linked to a two-dimensional number system: the complex and...
We show that the space of Euclid’s parameters for Pythagorean triples is endowed with a natural symplectic structure and that it emerges as a spinor space of the Clifford algebra R 21 , whose minimal version may be conceptualized as a 4-dimensional real algebra of “kwaternions.” We observe...
Maxwell’s equations with massive photons and magnetic monopoles are formulated using spacetime algebra. It is demonstrated that a single nonhomogeneous multi-vectorial equation describes the theory. Two limiting cases are considered and their symmetries highlighted: massless photons with...
In this paper the invariance criterion is applied for the nonlinear equation 0.1 where g ( u ) is a smooth function on u . Some particular set of Lie generators are given. In the case of inviscid Burger’s equation (1) 0.2 the Lie projectable symmetry algebra is determined, and the inviscid...
In hyperbolic complex space, the Clifford algebra is isomorphic to that of a corresponding Minkowski geometry. We define the hyperbolic imaginary unit j ( j 2 = 1, j ≠ ± 1, j * = − j ) to generate a class of Clifford algebras. We can introduce a class of non-Euclidean spaces and discuss the...
We establish the basis of a discrete function theory starting with a Fischer decomposition for difference Dirac operators. Discrete versions of homogeneous polynomials, Euler and Gamma operators are obtained. As a consequence we obtain a Fischer decomposition for the discrete Laplacian.
This is an implementation of the Fillmore–Springer–Cnops construction (FSCc) based on the Clifford algebra capacities (10) of the GiNaC computer algebra system. FSCc linearises the linear-fraction action of the Möbius group. This turns to be very useful in several theoretical and applied...
A “surable” is a category given by a special manifold of geometric algebra frames. It is a bale brought on by a surjective map the equivalence classes of which can constitute base elements of the associative algebra. It is also a stranded braid of idempotents based on a sheaf of base...
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