JOST A MON

Despite my stated intentions, I never did progress on my profiles of top-class women mathematicians ahead of the latest Fields Medals. The announcement has come and Maryam Mirzakhani has become the first woman to win this accolade.

As an 8-year-old, Maryam Mirzakhani used to tell herself stories about the exploits of a remarkable girl. Every night at bedtime, her heroine would become mayor, travel the world or fulfill some other grand destiny. ¹

Mirzakhani as a child (via Quanta magazine)

In 2010, I'd written that desis and women had another four years to go for their next pop at the award. It turns out Manjul Bhargava has also won, and although he's Canadian/American, I'm sure desis will be quite pleased to claim him as one of our own.



Reference

1. A tenacious explorer of abstract surfaces, by Erica Klarreich, Quanta Magazine, August 12, 2014.

In August this year, Seoul will host the latest International Congress of Mathematicians. It's that quadrennial time when gossips begin to natter about Fields Medallists. A lot of people are talking up Manjul Bhargava. It is his last chance to win - by the time the next ICM comes around he will be older than 40. I'm hoping at least one woman will be awarded the prize this time.

In the run-up to the last ICM (in Hyderabad), I'd blogged about potential candidates for that year. None of them won. I hope to profile a few new names ahead of this year's ICM.

One that has popped up (new to me, at least) is Laure Saint-Raymond. Here is a loose translation of an interview that appeared a few days ago in Le Journal: "Laure Saint-Raymond, la boss des maths", by Louise Mussat (April 28, 2014).

Elected to the Academy of Sciences in December 2013, this top-flight researcher will participate at the International Congress of Mathematicians in Seoul in August.

The blackboard that stretches across one of the walls of her office at the École normale supérieure (ENS) is whitened by clouds of intertwined mysterious symbols. These are equations, and they are unsurprising - Laure Saint-Raymond is a mathematician, a pearl in

her field of partial differential equations applied to problems of physics. A winner of the European Mathematical Society's Prize in 2008 and of the Joliot-Curie "Young Woman Scientist" award in 2011, she was elected to the Academy of Sciences when she was 38 years old, the youngest ever. And in August she will present a lecture in Seoul at the ICM, an unmissable event that attracts thousands of mathematicians every four years, during which the famous Fields medals are awarded.

Stratospheric Research

What does the daily life of a top mathematician look like? Wracking one's brains before a sheet of blank paper before scribbling equations on a blackboard? "Not exactly!" says this energetic researcher, smiling. "I spend a lot of time teaching, attending seminars, trawling the literature to help me keep abreast of what is happening, and communicating the results of my own research. But thinking in front of a blackboard is part of my work, and most of the time it's with my team. It's the creative phase, a mutual game during which we sometimes find nothing or sometimes not what we were looking for!"

What Laure Saint-Raymond seeks with her collaborators is to capture physical phenomena with mathematics. "My training in physics helps me to better understand the language of physicists and exchange ideas and develop an intuition for the phenomena we are trying to describe," she says. That was the year she graduated with a degree in applied mathematics from the University of Paris VI, the same year she obtained her degree in plasma physics.

All these strings to her bow have enabled her to tackle a tricky problem, first posed in 1900 and still unresolved: how does one effect a transition from a physical model to another, less complex and less precise? The scientist explains, "When we are in the presence of a rarefied gas, for example in the Earth's upper atmosphere, and we want to simulate the re-entry of a spacecraft, we use the so-called kinetic theory in which the state of the gas is characterised by a function of several variables, such as time as well as the positions of the atoms." Meanwhile, to describe the behaviour of gases around us, infinitely denser in atoms, it suffices to use a model wherein all the atoms are considered as a single continuous medium.

"What interests me is to understand how one moves from one to the other of these models: is there a smooth transition or, on the contrary, is there a discontinuity between the two descriptions which renders them both invalid in the intermediate regime?" says Laure Saint-Raymond.

Fundamental and stratospheric, this mathematics nevertheless has some physical applications: they allow us to understand better the entanglement phenomena at diferent scales of time and space, and to better account for the observed air flow around an aircraft, for example, or the formation of persistent whirlpools in the ocean.

From the Cello to the Sciences

Laure Saint-Raymond readily admits: as a child or a teenager, she had no particular attraction for mathematics. Her main thing was the cello. Then in high school, equations finally overcame the scales. "I had a facility with maths and physics, and my parents, both maths teachers, undoubtedly influenced my change in direction."

As a newly-minted bachelor in mathematics, she found the arms of the engineering industry extending towards her. However, she chose research, it being "intellectually more stimulating". In 1994, she joined the ENS where she began work on her thesis under the supervision of François Golse on the "kinetic theory of gases". To be clear: on mathematics applied to the movements of gases. Recruited by the CNRS in 2000, she spent two years as a Research Fellow before being appointed as a professor at the University of Paris VI. In 2007, she obtained a secondment at the ENS where she was made director of the department of Analysis. Currently, she is the deputy director of the Department of Mathematics and its Applications.

