Category Archives: Factoids

Field(s)notes: Mathematical Box Lunch on the Devil’s Staircase

20 August 2014

Joint lunchbreak with T in the freezing August sun. We sat on the steps of the Warwick Manufacturing Group eating our packed lunches and discussing Sard’s theorem (analysis) which says that for any properly smooth function on the Real line, the set of critical values (on the graph, those are the points at which the line is flat, parallel to the x-axis) has a measure zero (or something to that effect). That means that – although the number of critical values may easily be countably infinite, and even uncountably infinite – it is essentially zero when compared to the whole of the line.  Apparently, once you have too many critical values – imagine driving along a road and stopping everywhere – then the graph of the function is no longer smooth because it becomes broken down into too many infinitesimal segments. I imagine the line becoming very wiggly on a tiny scale – something like the 1.5-dimensional lines of fractals, though I’m not sure this is correct. A better example is the Devil’s Staircase which is very much non-smooth (so nothing prevents it, in theory, to have a large number of critical values).

Figure 1 The Devil's staircase (Source: https://www.math.hmc.edu/funfacts/figures/30003.3.1.gif)

Figure 1 The Devil’s staircase (Source: https://www.math.hmc.edu/funfacts/figures/30003.3.1.gif)

Sard’s theorem brought us to the question whether a flat function with one insanely tall and thin spike in the middle can be called a smooth function… and apparently it can’t, and is instead called a distribution. (http://en.wikipedia.org/wiki/Dirac_delta_function ) The cool thing about the distribution (aka Dirac delta function) is that it is essentially a function over functions, that is, it’s a function-like thing into which you can feed other functions and get results; it is hugely useful in physics. The annoying detail is that, while you can add distributions together, multiply a distribution with a number or even differentiate it, you can’t multiply it by other distributions (or by itself). So δ^δ or other useful operations such as powers or exponentiation make no sense.

"Dirac function approximation" by Oleg Alexandrov - self-made with MATLAB. Licensed under Public domain via Wikimedia Commons - http://commons.wikimedia.org/wiki/File:Dirac_function_approximation.gif#mediaviewer/File:Dirac_function_approximation.gif

“Dirac function approximation” by Oleg Alexandrov – self-made with MATLAB. Licensed under Public domain via Wikimedia Commons – http://commons.wikimedia.org/wiki/File:Dirac_function_approximation.gif#mediaviewer/File:Dirac_function_approximation.gif

This then brought me to an attempt to grasp the meaning of Martin Hairer’s contribution which brought him (and the Warwick Maths department) one of this year’s four Fields Medals. However, I won’t be summarising it here… I am far from being able to understand the essence of his 180-page manuscript which “must have been downloaded into his brain by a more intelligent alien race” (www.simonsfoundation.org/quanta/20140812-in-mathematical-noise-one-who-heard-music ) and lays out a neat theory of hitherto unknown regularity structures underpinning the previously unruly stochastic partial differential equations… In my defence, a friend who is a lecturer in computer science, said that of course he himself could’t understand the actual contribution either.

While T was drawing the naughty spiky delta nought function on the blackboard, Ian Stewart wandered past, unpacking some mail and recycling the package. I’ve learnt not to be star-struck when sitting around in the Warwick Maths department, mainly thanks to the fact that everyone is very friendly, but still, you don’t get Ian Stewarts wandering past you all the time (you do, if you work at Warwick Maths :p). In the book exchange box under the staircase in the middle of the Common Room, there is a German translation of Ian Stewart’s “Equations that changed the world”. I briefly considered nicking it and then got distracted by hugs at the blackboard (which, as I learnt yesterday from Christopher Zeeman’s essay http://www2.warwick.ac.uk/fac/sci/maths/general/institute/histories-small.pdf , are actually greenglass).

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How mathematics got its “Nobel” prize

Cool and informative article about the Fields Medal by Michael Barany

http://www.nytimes.com/2014/08/10/opinion/sunday/how-math-got-its-nobel-.html?smid=pl-share&_r=0

A computer passed a Turing test!

