Long- and short-time asymptotics of the first-passage time of the Ornstein-Uhlenbeck and other mean-reverting processes

The first-passage problem of the Ornstein-Uhlenbeck process to a boundary is a long-standing problem with no known closed-form solution except in specific cases. Taking this as a starting-point, and extending to a general mean-reverting process, we investigate the long- and short-time asymptotics using a combination of Hopf-Cole and Laplace transform techniques. As a result we are able to give a single formula that is correct in both limits, as well as being exact in certain special cases. We demonstrate the results using a variety of other models

Analytical approximation to the multidimensional Fokker--Planck equation with steady state

The Fokker--Planck equation is a key ingredient of many models in physics, and related subjects, and arises in a diverse array of settings. Analytical solutions are limited to special cases, and resorting to numerical simulation is often the only route available; in high dimensions, or for parametric studies, this can become unwieldy. Using asymptotic techniques, that draw upon the known Ornstein--Uhlenbeck (OU) case, we consider a mean-reverting system and obtain its representation as a product of terms, representing short-term, long-term, and medium-term behaviour. A further reduction yields a simple explicit formula, both intuitive in terms of its physical origin and fast to evaluate. We illustrate a breadth of cases, some of which are `far' from the OU model, such as double-well potentials, and even then, perhaps surprisingly, the approximation still gives very good results when compared with numerical simulations. Both one- and two-dimensional examples are considered.Comment: Updated version as publishe

The effect of free-stream turbulence on heat transfer to a strongly accelerated turbulent boundary layer

Free-stream turbulence effects on heat transfer to strongly accelerated turbulent boundary laye

An exactly solvable self-convolutive recurrence

We consider a self-convolutive recurrence whose solution is the sequence of coefficients in the asymptotic expansion of the logarithmic derivative of the confluent hypergeometic function $U(a,b,z)$. By application of the Hilbert transform we convert this expression into an explicit, non-recursive solution in which the $n$th coefficient is expressed as the $(n-1)$th moment of a measure, and also as the trace of the $(n-1)$th iterate of a linear operator. Applications of these sequences, and hence of the explicit solution provided, are found in quantum field theory as the number of Feynman diagrams of a certain type and order, in Brownian motion theory, and in combinatorics

Distribution of the time at which the deviation of a Brownian motion is maximum before its first-passage time

We calculate analytically the probability density $P(t_m)$ of the time $t_m$ at which a continuous-time Brownian motion (with and without drift) attains its maximum before passing through the origin for the first time. We also compute the joint probability density $P(M,t_m)$ of the maximum $M$ and $t_m$. In the driftless case, we find that $P(t_m)$ has power-law tails: $P(t_m)\sim t_m^{-3/2}$ for large $t_m$ and $P(t_m)\sim t_m^{-1/2}$ for small $t_m$. In presence of a drift towards the origin, $P(t_m)$ decays exponentially for large $t_m$. The results from numerical simulations are in excellent agreement with our analytical predictions.Comment: 13 pages, 5 figures. Published in Journal of Statistical Mechanics: Theory and Experiment (J. Stat. Mech. (2007) P10008, doi:10.1088/1742-5468/2007/10/P10008

Exploring connected learning spaces in teacher education

© 2013 Rachel Perry, Kimberley Pressick-Kilborn, Matthew Kearney. This paper reports on outcomes from a study that explored how connected learning spaces, mediated by videoconference technology, enabled real-world engagement in pre-service teacher education. Student teachers in drama and science education participated in the study, which involved varied connections with school children and their classroom teachers. Key themes that emerged were underpinned by a consideration of authentic learning: student teachers’ observations of teacher practices; enactment of multiple roles; and exposure to diverse and timely feedback. Implications for the design of discipline specific on-campus activities are considered in relation to how they inform effective integration of videoconference technology for real-world, professional engagement in teacher education

