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who introduced me to time series analysis and much more besides Abstract This thesis is about sampling theory and methods for analysing signals that have been sampled at irregularly spaced points. Irregular sampling may arise naturally (examples of its occurrence may be found in geophysics, tomography, astronomy, and laser anemometry). In many cases it(More)
The area swept out under a one-dimensional Brownian motion till its first-passage time is analysed using a Fokker-Planck technique. We obtain an exact expression for the area distribution for the zero drift case, and provide various asymptotic results for the non-zero drift case, emphasising the critical nature of the behaviour in the limit of vanishing(More)
Let s(N) denote the edge length of the smallest square in which one can pack N unit squares. A duality method is introduced to prove that s(6) = s(7) = 3. Let n r be the smallest integer n such that s(n 2 + 1) ≤ n + 1/r. We use an explicit construction to show that n r ≤ 27r 3 /2 + O(r 2), and also that n 2 ≤ 43. 1. Introduction. Erd˝ os and Graham [1](More)
We derive , the joint probability density of the maximum) , (m t M P M and the time at which this maximum is achieved, for a class of constrained Brownian motions. In particular, we provide explicit results for excursions, meanders and reflected bridges associated with Brownian motion. By subsequently integrating over m t M , the marginal density is(More)
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 nth coefficient is expressed as the (n(More)
Normalized functionals of first passage Brownian motion and a curious connection with the maximal relative height of fluctuating interfaces Abstract Direct variational methods are used to find simple approximate solutions of the Thomas–Fermi equations describing the properties of self-gravitating radially symmetric stellar objects both in the(More)
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.(More)
With the ever-increasing focus on obtaining higher device power conversion efficiencies (PCEs) for organic photovoltaics (OPV), there is a need to ensure samples are measured accurately. Reproducible results are required to compare data across different research institutions and countries and translate these improvements to real-world production. In order(More)
We analyse how the area swept out by a Brownian motion up to its first passage time correlates with the first passage time itself, obtaining several exact results in the process. Additionally, we analyse the relationship between the time average of a Brownian motion during a first passage and the maximum value attained. The results, which find various(More)