An exceptional journey

Her exceptional and rapid-as-lighting journey demands respect as it has happened in a largely male-dominated universe. "That is irrelevant," she counters. "My being a woman has never been a handicap, I have never been a victim of discrimination. And to do research today, it is no easier for a boy than for a girl, it's difficult for both: young people don't obtain a job till after several years of post-doctoral work, which, in some disciplines, means they have to wait till nearly 35 years of age before they are in a stable situation and able to start a family..."

Still, says Laure Saint-Raymond, her eyes shining and with enthusiasm in her voice, the game is worth the effort. "I will no doubt make a lot more money in the industry, but one's work cannot be reduced to a mere salary. Not one of my days as a teacher-researcher is like any other, and at the ENS, I have the immense freedom to conduct research that I want to do, and I enjoy myself thoroughly. This is priceless."

When she is not playing with partial differential equations, or not coaching one of her students, or not tutoring one of her six children (five boys and one girl, aged between 4 and 14 years!), Laure Saint-Raymond escapes to the mountains. Skiing in the winter or hiking in the summer, whichever, the key is to be outdoors all the time. "I really need to recharge my batteries, to disconnect completely from work, to return to the office with new ideas," she admits, and returns to whitening her blackboard again.

Spitalfields, the area just to the east of Liverpool Street station, is known for its thriving immigrant communities over the centuries. Jews, Huguenots, Bengalis; silk-weavers, bakers, brewers, artificers, opticians, and - would you believe it - mathematicians. In 1717, Thomas Middleton established the Spitalfields Mathematical Society, which was open to local residents, would serve to disseminate mathematical knowledge among them, and also encourage the members ('the square of 8' in number) to educate each other with questions and puzzles. Middleton himself was a teacher of navigational mathematics to sailors, and he started off the society by offering free lessons to his fellows. [1] 

The Society would meet weekly for three hours, during which questions would be posed, tutorials offered and information exchanged.  

One of the rules of the Institution, which had so humble an origin, observed for upwards of eighty years was, that one hour during the time of the meeting should be devoted to silent study. The Stewards were accustomed to put a sand-glass on the table, and no one was allowed, under penalty of a fine, to open his lips until the sand had run down. [2]

One of the cunning ways in which the society encouraged collaboration was by levying small fines on anyone who refused to attempt to solve a query raised by a colleague. Fines accumulated in a fund that was then used to buy books!

By the early 19th century, the society was offering public lectures (on topics such as 'galvanism', 'pneumatics', 'hydrostatics', and 'astronomy'). 

The members of this august association were not trained mathematicians for the most part, but some very able people did join. Simpson (of the Rule), Gompertz (of the law of mortality) and Dollond (the optician with Aitchison) were three of the famous men in the group. The others were less known but hardly slouches, and all appear to have been given to the use of mathematics in the solution of problems that they encountered in their professional lives.

An interesting example is the use of mathematics to design a special instrument  called the bent-lever balance to control the fineness of yarn. The theory of this instrument was discussed by William Ludlam in 1765, using mechanics, trigonometry and calculus. [3]

From the 1740s, the society also began to exhibit scientific instruments [4] and (no doubt inspired by the Royal Institution) conducted public experiments to much acclaim. Audiences of up to 500 people, paying sixpence per lecture, were reported. However, in the feverish period of the Napoleonic wars, the Seditious Meetings Acts passed in 1795 and 1799 began to cast a chill on societies such as this. They began to be closely monitored for discordant political views, and in 1809, experimental lectures were abandoned (ostensibly because lecturers were unavailable, but most likely for fear of prosecution). [5]

In 1846, faced with declining memberships, the society merged with the Royal Astronomical Society, and its library, archives and equipment were all handed over to the latter.

The Society occupied different premises during its lifetime - all in Spitalfields. It started at a pub, Monmouth's Head, moved to the White Horse in Wheeler Street eight years later, and then to the Ben Johnson's Head in Woodseer Street in 1735. During the 1770s and 1780s, it was based in the Black Swan, on Brown's Lane (later known as Hanbury Street). In 1793, it moved permanently to a room on Crispin Street, where it ended its life. The first pub became part of the Hanbury brewery; I can't tell what happened to the next two. Crispin Street lost much of itself when Spitalfields market expanded in the 1920s; the premises of the Society (No. 36/36A) were demolished in the same period.[6] 

For several years now, the London Mathematical Society has funded Spitalfields Days in honour of the old society: believing that it is important for recent developments in specialist topics to be made known to the general mathematical community, and, in particular, to research students ... provides funds to the organisers of these meetings so that they can provide a day of survey lectures, accessible to a general mathematical audience.[7]

References

I was working on some statistics the other day when the cleaner arrived. I left the work on the screen, and when she walked by, she noticed the mathematical formulae.

