We consider the motion of a Brownian particle trapped in an arbitrary bounded two or three-dimensional domain, whose boundary is reflecting except for a small absorbing window through which the particle can escape. We use the method of matched asymptotic expansions to calculate the mean first passage time, defined as the time taken for the Brownian particle to escape from the domain through the absorbing window. This is known as the narrow escape problem. Since the mean escape time diverges as the window shrinks, the calculation is a singular perturbation problem. We extend our results to include N absorbing windows of varying length in two dimensions and varying radius in three dimensions. We present findings in two dimensions for the unit disk, unit square and ellipse and in three dimensions for the unit sphere. The narrow escape problem has various applications in many fields including finance, biology, and statistical mechanics.

Communication security is a very important part of modern life. A crucial aspect of security is the ability to identify with near 100% certainty who is on the other side of a connection. This problem can be overcome through the use of random number generators, which create unique identities for each person in a network. The effectiveness of an identity is directly proportional to how random a generator is. The speed at which a random number can be delivered is a critical factor in the design of a random number generator. This thesis covers the design and fabrication of three ring oscillator based truly random number generators, the first two of which were fabricated in 0.13µ m CMOS technology. The randomness from this type of random number generator originates from phase noise in a ring oscillator. The second and third ring oscillators were designed to have a low slew rate at the inverter switching threshold. The outputs of these designs showed vast increases in timing jitter compared to the first design. The third design exhibited improved randomness with respect to the second design.

We consider the motion of a Brownian particle trapped in an arbitrary bounded two or three-dimensional domain, whose boundary is reflecting except for a small absorbing window through which the particle can escape. We use the method of matched asymptotic expansions to calculate the mean first passage time, defined as the time taken for the Brownian particle to escape from the domain through the absorbing window. This is known as the narrow escape problem. Since the mean escape time diverges as the window shrinks, the calculation is a singular perturbation problem. We extend our results to include N absorbing windows of varying length in two dimensions and varying radius in three dimensions. We present findings in two dimensions for the unit disk, unit square and ellipse and in three dimensions for the unit sphere. The narrow escape problem has various applications in many fields including finance, biology, and statistical mechanics.

Orientador: Elso Drigo Filho / Resumo: Tem-se como objetivo estudar a relação da equação de Fokker-Planck mapeada em uma equação tipo Schrödinger e assim usar supersimetria para resolução de alguns potenciais polinomiais encontrando sua distribuição de probabilidade P(x,t) e o tempo de passagem entre barreiras de potenciais e a partir destes dados compreender melhor o sistema físico proposto. / Abstract: The objective of this work is to study the relationship Fokker-Planck equation a Schrödinger-type equation . Thus, it is used supersymmetry for is to solve some polynomial potential in order to find the probability distribution, P (x, t), and the passage time between barriers of potential. These data prit us a better understand of the proposed physical system. / Mestre

Organisms have movements that are usually modeled by particles’ random walks. Under some mathematical technical assumptions the movements are described by diffusion equations. However, empirical data often show that the movements are not simple random walks. Instead, they are correlated random walks and are described by telegraph equations. This thesis considers telegraph equations with and without bias corresponding to correlated random walks with and without bias. Analytical solutions to the equations with absorbing boundary conditions and their mean passage times are obtained. Numerical simulations of the corresponding correlated random walks are also performed. The simulation results show that the solutions are approximated very well by the corresponding correlated random walks and the mean first passage times are highly consistent with those from simulations on the corresponding random walks. This suggests that telegraph equations can be a good model for organisms with the movement pattern of correlated random walks. Furthermore, utilizing the consistency of mean first passage times, we can estimate the parameters of telegraph equations through the mean first passage time, which can be estimated through experimental observation. This provides biologists an easy way to obtain parameter values. Finally, this thesis analyzes the velocity distribution and correlations of movement steps of amoebas, leaving fitting the movement data to telegraph equations as future work.

We consider the motion of a Brownian particle trapped in an arbitrary bounded two or three-dimensional domain, whose boundary is reflecting except for a small absorbing window through which the particle can escape. We use the method of matched asymptotic expansions to calculate the mean first passage time, defined as the time taken for the Brownian particle to escape from the domain through the absorbing window. This is known as the narrow escape problem. Since the mean escape time diverges as the window shrinks, the calculation is a singular perturbation problem. We extend our results to include N absorbing windows of varying length in two dimensions and varying radius in three dimensions. We present findings in two dimensions for the unit disk, unit square and ellipse and in three dimensions for the unit sphere. The narrow escape problem has various applications in many fields including finance, biology, and statistical mechanics. / Science, Faculty of / Mathematics, Department of / Graduate

