Terms Of Laurent Expansion Research Articles

New analytic solutions in f(R)$$ f(R) $$‐cosmology from Painlevé analysis

Using the singularity analysis, we investigate the integrability properties and existence of analytic solutions in ‐cosmology. Specifically, for some power‐law ‐theories of particular interest, we apply the ARS algorithm to prove if the field equations possess the Painlevé property. Constraints for the free parameters of the power‐law models are derived, and new analytic solutions are derived, expressed in terms of Laurent expansions.

Singularity analysis and analytic solutions for the Benney–Gjevik equations

Abstract We apply the Painlevé test for the Benney and the Benney–Gjevik equations, which describe waves in falling liquids. We prove that these two nonlinear 1 + 1 evolution equations pass the singularity test for the travelling-wave solutions. The algebraic solutions in terms of Laurent expansions are presented.

Static spherically symmetric Einstein-æther models II: Integrability and the modified Tolman–Oppenheimer–Volkoff approach

We investigate the existence of analytic solutions for the field equations in the Einstein-æther theory for a static spherically symmetric spacetime and provide a detailed dynamical systems analysis of the field equations. In particular, we investigate if the gravitational field equations in the Einstein-æther model in the static spherically symmetric spacetime possesses the Painlevè property, so that an analytic explicit integration can be performed. We find that analytic solutions can be presented in terms of Laurent expansions only when the matter source consists of a perfect fluid with a linear equation of state (EoS) μ=μ0+h−1p,h>1. In order to study the field equations we apply the Tolman–Oppenheimer–Volkoff (TOV) approach and other approaches. We find that the relativistic TOV equations are drastically modified in Einstein-æther theory, and we explore the physical implications of this modification. We study perfect fluid models with a scalar field with an exponential potential. We discuss all of the equilibrium points and discuss their physical properties.

Cosmological singularities and analytical solutions in varying vacuum cosmologies

We investigate the dynamical features of a large family of running vacuum cosmologies for which Lambda evolves as a polynomial in the Hubble parameter. Specifically, using the critical point analysis we study the existence and the stability of singular solutions which describe de-Sitter, radiation and matter dominated eras. We find several classes of Lambda (H) cosmologies for which new analytical solutions are given in terms of Laurent expansions. Finally, we show that the Milne universe and the R_{h}=ct model can be seen as perturbations around a specific Lambda (H) model, but this model is unstable.

Representations of Meromorphic Functions in Terms of Laurent Expansions and Partial Boundary Values

For any analytic function f on a Riemann surface S with an isolated singularity, we give a "good" formula representing f on a large domain containing any given point on which f is regular in terms of the coefficients of Laurent's expansion around any fixed isolated singular point using a conformal mapping of a doubly-connected domain. Representations of meromorphic functions in terms of partial boundary values will also be referred, clearly.

Applications of Laurent Expansions in Multivariable Control Systems

Many analysis and design problems for multivariable control systems can be handled more easily in terms of Laurent expansions. Others allow additional insight and allow a more natural extension from the scalar case if the interpretation in terms of Laurent expansion is considered. We consider the problems of determination of degree and order of poles and zeros, Smith-MacMillan form, cancellations in interconnections and representations, general, first degree and real factorizations, inversion of systems and the minimal design problem. In each case we discuss relevant control applications and go into the algorithmic and numerical aspects.