Statistics of the first passage time of Brownian motion conditioned by maximum value or area

We derive the moments of the first passage time for Brownian motion conditioned by either the maximum value or the area swept out by the motion. These quantities are the natural counterparts to the moments of the maximum value and area of Brownian excursions of fixed duration, which we also derive for completeness within the same mathematical framework. Various applications are indicated.Comment: 29 pages, 4 figures include

The first-passage area for drifted Brownian motion and the moments of the Airy distribution

An exact expression for the distribution of the area swept out by a drifted Brownian motion till its first-passage time is derived. A study of the asymptotic behaviour confirms earlier conjectures and clarifies their range of validity. The analysis also leads to a simple closed-form solution for the moments of the Airy distribution.Comment: 13 page

Does the use of intranasal oxytocin in the treatment of women with anorexia nervosa improve recognition of emotional expression compared to placebo?

OBJECTIVE: The objective of this selective EBM review is to determine whether or not “Does the use of intranasal oxytocin in the treatment of women with anorexia nervosa improve recognition of emotional expression compared to placebo?” STUDY DESIGN: A systematic review of three studies that were published in English between 2014-2018. DATA SOURCES: Three randomized controlled trials (RCTs), two with a crossover design, that were published in peer-reviewed journals and found via PubMed. Articles were chosen based on pertinence to the clinical question. OUTCOMES MEASURED: Improvement in the ability to recognize facial expressions were measured through the Reading the Mind in the Eyes Test (RMET) and attentional bias scores from a visual probe test. Images for the visual probe test were gathered from the Korean Facial Expressions of Emotion collection. RESULTS: Kim et al. conducted a cross-over designed study that found increased vigilance to anger during administration of intranasal oxytocin. The calculated t-score was -2.847 (df = 30). This calculation is statistically significant with a p-value of 0.008. A study performed by Leppanen et al. found a mean change 土 standard deviation in accuracy on RMET as 78.83 土 13.74 for women using a placebo. The change with regards to the intranasal oxytocin spray was 81.77 土 10.67. This study did not include a calculation for statistical significance. Russell et al. found in an RCT that the mean change in accuracy in performance on the RMET was of 1.6 土3.1 in the placebo group as contrasted to 0.1 土2.0 in the experimental group. This study also failed to provide a statistical significance calculation. CONCLUSION: Evidence presented in this review was inconclusive at determining whether intranasal oxytocin was effective at improving emotional recognition in women with anorexia nervosa. Statistical significance with a large treatment effect was shown in the study by Kim et al. in regards to increased vigilance of an angry expression. The studies by Leppanen et al. and Russell et al. did not provide a measure of statistical significance. Further research is needed to determine efficacy of intranasal oxytocin

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

On the Inelastic Collapse of a Ball Bouncing on a Randomly Vibrating Platform

We study analytically the dynamics of a ball bouncing inelastically on a randomly vibrating platform, as a simple toy model of inelastic collapse. Of principal interest are the distributions of the number of flights n_f till the collapse and the total time \tau_c elapsed before the collapse. In the strictly elastic case, both distributions have power law tails characterised by exponents which are universal, i.e., independent of the details of the platform noise distribution. In the inelastic case, both distributions have exponential tails: P(n_f) ~ exp[-\theta_1 n_f] and P(\tau_c) ~ exp[-\theta_2 \tau_c]. The decay exponents \theta_1 and \theta_2 depend continuously on the coefficient of restitution and are nonuniversal; however as one approches the elastic limit, they vanish in a universal manner that we compute exactly. An explicit expression for \theta_1 is provided for a particular case of the platform noise distribution.Comment: 32 page

Precise Asymptotics for a Random Walker's Maximum

We consider a discrete time random walk in one dimension. At each time step the walker jumps by a random distance, independent from step to step, drawn from an arbitrary symmetric density function. We show that the expected positive maximum E[M_n] of the walk up to n steps behaves asymptotically for large n as, E[M_n]/\sigma=\sqrt{2n/\pi}+ \gamma +O(n^{-1/2}), where \sigma^2 is the variance of the step lengths. While the leading \sqrt{n} behavior is universal and easy to derive, the leading correction term turns out to be a nontrivial constant \gamma. For the special case of uniform distribution over [-1,1], Coffmann et. al. recently computed \gamma=-0.516068...by exactly enumerating a lengthy double series. Here we present a closed exact formula for \gamma valid for arbitrary symmetric distributions. We also demonstrate how \gamma appears in the thermodynamic limit as the leading behavior of the difference variable E[M_n]-E[|x_n|] where x_n is the position of the walker after n steps. An application of these results to the equilibrium thermodynamics of a Rouse polymer chain is pointed out. We also generalize our results to L\'evy walks.Comment: new references added, typos corrected, published versio

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

Long Term Outcome of a Parenting Group Intervention: An Investigation Into the Generalisation and Maintenance of Treatment Effects Post Intervention

Abstract not provided