61 research outputs found
Analysis of NFIRS Data for Sensitivity to Foreclosure and Other Select Features
Arson is a grave threat to life and property. In the United States, fire information is collected and disseminated through the National Fire Incident Reporting System (NFIRS). Fire records obtained from NFIRS contain a full range of available information. This information includes the initial incident details in addition to investigative information regarding the cause of ignition and factors contributing to ignition. Combating the arson problem is accomplished in large part by understanding the motives and opportunities of those who commit arson. A common motive for arson is financial gain through insurance fraud. By connecting NFIRS data with mortgage and foreclosure information from RealtyTrac, insight into potential incidents of insurance fraud may be obtained.
Understanding the features that intentional fires have in common is necessary to assess the vulnerability of structures to intentional burning. One historically utilized method of predicting arson prone structures is linear discriminant analysis (LDA). LDA is a method of separating objects or events into two or more categories using a combination of features. Through feature analysis and selection, a discriminant function is proposed that incorporates foreclosure as an independent variable to classify fires as intentional or unintentional.
Additionally, graph theoretical algorithms for clustering are applied in support of the discovery of novel relationships between fires. In this thesis we leverage the paraclique algorithm, which has previously been applied to biological data, to help reveal latent associations within the NFIRS datasets
Topics in Graph Theory: Extremal Intersecting Systems, Perfect Graphs, and Bireflexive Graphs
In this thesis we investigate three different aspects of graph theory.
Firstly, we consider interesecting systems of independent sets in graphs, and the extension of the classical theorem of Erdos, Ko and Rado to graphs.
Our main results are a proof of an Erdos-Ko-Rado type theorem for a class of trees, and a class of trees which form counterexamples to a conjecture of Hurlberg and Kamat, in such a way that extends the previous counterexamples given by Baber.
Secondly, we investigate perfect graphs - specifically, edge modification aspects of perfect graphs and their subclasses. We give some alternative characterisations of perfect graphs in terms of edge modification, as well as considering the possible connection of the critically perfect graphs - previously studied by Wagler - to the Strong Perfect Graph Theorem. We prove that the situation where critically perfect graphs arise has no analogue in seven different subclasses of perfect graphs (e.g. chordal, comparability graphs), and consider the connectivity of a bipartite reconfiguration-type graph associated to each of these subclasses.
Thirdly, we consider a graph theoretic structure called a bireflexive graph where every vertex is both adjacent and nonadjacent to itself, and use this to characterise modular decompositions as the surjective homomorphisms of these structures. We examine some analogues of some graph theoretic notions and define a “dual” version of the reconstruction conjecture
A Mathematical Framework for Causally Structured Dilations and its Relation to Quantum Self-Testing
The motivation for this thesis was to recast quantum self-testing [MY98,MY04]
in operational terms. The result is a category-theoretic framework for
discussing the following general question: How do different implementations of
the same input-output process compare to each other? In the proposed framework,
an input-output process is modelled by a causally structured channel in some
fixed theory, and its implementations are modelled by causally structured
dilations formalising hidden side-computations. These dilations compare through
a pre-order formalising relative strength of side-computations. Chapter 1
reviews a mathematical model for physical theories as semicartesian symmetric
monoidal categories. Many concrete examples are discussed, in particular
quantum and classical information theory. The key feature is that the model
facilitates the notion of dilations. Chapter 2 is devoted to the study of
dilations. It introduces a handful of simple yet potent axioms about dilations,
one of which (resembling the Purification Postulate [CDP10]) entails a duality
theorem encompassing a large number of classic no-go results for quantum
theory. Chapter 3 considers metric structure on physical theories, introducing
in particular a new metric for quantum channels, the purified diamond distance,
which generalises the purified distance [TCR10,Tom12] and relates to the Bures
distance [KSW08a]. Chapter 4 presents a category-theoretic formalism for
causality in terms of '(constructible) causal channels' and 'contractions'. It
simplifies aspects of the formalisms [CDP09,KU17] and relates to traces in
monoidal categories [JSV96]. The formalism allows for the definition of 'causal
dilations' and the establishment of a non-trivial theory of such dilations.
Chapter 5 realises quantum self-testing from the perspective of chapter 4, thus
pointing towards the first known operational foundation for self-testing.Comment: PhD thesis submitted to the University of Copenhagen (ISBN
978-87-7125-039-8). Advised by prof. Matthias Christandl, submitted 1st of
December 2020, defended 11th of February 2021. Keywords: dilations, applied
category theory, quantum foundations, causal structure, quantum self-testing.
242 pages, 1 figure. Comments are welcom
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