I thought I would try to stick to at least one post per month and since I didn’t have time to research anything interesting this past month (three-body problem for master’s studies is going a bit slow and no new Ruby knowledge as of yet), I thought I would post something that I’ve done in the past.

This was a project I did for my honour’s studies last year. It is on elliptic curve cryptography and I will tell you the story of how it got started. Since my university has a noteable division between applied and pure mathematics, I got the idea to see how one of my favourite pure mathematics concepts got applied. Since I am a fan of abstract algebra (in particular group theory) and I wanted some general application, I went around Googling a bit and found the little gem that is elliptic curve cryptography.

Elliptic curve cryptography is relatively new area in cryptography (it’s been around since the 80′s or 90′s) there isn’t a lot of easily obtainable research on it. Most of the knowledge is hidden away in journal articles. I did find this one book (Cryptography: Theory and Practice by Douglas R. Stinson) that was a saviour and that I would recommend to anyone interested in cryptography (even if you don’t have a strong mathematical background).

Without going into any other details, here is my project on elliptic curve cryptography for you to enjoy.

And that is all for now, my minions. Until next time…

Cheers!

Those of use that are Amarok users in Linux and have jumped to Jaunty (Ubuntu 9.04) have seen the horror of Amarok 2. I will say this is my biased view, but in my searches and wanderings online I have seen half of the users react the same way. All of the blog posts and forums I searched were about getting Amarok 1.4 to work in Jaunty. I have found this little gem of a blog post – Amarok 1.4 in Jaunty (Ubuntu 9.04) by !mad.Works. Just a few edits and a few commands on the command line and presto! you have Amarok 1.4 working in Jaunty.

This will be all for now, minions.

Till next time!

Yay! My first purely scientific post.

As I said in the About Me page I am currently busy doing my masters dissertation on gravitational capture. In my research I have come across mostly three and four body problems to describe the gravitational capture phenomenon. In order to understand the motions of the celestial bodies I set myself the goal of solving the two body problem from scratch. Although it feels like I cheated a bit by reading about previous solutions online and in books, I felt that I needed the guidance a bit. The mathematics and concepts involved I haven’t touched or seen since my first year during my undergrad degree and I felt I could use the help while solving the two body problem as part of a refresher course.

So after about a month of reading and struggling, I finally have an acceptable generalised solution. Well, I say generalized but the solution is still a one variable function. However, my solution starts out in three dimensions and via mathematical proof moves to two dimensions and so on and so forth. I am posting my full solution here as it is now, but I might be updating it in future because there is still a lot of discourse and discussion left for my results.

I warn my non-mathematics and non-scientific readers that they will need to have an understanding of vector algebra, differential equations and mechanics to understand the solution to the full. But once you are able to see past the formulas, the simple solution (and mathematics behind it) is so beautiful and elegant in its nature that I can’t help loving mathematics more.

Before you get to the solution though, here is some background. I am not quoting any sites or any trusted resources or references and I am going from memory here. So if I am wrong in what I am saying here, I do apologize. The two body problem was originally solved by Sir Isaac Newton when he first formulated his gravitational law back in the 1600′s. The problem is usually stated as two bodies rotating around each other under their mutual gravity with no external forces influencing their motion. The problem of two bodies is to find a formula to determine their position at any given time. As far as I’ve read and seen this formula is not generally directly dependent on time, but on some angle made with one of the axis in the reference framework. However, it should not be to difficult to give a formula for this angle with respect to time. But that is a study for another time in the future or even the future of my dissertation. As I further my studies and results on my dissertation I will post my findings here. But without further wait, here is my solution to the two body problem.

Two body problem solution

As I said last time, my aim with this blog is to introduce and teach people about the wonderful world of mathematics, science and programming. I spent many a night thinking of what would be the perfect topic for my first post. Dynamical systems? Celestial mechanics? Game programming in Ruby? I figured that I still have to figure out a few things before I can do posts on those (I have to figure out how to get mathematical formulas from LaTeX into HTML. Still reading up on the latex2html app).

