Unified Functional Method for Solving General Polynomial Equations of Degree Less Than Five

A unified method for solving that incorporate a computational formula that relate the coefficients of the depressed equation and the coefficients of the standard polynomial equation is proposed in this study. This is to ensure that this method is valid for all It shall apply the undetermined parameter method of auxiliary function to obtain solutions to these polynomial equations of degree less than five in one variable. In particular, the result of our work is a unification and improvement on the work of several authors in the sense that only applicable for the case of polynomial equation of degree one. Finally, our results improve and generalize the result by applying standard formula methods for solving higher degree polynomials. It is recommended that the effort should be made toward providing other variant methods that are simpler and friendly.

Iterative stability analysis for general polynomial control systems

The main challenge of the stability analysis for general polynomial control systems is that non-convex terms exist in the stability conditions, which hinders solving the stability conditions numerically. Most approaches in the literature impose constraints on the Lyapunov function candidates or the non-convex related terms to circumvent this problem. Motivated by this difficulty, in this paper, we confront the non-convex problem directly and present an iterative stability analysis to address the long-standing problem in general polynomial control systems. Different from the existing methods, no constraints are imposed on the polynomial Lyapunov function candidates. Therefore, the limitations on the Lyapunov function candidate and non-convex terms are eliminated from the proposed analysis, which makes the proposed method more general than the state-of-the-art. In the proposed approach, the stability for the general polynomial model is analyzed and the original non-convex stability conditions are developed. To solve the non-convex stability conditions through the sum-of-squares programming, the iterative stability analysis is presented. The feasible solutions are verified by the original non-convex stability conditions to guarantee the asymptotic stability of the general polynomial system. The detailed simulation example is provided to verify the effectiveness of the proposed approach. The simulation results show that the proposed approach is more capable to find feasible solutions for the general polynomial control systems when compared with the existing ones.

Quadratic First Integrals of Time-dependent Dynamical Systems of the Form q¨a=-Γbcaq˙bq˙c-ω(t)Qa(q)

We consider the time-dependent dynamical system q¨a=−Γbcaq˙bq˙c−ω(t)Qa(q) where ω(t) is a non-zero arbitrary function and the connection coefficients Γbca are computed from the kinetic metric (kinetic energy) of the system. In order to determine the quadratic first integrals (QFIs) I we assume that I=Kabq˙aq˙b+Kaq˙a+K where the unknown coefficients Kab,Ka,K are tensors depending on t,qa and impose the condition dIdt=0. This condition leads to a system of partial differential equations (PDEs) involving the quantities Kab,Ka,K,ω(t) and Qa(q). From these PDEs, it follows that Kab is a Killing tensor (KT) of the kinetic metric. We use the KT Kab in two ways: a. We assume a general polynomial form in t both for Kab and Ka; b. We express Kab in a basis of the KTs of order 2 of the kinetic metric assuming the coefficients to be functions of t. In both cases, this leads to a new system of PDEs whose solution requires that we specify either ω(t) or Qa(q). We consider first that ω(t) is a general polynomial in t and find that in this case the dynamical system admits two independent QFIs which we collect in a Theorem. Next, we specify the quantities Qa(q) to be the generalized time-dependent Kepler potential V=−ω(t)rν and determine the functions ω(t) for which QFIs are admitted. We extend the discussion to the non-linear differential equation x¨=−ω(t)xμ+ϕ(t)x˙(μ≠−1) and compute the relation between the coefficients ω(t),ϕ(t) so that QFIs are admitted. We apply the results to determine the QFIs of the generalized Lane–Emden equation.

Finding Zeros of Quartic Polynomials by Using Radicals

We present a technique for finding roots of a quartic general polynomial equation of a single variable by using radicals. The solution of quartic polynomial equations requires knowledge of lower degree polynomial equations; therefore, we study solving polynomial equations of degree less than four as well. We present self-reciprocal polynomials as a specialization and additionally solve numerical example.

Finding Zeros of Quartic Polynomials by Using Radicals

We present a technique for finding roots of a quartic general polynomial equation of a single variable by using radicals. The solution of quartic polynomial equations requires knowledge of lower degree polynomial equations; therefore, we study solving polynomial equations of degree less than four as well. We present self-reciprocal polynomials as a specialization and additionally solve numerical example.

The radiation instability in modified gravity

We study the problem of the instability of inhomogeneous radiation universes in quadratic Lagrangian theories of gravity written as a system of evolution equations with constraints. We construct formal series expansions and show that the resulting solutions have a smaller number of arbitrary functions than that required in a general solution. These results continue to hold for more general polynomial extensions of general relativity.

Unique range sets - a further study

The purpose of the paper is to investigate the problems of unique range sets in the most general setting. Accordingly, we have studied sufficient conditions for a general polynomial to generate a unique range set which put all the variants of unique range sets into one structure. Most importantly, as an application of the main result we have been able to accommodate not only examples of critically injective polynomials but also examples of non-critically injective polynomials generating unique range sets which are for the first time being exemplified in the literature. Furthermore, some of these examples show that characterization of unique range sets generated by non-critically injective polynomials does not always demand gap polynomials which also complements the recent results by An and Banerjee-Lahiri in this direction. Moreover, one of the lemmas proved in this paper improves and generalizes some results due to Frank-Reinders and Lahiri respectively.