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

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

Editorial: summary for policymakers of the thematic assessment on pollinators, pollination and food production

Values of pollinators and pollination 1. Animal pollination plays a vital role as a regulating ecosystem service in nature. Globally, nearly 90 per cent of wild flowering plant species depend, at least in part, on the transfer of pollen by animals. These plants are critical for the continued functioning of ecosystems as they provide food, form habitats, and provide other resources for a wide range of other species. 2. More than three quarters of the leading types of global food crops rely to some extent on animal pollination for yield and/or quality. Pollinator-dependent crops contribute to 35 per cent of global crop production volume. 3. Given that pollinator-dependent crops rely on animal pollination to varying degrees, it is estimated that 5–8 per cent of current global crop production is directly attributed to animal pollination with an annual market value of $235 billion–$577 billion (in 2015, United States dollars1) worldwide

Ten policies for pollinators

Earlier this year, the first global thematic assessment from the Intergovernmental Science-Policy Platform on Biodiversity and Ecosystem Services (IPBES) evaluated the state of knowledge about pollinators and pollination (1, 2). It confirmed evidence of large-scale wild pollinator declines in northwest Europe and North America and identified data shortfalls and an urgent need for monitoring elsewhere in the world. With high-level political commitments to support pollinators in the United States (3), the United Kingdom (4), and France (5); encouragement from the Convention on Biological Diversity's (CBD's) scientific advice body (6); and the issue on the agenda for next month's Conference of the Parties to the CBD, we see a chance for global-scale policy change. We extend beyond the IPBES report, which we helped to write, and suggest 10 policies that governments should seriously consider to protect pollinators and secure pollination services. Our suggestions are not the only available responses but are those we consider most likely to succeed, because of synergy with international policy objectives and strategies or formulation of international policy creating opportunities for change. We make these suggestions as independent scientists and not on behalf of IPBES

On the structure of eigenfunctions corresponding to embedded eigenvalues of locally perturbed periodic graph operators

The article is devoted to the following question. Consider a periodic self-adjoint difference (differential) operator on a graph (quantum graph) G with a co-compact free action of the integer lattice Z^n. It is known that a local perturbation of the operator might embed an eigenvalue into the continuous spectrum (a feature uncommon for periodic elliptic operators of second order). In all known constructions of such examples, the corresponding eigenfunction is compactly supported. One wonders whether this must always be the case. The paper answers this question affirmatively. What is more surprising, one can estimate that the eigenmode must be localized not far away from the perturbation (in a neighborhood of the perturbation's support, the width of the neighborhood determined by the unperturbed operator only). The validity of this result requires the condition of irreducibility of the Fermi (Floquet) surface of the periodic operator, which is expected to be satisfied for instance for periodic Schroedinger operators.Comment: Submitted for publicatio

Mutations of the BRAF gene in human cancer

Cancers arise owing to the accumulation of mutations in critical genes that alter normal programmes of cell proliferation, differentiation and death. As the first stage of a systematic genome-wide screen for these genes, we have prioritized for analysis signalling pathways in which at least one gene is mutated in human cancer. The RAS RAF MEK ERK MAP kinase pathway mediates cellular responses to growth signals. RAS is mutated to an oncogenic form in about 15% of human cancer. The three RAF genes code for cytoplasmic serine/threonine kinases that are regulated by binding RAS. Here we report BRAF somatic missense mutations in 66% of malignant melanomas and at lower frequency in a wide range of human cancers. All mutations are within the kinase domain, with a single substitution (V599E) accounting for 80%. Mutated BRAF proteins have elevated kinase activity and are transforming in NIH3T3 cells. Furthermore, RAS function is not required for the growth of cancer cell lines with the V599E mutation. As BRAF is a serine/threonine kinase that is commonly activated by somatic point mutation in human cancer, it may provide new therapeutic opportunities in malignant melanoma

Developing European conservation and mitigation tools for pollination services: approaches of the STEP (Status and Trends of European Pollinators) project

