And now for something Completely different, Shakespeare (King John)

As a break from TMA's and maths I have decided to try and watch all the Shakespeare plays over the next year. An exercise I first did about 5 years ago when I invested in the superlative BBC TV series, first broadcast in the 1980's which covered all the then known Shakespeare plays  and which got me hooked when they were first broadcast. At the risk of sounding like Polonius from Hamlet I have grouped the plays as follows and this will be the order in which I watch them

History Part 1            (Comical Historical As Polonius would say)
King John
Richard II
Henry IV Part I
Henry IV Part II
The Merry Wives of Windsor
Henry V


Early Comedies                        (Just Comical)
Two Gentlemen of Verona
The Taming of the Shrew
The Comedy of Errors
Loves Labours Lost
A Midsummer Nights Dream

Early Tragedies (What Polonious would call Tragical Comical)
Titus Androndicus
Romeo and Juliet
Merchant of Venice (I refuse to class this as a comedy as was originally done)
Timon of Athens

History Part II                (Whilst this is the Chronological Order it should be noted that Shakespeare wrote  these before Richard II - Henry V )
Henry VI part 1
Henry VI part II
Henry VI part III
Richard III
Henry VIII.

Later Comedies                   
Much Ado about Nothing
As You like it
Twelfth Night

The 'Problem Plays' (Comical Problematical as Polonius might have said)
Troilus and Cressida
Measure for Measure
All's Well that Ends Well

The Roman Tragedies (Classical Historical as Polonius might have said)
Coriolanius
Julius Caesar
Anthony and Cleopatra

The Major Tragedies (The Big 4)
Macbeth
Othello
Hamlet
King Lear

The Pastoral (Autumnal) Plays
Pericles Prince of Tyre
A Winters Tale
Cymbeline
The Tempest.

Since the series was produced, Two Noble Kinsman, has now been attributed to Shakespeare.
Anyway last night I watched King John with Leonard Rossiter (Best known for his comedy series The rise and fall of Reggie Perrin) in the title role Despite this background there is no hint of comedy in this role at all. An unusual and not particularly well known play wirtten in about 1595.

It opens with King John being told that his claim to the throne is under threat from France (where else). He is asked to intervene in a dispute between two brothers the younger of which claims the inheritance as the elder is illegitmate. It turns out that the Elders father was no less than Richard the Lionheart, he (Faulconbridge) is persuaded to give up his claim and Join with John in the wars against France. Faulconbridge then becomes a 'chorus' commenting on the action to the audience as the play develops.

The second Act begins outside the walls of Angiers with John facing the King of France (Philip). Philip has been persuaded to take up the cause of Constance who's son Arthur (A young boy) has a claim to the English Crown as he is the son of John's other elder brother. The leader of Angiers says he will  swear allegience to whoever wins the battle. After two inconclusive battles John and Philp are persuaded by Faulconbridge to turn their arms against Angiers and then resume their fight. After some deft thinking the leader of Angiers persuades Philip and John to let John's niece Blanche and Philip's son the Dauphin marry thus sealing a lasting peace between France and England (and thus dissolving Arthurs claim to the throne). Blanche and the Dauphin are married and all seems well until a Papal legate enters the scene. The Pope is annoyed that King John has not appointed the Popes chosen bishop as Archbishop of Canterbury. John's reply to the Pope is quite amazing and worth quoting

                            " Thou canst not Cardinal, devise a name
                               So slight, unworthy, and ridiculous
                               To charge me to an answer, as the Pope.
                               Tell him this tale, and from the mouth of England
                               Add this more: that no Italian priest
                               Shall tithe or toll in our dominions
                               ...
                               So tell the Pope, all reverence set apart
                               To him and his usurped authority "

(It's a shame that David Cameron didn't cite that during the recent Pope's visit I digress)
Anyway to cut a long story short John is excommunicated, England and France resume their war with England winning. John takes away Arthur as a prisoner and persuades one of his lords, Hubert, to make sure he dies Act III ends with the Cardinal persuading the Dauphin that the way lies open for him to take the throne of England.

