Averages of the number of points on elliptic curves

If $E$ is an elliptic curve defined over $\mathbb Q$ and $p$ is a prime of good reduction for $E$, let $E(\mathbb F_p)$ denote the set of points on the reduced curve modulo $p$. Define an arithmetic function $M_E(N)$ by setting $M_E(N):= \#\{p: \#E(\mathbb F_p)= N\}$. Recently, David and the third author studied the average of $M_E(N)$ over certain "boxes" of elliptic curves $E$. Assuming a plausible conjecture about primes in short intervals, they showed the following: for odd $N$, the average of $M_E(N)$ over a box with sufficiently large sides is $\sim \frac{K^{\ast}(N)}{\log{N}}$ for an explicitly-given function $K^{\ast}(N)$. The function $K^{\ast}(N)$ is somewhat peculiar: defined as a product over the primes dividing $N$, it resembles a multiplicative function at first glance. But further inspection reveals that it is not, and so one cannot directly investigate its properties by the usual tools of multiplicative number theory. In this paper, we overcome these difficulties and prove a number of statistical results about $K^{\ast}(N)$. For example, we determine the mean value of $K^{\ast}(N)$ over all $N$, odd $N$ and prime $N$, and we show that $K^{\ast}(N)$ has a distribution function. We also explain how our results relate to existing theorems and conjectures on the multiplicative properties of $\# E(\mathbb F_p)$, such as Koblitz's conjecture.Comment: Corrected definition of K*(N

Regionalisation of Northern Territory land councils

The dispersal of the powers exercised and functions performed by the two major land councils has been a subject of debate and recommendations on a number of occasions since the Aboriginal Land Rights (Northern Territory) Act 1976 came into effect. The Reeves Review of the Act in 1998, and the subsequent Inquiry into that Review by the House of Representatives Standing Committee on Aboriginal and Torres Strait Islander Affairs (HORSCATSIA) this year, have raised the issue to prominence again and ensured that it will be dealt with in the coming round of statutory amendments. This Discussion Paper considers the steps that have been taken towards regionalisation under the current provisions of the Act, and compares models for further regionalisation proposed by David Martin, the two land councils, and HORSCATSIA. These proposals, while more moderate than that of Reeves in that they all presume the continued existence of the Northern and Central Land Councils, differ on a number of points. Regionalisation within, or outside, the existing land council structures, provision for local initiative in seeking devolution, and the role of the Minister, are among the matters at issue in an attempt to secure both increased local or regional autonomy and improved land council efficiency. Funding of regionalised bodies also demands attention, given the criticisms directed at this aspect of the Reeves model. This paper goes on to express concern that regionalisation has been accepted as a selfevidently desirable policy, and that insufficient critical attention has been paid to the advantages expected to flow from its implementation. We begin our critique by distinguishing between ‘administrative’ regionalisation and ‘decision-making’ regionalisation of land council functions and powers. We then separate out the real process of decision-making from the formal act of decision-taking in the scheme of the Act. Most importantly, we point to the already localised character of decisions by traditional owners under the informed consent provisions, and argue that the primary danger posed by regionalisation is that the regional decision-takers will trespass upon the decision-making prerogatives of the traditional owners. While in our view this problem is a threat to the fundamental distribution of authority under the existing Act, and is sufficiently serious to call into question the rationale for moves towards greater regionalisation, the breadth of opinion, including local Aboriginal sentiment, in favour of more localised autonomy, needs to be accommodated. We therefore argue for a number of measures in mitigation. Establishing regional areas of sufficiently large size, each represented by a committee or council of sufficiently small size, and serviced, in the case of internal land council regionalisation, by professional staff employed through the central organisation, are steps intended to protect the informed consent procedures of the Act. Some formal certification witnessing the adequacy of those procedures in each case should also be introduced as part of the conditions attaching to the affixing of the land council common seal to agreements. As only some of these measures are available in the case of independent, or ‘breakaway’, land councils, some caution is due in approving more of these, especially in assessing the spread and depth of popular support

Tidal Truncation of Circumplanetary Discs

We analyse some properties of circumplanetary discs. Flow through such discs may provide most of the mass to gas giant planets, and such discs are likely sites for the formation of regular satellites. We model these discs as accretion discs subject to the tidal forces of the central star. The tidal torques from the star remove the disc angular momentum near the disc outer edge and permit the accreting disc gas to lose angular momentum at the rate appropriate for steady accretion. Circumplanetary discs are truncated near the radius where periodic ballistic orbits cross, where tidal forces on the disc are strong. This radius occurs at approximately 0.4 r_H for the planet Hill radius r_H. During the T Tauri stage of disc accretion, the disc is fairly thick with aspect ratio H/r > 0.2 and the disc edge tapering occurs over a radial scale ~ H ~ 0.1 r_H. For a circular or slightly eccentric orbit planet, no significant resonances lie within the main body of the disc. Tidally driven waves involving resonances nonetheless play an important role in truncating the disc, especially when it is fairly thick. We model the disc structure using one dimensional time-dependent and steady-state models and also two dimensional SPH simulations. The circumplanetary disc structure depends on the variation of the disc turbulent viscosity with radius and is insensitive to the angular distribution of the accreting gas. Dead zones may occur within the circumplanetary disc and result in density structures. If the disc is turbulent throughout, the predicted disc structure near the location of the regular Jovian and Saturnian satellites is smooth with no obvious feature that would favor formation at their current locations.Comment: Accepted for publication in MNRA

