A canal surface in R 3 , generated by a parametrized curve C = m(t), is the Zariski closure of the envelope of the set of spheres with radius r(t) centered at m(t). This concept is a generalization of the classical notion of an offsets of a plane curve: first, the canal surface is a surface in 3-space rather than a curve in R 2 and second, the radius function r(t) is allowed to vary with the parameter t. In case r(t) = const, the resulting envelope is called a pipe surface. In this paper we develop an elementary symbolic method for generating rational parametrizations of canal surfaces generated by rational curves m(t) with rational radius variation r(t). This method leads to the problem of decomposing a polynomial into a sum of two squares over R. We discuss decomposition algorithms which give symbolic and numerical answers to this problem.

A canal surface S, generated by a parametrized curve m(t), in R 3 is the envelope of the set of spheres with radius r(t) centered at m(t). This concept generalizes the classical offsets (for r(t) = const) of plane curves. In this paper we develop elementary symbolic methods for generating a rational parametrization of canal surfaces generated by rational curves m(t) with rational radius variation r(t). In a pipe surface r(t) is constant.

We present an axiomatic approach to Gröbner basis techniques in free multi-filtered modules over a not necessarily commutative multi-filtered ring. It is shown that classical Gröbner basis concepts can be viewed as models of our axioms. Within this theory it is possible to prove a general theorem about the dimension of filter spaces in multifiltered modules. We use these ideas for computing the Hilbert function of finitely generated multi-filtered modules over difference-differential rings. Thus the presented method allows to compute a multivariate generalization of the univariate and the bivariate dimension polynomial considered in the papers of Winkler and Zhou.

We study the structure of one-generated semirings from the symbolical point of view and their connections to numerical semigroups. We prove that such a semiring is additively divisible if and only if it is additively idempotent. We also show that every at most countable commutative semigroup is contained in the additive part of some one-generated semiring.

The report is devoted to improvement of algorithms for separating functions by Bezier curve, which are based on the algorithm design with optimal classifying objects. The developed algorithm performs classification in the systems with complex order.