This paper describes a new symmetry-based approach to solving a given ordinary di¬erence equation. By studying the local structure of the set of solutions, we derive a systematic method for determining one-parameter Lie groups of symmetries in closed form. Such groups can be used to achieve successive reductions of order. If there are enough symmetries, the di¬erence equation can be completely ...

This paper considers control systems defined on Lie algebroids. After deriving basic controllability tests for general control systems, we specialize our discussion to the class of mechanical control systems on Lie algebroids. This class of systems includes mechanical systems subject to holonomic and nonholonomic constraints, mechanical systems with symmetry and mechanical systems evolving on s...

We apply nonlocal symmetry-like operators to systems of two first and two second-order ordinary differential equations to seek reduction to quadratures. The reduction of order of such systems is carried out with the help of analytic continuation of scalar equations in the complex plane. Examples are taken from the literature. Precisely it is shown how the reduction to quadratures of a system of...

We propose a new approach for detecting ellipses. The approach is based on the geometrical property that five points on an ellipse can determine the parameters of the ellipse, and the symmetry of ellipses is used to obtain these points. Using symmetry, we classify the edge points in an input image into several subimages. Ellipses with different symmetric axes will lie in different subimages. In...