Random graph asymptotics on high-dimensional tori. II. Volume, diameter and mixing time

For critical bond-percolation on high-dimensional torus, this paper proves sharp lower bounds on the size of the largest cluster, removing a logarithmic correction in the lower bound in Heydenreich and van der Hofstad (2007). This improvement finally settles a conjecture by Aizenman (1997) about the role of boundary conditions in critical high-dimensional percolation, and it is a key step in deriving further properties of critical percolation on the torus. Indeed, a criterion of Nachmias and Peres (2008) implies appropriate bounds on diameter and mixing time of the largest clusters. We further prove that the volume bounds apply also to any finite number of the largest clusters. The main conclusion of the paper is that the behavior of critical percolation on the high-dimensional torus is the same as for critical Erdos-Renyi random graphs. In this updated version we incorporate an erratum to be published in a forthcoming issue of Probab. Theory Relat. Fields. This results in a modification of Theorem 1.2 as well as Proposition 3.1.Comment: 16 pages. v4 incorporates an erratum to be published in a forthcoming issue of Probab. Theory Relat. Field

The Alexander-Orbach conjecture holds in high dimensions

We examine the incipient infinite cluster (IIC) of critical percolation in regimes where mean-field behavior has been established, namely when the dimension d is large enough or when d>6 and the lattice is sufficiently spread out. We find that random walk on the IIC exhibits anomalous diffusion with the spectral dimension d_s=4/3, that is, p_t(x,x)= t^{-2/3+o(1)}. This establishes a conjecture of Alexander and Orbach. En route we calculate the one-arm exponent with respect to the intrinsic distance.Comment: 25 pages, 2 figures. To appear in Inventiones Mathematica

Supplier qualification for high value goods and services in Nigeria: A comparison of qualified and non-qualified suppliers

This study set out to understand the capabilities of suppliers of high value products and services in Nigeria. Supplier capability data was collected from almost 500 potential suppliers across 28 product categories. The findings indicate that only a small minority of potential suppliers achieved the required levels of performance and that very few companies have suitable operational capabilities or corporate citizenship behaviour. Furthermore, significant differences were found between service suppliers and product suppliers with regard to performance. The findings suggest that the majority of suppliers of products and services in Nigeria need to improve their operational capabilities and corporate citizenship behaviours or else risk losing their ability to compete in a changing market place with new barriers to entry

Personal values, social capital, and higher education student career decidedness: a new 'protean'-informed model

This study investigates the role of personal values as motivational antecedents for understanding HE student career decidedness among university business school (UBS) students. We propose a new ‘protean’ informed HE student career decidedness model for theorizing how both personal values and social capital mediators (student social capital; personal, social and enterprise skills; access to resources) help in the student-centric and self-directed processes of career decision-making. A mixed methods study combines a (stage 1) survey of 308 UBS students from five (UK) university business schools, with results from (stage 2) four student focus groups, and (stage 3) two staff-student interactive seminars. From an employability perspective, arguably, the ultimate responsibility for becoming a ‘protean graduate’ rests with each UBS student, whilst the obligation of HE staff is to effectively facilitate and nurture all possible personal growth and skills development opportunities

Interviewer: 'Are women and girls ever responsible for the domestic violence they encounter?' Student: 'No, well, unless they did something really, really bad …'

Research shows the ‘gendered nature’ of domestic violence, with Women’s Aid (a UK-based charity) estimating that 1 in 4 women are affected (2014). This paper reports on a project - funded by Comic Relief, completed by Nottinghamshire Domestic Violence Forum (now known as Equation) and evaluated by Nottingham Trent University. The project adopts a Whole School Approach in seeking to prevent domestic violence. Students at three secondary schools attended between one and five blocks of work, and special events. There is evidence of positive developments - with young people showing understanding of domestic violence as well as the margins between healthy and unhealthy relationships. However, not all students could reply ‘never’ to the question of ‘are women and girls to blame for the domestic violence they experience?’, remarking that if the woman had done something ‘really, really bad’ then violence might be justified. We argue that young people’s uncertainties need to be situated within the gender-unequal socio-contexts of contemporary society, and further call for a WSA to domestic violence prevention to be a compulsory part of the UK national curriculum

A generation apart? Youth and political participation in Britain

Conventional wisdom holds that young people in Britain are alienated from politics, with some claiming that this reflects a wider crisis of legitimacy that should be met by initiatives to increase citizenship. This article addresses these areas, presenting both panel survey and focus group data from first-time voters. It concludes that, contrary to the findings from many predominantly quantitative studies of political participation, young people are interested in political matters, and do support the democratic process. However they feel a sense of anti-climax having voted for the first time, and are critical of those who have been elected to positions of political power. If they are a generation apart, this is less to do with apathy, and more to do with their engaged scepticism about ‘formal’ politics in Britain

Resolving new ultrastructural features of cytokinetic abscission with soft X ray cryo tomography

Mammalian cytokinetic abscission is mediated by the ESCRT membrane fission machinery. While much has been clarified on the topology and kinetics of abscission through high resolution microscopy, key questions regarding the mechanism of abscission remain open. Here we apply cryogenic soft X ray tomography to elucidate new ultrastructural details in the intercellular membrane bridge connecting cells undergoing abscission. In particular, we resolve defined ring like structures inside the midbody dark zone that have been inaccessible to EM, and identify membrane extrusions at the abscission sites. In cells at late stages of abscission we resolve a complex array of helical spirals, extending the structural information obtained by EM. Our results highlight the advantages of soft X ray tomography and emphasize the importance of using complementary approaches for characterizing cellular structures. Notably, by providing new structural data from intact cells we present a realistic view on the topology of abscission and suggest new mechanistic models for ESCRT mediated abscissio

Self-avoiding walks and connective constants

The connective constant $\mu(G)$ of a quasi-transitive graph $G$ is the asymptotic growth rate of the number of self-avoiding walks (SAWs) on $G$ from a given starting vertex. We survey several aspects of the relationship between the connective constant and the underlying graph $G$. $\bullet$ We present upper and lower bounds for $\mu$ in terms of the vertex-degree and girth of a transitive graph. $\bullet$ We discuss the question of whether $\mu\ge\phi$ for transitive cubic graphs (where $\phi$ denotes the golden mean), and we introduce the Fisher transformation for SAWs (that is, the replacement of vertices by triangles). $\bullet$ We present strict inequalities for the connective constants $\mu(G)$ of transitive graphs $G$, as $G$ varies. $\bullet$ As a consequence of the last, the connective constant of a Cayley graph of a finitely generated group decreases strictly when a new relator is added, and increases strictly when a non-trivial group element is declared to be a further generator. $\bullet$ We describe so-called graph height functions within an account of "bridges" for quasi-transitive graphs, and indicate that the bridge constant equals the connective constant when the graph has a unimodular graph height function. $\bullet$ A partial answer is given to the question of the locality of connective constants, based around the existence of unimodular graph height functions. $\bullet$ Examples are presented of Cayley graphs of finitely presented groups that possess graph height functions (that are, in addition, harmonic and unimodular), and that do not. $\bullet$ The review closes with a brief account of the "speed" of SAW.Comment: Accepted version. arXiv admin note: substantial text overlap with arXiv:1304.721