<div><div><p>Important cellular processes such as migration, differentiation, and development often rely on precise timing. Yet, the molecular machinery that regulates timing is inherently noisy. How do cells achieve precise timing with noisy components? We investigate this question using a first-passage-time approach, for an event triggered by a molecule that crosses an abundance threshold. We investigate regulatory strategies that decrease the timing noise of molecular events. We look at several strategies which decrease the noise: i) Regulation performed by an accumulating activator, ii) Regulation dues to a degrading repressor, iii) Auto-regulation and the presence of feedback. We find that either activation or repression outperforms an unregulated strategy. The optimal regulation corresponds to a nonlinear increase in the amount of the target molecule over time, arises from a tradeoff between minimizing the timing noise of the regulator and that of the target molecule itself, and is robust to additional effects such as bursts and cell division. Our results are in quantitative agreement with the nonlinear increase and low noise of <i>mig-1</i> gene expression in migrating neuroblast cells during <i>Caenorhabditis elegans</i> development. These findings suggest that dynamic regulation may be a simple and powerful strategy for precise cellular timing.</p><p>Autoregulatory feedback increases noise. Yet, we find that in the presence of regulation by a second species, autoregulatory feedback decreases noise. To explain this finding, we develop a method to calculate the optimal regulation function that minimizes the timing noise. Our method reveals that the combination of feedback and regulation minimizes noise by maximizing the number of molecular events that must happen in sequence before a threshold is crossed. We compute the optimal timing precision for all two-node networks with regulation and feedback, derive a generic lower bound on timing noise, and compare our results with the neuroblast migration during <i>C. elegans</i> development, as well as two mutants. We finds that indeed our model is aligned with the experimental findings.</p></div></div><div><p>Furthermore, we apply our framework of temporal regulation to explain how the stopping point of the migrating cells in <i>C. elegans</i> depends on the body size. Considering temporal regulation, we find the termination point of the cell for various larval sizes. We discuss three possible mechanisms: i) No compensation; here the migration velocity is constant across the mutants of <i>C. elegans</i>, and this results in the migration distance to be constant but the relative position to be different across various sizes; ii) Total compensation; here the velocity is compensated with body size, hence resulting in the same relative position of cells across mutants; and iii) Partial compensation; here the velocity of migration is correlated with body size to some degree, resulting in a non-linear relationship between termination point and body size. We find that our partial compensation model is consistent with experimental observations of cell termination.</p><p>Finally, we look at the detection of traveling waves by single-celled organisms. Cells must use temporal and spatial information to sense the direction of traveling waves, as seen in cAMP detection by the <i>amoeba </i><i>Dictyostelium</i>. If a cell only uses spatial information to sense the direction of the wave then the cell will move forward when the wave hits the front of the cell, and move backward when the wave hits the back of the cell, resulting in neutral movement. Cells must use temporal information along with spatial information to effectively move towards the source. Here we develop a mechanism by which cells are able to integrate the spatial and temporal information through a system of inhibitors. We find the optimal time to release the inhibitors for maximizing the precision of directional sensing.</p></div>

In a heterogenous environment, an animal will increase its search effort in areas where resources are abundant. This behavior can be detected in a path by a decrease in speed, an increase in tortuosity, or both. First passage time, the amount of time required for an animal to traverse a circle of a given radius, or buffer, is a common metric for quantifying spatial and temporal changes along a path. Historical methodology involving first passage time limits the utility of this metric. Here we instead follow the methodology put forth by Street et al. (2018) and use a power-law model to characterize the relationship between first passage time and the scale of the first passage time buffer radii. We then test the model’s applicability across multiple movement modes using simulated data and further explore its utility by applying it to a dataset of deer movement and the associated landscape data.

No description available.

A thesis submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Doctor of Philosophy. Johannesburg, December 2014. / Industrial units cannot operate without failure forever. When the operation of a unit deviates from industrial standards, it is considered to have failed. The time from the moment a unit enters service until it fails is its lifetime. Within reliability and often in life data analysis in general, lifetime is the event of interest. For highly reliable units, accelerated life testing is required to obtain lifetime data quickly. Accelerated tests where failure is not instantaneous, but the end point of an underlying degradation process are considered. Failure during testing occurs when the performance of the unit falls to some specified threshold value such that the unit fails to meet industrial specifications though it has some residual functionality (degraded failure) or decreases to a critical failure level so that the unit cannot perform its function to any degree (critical failure). This problem formulation satisfies the random signs property, a notable competing risks formulation originally developed in maintenance studies but extended to accelerated testing here. Since degraded and critical failures are linked through the degradation process, the open problem of modelling dependent competing risks is discussed. A copula model is assumed and expert opinion is used to estimate the copula. Observed occurrences of degraded and critical failure times are interpreted as times when the degradation process first crosses failure thresholds and are therefore postulated to be distributed as inverse Gaussian. Based on the estimated copula, a use-level unit lifetime distribution is extrapolated from test data. Reliability metrics from the extrapolated use-level unit lifetime distribution are found to differ slightly with respect to different degrees of stochastic dependence between the risks. Consequently, a degree of dependence between the risks that is believed to be realistic to admit is considered an important factor when estimating the use-level unit lifetime distribution from test data. Keywords: Lifetime; Accelerated testing; Competing risks; Copula; First passage time.