We consider interpolation in Lie groups. Based on points on the manifold together with tangent vectors at (some of) these points, we construct Hermite interpolation polynomials. If the points and tangent vectors are produced in the process of integrating an ordinary differential equation in terms of Lie-algebra actions, we use the truncated inverse of the differential of the exponential mapping...

We consider interpolation in Lie groups and homogeneous spaces. Based on points on the manifold together with tangent vectors at (some of) these points, we construct Hermite interpolation polynomials. If the points and tangent vectors are produced in the process of integrating an ordinary di erential equation on a Lie group or a homogeneous space, we use the truncated inverse of the di erential...

We show that every smooth closed curve Γ immersed in Euclidean space R satisfies the sharp inequality 2(P + I) + V ≥ 6 which relates the numbers P of pairs of parallel tangent lines, I of inflections (or points of vanishing curvature), and V of vertices (or points of vanishing torsion) of Γ. We also show that 2(P + I) + V ≥ 4, where P is the number of pairs of concordant parallel tangent lines....

We consider several examples of mechanical systems the configuration spaces which have form smooth manifolds with a unique singular point: two intersecting (or tangent) curves on two-dimensional torus, four four-dimensional torus common point, and cone (cusp) in $${{\mathbb{R}}^{6}}$$ . The main problem presented this paper is calculation (co)tangent space above point by means various theoretic...

Fulton’s algorithm is used to calculate the intersection multiplicity of two plane curves about a rational point. This work extends Fulton’s algorithm first to algebraic points (encoded by regular chains) and then, with some generic assumptions, to l many hypersurfaces. Out of necessity, we give a standard-basis free method (i.e. practically efficient method) for calculating tangent cones at po...

PSwarm was developed originally for the global optimization of functions without derivatives and where the variables are within upper and lower bounds. The underlying algorithm used is a pattern search method, more specifically a coordinate search method, which guarantees convergence to stationary points from arbitrary starting points. In the (optional) search step of coordinate search, the alg...

PSwarm was developed originally for the global optimization of functions without derivatives and where the variables are within upper and lower bounds. The underlying algorithm used is a pattern search method, more specifically a coordinate search method, which guarantees convergence to stationary points from arbitrary starting points. In the (optional) search step of coordinate search, the alg...