Pollinating insects form a key component of European biodiversity, and provide a vital ecosystem service to crops and wild plants. There is growing evidence of declines in both wild and domesticated pollinators, and parallel declines in plants relying upon them. The STEP project (Status and Trends of European Pollinators, 2010-2015, www.stepproject.net) is documenting critical elements in the nature and extent of these declines, examining key functional traits associated with pollination deficits, and developing a Red List for some European pollinator groups. Together these activities are laying the groundwork for future pollinator monitoring programmes. STEP is also assessing the relative importance of potential drivers of pollinator declines, including climate change, habitat loss and fragmentation, agrochemicals, pathogens, alien species, light pollution, and their interactions. We are measuring the ecological and economic impacts of declining pollinator services and floral resources, including effects on wild plant populations, crop production and human nutrition. STEP is reviewing existing and potential mitigation options, and providing novel tests of their effectiveness across Europe. Our work is building upon existing and newly developed datasets and models, complemented by spatially-replicated campaigns of field research to fill gaps in current knowledge. Findings are being integrated into a policy-relevant framework to create evidence-based decision support tools. STEP is establishing communication links to a wide range of stakeholders across Europe and beyond, including policy makers, beekeepers, farmers, academics and the general public. Taken together, the STEP research programme aims to improve our understanding of the nature, causes, consequences and potential mitigation of declines in pollination services at local, national, continental and global scales

Using citizen science to examine the nesting ecology of ground-nesting bees

Suitable nest sites are a crucial habitat requirement of ground nesting bees, but empirical studies of fossorial solitary bee nesting ecology in the UK are few in number. This study used a citizen science approach to overcome the logistical and temporal barriers associated with this type of research and to gather data on the abiotic environment associated with the nesting aggregations of four fossorial solitary bee species in the UK. Three hundred and ninety-four records were submitted by the public between March and November 2017. Sixty percent (236) of these records were verified as indicative of active nesting aggregations of the target species. Overall, the species in this study demonstrated the capacity to nest within a broad range of environmental variables. Although Colletes hederae (Schmidt and Westrich, 1993) was often reported from sloped, unshaded sites, and Andrena fulva (Müller in Allioni, 1766) was regularly associated with flat, shaded locations. This study demonstrated the efficacy of a citizen science approach in surmounting the intrinsic difficulties associated with studying solitary bee nest sites, which are both ephemeral and cryptic structures in the landscape

Flora nas fazendas.

Na atualidade, a conservação da biodiversidade e dos serviços ambientais representa um dos maiores desafios globais, em função do elevado nível das pressões antrópicas sobre os ecossistemas naturais. Neste contexto, a devastação da Caatinga vem se tornando cada vez mais preocupante não só para os amantes da conservação da natureza, mas também para os diversos setores produtivos que estão direta ou indiretamente ligados a exploração de seus recursos naturais. Apesar de a Caatinga ser o único bioma exclusivamente brasileiro, este ainda é pouco valorizado e carente de estudos. Assim, estamos perdendo um patrimônio único no mundo mesmo antes de conhecer seu potencial. Diante desse cenário de desconhecimento e destruição, ações voltadas para a sustentabilidade da Caatinga se fazem cada vez mais urgentes a fim de revertermos esse quadro. Nessa ótica, o projeto Manejo Sustentável de Fruteiras da Caatinga (SUFICA), em parceria com produtores da região do Vale do Submédio São Francisco, vem buscando identificar formas mais amigáveis de uso das áreas agrícolas, visando o desenvolvimento de uma agricultura sustentável. Nessa publicação, vamos falar sobre a vegetação da Caatinga e sua importância para a manutenção dos serviços ambientais nos agroecossistemas e apresentar parte dos resultados obtidos nas áreas de estudo do projeto

3. Bird Conservation

Expert assessors Tatsuya Amano, University of Cambridge, UK Andy Brown, Natural England, UK Fiona Burns, Royal Society for the Protection of Birds, UK Yohay Carmel, Israel Institute of Technology Mick Clout, University of Auckland, New Zealand Geoff Hilton, Wildfowl & Wetlands Trust, UK Nancy Ockendon, University of Cambridge, UK James Pearce-Higgins, British Trust for Ornithology, UK Sugoto Roy, Food and Environment Research Agency, DEFRA, UK Rebecca K. Smith, University of Cambridge, UK Wil..