Computability Models: Algebraic, Topological and Geometric Algorithms

Geometric algorithms for algebraic curves and surfaces

This work presents novel geometric algorithms dealing with algebraic curves and surfaces of arbitrary degree. These algorithms are exact and complete – they return the mathematically true result for all input instances. Efficiency is achieved by cutting back expensive symbolic computation and favoring combinatorial and adaptive numerical methods instead, without spoiling exactness in the overal...

Bouquets of geometric lattices: some algebraic and topological aspects

Matroid theory is in the center of Combinatorics, Finite Geometry, Lattice theory and Combinatorial Optimization. During the last decades, extensive search was done in order to find a good degree of generality which still preserves the validity of deep results known for matroids. One of such generalizations is the concept of bouquet of matroids introduced in 1983 by Deza, Frank1 and Laurent and...

Randomness, Computability, and Algebraic Specifications

ON ALGEBRAIC AND COALGEBRAIC CATEGORIES OF VARIETY-BASED TOPOLOGICAL SYSTEMS

Motivated by the recent study on categorical properties of latticevalued topology, the paper considers a generalization of the notion of topological system introduced by S. Vickers, providing an algebraic and a coalgebraic category of the new structures. As a result, the nature of the category   TopSys   of S. Vickers gets clari ed, and a metatheorem is stated, claiming that (latticevalu...

A topological view on algebraic computation models

We investigate the topological aspects of some algebraic computation models, in particular the BSS-model. Our results can be seen as bounds on how different BSS-computability and computability in the sense of computable analysis can be. The framework for this is Weihrauch reducibility. As a consequence of our characterizations, we establish that the solvability complexity index is (mostly) inde...