Amicable pairs and aliquot cycles for elliptic curves

An amicable pair for an elliptic curve E/Q is a pair of primes (p,q) of good reduction for E satisfying #E(F_p) = q and #E(F_q) = p. In this paper we study elliptic amicable pairs and analogously defined longer elliptic aliquot cycles. We show that there exist elliptic curves with arbitrarily long aliqout cycles, but that CM elliptic curves (with j not 0) have no aliqout cycles of length greater than two. We give conjectural formulas for the frequency of amicable pairs. For CM curves, the derivation of precise conjectural formulas involves a detailed analysis of the values of the Grossencharacter evaluated at a prime ideal P in End(E) having the property that #E(F_P) is prime. This is especially intricate for the family of curves with j = 0.Comment: 53 page

Googling the brain: discovering hierarchical and asymmetric network structures, with applications in neuroscience

Hierarchical organisation is a common feature of many directed networks arising in nature and technology. For example, a well-defined message-passing framework based on managerial status typically exists in a business organisation. However, in many real-world networks such patterns of hierarchy are unlikely to be quite so transparent. Due to the nature in which empirical data is collated the nodes will often be ordered so as to obscure any underlying structure. In addition, the possibility of even a small number of links violating any overall “chain of command” makes the determination of such structures extremely challenging. Here we address the issue of how to reorder a directed network in order to reveal this type of hierarchy. In doing so we also look at the task of quantifying the level of hierarchy, given a particular node ordering. We look at a variety of approaches. Using ideas from the graph Laplacian literature, we show that a relevant discrete optimization problem leads to a natural hierarchical node ranking. We also show that this ranking arises via a maximum likelihood problem associated with a new range-dependent hierarchical random graph model. This random graph insight allows us to compute a likelihood ratio that quantifies the overall tendency for a given network to be hierarchical. We also develop a generalization of this node ordering algorithm based on the combinatorics of directed walks. In passing, we note that Google’s PageRank algorithm tackles a closely related problem, and may also be motivated from a combinatoric, walk-counting viewpoint. We illustrate the performance of the resulting algorithms on synthetic network data, and on a real-world network from neuroscience where results may be validated biologically

Monotone graph limits and quasimonotone graphs

The recent theory of graph limits gives a powerful framework for understanding the properties of suitable (convergent) sequences $(G_n)$ of graphs in terms of a limiting object which may be represented by a symmetric function $W$ on $[0,1]$, i.e., a kernel or graphon. In this context it is natural to wish to relate specific properties of the sequence to specific properties of the kernel. Here we show that the kernel is monotone (i.e., increasing in both variables) if and only if the sequence satisfies a `quasi-monotonicity' property defined by a certain functional tending to zero. As a tool we prove an inequality relating the cut and $L^1$ norms of kernels of the form $W_1-W_2$ with $W_1$ and $W_2$ monotone that may be of interest in its own right; no such inequality holds for general kernels.Comment: 38 page

On the Mechanistic Origins of Toughness in Bone

One of the most intriguing protein materials found in nature is bone, a material composed of assemblies of tropocollagen molecules and tiny hydroxyapatite mineral crystals that form an extremely tough, yet lightweight, adaptive and multifunctional material. Bone has evolved to provide structural support to organisms, and therefore its mechanical properties are of great physiological relevance. In this article, we review the structure and properties of bone, focusing on mechanical deformation and fracture behavior from the perspective of the multidimensional hierarchical nature of its structure. In fact, bone derives its resistance to fracture with a multitude of deformation and toughening mechanisms at many size scales ranging from the nanoscale structure of its protein molecules to the macroscopic physiological scale.United States. Army Research Office (contract number W911NF-06-1-0291)National Science Foundation (U.S.) (CAREER award (contract number 0642545))Lawrence Berkeley National Laboratory (Laboratory Directed Research and Development Program)United States. Dept. of Energy (Office of Science, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering, contract number DE-AC02-05CH11231

Euclidean versus hyperbolic congestion in idealized versus experimental networks

This paper proposes a mathematical justification of the phenomenon of extreme congestion at a very limited number of nodes in very large networks. It is argued that this phenomenon occurs as a combination of the negative curvature property of the network together with minimum length routing. More specifically, it is shown that, in a large n-dimensional hyperbolic ball B of radius R viewed as a roughly similar model of a Gromov hyperbolic network, the proportion of traffic paths transiting through a small ball near the center is independent of the radius R whereas, in a Euclidean ball, the same proportion scales as 1/R^{n-1}. This discrepancy persists for the traffic load, which at the center of the hyperbolic ball scales as the square of the volume, whereas the same traffic load scales as the volume to the power (n+1)/n in the Euclidean ball. This provides a theoretical justification of the experimental exponent discrepancy observed by Narayan and Saniee between traffic loads in Gromov-hyperbolic networks from the Rocketfuel data base and synthetic Euclidean lattice networks. It is further conjectured that for networks that do not enjoy the obvious symmetry of hyperbolic and Euclidean balls, the point of maximum traffic is near the center of mass of the network.Comment: 23 pages, 4 figure

A quick approach for rheological evaluation of warm asphalt binders using response surface method

