Editorial, April 2018

Editorial, April 2018


On 15 March, the IMA and LMS jointly presented the 2017 David Crighton Medal to former IMA President Professor David Abrahams at the Royal Society. This medal is awarded every two years to an eminent mathematician for services both to mathematics and to the mathematical community. As Director of the Isaac Newton Institute for Mathematical Sciences in Cambridge and former Beyer Professor of Applied Mathematics at the University of Manchester, David is a worthy recipient. He followed this award with a fascinating lecture on Mathematics, Metamaterials and Meteorites.

Maria-Gaetana-Agnesi
Maria Gaetana Agnesi (1718–1799)

Two other notable one-day meetings are worth mentioning here. Firstly, the 12th annual conference IMA Mathematics 2018 was held in London and comprised several talks by prominent speakers including David, current President Professor Alistair Fitt and former Council Member Rear Admiral Nigel Guild. As in previous years, this was a popular and enjoyable occasion that offered welcome opportunities to mingle with other delegates. Secondly, the IMA Early Career Mathematicians’ Spring Conference was postponed due to adverse weather conditions, and will now be held on 21 April at Durham University. The current ECM chair is Dr Chris Baker who, after short spells in academia and industry, secured a Mathematics Teacher Training Scholarship and now teaches mathematics and computer science to secondary and A-level students.

I was delighted to hear that the judging panel decided to award the 2017 Catherine Richards Prize to Professor Alan Champneys for his article Boardmasters, which appeared in last August’s issue. He has published several excellent features as part of the popular Westward Ho! series that began a year ago and this article was particularly impressive. As always, I am grateful to the volunteer judges for their efforts.

Next month, we celebrate the 300th anniversary of the birth of Italian mathematician Maria Gaetana Agnesi. A remarkable person, she was born in Milan on 16 May 1718, was incredibly talented, had a distinguished career and lived to 80 years of age. She was the first woman appointed as a mathematics professor, indeed at the prestigious University of Bologna. She was also a philosopher, theologian and humanitarian, while one of her many siblings, Maria Teresa Agnesi Pinottini (1720–1795), was a famous composer and musician.

Maria was the first woman to write a mathematics handbook, Instituzioni Analitiche ad Uso della Gioventù Italiana (Analytical Institutions for the Use of Italian Youth), in 1748. She intended this to provide a systematic illustration of the different results and theorems of calculus, by combining algebra with analysis. This was the first book that discussed both differential and integral calculus, and appeared in two hefty volumes on finite quantities and infinitesimals. According to [1], the Académie des Sciences in Paris regarded Maria’s handbook as, ‘the most complete and best made treatise’. The website books.google.co.uk enables access to both the Italian original and an English translation by the Reverend John Colson in 1801. He was the Lucasian Professor of Mathematics at Cambridge University and also translated De Methodis Serierum et Fluxionum by Sir Isaac Newton from Latin to English in 1736.

The tale concerning Maria’s professorial appointment is remarkable. Pope Benedict XIV was knowledgeable in mathematics and was highly impressed with this published work [1], to the extent that he personally appointed her as an honorary reader at the University of Bologna. The Pope and the university’s president subsequently invited her to accept the chair of mathematics in 1750, though it is likely that she never actively engaged in this position as she became director of the Hospice Trivulzio of the Blue Nuns in Milan and remained there until her death on 9 January 1799. She was the second
woman ever to be granted a university chair, the first being Italian physicist Laura Bassi (1711–1778).

Although Maria’s writings were expository in nature rather than original theoretical developments, she tackled many challenging problems. Among those, she analysed a cubic curve that had been studied by Pierre de Fermat in 1630 and Guido Grandi in 1703. Grandi subsequently introduced the expression la versiera to describe this curve, based on a Latin word for rope, and Maria adopted this term. Colson seemingly translated this as the witch, based on the Italian noun l’avversiera. Consequently, this mathematical curve is now generally referred to as the witch of Agnesi. It can be defined as the hyperbolism [2] of a circle relative to one of its points and the tangent that is diametrically opposite to this point, as illustrated in Figure 1.

