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There is surely no more reliable way to kill enthusiasm and interest in a subject than to make it a mandatory part of the school curriculum.
This is just one quote from the superb article A Mathematician’s Lament. Go ahead. Read it, you know you want to.
Turns out I’m rather busy this (academic) year which probably explains my lack of posts so far. So to break the silence let’s review the courses I took last term.
Elementary Number Theory
The title of this course is misleading in two ways. Elementary certainly doesn’t mean easy, it supposedly refers to number theory using only elementary methods and let’s face it this is sometimes harder than using more advanced techniques. This incidentally is the other misleading aspect of its title. There’s a great deal of group theory and bits and pieces about rings and fields to be found in the course. Nonetheless this was actually my easiest course last term (simultaneously I know people who called this their hardest course). Overall I enjoyed this course. The lecturer was fairly entertaining and the material covered is pretty neat too.
Metric and Topological Spaces
Of all the courses I took last term this course has the best set of lecture notes. You can easily read through them and most of the course makes perfect sense. Furthermore most proofs are rather short and appear straightforward. On the other hand this course had quite likely some of the hardest exercises. There were a few questions on the problem sheets (assessed ones) that took me several hours to figure out. I don’t know if this is a general characteristic of analysis courses or whether I simply have more of a knack for algebraic courses, but I really have the impression that if you really understand everything in the lecture notes then questions in algebra are often really straightforward whereas questions in analysis can still take ages. Nonetheless, I thought this was a pretty good course, much better than expected (especially considering that the same lecturer gave the worst lecture of my second year when substituting for another lecturer).
Group Theory
This was a much anticipated course as we finally covered things such as conjugation which I had already seen in my first year project. The lecture notes were slightly disorganised, but overall I’m satisfied with the course. Though I still have to get the hang of semi-direct products.
Measure and Integration
This was by far the hardest course I took, not just last term but in my entire academic career. I still don’t fully get all the material in the course, but while revising over the holidays I started to understand a great deal of the material. Now I still have several exercise sheets to go over and see whether this new found understanding can be applied to the exercises.
Hopefully I’ll be able to find some time this term to ensure that I don’t forget everything that was covered in the last term. After all it’s always better to polish material over Easter rather than having to relearn everything. Though I already know that I’ll have to do a lot of work for Galois Theory this term as the lectures were pretty awful so far. Then again I’ve had so many ridiculously bad lecturers over the last few years that I got really good at teaching myself the relevant material.
Now that I’ve completed my second year here at Imperial College in
London I figured I’d update this thing again. Furthermore I’m doing it the lazy way. Since I like to keep my A-level maths teacher posted on how things are going I figured I’d mainly rip off the mail I sent him. So here we go:
This year we’ve had the following courses; Algebra II, Analysis II,
Probability and Statistics II, Numerical Analysis, Vector Calculus,
Differential Equations, Complex Analysis, Rings and Fields.
Algebra II was pretty straightforward as I had seen most of the group
theoretical part while working on my first year project (ie. normal
subgroups, even and odd permutations) and the part on vector spaces
was pretty all right too. So all in all it was one of the more
enjoyable courses.
Analysis II built on what we had done in Analysis I in the first year.
The idea of the course is to provide a rigorous foundation for the
integral and differential calculus. So there are things such as the
Intermediate Value Theorem, the Mean Value Theorem, etc. and the
course more or less ended with the Riemann integral. Next year I’ll
probably take Measure and Integration where we get to cover the
Lebesgue integral and other hopefully exciting measure theoretical
things.
Probability and Statistics II was not quite as boring as its
predecessor but still highly annoying. For some reason the stats
courses seem to have been designed for maximum confusion. While I had
a rough idea of what it was all about at A-level the lecturers have
now managed to make me lose touch with every aspect of the subject.
