Metrics

Hi. I wanted to avoid making this blog become little more than the diary of an ever-repeating cycle: I got behind schedule; I had a TMA due in and stayed up all night to finish; I got the results back. That certainly seems to be the way it's been going recently, so I have decided this time, to talk about some of the actual maths we've been up to lately, namely 'metrics'.

Why metrics you may ask if you're familiar with them. What are metrics you may ask if you're not. Good questions. The easy answer to the first is that I quite enjoyed them. The easy answer to the second is below. (It should probably be noted here, that I don't have the books with me, so I'm likely to make some mistakes...)

Metrics are a generalisation of the concept of distance between two points, that satisfy three axioms:
(M1) d(a, b) ≥ 0 with equality if and only if a = b.
(M2) d(a, b) = d(b, a)
(M3) d(a, c) ≤ d(a, b) + d(b, c)
So basically, and also quite obviously, the distance between two points a and b is always non-negative, and is 0 only if the points are in the same place. The distance from a and b is the same as from b to a. And yes, (M3) is our old friend the Triangle Inequality. The direct distance between two points a and c, is not more than it would be via a third point b.

Metrics have been my favourite part of the topology course thus far. To begin with, even within the highly abstract realm of topology, metrics have a very easy to grasp real-world basis, since after all, we talk in terms of distance all the time (as it goes, the basic distance we use is the 'Euclidian metric'). One of the first different ones presented in the text is the 'taxicab metric' (e₁(a,b) = |b₁ – a₁| + |b₂ – a₂|, the sum of the horizontal and vertical distances between a and b), which gives us the distance a taxi would travel to get from a to b in a city based on a block design, where roads only run in two perpendicular directions. Indeed, without knowing, we all use such an alternate measure of distance (albeit one that's a bit harder to define than the taxicab metric) which is the distance between two places via the roads. So the concept of different definitions of distance is really not all that alien.

With a particular metric, one can define an open ball B_d(a, r), which is the set of all points with distance less than r (the radius) from point a, a sphere S_d(a, r), which is the set of points with distance r from point a, and a closed ball B_d[a, r], which is the set of points with distance less than or equal to r from point a. Or, simply the union of the open ball and sphere. In two and three dimensions, things are pretty intuitive; in the plane, these are a disc, circle, and disc with boundary respectively, which fits nicely with our idea of balls and spheres as being round; in 3D, it can be pictured as something like the air inside a (round) balloon, the rubber of the balloon as the sphere, and then both together as the closed ball. Take it down to one dimension, on the real line, and things are not so round. The sphere is just two points, a –  r and a + r, but it makes sense considering the definition, because they're the only points with distance exactly r from point a. Well that was all with the Euclidian metric. But what about other metrics? Well, the definitions work just as well when we have another type of distance, another metric. With the taxicab metric, for example, balls and spheres end up as squares rotated by pi over 4. Squares balanced on a corner. And on it goes, getting less and less ball-like as we take less familiar metrics.

Which is the charming thing about metrics, the weirdness of abstraction: take for example, distances between infinite sequences of 1s and 0s, or distances between functions! Not something the average human considers on a daily basis, true, but defined appropriately, they can be shown to satisfy (M1) to (M3) and therefore qualify as metrics. If memory serves, the distance between infinite sequences of 1s and 0s was something like: given two such sequences a and b, then d(a, b) = 2^-n, where n is the number of the position where they first differ. So if a = 1,1,1,0,1,0,0,0..... and b = 1,1,1,1,1,0,0,0....., then they first differ at the 4th place, so their distance apart is 2^-4 = 0.0625. (Is that right? I can't remember, but it's something along those lines anyway).

