Colourings of cubic graphs inducing isomorphic monochromatic subgraphs

A $k$-bisection of a bridgeless cubic graph $G$ is a $2$-colouring of its vertex set such that the colour classes have the same cardinality and all connected components in the two subgraphs induced by the colour classes (monochromatic components in what follows) have order at most $k$. Ban and Linial conjectured that every bridgeless cubic graph admits a $2$-bisection except for the Petersen graph. A similar problem for the edge set of cubic graphs has been studied: Wormald conjectured that every cubic graph $G$ with $|E(G)| \equiv 0 \pmod 2$ has a $2$-edge colouring such that the two monochromatic subgraphs are isomorphic linear forests (i.e. a forest whose components are paths). Finally, Ando conjectured that every cubic graph admits a bisection such that the two induced monochromatic subgraphs are isomorphic. In this paper, we give a detailed insight into the conjectures of Ban-Linial and Wormald and provide evidence of a strong relation of both of them with Ando's conjecture. Furthermore, we also give computational and theoretical evidence in their support. As a result, we pose some open problems stronger than the above mentioned conjectures. Moreover, we prove Ban-Linial's conjecture for cubic cycle permutation graphs. As a by-product of studying $2$-edge colourings of cubic graphs having linear forests as monochromatic components, we also give a negative answer to a problem posed by Jackson and Wormald about certain decompositions of cubic graphs into linear forests.Comment: 33 pages; submitted for publicatio

Effect of packing on nasal mucosa of sheep

The effects of packing with ribbon gauze and neuropatties on the nasal mucosa was assessed using sheep as an animal model. Fourteen sheep either underwent ribbon gauze or neuropattie nasal packing. Trauma to nasal mucosa caused by ribbon gauze and neuropatties was compared to mucosa on the lateral aspect of the middle turbinate which was not in contact with any packing. This tissue was used as a control. Ribbon gauze packing resulted in significant loss of 68 per cent of the ciliated surface of the mucosa when compared with the control group with a 15 per cent loss of ciliated surface (p < 0.005). Neuropattie packing also resulted in significant loss of 50 per cent of the ciliated surface of the mucosa when compared with the control group (p < 0.005). There was no significant difference in loss of ciliated mucosa in the specimens packed with ribbon gauze or neuropatties (p = 0.25). Nasal packing results in a significant mucosal injury with loss of cilia. This may influence the mucociliary clearance of the nose in the post-operative healing phase. Pre-operative nasal packing should be used circumspectly and if possible avoided.Chi-kee Leslie Shaw, Robert B. Dymock, Allison Cowin and Peter-John Wormal

Endoscopic orbital decompression for Graves' ophthalmopathy

Graves’ disease may occasionally result in significant proptosis that is either cosmetically unacceptable or causes visual loss. This has traditionally been managed surgically by external decompression of the orbital bony skeleton. Trans-nasal endoscopic orbital decompression is emerging as a new minimally-invasive technique, that avoids the need for cutaneous or gingival incisions. Decompression of the medial orbital wall can be performed up to the anterior wall of the sphenoid sinus. This can be combined with resection of the medial and posterior portion of the orbital floor (preserving the infra-orbital nerve). This technique produces decompression which is comparable to external techniques. We present a series of 10 endoscopic orbital decompressions with an average improvement of 4.4 mm in orbital proptosis. There was an improvement in visual acuity in all patients with visual impairment. Endoscopic orbital decompression is recommended as an alternative to traditional decompression techniques.Desmond T. H. Wee, A. Simon Carney, Mark Thorpe and Peter J. Wormal

Sight Impairment registration due to stroke—A small yet significant rise?

Objectives In the United Kingdom, when an individual's sight falls to and remains at a certain threshold, they may be offered registration as sight impaired. Recent analysis of causes of registrable sight impairment in England/Wales indicated that visual impairment due to stroke had increased as a proportionate cause of sight loss. We aim to assess whether there is evidence of an increase in incidence of certification for sight impairment due to stroke in England/Wales between 2008 and 2014. Materials and Methods The number of certifications with a main cause of sight impairment being stroke was obtained from the Certifications Office London. Directly standardized rates per 100,000 were computed with 95% confidence intervals and examined. Poisson regression was used to assess evidence of trend over time. Results In the year ending 31st March 2008, 992 people were newly certified with stroke with an estimated DSR of 2.1 (2.0 to 2.2) per 100,000 persons at risk. In the year ending March 31st 2014, there were 1310 certifications with a DSR of 2.5 (2.4 to 2.7). Figures were higher for men than women. Poisson regression indicated an estimated incidence rate ratio of 1.03 per year with 95% confidence intervals of 1.028 to 1.051, P < .001. Conclusions These data suggest a small but statistically significant increase in the incidence of certifiable visual impairment due to stroke between 2008 and 2014. Figures are, however, considerably lower than estimated, perhaps suggesting that more should be done to address the visual needs of those who have suffered stroke

