We discuss wavelet frames constructed via multiresolution analysis (MRA), with emphasis on tight wavelet frames. In particular, we establish general principles and specific algorithms for constructing framelets and tight framelets, and we show how they can be used for systematic constructions of spline, pseudo-spline tight frames and symmetric biframes with short supports and high approximation orders. Several explicit examples are discussed. The connection of these frames with multiresolution analysis guarantees the existence of fast implementation algorithms, which we discuss briefly as well.

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Groups can be studied using methods from different fields such as combinatorial group theory or string rewriting. Recently techniques from Grobner basis theory for free monoid rings (non-commutative polynomial rings) respectively free group rings have been added to the set of methods due to the fact that monoid and group presentations (in terms of string rewriting systems) can be linked to special polynomials called binomials. In the same mood, the aim of this paper is to discuss the relation between Nielsen reduced sets of generators and the Todd-Coxeter coset enumeration procedure on the one side and the Grobner basis theory for free group rings on the other. While it is well-known that there is a strong relationship between Buchberger's algorithm and the Knuth-Bendix completion procedure, and there are interpretations of the Todd-Coxeter coset enumeration procedure using the Knuth-Bendix procedure for special cases, our aim is to show how a verbatim interpretation of the Todd-Coxete...

In this survey I should like to introduce some concepts of algebraic geometry and try to demonstrate the fruitful interaction between algebraic geometry and computer algebra and, more generally, between mathematics and computer science. One of the aims of this paper is to show, by means of examples, the usefulness of computer algebra to mathematical research. Computer algebra itself is a highly diversified discipline with applications to various areas of mathematics; many of these may be found in numerous research papers, proceedings or textbooks (cf. Buchberger and Winkler, 1998; Cohen et al., 1999; Matzat et al., 1998; ISSAC, 1988–1998). Here, I concentrate mainly on Gröbner bases and leave aside many other topics of computer algebra (cf. Davenport et al., 1988; Von zur Gathen and Gerhard, 1999; Grabmeier et al., 2000). In particular, I do not mention (multivariate) polynomial factorization, another major and important tool in computational algebraic geometry. Gröbner bases were introduced originally by Buchberger as a computational tool for testing solvability of a system of polynomial equations, to count the number of solutions (with multiplicities) if this number is finite and, more algebraically, to compute in the quotient ring modulo the given polynomials. Since then, Gröbner bases have become the major computational tool, not only in algebraic geometry. The importance of Gröbner bases for mathematical research in algebraic geometry is obvious and nowadays their use needs hardly any justification. Indeed, chapters on Gröbner bases and Buchberger’s algorithm (Buchberger, 1965) have been incorporated in many new textbooks on algebraic geometry such as the books of Cox et al. (1992, 1998) or the recent books of Eisenbud (1995) and Vasconcelos (1998), not to mention textbooks which are devoted exclusively to Gröbner bases, such as Adams and Loustaunou (1994),

The SymbolicData project has the following three main goals: 1. to systematically collect existing symbolic computation benchmark data and to produce tools to extend and maintain this collection; 2. to design and implement concepts for trusted benchmarks computations on the collected data; and 3. to provide tools for data access/selection/transformation using di erent technologies. SymbolicData has developed from a \grass root initiative" of a small number of people to a stage where it should be presented to, and evaluated and used by a wider community. In this paper we report about the current state of the project, i.e., we describe the main design principles and tools which were developed to realize our goals. 1

Abstract. We present a general method for constructing real solutions to some problems in enumerative geometry which gives lower bounds on the maximum number of real solutions. We apply this method to show that two new classes of enumerative geometric problems on flag manifolds may have all their solutions be real and modify this method to show that another class may have no real solutions, which is a new phenomenon. This method originated in a numerical homotopy continuation algorithm adapted to the special Schubert calculus on Grassmannians and in principle gives optimal numerical homotopy algorithms for finding explicit solutions to these other enumerative problems.

The paper has two main contributions: The first is a set of methods for computing structure and motion for m >= 3 views of 6 points. It is shown that a geometric image error can be minimized over all views by a simple three parameter numerical optimization. Then, that an algebraic image error can be minimized over all views by computing the solution to a cubic in one variable. Finally, a minor point, is that this "quasi-linear" linear solution enables a more concise algorithm, than any given previously, for the reconstruction of 6 points in 3 views. The second

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of the state of Rheinland-Pfalz). The centre is a scientific institution of