17,831 research outputs found
Towards a Convenient Category of Topological Domains
We propose a category of topological spaces that promises to be convenient for the purposes of domain theory as a mathematical theory for modelling computation. Our notion of convenience presupposes the usual properties of domain theory, e.g. modelling the basic type constructors, fixed points, recursive types, etc. In addition, we seek to model parametric polymorphism, and also to provide a flexible toolkit for modelling computational effects as free algebras for algebraic theories. Our convenient category is obtained as an application of recent work on the remarkable closure conditions of the category of quotients of countably-based topological spaces. Its convenience is a consequence of a connection with realizability models
A Convenient Category of Domains
We motivate and define a category of "topological domains",
whose objects are certain topological spaces, generalising
the usual -continuous dcppos of domain theory.
Our category supports all the standard constructions of domain theory,
including the solution of recursive domain equations. It also
supports the construction of free algebras for (in)equational
theories, provides a model of parametric polymorphism,
and can be used as the basis for a theory of computability.
This answers a question of Gordon Plotkin, who asked
whether it was possible to construct a category of domains
combining such properties
Topological Entropy and Algebraic Entropy for group endomorphisms
The notion of entropy appears in many fields and this paper is a survey about
entropies in several branches of Mathematics. We are mainly concerned with the
topological and the algebraic entropy in the context of continuous
endomorphisms of locally compact groups, paying special attention to the case
of compact and discrete groups respectively. The basic properties of these
entropies, as well as many examples, are recalled. Also new entropy functions
are proposed, as well as generalizations of several known definitions and
results. Furthermore we give some connections with other topics in Mathematics
as Mahler measure and Lehmer Problem from Number Theory, and the growth rate of
groups and Milnor Problem from Geometric Group Theory. Most of the results are
covered by complete proofs or references to appropriate sources
Some new directions in infinite-combinatorial topology
We give a light introduction to selection principles in topology, a young
subfield of infinite-combinatorial topology. Emphasis is put on the modern
approach to the problems it deals with. Recent results are described, and open
problems are stated. Some results which do not appear elsewhere are also
included, with proofs.Comment: Small update
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