Wednesday, 26 December 2012

Quantum Probability

Have being having yet another debate (or rather a continuation of one) about the meaning or otherwise of quantum mechanics on the physics fora. Those who have been reading this blog, know, that I think the simplest way to understand quantum mechanics is to see it as an extension of classical  probability, to embrace the concept of a probability amplitude which is a complex number, the modulus squared of which gives rise to a probability density function.  This is to take the Born Interpretation to it's logical conclusion, the standard view point is a curious mix of probability and the implications of this to a single particle or system. It is this switch between an essentially statistical idea and the claim that we can predict what will happen to one particle that I believe to leads what Karl Popper calls the great quantum muddle.

In a couple of previous posts, I have indicated how such a perspective clears up the interpretation of the two slit experiment. To recap the so called wave properties are essentially statistical applying to a population or ensemble. Thus questions which are essentially statistical tend to be confused with what we can say about individual particles. The claim being for example that a particle splits into two as it passes through the slits that magically recombines to form a single particle at the screen. The screen being alleged to 'collapse the wavefunction' so that it appears as a single point on the screen. Of course empirically what happens is that it is only after a significantly number of events have occurred that anything like a wave pattern can be discerned. This would imply that the 'wave properties' are related to the probabilistic aspects associated with an ensemble of photons and not to an individual particle. I show how this can be explained by the concept of complex probability amplitudes and the same applies to beam splitters etc in these posts

Anyway I am gratified to learn that there is a whole branch of mathematics called quantum probability which includes classical probability. On this view, the so called collapse of the wave packet is a mapping from the complex number space C^n to the real numbers. There is nothing physical about it all. Indeed those who did M338 will be familiar with the concept of a projection from a higher dimensional space to the real numbers.

Anyway a good introduction to the whole subject is given here

What is apparent is that it is clear that classical probability theory cannot account for many quantum phenomenon. The paper also makes it clear, that in so called Aspect experiments, where a correlation has been measured between the spins of two photons emitted from the same source and sent in opposite directions, The attempt to understand it's implications which has caused so much fuss over the past 20 -30 years, measurement of the spin of one particle does not affect the spin of the other. This is counter to so much of the prevailing wisdom (or at least that which dominates the debate in the media and the popular literature) that I really think this approach deserves further attention. What is gratifying about the quantum probabilistic approach is the way in which the correlation between two particles emitted from the same source just pops out of the formalism. I will expand on this in later posts.

The Open University physics project course includes one on quantum entanglement and I'm seriously contemplating doing this within the next year or so based on the above paper.  I'm swithering between doing the OU course SM358 in October, which I'm not sure I would need, or just diving in straight away.

Monday, 17 December 2012

MST326 Fluids TMA01

Well I've just finished the first TMA for MST326 Fluids although the course is on Fluid dynamics much of Block 1 is devoted to developing mathematical techniques

The first unit does cover fluids essentially hydrostatics with an interesting application of how lock gates work and also development of simple atmospheric models. The first question of this TMA covered this I confess I couldn't quite see how to do the later half of the first part which involved working out stability conditions for an inverted half open cylinder immersed in a liquidso left it, The second part was based on developing a simple atmospheric model of the pressure in the mesosphere. This I found tedious but quite straightforward.

The second unit covers techniques for solving second order differential equations specifically for those differential equations which have polynomial coefficeints in front of the derivatives not just constants . These included the Cauchy Euler method, the method of variation of parameters which enables you to generate a second solution to a differential equation if you know the first solution, order reduction another technique for generating the second solution to the differential equation from the first, Finally solution by series essentially you substitute a trial solution in the form of a polynomial into the differential equation and then generate recurrence relations for the coefficients. This method is probably the most powerful method for solving second order differential equations with polynomial coefficients. It leads to the so called special functions such as Bessel functions Legendre polynomials etc associated with the classic differential equations which crop up all the time in physics.

 The question covered all these techniques and I think I got most of it out apart from the last part which was quite tricky algebraically and I couldn't get the equation for the reduction of order to resemble anything like 
the equation in the hint so I gave up

The third unit is on techniques for solving first order partial differential equations this involves the method of characteristics. The question on this I found quite straightforward and am reasonably confident of getting a good mark for this one.

Finally the fourth unit on block 4 is Vector calculus, whilst this will be familiar to those who have done MST209 this is covered in more depth in this unit and the question in the TMA was correspondingly a lot more tricky. One part in particular involved calculating the curl of a vector field in spherical polar coordinates to show that it was zero. In order to do this involved quite a bit of playing around with trig identities. I was gratified to get it out, as also the last part which involved verifying Gauss's divergence theorem by calculating both a volume integral and a surface integral I was really pleased with myself when I showed that the volume integral was indeed the same as the surface integral

So overall two full questions more or less correct, and 4/5th s of another question and 3/5ths of the first question should be enough for a respectable grade 2 on this one.

In general the TMA is quite challenging and not just a repetition of exercises with slight modifications in the book. I enjoyed the intensity of it all even though slightly frustrated in parts. I hope by June to have got in a lot more practice on the past exam papers than I did for MS324 and realistically I can expect grade 2. But I need to spread the load a bit rather than let it all pile up in the last few days before the deadline.

Best wishes to you all Chris

Sunday, 9 December 2012

A224 Music TMA02 Pointless

Well I've just sent off what must have been the most pointless TMA I've ever done for an OU course.
This is for my music course. The exercise consists of listening to three short songs and trying to write some coherent sentences about them. The first was OK as it had the music and we were asked to identify it's phrase structure, key and other things including whether or not the word setting was syllabic or melisismatic.

However the other two did not and we had to identify the role various instruments played, the texture and so forth. It's really not clear (or at least to me) what the point of the exercise was. I mean apart from saying things like the flute plays some twiddly bits at 1"03 it gets louder at 2"43 what was the point. I'm afraid after having listened to the second song endless times and putting some half coherent sentences together I gave up. I really feel that this has been a total waste of time.

It's not as if we haven't done any thing which could have been tested such as some counterpoint exercises or basic harmony. I appreciate that say 1 third of the TMA could have been devoted to the exercise but why not also test the theoretical aspects as well. That is certainly what happend in A214.

The next TMA will involve setting a verse to music so hopefully I will find that a lot more interesting but really asking people to say something about music without the score in front so you can analyse it must be one of the most futile things I've ever been asked to do.

On a much brighter note at the invitatiion of a friend I heard the Scottish Ensemble play at Greyfriars Kirk a safe haven from the Christmas Bling of princes Street. The main piece was an arrangement of Bach's Goldberg variations for strings the concert was one of the most uplifting I've been to for a long time.

Next post I'll talk about the first TMA for the fluids course

(By  the way this is now post 100 so I'm just under 1 a week on average thanks to all my readers).

Wednesday, 28 November 2012

Women Bishops and the Anglican Church part 2 Thecla

As promised here is quite a surprise for those bought with the idea that the Church has never had female leaders. I refer to the tradition in the Syriac church regarding Thecla according to tradition she was an accomplice of Paul. The main tradition is based on a book called the Acts of the Apostle Paul and Thecla a summary of which can be found here.

Paul preached a doctrine of extreme ascetism, which was attractive to many women as it provided an escape from the traditional idea that the purpose of women was essentially to be married and to produce children (even though according to Timothy and the book of Genesis a woman was produced from a man :)). The early churches attitude to women  seems to have caused quite a stir amongst Roman society and the doctrine of ascetism for women  seems to have been regarded as a threat to society (sound familiar) . At about the turn of the 1st century there seems to have been a bit of a power struggle between men and women and it would seem that those men who didn't want women in the church eventually won the power struggle. There are many pictures of Thecla preaching alongside Paul but one of the most telling is this one,r:0,s:42,i:226

The painting shows Paul and Thecla preaching both are of the same height and both have their right hands lifted iconographically this would mean that both are seen as equal partners with the same amount of authority. But the image of Thecla has had her face defaced and an attempt has been made to obscure her right hand. This can be nothing but an attempt to airbrush the significance of Thecla out of history.     
Of course the ultimate irony is that she was canonized by the Catholic church which does not allow women as priests. Her feast day being 23rd September. Ascetism or abstinence from sexual activity now being seen as a good thing by a male patriachal society.

