[SADE] A Maple package for the Symmetry Analysis of Differential Equations

We present the package SADE (Symmetry Analysis of Differential Equations) for the determination of symmetries and related properties of systems of differential equations. The main methods implemented are: Lie, nonclassical, Lie-B\"acklund and potential symmetries, invariant solutions, first-integrals, N\"other theorem for both discrete and continuous systems, solution of ordinary differential equations, reduction of order or dimension using Lie symmetries, classification of differential equations, Casimir invariants, and the quasi-polynomial formalism for ODE's (previously implemented in the package QPSI by the authors) for the determination of quasi-polynomial first-integrals, Lie symmetries and invariant surfaces. Examples of use of the package are given

Noise-Tolerant Parallel Learning of Geometric Concepts

We present several efficient parallel algorithms for PAC-learning geometric concepts in a constant-dimensional space. The algorithms are robust even against malicious classification noise of any rate less than 1/2. We first give an efficient noise-tolerant parallel algorithm to PAC-learn the class of geometric concepts defined by a polynomial number of (d-1)-dimensional hyperplanes against an arbitrary distribution where each hyperplane has a slope from a set of known slopes. We then describe how boosting techniques can be used so that our algorithms\u27 dependence on {GREEK LETTER} and {DELTA} does not depend on d. Next we give an efficient noise-tolerant parallel algorithm to PAC-learn the class of geometric concepts defined by a polynomial number of (d-1)-dimensional hyperplanes (of unrestricted slopes) against a uniform distribution. We then show how to extend our algorithm to learn this class against any (unknown) product distribution. Next we defined a complexity measure of any set S of (d-1)-dimensional surfaces that we call the variant of S and prove that the class of geometric concepts defined by surfaces of polynomial variant can be efficienty learned in parallel under a product distribution (even under malicious classification noise). Furthermore, we show that the VC-dimension of the class of geometric concepts defined by a set of surfaces S of variant v is at least v. Finally, we give an efficient, parallel, noise-tolerant algorithm to PAC-learn any geometric concept defined by a set S of (d-1)-dimensional surfaces of polynomial area under a uniform distribution

On classification of some surfaces of revolution of finite type

In this article, we study the following problem of [5]: Classify all finite type surfaces in a Euclidean 3-space E3. A surface M in a Euclidean 3-space is said to be of finite type if each of its coordinate functions is a finite sum of eigenfunctions of the Laplacian operator on M with respect to the induced metric (cf. [1,2]). Minimal surface are the simplest examples of surfaces of finite type, in fact, minimal surfaces are of l-type. The spheres, minimal surfaces and circular cylinders are the only known exampls of surfaces of finite type in E3 and it seems to be the only finite type surfaces in E3 (cf. [5]). The first author conjectured in [2] that spheres are the only compact finite type surfaces in E3. Since then, it was prived step by sted and separately that finite type tubes, finite type ruled surfaces, finite type quadrics and finite type cones are surfaces of the only known examples (cf. [2,6,7,10].) Our next natural target for this classification problem is the class of surfaces of revolution. However, this case seems to be much difficult than the other cases mentioned above. We therefore investigate this classification problem for this class and obtain classification theorems for surfaces of revolution which are either of rational or of polynomial kinds (cf. §1 for the definitions). As consequence, further supports for the conjecture cited above are obtained

Image segmentation with adaptive region growing based on a polynomial surface model

A new method for segmenting intensity images into smooth surface segments is presented. The main idea is to divide the image into flat, planar, convex, concave, and saddle patches that coincide as well as possible with meaningful object features in the image. Therefore, we propose an adaptive region growing algorithm based on low-degree polynomial fitting. The algorithm uses a new adaptive thresholding technique with the L∞ fitting cost as a segmentation criterion. The polynomial degree and the fitting error are automatically adapted during the region growing process. The main contribution is that the algorithm detects outliers and edges, distinguishes between strong and smooth intensity transitions and finds surface segments that are bent in a certain way. As a result, the surface segments corresponding to meaningful object features and the contours separating the surface segments coincide with real-image object edges. Moreover, the curvature-based surface shape information facilitates many tasks in image analysis, such as object recognition performed on the polynomial representation. The polynomial representation provides good image approximation while preserving all the necessary details of the objects in the reconstructed images. The method outperforms existing techniques when segmenting images of objects with diffuse reflecting surfaces

Complete Algebraic Vector Fields on Danielewski Surfaces

We give the classification of all complete algebraic vector fields on Danielewski surfaces (smooth surfaces given by $xy=p(z)$). We use the fact that for each such vector field there exists a certain fibration that is preserved under its flow. In order to get the explicit list of vector fields a classification of regular function with general fiber $\mathbb{C}$ or $\mathbb{C}^*$ is required. In this text we present results about such fibrations on Gizatullin surfaces and we give a precise description of these fibrations for Danielewski surfaces.Comment: appearing in Annales de l'institut Fourie