The Spectre of Math

May 30, 2012

Visualizing complex singularities

Filed under: Hacking,Mathematics — jlebl @ 5:16 pm

I needed a way to visualize which t get hit for a polynomial such as $t^2+zt+z=0$ when z ranges in a simple set such as a square or a circle. That is, really this is a generically two-valued function above the z plane. Of course we can’t just graph it since we don’t have 4 real dimensions (I want t and z to of course be complex). For each complex z, there are generically two complex t above it.

So instead of looking for existing solutions (boring, surely there is a much more refined tool out there) I decided it is the perfect time to learn a bit of Python and checkout how it does math. Surprisingly well it turns out. Look at the code yourself. You will need numpy, cairo, and pygobject. I think except for numpy everything was installed on fedora. To change the polynomial or drawing parameters you need to change the code. It’s not really documented, but it should not be too hard to find where to change things. It’s less than 150 lines long, and you should take into considerations that I’ve never before written a line of python code, so there might be some things which are ugly. I did have the advantage of knowing GTK, though I never used Cairo before and I only vaguely knew how it works. It’s probably an hour or two’s worth coding, the rest of yesterday afternoon was spent on playing around with different polynomials.

What it does is randomly pick z points in a rectangle, by default real and imagnary parts going from -1 to 1. Each z point has a certain color assigned. On the left hand side of the plot you can see the points picked along with their colors. Then it solves the polynomial and plots the two (or more if the polynomial of higher degree) solutions on the right with those colors. It uses the alpha channel on the right so that you get an idea of how often a certain point is picked. Anyway, here is the resulting plot for the polynomial given above:

I am glad to report (or not glad, depending on your point of view) to report that using the code I did find a counterexample to a Lemma I was trying to prove. In fact the counterexample is essentially the polynomial above. That is, I was thinking you’d probably have to have hit every t inside the “outline” of the image if all the roots were 0 at zero. It turns this is not true. In fact there exist polynomials where t points arbitrarily close to zero are not hit even if the outline is pretty big (actually the hypothesis in the lemma were more complicated, but no point in stating them since it’s not true). For example, $t^2+zt+\frac{z}{n}=0$ doesn’t hit a whole neighbourhood of the point $t=-\frac{1}{n}$. Below is the plot for $n=5$. Note that as n goes to infinity the singularity gets close to $t(t+z) = 0$ which is the union of two complex lines.

By the way, be prepared the program eats up quite a bit of ram, it’s very inefficient in what it does, so don’t run it on a very old machine. It will stop plotting points after a while so that it doesn’t bring your machine to its knees if you happen to forget to hit “Stop”. Also it does points in large “bursts” instead of one by one.

Update: I realized after I wrote above that I never wrote a line of python code that I did write a line of python code before. In my evince/vim/synctex setup I did fiddle with some python code that I stole from gedit, but I didn’t really write any new code there rather than just whacking some old code I did not totally understand with a hammer till it fit in the hole that I needed (a round peg will go into a square hole if hit hard enough).

May 12, 2012

Filed under: Hacking,Linux,Technology — jlebl @ 5:02 pm

So … apparently searching an unordered list without any structure whatsoever is supposed to be better than having structure. At least that’s the new GNOME shell design that removes categories, removes any ordering and places icons in pages. The arguments are that it’s hard to categorize things and people use spatial memory to find where things are.

The spatial memory was here before with nautilus. It didn’t work out so great. No people don’t have spatial memory. For example for me, I use a small number of applications often, I put their launchers somewhere easy to reach. The rest of the applications I use rarely if never. No I do not remember where they are, I do not even remember what they are named. E.g. I don’t remember what the ftp client list, but I am not a total moron and I correctly guess to look for it in the “Internet” menu which is managable. Given I’ve used ftp probably once in a year, I do not remember where it is. Another example is when Maia (6 year old) needs a game to play. I never play games, but I have a few installed for these occasions. Do I want to look through an unordered list of 50-100 icons? Hell no. I want to click on “Games” and pick one. 95% or so of applications i have installed I use rarely. I will not “remember” where they are. I don’t want to spend hours trying to sort or organize the list of icons. Isn’t that what the computer can do for me? Vast majority of people (non-geeks) never change their default config, they use it as it came. So they will not organize it unless the computer organizes it for them. I have an android tablet, and this paged interface with icons you have to somehow organize yourself is totally annoying. One of the reasons why I find the tablet unusable (I don’t think I’ve turned it on for a few months now). That interface might work well when you have 10 apps, but it fails miserably when you have 100.

If I could remember that games are on page 4 (after presumably I’ve made a lot of unneeded effort to put them there) I can remember they are in the “Games” category. Actually there I don’t have to memorize it. Why don’t we just number all the buttons in an application since the user could remember what button number 4 that’s right next to button number 3 on window number 5 does. I mean, the user can use spatial memory right?

Now as for “that’s why there is search” … yeah but that only works when you know what you are searching for. I usually know what I am searching for once I found it. It’s this idea that google is the best interface for everything. Google is useful for the web because there are waaaaay too many pages to categorize. That’s not a problem for applications. Search is a compromise. It is a way to find things when there are too many to organize.

Finally, what if I lend my computer to someone to do something quickly. No I am a normal person, so I don’t create a new account. And even if I do create a new account, the default sorting of apps is unlikely to be helpful. If someone just wants to quickly do something that doesn’t involve the icons on the dash, they’re out of luck if I have lots of apps installed. Plus at work I will have a different UI, on my laptop I have a different UI, and any other computer I use will have a different UI. I can’t customize everyone of them just to use them.

As it is, if I had a friend use my computer with gnome-shell they were lost. If it’s made even less usable … thank god for XFCE, though I worry that these moves towards iphonization of the UI will lead to even worse categorization. There are already many .desktop’s with badly filled out Categories field, so there will be less incentive to do it correctly.

May 9, 2012

Determinants

Filed under: Mathematics,Teaching — jlebl @ 8:51 pm

I just feel like ranting about determinant notation. I always get in this mood when preparing a lecture on determinants and I look through various books for ideas on better presentation and the somewhat standard notation makes my skin crawl. Many people think it is a good idea to use

$\left\lvert \begin{matrix} a & b \\ c & d \end{matrix} \right\rvert$

instead of the sane, and hardly any more verbose

$\det \left[ \begin{matrix} a & b \\ c & d \end{matrix} \right]$     or     $\det \left( \left[ \begin{matrix} a & b \\ c & d \end{matrix} \right] \right)$.

Now what’s the problem with the first one.

1) Unless you look carefully you might mistake the vertical lines for brackets and simply see a matrix, not its determinant.

2) vertical lines look like something positive while the determinant is negative.

3) What about 1 by 1 matrces. $|a|$ is a determinant of $[a]$ or is it the absolute value of $a$.

4) What if you want the absolute value of the determinant (something commonly done). Then if you’d write

$\left\lvert\left\lvert \begin{matrix} a & b \\ c & d \end{matrix} \right\rvert\right\rvert$

that looks more like the operator norm of the matrix rather than absolute value of its determinant. So in this case, even those calculus or linear algebra books that use the vertical lines will write:

$\left\lvert \det \left( \left[ \begin{matrix} a & b \\ c & d \end{matrix} \right] \right) \right\rvert$

So now the student might be confused because they don’t expect to see “det” used for determinant (consistency in notation is out the window).

So … if you are teaching linear algebra or writing a book on linear algebra, do the right thing: Don’t use vertical lines.

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