Yesterday (which was a public holiday here in Germany) I was so excited when I caught a glimpse of a Guardian article which claimed that a computer had passed a Turing test for the first time. Today (which is no longer a holiday, so I should be reading more serious stuff, but this piece news was too exciting to let pass by unnoticed) I read a better journalistic summary in the Vice magazine of what actually happened. Of course, the Turing test is itself is not stable unobjective because the measure of success is a computer being able to fool a jury of human interrogators.  Turing wrote the following in his 1950 paper (quoted in Vice, you can read Turing’s original paper here):

We now ask the question, “What will happen when a machine takes the part of [the test subject] in this game?” Will the interrogator decide wrongly as often when the game is played like this as he does when the game is played between a man and a woman? These questions replace our original, “Can machines think?”

As Vice author Martin Robbins point out, “the key words here are “reliably” and “often.” Turing didn’t ask whether a machine could ever, on a single occasion, convince a human judge that it too is human; he asked whether a machine could do so reliably. “

So, the jury is still out… still bloody exciting, though!

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So what is mathematics about? (aka avoiding blogging about my fieldwork)

While I’m doing my fieldwork (interviewing mathematicians and computer scientists!), and far too busy to actually write about how it is going (urgently need sleep! Too into the process to decide what I can publicly write about, and what I can’t!), I’m going to be spamming with mathematical readings instead… sneaky plan, eh? So here is a nice and very readable article discussing ” what mathematics is actually about”. I think it explains the complicated notions of nominalism. Platonism and Aristotelian realism pretty well: http://aeon.co/magazine/world-views/what-is-left-for-mathematics-to-be-about/

Why do we (still) need to know more about female scientists?

Seemingly progressive posts such as “Eight women scientists that you need to know about” make my blood boil with anger. True, I had only heard about four of these eight scientists. You should read the article, it’s short, informative, and thought-provoking, and has inspiring historic portraits of the scientists at work – pictures which would have made at least one male relative of theirs mutter “she should be in the kitchen, not in the lab”. Their life histories made for an enjoyable and useful read on a Monday morning (and yes, that counted as research reading, aren’t I lucky). But this article was also a grim reminder of the fact that feminism has frozen in the first mile of a double marathon towards gender equality.

Why do we need to know more about female scientists?

Because of articles like this – how about some more female scientists? Are there really only eight?! And how about trying to read an article about the 800 or more male scientists who have made awesome discoveries that we also need to know about – I’m sure there’s lots that we don’t know about them? But no, that would not make for a nice news item or facebook trending post take because it would take longer than a Monday morning coffee break to read.

Because women who succeed in doing what they like and are good at (scientists or others) are still newsworthy. WTF?! Surely, not the ones who excel in anything related to the home, food or childrearing, that’s not news – women are just naturally good at it, haha.

Because becoming a scientist is a hard thing, and being a woman unfortunately continues to create more invisible barriers to a successful practicing of science than being a man does.

Because, if you take ten minutes to read their bios on Wikipedia, you will notice that most of them were at some point excluded, denied recognition, or discriminated against on the basis of being women. Nothing to do with their research, that was OK – in fact, it was good enough for others to gain credit for it sometimes. WTF?!

Because yesterday, when I heard that some friends have recently had a baby, I instinctively asked “girl or boy?” even though I can’t think of even one reason why the answer to that question should make a difference. But of course it will. (Un)helpful statistics, stereotypes, expectations, images and key words describing the likely life course, appearance, occupation, interests and possible futures open to persons of the male and female genders spring to mind immediately upon determining the sex of a newborn. Will people still care about that when that baby is an adult and wants, for example, to work as an astronomer? I hope not, but I’m afraid that they will.

Oh, just one thing <clambers onto soap box again>. Wikipedia has a special entry on “Female scientists before the 21st Century”. It seems to suggest that it has become easy enough – or at least relatively easier – for women to be scientists in the past couple of decades than it was previously. And it has. But it has not become equally easy to men, and it has not become sufficiently easy. 

And then, intersectionality. Combine class, race, wealth, disability, sexual orientation, and a bunch of other things that also mess up with our futures, making sure that the most talented, hard-working or brilliant people don’t  have a better chance.