Anyway the point is that is strong evidence that up until the middle of the second century women seemed to have been able to take leadership roles in the Church. The claim that women did not is just bogus thus there can be no justfication for the idea that leadership in the church was the sole prerogative of men.

What about the second premise that Jesus only appointed men as his apostles. The point is that the church was not founded by Jesus. Indeed it is highly doubtful whether or not Jesus saw him self as founding a new religion. That was done by Paul and I have given sufficient evidence to show that women played a leadership role in the church.

As always the current Church has ignored these uncomfortable facts. I must admit to being quite surprised, given that a few hours research on the internet or consultation of some standard textbooks uncovers plenty of evidence that women took leadership roles in the church, that none of these arguments seemed to have been raised in the interminable debates about whether or not women should be Bishops or not.  

Anyway I hope I have shown that the two main arguments usually raised against women taking leadership roles have very little justification. I sincerely hope no one in the church really believes that a woman was produced from a man. The sooner the Church allows women to rise to the highest position in the insitutions the better it will be for all. Until it does so the Church can only be seen as upholding  the patriachal elements of  society alongside the Republican party and large parts of the Conservative party. Indeed until the current situation is changed I don't think it's going to far to accuse the Church (or those churches which do not allow Women priests or Bishops) of institutionalised sexism and as such subject to justified ridicule and contempt. 

Tuesday, 27 November 2012

Women Bishops and the Anglican Church Part 1

As is well known the Anglican Church just missed moving with the times and finally allowing women to become bishops and take  leadership roles in the church. Some people might say what's the problem? the church is irrelevant, yet I feel that although I am no longer a church goer it sends all sorts of wrong signals about the possibility of a progressive form of religion, which even though I'm an agnostic, I would have more respect for than the current attempts  of a vociferous minority in the Church to move back to the dark ages and oppose every progressive development in society.  The current position just confirms my view that the Church has been hijacked by a combination of reactionary Anglo Catholics and Conservative Evangelicals who would otherwise have quite different theological views but have come together to prevent a long over due reform.

In this post I want to challenge some of the premises made by those who oppose women bishops and maybe introduce people to some quite novel ideas about the role of women in the early church that they might not have heard of before. There are two main premises I want to challenge

1) The Bible justifies the inferior status of women (a typical Evangelical argument)

2) Jesus selected twelve male apostles to found his Church thus women cannot have leadership roles (A typcial Anglo Catholic argument)

Most of what I have to say is summarised in Ehrman's book on the New Testament a standard college textbook and so representative of current scholarship. 

Ehrman B 2008 "The New Testament A Historical Introduction to the Early Christian Writings" 4th Edition

I will consider the first premise in this post "The Bible prohibits women taking positions of authority in the Church"

In an earlier post I have given my reasons why I don't believe that you can just read off ethical positions from the Bible written two thousand years ago and transplant them to positions in the 20th century.

However I want to challenge those who refer to the Bible to justify their obnoxious position. The main verses in the Bible used to justify the position that women should not have any authority in the church are those from the Pastoral letters in particular the second chapter of 1 Timothy (ch2 v11:15) which enjoins women to obey their husbands, keep silence in church  and not to have authority over men. More bizarely, the quite ludicrous claim, is made that woman was created from man and this justifies the inferior status of women. This goes against all biological facts, open to anyone who has had children or even who was born. However even if one accepts this ludicrous claim, there are still problems about the significance of these verses. The usual claim is that this is Paul's teaching, and the words given in Timothy were written by Paul. This goes against current thinking in mainstream academic theology departments (See eg Ehrman Chapter 24). At the time of Paul the church was in a fledging state and hardly institutionalised, indeed as the expectation of both Paul and Jesus was that the world was going to end soon. It is hardly likely that fully formed institutions were set up. The whole tone of the Pastoral Epistles is that of letters written to those who had faced up to the fact that the world was not going to end imminently and hence the move to making the church more acceptable to Roman society. Current datings have the pastoral letters written at about the year 100 AD well after the death of Paul.

What is of more relevance, is the fact that in the letters which Paul is thought to have written Paul himself refers to women leaders in the church, this is hardly consistent with the sentiments of the author of the letters of Timothy. For example Romans ch 16 mentions many women Phoebe a deacon, (vv 1-2), Prisca who is responsible for the Gentile mission and many others (for a full list see eg Ehrman Ch 25 pp 404). Thus those who would quote Timothy in favour of not allowing women to take leadership roles in the church are (as is usual with anyone who quotes the Bible to justify oppression) being highly selective to say the least. 

That should be enough to refute the premise, that the Bible says that women should have no authority in the Church. Some parts  of the Bible do, but other parts  do not, thus the Bible is at best ambiguous on this issue, but as the Pastoral letters were probably not written by Paul, those bits that do maintain the inferior status of women are arguably of less importance than those of Paul. (This includes parts of 1 Corintihians generally considered to be a later interpolation see Ehrman ch 25 p410).

In the next post I will write about a supressed tradition about a female called Thecla who is portrayed as preaching alongside Paul in some early paintings but who seems to have been air-brushed out of history.



Results Out

Well got my results as predicted Grade 4 for Topology and Grade 3 for Waves pretty disappointing really
will try and post reviews later on. Apologies for not posting for about a month but heard early in November that my mum had broken her hip and has severe dementia. Things are under control social services are looking out for her and she is in the best possible place hopefully she'll be able to move back in the home early next year neeedless to say haven't really haven't concentrated much on OU work.

Anyway I'll try and post reviews of both M338 topology and MS324 Waves in the next week or so. Must admit feeling a bit trapped by my OU commitments at the minute. The switch to October starts hasn't really helped either

Will probably just do M381 next year to recuperate and develop my piano and ABRSM skills.

So all in all a bit depressing really. Congratulations to those who have done better than me.

Will probably post a bit of a rant about the Anglican churches refusal to ordain women as bishops this weekend.


Saturday, 27 October 2012

A224 Intervals What do you call the musical Interval Db-D# ?

Well I have started in earnest on A224 music. One of the early Chapters is on the nomenclature used to describe muscial intervals. Eg given two notes how are we to describe them. The first starting point is to take the lowest note say C and then if the next note is G just count the letters CDEFG there are 5 so this is a fifth in fact a perfect fifth .
This is relatively straightforward. The problem comes when one of the letters is sharpened. In the case of C - G if we sharpen the G then we G# and because this is a semitone higher the fifth is said to be Augmented if the note was Gb a semitone lower then this would be a diminished 5th. However on a keyboard Gb is the same as F# so the same sounding interval has a different name C-F has four letters and so is a fourth (again a perfect fourth) so C-F# is now called an augmented fourth. The perfect intervals relate to only fourths and fiftths. Other intervals C-D a second or C-E a third and are both called major C-Db or C-Eb would be called minor intervals and C-D# is an augmented interval.

There are two main ways of looking at intervals the one in the text bases the intervals on which position they are in the scale. The lower note of an interval being treated as the tonic (First note of the scale) However I find this confusing as one has to remember all the sharps and flats in a scale. D-F# for example is a major third as the scale of D major has F# included in it. But D-F is a minor third. One can see how this could add to confusion.

