Solving the brachistochrone and other variational problems with soap films

We show a method to solve the problem of the brachistochrone as well as other variational problems with the help of the soap films that are formed between two suitable surfaces. We also show the interesting connection between some variational problems of dynamics, statics, optics, and elasticity.Comment: 16 pages, 11 figures. This article, except for a small correction, has been submitted to the American Journal of Physic

Preservice mathematics teachers’ professional noticing of students’ mathematical thinking with technology

The purpose of this study is to examine the ways in which preservice secondary mathematics teachers (PSMT) professionally notice middle school students’ mathematical thinking on a technology enhanced mathematical task. The middle school students’ work was captured as a videocase for PSMTs to examine. Findings show that every PSMT included a discussion of the middle school students’ interaction with the technology in their noticing prompts, demonstrating that PSMTs recognized that the middle school students’ mathematical understanding was tied to their interactions with the technology. Additionally, results from PSMTs’ justifications for their predictions of middle school students’ responses to the task, incorporated the middle school students language and described how the middle school students would interact with the technology

High-frequency monitoring of nitrogen and phosphorus response in three rural catchments to the end of the 2011–2012 drought in England

This paper uses high-frequency bankside measurements from three catchments selected as part of the UK government-funded Demonstration Test Catchments (DTC) project. We compare the hydrological and hydrochemical patterns during the water year 2011–2012 from the Wylye tributary of the River Avon with mixed land use, the Blackwater tributary of the River Wensum with arable land use and the Newby Beck tributary of the River Eden with grassland land use. The beginning of the hydrological year was unusually dry and all three catchments were in states of drought. A sudden change to a wet summer occurred in April 2012 when a heavy rainfall event affected all three catchments. The year-long time series and the individual storm responses captured by in situ nutrient measurements of nitrate and phosphorus (total phosphorus and total reactive phosphorus) concentrations at each site reveal different pollutant sources and pathways operating in each catchment. Large storm-induced nutrient transfers of nitrogen and or phosphorus to each stream were recorded at all three sites during the late April rainfall event. Hysteresis loops suggested transport-limited delivery of nitrate in the Blackwater and of total phosphorus in the Wylye and Newby Beck, which was thought to be exacerbated by the dry antecedent conditions prior to the storm. The high rate of nutrient transport in each system highlights the scale of the challenges faced by environmental managers when designing mitigation measures to reduce the flux of nutrients to rivers from diffuse agricultural sources. It also highlights the scale of the challenge in adapting to future extreme weather events under a changing climate

On the global hydration kinetics of tricalcium silicate cement

We reconsider a number of measurements for the overall hydration kinetics of tricalcium silicate pastes having an initial water to cement weight ratio close to 0.5. We find that the time dependent ratio of hydrated and unhydrated silica mole numbers can be well characterized by two power-laws in time, $x/(1-x)\sim (t/t_x)^\psi$. For early times $t < t_x$ we find an `accelerated' hydration ($\psi = 5/2$) and for later times $t > t_x$ a `deaccelerated' behavior ($\psi = 1/2$). The crossover time is estimated as $t_x \approx 16 hours$. We interpret these results in terms of a global second order rate equation indicating that (a) hydrates catalyse the hydration process for $t<t_x$, (b) they inhibit further hydration for $t > t_x$ and (c) the value of the associated second order rate constant is of magnitude 6x10^{-7} - 7x10^{-6} liter mol^{-1} s^{-1}. We argue, by considering the hydration process actually being furnished as a diffusion limited precipitation that the exponents $\psi = 5/2$ and $\psi = 1/2$ directly indicate a preferentially `plate' like hydrate microstructure. This is essentially in agreement with experimental observations of cellular hydrate microstructures for this class of materials.Comment: RevTeX macros, 6 pages, 4 postscript figure

Shift Towards P Limitation with N Deposition?

Atmospheric nitrogen (N) deposition is altering biogeochemical cycling in forests and interconnected lakes of the northeastern US, and may shift nutrient limitation from N toward other essential elements, such as phosphorus (P). Whether this shift is occurring relative to N deposition gradients across the northeastern US has not been investigated. We used datasets for the northeastern US and the Adirondack sub-region to evaluate whether P limitation is increasing where N deposition is high at two geographic scales, based on N:P mass ratios. Using a model- selection approach, we determined that foliar N for dominant tree species and lake dissolved inorganic N (DIN) increased coincident with increasing N deposition, independent of relationships between foliar N or lake DIN and precipitation or temperature. Foliar P also increased with N deposition across the northeastern US for seven of eight deciduous species, but changed less across the Adirondacks. Foliar N:P therefore declined at the highest levels of N deposition for most deciduous species across the region (remaining nearly constant for most conifers and increasing only for black cherry and hemlock), but increased across all species in the Adirondacks. Ratios between DIN and total P (DIN:TP) in lakes were unrelated to N deposition regionally but increased across the Adirondacks. Thus, nutrient limitation patterns shifted from N toward P for dominant trees, and further toward P for predominantly P-limited lakes, at the sub-regional but not regional scale. For the northeastern US overall, accumulated N deposition may be insufficient to drive nutrient limitation from N toward P; alternatively, elements other than P (e.g., calcium, magnesium) may become limiting as N accumulates. The consistent Adirondack foliar and lake response could provide early indication of shifts toward P limitation within the northeastern US, and together with regional patterns, suggests that foliar chemistry could be a predictor of lake chemistry in the context of N deposition across the region

Nonperturbative Effects from the Resummation of Perturbation Theory

Using the general argument in Borel resummation of perturbation theory that links the divergent perturbation theory to the nonperturbative effect we argue that the nonperturbative effect associated with the perturbation theory should have a branch cut only along the positive real axis in the complex coupling plane. The component in the weak coupling expansion of the nonperturbative amplitude, which usually includes the leading term in the weak coupling expansion, that gives rise to the branch cut can be calculated in principle from the perturbation theory combined with some exactly calculable properties of the nonperturbative effect. The realization of this mechanism is demonstrated in the double well potential and the two-dimensional O(N) nonlinear sigma model. In these models the leading term in weak coupling of the nonperturbative effect can be obtained with good accuracy from the first terms of the perturbation theory. Applying this mechanism to the infrared renormalon induced nonperturbative effect in QCD, we suggest some of the QCD condensate effects can be calculated in principle from the perturbation theory.Comment: 21 Pages, 1 Figure; To appear in Phys Rev

Towards a fullerene-based quantum computer

Molecular structures appear to be natural candidates for a quantum technology: individual atoms can support quantum superpositions for long periods, and such atoms can in principle be embedded in a permanent molecular scaffolding to form an array. This would be true nanotechnology, with dimensions of order of a nanometre. However, the challenges of realising such a vision are immense. One must identify a suitable elementary unit and demonstrate its merits for qubit storage and manipulation, including input / output. These units must then be formed into large arrays corresponding to an functional quantum architecture, including a mechanism for gate operations. Here we report our efforts, both experimental and theoretical, to create such a technology based on endohedral fullerenes or 'buckyballs'. We describe our successes with respect to these criteria, along with the obstacles we are currently facing and the questions that remain to be addressed.Comment: 20 pages, 13 figs, single column forma