We’re still far from a time when a person’s gender will not be the first thing we notice about them. Yes, there are numerous exceptions – but the point is, we have gone far in promoting exceptions, but we have not yet managed to create a world that supports a regularity, a world in which the particular set of sexual organs, secondary sexual characteristics and learned behaviours have no bearing to how well someone does their job.

Now, off that soapbox and back to my research desk before I evaporate in a puff of angry steam.

8-female-scientists

P.S. Maria Mitchell – first American professional astronomer who was female. And another reason why Quakers are cool. http://en.wikipedia.org/wiki/Maria_Mitchell

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Birthdays, binomial distributions, and romantic mathematicians

Suppose that four out of your 402 facebook friends have birthdays today. What’s the likelihood probability of that happening?  It’s not enough to just take (1/365)^3 because that doesn’t take into account how big your sample of facebook friends is. To find out, we need the binomial distribution: http://en.wikipedia.org/wiki/Binomial_distribution (the probability we need is the formula just after the table of contents with n = 402, k = 3, and p = 1/365). Plug the numbers in and you get 402!/6*399! * 48627125 * 0.997^399. How on earth does one calculate this by hand? I got stuck. But then my other half (who is a mathematician and who told I need the binomial distribution in the first place), wrote and emailed me a little Python program to calculate this. A mathematical/programming gift. Better than chocolates! And the likelihood is 0.0739604817154 – or just a bit more than 7%. That’s quite rare indeed, given that with 402 friends we have more people in need of birthdays than there are spare days in the year.

P.S. Actually, this formula is for sampling without replacement – but in this case we sample one friend at a time and then discard the name out of the pile, so  there is one name less each time – which means that the draws are not independent. So we actually get a hypergeometric distribution, instead of a binomial one. However, Wikipedia claims that “for N much larger than n, the binomial distribution is a good approximation, and widely used”, so I’m happy with that.

P.P.S. In the first draft I used “likelihood” as a synonym of  probability. As you see in this one, that was WRONG. Dammit, I’m such an imprecise social scientist.

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Maths, Math or Mathematics?

I hear different opinions. Is it merely a matter of preference, and what are the reasons behind the different preferences? It seems that “maths” is preferred by British speakers and “math” by American ones. I don’t know about Canadian and other English speakers. Any ideas?

By the way, the title of this blog is “matters mathematical” (a) to avoid both short versions of the word “mathematics” and (b) because “Coffee and Pi” was already taken. Oh, and it’s a quote from this song:

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Ook? Ook!

Brainfuck is an aptly named minimalist programming language based on the formal programming language P′′. It is as simple as its notation – cumbersome. It only uses 8 commands:

Character Meaning
> increment the data pointer (to point to the next cell to the right).
< decrement the data pointer (to point to the next cell to the left).
+ increment (increase by one) the byte at the data pointer.
– decrement (decrease by one) the byte at the data pointer.
. output the byte at the data pointer.
, accept one byte of input, storing its value in the byte at the data pointer.
[ if the byte at the data pointer is zero, then instead of moving the instruction pointer forward to the next command, jump it forward to the command after the matching ] command.
] if the byte at the data pointer is nonzero, then instead of moving the instruction pointer forward to the next command, jump it back to the command after the matching [ command.

http://en.wikipedia.org/wiki/Brainfuck

There is also an even more esoteric and minimalist Orang-outang version using only three words: Ook., Ook? and Ook! Here is how you would write “Hello, world!” in Ook:

Hello, world! program

Ook. Ook? Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook.
Ook. Ook. Ook. Ook. Ook! Ook? Ook? Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook.
Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook? Ook! Ook! Ook? Ook! Ook? Ook.
Ook! Ook. Ook. Ook? Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook.
Ook. Ook. Ook! Ook? Ook? Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook?
Ook! Ook! Ook? Ook! Ook? Ook. Ook. Ook. Ook! Ook. Ook. Ook. Ook. Ook. Ook. Ook.
Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook! Ook. Ook! Ook. Ook. Ook. Ook. Ook.
Ook. Ook. Ook! Ook. Ook. Ook? Ook. Ook? Ook. Ook? Ook. Ook. Ook. Ook. Ook. Ook.
Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook! Ook? Ook? Ook. Ook. Ook.
Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook? Ook! Ook! Ook? Ook! Ook? Ook. Ook! Ook.
Ook. Ook? Ook. Ook? Ook. Ook? Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook.
Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook! Ook? Ook? Ook. Ook. Ook.
Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook.
Ook. Ook? Ook! Ook! Ook? Ook! Ook? Ook. Ook! Ook! Ook! Ook! Ook! Ook! Ook! Ook.
Ook? Ook. Ook? Ook. Ook? Ook. Ook? Ook. Ook! Ook. Ook. Ook. Ook. Ook. Ook. Ook.
Ook! Ook. Ook! Ook! Ook! Ook! Ook! Ook! Ook! Ook! Ook! Ook! Ook! Ook! Ook! Ook.
Ook! Ook! Ook! Ook! Ook! Ook! Ook! Ook! Ook! Ook! Ook! Ook! Ook! Ook! Ook! Ook!
Ook! Ook. Ook. Ook? Ook. Ook? Ook. Ook. Ook! Ook.
http://esolangs.org/wiki/Ook!

If you want to know what the hell esoteric programming languages are, read this wiki http://esolangs.org/wiki/Main_Page.

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The year of code

Yearofcode

Did you know it was the Year of Code?

I can’t really code. But it makes life so much easier (and cheaper). If your job requires using computers for anything, then learning a bit about coding will help you do more stuff, not rely on others for help, be faster and more efficient on the computer, and, eventually, spend less time on it. And it’s fun because you get the computer to do things.  It’s like training a dog – only in fact you are training yourself, and not the dog. Strangely, I haven’t been able to find much research about the addictive potential of coding, apart from this now old book from 1989 by Margaret A. Shotton and this book about Hackers by Paul Taylor – although several friends who have done programming swear that it can be a highly addictive activity.

Well, it’s not that much fun, if you have health problems with your hands, arms, joints or back like me, so it is a bit of a Catch 22. This – and also the fact that I ended up working as a social scientist specialising in qualitative research – is why I don’t know much coding.  Thankfully, my friends do, and so does Google. By pestering friends and Google I’ve been able to do some small bits of HTML coding, and write hundreds of pages in LaTeX (without losing any work or ending up with hideous formatting – MSOffice, it’s your turn to blush). I tried to learn R last month and although it didn’t go very well, I’ll go back to it soon, because there are some awesome extensions for R that don’t exist on “button-based” data analysis programmes, made “especially” for us, social scientists… One in particular, TramineR, is so awesome and relevant for my work that I’m dreaming of being able to use it. Not to mention how often SPSS and NVIVO crash and how expensive they are for anyone who isn’t attached to a rich institution which can buy the packages for its employees. And- meh – they don’t work on Linux, while R and LaTeX have no problem with different platforms.

I really think that social science students and researchers in the UK, in general, could do with more knowledge about how to use computers to their own benefit. One reason why the existing packages are so, well, bad, is because the market is not educated enough. I’m told that the quality of coffee in the UK has soared in the last two decades. Why? Because consumers have become more demanding. I’m sure that one day when more social scientists and other people who need computers for their daily lives start being a bit more discerning about the software they use,  someone out there, or even one of us, will gather their wits and design better software.

It might be a better idea to get a keen pupil teach a class on coding, and not a teacher who is new to it, but hey. If “2014 – the year of code” succeeds in getting more students and teachers to learn code, then  with all its flaws it is a fantastic initiative (watch the video…but try not to headdesk when you realise that its director can’t code yet). Knowing some code it’s like knowing a bit of swimming – won’t hurt you (unless you have an underlying health condition, and even then can be beneficial under supervision), makes life more fun, and heck, it can even save your arse. So if like me you know little or no coding, do check out the Code Academy. And if your word document has ever crashed on you, have a peek at the marvellous thing called LaTeX [pronounced “leitek”].

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