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. nonidempotent 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 nonidempotent 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, nonidempotent hulls/closures, representation of
partial morphisms, onepoint 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 manypoint extension is
developed, which in many aspects parallels that of the onepointextension.
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 semiCauchy 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. WeckSchwarz.
φφ
