A splitter theorem for elastic elements in 33-connected matroids

An element $e$ of a $3$-connected matroid $M$ is elastic if ${\rm si}(M/e)$, the simplification of $M/e$, and ${\rm co}(M\backslash e)$, the cosimplification of $M\backslash e$, are both $3$-connected. It was recently shown that if $|E(M)|\geq 4$, then $M$ has at least four elastic elements provided $M$ has no $4$-element fans and no member of a specific family of $3$-separators. In this paper, we extend this wheels-and-whirls type result to a splitter theorem, where the removal of elements is with respect to elasticity and keeping a specified $3$-connected minor. We also prove that if $M$ has exactly four elastic elements, then it has path-width three. Lastly, we resolve a question of Whittle and Williams, and show that past analogous results, where the removal of elements is relative to a fixed basis, are consequences of this work.Comment: 22 pages, 2 figure

Spherical Formulation for Diagramatic Evaluations on a Manifold with Boundary

The mathematical formalism necessary for the diagramatic evaluation of quantum corrections to a conformally invariant field theory for a self-interacting scalar field on a curved manifold with boundary is considered. The evaluation of quantum corrections to the effective action past one-loop necessitates diagramatic techniques. Diagramatic evaluations and higher loop-order renormalisation can be best accomplished on a Riemannian manifold of constant curvature accommodating a boundary of constant extrinsic curvature. In such a context the stated evaluations can be accomplished through a consistent interpretation of the Feynman rules within the spherical formulation of the theory for which the method of images allows. To this effect, the mathematical consequences of such an interpretation are analyzed and the spherical formulation of the Feynman rules on the bounded manifold is, as a result, developed.Comment: 12 pages, references added. To appear in Classical and Quantum Gravit

Conformal Anomaly for Free Scalar Propagation on Curved Bounded Manifolds

The trace anomaly for free propagation in the context of a conformally invariant scalar field theory defined on a curved manifold of positive constant curvature with boundary is evaluated through use of an asymptotic heat kernel expansion. In addition to their direct physical significance the results are also of relevance to the holographic principle and to Quantum Cosmology.Comment: 8 pages. To appear in General Relativity and Gravitatio

Circuit-Difference Matroids

One characterization of binary matroids is that the symmetric difference of every pair of intersecting circuits is a disjoint union of circuits. This paper considers circuit-difference matroids, that is, those matroids in which the symmetric difference of every pair of intersecting circuits is a single circuit. Our main result shows that a connected regular matroid is circuit-difference if and only if it contains no pair of skew circuits. Using a result of Pfeil, this enables us to explicitly determine all regular circuit-difference matroids. The class of circuit-difference matroids is not closed under minors, but it is closed under series minors. We characterize the infinitely many excluded series minors for the class.Comment: 11 page

Puget Sound shoreline inventory and assessment using boat-based lidar

Boat-based lidar of Puget Sound shorelines collected by the Washington State Department of Ecology are developed to provide a comprehensive inventory, classification, and analyses of site conditions and variability. For example, quantitative metrics of shoreline characteristics are derived from DEMs such as bluff crest height, bluff slope, bluff toe elevation, beach slope, and shoreline armoring elevations. These metrics can then be compiled and compared within and among drift cells to determine regional variability such as differences between updrift and downdrift beaches and the effect of fetch, orientation, and other exposure variables. Certain features can also be correlated to characterize how the shoreline landscape may be affecting nearshore ecosystem services. For example, variability and gradients in beach slope and width may be correlated to proximity to feeder bluff activity and the position, length, and elevation of armoring relative to the shoreline and backshore. Upland development and shoreline modification may be correlated to the amount of overhanging vegetation, large woody debris, or beach wrack, and these findings can be compared to conditions at undeveloped shorelines. Details in the lidar point clouds, such as intensity values, can help identify groundwater seepage and potential bluff failure and erosion mechanisms. The complementary photos to the lidar point clouds provide additional documentation of bluff geology, stratigraphy, groundwater flow, and other characteristics to help assess relative bluff stability

Aspects of matroid connectivity and uniformity.

