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Institut für elegante Mathematik ... gruppe eM



abstracts:

Int'l Conference on Convergence Theory
Dijon, France, 1994

APPLICATIONS OF PRETOPOLOGIES IN OPTIMISATION
Rolf D. Brandt
BEB, Hannover, Germany
The category of pretopological spaces (i.e. non-idempotent hull operators) and continuous maps is the extensional topological hull (see talk by F. Schwarz) of the category TOP, of topological spaces and continuous maps [Herrlich 1988]. For this reason many solutions of optimization problems dealing with "convergence structures in the widest sense" have their fixpoint in pretopologies but not in topologies.

In the talk examples are given for applications of pretopologies for optimisation problems, i.e. algorithms using non-idempotent hull oprators. These algorithms are used for all kinds of "packing problems", e.g. optimal packing of wired circuit boards, packing problems in space technology, and in particular optimal handling of supply sources to meet demand in energy systems.

BEB is the largest producer of hydrocarbons in Germany. The above mentioned methods, in literature of computer science also called "bin packing algorithms", have been in practical use for many years, e.g. for optimising supply sources, and for "load balancing" calculations, i.e. determination of gas storage needs (working gas volume).
Keywords:
math.: pretopologies, non-idempotent hulls/closures, representation of partial morphisms, one-point extension, extensional topological hull, quasitopos, optimization, bin packing algorithms;
energy-/gas industry: optimal demand/supply balancing, net flow algorithms, optimal use of flexibilities, load balancing, working gas volume, underground gas storage.

Int'l Conference on Convergence Theory
Dijon, France, 1994

EXAMPLES OF EXTENSIONAL TOPOLOGICAL HULLS AND THEIR GENERAL CONSTRUCTION
Friedhelm Schwarz
University of Toledo, Ohio, U.S.A.
fschwarz@math.utoledo.edu
The objects X of the extensional topological hull of a topological category A are characterized, among the objects of any finally dense, extensional topological extension of A, by a test based largely on information intrinsically contained in X.

As a useful tool, the concept of a many-point extension is developed, which in many aspects parallels that of the one-point-extension. The characterization theorem is then applied to distinguish the extensional topological hull of the category of Cauchy spaces (resp. uniform limit spaces) from the category of semi-Cauchy spaces (resp. semiuniform limit spaces).

We give several examples of extensional topological hulls inside the category ofpretopological spaces and point out the importance of this hull for hereditary quotient maps.

Most of this is joint work with I.W. Alderton and S. Weck-Schwarz.
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