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The dynamic equations of the solar system outlined in relative coordinates provide the derivatives of the orbital angular momentum and the orbital energy with respect to time. From them, the perihelion precession rates as well as the variation rates of the orbital eccentricity of the planets of the solar system at J2000 are calculated under the approximation of elliptic orbits. The integration of the derivative of Earth's orbital angular momentum yields the planetary corrections to the equation of time. Venus' and Jupiter's corrections are the main ones, whose addition reaches up to $\pm 1$ minute.
Hypercomplex number systems are vector algebras with the definition of multiplication and division of vectors, satisfying the associativity and distributive law. In this paper, some new types of hypercomplex numbers and their fundamental properties are introduced, the Clifford algebra formalisms of hydrodynamics and gauge field equations are established, and some novel consistent conditions helpful to understand the properties of solutions to nonlinear physical equations are derived. The coordinate transformation and covariant derivatives of hypercomplex numbers are also discussed. The basis elements of the hypercomplex numbers have group-like properties and satisfy a structure equation $A^2=nA$. The hypercomplex number system integrates the advantages of algebra, geometry and analysis, and provides a unified, standard and elegant language and tool for scientific theories and engineering technology, so it is easy to learn and use. The description of mathematical, physical and engineering problems by hypercomplex numbers is of neat formalism, symmetric structure and standard derivation, which is especially suitable for the efficient processing of the higher dimensional complicated systems.
In this article, we consider deformed spheres as a new reference model for the geoid, alternatively to the classical ellipsoidal one. The parametrization of deformed spheres is furnished through the incomplete elliptic integrals. From the other side, the solutions for geodesics on those surfaces are given entirely via elementary analytical functions, contrary to the case of ellipsoids of revolution. We explicitly described algorithms (all necessary computational steps) for the solution of the direct and inverse geodetic problems on the deformed spheres. Finally, we presented a few illustrative numerical solutions of the inverse geodetic problems for two conceptual cases of near and far points. It had turned out that even in the non-optimized case we obtained the good agreement with the predictions of the World Geodetic System 1984's ellipsoidal reference model.
Despite the longstanding interest in the shapes of the eggs since the ancient time till nowadays, the available parametric descriptions in the modern literature are given only via purely empirical formulas without any clear relationships with their measurable physical parameters. Here we present a geometrical model of the eggs based on Perseus spirics which were known as well since the ancient time but their analytical parameterizations were absent in the meantime. Such parameterizations have been found recently and the present work is based on the idea to use the spirics as a geometrical model of the egg's shapes. Explicit formulas for the volume, surface area and the curvatures of the eggs are derived from the first principles and these have been compared with the available empirical formulas and experimental data.
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