Friday, 29 July 2011

Cambridge Computing Projects

Another distraction but one that is going to be fun (I hope)

The Cambridge Mathematics department every year publish a list of structured computing projects coverting all sorts of mathematical projects. I thought it would be quite fun and instructive to get a group  of my fellow OU students together to try and do some of them ourselves compare notes and code etc.

The department page is here

http://www.maths.cam.ac.uk/undergrad/catam/

The list of projects for the second year students is here

http://www.maths.cam.ac.uk/undergrad/catam/IB/index.html

The students are recommend to do 4 and the introductory project

Then the Bewildering list of projects for the third year is given here

http://www.maths.cam.ac.uk/undergrad/catam/II/

Of course no one does them all, I think it's about 3 or 4

The projects are not purely programming exercises or exercises in software engineering but seem to give a
real insight into the technique being used. I would estimate with all the other stuff TMA's exam revision etc
it would take 2 -3 months to complete and write a single project up. If any fellow OU student would like to join me in this venture I have set up a shared activities forum and if you send me your OU id (ie the short 5-6 letter code that the OU give you to let you into the system eg mine is cdaf2 Not your personal identifier)
I can let you in.

Tuesday, 26 July 2011

M208 TMAO5 away

Well posted TMA05 off this lunch time this covered the second part of group theory B and I'm afraid I didn't enjoy it at all. The questions seemed biased towards visualisation something I'm hopeless at. I totally bombed the last question as I couldn't get the results from the counting theorem to tie up with the pictures but having spent 3 frustrating hours on it last night I gave up in disgust.

Recap of questions

1 a long tedious question about symmetries of a 3d object and various conjugacy classes

2. A reasonably straightforward one on matrix groups and their subgroups

3. A question on homomorphisms the first part is straightforward, but the second part covers some really deep stuff on the relationship between the Image of a homomorphism and the quotient group which is the
normal subgroup formed from the group and the Kernel. I haven't fully digested all of the links but essentially the image of a homorphism f is isomorphic to G(f)/Ker(f) and aided by some solutions to past exam questions I was able to complete the TMA but definitely need to go back and understand why.

4. A relatively straightforward question on group actions and their orbits.

5 The tedious flag colouring question alluded to above.

In general I feel it unfair that this group theory question is biased to those who have visual skills. Also colouring flags made me feel like I was back at primary school. I'm expecting my lowest score for this assignment. Bring on the last part of the analysis course.

I must admit to being quite disappointed by the treatment of Group theory in M208. Far to many confusing examples which obscure the underlying structure. Why not provide us with a brief introduction to what Lagrange was trying to do with the symmetry groups of polynomials. Why not provide us with an introduction to the rotation group and the special unitary groups that play such an important part in physics. Instead we are expected to count ways of colouring squares or flags and mentally rotate 3 d objects in our head. Very disappointing.

Sunday, 24 July 2011

Big Bang for Dummies (Including Me)

Those who have been following this blog right from the start will be aware that one of my many long term goals is to understand the Big bang properly almost line by line This has been an on off project and quite often I get bogged down or TMA's and other bits of maths distract me. I last looked at this back in December 2010 when I got bogged down in trying to calculate the Riemann Tensor for the Robertson Walker Metric. I attempted to do this directly from the basic equations as this involves calculating 64 connection coefficients this is an exercise in tedium usually just quoted in many textbooks. The method whilst guaranteed to work is soul destroying to say the least and not much insight to the physics is given. There are more efficient methods based on Coordinate Free differential geometry but this takes a while to master.

Fortunately I remembered a method which I read about when I first got into General Relativity based on comparing the Equations of motion derivable from a Lagrangian and the equations for the Geodesics. I'll expand on this later I hope to work it out over August in the mean time for those new followers of mine I enclose a link to a summary of the main features of the big bang I worked out about 4 years ago. This amazingly shows that to capture the essential features of the Big Bang you do not need General relativity or sophisticated particle physics. It is somewhat surprising that an estimate of the hydrogen helium abundance can be got on the basis of Newtonian Physics and the Boltzmann distribution. Ok the interpretation of the Newtonian approach is a bit dubious but it still ends up with the Friedmann equations. Anyway I offer it to my fellow bloggers in the hope that they will find it instructive  it shouldn't require to much background in either physics or maths

http://dl.dropbox.com/u/16049029/Bigbang_simp.pdf

Enjoy

Chris.

