182 research outputs found
Field reduction and linear sets in finite geometry
Based on the simple and well understood concept of subfields in a finite
field, the technique called `field reduction' has proved to be a very useful
and powerful tool in finite geometry. In this paper we elaborate on this
technique. Field reduction for projective and polar spaces is formalized and
the links with Desarguesian spreads and linear sets are explained in detail.
Recent results and some fundamental ques- tions about linear sets and scattered
spaces are studied. The relevance of field reduction is illustrated by
discussing applications to blocking sets and semifields
On line covers of finite projective and polar spaces
An - of lines of a finite projective space (of a
finite polar space ) is a set of lines of (of
) such that every point of (of ) contains
lines of , for some . Embed in .
Let denote the set of points of lying on the
extended lines of .
An -cover of is an -dual -cover if
there are two possibilities for the number of lines of contained in an
-space of . Basing on this notion, we characterize
-covers of such that is a
two-character set of . In particular, we show that if
is invariant under a Singer cyclic group of then it is an
-dual -cover.
Assuming that the lines of are lines of a symplectic polar space
(of an orthogonal polar space of parabolic
type), similarly to the projective case we introduce the notion of an
-dual -cover of symplectic type (of parabolic type). We prove that an
-cover of (of ) has this dual
property if and only if is a tight set of an Hermitian variety
or of (of or of ). We also provide some interesting examples of -dual
-covers of symplectic type of .Comment: 20 page
Automorphisms and opposition in spherical buildings of classical type
An automorphism of a spherical building is called domestic if it maps no
chamber to an opposite chamber. In this paper we classify domestic
automorphisms of spherical buildings of classical type
Maximal subgroups of finite classical groups and their geometry
We survey some recent results on maximal subgroups of finite classical groups
Pictures of Processes: Automated Graph Rewriting for Monoidal Categories and Applications to Quantum Computing
This work is about diagrammatic languages, how they can be represented, and
what they in turn can be used to represent. More specifically, it focuses on
representations and applications of string diagrams. String diagrams are used
to represent a collection of processes, depicted as "boxes" with multiple
(typed) inputs and outputs, depicted as "wires". If we allow plugging input and
output wires together, we can intuitively represent complex compositions of
processes, formalised as morphisms in a monoidal category.
[...] The first major contribution of this dissertation is the introduction
of a discretised version of a string diagram called a string graph. String
graphs form a partial adhesive category, so they can be manipulated using
double-pushout graph rewriting. Furthermore, we show how string graphs modulo a
rewrite system can be used to construct free symmetric traced and compact
closed categories on a monoidal signature.
The second contribution is in the application of graphical languages to
quantum information theory. We use a mixture of diagrammatic and algebraic
techniques to prove a new classification result for strongly complementary
observables. [...] We also introduce a graphical language for multipartite
entanglement and illustrate a simple graphical axiom that distinguishes the two
maximally-entangled tripartite qubit states: GHZ and W. [...]
The third contribution is a description of two software tools developed in
part by the author to implement much of the theoretical content described here.
The first tool is Quantomatic, a desktop application for building string graphs
and graphical theories, as well as performing automated graph rewriting
visually. The second is QuantoCoSy, which performs fully automated,
model-driven theory creation using a procedure called conjecture synthesis.Comment: PhD Thesis. Passed examination. Minor corrections made and one
theorem added at the end of Chapter 5. 182 pages, ~300 figures. See full text
for unabridged abstrac
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