A much simpler way is to just count the number of semitones between each interval and use the following table

  • P1, d2 = 0 semitones
  • m2, A1 = 1 semitones
  • M2, d3 = 2 semitones
  • m3, A2 = 3 semitones
  • M3, d4 = 4 semitones
  • P4, A3 = 5 semitones
  • A4, d5 = 6 semitones
  • P5, d6 = 7 semitones
  • m6, A5 = 8 semitones
  • M6, d7 = 9 semitones
  • m7, A6 = 10 semitones
  • M7, d8 = 11 semitones
  • P8, A7 = 12 semitones
The abbreviations are P perfect, M major m minor d diminished A Augmented.
The number corresponds to the number of letters thus C-A is a 6th the notes between the letters are
C# D D# E F F# G  G# A which is 9 so this is a major 6th. Once one has got the hang of this it is quite straightforward.

Putting my (failed) mathematicians hat on the obvious thing to do would be to draw a sort of Cayley table listing the lower note in the columns and the upper note in the intervals count the number of semitones between each note and then use the table above to label the interval. Starting from C there will be 21 entries as each note will have three forms eg Cb-C-C# however once one does that it soon becomes apparent that there are gaps. Db -D # is a good example Db-Db is a perfect unison, Db-D has one semitone and so is an Augmented unision but what is Db-D# ? it must be a unision as it has the same letters but it can't be a perfect or augmented unison. So I propose to call this a doubly augmented unison.

A similar problem occurs for say Db-B# this must be a form of 6th. there are 11 semitones between Db and B# but the only sixth's that have a name are the Major 6th 9 semitones and the Augmented 6th which has 10 semitones. Thus again I propose to call this a doubly augmented 6th. This is not a term I've seen used in any music textbook that I have, so I might have invented a new musical term.

When I've completed the laborious task of labelling all the intervals in this manner I'll publish the results.

Thursday, 18 October 2012

MS324 Exam Debrief

Well I feel a lot happier about this one than last weeks topology. However I still think I'll be borderline grade 2/grade 3. OK so the questions

1) 3 first order differential equations I think I got most of this correct

2) A tricky one on Fourier transforms the first part was OK but the second and third parts were tricky I couldn't see how to do the second part although I'm sure it will seem obvious on reflection. Left this question and then went onto the second part

7) A question on Lagrangian mechanics a pendulum attached to a massive spring. This question had appeared before but was a bit fiddly still got most of the marks for this one

5) The inevitable question on waves on a rectangular boundary again this question appeared before. Straightforward but tedious and a long time to write out. Got say 3/4 of the marks for this one

6) One dimensional diffusion equation for a bar. Again this question has appeared before but only once (2006) so I guess some people might have been taken by surprise by this one. The past two years have been a heat conduction type equation for a cylinder or sphere and generally a lot easier to get full marks for. Time was running out and as the question involved a Fourier series I got the method but my constants came out all wrong. So reckon about 1/2 marks for this one

So back to part 1 question 3 A probability question on recurrence relations which I left and question 4 a tricky one on variational principles. The first part involving a tricky taylor expansion upto 4th order however the second part involved solving a fairly simple differential equation and in the last 10 minutes I think I got this out so say 5 marks

So overall estimate say
13 for question 1, 4 for question 2, 18 for question 7, 14 for question 5 10 for  question 6 and 4 for question 4 to give

13+4+18+14+10+4 = 63 so if the examiners are feeling generous they might bump up the marks a bit and I might just scrape grade 2.

Anyway it's over I'll post full reviews of both M338 and MS324 later on this month.

Managed to have a few pints in Diggers with another guy. One of the problems with this course is the lack of face -face tutorials so it is difficult to get to know people. He's doing fluids next year along with me so hopefully we can meet up again.

Sunday, 14 October 2012

MS324 Past Exams and Revision Strategy

Well there aren't that many past papers here only 4 But even within those papers there appears to be quite a strong variation.

Question 1 in earlier years seemed to be 3 part questions on First order differential equations.
This seems to have crystallised to a longish question on 1 differential equation. I was caught out by
a part question in 2005 covering a technique which I had forgotten

Question 2 is usually on Fourier Transforms and the translation theorem the later part of this question tends to be a bit tricky so probably best avoided

but again 2005 differed as it had a question on D'Alembert's solution to the Wave equation

Queston 3 is on probability alternating between a continuous probability distribution or a recurrence relation type question for random variables (One of my weak points)

Question 4 is on Euler Lagrange equations for a given Functional

Then Question 5 is on the Wave equation for a rectangular membrane, This type of question has occurred in all 4 papers and should be a banker question but it is actually quite time consuming

Question 6 is on the heat equation usually this is in cylindrical coordinates and reduces to 1 dimensional form
However in 2005 there was an exception as it asked for a Fourier Series type solution

Question 7 is on Lagrangian mechanics involving usually a pendulum with a variable support. (I think the past papers have exhausted all the possibilities ) so they might decide to give a completely different type of problem here.

We can get our marks from any part of the paper and those people who are quick could probably get over 100% However my practice seems to be showing that I'm quite slow. Some of the latter parts are really ludicrous for just a few marks.

So my strengths are
a) Differential Equations, Lagrangian Mechanics and the Euler Lagrange equation, The wave equation and the Heat Equation

So I'll do the part 1  questions on differential equations and the Euler Lagrange equation first. If I work quickly I should be able to do most of these questions  in the first 3/4 hour

Then the Lagrangian Mechanics Question, the 2 dimensional Wave equation and the Heat equation again 3 questions in hopefully no more than half hour per question, leaving 3/4 hour to tackle the other two questions. As it is easier to pick up marks in the earlier part of the questions If I find I'm getting stuck I'll move on. The Fourier Transform question is usually quite straightforward in the first part but gets trickier afterwards again similarly with the probability question. If it's a recurrence relation I'll probably leave it.

My first attempt this morning got me borderline grade 3 grade 2 as I mucked up the Equations of motion and made the mistake of getting bogged down with the Fourier transform question and made some stupid mistakes. I must admit this harum scarum test of ones ability is a bit unfair. It all hangs on whether you can do enough in the exam. I appreciate that one must have some guarantee that one has worked independently but the move away from taking the average of one's TMA score and the Exam when I first started my OU life 12 years ago to basing it on your worst score of the two is quite unfair.

I predict borderline grade 2 grade 3 for this one simply because I just cannot work quickly enough to answer the questions accurately enough.

Why do we put ourselves through this torment  ?

Friday, 12 October 2012

What Next after Topology

The question was asked in the Topology forums what happens next. What is the point of say Cauchy series and completeness and connectedness. Here is my personal take on where I want to go next.

The next step is to finally tackle functional analysis this is a combination of metric space theory and linear algebra. The concept of a vector space of linear algebra is extended to spaces of functions, orthogonality being defined by an integral over the functions. The simplest example being the trig functions which satisfy

$$\int^{\pi}_{-\pi}sin(nx)sin(mx) dx = 0$$ .

if m is not equal to  n or $$\pi$$ if m = n

The Integral can be seen as a type of scalar product or Norm for the trig functions.
Thus after suitable normalisation the set of functions sin(mx) m = 0,1.... can be said to form a set of orthonormal functions. Also sin(mx) is an eigenfunction of the differential operator

$$\frac{d^2}{dx^2}$$ with eigenvalue $$-m^2$$ so that the solution space mapped out by the differential equation

$$\frac{d^2y}{dx^2}+ m^{2}y = 0$$ is precisely the set of orthonormal trig functions.

The generalisation of this to all sorts of differential operators and the abstract study of such relations forms a large part of functional analysis.

The OU MSc course does a course on this and the recommended text book is by Maddox which is out of print. I managed to get a copy from Amazon but it is really dry, more importantly from my own perspective, namely trying to understand the applications of Pure Maths to physics there is nothing on which most physicists would call the essential point namely the spectral theorem for both finite and unbounded operators. This is important for understanding quantum mechanics.

