Archive for the 'Occasional' Category

The Ross-Littlewood Paradox

Published April 9, 2020 Occasional Leave a Comment
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Ross-Littlewood-01

Ross-Littlewood Paradox [Image from Steemit website:  here. ]

A most perplexing paradox appeared in Littlewood’s book A Mathematician’s Miscellany. It was later analysed in detail by Sheldon Ross in his 1988 book A First Course in Probability.

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Bang! Bang! Bang! Explosively Large Numbers

Published March 26, 2020 Occasional Leave a Comment
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Bang-1224578

Typical Comic-book `bang’ mark [Image from vectorstock ].

Enormous numbers pop up in both mathematics and physics. The order of the monster group, the largest of the 26 sporadic groups, is

\displaystyle 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000

which is approximately {8\times 10^{53}}. The number of atoms in the universe is estimated to be about {10^{80}}. When we consider permutations of large sets, even more breadth-taking numbers emerge.

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Samuel Haughton and the Twelve Faithless Hangmaids

Published March 12, 2020 Occasional Leave a Comment
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In his study of humane methods of hanging, Samuel Haughton (1866) considered the earliest recorded account of execution by hanging (see Haughton’s Drop on this site). In the twenty-second book of the Odyssey, Homer described how the twelve faithless handmaids of Penelope “lay by night enfolded in the arms of the suitors” who were vying for Penelope’s hand in marriage. Her son Telemachus, with the help of his comrades, hanged all twelve handmaids on a single rope.

Hangmaids-05-COL

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Zhukovsky’s Airfoil

Published February 27, 2020 Occasional Leave a Comment
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A simple transformation with remarkable properties was used by Nikolai Zhukovsky around 1910 to study the flow around aircraft wings. It is defined by

\displaystyle \omega = \textstyle{\frac{1}{2}}\displaystyle{\left(z +\frac{1}{z}\right)}

and is usually called the Joukowsky Map. We begin with a discussion of the theory of fluid flow in two dimensions. Readers familiar with 2D potential flow may skip to the section Joukowsky Airfoil.

Zhukovsky-Animation

Visualization of airflow around a Joukowsky airfoil. Image generated using code on this website.

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A Ring of Water Shows the Earth’s Spin

Published February 13, 2020 Occasional Leave a Comment
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Around 1913, while still an undergraduate, American physicist Arthur Compton described an experiment to demonstrate the rotation of the Earth using a simple laboratory apparatus.

Comptons-Generator-SciAm2

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The Rambling Roots of Wilkinson’s Polynomial

Published January 30, 2020 Occasional Leave a Comment
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Finding the roots of polynomials has occupied mathematicians for many centuries. For equations up to fourth order, there are algebraic expressions for the roots. For higher order equations, many excellent numerical methods are available, but the results are not always reliable.

Wilkinson-Polynomial

A 10th-order polynomial (blue) and a slightly perturbed version, with the coefficient of {x^9} changed by one part in a million.

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Adjoints of Vector Operators

Published January 23, 2020 Occasional Leave a Comment
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We take a fresh look at the vector differential operators grad, div and curl. There are many vector identities relating these. In particular, there are two combinations that always yield zero results:

\displaystyle \begin{array}{rcl} \mathbf{curl}\ \mathbf{grad}\ \chi &\equiv& 0\,, \quad \mbox{for all scalar functions\ }\chi \\ \mathrm{div}\ \mathbf{curl}\ \boldsymbol{\psi} &\equiv& 0\,, \quad \mbox{for all vector functions\ }\boldsymbol{\psi} \end{array}

Question: Is there a connection between these identities?

CurlGradAndDivCurl

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