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

The Zieschang-McCool method for generating algebraic mapping-class groups

Let g and p be non-negative integers. Let A(g,p) denote the group consisting of all those automorphisms of the free group on {t_1,...,t_p, x_1,...,x_g, y_1,...y_g} which fix the element t_1t_2...t_p[x_1,y_1]...[x_g,y_g] and permute the set of conjugacy classes {[t_1],....,[t_p]}. Labru\`ere and Paris, building on work of Artin, Magnus, Dehn, Nielsen, Lickorish, Zieschang, Birman, Humphries, and others, showed that A(g,p) is generated by a set that is called the ADLH set. We use methods of Zieschang and McCool to give a self-contained, algebraic proof of this result. Labru\`ere and Paris also gave defining relations for the ADLH set in A(g,p); we do not know an algebraic proof of this for g > 1. Consider an orientable surface S(g,p) of genus g with p punctures, such that (g,p) is not (0,0) or (0,1). The algebraic mapping-class group of S(g,p), denoted M(g,p), is defined as the group of all those outer automorphisms of the one-relator group with generating set {t_1,...,t_p, x_1,...,x_g, y_1,...y_g} and relator t_1t_2...t_p[x_1,y_1]...[x_g,y_g] which permute the set of conjugacy classes {[t_1],....,[t_p]}. It now follows from a result of Nielsen that M(g,p) is generated by the image of the ADLH set together with a reflection. This gives a new way of seeing that M(g,p) equals the (topological) mapping-class group of S(g,p), along lines suggested by Magnus, Karrass, and Solitar in 1966.Comment: 21 pages, 0 figure

THREaD Mapper Studio: a novel, visual web server for the estimation of genetic linkage maps

The estimation of genetic linkage maps is a key component in plant and animal research, providing both an indication of the genetic structure of an organism and a mechanism for identifying candidate genes associated with traits of interest. Because of this importance, several computational solutions to genetic map estimation exist, mostly implemented as stand-alone software packages. However, the estimation process is often largely hidden from the user. Consequently, problems such as a program crashing may occur that leave a user baffled. THREaD Mapper Studio (http://cbr.jic.ac.uk/threadmapper) is a new web site that implements a novel, visual and interactive method for the estimation of genetic linkage maps from DNA markers. The rationale behind the web site is to make the estimation process as transparent and robust as possible, while also allowing users to use their expert knowledge during analysis. Indeed, the 3D visual nature of the tool allows users to spot features in a data set, such as outlying markers and potential structural rearrangements that could cause problems with the estimation procedure and to account for them in their analysis. Furthermore, THREaD Mapper Studio facilitates the visual comparison of genetic map solutions from third party software, aiding users in developing robust solutions for their data sets

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

Peak reduction technique in commutative algebra

The "peak reduction" method is a powerful combinatorial technique with applications in many different areas of mathematics as well as theoretical computer science. It was introduced by Whitehead, a famous topologist and group theorist, who used it to solve an important algorithmic problem concerning automorphisms of a free group. Since then, this method was used to solve numerous problems in group theory, topology, combinatorics, and probably in some other areas as well. In this paper, we give a survey of what seems to be the first applications of the peak reduction technique in commutative algebra and affine algebraic geometry.Comment: survey; 10 page

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

The role of agri-environment schemes in conservation and environmental management.

Over half of the European landscape is under agricultural management and has been for millennia. Many species and ecosystems of conservation concern in Europe depend on agricultural management and are showing ongoing declines. Agri-environment schemes (AES) are designed partly to address this. They are a major source of nature conservation funding within the European Union (EU) and the highest conservation expenditure in Europe. We reviewed the structure of current AES across Europe. Since a 2003 review questioned the overall effectiveness of AES for biodiversity, there has been a plethora of case studies and meta-analyses examining their effectiveness. Most syntheses demonstrate general increases in farmland biodiversity in response to AES, with the size of the effect depending on the structure and management of the surrounding landscape. This is important in the light of successive EU enlargement and ongoing reforms of AES. We examined the change in effect size over time by merging the data sets of 3 recent meta-analyses and found that schemes implemented after revision of the EU's agri-environmental programs in 2007 were not more effective than schemes implemented before revision. Furthermore, schemes aimed at areas out of production (such as field margins and hedgerows) are more effective at enhancing species richness than those aimed at productive areas (such as arable crops or grasslands). Outstanding research questions include whether AES enhance ecosystem services, whether they are more effective in agriculturally marginal areas than in intensively farmed areas, whether they are more or less cost-effective for farmland biodiversity than protected areas, and how much their effectiveness is influenced by farmer training and advice? The general lesson from the European experience is that AES can be effective for conserving wildlife on farmland, but they are expensive and need to be carefully designed and targeted.This is the final published version. It first appeared from Wiley http://dx.doi.org/10.1111/cobi.1253

Examining the Polymorphisms in the Hypoxia Pathway Genes in Relation to Outcome in Colorectal Cancer

Introduction Colorectal cancer is a common malignancy. Identification of genetic prognostic markers may help prognostic estimations in colorectal cancer. Genes that regulate response to hypoxia and other genes that are regulated under the hypoxic conditions have been shown to play roles in cancer progression. In this study, we hypothesized that genetic variations in the hypoxia pathway genes were associated with the risk of outcome in colorectal cancer patients. Methods This study was performed in two phases. In the first phase, 49 SNPs from six hypoxia pathway genes (HIF1A, HIF1B, HIF2A, LOX, MIF and CXCL12) in 272 colorectal cancer patients were analyzed. In the second phase, 77 SNPs from seven hypoxia pathway genes (HIF1A, HIF1B, HIF2A, HIF2B, HIF3A, LOX and CXCL12) were analyzed in an additional cohort of 535 patients. Kaplan Meier, Cox univariate and multivariable regression analyses were performed to analyze the relationship between the SNPs and overall survival (OS), disease free survival (DFS) or disease specific survival (DSS). Since this was a hypothesis-generating study, no correction for multiple testing was applied. Results In phase I, one SNP (HIF2A rs11125070) was found to be associated with DFS in multivariable analysis; yet association of a proxy polymorphism (HIF2A rs4953342) was not detected in the phase II patient cohort. In phase II, associations of two SNPs (HIF2A rs4953352 and HIF2B rs12593988) were significant in both OS and DFS multivariable analyses. However, association of HIF2A rs4953352 was not replicated in the phase I cohort using a proxy SNP (HIF2A rs6706003). Conclusion Overall, our study did not find a convincing evidence of association of the investigated polymorphisms with the disease outcomes in colorectal cance