A more accessible book on Functional Analysis is Kreyszig

This has an introductory chapter on metric spaces, open sets compactness and convergence but also(eventually !)  proves the Spectral theorem for unbounded operators which is important in understanding quantum mechanics.

The definitive book on the applications to quantum mechanics being Von Neumann's book

But some understanding of Lebesgue Integration may be necessary the Old OU course used to be based on Weir

Unform convergence and Cauchy sequences as I discussed with Neil and the other Edinburgh M338 er's in the Diggers after the exam  play an important part of showing rigourously why Fourier Series converge. The MSc course in Approximation theory covers some of this. It's covered in Kreyszig as well.

Topology is of course used in General relativity especially the proof of the singularity theorems the definitive book on the subject being Hawking and Ellis.

However the book is extremely concise some familiarity with differential geometry in it's modern form is needed and an ideal book would seem to be the second edition of O'Neill's book on differential geometry

If that is considered to simple then either read chapter 2 of Hawking and Ellis or

A slightly more accessible account of the singularity theorems seems to be given by Wald

There appears to be an extension of the notion of compactness to differerntiable manifolds called para compactness
And this would appear to be an essential condition for defining a Riemannian Manifold.

Another area of study is the relationship between Lie Groups and Topology a book on which is given by Gilmore

And there are books on the application of Topology to physics the most accessible of which is probably by Nash and Sen

So plenty to build on from the topology course  is a stepping stone to understanding the pre-requisites to understand the pre-requisites for books like Hawking and Ellis or Von Neumann

I intend to tackle Kreyszig and O'Neill and possibly Weir next

In the mean time I shall have my pudding and get stuck into to some past papers for MST324 a weekend solving differential equations both partial and ordinary. Doing Fourier transforms and solving problems in Lagrangian Mechanics absolute Bliss. I hope I can get up to speed.

Wednesday, 10 October 2012

M338 Exam debrief

Well this will be quite short as predicted in the previous post I really did not do well on the exam time just run out and I got bogged down on basics. I managed 4 out of 8 questions in part 1 and 1 more or less full question in part 2 and bits of a question that should have been a doddle

OK question 1 was as expected a relatively straightforward question on the interior and exterior of two sets.
Think I got most of this but due to nerves etc I took 25 minutes on this

Question 2 was on whetther or not a set defined a topology on a finite sets again I think I got most of this out another 20 minutes on this

Question 3 was showing that certain edge equations could be reduced to canonical form. It looked like this would take ages to work out so I moved on, with the hope that I could return to it later

Question 4 was a question on a shape with a twist in it and you had to divide it by drawing veritices etc a question I never got the hang of so moved on

Question 5 was one which should have been straightforward involving the subdivision of a sphere in terms of octagons and triangles. However normally this type of question just involves one type of polygon. I Didn't have time to think through the implications of this so moved on

Question 6 was about compact sets again I left this as I was never that clear about this type of thing

Question 7 was about sequences in a topology think I got most of this correct

Question 8 was about similarity relations for a fractal again I think I got most of this correct

So maximum possible marks is 32 say 24 after dropping two points per question

Then part 2

Question  9 was on edge equations and reducing them to canonical form something that should have been a doddle. But the twist here was that the edge equations were separated out. So before one could even start one had to combine the equations to get a single equation. I did this then got the Euler characteristic but completely forgot how to obtain the boundary nuimber for this type of question. And I could not reduce to the equation to canonical form. Part 2 of this question was similar involving 4 equations. Again I could get the Euler characteristic but not much more. So a maximum of 4 marks here.

Question 11 was on metric spaces and sequences defined on the metric space. The metric was quite straightforward and I got most of the first two parts correct. Namely calculating the metric for two values for 2 marks and then showing that the metric was indeed a metric space. 8 marks. One then had to calculate the metric for various open balls defined on the metric a maximum of 5 marks so say 10 marks for this question overall.

Time ran out I couldn't be bothered to chase after the remaining marks
so a reasonable estimate is 24 + 4 + 10 = 38 marks. You need 40 marks to pass so I have definitely failed this one.

Hopefully I can move on and do better with my waves course. If I get offered a resit I will probably take it. In the long term scheme of things Topology doesn't matter. I'm a bit more familiar with the language of topology and that should give me the confidence to look at say the applications of topology to physics. But this has been a nightmare best forgotten.


Tuesday, 9 October 2012

Going to Fail M338

Well tomorrow is M338 exam day. I feel that I deserve to fail and I probably will. Still I'll turn up and see what happens. OK what went wrong. I basically switched off. I thought I might be able to salvage something by concentrating on Block B, But the exam questions seem to bear little or no resemblance to what is in the course material or if it does then there are very little guidelines as to what one is supposed to do, OK so a typical question will ask us to subdivide a solid with twists and deduce it's characteristic equations and say its boundary number. In order to calculate the Euler characteristic one must work out the number of vertices the number of edges and the number of face. The first two are relatively straightforward but the last one seems to defy all sorts of logic. Again there is normally a question on embeding a network within a rectangle representing say a torus or a projective plane but again  the course material gives no general guidance as to how this can be done apart from here is the diagram and here is the embedding. Fine I'll leave it to those who can intuitively understand what is going on for me it just leaves me baffled. Again in the second part there is usually a question relating the forms of surface described in connected sum form  to another descriprtion in connected sum form . Again little or no guidance has been given as to how to tackle this type of question.

My banker questions from part 2 will be deducing the form of a solid from vertex insertions and possibly one on Metric spaces and whether or not sequences defined on the metric space are Cauchy. But the examples apart from the one in 2010 are such that one has to really think on ones feet as to what the metric really means.

So I should get more or less full marks for one long question in part 2 that is say 14 marks. There are two possibly 3 part 1 questions I can get more or less full marks for thats 24 marks then I'll be really stretched to get more. I could aim to delibrately fail in the hope of getting a resit next year but there seems to be some ambiguity as to if I did that I would only qualify for a straight pass. So I might just scrape a pass but anything more I wont deserve. Fortunately I don't have to count this course for anything for my second open degree so it's best quietly dropped whatever the grade whilst I include other stuff.

Good luck to those who have got more out of this course than I have. But I never really got in to it and so deserve to fail.

Monday, 24 September 2012

Why I don't trust the Bible

I have been embroiled in a discussion with a work colleague who is an Evangelical Christian. He asked me to read a book by a Ms Amy Orr-Ewing called "Why I trust the Bible"  this is my response. There is a rise in Fundamentalist Christianity today I hope the following text and associated links will give some arguments for those people who get embroiled in such discussions. I should come clean in my first year at University I was a member of the Christian Union for a year. However the more I read around the subject the more, I became aware that the Evangelical View of things was just one particularly obnoxious view of Christianity placing more emphasis on individual salvation rather than say the fight for social justice. I abandoned Evangelical Christianity and became a liberal Christian. I have retained a modicum of interest in academic theology of which I was quite keen on in my twenties, but would now call myself agnostic, however given the rise of fundamentalist Christianity and traditional Catholicism I think it is worth trying to counter the obnoxious views prevalent in todays Churches. What follows is an attempt to redress the balance.