I'm starting to drone on a bit, but I wanted at least, to mention the thing that first made me ponder writing about metrics, and that was the interesting one we encountered recently, namely the 'vine-picker's metric', which was in our recent TMA. I figure it was long enough ago that it's not a problem to discuss it, and anyway, I'm not going to go any further into the question than the info on the assignment booklet. A week or so after the TMA was done, I cleaned some space on my blackboard and had a little play with the vine-picker's metric. It is defined as:
d(x, y) = |x₁ – y₁| if x₂ = y₂, or |x₁| + |x₂ – y₂| + |y₁| if x₂  ≠  y₂.
Below is a photo of my blackboard. On the left is the rule and graph, and on the right is some vine-picker distances worked out for points on the graph.

So, we end up with a situation that's as if there is something stopping you from moving in the up-down direction, unless you are above the origin, like there are impenetrable grape vines blocking your way, and the only gap between them runs vertically through the origin (for the record, my diagram simplifies it somewhat; in this case the points have integer coordinates, so the vines could be drawn in).  To travel from one point to another in vine-picker metric terms, you therefore have two cases: if the points are at the same "height", (x₂ = y₂), you just go directly horizontal, but if they're not at the same "height" (x₂  ≠  y₂), then you must walk horizontally to the axis, then up (or down) to the appropriate height, and then back out to the second point. The example on the blackboard demonstrates d((2, 3), (8, 3)) = 6 and d((5, 1), (3, 5)) = 12.

Of course, the vine-picker's metric satisfies (M1) to (M3). We had to prove some stuff about it in the TMA.

Anyway, that's metrics. Very cool indeed. It's only Unit A2 of Topology, so there's clearly plenty more fun where that came from. For what it's worth, it was the surfaces and shapes side of topology that I was really looking forward to (surely that was the case for most of us?). However, since I've been going back through (or indeed simply "through" in certain cases) and more carefully learning block A, I'm beginning to really enjoy the point-set variety.

In Our Time

I've been listening to Radio 4 for more than 5 years now, and (despite loving I'm Sorry I Haven't a A Clue, Laura Solon's Talking and Not Talking and various other 6.30-slot comedies) my favourite program remains In Our Time with Melvyn Bragg. Little can make you feel as sophisticated and intelligent as listening to Baron Bragg and his distinguished academic guests discussing the fall of Carthage, the Observatory at Jaipur, Schopenhauer, or some other deep topic.

Anyway, every now and then, we get treated to a maths topic for 45 minutes, usually involving Ian Stewart and Marcus Du Sautoy. I've heard discussions on Godel, infinity, the Calculus, the Poincare Conjecture, pi and many more subjects In My Time :)

After what seems to have been forever (though I could easily have missed some without noticing), it was a mathsy topic again this morning, namely Game Theory. I'm not overly interested in Game Theory myself, but it was entertaining enough, and gave me something to listen to while doing the washing up and tidying the kitchen. Ian Stewart was there as anticipated, and was as brilliant as ever. I'm quite an Ian Stewart fan for what it's worth. I made the (blindingly obvious) prediction to Kim, that John Nash would get a mention, and lo, at 18 past 9, he did. Other stars included von Neumann, and a tramp.

If anyone's interested, a cut-down-to-thirty-minutes version is usually repeated at 9.30pm tonight. It's also available on the Radio 4 website, as are all the episodes of In Our Time since 1998. Actually, I got rather good at Minesweeper a few years ago when I played it to keep myself visually stimulated (I tend to drop off eventually if the only thing I am doing is listening to something) while going through all the science and maths ones that caught my eye... and there were quite a few. I'm not very cultured, so I didn't bother with the art, philosophy, religion, &c, but I do listen to them if they're "live."

More maths please Melvyn!

Well, I got my assignments back over the last few days, and I'm really happy with the outcome.

I dropped one point on the M336 TMA, which is actually rather annoying, because I just didn't copy out the whole of an answer from my rough version. It was a question asking what transformation you need to do to move one of the 'motifs' on a freize pattern, onto another. The answer, remaining as vague as feasible, was in two parts: the type of transformation and the point about which you apply it. When I copied out my answer (and I did have it on my draft, because I dug out the sheets to check and lo, the full answer was there), I seem to have stopped the sentence short without fully explaining the point where to apply it. I don't begrudge this costing a mark, because I accept it being a vital bit of information. Just annoyed at myself is all. Though what can you really expect, with the last-minute panic to finish, the all-nighters, and all the other (admittedly self-induced) obstacles. Rather, it's surprising I only dropped one point.