This course was especially amazing in the sense that it managed to
include the worst aspects of pure and methods courses. On one hand we
are given the rigorous definitions, eg. that the probability space has
to be a sigma algebra but we never build on these rigorous definitions
and instead are just expected to apply a bunch of methods without
really knowing why they work or why we would use them in the first
place. So this is quite likely the last stats course I’ll ever take.
Numerical Analysis was a pretty good course. It’s mainly about
orthogonality and we covered things such as the Gram-Schmidt
algorithm, Givens rotations and QR factorization.
Overall it felt a lot like a pure maths course as everything was
proved and we weren’t subjected to any kind of hand waving.
Vector Calculus and Differential Equations were definitely the two
worst courses this year. Mainly because the lectures and lecture notes
were completely useless. Which is kinda hard to believe considering
that most of the content of these courses is pretty straightforward.
So I essentially had to teach myself these courses out of books, so
far the only courses where I was forced to rely on books. Other than
that I didn’t find them particularly interesting. I have a tendency to
find methods courses mainly pointless as they only teach us methods
without understanding why they work or why we would want to use them
in the first place. Hence it is no surprise that we won’t choose any
methods courses next year.
Complex Analysis turned out to be better than I remembered once I read
through the entire notes at the end of the Easter holidays. The main
problem with that course was the amount of repetition. There is so
much material that we had covered before in Analysis I and II, so it
is no surprise that I stopped paying attention in lectures early on.
Seriously, I can’t believe we wasted the first three lectures on an
introduction of the complex numbers after we covered them in at least
three courses in the first year.
Rings and Fields was the conceptually most difficult course so far but
at the same time this makes it one of the best courses we had the
pleasure of taking. Unfortunately Rings and Modules isn’t running next
year so the only way we get to build on the material we’ve covered in
this course is by taking Galois Theory, Group Representation Theory
and Algebraic Number Theory next year. Though it’s a small price to
pay.
Overall the summer exams went all right, except for Probability and
Stats II which was several orders of magnitude harder than the past
papers we had seen, also most of the other exams were somewhat more
difficult than usual. Nevertheless I am confident that I have passed
all of them.
After the exams we’ve had about four weeks to work on a group project.
When choosing the three areas we’d like to work in I chose Algebra,
Numerical Analysis and Analysis, thinking I really had chosen three
pure courses, but as it turned out choosing Numerical Analysis meant I
ended up doing a highly applied project with some Matlab coding. The
project title was Computing Phase Transition Phenomena in Wetting
Problems. I know a lot more about wetting problems than before but I
can’t really pretend that I understand any of it. At least we won’t
have any group projects anymore. The group I was in was for the most
part great, but the topic was a bit of a downer. The next time we’ll
get to do a project will be the fourth year project that counts for
1/4 of the final year mark and since I’m mainly choosing pure courses
I guess it’s safe to say that I won’t be doing an applied project then.
From next year on we don’t have any compulsory courses so my choices,
based on what courses are supposedly running next year, are:
Metric and Topological Spaces, Measure and Integration, Group Theory,
Elementary Number Theory, Functional Analysis, Galois Theory, Group
Representation Theory, Algebraic Number Theory. I might also be
checking out the lectures for the fluid dynamics courses after I’ve
seen some of the cool simulations the Applied Modelling and
Computation Group are coming up with.
Anyway, all in all it was a pretty good year, though I’m really
looking forward to the exciting courses I’m going to take next year.
It’s been some time since my last post. It’s like I found other ways to procrastinate rather than writing here. Anyway, while I could be complaining about the group project I’m working on, it’s easier to look to the future and ponder next year’s course options.
Let’s start with the awesome facts. From next year on we are talking about 100% choice. There might be some minor restrictions such as only being allowed to take one non-maths option but for all practical purposes we may see this as 100% sheer awesomeness.
I’ll start with the first term. I’ll be taking 4 courses in each term, so I’ll start by giving my 4 most likely choices followed by the B-list. So let’s hit it:
To start with there are 3 courses in the first term that I consider fundamental
Metric & Topological Spaces, Measure & Integration and Elementary Number Theory.
