Archive for April, 2023

Broken Symmetry and Atmospheric Waves, 1

Published April 27, 2023 Occasional Closed
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Part I: Vertically propagating Waves and the Stratospheric Window

Symmetry is a powerful organising principle in physics. It is a central concept in both classical and quantum mechanics and has a key role in the standard model. When symmetry is violated, interesting things happen. The book Shattered Symmetry by Pieter Thyssen and Arnout Ceulemans discusses many aspects and examples of broken symmetry.

In this article (and the following one) we look at some consequences of broken symmetry in atmospheric dynamics. In particular, we see how mountains (which are stationary!) can generate waves in the atmosphere that propagate towards the west. We will look at this unexpected breaking of symmetry and try to explain it.

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DLWP: A New Age of Weather Forecasting

Published April 20, 2023 Irish Times Closed
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Hurricane Frances

Before the age of computers, weather forecasters analysed observations plotted on paper charts, drew isobars and other features and — based on their previous knowledge and experience — constructed charts of conditions at a future time, often one day ahead. They combined observational data and rules of thumb based on physical principles to predict what would follow from a given state. The results were undependable for two main reasons: the data were sparse, and the empirical rules were unreliable  [TM244 or search for “thatsmaths” at irishtimes.com].

For the past sixty years or so, forecasts have been based on computer models that numerically solve the mathematical equations expressing the physical laws. This approach is radically different but, after a shaky start, the numerical weather prediction (NWP) models have become remarkably accurate, with forecasting skill increasing by about one day each decade. Now there are signs of a return to the analogue, data-driven methods.

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Numbers Without Ones: Chorisenic Sets

Published April 13, 2023 Occasional Closed
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Left: Count of elements of set {\mathbf{Xe}_{10}}. Right: partial density of set {\mathbf{Xe}_{10}}.

There is no end to the variety of sets of natural numbers. Sets having all sorts of properties have been studied and many more remain to be discovered. In this note we study the set of natural numbers for which the decimal digit 1 does not occur.

Google Translate on my mobile phone gives the Greek for “without ones” as {\chi\omega\rho\iota'\varsigma} {\varepsilon'\nu\alpha} or choris ena, so let us call a set of “oneless numbers” a chorisenic set. Continue reading ‘Numbers Without Ones: Chorisenic Sets’

Amusical Permutations and Unsettleable Problems

Published April 6, 2023 Occasional Closed
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John Horton Conway (1937–2020) in 2009 [Photo (c) Denise Applewhite, Princeton University]

In a memorial tribute in the Notices of the American Mathematical Society (Ryba, et al, 2022), Dierk Schleicher wrote of how he convinced John Conway to publish a paper, “On unsettleable arithmetical problems”, which included a discussion of the Amusical Permutations. This paper, which discusses arithmetical statements that are almost certainly true but likely unprovable, was selected for the 2014 edition of “The Best Writing on Mathematics,” published by Princeton University Press [Pitici, 2014]. Amusical Permutations was an attempt to find a simple sequence whose behaviour was undecidable.

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