                          Why Trust the Bible ? Amy Orr-Ewing A review


At the request of a work colleague I have been asked to state my opinions on the above book.This book is written by an unashamedly evangelical Christian who works for a Christian apologetic organisation As such it is a piece of bibliolatory, designed to convince sceptics that the Bible is to be trusted (whatever that means). Presumably Ms Orr-Ewing means that the Bible can be trusted as a reliable witness to the allegedly historical events that are described in it. That it’s strictures on ethics are reliable and that despite numerous discrepancies between say the four gospels, these can all be harmonised with each other. More importantly from her point of view Ms Orr-Ewing wants to get people to read the Bible so that they will become converted and accept Jesus as their Saviour, believe in the penal substitutionary theory of the atonement, a literal virgin birth and that the miracles of Jesus and the other fantastical stories in the Bible were historical events which actually happened.
Although Ms Orr-Ewing doesn’t explicitly state it ,if Ms Orr-Ewing is a typcial evangelical Ms Orr-Ewing believes that the Bible is the inerrant word of God and that this is the only way in which the Bible is to be interpreted. Liberal Christians and others who see the Bible as a piece of culturally significant piece of literature, some of which has merit for today, but much of it, especially large chunks of the Hebrew Bible such as the barbaric laws in Leviticus and Deuteronomy, is to be discarded, according to Ms Orr-Ewing are just wrong and cannot be ‘true believers’. The liberal secular view of the Bible despite it being that of most Academic theology departments, is anathema to her, as her whole chain of beliefs would come tumbling down.
One of the most dangerous things one can do, is to elevate any text and say it is beyond criticism and must be correct for all time. This is especially true of religious texts, including both the Bible and the Quran. This leads to the intolerance we are currently seeing in Muslim countries, where the smallest degree of criticism of the Quran has led to the murder of many people, death threats being laid on authors such as Salman Rushdie and journalists. In the past critics of the Bible, have been hung or burnt at the stake, thankfully we don’t, at least in the West live in such times but the tendency is still there. The Catholic Church seems to want to move Europe back to a theocracy and in America large numbers of states seem pretty close to a theocracy. Certainly in the South, anyone who dares challenge the alleged authority of the Bible would be regarded with deep suspicion. All of this is inimical to the democratic notion of free speech and open inquiry essential to the Modern Age. If the Evangelicals had their way they would also try and impose a degree of theocracy and go back to a time when the church had control over people’s lives, even if they weren’t Christians.

I don’t have time to deal with all of Ms Orr-Ewing’s points, but some will suffice, In her first chapter Ms Orr-Ewing claims to have dealt with the question of whether or not anyones understanding of the significance of the Bible is just a matter of interpretation. Unfortunately Ms Orr-Ewing has a tendency to include a lot of anecdotal stories about particular people who were previously sceptical but on reflection became Evangelical Christians, Ms Orr-Ewing doesn’t mention for example others such as Bart Ehrman who were evangelical Christians and then later rejected the evangelical interpretation of the Bible. All these anecdotes really do not help her case at all.

Her key point in this chapter, is that under the influence of postmodernism has arisen a suspicion of the use of key texts such as the Bible to maintain power. One of the main proponents of this view being Foucault. Ms Orr-Ewing attempts to dismiss this view as Foucault and other post modernists cannot avoid a charge of circularity as their texts also seek to impose a power structure on people. However the point is that whilst historically the Bible has been used to oppress people, particularly women, the poor, people of differing ethnicity and sexual orientation, no one uses Foucault’s or any other post modern writing to oppress people. Ms Orr-Ewing, despite having a degree in theology, would have us believe that the call for the critical examination of texts such as the Bible is a recent occurrence. Yet with the rise of Biblical criticism in the late 18th century, people have argued that the Bible should be subject to the same degree of criticism as say the works of Plato or any other classical literature. In the 1860’s a collection of essays written by eminent clerics and theologians advocated this very same point of view. [1]. The results of Biblical criticism namely that what we have in the Bible is an intertwining of different sources and that the Bible is essentially a collection of different books containing differing views about the relationship between God and man or differing interpretations of the significance of Jesus Christ is now seen as the main way of teaching Old and New testament theology in universities today [2].

Ms Orr-Ewing may not like it, but the fact is that the Bible has been interpreted in different ways by different people throughout history. Ms Orr-Ewing’s and the Evangelical view that the aim of the Bible is to get people to believe in the penal substitutionary theory of the atonement, to have faith in Jesus Christ and that the Bible is the inerrant word of God is by very definition a particular interpretation, not shared by most people including other Christians. Thus I cannot find her arguments in the first chapter at all convincing. Ms Orr-Ewing is guilty of the special pleading that she accuses other people of .

In her second chapter Ms Orr-Ewing tackles the question as to whether or not we can know anything about history. Again Ms Orr-Ewing seems to think (along with other writers such as Dever [3]) that the current post modernist climate is sceptical about history and Ms Orr-Ewing makes a rather tendentious claim, that those who deny that the Bible is historical are the same as Holocaust deniers. Doubting the historicity of certain texts and events described in the Bible, is not the same as denying that history occurred. We have evidence for the Holocaust, we do not have evidence for the Exodus, the miracles of Jesus, the conquest of Canaan by the Israelites, as opposed to a gradual settlement [3,4]and so forth. What is doubtful are the versions of history as presented in the Bible, not that historical events did not occur. I could give many examples lets just take a couple

First the dating of the destruction of Jericho [For more see the debate [5]) In the 1950’s Kathlyn Kenyon performed a series of excavations at a site generally considered to be that of Jericho. It uncovered evidence of a cataclysmic event and this was dated as 1550 BCE (Before Common Era). However if one takes the chronology of the Bible at face value (admittedly something quite dubious) The conquest of Canaan as described in the book of Judges took place 460 years before the reign of Solomon. This is reckoned to be about 1400 BCE thus there is 150 year discrepancy between the chronology of the Bible and the archaeological evidence. Current archaeological evidence casts great doubt on the historicity of much of the Old Testament. It is true that there is some debate between maximalists such as Dever [3] and minimalists such as Finkelstein [4], but both agree that there is little or no evidence for the Exodus stories and the conquest of Canaan as related in the book of Joshua.

Secondly, the conflicts in the Gospels surrounding the events of the crucifixion and the alleged resurrection of Jesus. If one compares the stories in the four Gospels with each other reading the Bible horizontally, as Bart Ehrman suggests. It immediately becomes clear that there are numerous discrepancies. The earliest versions of Marks Gospel do not have any appearances of Jesus to the disciples [6] it is generally considered that the appearance story that does appear in the version of Mark in today’s version is a later interpolation [7] Matthew has a fantastical story about 500 saints being resurrected and appearing in Jerusalem. The synoptic Gospels have Jesus being crucified after the Passover meal whilst John has him being crucified before the Passover meal. A chart cataloguing some of the differences in the stories can be accessed here [8]

Any open minded person would see these stories as not being straightforward history. The differences occur because the Gospel writers had different views of the significance of Jesus’ death and his alleged appearances to the disciples. More sinisterly the shift of the blame for the death of Jesus, gets transferred from Pontius Pilate to the Jews. In one of the most chilling passages in the Bible (Matthew 27:25) the author of the so called gospel of Matthew alleges that the Jewish crowd say "His death be on us and on our children." This verse has been used down the centuries to justify the persecution of the Jews and as the Gospels develop, there seems to be an increase in Anti-Semitism. Of course any normal person would say that this verse and all the anti semitic aspects of the Gospels should be disregarded. That option is not open to those such as Ms Orr-Ewing, as she believes that the Bible is the inerrant word of God.

It is true that there is a whole industry devoted to trying to reconcile the differences [9]. Typical arguments include such claims, as the Gospels only reflect partial accounts. The events happened twice or thrice and so forth. If there really were 500 saints that visited Jerusalem, as the author of Matthew claims, why wasn’t that considered significant by the other Gospel writers. Clearly if you want to go down that road you can reconcile anything, however that is not the mainstream academic view. The differences reflect the differing Christianities that were around at the time. It does not help an understanding of the significance of the Bible to gloss over these differences. Bart Ehrman has painstakingly documented all the other gospels that did not make it into the final canon [10]. These contain many other stories about Jesus, why is it that these are not considered historical, yet the ones that made it into the canon are ?