So it’s unlikely that I’ll drop any of these or postpone them to the 4th year as I consider them too important. The first two weren’t anywhere close to my A-list a year or so ago when I first started considering 3rd year options. To be fair, they don’t really look like fun, especially not the one on Metric Spaces as I know the lecturer to be quite awful. Nevertheless I believe their content to be rather important and as such they’re clearly on my A-list.
So what’s my 4th choice for the first term, you may wonder. It’s bound to be
Group Theory
Bit of weird course title, considering that we’ve covered group theory in two courses so far. I liked how when they presented next year’s course options they said: ‘If you enjoyed group theory so far you should definitely take this course. If you didn’t, you still should take it’.
So what’s on my B-list you may wonder. There are only 2 serious contenders
Discrete Maths and Games, Risks and Decisions
Discrete Maths is mainly about elementary coding theory. The lecturer is awesome and still, I probably won’t take it. One good reasons for not taking it is that chances are that this course and the one on Graphs and Optimization will get replaced by a proper course on Combinatorics. So that alone makes it worth waiting an extra year. That and the fact that the syllabus for Discrete Maths isn’t really mind blowing stuff. Games, Risks and Decisions used to be on my A-list for ages. It’s the only reason why I took Probability and Stats II this year and yet it got kicked off my A-list. Now frankly I believe it’s got some fun maths, but at the same time it’s also got some annoying probability distributions kind of stuff. Though the main reason simply is that I believe that I will benefit more from the other options.
So let’s move on to the second term. My choices are bound to be
Functional Analysis, Galois Theory, Group Representation Theory and Algebraic Number Theory.
You may have noticed a pattern in my choices. It’s pretty clear that all my choices are pure maths. There’s no applied, stats or methods courses on my A-list. Technically I don’t really know much about any of these courses. It’s one thing to read the syllabus description and look up stuff online but until one has covered and hopefully understood the material there’s no way of telling whether the courses are anywhere as interesting/fun/important as one thought they would be.
The only thing worth mentioning on the B-list is Scientific Computation. The cool thing about that course is that it’s 100% project work. This would mean one less exam in the summer term, but it would also mean one less awesome maths course.
So these are bound to my choices for all its worth. Choosing courses turned out to be much easier than I first anticipated based on the simple fact that a number of courses that would have been serious contenders aren’t running (for example Ring and Modules, Complex Analysis II and Linear Algebra and Matrices).
In case you’re desperate to see all the exciting courses I chose to miss out on you can find the guide to courses online.
Procrastination, oh sweet procrastination, thou shalst be my downfall.
Obviously revision time has always been the best time to indulge in the pleasures of procrastination. I’m sure the revisionistas will disagree but ’tis most important that one wastes significant portions of one’s day on the internets rather than squeezing obsolete knowledge on the mental hard drive. ’tis in this spirit that I bring you the first installment of this multipart-series – a best of procrastination, if you will.
Disclaimer: Procrastination should not be attempted by people who think it may interfere with them passing their exams.
Originally this was going to be a list of all the wonderful things you can find on the Internets with which to kill time and procrastinate like there’s no tomorrow, but the list was getting longer and longer and so I decided to have several installments, each for the categories under which I listed the links. Today you shall meet
Webcomics
We all know and love comics. Thanks to the web not only can we rss our favourite comics but we can stumble upon comics that are, in some cases, only published on the web, which is why we shall refer to them as webcomics.
Sinfest is my favourite webcomic. As much for the stories as for the drawing. I mean, how many comics feature god, the devil as well as talking cats and dogs? Obviously this comic and the next aren’t exactly politically correct. If you go to the archives and start with the very first episode you’ve got almost 3000 episodes to read through.
