Author Archives: Dr Christian P. H. Salas

In this note, I study a constrained maximisation approach seeking to clarify the growth history of microscopic water droplets which undergo runaway aggregation to become raindrops. Using this approach, and the setup outlined in a previous note about a 2016 paper in Physical Review Letters on the large deviation analysis of rapid-onset rain showers (referredContinue reading “Study of a Lagrangian function approach to analysing raindrop growth”

In a previous note, I outlined a mathematical model of rapid-onset rain formation published by Michael Wilkinson in Physical Review Letters, which will be referred to as MW2016 herein. (Full reference: Wilkinson, M., 2016, Large Deviation Analysis of Rapid Onset of Rain Showers, Phys. Rev. Lett. 116, 018501). In the present note, I want toContinue reading “Monomer injections leading to oscillatory behaviour in a mathematical model of rain formation”

Latin has played a significant role in the history and development of mathematics. For example, an interesting article by Richard Oosterhoff on Neo-Latin Mathematics indicates the extent to which Latin has helped to promote the prestige and circulation of mathematical work since medieval times. There is a huge amount of mathematics written in Latin freelyContinue reading “A note on mathematics in Latin”

A 2016 paper by M Wilkinson in Physical Review Letters suggests that large-deviation theory is a suitable framework for studying the phenomenon of unexpectedly rapid rain formation in collector-drop collision processes. Wilkinson derives asymptotic approximation formulae for a set of exact large-deviation functions in the collector-drop model, such as the cumulant generating function and theContinue reading “Accurate analytical approximation formulae for large-deviation analysis of rain formation”

The following is a record of some notes I made on the main mathematical developments in Michael Wilkinson’s 2016 Physical Review Letter on the large deviation analysis of rapid-onset rain showers (reference: Wilkinson, M., 2016, Large Deviation Analysis of Rapid Onset of Rain Showers, Phys. Rev. Lett. 116, 018501). This publication will be referred toContinue reading “A Phys. Rev. Letter by M Wilkinson on large deviation analysis of rapid-onset rainfall”

A paper by Sen and Balakrishnan pointed out that Lagrange’s polynomial interpolation formula can be used to construct relatively simple proofs of some complicated-looking sum/product identities. (Sen, A., Balakrishnan, N., 1999, Convolution of geometrics and a reliability problem, Statistics and Probability Letters, 43, pp. 421-426). In this note I want to make a quick recordContinue reading “Using Lagrange’s interpolation formula to prove complicated sum/product identities”

I needed to find an extremum of a function of the form by choice of , ideally obtaining a closed form solution for the critical value . The precise context is not relevant here – I just want to record an asymptotic approximation trick I was able to use to solve this problem. Differentiating andContinue reading “Using an asymptotic approximation to obtain a closed form solution”

In this note I want to quickly go through the motions of applying the Bromwich integral (i.e., inverse Laplace transform) to the cumulant generating function of an exponential variate with probability density function for . The cumulant generating function is the log-Laplace transform of , obtained as Note that we need to assume for theContinue reading “Applying the Bromwich integral to a cumulant generating function”

Let the Fourier transform of a function be The corresponding inverse Fourier transform would then be As an exercise, what I want to do in this note is derive the characteristic function of an exponential random variate with probability density function for . The required characteristic function is the Fourier transform of this density function.Continue reading “Inverse Fourier transform of a characteristic function”

In this note I want to record an interesting issue I have noticed with regard to how the form of the large deviation entropy function can `flip’ depending on the way the underlying cumulant generating function is formulated. The cumulant generating function of a random variable is the natural log of its moment generating function.Continue reading “Two versions of the exponential cumulant generating function”