"You are mathematic?" she said. 

She is not entirely fluent in English.

"You are a teacher?" she said.

"No, not a teacher," I said.

"But you are mathematic?" she urged.

"Yes," I said.

"I am economic," she said. "I study in Poland."

****

The boy sees me reading a history book. He has keenly noted my predilection for things past. He sits with me when I watch TV programmes on ancient civilisations and disappeared empires.

"I want to be historic, like Acha," he says.

"Not pre-historic, like the dinosaurs?" I tease.

"Stop it, Acha," he says.

It's been a while since I've CodeCogged, and I'm pleased to find that the site is still there. This brings out an intemperate desire to see some fancy mathematical typesetting, and an email from Springer provides the impetus. I get free previews for the occasional paper, you see, and it's always fun to look at its contents and realise, once again, that while I might understand every word in a sentence, I have no idea what the thing in its entirety means.

So, anyway, here's a theorem from Alexander Kuznetsov (2008) 'Lefschetz decompositions and categorical resolutions of singularities', Selecta Mathematica (New Series), Volume 13, Number 4, pp 661-696.

Theorem 1. The triangulated category with functors and is a categorical resolution of . Moreover, we have a semiorthogonal decomposition



Finally, if is Gorenstein and  then

•  is weakly crepant if
•  is strongly crepant if  

Kuznetsov goes on to say: 'Certainly, minimal categorical resolutions are the most interesting.'

One day I'll figure out how to inline the LaTeX so that it doesn't appear off the rest of the text. Good thing to have an occasional challenge, eh?

I swear I started with every good intention. Why not do a brain dump into the Hindi Wikipedia of all the good stuff I know about differential equations? I still recall my twelfth grade Mathematics final exam that had questions in Hindi and English. Solve the following differential equation, it said, and beneath it, unforgettably, निम्नलिखित अवकल समीकरण को हल कीजिये. Ooh, I was all aquiver. 

And I was dashed. Dashed, I tell you. I could make no progress whatsoever. For one thing, there already is an article on differential equations. For another, formatting Hindi text was bad enough - mathematical text did my head in. 

So I caved. Mamta Mohandas now has a Hindi Wikipedia page. This is clearly a quixotic effort. Who in North India even cares about Malayalam cinema? And which of these worthies will read up on her in Hindi, when the English page is equally comprehensive? I don't know.

Will someone please take a look at the page and let me know if I've made any grammatical or semantic errors? It's been years since I've read or written Hindi. 

Help!

A fairly nondescript day, you might think, 18 September. A vacant yet short-tempered fellow called Joseph of Cupertino, Italy, levitated seventy times in the seventeenth century and was first Inquisitioned and then raised to sainthood. This is his day. 

It is also the day Leonhard Euler died in 1783. 

Jean-Baptiste-Leon Foucault the physicist was born the same day in 1819. 

Then it was Joseph Locke's turn to pass on (1860) - he was a civil engineer. 

Greta Garbo arrived in 1905. 

Nobel-winning physicist John Cockcroft died on 18 September 1967. 

Then Jimi Hendrix departed this earthly vale of tears three years later.

So now you know.

A taxicab

You've probably heard this story before. The English mathematician G.H. Hardy once visited the Indian mathematician S. Ramanujan in hospital. Hoping to the cheer up the patient, Hardy said he'd arrived in a taxi numbered 1729, which struck him as a rather boring number.

Ramanujan demurred. It is an interesting number, he said, because it's the smallest number that can be expressed a sum of two cubes in two different ways.

That is to say, 1729 = 9³ + 10³ = 1³ + 12³

And there's a galaxy (in Orion) with the catalogue number NGC 1729. (A supernova was detected there on February 20, 2012. Check it out here.) We could conceivably name it after Ramanujan, I daresay?

But 1729 is interesting in several other ways too. Sure, these are mere numerical coincidences. Numerologists base their entire lives on the like. I am inclined to point out some of these below because it's a slow day and it's raining and Usain Bolt has done his thing and I'm bored.

If you go to Mandalay, your eye will oftentimes be drawn to the enormous Mandalay Hill. Pagodas and shrines surround the long path that leads up to it. How many steps on that path? 1729.

There's a baroque musical group called Ensemble 1729. I think they're a bunch of Canadians. They are inspired by Ramanujan and by the fact that the year 1729 has several historical coincidences - you can see what they were on the group's website.

Gideon Rubin: Louis XV

If you fancy a bit of champagne, you'll be pleased to hear that its first house was also established in the year 1729. This was the house of Ruinart, which received the heraldic crest in 1817 that it has used ever since. Ruinart likes to promote contemporary art. In Basel last year they came up with an exhibition of works by Gideon Rubin who based his paintings on photographs and drawings from the Ruinart archives.