To summarise then there is more than enough evidence to show that the Bible cannot be seen as straightforward history. This realisation occurred in Western Europe over 250 years ago with the rise of Biblical Criticism, Evangelicals such as Ms Orr-Ewing can stick their heads in the sand and ignore this but they will end up ignoring the fruits of academic research and hence despite a popular appeal will become intellectually irrelevant.

In the third chapter of her book, she discusses the establishment of the Bible as we have it to day. Glossing over all the difficulties in getting to the original documents, she blithely claims that the canonical formation of the Bible was unproblematic. Yet as Dale Martin points out in his second lecture [11], there were many variations of the alleged canonical books right up until the 5th and 6th centuries. These local variations persist today, furthermore there were many alternative gospels which were rejected as they did not fit the theological prejudices of Eusebius. Ms Orr-Ewing makes no mention of the fact, that it was only when Christianity became the State religion that anything like an agreed set of texts was established. Prior to the adoption of Christianity by Constantine, there was not just one Christianity but many. What became orthodoxy in the West was a fairly arbitrary selection of traditions, which suited the agenda of Constantine and his Bishop, Eusebius. Ms Orr Ewing would have us believe that the Evangelical interpretation of the Bible is the only one possible, yet even the Evangelical interpretation is a development of doctrine after Nicea, It’s claim to orthodoxy being a mirage, as there was never an orthodoxy in the first place.

Finally Ms Orr-Ewing discusses ethical matters, showing the Evangelical pre-occupation with sexual matters, instead of say social justice. She discusses the status of women in the Bible and the sanctions of the Bible against homo-sexuality. Here Ms Orr-Ewing seems to be rather selective to say the least. She rightly points out that the early church, had many women as leaders, but she conveniently ignores the passages in the Bible such as the Pastoral letters which enjoin women to stay silent in the church and so forth. The fact is that the church today especially the Catholic Church and the Republican right has continually opposed the long struggle for equality for women. Indeed the Catholic church persists in not allowing women any authority in the church and even the Anglican church still agonises over whether or not to ordain women as bishops, bending over backwards, to accommodate those misogynist clergy who oppose the ordination of women on allegedly theological grounds.

If Ms Orr-Ewing took the liberal view, namely that these verses are just outdated and should be dropped, then it would be perfectly fine for her to ignore those verses which insist on the inferiority of women, but again her adherence to the view that the Bible is the inerrent word of God precludes her taking this option.

In another chapter Ms Orr-Ewing tackles the issue of homo-sexuality. She takes the traditional line quoting Paul that homo-sexuality is sinful and obviously opposes any notion of civil partnership or marriage. Then she wants to plead, that she is not bigoted or intolerant. Unfortunately Ms Orr-Ewing has to put up with the fact that the Church in many places is being extremely prejudiced to say the least. Indeed when children of the age of 4 are applauded in church for making up ditties that homosexuals wont get to heaven [13] something has seriously gone wrong. Ms Orr-Ewing no doubt would claim the sanctity of biblical marriage, but of course she doesn’t mention the fact that books such as Deuteronomy and Leviticus have views on marriage which would seem abhorrent to any reasonable person. To take one example (of many) Deuteronomy [22 Vs 28-29] claims that a victim of rape must marry the rapist and all the rapist has to do is pay the Father of the woman 50 shekels and not divorce her. More on just how obnoxious the Old Testament view of Biblical marriage can be seen here. [13]. Ms Orr-Ewing may say that is the Old Testament, no longer relevant to today, but that again would contradict her position that the Bible is the inerrant word of God .

There is a general point about using the Bible to uphold any ethical principle, in that the idea that one can transplant views written over 2000 years ago to the ethical dilemmas of today’s society is just incredible. The more one closely investigates some of the strictures of the Bible on ethical positions, the more irrelevant they seem to today. It is far better I would argue to look at the philosophers and their views on current ethical dilemmas. Philosophy enjoins people to argue their case as best they can using, as far as possible reason. Indeed given that the study of ethics independent of any religious bias started with the Greeks 400 years before Christianity started, the idea that our Western system of ethics derive from Christianity is just wrong.

The world lives in the grey, not in the black and white of Fundamentalist religion and requires thoughtful and critical analysis. Just blindly adhering to an ethical position, because it’s in the Bible or any other religious text, isn’t really going to help matters. Historically the Church especially the Catholic Church, has always opposed progressive social change. It opposed the ideals of the French revolution, it supported the Fascists in the Spanish Civil war and it came to an accommodation with Mussolini and Hitler. It continues to oppose equality rights bills such as Harriet Harman’s recent bill. In America the Evangelical churches support the Republican Party. Of course there are exceptions, the liberation theology of the Latin American church in the late 1970’s which attempted to resist Latin American dictatorships [14] and the Black churches campaign for Civil rights in America, But liberation theology was suppressed by the Catholic church and by definition Black churches are still not mainstream in American society.

To summarise then I have given reasons why I don’t think the Bible is to be trusted, it cannot be trusted historically, and it certainly can not be used as a guide to Ethics. It has a place in society as a piece of culturally significant literature and should be treated, as we treat any other piece of literature, such as say the works of Plato or Aristotle or Shakespeare. Taking the Bible as literally true, or as the inerrant word of God, is just not credible, as we learn more and more about it’s historical context. I long for the day when people no longer use the Bible blindly to justify discrimination against women, people of ethnic minorities and sexual orientation. I long for the day when the Bible is not used to justify oppression and I long for the day when the barbarity of Old testament books such as Deuteronomy and Leviticus are seen for what they are and rejected by the churches.
References and External Links

[1] Essays and Reviews 1860 An online version is available here.

Some of the background to the book is given here


Despite it’s age it is surprising how modern some of the views are. Contrary to Ms Orr –Ewings sentiments I think it is a real tragedy that the views expressed did not become mainstream, thus allowing for a more mature assessment of the significance of the Bible to be made.

[2] A convenient summary of current academic views on the New Testament and the Hebrew Bible can be found in the Yale Open Course website.

What is clear from these courses is that the current Bible cannot be seen as a unified text perpetrating Christian orthodoxy but is a collection of diverse literature with differing perspectives on the nature of God and whose historical accuracy is to be questioned. Whatever your religious views or none, these courses are very informative.

[3] Dever W G ‘What did the Bible Writers Know and when did they Know it’ WM B Edermans Publishing Co Grand Rapids/ Cambridge[2001]

[4] Finkelstein I and Silberman‘’The Bible Unearthed ‘ Simon and Schuster 2001


The whole website has a series of articles refuting the claims of those who say either the Bible has no contradictions or that it can be taken as historically accurate.

[6] See Dale Martin lecture on the Gospel of St Mark lecture 6

[7] See above


(NB I should point out that whilst the chart is a useful reference I do not endorse the sentiments of the host site. However I can sympathise with the attempt to resist conversion of Jews to Christianity).
[9] I’m grateful to my work colleague for giving me a document ‘Apparent Contradictions in the Bible’ by a Mr Charlie Campbell who makes a living trying to refute the claims of academic scholarship.

[10] Ehrman B ‘Lost Christianities’ Oxford University Press 2003.

[11] Martin D Lecture 2 From Stories to Canon.


(NB In the above two references I do endorse the sentiments of the host site which is to get people to look at the Bible in its historical context and to oppose the use of the Bible to justify bigotry).

[14] Gutierrez Liberation Theology. SCM Press 1974.

The concerns of liberation theology were the fight for social justice, it showed how parts of the Bible could be made relevant to contemporary issues, by reminding people that one of the concerns of the Old Testament prophets was for social justice and that Jesus could be seen as siding with the poor against the powerful. Of course such a message is alien to Evangelicals, such as Ms Orr Ewing, who seem more concerned with sexual morality instead of social justice.