My tutor was very kind in her comments, and gave me some useful advice on what to put more effort into in that block, &ct. I've not met her yet, because I was too busy working on catching up with the material, that I had to skip the tutorial last month. Actually, I had to miss the tutorial for both courses as they were on the same day (they were, however, at different times, which was nice as it meant we can go to both). It was a shame to miss the tutorials, as I do like to go, even if it is a bit of a trek to Leeds, and the fact that I can never find Leeds Metropolitan University when I get there. The problem was, no matter how useful they would be, the 7 hours I spent going to them would not get me any further through the study materials, which was my main concern at the time.

As for M338, well, it was nowhere near as bad as I expected, in fact, I just about pipped into the 80s! Not quite distinction territory, but easily high enough to put me in a good mood all day after I opened the envelope (...Kim opened it actually, as I usually make her check the results, when I'm worried about it). I tend to expect the worst about most things, but this time, I really went overboard! I mean, I was even trying to work out what I needed to do, to make up for it, if I only got 50!

In his comments, my tutor said it is a really tough TMA and that I should be pleased with myself for doing as well as I did. Of course, it's his job to encourage, but regardless of motivation, it made me feel good about myself. I kind of figured, if I managed to do okay under the conditions I was in at the time, I can assume that I might have done very well otherwise. Either way, he did warn us back at the start of the course, that Block A is hardest, whereas Block B is easy, and Block C is just "hard."

For what it's worth, I'm really enjoying the surfaces unit, B1. In fact, I will hopefully get through most of the last part this afternoon/evening. It certainly is a chance to breathe after the ultra-dense Block A. In fact, I am going to go and work on it now. I've already spent enough of the day watching snooker, reading (Rendezvous with Rama by Arthur C. Clarke - very cool), and general pottering around on the Internet. It's time to get productive, I reckon.

By the way, for anyone that didn't catch it, Dara O'Briain's School of Hard Sums, that I mentioned last time, was fairly entertaining. Not ground-breaking TV or anything, but hey, it was worthy enough of watching again tomorrow at least. Warning though, the puzzles aren't particularly hard (and I mean, for "normal" people as well as us mathsy types), but titling inaccuracies aside, it was an enjoyable enough 40 minutes, and I have little else on at eight on a Monday.

School of Hard Sums

Hello again,

Just a quickie for anyone who's interested, and doesn't know already.
The practically ubiquitous TV panel show comedy bloke, Dara O'Briain has a new programme tomorrow night on Dave: School of Hard Sums.
http://uktv.co.uk/dave/series/tvseries/257755
I heard him being interviewed about it on Radio 4 t'other day, and it sounded quite interesting.
And look, there's half of Melvyn Bragg's In Our Time maths brigade, Marcus Du Sautoy too...
Note how the set looks like it's right out of an OU video :)

Neil

Edit: Sorry, I just thought. As wrong as I no doubt am, it's wrong of me to give my answer. Deleted.

The best laid plans, eh

Hi

Sorry for being so rubbish at writing in my blog lately. Actually, this is a symptom of a much wider issue I'm going through of being rubbish at doing anything.

Whether it's been work, study, or just life in general. I've had almost zero motivation and very little capacity to stick at something for any useful period of time. Well, that's not totally true I guess, my capacity to get through Twin Peaks on DVD recently has been admirable, but that's hardly productive...

2012 has been tainted by a general malaise so far, and the worst effect it has had is on my studying. Although I've been equally unmotivated with work, I'm still meeting all my deadlines, being too responsible to do otherwise, and too courteous to refuse work when allocated to me. So despite the many times I've not even got started until 4pm, or when I've had three days to do a job but not started until the final day, work has been mostly unaffected.

