## Friday, 24 December 2010

### Happy Christmas to you all (Bah Humbug)

Not having a family or partner I shall be spending Christmas on my own however whilst those of you forced to entertain the in laws, spend a vast fortune on presents for your kids which will be discarded within a couple of days, and eating tasteless turkey/ I shall be watching Wagner's Ring cycle a rough time table is as follows

Tonight from 7 : pm - 9:30

Das Rheingold

10:30 to midnight approx

Act 1: Das Valkyrie

Christmas Day

10:00 am to 12:30 pm Act II Das Valkyrie (the Crucial act in my opinion)

2:00 - 3:30 Act III Das Valkyrie (Means I wont be watching the queen';s speech oh dear  what a pity ;))

Cook my special dinner

6:00 - 7:30 Act I Siegfried

9:00 - 10:30 Act II Siegfried

Boxing Day

10: 11:30 Act III Siegfried

1:00 - 3:30 Act I Gotterdammerung

5:00 - 6:30 Act II Gotterdammerung

8:00 - 9:30 Act III Gotterdammerung

So I've stocked up on food and booze and if I get through that timetable I shall be as high as a kite and pissed as a newt and sufficiently energised to get back to some maths after that. I'll give a post on what I think the ring cycle means

Anyway whatever you are doing over the festive season have a good one

Best wishes to you all Chris

## Saturday, 18 December 2010

### Test of Dropbox

Thanks to Nilo he pointed me to a way in which files can be shared called dropbox. As a test of whether or not it works I enclose a link to my simplified big bang calculation

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

As stated in an earlier post, it is quite surprising just how a simple calculation like this can get close to the results for the real calculation. I know that there is speculation that the extrapolation from the 1st 100th second backwards breaks down, however this bit is I would claim robust physics, in that it is the only convincing explanation of the relative abundances of Hydrogen and Helium in the early universe. A large piece of the jigsaw. Given the relative simplicity of the calculation it should be accessible to any one with a mathematical knowledge equivalent to MS121 and physics at the level of S207 or anyone with A level/higher physics and A level/higher mathematics. As such I believe it should be more widely known. If the dropbox link works I'll post a whole lot of other files here. At the minute this is a pdf of a word document I hope over the next year or so to migrate to LaTEX.

Best wishes Chris.

## Thursday, 16 December 2010

### Results In

Hi I'll post more later but my results for Both A211 Philosophy and the Human Condition and MST221 are both in. I managed to get 85% for both exams so just scraped a distinction for both courses. I was really surprised for A211 as my essays only averaged 78% well within grade 2 territory. They must have applied some discretion. Anyway needless to say I'm chuffed to bits. I apologise for not posting much before a combination of a severe cold and the bad weather. I'm currently looking at solving differential equations by series solution of which more later
I'll also post my thoughts on A211 in the next few days

## Saturday, 20 November 2010

### Decisions Decisions

As many people reading this blog will know one of the endless games one gets into as an OU student is planning ahead. This involves a combination of planning ahead and keeping an eye on what will be available years ahead, also constraints of time and finances etc.  My problem is compounded as I have ambitions in both philosophy and mathematics. The maths plan is fairly straightforward although I was dithering as to its final form. Anyway I think its more or less finalised. I aim to complete a degree in Mathematics and Statistics.
by end of 2012 and then embark on the MSc

To this end I need the following compulsory courses

MST121 and MS221 of which I've completed both see earlier blog 30 points each.

Statistics 4 courses
Level 2
M248 Analysing Data                           30  points     Completed 2008   Grade 3 pass
M249 Applied Statistics                       30 points       Completed 2008  Grade 2 pass
Level 3
M343 Probability                                  30 points     Completed 2009  Grade 4 pass (bombed exam completely)
M346 Linear Statisitical Modelling         Still to do.

