A chain theorem for internally 4-connected binary matroids

This is the post-print version of the Article - Copyright @ 2011 ElsevierLet M be a matroid. When M is 3-connected, Tutte’s Wheels-and-Whirls Theorem proves that M has a 3-connected proper minor N with |E(M) − E(N)| = 1 unless M is a wheel or a whirl. This paper establishes a corresponding result for internally 4-connected binary matroids. In particular, we prove that if M is such a matroid, then M has an internally 4-connected proper minor N with |E(M) − E(N)| at most 3 unless M or its dual is the cycle matroid of a planar or Möbius quartic ladder, or a 16-element variant of such a planar ladder.This study was partially supported by the National Security Agency

Towards a splitter theorem for internally 4-connected binary matroids

This is the post-print version of the Article - Copyright @ 2012 ElsevierWe prove that if M is a 4-connected binary matroid and N is an internally 4-connected proper minor of M with at least 7 elements, then, unless M is a certain 16-element matroid, there is an element e of E(M) such that either M\e or M/e is internally 4-connected having an N-minor. This strengthens a result of Zhou and is a first step towards obtaining a splitter theorem for internally 4-connected binary matroids.This study is partially funded by Marsden Fund of New Zealand and the National Security Agency

Democracy and early childhood: diverse representations of democratic education in post-dictatorship Portugal

Education systems are often expected to play a key role in developing, maintaining and promoting democratic values and behaviours. This is particularly apparent in Portugal where, after nearly half a century of dictatorship ending in 1974, democracy emerged as a central national aspiration, especially within Early Childhood Education (ECE). However, the ambiguity of the term “democracy” has allowed policymakers, academics and educators alike to promote diverse understandings of its meaning. This article delves into the ambiguities of democracy, revealing its flexible and context-dependent nature through the diverse representations of democracy encountered in three early years settings in Portugal: a public (state) kindergarten, a not-for-profit kindergarten and a private kindergarten. Using interviews, documentary analysis and observations we illustrate the diverse ways in which democracy is represented in each setting. We classify these representations of democracy as structural, individual and collective. We argue that these reflect the different ideologies of the three types of setting that draw upon particular theoretical understandings and sociopolitical developments in Portugal’s educational history

Experimentally estimated dead space for GaAs and InP based planar Gunn diodes

The authors would like to thank the staff of the James Watt Nanofabrication Centre at the University of Glasgow for help in fabricating the devices which is reported in this paper. ‘Part of this work was supported by ESPRC through EP/H011862/ 1, and EP/H012966/1.Peer reviewedPublisher PD

Treatment resistant depression in primary care: Co-constructing difficult encounters

Many patients with depression do not recover despite medication or therapy. Individuals with treatment resistant depression often have co-morbid anxiety, personality difficulties and drug or alcohol misuse and have been characterised as difficult, heartsink or problem personalities by general practitioners. Yet critical studies of interaction in medical settings suggest that the context may have a role in constructing the patient. A total of 12 audio-recorded routine consultations were analysed following guidelines for qualitative analysis of medical discourse. The interpretation focused on ways in which the context and structure of primary care consultations in a UK setting construct difficult encounters, which may lead to patients with treatment resistant depression being seen as difficult to manage in various ways. Three overarching observations were that presentation of multiple problems in multiple domains clash with the consultation format; that patients? atypically high level of activity in a time-limited setting prevents patient-centred work; that the question and answer format restricts multifaceted discussions of social and emotional problems, preventing shared understandings emerging. However, although interactions appear uneasy, they are repaired and may be moderately palliative. Suggestions are made for re-orienting general practitioner work with treatment resistant depression towards long-term goal setting outside of the traditional consultation structure in order to develop shared understandings

A class of non-binary matroids with many binary minors

It is a well-known result of Tutte, A homotopy theorem for matroids, I, II, Trans. Amer. Math. Soc. 88 (1958) 144-174. that U2,4 is the only non-binary matroid M such that, for every element e, both M\e and M/e are binary. Oxley generalized this result by characterizing the non-binary matroids M such that, for every element e of M, the deletion M\e or the contraction M/e is binary. We characterize those non-binary matroids M such that, for all elements e and f, at least two of M\e, f; M\e/f; M/e\f; and M/e, f are binary. © 1999 Elsevier Science B.V. All rights reserved

Zero-free regions for multivariate Tutte polynomials (alias Potts-model partition functions) of graphs and matroids

The chromatic polynomial P_G(q) of a loopless graph G is known to be nonzero (with explicitly known sign) on the intervals (-\infty,0), (0,1) and (1,32/27]. Analogous theorems hold for the flow polynomial of bridgeless graphs and for the characteristic polynomial of loopless matroids. Here we exhibit all these results as special cases of more general theorems on real zero-free regions of the multivariate Tutte polynomial Z_G(q,v). The proofs are quite simple, and employ deletion-contraction together with parallel and series reduction. In particular, they shed light on the origin of the curious number 32/27.Comment: LaTeX2e, 49 pages, includes 5 Postscript figure

Feynman graph polynomials

The integrand of any multi-loop integral is characterised after Feynman parametrisation by two polynomials. In this review we summarise the properties of these polynomials. Topics covered in this article include among others: Spanning trees and spanning forests, the all-minors matrix-tree theorem, recursion relations due to contraction and deletion of edges, Dodgson's identity and matroids.Comment: 35 pages, references adde

A survey of common rabbit handling methods and reasons for their use

Rabbits are both popular pets commonly seen within veterinary practices, and frequently used in laboratories in the United Kingdom and elsewhere. However, to date there is minimal research that has investigated why and how people handle rabbits and why they choose to use such methods. In addition, few studies have considered the sources individuals use to learn about rabbit handling and their reasons to not use specific handling methods