So I finally decided on a topic that everybody is interested in – Neural Networks – that would be the stepping stone into my world and is also a combination of most of the fields that I am interested in. It can also serve as a great divergence point for each of the fields (mathematics, science, programming). Anyway, let’s get down to it.

The human brain is the most powerful computational tool known to man, with the ability to do millions upon billions of computations per second. Artificial neural networks are way to emulate and model the brain’s way of doing computations and are based on the actual biological neural networks in our brains.

Neurons (also called “nodes”) are the basic processors in the brain. Neurons are connected to each other to form a neural network. The neuron receives input which is weighed and if successful activates the neuron, which in turn sends a electrical signal (as output) to another neuron or to some other part of the brain and/or body.

Somewhere in the 1950s or ’60s some mathematician tried to apply this process to computational problems and so the artificial neural network was born. The basic artificial neuron can be modelled as a nonlinear multi-input device with weighted interconnections (also called synaptic weights or strengths). The cell body (soma) is presented by a nonlinear limiting or threshold function.

The simplest ANN is a perceptron. I decided to implement the perceptron (emulating the logical AND-gate without a learning algorithm) in Ruby. The class looks as follows:

class Perceptron

  attr_accessor :weights, :threshold, :inputs

  def initialize( input_array = [], weight_array = [], threshold_val = 0 )
    @weights = weight_array
    @inputs = input_array
    @threshold = threshold_val
  end

  def calculate_output
    value = @inputs.zip(@weights).inject(0) { |sum, i| sum + i[0]*i[1] }
    value < @threshold ? false : true
  end

  def self.run( input_array = [] )
    p = Perceptron.new( input_array, [1,1], 2 )
    puts p.calculate_output
  end

end

The class takes input and calculates the output via the calculate_output method (seeing if the artificial neuron fires). It takes the inputs, weighs them and evaluates them against a chose threshold value. As an example, the following

Perceptron.run( [0,0] )
Perceptron.run( [0,1] )
Perceptron.run( [1,0] )
Perceptron.run( [1,1] )

would simulate the logical AND-gate. This is one of the simplest neural networks out there. From here on they progress to include learning algorithms (Hopfield and Kohonen networks) and include hidden layers. But those are topics for other future posts.

I hope you enjoyed this brief introduction to neural networks and your brief time in my domain. Until next time!

P.S.: Here are some other useful links about neural networks:

• Neural Networks in Ruby
• Wikipedia article on neural networks
• The Nonlinear Workbook by W.H. Steeb – this textbook was my introduction to neural networks during my undergrad studies.

This is your universal ruler Xerg. I welcome you all to my alter-ego’s domain where discussions of mathematics, science and programming are common place.

Ah heck! Just joking about the minions and universal ruler (although I am unsure of the multiple personalities – people normally associate me with the word crazy). However I was being serious about the mathematics, science and programming.

I am writing this blog to share my knowledge and to teach people about the beautiful world of mathematics and programming. Quite a lot of people dislike mathematics and I aim to change this by showing what a powerful tool mathematics can be. As I am currently doing a master’s dissertation in Applied Mathematics, you, my dear Reader, can expect some of that work and information to be shown here. But since most of my research covers celestial mechanics, I will also try to do a few posts about more known topics in the programming world (neural networks, genetic programming, optimization, etc.).

As for the programming, I had a course in C++ during my undergraduate studies – mostly theoretical stuff without major real world application. About a year and a half ago I started working for a company whose major development uses Ruby and Ruby on Rails. This is where I mostly learned about how programming is applied to real life and in a business world. So I will be looking to do posts about my experiences in both these languages and whatever new things I learn in them. There will be lots for me to learn and post as I am doing a few programming projects on the side, apart from my academic research and work-related research.

You, my dear Reader, can look forward to posts on Merb, CouchDB and intergrating the one with the other, introductions to celestial mechanics, the Ruby programming language (I think I will put I nice spin on this because I come from a scientific background), and some dynamical systems (which will later hopefully lead to a few posts on chaos theory).

But that is all for now, dear Readers.

Until next time, cheers!