Act IV begins with Arthur in Prison and Hubert threatening to put out his eyes and murder him. However he relents and heads back to John to try and persuade him that Arthur is indeed dead. The lords revolt especially when the rumour abounds that Arthur has indeed been murdered. John in order to save his skij upbraids Hubert as he believes wrongly that he has murdered eventually Hubert tells John the truth that he has put the boy in hiding. John then persuades the Lords to see for themselves, but alas in the intervening period Arthur has tried to escape and killed himself by jumping from the Tower. The Lords think he has been murdered War ensues and the Dauphin lands on the shore of England. In the meantime John asks the Cardinal for forgiveness and John's excommunication is revoked.  The Cardinal withdraws his support from the Dauphin who has lost all his supplies. He begins to withdraw, but news arrives that John has been poisoned by a monk and the play ends with Johns death.

An interesting play, the context is obviously the continued threat of invasion from Spain, hence the Anti Catholic sentiment. There has been a recent attempt by some scholars to make out that Shakespeare was a closet Catholic I find this hard to believe especially given the Anti-Catholic sentiment in this play
Unlike other of Shakespeares histories (Richard II - Henry V) there is no comic relief and it is highly unlikely that the play is an accurate representation of history. Indeed it is surprising that no mention is made of the Magna Carta or Robin Hood.

Still as one rarely performed it is interesting to see it now and again. I remembered hardly any of this play since I last saw it so obviously not a show stopper. For those 'completists' like myself it is worth seeing but not the best introduction to Shakespeare and probably best got out the way quickly. Richard II next which should be much better I'll keep you informed.

M208 TMA05 Back

Well I was dreading this but I got one of my highest scores so far. I do think my tutor Alan has been quite generous to my astonishment I only dropped 1 mark on the last question. Still my rant against the emphasis on visualisation seems to have sparked off some interest in other people who have been inspired to come up with a non visual way of dealing with rotating flags etc.

The essence appears to be

1) Work out the effects of the symmetry operation of the group in terms of permutations of the areas

2) A colouring can be seen as a mapping from each square to the set of N colours {B,W...}

3) So and this has yet to be clearly defined the number of fixed objects is related to the cycle stucture of the permutation raised to the power of the colours. But you must include all the cycles.

4) One of the guys on the M208 forum Toby has suggested that the nmber of fixed points for a given symmetry operation is
$$ |C|^{nc}$$
where |C| is the number of cycles associatied with each permutation and nc is the number of colours others have been trying to prove or disprove this. My mate Neil thinks he's close to finding a general algorithm
Whatever  the outcome, it's good that people are 'playing' with the Maths and not just desparately trying to find answers to TMA questions.  

M337 Complex Variables TMA03 Away

Well posted the third TMA for M337 Complex Variables and just like the fifth TMA for M208 there were bits in the last question which had me stumped. Anyway a brief resume

1) Questions on Calculus of residues to solve Integrals and series
     Relatively straightforward and a joy to do

2) Questions on zero's, maxima of functions and Inverse Taylor series

     Again tricky but quite straightforward

3) a) A question showing that two series are direct analytic continuations of each other
        What you have to do is show that if the two series have a common region and they take the same values on the region then the series with the larger region can be seen as a diect analytic continuation of each other. However whilst I could see that they had a common region pages and pages of scribbling failed to convince me that the two series were equal on this region.

b) A straightforward application of the residue theorem to solve an Integral

c) The use of Wierstrass's theorem to show that a series is convergent
    Again the method is to a) show that the series is bounded by a sequence of positive terms
                                       b) show that the sequence of positive terms is convergent
   a) was straightforward but I couldn't show that the series of positive terms was convergent it failed as far as
I could see the ratio test and the comparison test.

d) Some straightforward questions on the Gamma function

So should be close to getting full marks for 1 and 2 and about half the avaiable marks for question 3. As there are only three questions I should just scrape a grade two pass.

So I'll give myself a week off to concentrate on
                     a) Cambridge Computing projects
                     b) The variational method in General relativity and it's application to the Robertson
                         Walker metric
                     c) Cubics, quartics and quintics
    I;ve found (yet another) book on Galois theory which takes a nice historical perspective and doesn't get to bogged down in formal detail until the end

http://www.galois-theorie.de/galois-theory.htm

I would probably reccomend this book as the best starting point.

Finally so far have got 8 recruits to the shared activity forum on the Cambridge Computing projects forum

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.

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.
 

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 )

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.