The growth and hydrodynamic collapse of a protoplanet envelope

We have conducted three-dimensional self-gravitating radiation hydrodynamical models of gas accretion onto high mass cores (15-33 Earth masses) over hundreds of orbits. Of these models, one case accretes more than a third of a Jupiter mass of gas, before eventually undergoing a hydrodynamic collapse. This collapse causes the density near the core to increase by more than an order of magnitude, and the outer envelope to evolve into a circumplanetary disc. A small reduction in the mass within the Hill radius (R_H) accompanies this collapse as a shock propagates outwards. This collapse leads to a new hydrostatic equilibrium for the protoplanetary envelope, at which point 97 per cent of the mass contained within the Hill radius is within the inner 0.03 R_H which had previously contained less than 40 per cent. Following this collapse the protoplanet resumes accretion at its prior rate. The net flow of mass towards this dense protoplanet is predominantly from high latitudes, whilst at the outer edge of the circumplanetary disc there is net outflow of gas along the midplane. We also find a turnover of gas deep within the bound envelope that may be caused by the establishment of convection cells.Comment: 16 pages, 16 figures. Accepted for publication in MNRA

Computation of Light Scattering in Young Stellar Objects

A Monte Carlo light scattering code incorporating aligned non-spherical particles is described. The major effects on the flux distribution, linear polarisation and circular polarisation are presented, with emphasis on the application to Young Stellar Objects (YSOs). The need for models with non-spherical particles in order to successfully model polarisation data is reviewed. The ability of this type of model to map magnetic field structure in embedded YSOs is described. The possible application to the question of the origin of biomolecular homochirality via UV circular polarisation in star forming regions is also briefly discussed.Comment: Accepted by The Journal of Quantitative Spectroscopy and Radiative Transfer. Replaced version corrects an error in the definition of the sense of Cpol in the published version and other minor errors found at the proof stag

Planetesimal-driven planet migration in the presence of a gas disk

We report here on an extension of a previous study by Kirsh et al. (2009) of planetesimal-driven migration using our N-body code SyMBA (Duncan et al., 1998). The previous work focused on the case of a single planet of mass Mem, immersed in a planetesimal disk with a power-law surface density distribution and Rayleigh distributed eccentricities and inclinations. Typically 10^4-10^5 equal-mass planetesimals were used, where the gravitational force (and the back-reaction) on each planetesimal by the Sun and planetwere included, while planetesimal-planetesimal interactions were neglected. The runs reported on here incorporate the dynamical effects of a gas disk, where the Adachi et al. (1976) prescription of aerodynamic gas drag is implemented for all bodies. In some cases the Papaloizou and Larwood (2000) prescription of Type-I migration for the planet are implemented, as well as a mass distribution. In the gas-free cases, rapid planet migration was observed - at a rate independent of the planet's mass - provided the planet's mass was not large compared to the mass in planetesimals capable of entering its Hill sphere. In such cases, both inward and outward migrations can be self-sustaining, but there is a strong propensity for inward migration. When a gas disk is present, aerodynamic drag can substantially modify the dynamics of scattered planetesimals. For sufficiently large or small mono-dispersed planetesimals, the planet typically migrates inward. However, for a range of plausible planetesimal sizes (i.e. 0.5-5.0 km at 5.0 AU in a minimum mass Hayashi disk) outward migration is usually triggered, often accompanied by substantial planetary mass accretion. The origins of this behaviour are explained in terms of a toy model. The effects of including a size distribution and torques associated with Type-I migration are also discussed.Comment: 37 pages, 17 figures, Accepted for publication in Icaru

Conformational changes of calmodulin upon Ca2+ binding studied with a microfluidic mixer

A microfluidic mixer is applied to study the kinetics of calmodulin conformational changes upon Ca2+ binding. The device facilitates rapid, uniform mixing by decoupling hydrodynamic focusing from diffusive mixing and accesses time scales of tens of microseconds. The mixer is used in conjunction with multiphoton microscopy to examine the fast Ca2+-induced transitions of acrylodan-labeled calmodulin. We find that the kinetic rates of the conformational changes in two homologous globular domains differ by more than an order of magnitude. The characteristic time constants are ≈490 μs for the transitions in the C-terminal domain and ≈20 ms for those in the N-terminal domain of the protein. We discuss possible mechanisms for the two distinct events and the biological role of the stable intermediate, half-saturated calmodulin

Changing the Face of an Institution: Creative Partnerships for Women’s Professional Development

Purpose of program/study/research:To accelerate the advancement of women professionals at an academic medical center through creative collaboration. Methodology (including study design, analysis, and evaluation):The UMass Medical School Women’s Faculty Committee (WFC) initiated a partnership with the medical library to compete successfully to host the traveling exhibition “Changing the Face of Medicine: Celebrating America’s Women Physicians.” Concurrent with the 6-week exhibition, fifteen events brought local and nationally prominent women together for mentoring activities, an original dramatic production, and an awards luncheon for women faculty. Women featured in the exhibit as well as its Visiting Curator spoke at UMMS about their research and career challenges, read from their published works, and were featured at graduation and a regional medical society event. Results:Application-writing and event planning sessions forged robust working relationships among top-ranking administrators, senior and junior faculty, and staff. The exhibition increased opportunities for administration, faculty, and students to understand the impact of women in medicine, their leadership potential, and historical contributions. It also generated new mentor/mentee relationships and grant-writing collaborations. The exhibition’s national recognition helped draw a larger, more diverse and gender-balanced audience (575+ people) to the events enhancing the visibility of the WFC, as evidenced by institutional funding for women faculty to attend the AAMC WIM professional development workshops and ELAM for the first time. Conclusion(s):Creative partnerships, motivated by the opportunity to host “Changing the Face of Medicine,” produced greater than expected gains for women faculty, generating new awareness and understanding of women’s accomplishments and leadership potential. This partnership allowed for a wide range of multi-disciplinary efforts, strengthening networking across silos, and advancing the goals of women in an academic medical center. Presented October 29, 2006 at AAMC 2006 Annaul Meeting, Seattle, WA