I've been looking forward to M338 since before I even started MST121, so it's quite upsetting to be so far behind so soon. I even bought the first three units back in December so I could get started... only I didn't. So where exactly am I? Well, I finished my first TMA, more or less on time, and got 90-odd. I was not expecting there to be a TMA based on the first unit of the course only, so it was lucky that my tutor who, coincidentally is not only from Portsmouth as I am, but did his undergraduate degree at Sheffield Uni, like me... (I wonder if it was at the same time because he looks fairly young; I should ask!), emailed to remind us. The comments on it were very encouraging. However, I had to then switch to M336 to catch up for a while, and somewhere along the way I got to last week, at which point the situation was thus:

M336 TMA due 12th April - relevant units complete
M338 TMA due 13th April - only Unit A2 complete (with the TMA based on A2 - A4)

Well, I got a good start on the M336 TMA over the weekend, but had to put it aside for a bit to finish a big job (it was legal, a contract, and a right pain to complete - involving an all-nighter from Monday, until the deadline at 9am Tuesday). I finally finished the work itself on that TMA sometime Tuesday afternoon/evening. But as I was so knackered from the all-nighter right before, I opted to quit for the day and go to bed early, at ten. Wednesday morning, after driving Kim to uni (it's the holidays but she wanted to go in and do some work), I wrote up the M336 TMA in neat. Of course that took hours, as it always does for me, and I got it to the Post Office at about 2.45 on the 11th. Pretty sure it arrived on time though; it usually does.

So then I found myself faced with M338 TMA02, at around 3pm the day before I had to send it, and only a third of the material covered. Not good! Rather than just dive into it head first, I thought I'd speed read those two units before starting. So from 4 to 6, I blitzed my way through A3 on Topological Spaces, and after dinner, from about 7 to 9 on A4, Closed Sets. I can't deny that I took very little of it in... but I was up for a challenge.

It took me from 9.30 until about 12.30 to do the stuff on A2, Metric Spaces, but I'm fairly confident of that part. In fact, I was pretty pleased with myself over how I cracked the question on proving the third metric space axiom is satisfied by the "vine-picker's metric", though I suspect I went the long way round. By now, putting aside the prospect of another all-nighter so soon, I was feeling pretty good. I figured, well it took me three hours to do the part I knew. Maybe it'll only take twice that to do the rest, so six hours on A3 and six on A4. Actually, fast forward a bit and I found myself only 15 points from the end (though with a 9-pointer skipped), at 5.30am. So I actually went to bed for two hours! Finished those fifteen points by about 10am. Took me just over four hours to write up. Leaving me at 2.30 with the question I'd skipped left to try and do, and hopefully get to the Post Office at 4-ish.

Unfortunately, by this point, I could barely hold a pen and write, let alone be capable of abstract thought. A few false starts, then I just decided to get it to sent.

I'm not expecting a very good result. But, I think, perhaps, that this might have been the kick up the bum I needed. I actually came this close to picking up my text books again later that evening, despite the epic burn-out of the week so far. I talked myself out if it and opted to just lay on the sofa that evening with B'elanna our dog, watching Vinnie Jones as a butcher killing people on the underground (in Midnight Meat Train, based on a Clive Barker short story). Not bad actually, but the makers need to realise that CGI gore looks silly; the flying eyeball put me in mind of the health meter on the first few series of the (excellent) kid's show Knightmare.

My aim for catching up is to continue working through the material from where I actually should be by now, and working on getting through the previous stuff at the same time. I do not think it's in my interests to catch up first, as from experience, I know I'll just end up a unit or two behind throughout. Meanwhile, I think I'm still on schedule with M336, and with some luck, I reckon I can keep it that way.

I'm sure either you've given up reading by now, or would like me to finish soon. So I shall.
Hopefully, by the next time I post, I'll be up-to-date on everything, getting great results back, and just generally having a good time. I doubt things will change quite that quickly, but I reckon I'm over the hill now and will start sorting out, whatever it is that's been going wrong... watch this space!

