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

Evaluation of activated high volume fly ash systems using Na2SO4, lime and quicklime in mortars with high loss on ignition fly ashes

In general, concretes made with blended Portland cement containing high volumes of fly ash provide an alternative to conventional Portland cement concrete to enable carbon footprint reduction. This study evaluates the chemical activation of four fly ashes in blends with Portland cement, by assessing their effects on hydration and compressive strength. In this study, a sieving process is used to regulate the fly ash composition, which has an effect in the chemistry and reaction of the mix. The results show the importance of the amorphous content of the fly ash with respect to achieving a high compressive strength. The effect of sodium sulfate, added as an activator, is significant in terms of compressive strength at early age for two of the fly ashes studied; in this case, the parameter used to correlate with the compressive strength evolution is the amount of portlandite consumed through pozzolanic reactions. However, sodium sulfate does not have the same effect on fly ashes with a high amount of Fe2O3, in which portlandite consumption is much lower

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

Assessment of MRI issues at 3-Tesla for metallic surgical implants: findings applied to 61 additional skin closure staples and vessel ligation clips

<p>Abstract</p> <p>Purpose</p> <p>Metallic skin closure staples and vessel ligation clips should be tested at 3-Tesla to characterize MRI issues in order to ensure patient safety. Therefore, metallic surgical implants were assessed at 3-Tesla for magnetic field interactions, MRI-related heating, and artifacts.</p> <p>Methods</p> <p>A skin closure staple (Visistat Skin Stapler, staple, Polytetrafluoroethylene, PTFE, coated 316L/316LVM stainless steel; Teleflex Medical, Durham, NC) and a vessel ligation clip (Hemoclip Traditional, stainless steel; Teleflex Medical, Durham, NC) that represented the largest metallic sizes made from materials with the highest magnetic susceptibilities (i.e., based on material information) among 61 other surgical implants (52 metallic implants, 9 nonmetallic implants) underwent evaluation for magnetic field interactions, MRI-related heating, and artifacts using standardized techniques. MRI-related heating was assessed by placing each implant in a gelled-saline-filled phantom with MRI performed using a transmit/receive RF body coil at an MR system reported, whole body averaged SAR of 2.9-W/kg for 15-min. Artifacts were characterized using T1-weighted, SE and GRE pulse sequences.</p> <p>Results</p> <p>Each surgical implant showed minor magnetic field interactions (20- and 27-degrees, which is acceptable from a safety consideration). Heating was not substantial (highest temperature change, ≤ 1.6°C). Artifacts may create issues if the area of interest is in the same area or close to the respective surgical implant.</p> <p>Conclusions</p> <p>The results demonstrated that it would be acceptable for patients with these metallic surgical implants to undergo MRI at 3-Tesla or less. Because of the materials and dimensions of the surgical implants that underwent testing, these findings pertain to 61 additional similar implants.</p