A comparison theorem for the isoperimetric profile under curve shortening flow

We prove a comparison theorem for the isoperimetric profiles of simple closed curves evolving by the normalized curve shortening flow: If the isoperimetric profile of the region enclosed by the initial curve is greater than that of some `model' convex region with exactly four vertices and with reflection symmetry in both axes, then the inequality remains true for the isoperimetric profiles of the evolved regions. We apply this using the Angenent solution as the model region to deduce sharp time-dependent upper bounds on curvature for arbitrary embedded closed curves evolving by the normalized curve shortening flow. A slightly different comparison also gives lower bounds on curvature, and the result is a simple and direct proof of Grayson's theorem without use of any blowup or compactness arguments, Harnack estimates, or classification of self-similar solutions.Comment: 32 pages, 6 figure

Py-Tetrad and RPy-Tetrad: A New Python Interface with R Support for Tetrad Causal Search

We give novel Python and R interfaces for the (Java) Tetrad project for causal modeling, search, and estimation. The Tetrad project is a mainstay in the literature, having been under consistent development for over 30 years. Some of its algorithms are now classics, like PC and FCI; others are recent developments. It is increasingly the case, however, that researchers need to access the underlying Java code from Python or R. Existing methods for doing this are inadequate. We provide new, up-to-date methods using the JPype Python-Java interface and the Reticulate Python-R interface, directly solving these issues. With the addition of some simple tools and the provision of working examples for both Python and R, using JPype and Reticulate to interface Python and R with Tetrad is straightforward and intuitive.Comment: Causal Analysis Workshop Series (CAWS) 2023, 12 pages, 4 Figures, 2 Table

Inducing Sets: A New Perspective for Ancestral Graph Markov Models

Directed acyclic graphs (DAGs) and their corresponding Markov models have become widely studied and applied in the fields of statistics and causality. The simple directed structure of these models facilitates systematic learning procedures and provides an interpretable representation for causal relationships. However, DAGs are ill-equipped to handle latent variables without explicitly invoking them. This manifests as a lack of stability under marginalization and conditioning and a disparity between statistically and causally valid models. Meanwhile, latent confounding and selection effects occur with some regularity in many domains. The family of maximal ancestral graphs (MAGs) extends the family of DAGs by implicitly taking latent variables into account. In fact, the family of MAGs constitutes the smallest superset of the family of DAGs that is stable under marginalization and conditioning. Accordingly, MAGs and their corresponding Markov models---ancestral graph Markov models---provide a natural choice for statistical and causal modeling in systems with latent confounding and selection effects. In this work we introduce inducing sets as a new perspective for reasoning about ancestral graph Markov models. In particular, we derive and study m-connecting sets which are a special case of inducing sets and provide an alternative representation for MAGs. We show that m-connecting sets admit a characterization of Markov equivalence for MAGs and a factorization criterion equivalent to the global Markov property for directed MAGs. Using the factorization criterion, we formulate a consistent probabilistic score with a closed-form for the Markov models of directed MAGs. Ultimately, we design a local causal discovery algorithm called the ancestral probability (AP) procedure which estimates the posterior probabilities of ancestral relationships. We evaluate the AP procedure on synthetically generated data and a real data set measuring airborne pollutants, cardiovascular health, and respiratory health

Synthesis and characterisation of lanthanide complexes for application as responsive probes

A range of new lanthanide complexes have been synthesised. The ligands and ligand precursors were characterised by NMR (1H, 13C), electrospray mass spectrometry, UV-Vis and IR spectroscopy. Several ligands or ligand precursors have been crystallised and analysed by X-ray crystal diffraction. The complexes were characterised by electrospray mass spectrometry, UV-Vis, IR spectroscopy and luminescence spectroscopy. In Chapter 2 a quinoxaline chromophore was incorporated in a macrocyclic ligand and it was shown that quinoxaline was capable of sensitising Ln(III) luminescence in the visible and NIR region. The Eu(uT) emission intensity and lifetime were shown to be responsive to pH. In Chapter 3 two N-(2-nitrophenyl)acetamide-derived chromophores were incorporated into macrocyclic ligands and shown to be capable of sensitising Ln(III) emission. A combined structural, spectroscopy and computation study was undertaken to investigate the spectral differences between the ligands. In Chapter 4 a synthetic strategy towards the synthesis of metal-ion responsive lanthanide complexes is detailed. The Eu(III) complexes were titrated against various metals and it was shown that the metal-based luminescence was sensitive to the concentration of Hg(II) and Cu(II). Hydroxyquinoline, aminoanthracene, amidopyrene, amidoquinoline and amidoanthraquinone chromophores were incorporated into macrocyclic ligands in Chapter 5 in order investigate the feasibility of adapting the synthetic strategy presented in Chapter 4 to synthesise responsive probes containing long-wavelength absorbing chromophores. Binding studies demonstrated the potential for these complexes to respond to the presence of Group 12 metals with changes in the overall emission intensity, relative intensity of hyperfine transitions and luminescence lifetimes. In Chapter 6 the synthetic strategy presented Chapter 4 was utilised to synthesis a ligand capable of forming bimetallic complexes with Ln(III) ions. Combined luminescence and relaxivity studies indicated that the binding of Hg(II) resulted in a change in Ln(III) coordination environment, whereas the binding of Cu(II) caused quenching of emission without increasing q.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

The m-connecting imset and factorization for ADMG models

Directed acyclic graph (DAG) models have become widely studied and applied in statistics and machine learning -- indeed, their simplicity facilitates efficient procedures for learning and inference. Unfortunately, these models are not closed under marginalization, making them poorly equipped to handle systems with latent confounding. Acyclic directed mixed graph (ADMG) models characterize margins of DAG models, making them far better suited to handle such systems. However, ADMG models have not seen wide-spread use due to their complexity and a shortage of statistical tools for their analysis. In this paper, we introduce the m-connecting imset which provides an alternative representation for the independence models induced by ADMGs. Furthermore, we define the m-connecting factorization criterion for ADMG models, characterized by a single equation, and prove its equivalence to the global Markov property. The m-connecting imset and factorization criterion provide two new statistical tools for learning and inference with ADMG models. We demonstrate the usefulness of these tools by formulating and evaluating a consistent scoring criterion with a closed form solution