Neil

Edit: Mentioning Knightmare above made me think.... do you guys remember The Adventure Game in the early 80s, set on the planet Arg? That must've been the highlight of my week when I was little....

Edit 2: The attentive will note that I've been discussing M336, not S282 Astronomy. Well spotted. Actually, I changed my choice within a week or so, due to concerns over the transitional fee rules.

The Hiatus is Over

Hi all!

Gosh, I really have been away for a long time. I knew I'd left it a while since last posting, but only just now realised that it was way back on 4th July that I last wrote anything.

Many things have happened in the meantime, the most significant of which is that I got married on 12th August. It was a great day except for the New Forest weather which tried to ruin things right around the time the photos were to be taken. Some of you may appreciate that me and Kim have Mobius wedding rings. While it doesn't do it justice, I've added a picture of mine below (please excuse the state of my fingers, I had a fight with a ferret a couple of weeks ago...)

My M381 TMA was, unfortunately, due around this time. My tutor was kind enough to give me an extension because of the wedding, which certainly helped matters, as I had a few large projects to get finished in my work as well. The M337 TMA was due a few weeks earlier, but again, poor timing on my behalf meant I spend the whole weekend preceding, camping with my brother and a few friends. So I had to get that one out over an intensive all-nighter after getting back. Fortunately, I got good marks on both of these.

In September, we went to New Orleans for a week for our honeymoon. Again, TMA deadlines almost perfectly coincided with our departure, and as you probably know, there is no scope for extension on the final assignments. I managed to complete my M337 TMA in Portsmouth after driving our dog there so my parents could doggy sit while we were away; I only got about two hours sleep though because I spent the whole time trying to get my head around conformal mappings and fluid flows (the Mandelbrot set questions were easier at least), which made the prospect of a four hour drive back a little worrying. Alas, this left me little time to finish my final TMA for M381; I was actually working on it at the airport on the morning we left, but had to post it more than 2/3 unfinished. I thought it would be better at least, to send something, however short, than to skip it altogether. I was quite upset at having to do this, but tried to put it out of my mind, and focus on the fun and exciting (but hot and expensive) week ahead. I was under the slightly deluded impression that I'd be able to study while out there, so I took practically all my textbooks with me, as well as past papers and previous TMAs. Ultimately this just added about 15kg to our luggage for nothing, as I did not one minute of study.

After getting back, it was time to begin thinking about the exams. I am rather annoyed at myself in this regard as I put it off and put it off, and started much later than I should have done. The exams themselves were very hard, though I doubt starting my revision earlier would have changed that too much.

The M337 exam started out okay, but I was behind schedule by the time I got to Part 2, and when faced with the optional questions, quickly realised that I was only really qualified for one of them. You'll have to forgive me because I cannot remember now which that was. My second choice was on generalised circles and Mobius transformations, and I doubt I got many points on that one at all. I left it thinking that I'd probably got a Grade 2, but would not be overly surprised by a Grade 3. A twinkle of hope for a Distinction arose in my mind a few times over the ensuing month and a half but I made a point of quickly extinguishing that each time, as it would only lead to tears.

The M381 exam started out badly and then got worse, or at least that's how it felt. I probably did better than I felt I was doing at the time, when in the throes of mid-exam despair. I got off to a bad start when I took almost half an hour over a question involving a proof by induction. The problem was that I knew I could do it. I knew that if I fiddled long enough I could get the two sides of the induction step to match and therefore complete the proof, so I kept on going until I did. Not the most efficient way of tackling an exam, that's for sure. By the time I'd finished, I had only done eight questions, and since they take your best nine answers across the two topics, that did not bode well either. I can't really recall much of the content; probably a psychological defence mechanism. But I do remember my final question being one where an incomplete formal proof is given and you have to fill in which assumptions (or was it the rules, or both, I'm not sure?) were in use at each step. By the time I got to that point, I had about two minutes left. I tried to to ignore the time and tackle it normally, as I was quite good at this when I did similar questins before. But in the end I had to quickly scrawl out some answers in the hopes I'd score some points by luck, or indeed by hoping my subconscious would do the work for me. Anyway, I left that exam very very depressed. I've not had an exam go so badly since my module on Modern Japanese History in my 2nd year at Sheffield University... at least I didn't leave the M381 exam half-way through knowing that even what I did write was little better than gibberish! So I was hoping for a Grade 2, but expecting a Grade 3 (remember also, I'd only done a small part of the final TMA, and as useful as the substitution rule is, it can't fully counteract something like that!).

