July 20, 2012

Determinant of the Wronskian

The Wronksian of n functions f_1, \dots, f_n is the matrix determinant \begin{vmatrix}f_1&\dots&f_n\\\dots&\dots&\dots\\f_1^{(n-1)}&\dots&f_n^{(n-1)}}\end{vmatrix}. Its derivative is the matrix determinant \begin{vmatrix}f_1&\dots&f_n\\\dots&\dots&\dots\\f_1^{(n)}&\dots&f_n^{(n)}\end{vmatrix} (that is, the previous matrix with a different bottom row). It’s an interesting exercise to prove this, so let’s do that.

We proceed by our old friend, induction. For n=1 (or 0), the case is obvious. Let it be true through n-1. Expand by the bottom row:
\begin{vmatrix}f_1&\dots&f_n\\\dots&\dots&\dots\\f_1^{(n-1)}&\dots&f_n^{(n-1)}\end{vmatrix} = f_1^{(n-1)}\begin{vmatrix}f_2&\dots&f_n\\\dots&\dots&\dots\\f_2^{(n-2)}&\dots&f_n^{(n-2)}\end{vmatrix} + \dots + f_n^{(n-1)}\begin{vmatrix}f_1&\dots&f_{n-1}\\\dots&\dots&\dots\\f_1^{(n-2)}&\dots&f_{n-1}^{(n-2)}\end{vmatrix}
We take the derivative, applying our induction assumption, obtaining \begin{vmatrix}f_1&\dots&f_n\\\dots&\dots&\dots\\f_1^{(n-2)}&\dots&f_n^{(n-2)}\\f_1^{(n)}&\dots&f_n^{(n)}\end{vmatrix}+\left( f_1^{(n-1)}\begin{vmatrix}f_2&\dots&f_n\\\dots&\dots&\dots\\f_2^{(n-1)}&\dots&f_n^{(n-1)}\end{vmatrix} + \dots + f_n^{(n-1)}\begin{vmatrix}f_1&\dots&f_{n-1}\\\dots&\dots&\dots\\f_1^{(n-1)}&\dots&f_{n-1}^{(n-1)}\end{vmatrix}\right)
But the bracketed part is just \begin{vmatrix}f_1&\dots&f_{n-1}\\\dots&\dots&\dots\\f_1^{(n-1)}&\dots&f_{n}^{(n-1)}\\f_1^{(n-1)}&\dots&f_n^{(n-1)}\end{vmatrix}, which is zero as a matrix with repeated rows is singular. We are done.

June 17, 2012

Final entry for Warwick Skills Portfolio Award

Final entry, in which I look back over my posts in an attempt to remember what went on at each of these workshops, spread out somewhat evenly over the year. Let’s do it chronologically, and conclude at the end:

P1: I suppose this is “An Introduction to Skills Development” in the same manner as “An Introduction to Topology”, this workshop stands on its own. I feel that the things I learnt about skills development as opposed to actual skills were merely side-effects of the process - for example, setting goals that actually have benchmarks with which they can be measured (a goal that I learnt several times over across all the workshops and may even only now be beginning to stick). In short, I feel this was more… solid than the meta-skills workshop I was expecting. Then again, the best way to learn the meta-skills is through experience, and you can’t really make action points about something theoretical - “I will make an action point that is measurable” doesn’t really work. I don’t recall SMART goals being empasized here, though (I believe they were in P6) and it does seem like a rather good place for them.

A4: This was the first workshop I attended that I was able to observe a positive long-term result from - the improvement of my ability to listen (and recall) (and learn). It was also nice in that I started something and utterly failed, instead succeeding in the harder task. As a result of my failure (and other things) I got hold of a memory book, which was certainly an enjoyable (and helpful) read.

P4: This was far and away the most enjoyable workshop, and the first I was willing to actively recommend attending (over, say, reading about things on the internet)! More like active training than a seminar (in concordance with the subject matter, I rather liked (and instinctively feared) the “Did you think you’d be sitting at a desk while I lectured you on presentation techniques?”). Bev was very professional and authoritarian from the outset; this was also the only workshop (I attended) that felt like one you’d actually pay to attend. I attended this one just before the holidays so the bulk of the improvement came from the workshop itself - even with that caveat, I think the gain was comparable to months of reflection/application from other workshops - it was very concentrated, and you could see the benefit on others during the presentations given.