These 4 courses form a diploma in Statistics

In 2012 there will be another statistics course M347 called mathematical statistics. However after a long correspondence on the First Class conference I found out that whilst M346 was deemed necessary for the degree M347 is optional. This somewhat dismayed me as M346 is essentially a computer based course with little or no explanation of the underlying mathematics as incidentlally M249 was. Also the programme used is GenStat the  licence of which runs out after the course so any skills learnt are not transferable. However after much soul searching bearing in mind I want the degree finished in two years to start the MSc in 2013. I've decided to do it this year. I will get a diploma at the end of 2011 and be put in a strong positon to finish the degree in 2012.

Then there is an option to pursus an Applied or Pure Track. For me the choice was obvious I need to understand pure maths for my other ambitions. Also I'm familiar with much of the content of the OU applied maths/ physics courses.

So its M208 Pure Maths (60 points)
M337 Complex Analysis (30 points)
I have to do Complex Analysis this year as it's only presented in even years. So that with M346 fills my quota for the year (You are only allowed 120 points per year)

That leaves 2012 The other two Pure maths courses I can do this year are
M336 Group Theory                                                           30
M338 Topology (The last time its going to be presented)     30

I could then cheat and include one of my second level arts courses as a free option and get the degree but it seems a Cheat and also I want to revise some of  my mathematical modelling skills. So the obvious course to do is M326 Mathematical methods and Fluid dynamics. However after 2011 it doesn't run again until October 2012. That suits me as for reasons to be explained below I want to do the 3rd level Philosophy course Philosophy of Mind which would round off my OU Arts studies nicely. Technically that would mean doing 150 points in 2012 but given the shift of starts I'm sure I'll get away with it. So thats the Maths degree sorted.

Now for my philosophy ambitions to tidy up the Open university Arts Options

My situation is as follows I first started Open university studies 10 years ago by doing the diploma in music the courses I studied were

A214           Understanding  Music     2nd Level    Grade 2 pass  60 points    2001
AA314         Studies in Musc             3rd Level     Grade 3 pass  60 points    2002

This led to a diploma in music. I then toyed with the idea of doing their MA in music but was put off by the fact that the MA did not allow for analysis of musical works but concentrated on the history of performance and the reception of works. Not really what I wanted. Due to other pressures I stopped my OU studies made some fitful starts on some courses but didn't finish them.

I then had the idea of doing the degree in Philosophy and Psychological Studies as a preliminary to doing postgraduate research in philosophy. I'll talk about this in more detail later but I've alternated as to what would be the best course for me to pursue this ambition within a reasonable time scale. Anyway as a result I did the Foundation course in Social sciences

DD100      Foundation Course in Social Sciences  (October 2006 - 2007)         Pass
DSE212    Introduction to psychology                    October 2007                      Grade 3

And then I would have alternated between Philosophy and the Human Condition, Cognitive Psychology and
Philosophy of the mind. Howver as a result of the psychology course which I didn't really enjoy I got into statisitcs which I did enjoy so started the Diploma in Statistics . Also I became suspicious of the relevance of the Open University philosophy courses to what I really wanted to do (a suspicion I still have but I've learnt to live with it). Ok so having abandoned psychology I concentrated on Statistics which lead to the fully fledged maths degree. I had planned to embark on the external London BA in Philosophy in 2008

With tuition available from Pathways

http://www.philosophypathways.com/programs/lond.html

And if I were starting this journey again that would be the Path I would follow. However again time procrastination and so forth led to me putting it off. Also as it was easier to continue doing the OU courses I decided to do A211  Philosophy and the Human condition whilst concurrently building up a portfolio of courses which count as 10 points towards accreditation from the Depatment of continuing education at Edinburgh and Oxford University.

http://www.conted.ox.ac.uk/courses/online/short/subject.php?course_subject=Philosophy

So far I've completed 30 points

Philosophy of language   Edinburgh     Jan   - Mar        2009
Metaphysics                  Oxford          Apr - August     2009
Ethics                            Edinburgh      Sept - Dec        2009

And then did A211 this year.