Anyway, as some of you must know, the results came out on the night of the 30th. That's almost two weeks earlier than my M208 one last year! I was relieved to find that I'd managed a Grade 2 in both cases. I'm in shock regarding the result of M381, and I keep checking back to make sure they haven't changed their minds on me. I like to think they're not allowed to do that, but I still worry.

I've just signed up, over the phone, for M338 Topology, and as a bit of a diversion (as well as a step back, hopefully, in difficulty) S282 Astronomy. I may even be tempted to get out my telescopes over the winter to get me in the mood; there's already been some very starry nights here. I expect Topology to make M337 and M381 look like child's play, but I've been looking forward to the course itself for a long time, as the topic really interests me. Hopefully that, along with the (presumable) lack of weddings and honeymoons next summer, will help me to do well in it.

I hope everyone else who was awaiting results, has got what they hoped for!
If I don't post again in the meantime, have a great Christmas.

Neil

Patterns of study

I had not thought this through beforehand, but somewhere along the line I fell into the pattern of completing a whole unit of one of my courses at a time. No one day one one course, the next day the other, then repeat, nor a bit of each course every day. Nope, I work right through one unit of a course for however many days it takes, then work right through the corresponding unit in the other course (though that's not strictly true, as I'm not at the stage of "corresponding units" any more; not since I cut my losses on M381 and accepted that I'd be behind until the end, but you get the idea).

Pro: I get to keep my mind on one area of maths, e.g. Complex Analysis, for a few days without disrupting my flow by having to get my head around Church's Thesis slap-bang in the middle. Then, within Complex Analysis itself, I get to cover an entire unit, which basically equates into a single topic, residues, Laurent series, or whatever, to its conclusion before switching to something completely different. And these courses are different! Psychologically, this way also helps me feel like I'm getting somewhere, you know? Because I can tick off another completed unit fairly quickly. I'm well aware that this is silly, because technically, I should progress at the same pace either way, but it feels effective at least. A schedule placebo of sorts.

Con: I am prone to not remembering diddly when I get back to one course after maybe a week on the other one. This is worse with M381 because of the fact it alternates Logic / Number Theory / Logic.. It could be three weeks by the time I get back to number theory after our last encounter. That can't be too helpful for driving the material home, can it? Would I have had to look up Euclid's Lemma what seems like a hundred times already, if I were chip-chip-chipping away at M381 every day? Who knows?

Anyway, why am I telling you this? I don't know. It just strikes me as curious, that beavering away in the solitude of our homes, we've probably all fallen into patterns of study quite unique to ourselves. I assume they vary wildly. I'd certainly not be surprised to find that any of my fellow OU students are doing a couple of hours of each course per day. But it doesn't work for me. I'm fairly sure that I started off with the intention of doing it that way. And I didn't make a conscious effort to stop either; but it clearly happened at some point.

Until this year, I'd not had two courses at the same time, so I've never had to tackle differing topics concurrently. I opted to do MST121 and MS221 with an overlap, starting one in the autumn and the other in February, but it wasn't quite the same. Not only were there long periods of one course only, but they also complemented each other in terms of content. Then of course, M208 was a 60-pointer, so it was not an issue there either. In fact, that's the one thing I find attractive about the upcoming level 3 pure maths course; at least in terms of scheduling, there'll be no need to work out how to go about it. Though, I'm not interested in any other way than that, and I'm so glad I started with the OU when I did, so I'm still on for M338 Topology next year before it goes bye-bye.

Neil H