52 research outputs found
Topologies on Quantum Effects
Quantum effects play an important role in quantum measurement theory. The set
of all quantum effects can be organized into an algebraical structure called
effect algebra. In this paper, we study various topologies on the Hilbert space
effect algebra and the projection lattice effect algebra
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
Mappings between distance sets or spaces
[no abstract
Pattern-Equivariant Homology of Finite Local Complexity Patterns
This thesis establishes a generalised setting with which to unify the study
of finite local complexity (FLC) patterns. The abstract notion of a "pattern"
is introduced, which may be seen as an analogue of the space group of
isometries preserving a tiling but where, instead, one considers partial
isometries preserving portions of it. These inverse semigroups of partial
transformations are the suitable analogue of the space group for patterns with
FLC but few global symmetries. In a similar vein we introduce the notion of a
\emph{collage}, a system of equivalence relations on the ambient space of a
pattern, which we show is capable of generalising many constructions applicable
to the study of FLC tilings and Delone sets, such as the expression of the
tiling space as an inverse limit of approximants.
An invariant is constructed for our abstract patterns, the so called
pattern-equivariant (PE) homology. These homology groups are defined using
infinite singular chains on the ambient space of the pattern, although we show
that one may define cellular versions which are isomorphic under suitable
conditions. For FLC tilings these cellular PE chains are analogous to the PE
cellular cochains \cite{Sadun1}. The PE homology and cohomology groups are
shown to be related through Poincar\'{e} duality.
An efficient and highly geometric method for the computation of the PE
homology groups for hierarchical tilings is presented. The rotationally
invariant PE homology groups are shown not to be a topological invariant for
the associated tiling space and seem to retain extra information about global
symmetries of tilings in the tiling space. We show how the PE homology groups
may be incorporated into a spectral sequence converging to the \v{C}ech
cohomology of the rigid hull of a tiling. These methods allow for a simple
computation of the \v{C}ech cohomology of the rigid hull of the Penrose
tilings.Comment: 159 pages, 8 figures, PhD thesi
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