134,883 research outputs found
Approximation in quantale-enriched categories
Our work is a fundamental study of the notion of approximation in
V-categories and in (U,V)-categories, for a quantale V and the ultrafilter
monad U. We introduce auxiliary, approximating and Scott-continuous
distributors, the way-below distributor, and continuity of V- and
(U,V)-categories. We fully characterize continuous V-categories (resp.
(U,V)-categories) among all cocomplete V-categories (resp. (U,V)-categories) in
the same ways as continuous domains are characterized among all dcpos. By
varying the choice of the quantale V and the notion of ideals, and by further
allowing the ultrafilter monad to act on the quantale, we obtain a flexible
theory of continuity that applies to partial orders and to metric and
topological spaces. We demonstrate on examples that our theory unifies some
major approaches to quantitative domain theory.Comment: 17 page
Solve[order/topology == quasi-metric/x, x]
AbstractIn the study of the semantics of programming languages, the qualitative framework using partially ordered sets and the quantitative framework using pseudo-metric spaces have existed separately for years. Smyth however noticed that both concepts can be unified by means of quasi-metric spaces.Recent literature concerning these “quantitative domains”, lacks the canonicity which is so typical for the relationship between topological techniques and theoretical computer science in the classical settings mentioned above. On the one hand, this yields the use of structures which could be considered “ad hoc” from a categorical point of view, such as continuity spaces by Flagg and Kopperman. On the other hand, this yields “incomplete structures”, which essentially belong to one of both classical settings, such as the generalized Scott topology by Bonsangue e.a.We shall discuss a natural generalization of the symbiosis between ordered sets and topology to an analogous relationship between quasi-metric spaces and approach spaces. Approach spaces seem to be an important tool in the study of certain aspects concerning quantitative domains
Quantitative Homogenization of Elliptic PDE with Random Oscillatory Boundary Data
We study the averaging behavior of nonlinear uniformly elliptic partial
differential equations with random Dirichlet or Neumann boundary data
oscillating on a small scale. Under conditions on the operator, the data and
the random media leading to concentration of measure, we prove an almost sure
and local uniform homogenization result with a rate of convergence in
probability
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