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During his Hamilton lecture in Dublin recently, Fields medalist Martin Hairer made a passing mention of the “Two Envelopes Paradox”. This is a well-known problem in probability theory that has led to much misunderstanding. It was originally developed in 1912 by the leading German number theorist Edmund Landau (see Gorroochurn, 2012). It is frequently discussed on the web, with much misunderstanding and confusion. I will try to avoid adding to that.
Newton’s law of gravitation describes how two celestial bodies orbit one another, each tracing out an elliptical path. But this is imprecise: the theory of general relativity shows that two such bodies radiate energy away in the form of gravitational waves (GWs), and spiral inwards until they eventually collide.
Warning sign, described by Thomas Moore as a “geeky insider GR joke” [image from Moore, 2013].
The rational numbers are countable: they can be put into one-to-one correspondence with the natural numbers. In previous articles we showed how the rationals can be presented as a list that includes each rational precisely once. One approach leads to the Farey Sequences. A second, related, approach gives us the Stern-Brocot Tree. Here, we introduce another tree structure, The Calkin-Wilf Tree.
Continue reading ‘Listing the Rational Numbers III: The Calkin-Wilf Tree’
At 2:00 AM on Sunday 28 October the clocks throughout Europe will be set back one hour, reverting to Standard Time. In many countries, the clocks are put forward one hour in Spring and set back to Standard Time in the Autumn. Daylight saving time gives brighter evenings in Summer.
In Summer, the mornings are already bright before most of us wake up but, in Winter, the mornings would be too dark unless we reverted to Standard Time.
Continue reading ‘Saving Daylight with Hip-hop Time: a Modest Proposal’
The rational numbers are countable: they can be put into one-to-one correspondence with the natural numbers. But it is not obvious how to construct a list that is sure to contain every rational number precisely once. In a previous post we described the Farey Sequences. Here we examine another, related, approach.
Continue reading ‘Listing the Rational Numbers II: The Stern-Brocot Tree’
We know, thanks to Georg Cantor, that the rational numbers — ratios of integers — are countable: they can be put into one-to-one correspondence with the natural numbers.
Continue reading ‘Listing the Rational Numbers: I. Farey Sequences’
Walking in Hyde Park recently, I spied what appeared to be a huge red pyramid in the middle of the Serpentine. On closer approach, and with a changing angle of view, it became clear that it was prismatic in shape, composed of numerous barrels in red, blue and purple.
Changing perspective on approach to the Mastaba
