On the intersection of free subgroups in free products of groups

Let (G_i | i in I) be a family of groups, let F be a free group, and let G = F *(*I G_i), the free product of F and all the G_i. Let FF denote the set of all finitely generated subgroups H of G which have the property that, for each g in G and each i in I, H \cap G_i^{g} = {1}. By the Kurosh Subgroup Theorem, every element of FF is a free group. For each free group H, the reduced rank of H is defined as r(H) = max{rank(H) -1, 0} in \naturals \cup {\infty} \subseteq [0,\infty]. To avoid the vacuous case, we make the additional assumption that FF contains a non-cyclic group, and we define sigma := sup{r(H\cap K)/(r(H)r(K)) : H, K in FF and r(H)r(K) \ne 0}, sigma in [1,\infty]. We are interested in precise bounds for sigma. In the special case where I is empty, Hanna Neumann proved that sigma in [1,2], and conjectured that sigma = 1; almost fifty years later, this interval has not been reduced. With the understanding that \infty/(\infty -2) = 1, we define theta := max{|L|/(|L|-2) : L is a subgroup of G and |L| > 2}, theta in [1,3]. Generalizing Hanna Neumann's theorem, we prove that sigma in [theta, 2 theta], and, moreover, sigma = 2 theta if G has 2-torsion. Since sigma is finite, FF is closed under finite intersections. Generalizing Hanna Neumann's conjecture, we conjecture that sigma = theta whenever G does not have 2-torsion.Comment: 28 pages, no figure

Determinants of Students’ First Impressions of Instructors and Courses

Students evaluated instructors and courses in the first two weeks of the fall semester to determine the factors that form impressions in the early stages of the semester. Results indicate differences exist between upper and lower division courses with presentation of material and perceived workload as key factors that students use to form first impressions.Teaching/Communication/Extension/Profession,

THE USE OF BIOMECHANICALLY ADJUSTABLE CYCLING PEDALS IN THE DEVELOPMENT OF REHABILITATION PROTOCOLS FOR ANKLE AND KNEE INJURIES

This paper examines the use of stationary cycling with a biomechanically adjustable pedal (Biopedal) in the development of rehabilitation protocols for the knee and ankle. This paper draws on previous research which examines the role of the foot pedal interface and cycling pathomechanics, as well as research which evaluates the differences in E.M.G. activity utilizing an orthopaedically adjustable pedal. As a result of this research and clinical findings, rehabilitation protocols for selected ankle and knee injuries were developed utilizing the three-plane adjustable pedal and stationary exercise bicycle. These protocols have been clinically tested and are presented. The results indicate that the adjustable pedal can be used to reduce trauma to connective tissue structures. It can be used in rehabilitation to minimize risk of injury during active recovery and reinforce the integrity of and support injured structures

Advantages and challenges of microalgae as a source of oil for biodiesel

Microalgal oil is currently being considered as a promising alternative feedstock for biodiesel. The present demand for oil for biofuel production greatly exceeds the supply, hence alternative sources of biomass are required. Microalgae have several advantages over land-based crops in terms of oil production. Their simple unicellular structure and high photosynthetic efficiency allow for a potentially higher oil yield per area than that of the best oilseed crops. Algae can be grown on marginal land using brackish or salt water and hence do not compete for resources with conventional agriculture. They do not require herbicides or pesticides and their cultivation could be coupled with the uptake of CO2 from industrial waste streams, and the removal of excess nutrients from wastewater (Hodaifa et al., 2008; An et al., 2003). In addition to oil production, potentially valuable co-products such as pigments, antioxidants, nutraceuticals, fertilizer or feeds could be produced (Mata et al., 2010; Rodolfi et al., 2009)

Are prices of new dwellings different? A spectral analysis of UK property vintages

The work makes two contributions to Hui’s (2011) dynamic house price classification. First, a house price ripple in cycles from Modern to Older dwellings is revealed and, second, as New housing is shown to have lower volatility than the other two. Using spectral analysis, it is argued that there is a 7½-year repeat buyer-second-hand cycle and a five year, first time buyer-New housing cycle, common to three house price vintages. These cycles reinforce each other every fifteen years, which corresponds with a Minsky super-cycle in housing finance. The equity of the owner-occupier is fortified by higher house prices whereas new builds extract embedded equity from the market. Government should support builders and facilitate access to market to first time buyers and through programmes like Help-to-Buy 1. However, to address the greater price instability that should follow, Government should impose a capital gains tax on the house seller

Management and drivers of change of pollinating insects and pollination services. National Pollinator Strategy: for bees and other pollinators in England, Evidence statements and Summary of Evidence

These Evidence Statements provide up-to-date information on what is known (and not known) about the status, values, drivers of change, and responses to management of UK insect pollinators (as was September 2018). This document has been produced to inform the development of England pollinator policy, and provide insight into the evidence that underpins policy decision-making. This document sits alongside a more detailed Summary of Evidence (Annex I) document written by pollinator experts. For information on the development of the statements, and confidence ratings assigned to them, please see section ‘Generation of the statements’ below. Citations for these statements are contained in the Summary of Evidence document

Experimental and modelling evidence for structural crossover in supercritical CO₂

Physics of supercritical state is understood to a much lesser degree compared to subcritical liquids. Carbon dioxide in particular has been intensely studied, yet little is known about the supercritical part of its phase diagram. Here, we combine neutron scattering experiments and molecular dynamics simulations and demonstrate the structural crossover at the Frenkel line. The crossover is seen at pressures as high as 14 times the critical pressure and is evidenced by changes of the main features of the structure factor and pair distribution functions

Estimating the risk of species interaction loss in mutualistic communities

Funder: Royal Commission for the Exhibition of 1851 (GB)Funder: Cambridge TrustFunder: Cambridge Depatment of ZoologyFunder: Grantham Foundation for the Protection of the Environment; funder-id: http://dx.doi.org/10.13039/100008118Funder: Kenneth Miller TrustFunder: ArcadiaInteractions between species generate the functions on which ecosystems and humans depend. However, we lack an understanding of the risk that interaction loss poses to ecological communities. Here, we quantify the risk of interaction loss for 4,330 species interactions from 41 empirical pollination and seed dispersal networks across 6 continents. We estimate risk as a function of interaction vulnerability to extinction (likelihood of loss) and contribution to network feasibility, a measure of how much an interaction helps a community tolerate environmental perturbations. Remarkably, we find that more vulnerable interactions have higher contributions to network feasibility. Furthermore, interactions tend to have more similar vulnerability and contribution to feasibility across networks than expected by chance, suggesting that vulnerability and feasibility contribution may be intrinsic properties of interactions, rather than only a function of ecological context. These results may provide a starting point for prioritising interactions for conservation in species interaction networks in the future