When Sage also fails

+Patrick Honner wrote a post titled: 'When Desmos Fails' which you should go read. In it he shows a quirk about Desmos (a free online graphing calculator) that seems to not be able to correctly graph around the removable discontinuity  $(-2,-1)$ of the following function:

$$f(x)=\frac{x+2}{x^2+3x+2}$$

People who understand this better than me say it might have something to how javascript handles floats...

Anyway I thought I'd see how +Sage Mathematical Software System could handle this. Here's a Sage cell with an interact that allows you to zoom in on the point (click on evaluate and it should run, it doesn't seem to fit too nice embedded in my blog so here's a link to a standalone Sage cell: http://goo.gl/WtezZ4):

It looks like Sage doesn't have the same issues as Desmos does. This is probably not a fair comparison, and needed a bit more work than Desmos (which I can't say I've used a lot) to get running but I thought it was worth taking a look at :)

EDIT: IF you Zoom in more you do get the same behaviour as Desmos! I thought I had zoomed in to the same level as +Patrick Honner did but perhaps I misjudged from his picture :)

Here's the same thing in Sage (when setting $z=10^7$ in the above code):

Plotting complex numbers in Sage

I had a student email me overnight asking how to plot complex numbers in +Sage Mathematical Software System.

I spent a couple of minutes googleing and found various command that would plot complex functions:

f = sqrt(x) + 1 / x complex_plot(sqrt, (-5,5), (-5, 5))

This gives the following plot:

This was however not what my student was asking. They wanted to know how to plot a given set of points in the complex plain (referred to as the Argand plane). A quick google to check if there was anything in Sage pre built for this brought me to this published sheet by +Jason Grout.

I tweaked it slightly so that it was in line with the commands my students have learnt so far and also to include axes legends and put the following in to a function:

def complex_point_plot(pts): 
    """ 
    A function that returns a plot of a list of complex points. 
    Arguments: pts (a list of complex numbers) 
    Outputs: A list plot of the imaginary numbers 
    """ 
    return list_plot([(real(i), imag(i)) for i in pts], axes_labels = ['Re($z$)', 'Im($z$)'], size=30)

This function simply returns a plot as required. Here is a small test with the output:

complex_point_plot([3*I, e^(I*pi), e^(I*3*pi/4), 4-4*I])




Here is some code that will plot the unit circle using the $z=e^{i\theta}$ notation for complex numbers (and the Sage srange command):

pts = [e^(I*(theta)) for theta in srange(0, 2*pi, .1)] 
complex_point_plot(pts) 


Here is the output:

I published all this in a worksheet on our server so it's now immediately available to all our students. I'm really enjoying teaching +Sage Mathematical Software System to our students.

A Sage cell with the above code (that you can run in your browser) can be found here: http://goo.gl/jipzxV

EDIT: Since posting this +Punarbasu Purkayastha pointed out on G+ that list_plot can handle complex points right out of the box :) So the above can just be obtained by typing:

pts = [e^(I*(theta)) for theta in srange(0, 2*pi, .1)] 

list_plot(pts, axes_labels=['Re($z$)','Im($z$)'], size=30)

Learn something new everyday...

What I have in my backpack: my academic toolbox

+Rodolphe D'Inca has started a series of posts on G+ looking at his 'toolbox' as a scientist:

1. Reading Science
2. Building a bibliography

In this post I thought I'd join in and briefly share the kit I carry around in my backpack and the stuff I use on a day to day basis software wise.

Hardware

Here is a picture of the usual contents of my backpack:

In there you see:

- My tablet (a nexus 7). I use this mainly to read papers and books but also for some teaching stuff where I use it to tick student progress on +Google Drive.
- My smartphone (a nexus 4). I'm kind of addicted to my phone. I use it for email, +Google+, Google Keep, and to remember where I have to be and when (if it's not in my calendar I won't be there: the reverse not necessarily being guaranteed either...).
- My 11" macbook air. This I probably care about more than my wife. I'll talk about software below but it's really from a hardware point of view that I love this machine. So portable and just the right size. I've been really tempted by the +Dell XPS 13 Developer edition laptop but the screen size (13") puts me off a bit. I love my big monitors for my desktop but for a laptop I think 11" is perfect (the fact that +Linus Torvalds uses it is also kind of cool).
- The dongle to let me connect my macbook to a projector (kind of annoying that you need this but oh well...).
- An actual paper on paper (every now and then I have a paper in my bag as opposed to reading on my tablet: I happened to have one in there today). I've just done a screencast about the particular paper I'm looking at.
- A 250GB hard drive for backups (I'm pretty paranoid about backups).
- My +Moleskine notebook. For the past 3 or 4 years I've only been using this notebooks and make sure I date them so that I've got a tidy record of all my scribblings on my bookshelf in the office. I got this one autographed by Jorge Cham of +PHD Comics:

- My 4 color Bic pen. I used these throughout highschool but 'lost my way' during University. Just started using them again and remember how awesome they are: always work, nice to have the colours and perfect balance for finger spinning (it might be because I spent most of highschool learning how to finger spin with them).
- A usb stick but I very rarely use it (it was in my backpack when I wrote this).

EDIT: After +Rodolfo Carvajal asked on G+ here's a photo of the backpack itself:

I've had it for about 4 years now and it's an Eastpack (I pretty much try to refuse to use anything else as I learnt to love how durable they were during highschool). It's got a nice padded section for my laptop and straps that I can hook my water bottle too but also that I can use to compress the bag when it's empty (the stuff I carry in it does not take much space). It will be a very sad day when (if?) this bag dies as it's by far my favourite backpack ever (pretty much just being compared against other Eastpacks). There are various other tiny bits of kit that I'm missing here (laser pointer, portable battery) but that's because I've left them somewhere and weren't in the backpack when I got the stuff out to take the photo...

2nd EDIT: They still make the backpack! :) http://goo.gl/Vv4STR

Software

I've posted about software and my preferences of *nix environments before but in general I use the following software:

- Vim for editing pretty much everything (LaTeX, +Python, +Sage Mathematical Software System, md);
- +Sage Mathematical Software System for math calculations;
- +Python for more general programming;
- Git for version control;
- Rstudio for R code editing;
- Zotero for reference management (I blogged about using Zotero with +Dropbox here)
- +Dropbox (big fan)

Anyone got anything I should know of and use?

Selfish behaviour in queues and some open source graphical simulation software

In 1969 Naor, wrote a really nice paper called 'The Regulation of Queue Size by Levying Tolls'. In this paper Naor considered a system with a single server queue:

• With an arrival rate $\lambda$ (customers per time unit)
• A service rate $\mu$ (customers per time unit)
• and a reward and cost for service that can actually just be considered as a "value for service": $\beta$

Naor then considered two types of customers: Selfish and Optimal.

It is relatively straightforward to see that Selfish customers should join if and only if:

$$\frac{n+1}{\mu}\leq \beta$$

where $n$ is the number of other customers in the system upon arrival.

What is slightly less straightforward is that Optimal customers should join if and only if $n\leq n^*$ where:

\[\frac{n^*(1-\rho)-\rho(1-\rho^{n^*})}{(1-\rho)^2}\leq \beta \mu < \frac{(n^*+1)(1-\rho)-\rho(1-\rho^{n^*+1})}{(1-\rho)^2}\]

(where $\rho=\lambda/\mu$)

It's a really cool result and one that has given rise to a lot more research (including what I mainly enjoy looking at).

I was asked recently be a colleague to give a 15 minute talk about my research to her second year OR class who will have just seen some queueing theory. I decided to talk about Naor's paper and thought that it would be nice if I could give a graphical representation of the customers arriving at the queue (similar to the DES package: +SIMUL8). So I spent some time writing a simulation engine and using the in built +Python Turtle library to get some graphics. A part from some of the optional plotting (matplotlib), this only uses base python libraries. Here's a gif from an early prototype:

Here's a video discussing Naor's result and showing demonstrating everything with my simulation model:

The code is all up on github and it really could do with some improving so it would be great if anyone wanted to contribute: https://github.com/drvinceknight/Simulating_Queues