Next spring I hope to do a course on Hume at Edinburgh and then either an Oxford course or some of the Pathways courses

http://www.philosophypathways.com/programs/pak2.html

which although not credited have got quite a reputation. I want to study metaphysics and the philosophy of language in more depth.

I'll round all the philosophy off by doing the level three OU course in the philosophy of mind in 2012.That would mean my OU Arts courses would combine to give me an Open degree. If I were vain enough I could turn an Open degree into a named degree in humanities with music and philosophy but I would have to do another Arts course.

I should then be in a  reasonable position to formulate a research proposal (of which more later) even though I would not have a fully fledged philosophy degree.  I hope to  start part time research in either 2013 or 2014. I might take advantage of Geoffrey Klempner's associateship and fellowship awards of the ISFP to help me formulate my research proposals better. The field I want to explore would be the relationship between current ideas in the philosophy of language epistemology and the interpretation of quantum mechanics. Specifically for those who know about it the research would explore the connection between Dummett's claim that realism is commited to bivalence (ie something is either true or false)  and the fact that the current mathematical formalism of quantum mechanics is not bivalent and hence intrinsically non-realist.  I'll expound more in another post. After long agonising this seems the best way forward, if I were to do the London BA it would be at least another five years before I could complete it and that would only postpone the ambition to do research in philosophy. I do feel slightly guilty that I wont be doing the best available qualification in philosophy available to me but I'm aware of its contents and have familiarised myself with the exams  and reading lists so I'll pursue those on my own anyway.

## Sunday, 14 November 2010

### Preparation for M208 , and a bit on Latex and Mathbin

As part of my preparation for M208 I've been trying to grapple with the epsilon - delta definition of continuity. Open up any Old style textbook on analysis and you will find a definition which is almost incomprehensible anyway things have improved greatly. At the risk of boring those people I know from the OU forums I've been using a book by Brannan called A first course in mathematical Analysis published by Cambridge University Press. This covers the analysis part of M208 and I would recommend anyone who is contemplating doing M208 to invest in it.

http://www.amazon.co.uk/First-Course-Mathematical-Analysis/dp/0521684242/ref=sr_1_1?ie=UTF8&s=books&qid=1289762572&sr=8-1

I think I finally understand (after 30 years) how to use the definition of continuity  at least for simple cases. I've enclosed a basic summary on Mathbin

http://mathbin.net/55459

Duncan introduced me to Mathbin  as a consequence of some  work we are doing on the Cambridge Maths departments  example sheets for rotational motion of which more later.

Mathbin is great for exchanging posts which require mathematics. Its based on Latex which Nilo has kindly given me some hints how to use.  However it has it's own idiosyncracies in that for Internet explorer the previewer doesn't work and Latex copy doesn't copy across as mathematical text  is bracketed with [EQN]   [/EQN] instead of  and it is a real pain to adjust LATEX files so that they work first time on Mathbin.  It took me about 10 iterations before I got it correct. Fairly soon following the hints that Nilo gave me I hope to incorporate Latex on this blog. For now as I'm just getting used to the whole Latex thing  then this hash will have to do. Eventually for my large projects I will use something like ScribTEX.

## Monday, 8 November 2010

### Review of MS221 and MST121

Hi first let me apologise for not posting earlier I'm currently working on the so called 'Straightforward but Tedious' process of deriving the Friedmann equations (the ones that form the basis of modern cosmology) from Einstein's Theory of relativity. The process is certainly tedious whether or not it is straightforward is another matter. I'll keep you posted. Anyway in the mean time here are my thoughts on the two maths courses I did this year. These were

MST121 Using mathematics and MS221 Exploring Mathematics.

As essentially entry courses for the degree in mathematics they try to cover most of the stuff that would be expected for an entry into a typical university. So roughly equivalent to the old A level Pure and Furthe Pure maths courses with a little mechanics and statistics thrown in. Without boasting for me most of the course material was revision  mainly to get back into thinking about stuff which I hadn't done for years. My colleagues Neil and Nilo have already described these courses in some detail and there is not much I can add to their descriptions. For those who want a blow by blow description I refer to their blogs.

