A combinatorial Li-Yau inequality and rational points on curves

We present a method to control gonality of nonarchimedean curves based on graph theory. Let k denote a complete nonarchimedean valued field.We first prove a lower bound for the gonality of a curve over the algebraic closure of k in terms of the minimal degree of a class of graph maps, namely: one should minimize over all so-called finite harmonic graph morphisms to trees, that originate from any refinement of the dual graph of the stable model of the curve. Next comes our main result: we prove a lower bound for the degree of such a graph morphism in terms of the first eigenvalue of the Laplacian and some “volume” of the original graph; this can be seen as a substitute for graphs of the Li–Yau inequality from differential geometry, although we also prove that the strict analogue of the original inequality fails for general graphs. Finally,we apply the results to give a lower bound for the gonality of arbitraryDrinfeld modular curves over finite fields and for general congruence subgroups Γ of Γ (1) that is linear in the index [Γ (1) : Γ ], with a constant that only depends on the residue field degree and the degree of the chosen “infinite” place. This is a function field analogue of a theorem of Abramovich for classical modular curves. We present applications to uniform boundedness of torsion of rank two Drinfeld modules that improve upon existing results, and to lower bounds on the modular degree of certain elliptic curves over function fields that solve a problem of Papikian

Brief of requirements of the dairy cow

This report lists the brief of requirements of the dairy cow, based on her needs (also listed). The BoR indicates the actor’s needs with regards to the animal husbandry system. BoR of the main actors are incorporated in the redesign of a dairy husbandry system in the project Cow Powe

Fostering research engagement in partnership schools: networking and value creation

© 2017 Informa UK Limited, trading as Taylor & Francis Group. The call for teachers and schools to become more research-engaged is resonating stronger than ever with government efforts to improve research impact and educational quality in the United Kingdom (UK) and many other countries. In these endeavors strengthening the social network structure and collegial relationships that enable collaborative research and learning in schools are often overlooked. This longitudinal, multi-method case study focused on understanding this pivotal social dimension. It examined in what way and to what extent research-engaged relationships are developed and value for practice is created among school colleagues. The context was a secondary school that is part of a longstanding school–university research partnership in the UK. Social network and survey data, school documents, individual narratives, and interview data were collected and analysed during one academic year. Results reveal that school leadership adopted an approach in which they combined the development of formal structures and informal networking to foster a significant increase in research-engaged interactions among school staff. Two key elements of the school–university research partnership were distinguished as powerful drivers of fostering research engagement. Outcomes show that the increased research engagement led to diverse types of value creation in both the core and periphery of the research network of school staff.Marie Curie Fellowship fundin

Degenerating families of dendrograms

Dendrograms used in data analysis are ultrametric spaces, hence objects of nonarchimedean geometry. It is known that there exist $p$-adic representation of dendrograms. Completed by a point at infinity, they can be viewed as subtrees of the Bruhat-Tits tree associated to the $p$-adic projective line. The implications are that certain moduli spaces known in algebraic geometry are $p$-adic parameter spaces of (families of) dendrograms, and stochastic classification can also be handled within this framework. At the end, we calculate the topology of the hidden part of a dendrogram.Comment: 13 pages, 8 figure

Performance reporting

__Abstract__ Public organizations account for their performance through making public sector performance information publicly available, both to politicians through performance reporting, and to citizens through rankings, websites, and performance reports. This chapter reviews whether performance reporting makes public organizations more accountable: Do citizens and politicians actually consult and use performance information, and does this information change their decisions and behaviours? The chapter first looks at the use of performance metrics in political decision making, drivers of this use, and differences in use across groups. It subsequently reviews the literature on whether citizens use publicly available performance indicators and rankings to make an informed choice between alternative service providers. The focus is on school and hospital performance data. The chapter ends by discussing implications on equity, power relations, and the internal dynamics of organizations