This paper describes a quick approach for quantification of the effects of a chemical warm named Rediset, and its interactions with temperature and aging on the rheological properties of asphalt binders using Response Sur-face Method. The central composite method was applied to design experimental programs for three test temperature conditions, namely; very high temperature (120–180 °C), high temperature (46–82 °C), and intermediate temperature (19–31 °C). Rotational viscosity, G*/sin δ and G*sin δ were selected as parameters to assess the effects of the chemical warm additive on the rheological properties of asphalt binders for different aging conditions. Evaluation of the effects of this additive on the transformed value of G*/sin δ at high temperatures indicates that additive content has significant effect on Ln(G*/sin δ). The results for intermediate temperatures show that this additive has a positive effect on G*sin δ of asphalt binders

Numerical evidence toward a 2-adic equivariant ''Main Conjecture''

International audienceWe test a conjectural non abelian refinement of the classical 2-adic Main Conjecture of Iwasawa theory. In the first part, we show how, in the special case that we study, the validity of this refinement is equivalent to a congruence condition on the coefficients of some power series. Then, in the second part, we explain how to compute the first coefficients of this power series and thus numerically check the conjecture in that setting

Aerial wetting contact angle measurement using confocal microscopy

A method is presented in which the wetting contact angle of a sessile drop is acquired aerially using confocal techniques to measure the radius and the height of a droplet deposited on a planar surface. The repeatability of this method is typically less than 0.25°, and often less than 0.1°, for droplet diameters less than 1 mm. To evaluate accuracy of this method, an instrument uncertainty budget is developed, which predicts a combined uncertainty of 0.91° for a 1 mm diameter water droplet with a contact angle of 110°. For droplets having diameters less than 1 mm and contact angles between 15° and 160°, these droplets approach spherical shape and their contact angles can be computed analytically with less than 1% error. For larger droplets, gravitational deformation needs to be considered

Health promoting settings in primary health care - "hälsotorg": an implementation analysis

Background Sweden, like many other western countries, faces increasing rates of lifestyle related diseases and corresponding rise in costs for health care. To meet these challenges, a number of efforts have been introduced at different societal levels. One such effort is "Hälsotorg" (HS). HS is a new health promotion setting that emerged in collaboration between the Swedish County Councils and Apoteket AB, a state-owned pharmacy company. HS's overall aim was to improve population health and facilitate inhabitants' responsibility for self-care. A new National Public Health Policy, introduced in 2008, emphasizes more focus on individual's needs and responsibility as well as strong need for county councils to provide supportive environment for individual-centred health services and increased health literacy among the population. In light of this policy, there is a need to examine existing settings that can provide supportive environment for individuals at community level. The aim of this study was to explore HS's policy implementation at local level and analyse HS's activities, in order to provide a deeper understanding of HS's potential as a health promoting setting. Methods Materials included a survey and key documents related to the development and nature of HS on local and national levels. A policy analysis inspired by Walt and Gilson was used in data analysis. In addition, an analysis using the principles of health promotion in relation to HS policy process and activities was also carried out. Results The analysis illuminated strengths and weaknesses in the policy process, its actors, contextual factors and activities. The health communication approach in the analysed documents contained health promoting intentions but the health promoting approach corresponding to a health promoting setting was neither apparent nor shared among the stakeholders. This influenced the interpretation and implementation of HS negatively. Conclusions The analysis indicates that HS has potential to be a valuable health promotion setting for both population and individuals, given the strong intentions for a health and empowerment building approach that is expressed in the documents. However, for a more sustainable implementation of HS, there is need for an in- depth understanding of the health promotion approach among HS stakeholders

Provision of mental health services in resource-poor settings: a randomised trial comparing counselling with routine medical treatment in North Afghanistan (Mazar-e-Sharif)

<p>Abstract</p> <p>Background</p> <p>Psychosocial stress caused by war, ongoing conflict, lack of security, and restricted access to resources promotes mental suffering and diseases in many resource-poor countries. In an exemplary setting, the present study compares the efficacy of psychosocial counselling with routine pharmacological treatment in a randomised trial in Mazar-e-Sharif (Afghanistan).</p> <p>Methods</p> <p>Help seeking Afghan women (N = 61), who were diagnosed with mental health symptoms by local physicians either received routine medical treatment<b/>(treatment as usual) or psychosocial counselling (5-8 sessions) following a specifically developed manualised treatment protocol. Primary outcome measures were symptoms of depression and anxiety assessed before treatment and at follow-up using the Hopkins Symptom Checklist and the Mini-International Neuropsychiatric Interview. Secondary outcome measures were psychosocial stressors and coping mechanisms.</p> <p>Results</p> <p>At 3-month follow-up, psychosocial counselling patients showed high improvements with respect to the severity of symptoms of depression and anxiety. In addition, they reported a reduction of psychosocial stressors and showed an enhancement of coping strategies. At the same time, the severity of symptoms, the quantity of psychosocial stressors and coping mechanisms did not improve in patients receiving routine medical treatment.</p> <p>Conclusion</p> <p>These results indicate that psychosocial counselling can be an effective treatment for mental illnesses even for those living in ongoing unsafe environments.</p> <p>Trial registration</p> <p><a href="http://www.clinicaltrials.gov/ct2/show/NCT01155687">NCT01155687</a></p