Speaking of champagne, Baltimore, Maryland, was founded in 1729 and scarcely seventeen years later, its first manufacturing industry was beer.

I'm forced to admit that this litany of 1729-hood is rapidly devolving into a list of dates, which is not interesting at all. Hence a rapid exit is indicated.

Impossible, then, for travellers to know the exact day of a religious festival. They could already feel the tongues of hell-fire touching their feet.


David Gans considered the issue serious enough to request the Holy Roman Emperor Rudolf to help, and the likes of Johannes Kepler got involved: "After they considered these questions for several days, and debated with me, they admitted and were not ashamed to say that they had not attained a correct and satisfactory answer."


The solution eventually was one of pragmatism and convenience: the Sabbath is what the local custom says it is.


Now, consider the problem of facing Mecca - the Qibla direction - when a Muslim wants to pray. The faithful not too far from Arabia had a pretty fair idea of the direction of the holy city. They knew that twice a year (on May 28 and July 16) the sun is directly overhead on Mecca at noon. So all they needed to do was to look at the direction of the sun at the local time corresponding to the Meccan noon, and orient their mosques thither. (I assume they knew how many hours Mecca was ahead or behind them.)


It turns out that the direction they gazed at was along a great circle, a geodesic, the shortest distance between them and Mecca.


Now imagine the situation of a Muslim in North America. A flat earth map would indicate that the Qibla is south-east. But the earth is round, and so the geodesic from North America to Mecca goes almost via the North Pole. The Muslim, in other words, has to face nearly north - quite counterintuitive.


Still, they found a solution - using the same methodology as their brethren did nearer to Mecca:

It has been observed that around noon time of Makkah, it is about 6 am in Nova Scotia, Canada and Maine, USA. The sun rises in those locations as it comes overhead Makkah at local noon time. Facing the sun on those two dates around 6 am gives the correct direction of Qibla from North America. Those who had observed this confirmed that they saw the sun in North East direction at the specified time and date. Therefore, it is correct to say that Qibla from North America is generally North-East, except from Alaska and California where it is close to North direction. ²

References

1. Paul Kriwaczek, Yiddish Civilisation: the Rise and Fall of a Forgotten Nation (London: Weidenfeld and Nicolson, 2005)
2. Qibla Direction.

In 1236, Walter (Gautier) of Coincy wrote a vernacular poem for a largely non-literate audience. In it, he used the expression 'ciffres en augorisme' to mean a vacuous person. In other words, a zero. Clearly, if even an unlearned audience was meant to understand the reference, the cipher, or the Hindu-Arabic numeral, was already well-established in Europe.

Scientific, as opposed to mathematical, knowledge had already started flowing westwards in the preceding century. "After 1100, Euclid’s Elements gained increased prominence; in 1126 Adelard of Bath brought Al-Khwarizmi’s trigonometry to the West; in 1145 Robert of Chester translated Al-Khwarizmi’s Algebra; Ptolemy’s Almagest was translated from the Greek in 1160." ^[1]

How did the Europeans gain access to the concept of zero? As widely documented, the good news came from the East, via the Arabs.

I have read previously that Leonardo da Pisa (or Fibonacci)'s learned work Liber Abaci of 1202 was the main mechanism of transmission. Fibonacci had travelled extensively in North Africa training to become a merchant, had come in contact with Arabs there, and learned their sciences and mathematics.

Sefer ha-Mispar of Rabbi ben Ezra

As evidenced by Fibonacci, the main impetus to arithmetic appears to have been mercantile. The widespread use of the abacus had already introduced the notion of 'place value' to the Europeans, but they persisted in using Roman numerals in their documentation. Indeed, even innovators in business such as the English Exchequer and the Medici Bank decried the use of the new-fangled numerals. The Florentine guild of bankers required its members to “write openly and at length, using letters” — the fact that the ordinance had to be repeated three more times meant that by 1299, bankers in Florence had found it faster and more convenient to use the Hindu-Arabic numerals rather than write “at length” in the old script." ^[2]

I learn now, though, that an even earlier book had introduced the concept of zero to the Europeans.^[3] The Spanish rabbi Abraham ben Ezra wrote about the Hindu-Arabic numerals in his book Sefer ha-Mispar (Book of Number) while visiting Verona in 1146. He used the first 9 letters of the Hebrew alphabet to represent the numbers 1 to 9, and made a small circle that he called galgal (Hebrew for 'wheel') for zero. (The Arabs used a dot.)