Pharmacological treatments for Friedreich ataxia

BACKGROUND: Friedreich ataxia is a rare inherited autosomal recessive neurological disorder, characterised initially by unsteadiness in standing and walking, slowly progressing to wheelchair dependency usually in the late teens or early twenties. It is associated with slurred speech, scoliosis, and pes cavus. Heart abnormalities cause premature death in 60% of people with the disorder. There is no easily defined clinical or biochemical marker and no known treatment. This is the second update of a review first published in 2009 and previously updated in 2012. OBJECTIVES: To assess the effects of pharmacological treatments for Friedreich ataxia. SEARCH METHODS: On 29 February 2016 we searched The Cochrane Neuromuscular Specialised Register, CENTRAL, MEDLINE, EMBASE and CINAHL Plus. On 7 March 2016 we searched ORPHANET and TRIP. We also checked clinical trials registers for ongoing studies. SELECTION CRITERIA: We considered randomised controlled trials (RCTs) or quasi-RCTs of pharmacological treatments (including vitamins) in people with genetically-confirmed Friedreich ataxia. The primary outcome was change in a validated Friedreich ataxia neurological score after 12 months. Secondary outcomes were changes in cardiac status as measured by magnetic resonance imaging or echocardiography, quality of life, mild and serious adverse events, and survival. We excluded trials of duration shorter than 12 months. DATA COLLECTION AND ANALYSIS: Three review authors selected trials and two review authors extracted data. We obtained missing data from the two RCTs that met our inclusion criteria. We collected adverse event data from included studies. We used standard methodological procedures expected by Cochrane. MAIN RESULTS: We identified more than 12 studies that used antioxidants in the treatment of Friedreich ataxia, but only two small RCTs, with a combined total of 72 participants, both fulfilled the selection criteria for this review and published results. One of these trials compared idebenone with placebo, the other compared high-dose versus low-dose coenzyme Q10 and vitamin E (the trialists considered the low-dose medication to be the placebo). We identified two other completed RCTs, which remain unpublished; the interventions in these trials were pioglitazone (40 participants) and idebenone (232 participants). Other RCTs were of insufficient duration for inclusion. In the included studies, the primary outcome specified for the review, change in a validated Friedreich ataxia rating score, was measured using the International Co-operative Ataxia Rating Scale (ICARS). The results did not reveal any significant difference between the antioxidant-treated and the placebo groups (mean difference 0.79 points, 95% confidence interval -1.97 to 3.55 points; low-quality evidence). The published included studies did not assess the first secondary outcome, change in cardiac status as measured by magnetic resonance imaging. Both studies reported changes in cardiac measurements assessed by echocardiogram. The ejection fraction was not measured in the larger of the included studies (44 participants). In the smaller study (28 participants), it was normal at baseline and did not change with treatment. End-diastolic interventricular septal thickness showed a small decrease in the smaller of the two included studies. In the larger included study, there was no decrease, showing significant heterogeneity in the study results; our overall assessment of the quality of evidence for this outcome was very low. Left ventricular mass (LVM) was only available for the smaller RCT, which showed a significant decrease. The relevance of this change is unclear and the quality of evidence low. There were no deaths related to the treatment with antioxidants. We considered the published included studies at low risk of bias in six of seven domains assessed. One unpublished included RCT, a year-long study using idebenone (232 participants), published an interim report in May 2010 stating that the study reached neither its primary endpoint, which was change in the ICARS score, nor a key cardiological secondary endpoint, but data were not available for verification and analysis. AUTHORS' CONCLUSIONS: low-quality evidence from two small, published, randomised controlled trials neither support nor refute an effect from antioxidants (idebenone, or a combination of coenzyme Q10 and vitamin E) on the neurological status of people with Friedreich ataxia, measured with a validated neurological rating scale. A large unpublished study of idebenone that reportedly failed to meet neurological or key cardiological endpoints, and a trial of pioglitazone remain unpublished, but on publication will very likely influence quality assessments and conclusions. A single study of idebenone provided low-quality evidence for a decrease in LVM, which is of uncertain clinical significance but of potential importance that needs to be clarified. According to low-quality evidence, serious and non-serious adverse events were rare in both antioxidant and placebo groups. No non-antioxidant agents have been investigated in RCTs of 12 months' duration

Sea-Level Rise Tipping Point Of Delta Survival

The estimated rate of global eustatic sea-level rise (RSLR) associated with the formation of 36 of the world\u27s coastal deltas was calculated for the last 22,000 years. These deltas are located in a variety of environmental settings with respect to tidal range, isostasy, and climate. After correcting the original uncalibrated radiocarbon age estimates to calibrated years, 90% of the deltas appear to have formed at an average age of 8109 +/- 122 before present (BP) and a median age of 7967 BP. This age corresponds to a period of significant deceleration in the RSLR to between 5 mm y(-1) and 10 mm y(-1), and is in agreement with two regional estimates of vegetation growth limits with respect to RSLR. This RSLR tipping point for delta formation can be used to inform forecasts of delta resiliency under conditions of climate change and concomitant SLR. The RSLR is accelerating and will likely be several times higher than the formation tipping point by the end of this century. Hence, the world\u27s deltaic environments are likely to be lost within the same time frame

Area distribution and the average shape of a L\'evy bridge

We consider a one dimensional L\'evy bridge x_B of length n and index 0 < \alpha < 2, i.e. a L\'evy random walk constrained to start and end at the origin after n time steps, x_B(0) = x_B(n)=0. We compute the distribution P_B(A,n) of the area A = \sum_{m=1}^n x_B(m) under such a L\'evy bridge and show that, for large n, it has the scaling form P_B(A,n) \sim n^{-1-1/\alpha} F_\alpha(A/n^{1+1/\alpha}), with the asymptotic behavior F_\alpha(Y) \sim Y^{-2(1+\alpha)} for large Y. For \alpha=1, we obtain an explicit expression of F_1(Y) in terms of elementary functions. We also compute the average profile < \tilde x_B (m) > at time m of a L\'evy bridge with fixed area A. For large n and large m and A, one finds the scaling form = n^{1/\alpha} H_\alpha({m}/{n},{A}/{n^{1+1/\alpha}}), where at variance with Brownian bridge, H_\alpha(X,Y) is a non trivial function of the rescaled time m/n and rescaled area Y = A/n^{1+1/\alpha}. Our analytical results are verified by numerical simulations.Comment: 21 pages, 4 Figure

A study of multiplex data bus techniques for the space shuttle

A comprehensive technology base for the design of a multiplexed data bus subsystem is provided. Extensive analyses, both analytical and empirical, were performed. Subjects covered are classified under the following headings: requirements identification and analysis; transmission media studies; signal design and detection studies; synchronization, timing, and control studies; user-subsystem interface studies; operational reliability analyses; design of candidate data bus configurations; and evaluation of candidate data bus designs