Jesus and Mo is most certainly blasphemous, politically incorrect and probably extremely evil. Don’t look at it if you’re easily offended, heck if you’re easily offended you should turn your computer off – don’t you know how dangerous the Internet is? Anyway, back to the comic. It isn’t drawn that well, but the concept is rather interesting; Jesus and Mohammed are roommates and they discuss various things, many of which are of a religious nature.
Diesel Sweeties is kinda weird and includes a girl with a robot boyfriend. The look of the comic reminds one of the computer games of the distant past when computer screens had a much lower resolution. Nevertheless it’s got its amusing moments. Obviously you shouldn’t read that kind of comic if robot sex offends you.
Cyanide and Happiness is alternating between offensive and just plain gross with the occasional silly moment thrown in for good measure.
Least I Could Do is hard to describe. Basically the main character Rayne is the most egocentric person ever and he does some outrageously silly things. Like using ninja stars to send invitations to a party.
Boy on a Stick and Slither and Count Your Sheep is what I’d call really cute stuff. And we all know that every once in a while what one needs is cute stuff.
xkcd describes itself as a webcomic of romance, sarcasm, math, and language. I really like their disclaimer which reads:
Warning: this comic occasionally contains strong language (which may be unsuitable for children), unusual humor (which may be unsuitable for adults), and advanced mathematics (which may be unsuitable for liberal-arts majors).
If the classics are more what you’re after check out Comics.com which labels itself the home of comics on the web.
This should be enough to keep you busy for a while. The next installment will follow whenever it’s ready. In the meantime I have to cover a fair share of maths for the group project which started yesterday. I’ll tell you what it’s about once I figure it out.
Regular readers will have noticed that there was nothing regular to read in the last few weeks. Some may have wondered whether I had been cast into the dark pits of hell. The reason for my absence was only marginally more entertaining; I was busy revising and sitting exams.
Fortunately today was my last exam but instead of going out and partying I will grace these pages with a summary of these very same exams. Being extremely original I will summarise them in the order in which they happened.
Probability and Statistics II
This exam was just insane. It looked like nothing one would have expected. Not like the coursework and not like any of the past papers. Every question was either really obscure or if it looked familiar it was much harder than usual. Everybody hated this exam and I doubt anyone got close to acing it. It was so bad someone should have broken down and cried during the exam. I’m sure most of us felt that way but having someone actually break down makes great fodder for four o’clock tea conversations. The bittersweet thing is that just before the exam I was thinking that it was kind of sad that I wasn’t going to take any more stats courses next year now that I actually really started to understand the material. I guess I didn’t really understand it after all. On an upside; from next year on I’m only taking real maths courses.
Vector Field Theory
This was a reasonably hard paper which begs the question whether the lecturer acted on his threats. He used to complain that no one showed up to the problem solving classes (which are optional) and threatened to make the exam harder if this was to stay that way (thinking that no one showing up was an indication that everyone finds the course easy). Which was stupid on so many levels. First off there was a gap of something like 4-6 weeks in which he didn’t hand out any problem sheets so if we don’t have any problems to solve why would anyone show up to problem solving class? (To clarify material covered in lectures, you say? But no, the lectures and lecture notes were completely useless, no one in their right mind would go to the source to clarify the material) As an aside I and a few of my colleagues like to refer to these simply as problem classes and say they’re only for people who have problems. Anyway, at the end of the day the course covers mostly trivial material so even though the exam was harder than usual it was still mostly straightforward if you understand what you’re doing (having done complex analysis makes understanding some of the material even easier). The only strange bit was question four and I still managed to frantically figure out most of it in the last 5 minutes of the exam.
Differential Equations
That exam was pretty standard. As a matter of fact if you look at the last 5-6 years worth of past papers you always see the same type of questions. So preparing for it wasn’t too difficult. Unfortunately he chose the most annoying ODE for the 5th question (seriously, the Bessel equation is such a pain in the neck). I might’ve still scammed my way to half the marks on that question. The rest was pretty easy.
