November 17, 2011

A minor celebration today…I’ve finally had my first paper accepted for publication.  As a measure of how long and convoluted this process has been, consider the fact that this is the very same* paper that I discussed in this post.  That was written in October 2010, and was titled “A year’s work, lessons learnt”.  Which means, according to my calculations, that the time between beginning work on this project, and actually getting something published has been over 2 years!

Of course, most of this time was not spent actually doing anything related to that particular paper.  In fact, the majority of the time was spent waiting for referees to get round to reading the thing.  Actually “waiting” is the wrong word, as I have come to realise that the best strategy when submitting papers to journals is not to wait, but to completely put it out of your mind (unfortunately this doesn’t help when you then have to revise it months later), and perhaps set some kind of reminder to get in touch with the editor one year in the future and ask exactly what is going on.  I currently have two other papers “under review”, one of which has been “with editor” (I assume this to mean that the editor hasn’t got around to actually looking at it, let alone passing it to a referee) since May, and the other which, perhaps thankfully, I have no way of knowing what is happening with.

There is intermittent hand-wringing about the peer-review system in mathematical circles, and in academia in general.  Like exams, and job interviews, it seems to be grudgingly accepted to be the least bad form of evaluation.  Recently Timothy Gowers raised the possibility of an alternative system on his blog, which led to much fevered debate (I have just noted that I am at least the the seventh blog to have linked to that particular post, so it is safe to assume the debate sparked by it stretches much further than that particular lengthy list of comments!).

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Mathematicians and religion

September 26, 2011

Where was I?  Well, last week* we established, among other oddities, that diclofenac is bad for the religion of Zoroastrianism.  That post didn’t really have anything to do with mathematics (although I did at least attempt to tenuously link it chaos theory), so I will make up for it by at least mentioning some mathematicians this week, if not actual mathematics.  However, I will stick with the topic of religion for the time being.

This is partly inspired by a book I’ve just read: Galileo’s Daughter, by Dava Sobel.  It doesn’t really match up to Longitude, but is a good read nonetheless.  It is really about the life and work of Galileo Galilei, although Sobel gives us the hard science and history in a more easily digestible form, by interweaving commentary on his relationship with his daughter.  She seems to have been a quite extraordinary woman: sent to a convent at age thirteen due to her illegitimacy (and hence lack of marriage prospects), she spent her whole life in extreme poverty within those walls,  but still managed to be a doctor, playwright, composer, musician and prolific correspondent in the little time she had which wasn’t dedicated to prayer, labour and general suffering.

Anyway, one of the things which struck me most about Galileo’s life was his relationship with the all-powerful Catholic church at this time.  He was a very devout Catholic: publicly, of course (claiming Catholicism is, after all, preferable to torture and painful death), but more surprisingly, given the utter ignorance and persecution he suffered at the hands of the Inquisition, he remained privately devoted to the church.  He even said, near the end of his life:

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March 19, 2010

You may or may not know that Sunday this week was Pi Day in the US.  This occurs on March 14th every year, is so named because in the American system* the date reads 3.14(\pito two decimal places), and in the past has been little more than an excuse to eat lots of pie in a knowing manner.  But this all changed last year, when the U.S. House of Representatives passed a non-binding resolution – whatever that means – designating it as “National Pi Day”, capital letters and all.  The curmudgeons among you might argue that they should have more important things to do with their time…well, my aim in this post is to persuade you otherwise!

