On the Quadratization of the Integrals for the Many-Body Problem

A new approach to the construction of difference schemes of any order for the many-body problem that preserves all its algebraic integrals is proposed herein. We introduced additional variables, namely distances and reciprocal distances between bodies, and wrote down a system of differential equations with respect to the coordinates, velocities, and the additional variables. In this case, the system lost its Hamiltonian form, but all the classical integrals of motion of the many-body problem under consideration, as well as new integrals describing the relationship between the coordinates of the bodies and the additional variables are described by linear or quadratic polynomials in these new variables. Therefore, any symplectic Runge–Kutta scheme preserves these integrals exactly. The evidence for the proposed approach is given. To illustrate the theory, the results of numerical experiments for the three-body problem on a plane are presented with the choice of initial data corresponding to the motion of the bodies along a figure of eight (choreographic test).

Effects of Environmental and Electrical Factors on Metering Error and Consistency of Smart Electricity Meters

The smart electricity meter (SEM) is an important part of smart power grid, and the accuracy of SEMs is the basis for power grid operation control and trade settlement between power supply and electricity consumption, but the evolution behaviors of metering error of SEMs under field operation conditions have not yet been identified. The SEMs were installed and operated on site, metering error data were collected under various temperature and current conditions. The influences of current, power coefficient, and temperature on metering error and consistency were analyzed separately with the help of quadratic polynomials, and then an integrated model elaborating the joint effects of multi-stress was developed based on a binary quadratic polynomial. We find that a lower temperature and a larger current result in a higher metering error of SEMs; however, the effects of current on metering error are determined by power coefficients. The results have reference value for remote metrological verification, error monitoring, and the optimization of the operation and maintenance scheme of SEMs.

On Polynomial Solutions of Pell’s Equation

Polynomial Pell’s equation is x 2 − D y 2 = ± 1 , where D is a quadratic polynomial with integer coefficients and the solutions X , Y must be quadratic polynomials with integer coefficients. Let D = a 2 x 2 + a 1 x + a 0 be a polynomial in Z x . In this paper, some quadratic polynomial solutions are given for the equation x 2 − D y 2 = ± 1 which are significant from computational point of view.

The Pintz–Steiger–Szemerédi estimate for intersective quadratic polynomials in function fields

Let [Formula: see text] be the polynomial ring over the finite field [Formula: see text] of [Formula: see text] elements. For a natural number [Formula: see text] let [Formula: see text] be the set of all polynomials in [Formula: see text] of degree less than [Formula: see text] Let [Formula: see text] be a quadratic polynomial over [Formula: see text] Suppose that [Formula: see text] is intersective, that is, which satisfies [Formula: see text] for any [Formula: see text] with [Formula: see text] where [Formula: see text] denotes the difference set of [Formula: see text] Let [Formula: see text] Suppose that [Formula: see text] and that the characteristic of [Formula: see text] is not divisible by 2. It is proved that [Formula: see text] for any [Formula: see text] where [Formula: see text] is a constant depending only on [Formula: see text] and [Formula: see text]

How Accurately Can a Spherical Cap be Represented by Rational Quadratic Polynomials?

This paper discusses the incapability of a tensor product rational quadratic patch to accurately represent a spherical cap. It was analytically found that there is no combination of control points and associated weights to accurately represent the spherical cap. On top of that, an optimization technique has revealed that for a unit sphere the computed radii in the parametric space may reduce within the interval [0.999999994, 1.000104146]. This study makes sense as a preparatory stage in relation with the isogeometric analysis (IGA), which may be applied in conjunction with either the Finite Element Method (FEM) or the Boundary Element Method (BEM).

The bifurcation locus for numbers of bounded type

We define a family $\mathcal {B}(t)$ of compact subsets of the unit interval which provides a filtration of the set of numbers whose continued fraction expansion has bounded digits. We study how the set $\mathcal {B}(t)$ changes as the parameter t ranges in $[0,1]$ , and see that the family undergoes period-doubling bifurcations and displays the same transition pattern from periodic to chaotic behaviour as the family of real quadratic polynomials. The set $\mathcal {E}$ of bifurcation parameters is a fractal set of measure zero and Hausdorff dimension $1$ . The Hausdorff dimension of $\mathcal {B}(t)$ varies continuously with the parameter, and we show that the dimension of each individual set equals the dimension of the corresponding section of the bifurcation set $\mathcal {E}$ .

An Alternative Approach to Solving Cubic and Quartic Polynomial Equations

Aims: The aim of the research study was to develop a more direct and intuitive approach for the solution of polynomial equations of degree 3 and four. Study Design: The study employed equivalent polynomial substitution that is more intuitive and direct to formulate than the traditional formulations and one that is easily solvable. Place and Duration of Study: The study has been undertaken by the author at the university of Eswatini in the period from February to March 2021. Methodology: Two alternative procedures have been presented for the analytical solution of cubic and quartic equations and demonstrated with worked examples. The solution is derived through a direct procedure without involving intermediate variable substitution. Results: For cubic equations, the solution provides explicit expression of an equivalent cubic that is formed directly in terms of the original variable x. As such, the formula is intuitive and simple to derive or understand as well as apply. For the quartic equations, the same decomposition form is used as that of the cubic equation using two quadratic polynomials that have symmetric form thus making it easy to develop the solution as well as solve the equations Conclusion: The alternative formula is easy to formulate and solve and provides a more intuitive basis for understanding and solving polynomial equations.