This page represents only my own views, and not those of any university or other body.
Posted Thursday 29th October 2009 at 5.46pm
Pianos and spanning trees
There's a graduate lecture course running at Bath (and, thanks to the taught course centre, simultaneously at Oxford, Warwick and Bristol) this term called "The Uniform Spanning Tree and related models", run by Antal Jarai. I always find it very hard to concentrate for these things, because they go on for two hours in one warm, quiet room... and the high-tech TCC lectures are worse than the old-school ones because generally no-one dares speak except for the lecturer. This term I have too much to do already, so I haven't had time to catch up with the lecture notes out of the lectures - basically I've just been going along, sitting there and trying to absorb as much as possible. It turns out (I decided today) that this isn't a problem for this course. Lots of the proofs seem, when you're sitting there trying to stay awake in an environment tailor-made for sleep, quite boring. But if you think about them, you realise that if you were given this setup and told to prove one of the main theorems, you would have no idea where to start. The methods for the proofs are almost off-the-wall. It's just that once you see how the proof's going, there's a fair bit of messing about (and the TCC room doesn't help there with its high-tech board that's obviously better than a blackboard for telling someone miles away what's going on, but also far better than a blackboard at slowing everything down by being generally lame and getting things wrong) and concentrating on the details is a bit of a killer. So I've started trying to get the idea of the proof at the start, and then wondering for a while how one might have come up with that method, before more or less switching off till the proof gets to the end. It means I'm able to last to the end of the lectures without completely blanking out, and get a decent overview of what's clearly a really nice subject with a huge variety of ideas.
Something else came back to me during today's lecture. I suddenly remembered being sat in Irena Borzym's office in Catz College, in a very easy Topics in Analysis supervision, and complaining about how my director of studies had made me do all the courses in the second year when people from other colleges (like Catz) only had to do slightly over half of them (which was a much better way of getting a good mark in the second year exams). Anyway, she just laughed and said that I'd be grateful one day. And today, as Antal went through a proof that used a load of linear algebra, analysis and probability, with a sprinkling of geometry and numerical analysis, I thought: well, I guess she was right.
In completely unrelated news (except that it's a bit random), the BBC website keeps telling me about semi-humorous (I think) but also semi-useful or informative articles. There was one yesterday about the four colour theorem, and another today about "viewing and purchasing an upright piano". All good stuff, I remember having a reeeeaaaaally crap piano for years when I was learning. Unfortunately the internet didn't really exist back then so I didn't know this stuff. Any seven-year-olds reading my blog and wanting to know how to choose a piano - well, you could do worse than reading this article!
Finally, just when I though PhD comics was past its best, he's come up with another good one:
Posted Thursday 29th October 2009 at 11.39am
Mad car parking skillz
This problem is taken almost word-for-word from David Williams' book "Weighing the Odds". Simon, my supervisor, was going to put it on the third problem sheet for the second year probability course but decided against it - it might appear on a future sheet but there's no reason why the students shouldn't be able to see it here beforehand so here goes:
There are sites 1,2,...,n spaced a car's length apart. A car occupies the space between two numbers, so the interval [i,i+1] for some i=1,2,...,n-1. Once a car is parked in [i,i+1] no other cars can then park in [i-1,i] or [i+1,i+2].
The car park starts empty. Then the first driver comes along and chooses one of the n-1 available spaces uniformly at random (so with probability 1/(n-1) each). Then another driver comes along, and chooses one of the remaining available spaces uniformly at random. Then the third driver comes along, and so on until sites no good for parking are left.
Show that if n=5,
P(cars parked in [1,2] and [3,4]) = P(cars parked in [2,3] and [4,5]) = 3/8
and
P(cars parked in [1,2] and [4,5]) = 1/4.
Now, for general n, let p(n) = P(right-most site ends up isolated), that is p(n) is the probability that at the end no-one has parked (indeed no-one can park) in [n-1,n]. Show that
(n-1)p(n) = p(1) + p(2) + ... + p(n-2)
and deduce that
p(n) = 1 - 1 + 1/2! - 1/3! + 1/4! - 1/5! + ...
which converges to exp(-1).