The Jews, of course, already had large trade networks across the Mediterranean and the Levant and deep into the Muslim lands. Their affinity for new ideas and business acumen meant that they had a long-standing advantage over their Gentile competitors. Indeed, as we have seen, the Christians were not loath to shoot themselves in the foot with proscriptions against new (or heathen) techniques.

It was not till the 15^th century that the Church relented and allowed the use of the numerals. And they had the temerity all along to accuse of Jews of taking advantage of Christians and making money off the honest faithful. All I have to say is - pillocks.

References

1. Stephen E. Sachs, New Math: The‘Countinghouse Theory’and the Medieval Revival of Arithmetic (here)
2. Alexander Murray, Reason and Society in the Middle Ages (Oxford: Clarendon Press, 1978)
3. Paul Kriwaczek, Yiddish Civilisation: the Rise and Fall of a Forgotten Nation (London: Weidenfeld and Nicolson, 2005)

Last weekend, the boy and I sat down to do some homework. It dealt with paying for purchases and calculating the change due. This tied in neatly with the boy's inability to collect change from his weekly 'Tuck Shop' purchases.

'But the Tuck Shop is free,' he protested.

'Why do you say that?' I said.

'Because they don't give me any money back,' he said.

At this point, we needed to established definitions.

'If you give money for something, then it is not free,' I said. 'Do you give money at the Tuck Shop?'

'Yes,' he said.

'Well then, it's not free,' I said.

We then looked at his exercise sheet.

'Okay,' I said. 'Suppose I am a shopkeeper and you come into my shop to buy an apple.'

'I want to be the shopkeeper,' said the boy.

'All right,' I said, being an agreeable sort. 'I come to your shop to buy an apple.'

'What shop is it?' asked the boy. 'Is it Marks, or is it Tesco, or is it Sainsbury's?'

'It doesn't matter,' I said.

'I like Sainsbury's,' said the boy, with some relish.

'The apple's price is 5p,' I said. 'How much should I pay you for it?'

The boy looked at me, puzzled. I had a vivid recollection of Swami from Malgudi Days asking his father how big the fruit was when asked to work out its price.

'I should pay you the price of the apple, shouldn't I?' I urged.

He nodded.

'I can't pay you less than 5p, can I?' I asked.

He shrugged in a noncommittal way.

'If I paid only 3p and took the apple, that would be stealing, wouldn't it?' I said.

He sat up, interested.

'You must not steal,' he intoned. 'Otherwise, the police will come.'

'Yes,' I said.

'Let them come,' he said, seized with a sudden fit of bravado. 'I'll punch them on the nose.'

I sighed.

'Try to focus,' I said. 'Do you agree that if something costs 5p, you should not pay less than that?'

He yawned. He nodded.

'Okay,' I continued. 'Now what if I pay you more than 5p? Suppose I paid you 10p? What will you do?'

He stared.

'You should give me some money back, shouldn't you?' I said.

'I don't have any money,' he said.

'You are a shopkeeper. Of course you have some money,' I said.

'Did I get it from the bank?' he said.

'And from the other customers who came to buy things before me,' I said.

'Did they buy oranges, or did they buy sweets, or did they buy a Munch Bunch?' he said.

'It doesn't matter,' I said, feeling as though I had walked into a cloud. 'Let us worry about the apple I'm trying to buy, shall we?'

'I don't like apples,' whispered the boy in my ear.

'I know,' I said. 'So, anyway. The apple costs 5p, and I paid you 10p. I paid you more than I should have, so you should give me some change back. How do we work out the change?'

'Suppose you gave me 1 pound?' said the boy, his eyes as round as a pound.

'We'll worry about pounds later,' I said. 'Shall we stick to pence for now?'

'Pens?' said the boy, completely confused.

'Pence,' I said. 'You know, pennies.'

'Okay,' said the boy.

We looked at each other.

'I don't know,' he said, agonised. 'It's too hard.'

'Don't worry,' I said. 'I'll explain. If I pay you 10p for something that costs 5p, you should give me some money back. How much money? Well, you should return to me what I paid you minus the price.'

'What is minus?' said the boy.

'I meant 'take-away',' I said, correcting myself.

'Take-away' is the modern English method of teaching subtraction.

We looked at each other again.

'Did you get that?' I said. 'You give me back what I paid you take away the price.'

'Okay,' he said.

'So - what did I pay you?'

His eyes glazed over.

'I forgot,' he said.

'I paid you 10p. Please pay attention,' I said. 'How much did the apple cost?'

'5p,' said the boy.

'So what do you do?' I said.

After some more to-ing and fro-ing, the boy finally said, 'Ten take-away five.'

It took him several seconds to work this out in his head and on his fingers.

'I will give you 5p,' he said.

'Excellent,' I said.

I don't think he has quite got it yet. Even when he does, it will probably strike him as some kind of magic formula. How do I explain it to him so it appears natural and reasonable? I have no idea.

Exhausting stuff.