However here is a quick overview. I think the courses are best taken together as a unit with MST121 in October and MS221

The courses are split into 4 blocks and the material in the MS221 blocks A, B and C build on the corresponding blocks in MST121. Whereas block D in MST121 covers basic statistics and block D in MS221 provides an introduction to pure maths of which I will be getting a lot more of when I do M208.

Block A is concerned with sequences, functions, coordinate geometry trigonometry and recurrence relations
the stuff on recurrence relations was new to me and one of the more interesting parts of the course especially the stuff about Fibbonacci series and the golden ratio. It's that sort of maths which on the one hand is quite straightforward but has amazing applications. However solving a second order recurrence relation can be quite tricky to get correct and I'm sure I lost a couple of marks in the exam.  For those contemplating doing MST221 next year it really is worth mastering the procedure for rotation of a conic section as this question always seems to crop up in the second half of the exam. (Of course I didn't heed this advice of which more later)

Block B is concerned with vectors matrices and their application to geometric  transformations and also another topic which was new to me the fixed points of an equation and how they govern the behaviour of a function when it is iterated. Again a fixed point question seems to come up all the time and it is worth getting to the stage where you can invert a 2x2 matrix, find it's inverse, eigenvalues and corresponding eigenvectors.

Block C on calculus was my favourite but that is because I have loved calculus ever since I first came across it when I was about 15 or 16. I think anyone who hasn't got a basic understanding of calculus at the level of this course doesn't deserve to be called properly educated. A bit strong perhaps but without a grasp of calculus you have no idea how physics or engineering works. Those in the Arts world often call  physicists and engineers philistines as they aren't familiar with the main works of Shakespeare are just as philistine themselves if they don't know how calculus works . If I have a slight criticism of this unit I feel that they could have made an attempt to include material on Second order differential equations instead of Taylor series. It is second order differential equations which form the backbone of most physics modelling enshrined in the simple harmonic oscillator, The thing that made me really interested in mathematical physics was the realisation that the same basic differential equation described both mechanical and electrical oscillations. Taylor series whilst important could really be left to a level two course whereas an introduction to second order differential equations with an explanation of some of their applications would be much more useful. Especially for those people for whom MS221 will be their last maths course.  Anyway those who want a taste should down load the units of MST209 available on Open Learn. Anyway despite getting my best marks for the TMA's in this part I totally bombed the long exam question as I had a mental block. So whilst calculus is interesting doing it under pressure is not so good and it is not recommended to do the long calculus question.

MST121 Block D Statistics Most people found this boring however I have to say I thought it gave a good overview of the process of hypothesis testing. This has had quite significant social implications in recent years especially when it comes to testing the efficacy of certain proposed medical statistics. In brief hypothesis testing is a method whereby tests can be devised to test the validity or otherwise of a certian proposed treatment. It enables a statistic such as the mean from two samples to be compared, In medical tests two groups of people of  the same size are compared one is given a new drug say whilst the others are given a placebo. If the number of people who improve after being given the drug is significantly higher than those who are just given the placebo then the efficacy of the drug treatment is deemed worthy of further investigation. More often or not there is no statistically significant difference between the two samples and thus the treatment can be ruled out. In the current climate their appear to be all sorts of quack remedies such as homeopathy or other so called alternative medical treatments. By performing a hypothesis test the validity of these claims can be rigourously checked, Needless to say when these tests are performed the treatment is generally deemed statistically insignificant. The point is that this methodology is a powerful tool and politicians and people who make policy decisions should be aware of this. Had a proper analysis of the claims of Dr Wakefield who claimed that there was a link between the MMR vaccination and autism been undertaken much needless scaremongering and agony for many parents would have been avoided. However as a consequence of the claim many people were put off subjecting their children to the MMR jab and as a consequence we are now facing the rise in measeles a disease which had been effectively erradicated.
The point being that with tools like hypothesis testing soft sciences such as the social sciences can be raised to the status of real science as they have a method of falsifying claims. Unfortunately the unit in MST121 did not really mention this application of statistics. Had it done so then perhaps people would not have found it so boring.

