171 research outputs found
Recommended from our members
Tilings and other combinatorial results
In this dissertation we treat three tiling problems and three problems in combinatorial geometry, extremal graph theory and sparse Ramsey theory.
We first consider tilings of . In this setting a tile is just a finite subset of . We say that tiles if the latter set admits a partition into isometric copies of . Chalcraft observed that there exist that do not tile but tile for some . He conjectured that such exists for any given tile. We prove this conjecture in Chapter 2.
In Chapter 3 we prove a conjecture of Lonc, stating that for any poset of size a power of , if has a greatest and a least element, then there is a positive integer such that can be partitioned into copies of .
The third tiling problem is about vertex-partitions of the hypercube graph . Offner asked: if is a subgraph of such is a power of , must , for some , admit a partition into isomorphic copies of ? In Chapter 4 we answer this question in the affirmative.
We follow up with a question in combinatorial geometry. A line in a planar set is a maximal collinear subset of . P\'or and Wood considered colourings of finite without large lines with a bounded number of colours. In particular, they examined whether monochromatic lines always appear in such colourings provided that is large. They conjectured that for all there exists an such that if and does not contain a line of cardinality larger than , then every colouring of with colours produces a monochromatic line. In Chapter 5 we construct arbitrarily large counterexamples for the case .
We follow up with a problem in extremal graph theory. For any graph, we say that a given edge is triangular if it forms a triangle with two other edges. How few triangular edges can there be in a graph with vertices and edges? For sufficiently large we prove a conjecture of F\"uredi and Maleki that gives an exact formula for this minimum. This proof is given in Chapter 6.
Finally, Chapter 7 is concerned with degrees of vertices in directed hypergraphs. One way to prescribe an orientation to an -uniform graph is to assign for each of its edges one of the possible orderings of its elements. Then, for any -set of vertices and any -set of indices , we define the -degree of to be the number of edges containing vertices in precisely the positions labelled by . Caro and Hansberg were interested in determining whether a given -uniform hypergraph admits an orientation where every set of vertices has some -degree equal to . They conjectured that a certain Hall-type condition is sufficient. We show that this is true for large, but false in general.EPSR
Automatic continuity, unique Polish topologies, and Zariski topologies on monoids and clones
In this paper we explore the extent to which the algebraic structure of a
monoid determines the topologies on that are compatible with its
multiplication. Specifically we study the notions of automatic continuity;
minimal Hausdorff or Polish semigroup topologies; and we formulate a notion of
the Zariski topology for monoids.
If is a topological monoid such that every homomorphism from to a
second countable topological monoid is continuous, then we say that has
\emph{automatic continuity}. We show that many well-known monoids have
automatic continuity with respect to a natural semigroup topology, namely: the
full transformation monoid ; the full binary relation
monoid ; the partial transformation monoid ;
the symmetric inverse monoid ; the monoid Inj
consisting of the injective functions on ; and the monoid
of continuous functions on the Cantor set.
We show that the pointwise topology on , and its
analogue on , are the unique Polish semigroup topologies on
these monoids. The compact-open topology is the unique Polish semigroup
topology on and . There are at least 3
Polish semigroup topologies on , but a unique Polish inverse
semigroup topology. There are no Polish semigroup topologies
nor on the partitions monoids. At the other extreme, Inj and the
monoid Surj of all surjective functions on each have
infinitely many distinct Polish semigroup topologies. We prove that the Zariski
topologies on , , and Inj
coincide with the pointwise topology; and we characterise the Zariski topology
on . In Section 7: clones.Comment: 51 pages (Section 7 about clones was added in version 4
Developments in entanglement theory and applications to relevant physical systems
This Thesis is devoted to the analysis of entanglement in relevant physical
systems. Entanglement is the conducting theme of this research, though I do not
dedicate to a single topic, but consider a wide scope of physical situations. I
have followed mainly three lines of research for this Thesis, with a series of
different works each, which are, Entanglement and Relativistic Quantum Theory,
Continuous-variable entanglement, and Multipartite entanglement.Comment: Ph.D. Thesis, April 2007, Universidad Autonoma de Madri
- …