Welcome, folks, to the LXXVII edition of the Mathematics Blog Carnival. We have a wide-ranging litany of articles, although - despite our best efforts - not seventy-seven of them. Still, quite a few to whet an appetite or three.

According to custom, we must start with the oddities of the number. Instead, we'll just intersperse the facts amongst the various articles.

Sol Lederman presents Curve stitching with Mathematica posted at Playing With Mathematica.

Ever heard of curves with infinite perimeter and zero area? Read That's Impossible! One Giant Nerdgasm at Consumed By Wanderlust.

What does the Great Pyramid tell us about ancient Egyptian mathematics? Dave Richeson reveals an interesting consequence in Division by Zero.

Alexander Bogomolny presents Areas on the Graphs of Power Functions posted at CTK Insights.

At Travels in a Mathematical World,  Peter Rowlett presents a collection of podcasts and videos from the Math/Maths Week 2010, and from Young Researchers in Mathematics 2011.

Mike Croucher ponders whether graphical calculators have outlived their usefulness at Walking Randomly.

Speaking of calculators, did you know you could multiply on your fingers? I didn't, but Math and Multimedia reveals some tricks.

While the little folk do elementary mathematics on their fingers, the powerhouses of the discipline get their breakthroughs at the most peculiar places and moments in time. Dick Lipton lists some of them in Godel's Lost Letter and P = NP.

Pat Ballew highlights the quotation 'old mathematicians don't die, they just go off on a tangent', and illustrates nicely the properties of tangents to a cubic at Pat's Blog.

At Short Sharp Science, Catherine de Lange reveals how tattoos (unsightly at the best of times, heheh) become even unsightlier with age. Mathematicians have developed a model that describes the aging of tattoos. (Do you think the picture below looks like a tattoo of 77? No? Dash it.)

And IT History has a little piece on the beginnings of computer user groups - all the way back in 1952!

Speaking of beginnings, it's the centenary of IBM. Take a look at this celebratory post at Antipodes: Reflections from an Australian Expatriate in France?

"Wannabe professional gambler" Zac mixes up probability and ethical humanism in his post Gambling Theory at Zac Sky.

SquareCircleZ ponders what is the correct graph of arccot(x)?

Alex Bellos discovers that there are more to triangle centres than he had previously imagined (and revealed to us in his book Alex's Adventures in Numberland).

Roice has some clever Geodesic Saddles.

And Joe Manausa shows how Tallahassee residents need to wait till 2018 for their house prices to return to equilibrium in his case study. Long time to wait, eh?

And just so that we Anglospeakers don't feel too alone, we are pleased to reveal that the Spanish blogosphere has its own Carnival of Mathematics. The latest installment is by Juan Martínez-Tébar Giménez at Los Matemáticos no son gente seria, and it showcases entertaining pieces on, among a couple of dozen other things, Tartaglia and Cardano, the Nash conjecture, the decipherment of a wartime diary, and the centenary of the Royal Spanish Mathematical Society.

That's it for this month, people. Please do take a look at our sister carnival - Math Teachers at Play -  and also note that you can follow the Carnival of Math on Twitter: @Carnivalofmath. The next Carnival of Mathematics should come up around Jun 3, 2011. Please send in your submissions here.

I have learned over the years that what is amusing to one person is far from amusing to another. This is not surprising. Many people - some as young as five years old (see yesterday's post) - have told me that what I find interesting is less than crudworthy to them. Neither trivia nor humour really translates well.

And I do not even mean translating from one language to another. In a recent paper on an algebraic formula for the computation of p(n), the number of partitions of an integer n (a partition of n being any non-increasing sequence of positive integers that add up to n) - by all accounts a major development in number theory, having been an open problem for nearly 80 years - the authors write "We give an amusing proof of the fact that p(1) = 1."

When I saw the proof, I laughed. Hollowly. Here it is (following the statement of their main theorem, for which, see original paper):

In this case, we have that 24n - 1 = 23, and we use the G₀(6)-representatives

The corresponding CM points are

Using the explicit Fourier expansion of P(z), we find that

Using these numerics, we can prove that

We have that Tr(1) = 23, confirming that p(1) = Tr(1)/23 = 1.

Quod bloody erat freakin' demonstrandum.

The other day, I was trying to explain odd and even numbers to the boy. After some playing around with coins to explain how pairs are formed, the little chap appeared to understand the concept. Carried away by the breakthrough, I said:

"Hey, can I tell you something else that's interesting?"

"I don't think it will be interesting," said the boy.

"What!" I said. "Didn't you think odd and even numbers were interesting?"

"Ye-e-es," he said, somewhat doubtfully. "But I don't think whatever else you'll tell me is interesting."