Now let p(i,n) = P(site i ends up isolated), that is the probability that by the end there are no cars in [i-1,i] or [i,i+1].
"Argue convincingly" that p(i,n) = p(i)p(n-i+1)
(which is close to exp(-2) for almost all i when n is large).
This obviously depends on your interpretation of "argue convincingly". The general idea is quite easy to see after thinking for a while; proving it rigorously is slightly harder but doesn't use anything a first year student (in fact most A-level students) wouldn't know. It's fun though.
Also fun is this parking game. I remember playing it against my housemates in my third year of undergrad. Good times. I scored 384 just now.
Posted Wednesday 21st October 2009 at 5.12pm
Bikes
There seem to be some bike thieves round where I live. A few weeks ago I was walking home and found my bike about 20 yards down the road from my house; slightly confused, I picked it up and put it back where I had left it. Then yesterday, while I was still at work, one of my neighbours knocked on the door and told my housemates that they had seen some people who they thought were trying to steal my bike, and that my bike was now lying on the pavement outside our house. My housemates thankfully brought the bike inside. I had been leaving the bike (just locked to itself) in the front garden, so from now on I'll put it round the back instead where it's harder to get to. But it does beg the question: how incompetent can bike thieves be? I mean, bike theft is pretty easy to do if you've got some kind of lock-cutting device. You walk up to a bike when there's no-one around, cut the lock, and cycle off. Even if anyone tries to stop you, you then have the advantage that you now have a bike and they don't. What you don't do is pick up the bike, move it a few yards to alert everyone's attention, then go off home and get your lock cutters, giving the owner time to rescue his bike.
These photos of comedy cycle lanes also made me laugh.
Posted Sunday 18th October 2009 at 9.09pm
Ideas
I have this joke that I often use in my talks, about how my supervisor says spine martingales make everything look easy and make people think "what's he been pissing around at for three years?" (Maybe I have to think of a new joke now I've used that one up here.) Well, I'm currently rewriting our "unscaled growth along paths" paper - it's almost done but I've been trying to come up with some interesting examples to put the results into context. I think it's almost done now, I've just been trying to finalize one last thing which involves going back through the proofs in a certain critical case. And I got to part of the proof which I hadn't really thought about in a while, and tried to get my head around it in this special case, and I was like, how the hell did I ever think that up? I tried to take a shortcut with the special case and couldn't do it at all, so how did I ever get the general case to work? So I thought back and I remember vaguely standing in Simon's office, and we had this drawing on the board of some curve with straight lines drawn all over the place, step functions below it, step functions above it, and both of us going "yeah, it ought to work...", and then I remember sitting in my office trying to turn that picture (or a subpicture of that picture, or an alteration of that picture) into maths. And then I thought, well, yeah. That's how ideas come. Pictures and talking and hard work. And occasionally a eureka moment. But even the eureka moments usually come after lots of pictures and talking and hard work.
PS I happened upon this video of a Flaming Lips gig. It does get a bit of the feeling of being at a Flaming Lips gig across, I think. There are some nice balloon-based moments, like the kid in front catching a balloon and giving it to his short-arse girlfriend to punch, and the guy with the camera suddenly breaking out of his calm camera-holding to punch a balloon himself. (Pity Wayne's voice is shot - although it does get slightly better as it goes along!)
Posted Wednesday 14th October 2009 at 10.58pm
Thoughts for today
So, everyone seems to think Obama shouldn't have got the peace prize. I think he should: what did more for world peace recently than putting a sensible guy in charge of the USA? (I guess by that argument we should give the prize to the American people, but splitting up the cash wouldn't leave much per person.)
I had French class again today. It was a million times better than last week. Most of the already-fluent speakers weren't there. I still panic every time I try to speak, and my brain goes blank and I can't think of any words or what order to put them in or what the correct endings are. But this week it was actually fun (bar the times I tried to speak and ended up stammering / giving up / blushing).