P6: This was where I first found SMART goals - something that would probably have helped to come up in P1 (and if it did, to be more emphasized). It was also the first in which I had an honestly negative experience from a change (or at least one that I noticed) that made me fear, for the first time, that I could be doing real damage to myself, in the same vein as being indoctrinated into a religious order, or being taught to think critically about things without being reminded that you should also apply this to your own thought processes. I found I was suppressing my own sadness (and have been for years) but I could remember the previous day, when I was sad, and then I thought “perhaps if I take myself back there, I can recall the emotion”. And so I did, and it worked, and then there I was, sitting there, feeling the weight and then the worry hit me - this suppression was a skill I’d developed, it was useful, I wouldn’t want to lose it. So I suppose this is something the health of which we’ll have to agree to disagree on.

A7: Speed reading! This was probably the second most enjoyable workshop. It was interesting in that on attending I found that I was already using most of the techniques and - to my great surprise - I found that other people weren’t. I was honestly unaware that it was possible for people to read so slowly! I was hoping to increase my reading speed because it seems like a rather useful skill, so it was something of a downer to find out that I was already rather fast. Contained the excellent quote “I wouldn’t have thought you’d do much reading in Maths” (my library card begs to differ :P). Contained an important and obvious lesson (to get better at something, do it a lot). Learning the techniques meant I was able to help one friend with his slow reading, which is always nice.

A1: Attended this one after my lectures were over, which wasn’t the best plan. Still useful, though. Found a method that seemed obvious in retrospect, which is a nice indication it’s a good one. Came with the hint to make a booklist, which is a nice thing to have. Also introduced me to what skimming and scanning actually are - and I, who could already read quickly, had a hell of a time trying to put them into practice. Was I already doing them? How could I tell? Tricky business.

The workshop that I got the most from (discounting any reflection/blogging) was P4, followed by A1. The workshop that I got the most from reflection/blogging was A4: I was able to implement a slight change that lead to a long-term improvement that’s still going. In the meta sense, reflection was also especially helpful in P1 and P6, due to the side-effects mentioned as opposed to actual progress with the plan. Overall, I’d say that the side-effects that occurred as a result of implementing the points were more important and helpful than the results of implementing the points themselves - I feel I’ve learnt more about myself, my thoughts and my actions from them.

I decided to go for it on a whim, but looking back I don’t see any better (realistic) way I could have spent my time. The investment was comparatively small for the results obtained, and I’m glad I made it.


June 15, 2012

Experiments: Game Theory

Follow-up to To ensure that you are honest… from Midgley, Christopher - Pointless twaddle and meaningless diatribes

Realised I could model the previous two cases of product purchase as games. So let’s do that.

Game 1:
  1. You make a bid.
  2. A price is randomly generated.
  3. If the price is below your bid, you pay the price and get the item; otherwise you pay nothing (and don’t get the item).
Model as a game:
  • Price to play the game is X
  • You consider the product worth Y
  • We assume the RNG creates numbers between 1 and 100 inclusive.
  • X% of the time, you make Y+i, where i is the number generated minus 1.
  • (100-X)% of the time, you make X.
  • Your expected reward is one hundredth of (100-X)X+\sum_{i=0}^{X-1}{Y+i}=-\frac{X^2}{2}+XY+\frac{199X}{2}.
  • For payoff we subtract the bid.

At 0, we make nothing. At 100, we make 49.5+Y-X. Our maximum here occurs at X=Y+1/2 which, as we’re limited to integer solutions, still returns the same result.

Game 2:
  1. You make a bid.
  2. A price is randomly generated.
  3. If the price is below your bid, you pay your bid and get the item; otherwise you pay nothing (and don’t get the item).
Model as a game:
  • Price to play the game is X
  • You consider the product worth Y
  • We assume the RNG creates numbers between 1 and 100 inclusive.
  • X% of the time, you make Y.
  • (100-X)% of the time, you make X.
  • Your expected reward is one hundredth of XY+(100-X)X=100X+XY-X^2.
  • For payoff we subtract the bid.

At 0, we make nothing. At 100, we make Y-X. At Y=X, we make X, which is our best.

So in conclusion we have that both have the same maxima, but the former has a higher payoff, which I suppose you’d expect.


June 14, 2012

To ensure that you are honest…

Got back into attending experiments given a brief lull in exams. One psychology, one economics. Tried to do another psychology but I’d done a similar one so that was a no-go.

The psychology one was on coffee and mountains. Mountains are, unfortunately, not my area of expertise. We were asked the height of Mount Everest and questions such as “how high would a mountain have to be to be in the top 10% of mountains?”. Perhaps it was only the 100 tallest mountains, because there are a lot of mountains.

Attached was the following statement:
“To ensure honesty in this, you will be paid according to how accurate you are.”