Unit D MS221 Introduction to Pure Maths this covered complex numbers a brief introduction to number theory group theory and methods of proof including induction

This is an attempt to introduce some of the thinkning behind pure mathematics I was familiar with complex numbers, group theory and  mtethods of proof. As a taster the group theory was Ok but (and I suspect) this will become more so when I do M208 and M336 there was no real rationale behind why group theory is important. Yet it has had a profound impact in physics via the use of representation theory as applied to particle physics and quantum mechanics. However the part that is necessary for this is representation theory and this I believe is not taught in any of the OU maths courses. However group theory as a mathematical structure is quite profound in iteself I refer you to Nilo's site for those interested in this aspect.

As for number theory I really don;t get it and the course did not really convince me give me a complicated integral and I'll get my pen out and do my best to solve it. Show me the analogy between vector spaces and function spaces (of which more later ) and I'll be really impressed, On the other hand I just can't get excited by relationships between prime numbers or congruences. The application to coding given in this unit was quite tricky easy to make mistakes and fiddly. I'm afraid I couldn't resist at a tutorial getting my tutor to work through an example on the board I was relieved to find that he made more or less the same errors that I would have done.

Ok so overall impressions as a warm up to the real stuff and practice in doing TMA's these courses are great however the real stuff starts at level two I can't wait. As a sideline I was really lucky in having a really good tutor and a friendly tutorial group I have made lots of friends including Nilo (who I got to know via a link on Neils blog) , Duncan and Neil this is the first OU course where I became convinced of the value of tutorials and the friendships that follow. I would like to thank my tutor Alan Borthwick and all the friends I made on MS221.

Block B

## Wednesday, 27 October 2010

### Physics Projects 1 Big Bang

Outside of my OU work I have a number of physics projects I have been working on, on and off over the past 20 years or so. This post outlines my plan to fully understand the establishment of the big bang model in the late 1960's to early 1970's. When life was relatively simple and there was no dark matter, dark energy and only one universe to worry about. Open any book on cosmology, popular or otherwise and in the section entitled Big Bang Nuclear synthesis (BBN for short)  there will be a couple of graphs plotted showing the abundances of Helium and Hydrogen as they evolved in time from about 100th of a second to about 3 minutes after the big bang. Another graph will show the abundances of other light elements such as Lithium and Beylium. However what they will not tell you in any great detail is how these graph's were obtained. In those crucial moments the temperature of the universe (according to the story) was such that stable nuclei were able to be formed and in a few minutes all of the Hydrogen and Helium that we observe in the universe was created. The person who was able to synthesise his knowledge of both nuclear physics, statistical physics and  cosmology to give a coherent story which fits the facts was P J E Peebles ('Primordial Helium abundance and the Primordial Fireball II Astrophysical Journal 146 pp 545-552 1966). The significance, as readers of this blog will probably know, is that a number of cosmologists such as Fred Hoyle and Hermann Bondi had hoped that they could account for the Hydrogen and Helium abundances from Stellar explosions. Peebles calculation was what physicists//engineers refer to as a back of the envelope calculation and it wasn't long before more sophisticated calculations, based ironically on the codes that Hoyle had written to study stellar explosions were able to come up with more accurate predictions. This line of research culminated with the work of   Kawano in the early 1990's with a code written in FORTRAN (like all good scientific codes) which considers a staggering 88 basic nuclear reactions and is able to predict the abundances of all the light elements based on a knowledge of their reaction rates, a general relativistic cosmological model and relativistic statistical physics. Kawano's manual for the code he wrote is accessible from Fermi lab

http://lss.fnal.gov/archive/test-preprint/fermilab-pub-92-004-a.shtml

and it is one of my long term ambitions to put all the pieces together in one place concentrating in detail on all the in's and out's of the calculation and culminating in my own code which is able to reproduce at least in rough outline the same results that Peeble's got. Now that I have a bit of spare time between courses I hope to be able to give this project a well deserved kick up the backside and rejuvenate it.