\piis probably the most famous of the mathematical constants (fixed numbers)…indeed a sure sign of its importance and ubiquity is that it is one of the few mathematical constructs that most people have actually heard of.  Constants like\piandehave always fascinated mathematicians and non-mathematicians alike, and for good reason.  Mathematics is arguably the most pure and direct way we have of describing physical reality, and the pivotal role these numbers play in the subject hints strongly at some deep significance to us.  It has been suggested a number of times in popular culture – including in Darren Aronofsky’s film\piand Carl Sagan’s novel Contact – that there is a message hidden in the digits of\piwhich holds the key to understanding the nature of the universe.  And it is easy to see why this is such an enduring idea:\piis definite, fixed…yet mysterious and unknowable (in fact we will never know exactly what is).  It seems like such an arbitrary number, but at the same time could not really be anything else (circles would not be circles otherwise!).  Like the distribution of the prime numbers, there is no pattern contained in the digits of\pi ,but this doesn’t stop people looking for one anyway (you might say it positively encourages some).   As a race we have computed it, analyzed it, memorized it, and generally celebrated its existence ever since we first really thought about circles.

To date,\pihas been calculated to 2.7 trillion digits – far more than the average computer even has hard disk space to store.  And the record for reciting the number from memory is held by a Chinese graduate named Lu Chao, who took 24 hours and 4 minutes to recite\pito the 67,890th place.  This record has recently been challenged by a Japanese engineer named Akira Haraguchi, who has claimed to have recited 100,000 places – this is yet to be verified.  One method of memorizing large numbers is to use a mnemonic in which each digit is replaced with a word containing that number of letters.  Here is a well-known example for the first 15 digits of\pi ,attributed to the physicist Sir James Jeans:

“How  I need a drink (alcoholic of course), after the heavy lectures involving quantum  mechanics!”

There is even an entire adaptation of Edgar Allen Poe’s The Raven written in this rather constraining language…quite a feat, especially as it actually reads very well.

So I suppose I should explain what\piactually is before I go any further!  It is simply the ratio of the circumference of a circle to its diameter.  In other words, the circumference of a circle is\pitimes its diameter.   Or, if you prefer,

\pi=3.14159 26535 89793 23846 26433 83279 50288...

…etc.  The decimal expansion never ends, because\piis irrational.  This means that it cannot be expressed as a fraction, and that it has an infinite non-repeating decimal expansion.

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The Cardinality of the Continuum

February 24, 2010

A nice grand title to pique your interest!  After some thought and a couple of conversations, I have decided to keep this blog very much aimed at the layman; the thinking being that I don’t particularly want to write hard maths in my spare time, mathematicians don’t particularly want to read hard maths in their spare time, and non-mathematicians definitely don’t want to read hard maths ever.

My PhD supervisor recently appeared on a BBC programme about infinity, which, while good viewing, was rather over-ambitious, and so had to skip over some interesting stuff.  I thought I’d fill in some of the gaps in this post.  So what is this continuum?  Technically a continuum can be anything that is continuous – that is it goes through smooth, infinitesimally gradual transitions, and has no discontinuities or “jumps”.  But in reality the word is rarely used outside of Star Trek and mathematics.  The continuum I will write about is not the space-time continuum, but the real numbers,\mathbb{R}.  In my post onp-adic numbers, I mentioned that completeness is an important mathematical attribute for a space of numbers to possess.  The notion of continuity is even more fundamental, and people often refer to mathematics as having 2 distinct branches: continuous mathematics, and discrete – or discontinuous – mathematics.

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Mathematics in Music

February 12, 2010

Leaving aside the topic ofp-adic numbers (I feel as though I should learn more about it myself before I make any mistakes), I’m going to get back to a subject I hinted at a couple of posts back: the role of mathematics in music.  When I tell people that I studied music before switching to mathematics, they often say something along the lines of that the subjects are very similar/interconnected/both use “the left side of the brain”.   This isn’t really quite as true as seems to be commonly thought: you can certainly find a lot of mathematical patterns and structure in music; but so can you in any art, and indeed – arguably – in anything if you look hard enough!  And while I am personally averse to “side of the brain” arguments, if we are stooping to that level then I would argue that there is a creative right-side element integral to the creation and appreciation of music which is largely absent from mathematics.

However, it is true that many mathematicians are also involved in music – especially classical music – to some degree .  And it is true that the mathematical and logical structure of music is much more apparent and easier to appreciate than with other art-forms.   One of the best ways to see this is to look at tuning systems.

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