The measurements themselves were copious and exhaustive, incorporating - for the head alone, for example - such characteristics as cephalic index, head length, head breadth, nasal length, nasal breadth, and nasal index. Simple statistics such as the average and the standard deviation were clearly not sufficient to distinguish between one group and another, particularly when the measurement error alone resulted in overlapping values for two classes. In a brilliant series of papers published in a new journal called Biometrika, Karl Pearson and associates showed there were multidimensional measures whereby anthropological data could be analysed. More specifically, they could be used to 'assess similarity or dissimilarity between two populations' ¹.


An example of a prevalent question at the time was whether Ancient Egyptians had any social affinity with Hindus. Some researchers had claimed that the physical resemblance between the two peoples implied a partial colonisation from one country to the other. Karl Pearson introduced the coefficient of racial likeness, or CRL, to analyse this issue, 'one of the first quantitative procedures to measure the admixture proportions, or the proportions which 'hybrid' populations derive from their various ancestors'².


Given two populations of size n and n', with the mean of the ith measurement in the first population m_i, and that in the second population m'_i, and s_i² the pooled variance in the ith measurements, the CRL C is given by:





Almost immediately, the CRL was used by M. L. Tildesley in her analysis of Burmese craniums in 1921³. Shortly thereafter, critiques of the methodology began to appear in the literature - not only by anthropologists but also statisticians. A chief criticism was that the method assumed that the various metrics (e.g. head length, head weight, nose length) were independent of each other, a rather generous assumption. Another complaint was that the CRL was dependent on the sample size of observations, and that it provided only a degree of certainty that there was divergence between the groups, but could not quantify exactly how much divergence there was.


In 1925, an Indian statistician named P. C. Mahalanobis began an investigation into the question of racial differentiation in his native state of Bengal. He looked at the results of an anthropometric survey of Anglo-Indians (people of mixed European and Indian ancestry) conducted in 1891 to answer (in his words) the following questions:

How are these 200 Anglo-Indians in Calcutta related to the different caste-groups in Bengal? Are they more closely allied with the Hindus or the Mohammedans? Do they show a greater affinity with the higher castes of Bengal or with the lower castes? ... ⁴

In order not to be swayed by size differences in the various characteristics he used, Mahalanobis computed standardised values for them.

The characteristics differed by scale and variability. That is, Mahalanobis might have considered a half-inch difference in nasal length between two groups of skulls a significant difference whereas he considered the same difference in head length to be insignificant. Mahalanobis normalized differences in each characteristic by the characteristic’s standard deviation and then squared and summed the normalized differences, thus generating one composite distance measure that was invariant to the variability of each dimension. ⁵

This first Mahalanobis metric suffered from the same defect as Pearson's as it didn't consider the correlation between the various characteristics. In 1936, he introduced his famous concept of statistical differentiation that came to be known after him, the Mahalanobis distance.


If the human skull can be described by n characteristics that are measurable, an individual's skull can be represented as an n-dimensional vector. Now take a set of measurements belonging to one anthropological unit (say, Calcutta Brahmins), and compute its centre, or mean vector m, and its covariance matrix S. Then, if we want to classify a hitherto unclassified skull measurement y, what we do is compute its Mahalanobis distance D from the Calcutta Brahmin set's centre:




And we compute the distance against the centres of other classes, say, Calcutta Muslims or lower castes, and we decide that our unknown skull falls into that class from which it has the least distance D.


To simplify, assume that a skull can be characterised by two metrics, skull length and skull breadth. Then each skull can be represented by a point in two-dimensional space. If we plot our data of Calcutta Brahmin skulls and Calcutta Muslim skulls, we might find (if these are indeed two distinct anthropological classes) that our graph has two distinct clusters in it (Figure 1 from Kritzman and Li (reference below)):


In Kritzman and Li's words, then:

Suppose we compare a skull of unknown origin, represented by the square in Figure 1, with the two groups and categorize it. In terms of Euclidean distance, it lies closer to the center of Group 2 than to the center of Group 1. The Mahalanobis distance, however, would consider this skull more similar to Group 1 because its characteristics are less unusual in light of the more inclusive scatter plot of Group 1’s characteristics.

So what did Mahalanobis conclude from his investigation? First of all, he said, the Anglo-Indians in his sample derived (on the Indian side) from Biharis, Lepchas (of Sikkim), possibly from the Punjab, and none at all from the Northwest Frontier or the Chotanagpur tribals. He also noted that they seemed to derive from unions of higher-caste Indians and Europeans, adding that 'cultural status evidently played a large part in determining Indo-European Union.'


