### A- and B-Stability for Runge-Kutta Methods - Characterizations and Equivalence.

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In this paper, the existence and the localization result will be proven for vector Dirichlet problem with an upper-Carathéodory right-hand side. The result will be obtained by combining the continuation principle with bound sets technique.

In his famous tetralogy, Space Odyssey, A. C. Clarke called the calculation of a motion of a mass point in the gravitational field of the massive cuboid a classical problem of gravitational mechanics. This article presents a proposal for a solution to this problem in terms of Newton's theory of gravity. First we discuss and generalize Newton's law of gravitation. We then compare the gravitational field created by the cuboid -- monolith, with the gravitational field of the homogeneous sphere. This...

We study the integrability of two-dimensional autonomous systems in the plane of the form $=-y+{X}_{s}(x,y)$, $=x+{Y}_{s}(x,y)$, where Xs(x,y) and Ys(x,y) are homogeneous polynomials of degree s with s≥2. First, we give a method for finding polynomial particular solutions and next we characterize a class of integrable systems which have a null divergence factor given by a quadratic polynomial in the variable ${({x}^{2}+{y}^{2})}^{s/2-1}$ with coefficients being functions of tan−1(y/x).

The problem of asymptotic stabilization for a class of differential inclusions is considered. The problem of choosing the Lyapunov functions from the parametric class of polynomials for differential inclusions is reduced to that of searching saddle points of a suitable function. A numerical algorithm is used for this purpose. All the results thus obtained can be extended to cover the discrete systems described by difference inclusions.

We apply the well-known homotopy continuation method to address the motion planning problem (MPP) for smooth driftless control-affine systems. The homotopy continuation method is a Newton-type procedure to effectively determine functions only defined implicitly. That approach requires first to characterize the singularities of a surjective map and next to prove global existence for the solution of an ordinary differential equation, the Wazewski equation. In the context of the MPP, the aforementioned...

We give an estimate for the distance between a given approximate solution for a Lipschitz differential inclusion and a true solution, both depending continuously on initial data.

The purpose of this article is to find the 7th order formulas with rational parameters. The formulas are of the 11th stage. If we compare the coefficients of the development ${\sum}_{i=1}^{\infty}\frac{{h}^{i}}{i!}\frac{{d}^{i-1}}{d{x}^{i-1}}\mathbf{f}\left[x,\mathbf{y}\left(x\right)\right]$ up to ${h}^{7}$ with the development given by successive insertion into the formula $h.{f}_{i}({k}_{0},{k}_{1},...,{k}_{i-1})$ for $i=1,2,...,10$ and $k={\sum}_{i=0}^{10}{p}_{i},{k}_{i}$ we obtain a system of 59 condition equations with 65 unknowns (except, the 1st one, all equations are nonlinear). As the solution of this system we get the parameters of the 7th order Runge-Kutta formulas as rational numbers.