(PS Apologies for my Grasshopper mind I'll probably put Galois theory on the back burner for now as I've now got another obsession )

Monday, 18 July 2011

cubics quntics and progress on M208 and M337.

Well another month has passed and I'm about two thirds of the way through TMA05 for M208 on Group Theory and TMA03 for M337 Complex Analysis it's all coming together I really enjoyed the Calculus of Residues unit C1 on M337 it's amazing how productive the residue theorem is, all sorts of complicated integrals and series can be reduced to a simple contour integration and the calculation of a few residues for once I didn't feel I was having to scratch my hand vis a vis the 1st TMA question. Still the other two questions are a bit more stretching. Anyway Complex analysis is really clever and I hope to do it justice for the rest of the course having learnt some of the mechanics I hope to go back and look at it in more depth.

The M208 group theory was quite tricky in parts especially as it involves visualisation of symmetries of an object. Fortunately I found a way of systemising the symmetries as an adaption of the way Chemists treat molecules there are a few key points

1) The number of direct symmetries is identical to the number of indirect symmetries
Got my wrist slapped for pointing this out on the M208 forum rather unfairly I thought but heigh ho.

2) If there is one principal axis of rotation then the composite symmetries involve a rotation about the principal axis followed by a reflection in a plane orthogonal to the axis.

3) When finding normal subgroups as unions of conjugate classes you can immediately discount any subgroup which consists entirely of indirect symmetries as the composite of two indirect symmetries is a direct symmetry so that a proposed subgroup consisting entirely of indirect symmetries plus the identity will not satisfy closure so can't be a subgroup.

Would probably get my wrists slapped again for pointing these simple facts out on the M208 forum so I'll point them out here.

As well as that continuing with my distraction in trying to understand from a purely algebraic point of view the invariants of polynomials as a prelude to getting into Galois Theory. Those readers who want to amuse themselves might like to try to derive the Cardano formula for a cubic by carrying out the following steps

1) Reduce a cubic of the form

$$x^3 + bx^2 + cx +d = 0$$ to the form

$$y^3 + py +q = 0$$

by making the transformation

$$y = x -\frac{b}{3}$$

You should be able to show that

$$p = \frac{-b^2}{3} + c$$
and
$$q = \frac{2b^3}{27} - \frac{bc}{3} + d$$

Then in the equation for y make the substitution

$$y = z - \frac{p}{3z}$$

to transform the equation for y into a sextic for z

$$z^6 + qz^3 - \frac{p^3}{27} = 0$$

This may appear to have made matters worse but if we substitute
$$s = z^3$$ then the sextic becomes a quadratic in s

$$s^2 + qs - \frac{p^3}{27}=0$$

with solutions

$$s1 = \frac{1}{2}(-q + \sqrt{q^2 + \frac{4p^3}{27}})$$

and
$$s2 = \frac{1}{2}(-q - \sqrt{q^2 + \frac{4p^3}{27}})$$

from which the original cubic can be solved.

This is relatively straightforward however as is well known applying such techniques to the quartic get a bit more complicated and Galois showed that such general formula could not be obtained for higher polynomials such as quntics. However whilst in general such general solutions are not available nevertheless a procedure was developed as an extension of the above techniques for deciding whether or not a quintic is solvable and if it is how can a solution be obtained. One such procedure was given by Watson and a summary of his studies which builds on the work of Cayley, P C Young and others is given here

http://www.math.carleton.ca/~williams/papers/pdf/244.pdf

Some of the expressions look horrendous but as it's 'just algebra' it would be a relatively straightforward exercise to verify these expressions. All of this makes no reference to Galois theory as it has developed and group theory is only hinted at so it should be an amusing exercise to try and verify the expressions and even maybe write a computer program to solve a quintic I'll let you know how I get on.