Editorial-April-2018-figure-1
Figure 1: Construction method for the witch of Agnesi

The Cartesian equation for this curve is

    \begin{equation*} y = \frac{1}{x^2 + 1} \end{equation*}

and it has some intriguing properties. In particular, it is the derivative of the arctangent function and approximates the spectral energy distribution of X-ray lines, the surface shape of a single water wave and the cross-sectional profile of a smooth hill. It also defines the probability density function (PDF) of the standard Cauchy distribution

    \begin{equation*} f(x) = \frac{1}{\pi(x^2 + 1)}, \quad x \in \mathbb{R}, \end{equation*}

which is equivalent to Student’s t(\nu) distribution with \nu = 1. Its PDF is displayed in Figure 2 with those of the t(3) and standard normal distributions for comparison: the last of these is equivalent to t(\infty). The median and mode of the t(\nu) distribution are both zero for \nu > 0. Although the mean is also zero for \nu > 1, amazingly it is undefined for \nu \le 1 (including the Cauchy distribution) despite the PDF’s obvious symmetry and unimodality.

Editorial-April-2018-figure-2
Figure 2: Cauchy, t and normal PDFs

I encountered another interesting probability distribution last November at a conference in the beautiful city of Samsun on the Black Sea coast of Turkey. This event was organised by my first PhD student, Mehmet Ali Cengiz, who is now a professor and vice rector at Ondokuz Mayıs University. One of the keynote speakers was Professor Narayanaswamy Balakrishnan of McMaster University (Canada), who presented a lecture on flexible cure models for medical research. What particularly caught my attention was his application of the surprisingly underused COM-Poisson distribution [3] with probability mass function (PMF):

    \begin{equation*} p(x) = \frac{1}{Z(\lambda ,\nu)}\frac{\lambda ^x}{(x!)^\nu}, \quad x\in\{0,1,\ldots\}, \end{equation*}

where Z(\lambda,\nu) is a normalising constant, an intractable function of parameters \lambda > 0 and \nu \ge 0 subject to \lambda<1 if \nu = 0.

Figure 3: Bernoulli, geometric and Poisson PMFs

This generalises the Poisson distribution, which corresponds to the special case with \nu = 1 but is often too restrictive as its mean and variance are equal. Although the negative binomial distribution resolves problems of over-dispersion, the COM-Poisson distribution can also model under-dispersion. Moreover, it reduces to the Bernoulli distribution in the limit as \nu \to \infty and the geometric distribution on setting \nu = 0. These convolve to binomial and negative binomial distributions respectively, so this model offers a flexible alternative for analysing count data. Figure 3 displays illustrative PMFs for these special cases.

Finally, I cannot resist mentioning our adorable new puppy, Monty. He chews pretty much anything and can undo shoelace bows and the zip on his soft kennel, though at least he responds well to recall and fetch commands. My mother already spoils him with titbits when we visit.

Anyway, I shall try to read April’s issue of Mathematics Today before he gets his gnashers on it and I hope that you too enjoy reading the various notices, reviews, features and puzzles on offer. I leave you with a simple limerick over which to mull:

My name is Maria Agnesi,
And some folks are driving me crazy.
They call me a witch;
It’s a bit of a hitch;
I’m really a saint Milanese.

David F. Percy CMath CSci FIMA
University of Salford

References

  1. O’Connor, J.J. and Robertson, E.F. (1999) Maria Gaëtana Agnesi, MacTutor History of Mathematics, University of St Andrews, www-history.mcs.st-andrews.ac.uk/Biographies/Agnesi.html (accessed 28 February 2018).
  2. Ferréol, R. (2017) Hyperbolism and antihyperbolism of a curve: Newton transformation, Mathcurve, www.mathcurve.com/courbes2d.gb/hyperbolisme/hyperbolisme.shtml (accessed 28 February 2018).
  3. Shmueli, G., Minka, T.P., Kadane, J.B., et al. (2005) A useful distribution for fitting discrete data: revival of the Conway–Maxwell–Poisson distribution, J. R. Stat. Soc. Ser. C, vol. 54, pp. 127–142.

Reproduced from Mathematics Today, April 2018

Download the article, Editorial, April 2018 (pdf)

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