Complex Analysis
Considering I only started really understanding this course a few weeks ago it could have been much worse. I could do most of the questions though there were always a few tiny things I couldn’t complete. In hindsight I should’ve tried to understand the course significantly earlier. It wouldn’t have necessarily made much of a difference for that particular exam but really understanding the material makes revision so much easier. (This is a non-trivial statement no matter what you think) Furthermore once I started understanding it I also started appreciating it. So let me say this; Complex Analysis is actually really neat stuff.
Analysis II
That exam went all right even though I know of at least two mistakes I’ve made. One of which I realised while waking up two days later, which is an odd way to notice your mistakes (maybe I should nap during exams). The problem with this course was that I really understood it in the first term and didn’t look at it during the second term. Shockingly enough that was enough to forget quite a bit and no longer be as fluent in it as I used to be.
Algebra II
This is the only exam where I am reasonably happy with my performance. But then again this is my favourite subject and the one I’m best at, so I could be expected to do well. The exam was fairly straightforward even though it included a tricky question which I’ve managed to get wrong (after being half way there that is), though it should only be worth a few marks. So all is well, if you disregard an imperfection in one of my proofs which I’ve also realised a few days later when waking up.
Rings and Fields
This exam was mostly all right. Though I was surprised at how much of this material I managed to forget over Easter. Since I used to really understand it at the end of last term I didn’t spend much time revising it over Easter and unfortunately managed to lose touch with it enough to suffer the consequences at least a little bit. Trivia: This was the only exam where all 5 questions fit on one side.
Numerical Analysis
When I took this course in the first term I thought this was going to be one of the easiest exams to do well in. Especially since the lecturer said that one usually only has to ace 4 of the 5 questions to get full marks on the paper. When looking at past papers I quickly realised why this was the case. That exam is notoriously difficult to complete in two hours. Take the first question for example. The second part alone takes ages to compute. It’s just using Givens rotation matrices to solve an overdetermined system. Now considering that we’re not allowed calculators it seems to take forever to do the matrix computations (heck, just writing down all these matrices takes ages). The other questions aren’t much better. They’re crammed with stuff you need to do and none of it is really trivial. To make one more point; I needed an extra booklet for Rings and Fields, whereas for this I almost needed two extra booklets, the only reason I didn’t was that I seriously started cutting down on readability and careful arguments, still could have done with extra time though.
In closing I notice the same thing I did last year. There’s not one exam I’m really happy about. There was only one in which I was done early (Rings and Fields – where early means 10 mins before the end).
So what are the lessons we’ve learned?
Don’t take Stats. Keep revising throughout the year. Look at past papers early as it may influence how to revise more efficiently.
One extra thing I’ve learned which isn’t really related to exams per se is to really understand a subject. I mean really understand it. Take Algebra II for example. I thought I really understood it. I hardly did any revision and it’s still bound to be my best result. Nonetheless there are a number of results in the lecture notes that I do not fully grasp. For the most part you could simply consider them subtleties. But in order to fully understanding a subject you also need to grasp its subtleties. So from next year on I’ll try to really understand the courses I’m taking. Which shouldn’t be impossible, after all it’s not like I’m choosing some nonsense like Stats.
If you thought that posting about these cool plots on Sunday and Monday was going to be enough you were wrong. Though the good news is that I have figured out why the plots look the way they do.
The short answer is: continuity. The long answer is: continuity, obviously.
First for the poor souls out there who haven’t had the pleasure of learning Analysis I will give you the definition for a continuous function:
A function 
Intuitively this means that small changes in the input result in small changes in the output. Now how does this relate to our plots?
First let me tell you what we’re exactly plotting in case you haven’t perused Foxy’s post with which this madness has been set loose.
