18 February 2011

Paradoxial games

This was a free lecture to anyone in Greenwich or out. The Greenwich MathSoc set this up with help from @NoelAnn and @Tony_Mann. When I was doing my last year at Greenwich Eduardo Cuntin did a project on paradoxical games giving the history off it and interviewing the man himself. So a couple of months later MathSoc asks him to come and then we have an event. A couple of people on twitter and facebook spread the word of the event. Then we are brought to the day of the event.

WARNING: if you are going to attend a lecture by Juan M.R. Parrondo do not read any further, if your interested in the maths contine.

The man who everyone came to see is Juan M.R. Parrondo was introduced by Noel-Ann Bradshaw and Eduardo Cuntin. This problem is apparently 10 years old. The idea is there is 2 games which if played individually will end up with you losing money (gambling is easier to visualise) but if you alternate between the 2 games randomly then you will win. Confused? read on.
He gave an example, Game A is a simple game you win with probability p1 and lose with p2. The second game Game B is more complicated if your capital is divisible by 3 then you win with prob p3 and lose with p4, if your capital is not divisible by 3 then you win with prob p5 and lose with p6. This should help visualise it

This is what happens when you play the games randomly or the games separate

As you can see playing randomly turns to be profitable (The above was the game being played 5000 times).

Apparently the trick to working out this paradox comes from something called Brownian motor and the probability of each game.

Game A has a small positive gradient and Game B is a mixture of 2 gradients 1 big positive gradient and one shallow negative gradient. Switching between the 2 games means you increase your profits. This website may explain the Brownian motor better.

There was some criticism of the paradox in 2004. They questioned the paper and said if it was not random but they got to pick which game they chose then their odds and profit would go up. This is true but if you deal with more than one person choosing between which game to play i.e. democracy then you will end up losing but random still triumphs.

The lecture ended.
Parrondo's paradoxical games homepage

15 February 2011

True Grit

True Grit was actually filmed in 1969 and then re envisioned in 2010. It was based on a novel by Charles Portis. The film is very good, but Jeff Bridges character has a mumble and a deep voice which makes it hard to understand what he's saying. His character however is brilliantly done, looks like a drunk who smokes a heck of a lot and doesn't enjoy his job. The language is old or like the 1969 film, not sure if its like the book, haven't read it. Trying to understand what is happening is hard. I would give the film 2 out of 5. If the language was more modern I would give it 4 out of 5.

According to Rotten Tomatoes I'm wrong. IMDb link

pre dated for yesterday

The Green Hornet

I went yesterday to the cinema to watch this. I actually watched it in 3D, it is not worth paying the extra money for 3D, the film is already good in 2D. The main idea of the film is its a hero film, I don't think they have superpowers but that's debatable (watch film and tell me what you think). I didn't know about this before but they wanted to frame themselves as the villains as apparently this is every heroes weakness. Also it means only the good guys would come after them and if they got caught, prison would mean they get respect rather than abuse. Seth Rogen would never be chosen for a hero movie (no offence) but this is perfect. The main character acts like he's 18 and in college i.e. party, booze and everything else. Then he has to change and become "The Green Hornet" which is a lot of fun to watch. The gadgets on the car by the way is amazing, might be a reference to a couple of James Bond movies.

The other thing I liked about this film was they got the crime locations off someone (no spoilers so far (I think)). Now good news is within the last week or so there is a new website set up that displays crime on a map. Which is a good way of spotting the troublesome areas, the police would more than likely advise against using spandex to patrol the streets and trying to be a hero.

The downside of the map is it's not accurate enough but there is data so if you wanted to build your own app, or analyse the numbers, why not check it out.

IMDb and Rotten Tomatoes as references, if you don't trust my review check out the rotten tomatoes link.

This should be pre dated to 3rd of February 2011

Unstructured Meshes

Last term for me I was studying meshes and the physics of what goes on when the radiator is on. The most common mesh is structured meshes, like calendars, all squares all evenly distributed. i.e.

