Recognition Problems for Connectivity Functions

A connectivity function is a symmetric, submodular set function. Connectivity functions arise naturally from graphs, matroids and other structures. This thesis focuses mainly on recognition problems for connectivity functions, that is when a connectivity function comes from a particular type of structure. In particular we give a method for identifying when a connectivity function comes from a graph, which uses no more than a polynomial number of evaluations of the connectivity function. We also give a proof that no such method can exist for matroids

Monadic transductions and definable classes of matroids

A transduction provides us with a way of using the monadic second-order language of a structure to make statements about a derived structure. Any transduction induces a relation on the set of these structures. This article presents a self-contained presentation of the theory of transductions for the monadic second-order language of matroids. This includes a proof of the matroid version of the Backwards Translation Theorem, which lifts any formula applied to the images of the transduction into a formula which we can apply to the pre-images. Applications include proofs that the class of lattice-path matroids and the class of spike-minors can be defined by sentences in monadic second-order logic

Phase transition in the Ising model on a small-world network with distance-dependent interactions

We study the collective behavior of an Ising system on a small-world network with the interaction $J(r) \propto r^{-\alpha}$, where $r$ represents the Euclidean distance between two nodes. In the case of $\alpha = 0$ corresponding to the uniform interaction, the system is known to possess a phase transition of the mean-field nature, while the system with the short-range interaction $(\alpha\to\infty)$ does not exhibit long-range order at any finite temperature. Monte Carlo simulations are performed at various values of $\alpha$, and the critical value $\alpha_c$ beyond which the long-range order does not emerge is estimated to be zero. Thus concluded is the absence of a phase transition in the system with the algebraically decaying interaction $r^{-\alpha}$ for any nonzero positive value of $\alpha$

What is a 4-connected matroid?

The breadth of a tangle $\mathcal{T}$ in a matroid is the size of the largest spanning uniform submatroid of the tangle matroid of $\mathcal{T}$. The matroid $M$ is weakly 4-connected if it is 3-connected and whenever $(X,Y)$ is a partition of $E(M)$ with $|X|,|Y|>4$, then $\lambda(X)\geq 3$. We prove that if $\mathcal{T}$ is a tangle of order $k\geq 4$ and breadth $l$ in a matroid $M$, then $M$ has a weakly 4-connected minor $N$ with a tangle $\mathcal{T}_N$ of order $k$, breadth $l$ and has the property that $\mathcal{T}$ is the tangle in $M$ induced by $\mathcal{T}_N$. A set $Z$ of elements of a matroid $M$ is 4-connected if $\lambda(A)\geq\min\{|A\cap Z|,|Z-A|,3\}$ for all $A\subseteq E(M)$. As a corollary of our theorems on tangles we prove that if $M$ contains an $n$-element 4-connected set where $n\geq 7$, then $M$ has a weakly $4$-connected minor that contains an $n$-element $4$-connected set.Comment: 34 page

A longitudinal assessment of the responsiveness of the ICECAP-A in a randomised controlled trial of a knee pain intervention

PURPOSE: The ICECAP-A is a simple measure of capability well-being for use with the adult population. The descriptive system is made up of five key attributes: Stability, Attachment, Autonomy, Achievement and Enjoyment. Studies have begun to assess the psychometric properties of the measure, including the construct and content validity and feasibility for use. This is the first study to use longitudinal data to assess the responsiveness of the measure. METHODS: This responsiveness study was completed alongside a randomised controlled trial comparing three physiotherapy-led exercise interventions for older adults with knee pain attributable to osteoarthritis. Anchor-based methodologies were used to explore the relationship between change over time in ICECAP-A score (the target measure) and change over time in another measure (the anchor). Analyses were completed using the non-value-weighted and value-weighted ICECAP-A scores. The EQ-5D-3L was used as a comparator measure to contextualise change in the ICECAP-A. Effect sizes, standardised response means and t tests were used to quantify responsiveness. RESULTS: Small changes in the ICECAP-A scores were seen in response to underlying changes in patients' health-related quality of life, anxiety and depression. Non-weighted scores were slightly more responsive than value-weighted scores. ICECAP-A change was of comparable size to change in the EQ-5D-3L reference measure. CONCLUSION: This first analysis of the responsiveness using longitudinal data provides some positive evidence for the responsiveness of the ICECAP-A measure. There is a need for further research in those with low health and capability, and experiencing larger underlying changes in quality of life

An analysis of the complementarity of ICECAP-A and EQ-5D-3L in an adult population of patients with knee pain

BACKGROUND: The ICECAP measures potentially offer a broader assessment of quality of life and well-being, in comparison to measures routinely used in economic evaluation, such as the EQ-5D-3 L. This broader assessment may allow measurement of the full effects of an intervention or treatment. Previous research has indicated that the ICECAP-O (for older people) and EQ-5D-3 L measure provide complementary information. This paper aims to determine similar information for the ICECAP-A (for the entire adult population) in terms of whether the measure is a substitute or complement to the EQ-5D-3 L. METHODS: Data from the BEEP trial - a multi-centre, pragmatic, randomised controlled trial - were used. Spearman rank correlations and exploratory factor analytic methods were used to assess whether ICECAP-A and EQ-5D-3 L are measuring the same, or different, constructs. RESULTS: A correlation of 0.49 (p < 0.01) was found between the ICECAP-A tariff score and the EQ-5D-3 L index. Using the pooled items of the EQ-5D-3 L and the ICECAP-A a two factor solution was optimal, with the majority of EQ-5D-3 L items loading onto one factor and the majority of ICECAP-A items onto another. CONCLUSION: The results presented in this paper indicate that ICECAP-A and EQ-5D-3 L are measuring two different constructs and provide largely different, complementary information. Results showed a similarity to results presented by Davis et al. using the ICECAP-O. TRIAL REGISTRATION: ISRCTN 93634563

In Honor of Angela Harris: Finding Breathing Space, Embracing the Contradictions, and Education Work

In Honor of Angela Harris: Finding Breathing Space, Embracing the Contradictions, and “Education Work,” serves as a tribute to Angela Harris. This essay explores Harris’ concept of education work, the work that people of color in predominantly white settings must do to maintain their own integrity and to help their white colleagues to build inclusive communities. Part I explores this idea of education work and suggests that whites need to undertake part of the load of this work. Education work by whites provides an opportunity to create allies and to work across racial lines. Part II addresses the idea of incorporating mindfulness into legal education. Again, Harris has been a pioneer in this area through her scholarship on reframing legal education. The essay includes breathing exercises in the yoga tradition and concludes with a recipe for chocolate chip cake, recognizing Harris’ and my love of chocolate