Cubic algebraic equations in gravity theory, parametrization with the Weierstrass function and nonarithmetic theory of algebraic equations

Abstract

A cubic algebraic equation for the effective parametrizations of the standard gravitational Lagrangian has been obtained without applying any variational principle. It was suggested that such an equation may find application in gravity theory, brane, string and Rundall-Sundrum theories. The obtained algebraic equation was brought by means of a linear-fractional transformation to a parametrizable form, expressed through the elliptic Weierstrass function, which was proved to satisfy the standard parametrizable form, but with g 2 and g 3 functions of a complex variable instead of the definite complex numbers (known from the usual (arithmetic) theory of elliptic functions and curves). The generally divergent (two) infinite sums of the inverse first and second powers of the poles in the complex plane were shown to be convergent in the investigated particular case, and the case of the infinite point of the linear-fractional transformation was investigated. Some relations were found,which ensure the parametrization of the cubic equation in its general form with the Weierstrass function.

Non-Minimally Conformally Coupling Cosmology with Multiple Vacua Potential with Cubic-Quintic-Septic Duffing Oscillator Properties

“…These cycloid oscillations occur due to the presence of the Jacobi elliptic function which is a particular solution of the non-linear Klein-Gordon wave equation. We note that these functions were very useful in solving tricky problems in cosmology and astrophysics [60][61][62][63]. The main difference between both previous cases discussed concerns first the crossing of the phantom divide line and besides the scale field and the scale factor oscillates with comparable amplitude, whereas for the previous case, the scalar field oscillates with amplitude larger than that of the scalar field.…”

Elliptic Curves, Algebraic Geometry Approach in Gravity Theory and Uniformization of Multivariable Cubic Algebraic Equations

“…In [9,10], instead of seaching new solutions for special cases, a more general approach has been proposed. Concretely, it has been shown that the Einstein's vacuum equations and the gravitational Lagrangian can be represented in the form of multivariable cubic algebraic equations.…”

“…the algebraic variety of the differentials dX i ) of the cubic algebraic equation (1.4) (in the limit d 2 X k = 0). The applied method has been proposed first in [9] but here it will be developed further and applied with respect to a sequence of algebraic equations with algebraic varieties, which are embedded into the initial one. This means that if at first the algorithm is applied with respect to the three-dimensional cubic algebraic equation (1.4) and a solution for dX 3 (depending on the Weierstrass function and its derivative is found), then the same algorithm will be applied with respect to the two-dimensional cubic algebraic equation with variables dX 1 and dX 2 , and finally to the one-dimensional cubic algebraic equation of the variable dX 1 only.…”

Reconstruction of the equation of state for the cyclic universes in homogeneous and isotropic cosmology

“…[113]. Furthermore, it is suggested [114][115][116][117] that the Weierstrass ℘( ), ζ( ) and σ ( )-functions and the Jacobian elliptic functions can play an important role to examine astrophysical and cosmological problems (for recent studies of the applications, see, e.g., [118,119]). In particular, in Ref.…”