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We study almost Riemann solitons and almost Ricci solitons in an $(\alpha,\beta)$-contact metric manifold satisfying some Ricci symmetry conditions, treating the case when the potential vector field of the soliton is pointwise collinear with the structure vector field.
The differential equation in polar coordinates of the Moon's orbit is outlined from the first-order approximation to the Lagrange equations of the Sun-Earth-Moon system expressed with relative coordinates and accelerations. The orbit of the Moon calculated this way is similar to Clairaut's modified orbit and has better parameters than those previously published. An improvement to this orbit is proposed based on theoretical arguments. With help of this new orbit, the variations in the draconic, synodic and anomalistic months are also computed showing very good agreement with observations.
In a previous paper, the notion of Gibbs state for the Hamiltonian action of a Lie group on a symplectic manifold was given, together with its applications in Statistical Mechanics, and the works in this field of the French mathematician and physicist Jean-Marie Souriau were presented. Using an adaptation of the cross product for pseudo-Euclidean three-dimensional vector spaces, we present several examples of such Gibbs states, together with the associated thermodynamic functions, for various two-dimensional symplectic manifolds, including the pseudo-spheres, the Poincaré disk and the Poincaré half-plane.
Here we derive explicit formulas that parameterize the Cassinian ovals based on their recognition as the so called spiric sections of the standard tori in the three-dimensional Euclidean space which was suggested in the ancient time by Perseus. These formulas derived originally in terms of the toric parameters are expressed through the usual geometrical parameters that enter in the present day definition of the Cassinian curves. All three types of morphologically different curves are illustrated graphically using the corresponding sets of parameters and respective formulas. The geometry of the ovals can be studied in full details and this is done here to some extent. As examples explicit formulas for the embraced volume and the surface area of the dumbbell like surface generated by the oval are presented. Last, but not least, new alternative explicit parameterizations of the Cassinian ovals are derived in polar, and even in non-canonical Monge forms.
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