Archive for the ‘Mathematics’ Category
Friday, May 22nd, 2009
As physicists, we learn that the Poincaré algebra has two Casimirs, [tex]p_\mu p^\mu[/tex] and [tex]W_\mu W^\mu[/tex], together describing the mass and spin of a particle. A standard question is then, "why does the algebra have two Casimirs?" and the standard answer is, "because it is a rank 2 algebra, for ...
Posted in Mathematics, Physics | 1 Comment »
Thursday, February 14th, 2008
Proof that 1 = -1:
[tex] \frac{1}{\sqrt{i}} = \sqrt{\frac{1}{i}} = \sqrt{-i} = \sqrt{-1} \sqrt{i} = i \sqrt{i}[/tex]
[tex] \Rightarrow 1 = i \sqrt{i} \sqrt{i} = i^2 = -1 [/tex]
This function should be taken outside and shot.
Posted in Mathematics | 2 Comments »
Saturday, January 5th, 2008
The Legendre transform is a simple and useful tool in some branches of physics such as Thermodynamics and Mechanics. In my experience though, this transform is usually explained in a very confusing way.
I want to give my own derivation, which I hope is clear and straightforward. If you're already ...
Posted in Mathematics, Physics | 2 Comments »
Monday, September 17th, 2007
Last time I described the first pitfall I encountered when solving a PDE -- an inherent instability in the partial derivatives. This time I'll talk about the second pitfall, which is simpler conceptually, but has wider implications for programming in general.
In my first implementation of the solution I used a ...
Posted in Mathematics, Physics | No Comments »
Tuesday, May 1st, 2007
I just finished working on a numerical simulation of a set of partial differential equations (PDE). I developed these equations for a physics research project I'm involved in. The equations did not seem to be solvable analytically, so I had to do it numerically. This was my first attempt at ...
Posted in Mathematics, Physics, Uncategorized | 1 Comment »
Saturday, March 17th, 2007
Here's a question that was asked in a recent oral exam for a Master's degree in Mathematics.
Let [tex]f : [0,1] \rightarrow \mathbb R[/tex] be a real function such that [tex]f[/tex] has a limit at each point. Does [tex]f[/tex] have at least one continuity point?
Everything you need to solve this question ...
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Saturday, November 11th, 2006
Long time no blogging! A lot has happened since my last post. I got married, was out on a honeymoon, left my job and returned to school to finish my degree. But now I'm back, so read on and enjoy!
We were in the office a while back, and we started ...
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Sunday, September 24th, 2006
The Ising model is a simple model used to describe a magnet: It consists of a grid of spins. Each spin can be thought of as a tiny magnet, and can have the value +1 or -1 (called 'up' and 'down'). Two neighboring spins interact, having positive energy if they ...
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Thursday, September 14th, 2006
Benford discovered that in many sets of data, the leading digit is much more likely to be '1' than any other digit. Take, for example, the population counts of nations. The most significat digit probabilities there are as follows:
MSDProbability
10.26
20.20
30.10
40.13
50.07
60.08
70.07
80.05
90.04
The MathWorld article offers an explanation in terms of distributions that are ...
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Thursday, July 27th, 2006
Took my last test of the semester today, in Topology. It came after four days of intensive studying. Sometime during the second day I realized that I had underestimated the breadth of the material, and that I needed more time. So I went into a blitz that involved little sleep ...
Posted in Mathematics, Personal | 1 Comment »
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