Geographical variation in certification rates of blindness and sight impairment in England, 2008-2009

To examine and interpret the variation in the incidence of blindness and sight impairment in England by PCT, as reported by the Certificate of Vision Impairment (CVI). Design: Analysis of national certification data. Setting: All Primary Care Trusts, England. Participants: 23 773 CVI certifications issued from 2008 to 2009. Main Outcome measures: Crude and Age standardised rates of CVI data for blindness and sight loss by PCT. Methods: The crude and age standardised CVI rates per 100 000 were calculated with Spearman's rank correlation used to assess whether there was any evidence of association between CVI rates with Index of Multiple Deprivation (IMD) and the Programme Spend for Vision. Results: There was high-level variation, almost 11-fold (coefficient of variation 38%) in standardised CVI blindness and sight impairment annual certification rates across PCTs. The mean rate was 43.7 and the SD 16.7. We found little evidence of an association between the rate of blindness and sight impairment with either the IMD or Programme Spend on Vision. Conclusions: The wide geographical variation we found raises questions about the quality of the data and whether there is genuine unmet need for prevention of sight loss. It is a concern for public health practitioners who will be interpreting these data locally and nationally as the CVI data will form the basis of the public health indicator ‘preventable sight loss’. Poor-quality data and inadequate interpretation will only create confusion if not addressed adequately from the outset. There is an urgent need to address the shortcomings of the current data collection system and to educate all public health practitioners

High population density recruitment and survival of a harvested coral reef fish

A negative relationship between population growth and population density (direct density dependence) is necessary for population regulation and is assumed in most models of harvested populations. Experimental tests for density dependence are lacking for large-bodied, harvested fish because of the difficulty of manipulating population density over large areas. We studied a harvested coral reef fish, Lutjanus apodus (schoolmaster snapper), using eight large, isolated natural reefs (0.4–1.6 ha) in the Bahamas as replicates. An initial observational test for density dependence was followed by a manipulation of population density. The manipulation weakened an association between density and shelter-providing habitat features and revealed a positive effect of population density on recruitment and survival (inverse density dependence), but no effect of density on somatic growth. The snappers on an individual reef were organized into a few shoals, and we hypothesize that large shoals on high-density reefs were less vulnerable to large piscivores (groupers and barracudas) than the small shoals on low-density reefs. Reductions in predation risk for individuals in large social groups are well documented, but because snapper shoals occupied reefs the size of small marine reserves, these ecological interactions may influence the outcome of management actions

An algorithm for counting circuits: application to real-world and random graphs

We introduce an algorithm which estimates the number of circuits in a graph as a function of their length. This approach provides analytical results for the typical entropy of circuits in sparse random graphs. When applied to real-world networks, it allows to estimate exponentially large numbers of circuits in polynomial time. We illustrate the method by studying a graph of the Internet structure.Comment: 7 pages, 3 figures, minor corrections, accepted versio

Magnetic signatures of domain walls in s+is and s+id superconductors: Observability and what that can tell us about the superconducting order parameter

One of the defining features of spontaneously broken time-reversal symmetry (BTRS) is the existence of domain walls, the detection of which would be strong evidence for such systems. There is keen interest in BTRS currently, in part, due to recent muon spin rotation experiments, which have pointed towards Ba1−xKxFe2As2 exhibiting a remarkable case of s-wave superconductivity with spontaneously broken time-reversal symmetry. A key question, however, is how to differentiate between the different theoretical models which describe such a state. Two particularly popular choices of model are s+is and s+id superconducting states. In this paper, we obtain solutions for domain walls in s+is and s+id systems, including the effects of lattice anisotropies. We show that, in general, both models exhibit spontaneous magnetic fields that extend along the entire length of the domain wall. We demonstrate the qualitative difference between the magnetic signatures of s+is and s+id domain walls and propose a procedure to extract the superconducting pairing symmetry from the magnetic-field response of domain walls

Sudden emergence of q-regular subgraphs in random graphs

We investigate the computationally hard problem whether a random graph of finite average vertex degree has an extensively large $q$-regular subgraph, i.e., a subgraph with all vertices having degree equal to $q$. We reformulate this problem as a constraint-satisfaction problem, and solve it using the cavity method of statistical physics at zero temperature. For $q=3$, we find that the first large $q$-regular subgraphs appear discontinuously at an average vertex degree c_\reg{3} \simeq 3.3546 and contain immediately about 24% of all vertices in the graph. This transition is extremely close to (but different from) the well-known 3-core percolation point c_\cor{3} \simeq 3.3509. For $q>3$, the $q$-regular subgraph percolation threshold is found to coincide with that of the $q$-core.Comment: 7 pages, 5 figure