We compute 
Let me give you an example. Let’s say that 
Now as 
A great way to show what this means is to draw a 3 dimensional plot 
Let’s take a really cute function to illustrate this; on the left is our traditional plot and on the right its 3-dimensional brother (click on it to see it in its full glory):
So a simple way to think about it is that depending on the height of our graph we draw it in a different colour. It’s like comparing it to a skyscraper. The ground floor is drawn in black, the first floor is red, the second green, the third is magenta and the fourth is black again and we start the cycle anew. As the functions we’ve used to plot so far have been predominantly continuous on the whole of 
So from this point of view our Contour Maps of Death have been similar to how hills are indicated on maps by Contour lines.
The great thing is that now that we have figured this out it makes it easier to create certain types of plots and achieve certain designs. I mean, if you have thoroughly digested the information we presented in this post you might easily see which function we’ve plotted in the example we gave.
Okay, given the small interval it’s difficult to tell, so let’s make this more fun. I’ll give you another graph of the same function over a significantly bigger interval but this time instead of plotting
This is a follow-up to yesterday’s post. As I was trying to fall asleep yesterday I was thinking; those plots look pretty good, but they’d look much better if we could make them more colourful. So today I’ve played around with my code and this is the result:
all I had to do to get such a really neat result was changing a few lines of the code I posted yesterday. For simplicities sake I will post the entire new Matlab code; so here it is
a = 3;
k = 0.015;for x = -a:k:a
for y = -a:k:a
z = x+y*i;
f = z*(z-(1+i))*(z-2)*(z-2.5*i);
g = abs(f);
frac = g – fix(g);
h = fix(10*frac);
h2 = mod(h,8);
if h2 == 1
plot(x,y,’.k’,'MarkerSize’,4.5)
hold on;
elseif h2 == 3
plot(x,y,’.r’,'MarkerSize’,4.5)
hold on;
elseif h2 == 5
plot(x,y,’.g’,'MarkerSize’,4.5)
hold on;
elseif h2 == 7
plot(x,y,’.m’,'MarkerSize’,4.5)
hold on;
end
end
end
And here are a few more really cool plots. Just click on the thumbnails to see them in their full size.
Insanely cool and we all know it. I’ve previously told you how much I like the Colbert Report so it doesn’t really come as a surprise that the Daily Show is also one of my favourite shows. And as I really enjoyed Stephen Colbert being interviewed at Harvard’s Kennedy School of Government it was to be expected that I would enjoy Jon Stewart being interviewed by Bill Moyers. As much as I love those two shows on Comedy Central it is also rather nice to see Jon and Stephen talking about their work.
I think it’s something that I just enjoy in general; people talking about their work, be it comedians, writers or scientists, it’s really cool to get to find out more about their work, about how they do it, getting a sneak peak into their mind. Incidentally this is pretty much the only thing I liked about the documentary Fermat’s Last Theorem: mathematicians talking about their work, other than that there wasn’t much in that tv documentary (rather thin on the maths content side, but then again it’s supposed to be accessible – the book on the other hand was much better).
Anyway, sorry for the quick detour, back to what we were talking about; thank PBS and see Bill Moyers interviewing Jon Stewart here.
First off you’ll want to check out Foxy’s post called Contour Maps of Death
to know what this is all about.
The main reason why I post this tribute to his post is that the code he included to create this type of cool plot is for Mathematica and I unfortunately only have Maple and Matlab. So I set out to translate his code so I could create these plots too. I found a simple enough way to do it in Matlab and this is the code of the m-file
a = 3;
for x = -a:0.015:a
for y = -a:0.015:a
z = x+y*i;
f = z*(z-(1+i))*(z-2)*(z-2.5*i);
g = abs(f);
frac = g – fix(g);
h = fix(10*frac);
h2 = mod(h,2);
if h2 == 1
plot(x,y,’.k’,'MarkerSize’,4.5)
hold on;
end
end
end
I haven’t figured out a way to do the same thing with Maple. I believe the approach would have to be slightly different. Still, if you use either Matlab or Mathematica you now have the code to create these cool plots yourself.
On a related note check out Foxy’s blog FoxMaths where you can find plenty of really cool entries on math, be it experimental or else.
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