I've been listening to the new Mountain Goats album today. It's good. Also tried listening to the new Flaming Lips again yesterday as it's getting a load of good reviews. It grew on me a bit but I still don't really get it. I remember trying to put a CD on in the car on some family outing once, and everyone complaining, and my mum saying "sometimes you just want a nice tune", which I took as a real blow, because that's what I'm all about, the nice tunes. Call me boring, closed-minded, one-dimensional, I don't care. But don't tell me I'm not all about the nice tunes.
To kill one Pitchfork review with another:
"There's a moment in last year's documentary, The Fearless Freaks, where Wayne Coyne is playing a song he was writing during the time of the Clouds Taste Metallic sessions. With only his strumming to accompany him, Wayne sings, "Cats killing dogs, pigs eating rats..." The song is "Psychiatric Explorations of the Fetus With Needles", and on its way to actualization, it will acquire a weird intro and a stranger instrumental bridge, and will be puffed up large and colorful enough to suit the rest of that big, glowing album. But even as Coyne plays it alone on guitar, you can hear something special.
Listening to At War With the Mystics-- the Flaming Lips' first new album in almost four years, and the product of many months in the studio-- it's difficult to imagine a similarly inspiring glimpse into one of these songs' construction. Much of the record sounds like chords and melodies were written later, as an afterthought to flesh out production experiments. The goofy noises, glitches, and wafts of Wilsonian harmony in "Haven't Got a Clue", for example, seem to be more central to the track's focus than the melody (of which there is almost none) or the lyrics ("Every time you state your case/ The more I want to punch your face"). But the sounds are certainly interesting."
What there doesn't also apply to Embryonic? OK, so the lyrics are probably better on Embryonic than Mystics. But where are the tunes, Wayne?
Posted Saturday 10th October 2009 at 3.35pm
Ladybirds invaded my room today.
Bath uni offers free language courses for postgrads. I started on the advanced course today - I did the improvers and intermediate courses in my first and second years, and so advanced is the only one left to do. Boy was it hard work. Most of the people there had, I think, already spent some time living in France, and their French was far more fluent than mine. It was a real effort to keep track of what people were saying - I was just about able to keep up most of the time but got lost several times. And as for speaking, I'm just so slow! It takes me ages to form sentences - I have to go through "decide what I want to say in English, translate into French, try to remember that word that means ______, try to think of synonyms for _______ that I might be able to translate, check through grammar and correct" before I say anything. I think I'm technically reasonably capable: when I saw a paragraph one of the other attendees had written, I thought to myself "that word should be that, and that should be that" - and then the teacher came over and said exactly the same thing. So I can do it - but I need to do it a hell of a lot faster!
I think I know now what it feels like to not understand in school! I can sympathise with the kids who messed about in lessons and made life hell for the teachers - when you're struggling to keep up it's so tempting just to give up. By the end of the two hours my brain was fried, and that's with barely saying a word myself, just trying to keep track of what the teacher (and others) were talking about.
So should I give up? That'd be the easy option. I have plenty of excuses, the biggest being that I don't really have the time to be taking two hours (plus recovery time!) out every Wednesday. I'm moving to Paris in January and it feels like it would be less embarrassing to learn as I go along over there - at least the French have good reason to be a million times better than me at French! But this latter reason is obviously flawed - if I give up now what makes me think I'll be able to cope when I have people speaking French around me almost all the time, rather than just two hours a week? I think, for now, I'll keep going - it'll make me feel superior to the secondary school class clowns, and what have I got to lose by embarrassing myself with my sssssllllooooowwwww speaking for a few more hours? I just hope I can motivate myself to go along when it does come around again next week. The other option is to find some other people who speak just as slowly as me... maybe I could seek out some stutterers... or some French whales...
PS I made some spicy beef and noodle soup tonight. Weird thing was, it tasted strongly of fish sauce, but I didn't put any fish sauce in because I didn't have any. Maybe Thai-style food is just destined to taste of fish sauce, no matter what - the Thai are just going with the flow.