Sunday, 23 September 2012

Interlude OCAS for M338 and MST324

Well got both marks back for M338 and M324 At  the high end of grade 2 for M338 and  a distinction for MST324. Overall that means my OCAS for both courses is at the high end of grade 2 being only a couple of marks short of distinction. So it's all to play for if I do really well in the exams I might just get a distinction as happened for MS221 a couple of years back however whilst that seems feasible for MST324 but I have my doubts whether it will work for M338

 One thing about 30 point courses is that a bad TMA has so much more weight than a 60 point course. So if you do badly as I did with the 3rd TMA for MST324 then despite substitution you cant really make it up. Revision has started in earnest for Topology maybe I'm a bit overconfident vis a vis MST324 but unlike Topology the waves stuff is quite familiar so I think a few past papers in the last week should be enough to get me through. With Topology it really is difficult to remember what you did in the past. I'm afraid I'll be taking the path of least resistance focusing on Block B and doing enough of the basic definitions to answer the short questions. The format of the exam is 8 short questions 4 of which tend to be on the hard stuff and the other 4 on relatively straightforward stuff and then two long questions from 4. Two of which tend to be on Block B stuff. So that will be my focus. I havent yet looked at the past papers in any detail but am concentrating on getting the basics of Block B sorted out. I will be rather busy over the weekends before 10th October when the exam is but am taking a couple of days off work before the exam. So hopefully I'll be able to consolidate things then, Then that gives me another 4 days to concentrate on MST324 the weekend of the 12th-14th and two days before the exam on the 18th. I past paper per day should nail it.
So predictions I think M338 will be a grade 3 if I'm realistic and grade 2 if I'm lucky and MST324 should be a solid grade 2 with grade 3 if I'm unlucky.

On another note I've got my course materials for both the Fluids course and the Music course which I've not really had a chance to look at in any depth. However as both courses seem quite familiar I doubt whether they will cause me as much grief as M338 has. One very interesting thing about the Music course is that it tries to teach composition of songs something A214 the earlier course didn't. I hope this will lay the foundation for my eventual courses in composition via the OCA. Unfortunately my schedule for embarking on that is getting pushed further and further back. I can't see it being feasible financially or timewise until at least June 2014.  Especially as I want to do M381 (Number theory and logic) and the level course on the philosophy of mind (AA308 I think) in October. For music I will concentrate on the piano and the higher level grade theory exams which again will complement the OU music. I get the impression that whilst the OU covers a lot of stuff in music it does skim on assessing your basic knowledge of the theory. In contrast the Grade 5-8 theory exams do test rigourously your basic understanding. I have to say I'm impressed with the ABRSM framework they have consistently maintained standards over the past 150 years unlike say school exams. Perhaps Michael Gove should take a lesson from how the ABRSM exam system is structured. You could split each subject into 8 levels with 1-5 being equivalent to O level and 6-8 equivalent to A level and on average most pupils could sit 1 level per year. So that it doesn't come as a shock to the system.

As for the Fluids course it covers complementary ground to MS324 and develops the mathematical techniques further. including an introduction to the solution of differential equations by series, waves in fluids and Laplaces equation in spherical polar coordinates and an introduction to Legendre polynomials. One slightly annoying thing is that the revision book has to be downloaded. I do think this is a bit mean of the OU which seems to be moving to putitng more and more stuff   online. Fortunately I live in Scotland so am not subject to the tremendous hike in fees that those in England have to suffer. I'm afraid I still like hard copies and I would really question the value of paying £1250 for a 30 point module which skimps on hardcopies of the material. I can see that this is just going to get worse and worse as time moves on.

Sunday, 9 September 2012

MST324 TMA04 Finished

Woohoo thats my last maths assignment away for this batch of courses. Of course with the switch to the October starts it wont be the last maths assignment of 2012.

So the focus of the last block of MST324 is the caclulus of variations. In my undergraduate lectures we had about 1 lecture on this fascinating subject. A typical problem that can be solved by this technique is what is the shape formed by a rope or cable under it's own weight when slung between two points. It has become a key technique in many branches of physics, especially quantum physics and the development of the path integral formalism by Feynman. Of course none of this heady stuff is covered here. However the applications to mechanical problems is of interest and it culminates in a new way of presenting mechanics based on the minimisation of the Energy known as Lagrangian mechanics. This provides a simpler way of obtaining the equations of motion for complicated systems instead of analysing the forces on a body. Any of my readers who have done MST209 will know just how confusing it can be to set up the forces even for the simplest of mechanical systems.

The essential part of the calculus of variations is to see what happens when small variations are made to a function under an Integral. Going back to the example of a cable hung between two points there are many possible functions which could pass between the two points and so one considers a collection of functions a functional rather than just one function.  By considering small variations in the Functional it is possible to obtain a set of differential equations for the Function which minimises the variation. These are called the Euler Lagrange Equations.

So to a brief overview of the TMA

The first question involved  caclculating the effect of small variations on a function. Seeing what happened if parts of the function were set to a constant. This involved using the Binomial theorem to expand a function under a square root sign and calculating the difference. Comparison with a generalised Taylor theorem then gives certain conditions. Whilst in principle straightforward this was a bit tricky algebraically (indeed that is a theme of this TMA). Still I think I got this one out

Questions 2 and 3 were concerned with obtaining the Euler lagrange equations for various functionals. Question 2 was again straightforward but applying the boundary conditions to the Functional led to quite a messy solution. Question 3 was concerned with showing that a transformation applied to a given Functional gave a simpler form of the  Euler Lagrange Equations for the original Functional. This was probably the easiest question of the TMA. Again I think I did justice to the questions.

Finally question 4 a question of applying Lagrangian Mechanics to the problem of a bead on a rotating parabolic Wire. There was a whopping 40 marks for this question

The first step is to obtain expressions for the Kinetic Energy and the Potential energy in terms of coordinates which are easier to express the problem in. The so called generalised coordinates.
This involved setting up a conversion between the postion of the particle expressed in Cartesian coordinates and the more convenient coordinates. The Lagrangian is the difference between the Kinetic energy and the Potential Function. The equations of motion are then derived for the generalised coordinates by applying the Euler Lagrange Equations. For this Lagrangian the equations of motion for one of the coordinates seemed quite complicated. The rest of the question then involved solving the equations of motion to obtain the angular frequency of oscillations of the bead for various conditions. This was algebraically quite involved using Taylor series and dropping terms of higher order. As there was no show that the solutions are of the form ... This made the question quite tricky to see if you were on the right track. However as the last part asked us to comment on the relation between the various angular frequencies and as my answer showed quite a simple relationship between them. I'm reasonably confident I got the algebra correct. The thing with complicated algebra is to just keep going and not to panic.

So overall I'm reasonably confident of a high mark for this one. Just as well as I need to make up for the last TMA.

I'm going to take a complete break from maths this week. The past few weeks have been a nightmare as I desperately tried to complete the TMA's . I will start revision in earnest after next Monday. For now I just want to chill.

Wednesday, 29 August 2012

M338 TMA04

So I managed to finish the final TMA for M338 and will post it off tomorrow. I have to say conceptually this unit and block have been the hardest course I have ever done. However things improved from my initial wtf about block C at the start of the month to the stage where I'm begining to see how some of the definitions hang together and what they can be used for.

So an outline of the TMA

Question 1 was a question about whether or not subsets of a set with a given topology were disconnected or not. This involved the union of Z (the set of all integers) with N the set of all integers > 1. Also whether or not the topology is path connected. I think I managed to answer most of this question

Question 2 was concerned with the notion of compactness essentially an extension of the concept of finiteness to topological spaces. A set is compact if for each cover of a set X (ie a collection of sets wich contains X) a finite subcover can be found. It's a generalisation of the idea that whilst a closed interval of the real line [a,b] can be covered by a set of finite intervals an open interval (a,b) cannot be, as there will always be a gap between a and any finite interval of the form a-1/n as a is not in the interval. Despite a slightly counterintuitive part I think I got the first part correct. The second part of question was quite straightforward essentially sketching two sets and showing that that their union was closed and bounded.