This was taken from my iCal on my mac. (note sure who I ask permission to use this image, if you know can you reply in the comments)

Structured meshes are good for rectangle type of shapes but what if they are not rectangle? For this type of problem we can triangles and view this as an unstructured mesh.
An example would be a rectangle split into triangles (because i can claim its my own work and its easy to draw)

This image was made by the software Paintbrush
Now before we get into the how a computer works with unstructured meshes we first need to know each triangles neighbour.
1: has neighbours 2 & 3.
2: has neighbours 1 & 4.
3: has neighbours 1 & 6.
4: has neighbours 2 & 5.
5: has neighbours 4, 6 & 7.
6: has neighbours 3 & 5.
7: has neighbours 5 & 8.
8: has neighbour 7.

Now we create three vectors, one will be the neighbours so something like
The second will be to determine how many where the neighbours finish for that triangle i.e. for triangle 3 it would be 6, the vector will look like:
The last matrix will be the coefficients for those triangles so something like:

Then you can use Successive over relation or Gauss Seidel or even the Jacobi method. Other iterative methods are available.

One last bit of advice do not use my triangle arrangement to solve problems mainly because it is not evenly spread out and so any results you get will be wrong or at least not very accurate.

This should be pre dated to 30th January 2011

Math/Maths Live at University of Greenwich

Math/Maths podcast is hosted by Pulse Project starring Peter Rowlett (@PeterRowlett) and Samuel Hansen (@Samuel_Hansen).
I was Meant to upload this sooner, but coursework got in the way. Anyway on Pulse Project website you can listen to the audio of the podcast, but you never get to see them, so I recorded them doing their thang (<--deliberately spelt wrong).

At the start of this event they gave a talk of Peters and Samuels trip to Nottingham and mathematicians that lived there. After the Math/Maths recording a group of us went up to explore the Greenwich Observatory and came across a unique sundial:

This should be pre dated to 4th of December 2010

Recreational Mathematics

At Greenwich today the maths society (MathSoc) arranged a lecture on the subject of "Recreational Mathematics" which was given by David Singmaster. Unfortunately after my first picture my iPhone lost internet connection. Here is my first pic:

If you did read his wiki article, you would know he is famous for his Rubik cube solution, so I was glad when he brought that out:

Unfortunately my memory sucks so I can only recoil memories from the pictures. So somehow we managed to get to a bit about labyrinths and he found one underground, this was him getting into the cave:

And the he presented the cave itself and a drawing of the labyrinth

We then moved on to the Fibonacci series and there was a weird pattern in there:

Its a bit hard to see but if you look at the 15th line where it begins 14,1,....., 1001, 2002, 3003.... Huh this is weird but wait, because of this the 16th line reads 15, 1,.....,3003,5005..... and again on the 17th line we get 16,1,...., 8008,..... Strange phenomenon, its uncommon and doesn't happen often.

He then showed us fibonacci work (I believe) and the numbers we now use today which he got when he travelled abroad.

We then looked at another bit of Fibonacci's work, which included work that showed how to use the numbers for adding and multiplying. No picture here as it seemed a bit boring.

Now we looked at problems and teapots. I'll get to the teapots in a minute. One problem we looked at was:

If you can't read it, it says
"As I was going to St. Ives, I met a man with seven wives, each wife had seven sacks, each sack had seven cats, each cat had seven kits. Kits, cats, sacks and wives. How many were going to St. Ives?"
The answer wonderfully is none. Or one if you count the man himself.
There was other puzzles but I forgot them and forgot to take a picture of the puzzles. Shame apparently those puzzles perplexed even the greatest mathematicians.

Okay now the teapot. David Singmaster brought one with him:

As you may notice although the picture isn't that good, it has no lid to speak of, so how do you use it I hear you ask. Well glad you did, because I wondered as well and here's how:

You fill it up from the bottom and then flip it over, wonderful.

Next we had a problem:
"Can you draw three rabbits with three ears between them and that each rabbit looks like they have two ears each"
Easy enough draw them in a triangle formation like so:

This is all well and good but this image has been used in loads of different places:

Then the lecture finished:

for the next hour or so any people who were left got to play around with some stuff he brought along i.e. teapot, rubik cube (3*3*3), rubik cuke (7*7*7), Rubik cube (1*3*3), bolts that were together, sheet of paper that looked like it had space in side of it and some more things I can't remember.
Actually one thing i did remember was a magic trick (apparently it actually something to do with physics) which was cool here is the 10 sec video:
Wordpress wants me to upgrade to put video on here so hello Twit Vid

This should be back dated to 10th of November 2010