In approaching a combinatorial problem, it is often desirable to be armed with a notion asserting that some objects are more highly structured than others. In particular, focusing on highly structured objects may avoid certain degeneracies and allow for the core of the problem to be addressed. In matroid theory, the principle notion fulfilling this role of “structure” is that of connectivity. This thesis proves a number of results furthering the knowledge of matroid connectivity and also introduces a new structural notion, that of generalised uniformity. The first part of this thesis considers 3-connected matroids and the presence of elements which may be deleted or contracted without the introduction of any non-minimal 2-separations. Principally, a Wheels-and-Whirls Theorem and then a Splitter Theorem is established, guaranteeing the existence of such elements, provided certain well-behaved structures are not present. The second part of this thesis generalises the notion of a uniform matroid by way of a 2-parameter property capturing “how uniform” a given matroid is. Initially, attention is focused on matroids representable over some field. In particular, a finiteness result is established and a specific class of binary matroids is completely determined. The concept of generalised uniformity is then considered more broadly by an analysis of its relevance to a number of established matroid notions and settings. Within that analysis, a number of equivalent characterisations of generalised uniformity are obtained. Lastly, the third part of the thesis considers a highly structured class of matroids whose members are defined by the nature of their circuits. A characterisation is achieved for the regular members of this class and, in general, the infinitely many excluded series minors are determined

Profitability of owning a professional sports team in the United States

Eight years ago, during an interview with CNN Business one of the more infamous NBA team owners, Mark Cuban, claimed that owning a sports team was no longer a trophy asset. He said many of the new ownership groups are a consortium with multiple investors and they're seeking to make a strong financial return. It was no longer about wealthy businessmen and women wanting to own the local team but about firms and wealthy investors seeking to turn a profit. I want to explore the truth of this claim and see if these teams actually are strong sound investments that private equity firms or individual family offices should be pursuing

Upregulation of Heme Oxygenase-1 Combined with Increased Adiponectin Lowers Blood Pressure in Diabetic Spontaneously Hypertensive Rats through a Reduction in Endothelial Cell Dysfunction, Apoptosis and Oxidative Stress

This study was designed to investigate the effect of increased levels of HO-1 on hypertension exacerbated by diabetes. Diabetic spontaneously hypertensive rat (SHR) and WKY (control) animals were treated with streptozotocin (STZ) to induce diabetes and stannous chloride (SnCl2) to upregulate HO-1. Treatment with SnCl2 not only attenuated the increase of blood pressure (p<0.01), but also increased HO-1 protein content, HO activity and plasma adiponectin levels, decreased the levels of superoxide and 3-nitrotyrosine (NT), respectively. Reduction in oxidative stress resulted in the increased expression of Bcl-2 and AKT with a concomitant reduction in circulating endothelial cells (CEC) in the peripheral blood (p<0.005) and an improvement of femoral reactivity (response to acetylcholine). Thus induction of HO-1 accompanied with increased plasma adiponectin levels in diabetic hypertensive rats alters the phenotype through a reduction in oxidative stress, thereby permitting endothelial cells to maintain an anti-apoptotic environment and the restoration of endothelial responses thus preventing hypertension

Perturbative Evaluation of Interacting Scalar Fields on a Curved Manifold with Boundary

The effects of quantum corrections to a conformally invariant scalar field theory on a curved manifold of positive constant curvature with boundary are considered in the context of a renormalisation procedure. The renormalisation of the theory to second order in the scalar self-coupling pursued herein involves explicit calculations of up to third loop-order and reveals that, in addition to the renormalisation of the scalar self-coupling and scalar field, the removal of all divergences necessitates the introduction of conformally non-invariant counterterms proportional to $R\Phi^2$ and $K\Phi^2$ in the bare scalar action as well as counterterms proportional to $RK^2$, $R^2$ and $RK$ in the gravitational action. The substantial backreaction effects and their relevance to the renormalisation procedure are analysed.Comment: 25 pages, 1 figure. Minor elucidations in the Appendix regarding the cut-off $N_0$ and in p.4 regarding the gravitational action. Certain reference-related ommission corrected. To appear in Classical and Quantum Gravit