The first part of Question 3 Involved Cauchy sequences and it was good to get back to something a bit more concrete. A Cauchy sequence of functions is one for which given an epsilon > 0 and integers n,m > N the distance function between a sequence of the form an is such that d(an,am) < epsilon whenever n,m > N
This was relatively straightforward. The second part of the question involved showing that a union of sets was closed and compact. This involved a lot of epsilon delta and upper bounds of sequence type arguments it felt like the more trickier parts of the analysis parts of M208 but I think I got there in the end

Question 4 involved the contraction mappting theorem and its use to find the size of an interval containing the zero of f. There was a little bit of calculus here but not enough for my tastes. The contraction mapping theorem has an application to the proof of the existence of solutions to a given type of differential equations but alas this part is not assessed.

Question 5 involved similiarity transformations for Fractals and is related to the contraction mapping theorem. As there were similar examples in the exercise book to the question, it wasn't to difficult to solve the problem. In fact question 5 is probably the easiest question of the whole course.

So all in all, I think I've done reasonably well I may not have phrased my answers quite correctly and I certainly cant claime to understand all the chain of definitions and proofs that this intricate subject requires but I feel reasonably confident about the TMA and the ability to answer the exam questions with a bit of practice. But again jsut as with M337 and M208 I can only feel I've scratched the surface of this subject and would appreciate the time to revisit it again.

The past few weeks have been quite intensive as I've been to quite a few concerts at the Edinburgh Festival. I'll give a review of the concerts in the next week or two. But there is no rest for the wicked as I have to do the TMA for MST324 withing the next week. I'm still plugging away at my scales but due to the festival and finishing the TMA for M338 not as much as I could be.


Sunday, 12 August 2012

M338 Block C

Ok momentary panic over. I've decided after  the tutorial to stick with the Topology. I get the impression that we are all struggling with it. Even our tutor Alan says he's not sure about some bits of it. Anyway the tutorial clarified some issues or at least showed how to go about tackling questions. My mate Neil showed a brilliant insight into one of the questions involving whether there was a path in Z between all elements of Z for  a topology which consisted of {0,z} U (Subsets of even numbers). Our tutor had showed that there was indeed a path between odd and even numbers and was about to go through the same tortuous reasoning to show a path between the even numbers and a path between the Odd numbers. When Neil interjected and said that if there is a path between an odd and an even number then there is also a 2 step path to an even number or odd number if P(a,b) is a path from a to b where a is odd and b is even there is also another path
P(b,c) from b an even number to c an odd number so there is also a path from a to c which is the sum
P(a,c) = P(a,b) + P(b,c) and as a and c are both even or odd then this shows that there is a path for all elements of Z. Anyway it's battling on with the TMA this week. I feel a bit more confident and hope to finish by the end of next week. One of the problems as Alan admitted is that the course material gives very few examples to illustrate the definitions and there does seem to be little or no motivation for them. I will probably do M303 the new pure maths course when it comes on the scene even though it is an excluded combination as it will help me consolidate my meagre understanding of the course material for M338, also as I wont be doing the group theory course M336 then its a chance to cover that also M303 contains an introduction to groups rings and fields. So I will put this into my third (!) open degree. I'll be concentrating on Music for the next 4-5 years so don't think I want to embark on postgraduate work for either maths or philosophy. I want to do at least the new philosophy 3rd level course and there is going to be a new third level music course appearing on the scene as well. I might also include the two third level astrophysics courses and the second level Chemistry course we'll see.

The Festival starts for me on Wednesday and I'm going to quite a few of the concerts in the Usher Hall I will also be taking a few days off. I'll give reviews of the concerts I'll be going to over the next few weeks. The itinery for this week is

Wednesday   15th  Tristan and Isolde
Thursday       16th Syzmanowski Symphony No 1 and Brahms No 1 this includes a performance with Nicola Benedetti
Friday           17th Syzmanowski Symphony No 2 and Brahms No 2
Saturday       18th Syzmanowski  Symphony No 3 and Brahms No 3
Sunday         19th Syzmanowski  Symphony No 4 and Brahms No 4

I must confess I don't really know the Syzmanowski Symphonies and so I'm looking forward to getting to know these pieces.

Anyway I may bump into some of you who read this blog there. I'll usually be in the Traverse theatre bar after the concerts.

The piano practice is coming on I'm doing at least 1/2 hour per day before I get to work, This week I want to consolidate what I have done so far namely scales and broken chords for C major, D major and G major. The left hand and right hand parts for two of the Grade 1 pieces separately and a rather pathetic attempt to put them together and Chapters 1 - 4 of Fanny Waterman's book volume 1.

Next post I'll tell you how the Key system of Western music and the circle of 5ths can be reduced to modulo arithmetic.

Monday, 6 August 2012

Do I or don't I

Well just like other people who indulge in  this OU maths blogging lark I'm in a bit of a dilemma vis a vis whether or not I should continue with a course. The course in question is M338. So far I've done hardly anything for the last TMA and I really can't motivate myself to continue wading through definition after definition wondering what the point of it all is. On the other hand I've come so far 3 TMA's all with a respectable grade two passes and it's only 1 more TMA, the last part of which looks relatively straightforward and I guess the exam wont ask anything too challenging  It's just getting a feel for the definitions of compactness and connectedness which I'm really stuck on. So it would seem on the face of it stupid to give up now. A grade 2 pass should be within my grasp. I have a tutorial next week and so I'll leave it till then.

Life would so much simpler if I jacked it in but it would also be an admission of defeat.

On a brighter note I seem to be slowly getting through the keyboard/piano practice and can do (single handedly) a reasonable attempt at the scales of C major, G major and D major. Also my broken chords seem to be coming together. The left hand is a bit more tricky especially for scales ascending. It's not just a question of playing the correct notes but also making sure the fingering is correct. I'm slowly beginning to develop an intuition when something is not quite right for example in the scale of G major which has one sharp F# when using the left hand the second finger must always hit the black key. If it doesn't then you know you have your fingering wrong and it can be quite frustrating at times. I'm manging to put in about 1/2 an hour before I go to work and about an hour when I get back something seems to have been unleashed here. If I carry on this rate with suitable guidance I would hope to put myself forward for the grade 1 exams in March of next year and ideally Grade 2 in the summer and Grade 3 in winter certainly 1 want Grade 1 and 2 by the end of 2013.  I will also do the Grade 5 theory exam (a prerequisite for going beyond grade 5 practical) in 2013 probably after I've completed my  OU music course which covers grade 5 and a lot more. 

I've decided to only do 1 OU maths course per year from October 2012 and due to the timing of the presentations it is going to have to be Fluids and mathematical methods for 2012 and Number theory and Logic for 2013, which has been given another years leave of absence before the doors finally close.  I really wish the open university could be a bit more clearer as to when the last presentation of a course is going to be. I felt bounced into registering for M381 as it was going to be it's last presentation but then all of a sudden they change their minds.

I wonder if certain friends (I'm talking about you Duncan) might be tempted to do number theory  in 2013 as well. Anyway Fluids and mathematical methods along with music and piano practice should be more than enough to keep me busy for 2013.

Finally for those who are interested I got my third TMA for MST324 in the mid sixties as expected. If I get a distinction for the final one and  do well in the exam I still qualify for at least grade 2 and maybe even distinction. That incidently is another reason for me to contemplate abandoning topology as I would only have one exam to concentrate on.