Precisely, it was something like 30p for 30% correct, 10p for 10% correct. But entering with no information at all (I didn’t know how tall Everest was, so I didn’t even have an upper bound – or a lower one), I decided it was to my advantage to be dishonest: I picked a number (3000m) and entered it for all the questions, thinking that it was at least somewhat likely to show up somewhere. It didn’t, and I didn’t get anything, but I’m not sure how paying for accuracy ensures you put down what you honestly think :P.

The coffee one also had a variant on the “how much would you pay for product X thing”. It normally goes like this:
  1. You make a bid.
  2. A price is randomly generated.
  3. If the price is below your bid, you pay it and get the item; otherwise you pay nothing (and don’t get the item).
    The variant was that you pay your bid instead of the randomly generated price. The former seems to encourage “what is the maximum you’d be comfortable with paying for X”, the latter “what is the average you’d like to pay for X”, with the psychological annoyance of having the random price much lower than what you bid. I wonder whether a series of ‘games’ like that would show any difference in bidding…

The second was an economics one, which are always nice, and can often pay quite well. This one paid you depending on how good you were at simple arithmetic and IQ tests, so rather convenient. In doing the IQ tests, though (Raven but with 30 seconds per question), I realised how much of an advantage it was to have done them before (I had not), as while staring at them I had no prior methods with which to work from, and glanced blindly around. Columns? Rows? Patterns? I did not know. The first question (and they appeared to be in order of difficulty) took me about 10 seconds to figure out that the pattern went down the columns, and about the remainder of the time to pick the correct answer out of the similar boxes. It was fairly fun, though.


May 31, 2012

A1: Final Reflection and a Tired, Tired Trend

Dr Laura Davies; 3rd May.

Missed last week, deadline is tomorrow, suppose 2 mid-entries is okay according to the spec.

First consideration: skimming is much harder on a screen - to do it effectively you need to be able to see as much as possible at the same time. Scrolling does no good. All that’s really happened with this action point is I’ve found a lot of things to not do, and can get something decent working under certain conditions.

In the same vein as a fair few of the other workshops, I started off with a ‘bright idea’ that simply didn’t work out as I’d hoped, so I decided to go back to how it was before. I suppose one could say the experimentation itself was the important thing.

A book list has been made, and hopefully this can be a port of call in remembering where I found something. Unfortunately it lacks a ‘find’ function - if I haven’t written it in the comments section, it’s unlikely to be found. I find opening the PDFs I read in Google Docs adds them to my account, so I can search through those there as well. Additionally, the list is unlikely to get substantially off the ground until exams are over and I can start reading more widely.

These action points remind me of how A4 went - one that didn’t work, one that I’ll continue with, and one that meandered around and possibly had some effect but I don’t really know.

As for what I would have done differently—I suspect attending a workshop on taking better lecture notes (and other things) after all my lectures had finished wasn’t the wisest of moves :).


May 19, 2012

A1: Note–Making and Reading but Rather More of the Latter

Follow-up to A1: Reading and Reading and More of the Same from Midgley, Christopher - Pointless twaddle and meaningless diatribes

Dr Laura Davies; 3rd May.

Recording publishing date may be interesting if just so I can see how old all the books I’m reading are. Decided to go for original date in terms of republishing if the edition hasn’t been updated.

Penwise: Looking at this, it’s just not my preference. I like to have a solid goal in mind: “record all theorems, definitions in a list” for instance. Without anything proper it’s far too easy to just have a set of disorganized (or generally notes that don’t add anything to the course notes) notes heading linearly through the course. I can do well without writing down anything extra, so when I do I want to have a plan, for an unattractive set of notes simply won’t be looked at again :)

Skimming: for material I’m relatively familiar with (in general), it seems to go okay - but for unfamiliar material I’m unable to effectively link and recall the headings unless I slow down and focus on recitation. I could note the headings as I move past them, but that would seem to rather defeat the point :)


May 12, 2012

A1: Reading and Reading and More of the Same

Follow-up to A1: Note–Making and Reading but Rather More of the Former from Midgley, Christopher - Pointless twaddle and meaningless diatribes

Dr Laura Davies; 3rd May.

I haven’t been reading anything (major) from a screen, so point 2 has nothing done :).

books.txt has been created and already looks unwieldy, but I don’t like the loadtime of a proper xls so I’ll probably just try to keep it consistently csv-like for now. Currently ISBN,Name,Author,Comments. ISBN for lookup, name and author so I know what book I’m talking about, comments for self-explanatory.

