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

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

An engineered Tetrahymena tRNA(Gln) for in vivo incorporation of unnatural amino acids into proteins by nonsense suppression

A new tRNA, THG73, has been designed and evaluated as a vehicle for incorporating unnatural amino acids site-specifically into proteins expressed in vivo using the stop codon suppression technique. The construct is a modification of tRNAGln(CUA) from Tetrahymena thermophila, which naturally recognizes the stop codon UAG. Using electrophysiological studies of mutations at several sites of the nicotinic acetylcholine receptor, it is established that THG73 represents a major improvement over previous nonsense suppressors both in terms of efficiency and fidelity of unnatural amino acid incorporation. Compared with a previous tRNA used for in vivo suppression, THG73 is as much as 100-fold less likely to be acylated by endogenous synthetases of the Xenopus oocyte. This effectively eliminates a major concern of the in vivo suppression methodology, the undesirable incorporation of natural amino acids at the suppression site. In addition, THG73 is 4-10-fold more efficient at incorporating unnatural amino acids in the oocyte system. Taken together, these two advances should greatly expand the range of applicability of the in vivo nonsense suppression methodology

Maximum relative height of one-dimensional interfaces : from Rayleigh to Airy distribution

We introduce an alternative definition of the relative height h^\kappa(x) of a one-dimensional fluctuating interface indexed by a continuously varying real paramater 0 \leq \kappa \leq 1. It interpolates between the height relative to the initial value (i.e. in x=0) when \kappa = 0 and the height relative to the spatially averaged height for \kappa = 1. We compute exactly the distribution P^\kappa(h_m,L) of the maximum h_m of these relative heights for systems of finite size L and periodic boundary conditions. One finds that it takes the scaling form P^\kappa(h_m,L) = L^{-1/2} f^\kappa (h_m L^{-1/2}) where the scaling function f^\kappa(x) interpolates between the Rayleigh distribution for \kappa=0 and the Airy distribution for \kappa=1, the latter being the probability distribution of the area under a Brownian excursion over the unit interval. For arbitrary \kappa, one finds that it is related to, albeit different from, the distribution of the area restricted to the interval [0, \kappa] under a Brownian excursion over the unit interval.Comment: 25 pages, 4 figure

The comprehensive cohort model in a pilot trial in orthopaedic trauma

Background: The primary aim of this study was to provide an estimate of effect size for the functional outcome of operative versus non-operative treatment for patients with an acute rupture of the Achilles tendon using accelerated rehabilitation for both groups of patients. The secondary aim was to assess the use of a comprehensive cohort research design (i.e. a parallel patient-preference group alongside a randomised group) in improving the accuracy of this estimate within an orthopaedic trauma setting. Methods: Pragmatic randomised controlled trial and comprehensive cohort study within a level 1 trauma centre. Twenty randomised participants (10 operative and 10 non-operative) and 29 preference participants (3 operative and 26 non-operative). The ge range was 22-72 years and 37 of the 52 patients were men. All participants had an acute rupture of their Achilles tendon and no other injuries. All of the patients in the operative group had a simple end-to-end repair of the tendon with no augmentation. Both groups then followed the same eight-week immediate weight-bearing rehabilitation programme using an off-the-shelf orthotic. The disability rating index (DRI; primary outcome), EQ-5D, Achilles Total Rupture Score and complications were assessed ed at two weeks, six weeks, three months, six months and nine months after initial injury. Results: At nine months, there was no significant difference in DRI between patients randomised to operative or non-operative management. There was no difference in DRI between the randomised group and the parallel patient preference group. The use of a comprehensive cohort of patients did not provide useful additional information as to the treatment effect size because the majority of patients chose non-operative management. Conclusions: Recruitment to clinical trials that compare operative and non-operative interventions is notoriously difficult; especially within the trauma setting. Including a parallel patient preference group to create a comprehensive cohort of patients has been suggested as a way of increasing the power of such trials. In our study, the comprehensive cohort model doubled the number of patients involved in the study. However, a strong preference for non-operative treatment meant that the increased number of patients did not significantly increase the ability of the trial to detect a difference between the two interventions

Phonon-drag effects on thermoelectric power

We carry out a calculation of the phonon-drag contribution $S_g$ to the thermoelectric power of bulk semiconductors and quantum well structures for the first time using the balance equation transport theory extended to the weakly nonuniform systems. Introducing wavevector and phonon-mode dependent relaxation times due to phonon-phonon interactions, the formula obtained can be used not only at low temperatures where the phonon mean free path is determined by boundary scattering, but also at high temperatures. In the linear transport limit, $S_g$ is equivalent to the result obtained from the Boltzmann equation with a relaxation time approximation. The theory is applied to experiments and agreement is found between the theoretical predictions and experimental results. The role of hot-electron effects in $S_g$ is discussed. The importance of the contribution of $S_g$ to thermoelectric power in the hot-electron transport condition is emphasized.Comment: 8 pages, REVTEX 3.0, 7 figures avilable upon reques

Story in health and social care

This paper offers a brief consideration of how narrative, in the form of people‟s own stories, potentially figures in health and social care provision as part of the impulse towards patient-centred care. The rise of the epistemological legitimacy of patients‟ stories is sketched here. The paper draws upon relevant literature and original writing to consider the ways in which stories can mislead as well as illuminate the process of making individual treatment care plans