Posted Monday 5th October 2009 at 11.20pm
Green like a geisha gown
My sister, Corran, has just started university. She's studying maths at Birmingham, and she rang me yesterday to say she'd done all the questions on her first problem sheet except one: how many (natural) numbers less than a million contain a 4 or a 6? I didn't tell her the answer - in fact at first I just said she should think long and hard about it. A few hours later I rang her back and gave her two clues that I said were applicable to many maths problems, not just the one she was struggling with. One: if you can't answer a question, think about whether you can answer any related questions - like the "opposite" question (a first year's intuitive grasp of "opposite" being enough that we needn't worry about exactly what we mean). So for example, can we answer "how many natural numbers less than a million don't contain a 4 or a 6"? And two: if your question can be simplified, can you answer the simpler question? Do we know how many natural numbers less than 10 (don't) contain a 4 or a 6? How about less than 100? 1000?
Anyway, I don't know if she managed to solve the problem in the end but it got me thinking: should I be giving her these clues? All I know is these are now two of my natural reactions when presented with a problem. I don't know how I got those reactions.
More generally, how do we learn to do maths? Clearly it's important to solve lots of problems in a non-prescriptive way, by which I mean that you shouldn't know what method is going to work before you start - the A-level approach of giving you a method and getting you to practise it 20 times, then moving on to the next method, won't do you much good at this stage. But why am I now able to solve a problem in 10 seconds while holding down a conversation over the phone, where seven years ago (I can't believe it's been seven years!) it would have taken me a lot of hard thought and several failed attempts? Is it just because I've seen equivalent problems solved so many times by myself and others, or is it that I have more general problem-solving skills hard-wired into my brain by virtue of having done maths nearly every day for seven years? Maybe it's a bit of both.
Can it be a bad thing to be telling my sister, and my tutees in Bath, general ways of solving problems? Should they be discovering them for themselves? I remember one of my lecturers once claiming that it was important for students not to be able to do all the questions, because otherwise they might get the false impression that maths was easy. I agree with that (although I think the comment was actually after someone had pointed out that one of the lecturer's problems was in fact impossible to solve). But I think we should try to teach ways of solving problems. We should try our best to teach them enough general methods to make the problems as easy as possible. And if we can teach them enough to make the problems easy, then we should be giving them harder problems!
Posted Sunday 27th September 2009 at 10.16pm
What the...?
Posted Wednesday 23rd September 2009 at 8.09pm
Mr and Mrs Akhund
Many congratulations and best wishes to Immad and Fatema, who married yesterday.
Posted Monday 14th September 2009 at 9.39pm
Bangers and mash with spicy mango and apple gravy
No progress on the maths post I promised, still looking for time to go through it properly. So in the meantime I thought I'd post another recipe that I made up today. As usual I just threw things in without bothering to measure, so it's all approximate. More guidelines than actual rules. The below makes enough for one hungry student - adjust as you see fit!
2 sausages
1 large potato
Rosemary (preferably fresh - way nicer than dried)
Lump of butter
Dash olive oil
1 medium-sized onion
Half a leek
1 carrot
1 red chilli
1 chicken stock cube (I recommend the Knorr ones... they're awesome)
Half a small glass of apple juice
1 heaped tablespoon of mango chutney
Other veg (I used a few mangetout, green beans and peas).
Put the sausages in the oven, and the potatoes on to boil. Chop the carrot into big chunks and add to the pan with the potatoes. After 5 mins or so (use the 5 mins to chop the onion, leek, chilli and rosemary) heat up a dash of olive oil and start frying the onion and leek. Fish the carrot out of the boiling water and add to the frying pan. Once the onion starts looking vaguely cooked, add the chilli (you can add this earlier if you want it spicier), then the stock cube and keep adding water a bit at a time (just pour some out of the potato pan). Once the stock cube has been absorbed add the apple juice and mango chutney. Keep stirring and adding water until you've got a sauce-like consistency - you can add some flour to thicken it up if needed.
After around 20 mins of boiling the potatoes should be done, so drain them, add the butter and rosemary, and mash. Add the veg to the gravy for a couple of minutes (or longer if needed) then serve.
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