We present some methods and results in the application of algebraic geometry and computer algebra to the study of algebraic vector bundles, foliations and zeta functions. A connection of the methods and results with noncommutative geometry will be consider.

Interactive exercise assistants support students in practicing exercises, and acquiring procedural skills. Many mathematical topics can be practiced in such assistants. Ideally, an interactive exercise assistant not only validates final answers, but also comments on intermediate steps submitted by a student, provides hints on how to proceed, and presents worked-out examples. For these purposes,...

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We show the following two universality statements on the entry-ranges and Markov bases of spaces of 3-way contingency tables with fixed 2-margins: (1) For any finite set D of nonnegative integers, there are r, c, and 2-margins for (r, c, 3)-tables such that the set of values occurring in a fixed entry in all possible tables with these margins is D. (2) For any integer n-vector d, there are r, c...

The paper presents a possibility how to accomplish numerical and symbolic computations in online applications and in web environment. For this purpose the computer algebra system Maxima was used. The proposed solution is illustrated on the corresponding examples that are oriented to the area of automation and control.

The iterative process o-n+i = (an + 36„)/4, i>„+i = (\Janbn + b„)/2 is studied in detail. The limit of this quadratically converging process is explicitly identified, as are the uniformizing parameters. The role of symbolic computation, in discovering these nontrivial identifications, is highlighted.

It is uncommon to apply computer algebra software to proof, especially for topics which are rarely touched by computer algebra. In order to illustrate the efficacy of symbolic computation in these respects, DERIVE is used in a statistical context to prove the Central Limit Theorem and some advantages and disadvantages of using the techniques outlined here are assessed. The proof is illustrated ...

Derivatives arise frequently in graphics and scientific computation applications. As GPU’s become more widely used for scientific computation the need for derivatives can be expected to increase. To meet this need we have added symbolic differentiation as a built in language feature in the HLSL shading language. The symbolic derivative is computed at compile time so it is available in all types...

a r t i c l e i n f o a b s t r a c t A new approach is proposed for computing a comprehensive Gröb-ner basis of a parameterized polynomial system. The key new idea is not to simplify a polynomial under various specialization of its parameters, but rather keep track in the polynomial, of the power products whose coefficients vanish; this is achieved by partitioning the polynomial into two parts...