Skimming: eh, I tried. However, when I see something interesting, the inclination is just to read that immediately, and then get distracted and carry on from there. Ah well, more time is available.


May 04, 2012

Cauchy Condensation Test

The Cauchy condensation test is a convergence test. It’s also, from what I can see, one of the few we didn’t cover in Analysis. Which is a shame, because it’s rather nice.

For a positive non-increasing sequence f(n), the sum \sum_{n=1}^{\infty}f(n) converges if and only if the sum \sum_{n=1}^{\infty}2^nf(2^n) converges.

A sketch proof in one direction should be rather evident: as the sequence is non-increasing, we can replace every group of length 2^n by its initial value. For the reverse, the idea is similar.

Let us consider our old friend \sum_{n=1}^{\infty}\frac{1}{n^p}. Consider \sum_{n=1}^{\infty}2^n(\frac{1}{2^n})^p=\sum_{n=1}^{\infty}2^{n-np}=\sum_{n=1}^{\infty}2^{n(1-p)} which is a geometric series and convergent if and only if p>1.


A1: Note–Making and Reading but Rather More of the Former

Dr Laura Davies; 3rd May.

Looks like a slight (major :P) adaptation of the Cornell method could be useful; in particular having a summary of every page would be nice. Unfortunately I’ve no longer lectures so this won’t be something I’ll be able to put into practice as such. My current favoured techniques are plain linear on lined paper for lectures, and a sort of “patchwork” or “tortoiseshell” method, which takes a blank sheet of A4, starts writing at the top left and, when a section is completed, draws a line around it and continues writing. It’s rather dense, quick and fun to write, but not too simple to read quickly. I tend to transfer notes made this way to the computer afterwards.

One interesting point brought up was noting the books you read: where you read them, and interesting points. This seems like a good idea - I recently found a message in which I’d commented that the book in question contained a countable covering of R2, but hadn’t noted which book (it was probably Apostol’s Analysis, based on the person I was talking to). This seems a good idea, although I wonder how to do it nicely - it would seem easy to get cluttered.

One point made was having a pen in hand to encourage specificity and focus. While reading from a screen this seems a decent idea, while reading from a book I doubt I’ll do it (my specialty is certainly reading from a screen).

The last of interest were comments (however brief) on skimming/scanning. Scanning seems to be what the Survey part of SQ3R intends, instead of “contemplate the issue”, so that’s something to try. My current outlook on the issue is that of the fellow who has learnt to move his hands at ridiculous speeds and so has never learnt of touch-typing, as his current methods allow him to type perfectly well (to use a somewhat strenuous metaphor) - that is, my default reading speed is rather fast, so I’ve never had to skim to be able to read something within a time limit.

So, let us:
#Note books (probably in a .txt, I like those files)
#Pen in hand, paper ready
#SkimQueryReadReciteReview


April 11, 2012

Speed Reading: Finality

Follow-up to Speed Reading: Reflections from Midgley, Christopher - Pointless twaddle and meaningless diatribes

Tutor was Han-Na Cha.

Likely I’ll continue with the nonlinearity, or at least try to, as part of the SQ3R thing that was recommended in the memory book.

Unlikely I’ll continue with the reading more slowly – in particular I’m not particularly sure how to read more slowly. I can’t consciously chunk it less, because this kills my understanding, and any slower speed seems to involve smaller chunks. I’ve come to reading a sentence, then trying to recite it to make sure I actually remembered it. This isn’t actually ‘reading slower’ but it’s close enough, I feel, to the original intent. This was a point that seemed like there should be an obvious way of doing it (surely it should be easier to read slowly instead of quickly?); but I can’t get it to work reasonably. I’ve developed a system that’s worked fairly well so far, and it comes easily to me, so I think I’ll stick with it until circumstances prove me wrong.

Reading symbols is part of doing mathematics so that’s not something I can ignore :D

Hmm, I put most (read: all) of my “aha!” moments for the extant action points into last week’s post..time for some other miscellaneous commentary.

I still feel it’s too easy to let my attention drift away from what I’m reading. On the plus side, I no longer wake up and realise I have no idea where I am – I just stop, dead. I found a few ways of getting around that. The first was timing myself, and reading for a fixed length of time, which I thought worked well until I found out I was somehow getting less done in a greater length of time than before timing. It did, however, get me to stop taking absurdly long breaks (the enjoyment returns are strictly diminishing after, say, 20 minutes off).

The other way was to give up on reading and try reciting what came before. If I can’t remember, that’s a hint to go back :). Otherwise it’ll improve retention without risk of boring further.


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