From a broader investigation into the anthropological classes of Bengal, Mahalanobis was able to arrive conclude: 

Summing up we find that intermixture within Bengal, i.e. intra-provincial intermixture has varied with the degree of cultural proximity, so that for Brahmins the amount of intermixture with other castes has been in proportion to the social standing of the caste concerned. Influence from outside Bengal, i.e., inter-provincial intermixture has followed two well-defined and clearly distinguished streams, one from the castes of Northern India (chiefly from Bihar and the Punjab) and the other from the aboriginal tribes of Chotanagpur. The influence of the Northern Indian castes decreases and that of the aboriginal tribes of Chotanagpur increases as we go down to the social scale... . None of the castes analysed here show much resemblence with any of the aboriginal tribes of the east... . Mohammedans (also) show a highly mixed character. They appear to be originally largely derived from Bihar but have intermixed extensively in Bengal; they do not show any resemblance with the Punjab Pathans. ⁶

As it happens, not all the anthropological conclusions of that 1925 paper are held valid today. Mahalanobis was correct in his assertion that Bengal Brahmins resemble other Bengal castes more than Brahmins elsewhere in India. However, later datasets have invalidated his claim that only the Brahmins among the people of Bengal have admixtures from the Punjab. 'Moreover, as far as the Anglo-Indian community is concerned, it is now believed that Mahalanobis had probably confined his study to a sample from the upper stratum of the community, and hence his conclusion of resemblance to upper caste Hindus is applicable to the upper class Anglo-Indians only'. ⁷


These days, Mahalanobis is venerated by many people not for his anthropological research or its conclusions; rather, it is the methodology he developed that is considered his greatest contributions to the sum of human knowledge. Even today, the Mahalanobis distance is part of the armoury of every scientist who needs to classify multidimensional data. As you can probably infer from the reference list, this includes not just statisticians, but also anthropologists, social scientists, and financial engineers. 


Quite a lot of good, in short, has come out of what once was eugenics research.


References


1. S. Dasgupta, "Evolution of the D²-Statistic of Mahalanobis", Sankhyā: The Indian Journal of Statistics, 1993, Special Volume 55, Series A, Pt 3, p 442.
2. M. Tapper, In the Blood: Sickle Cell Anemia and the Politics of Race, University of Pennsylvania Press, 1999.
3. M. L. Tildesley, "A First Study of the Burmese Skull", Biometrika, 13, 1921, 247-251
4. P. C. Mahalanobis, "Analysis of Race-mixture in Bengal", Journal of the Asiatic Society of Bengal, 23, 301-333.
5. M. Kritzman and Yuanzhen Li, "Skulls, Financial Turbulence, and Risk Management", Financial Analysts Journal, 66(5), 2010.
6. S. Dasgupta, as above, p 447.
7. J. K. Ghosh, "Mahalanobis and the Art and Science of Statistics: The Early Days", Indian Journal of History of Science, 29(1), 1994.

It appears that a man called Albert of Saxony wrote a little treatise in 1350 consisting of various proofs and assertions on the possibility of squaring the circle. His book was called Quaestio de quadratura circuli (Question on the Quadrature of the Circle), and he made some pithy observations such as the following:

Now I won't claim that this is absolutely rigorous, although it's fairly clear that if you assume that a square can be stretched continuously, then a square inscribed in a circle will at some point exceed the size of circle, and a fundamental consequence of continuity is that it will do so without any sudden breaks. So, in fact, there is a square with exactly the same area as a given circle.

We know this is true. If the area of a circle with radius r is π r², then the side of the square with the same area will be r√π. Every real number has a square root, so we are good here.

The question that Albert was hoping to answer - I suspect - was the long-standing one of if it's possible to construct the equivalent square using only a straightedge and a compass. This was one of the classical problems of mathematics, known as far as back as the Greeks (Archimedes had provided an incorrect solution), and possibly even earlier.

A century or so after Albert, Leonardo da Vinci put his fecund imagination to the problem. He had constructed very clever mechanical means for squaring the circle,  but as his friend (and math teacher) Luca Pacioli pointed out, these were mere approximations, and not true constructions. In fact, they were not even original. Stung by this criticism, Leonardo (sometime in 1503) decided to solve the problem once and for all. From his notebooks, it is evident that he had spent time before this on the issue, trying out one mechanical method after another. That night in November, however, he resolved not to get up from his desk until he had settled the question.

The reasons for his failure (and the failure of every other mathematician or charlatan who attempted it (and continue to do so to this day)) were to become apparent only 400 years later. In 1892, Lindemann proved that π is a transcendental number. In other words, there is no algebraic equation that has π as its root. Every straightedge-and-compass construction can be translated into an algebraic equation. Therefore, there is no straightedge-and-compass construction that squares a circle.

References

1. Mathematical Intelligencer, Volume 1, Number 3, 1978/79.

2. Paul Strathern, The Artist, The Philosopher and The Warrior, Vintage Books, 2010, London.

3. Tom Pastorello, Leonardo Squared The Circle! – Da Vinci’s Secret Solution in the Vitruvian Man Decoded,