The current status is as follows

1) A general overview based on a classical approximation.  (status Complete)

Some what bizarrely a good approximation to the results of Peeble's can be obtained on the basis of Newtonian Physics and classical statistical physics as enshrined in the Boltzmann distribution. I have summarised this work in a word document which provides a good overview hopefully understandable by anyone who has completed MS221 and possible S207. If people would like a copy please request one from my home e-mail

The core background to understand the calculation fully draws on three main topics

a) General Relativity and it's application to Cosmology in particular the Friedmann equations.

b) Fermi's theory of the Weak Interaction and a calculation of the Nuclear Decay rate which is the  key parameter in the theory. One can always cheat by just quoting the experimental value, however one misses out on a key insight as to how particle physics and Cosmology combine together.

c) Relativistic Statistical Physics

I've more or less completed some notes on neutron decay and I've got backgtound notes on General Relativity my aim over the next couple of months is to tidy these notes up and then make a start on relativistic statisitical physics. Statistical physics has never been one of my strong points when I was at Exeter a certain Dr Jones tried to teach our class it, but he taught it in an idiosyncratic manner to say the least and so I was always put off it. I hope to remedy this.

When all that is under my belt, I can start given the guidelines in Kawano's manual writing my own code.

So if I'm lucky this project should see completion in the next 2-3 years but I do have to stop procrastinating.
﻿
﻿

## Sunday, 24 October 2010

### Introduction

Hi and welcome to this blog inspired by a number of my fellow Open University students such as Neil Anderson
http://neilanderson.freehostia.com/

and Nilo de Rook.

http://mathematics-diary.blogspot.com/

I decided to create my own.

This blog will include my feelings about my Open University experiences but also a lot more. I've just finished my open university courses for this year these were:

MS211: Exploring Mathematics

A211:    Philosophy and the Human Situation

Earlier in this year I also did MST121 Using Mathematics.

I have an MSc in Theoretical Physics focusing on particle physics for which I did the exams in 1979 - 80 and completed the disseration in 1984. I also have a BSc in the same subject which I got from Exeter in 1978. Since then I have acquired a postgraduate certificate in Further Education and a diploma in Optoelectronics from Heriot Watt University Edinburgh. I have also had two postgraduate research associateships one in Theoretical Chemistry at Bristol university where I spent an admittedly frustrating year trying to get a code to model the photodissociation of molecules working and one in image processing at Heriot Watt university where I did a comparative study of three image processing techniques. Since 1990 I have worked in the antenna departments of GEC Marconi Chelmsford and since 1994 I have been working in Edinburgh again in the antenna department of a company which has changed its name from GEC to BAE Systems and is now called SELEX - Galileo. Despite all these changes my job has essentially remained the same which is to model the effects of the nose of an aircraft (the radome) on the radiation pattern of the antenna and investigate the effects of the radome on clutter. This involves quite intensive computational electromagnetics.

Despite the switch to Applied Physics since the mid eighties I have retained my interest in pure physics and I was most happiest when doing my part time MSc at Southampton. There I was introduced to the exciting world of particle physics and the main tools that physicists use to investigate this subject namely relativistic quantum field theory and symmetries based on Lie Group theory. I came just as the Standard model of particle physics was being finalised. I think that this is probably the greatest intellectual achievement of the last quarter of the 20th Century. I also think that given the lack of data, extensions to the Standard model of physics such as superstrings M theory and so forth whilst interesting mathematics can only be speculation. I have at present no desire to pursue these topics as I really don't think they will lead anywhere. In the words of Dirac/Pauli because they can't be tested they are not even wrong. One of my aims to be discussed further in other posts is to get to the stage where I can do what are called radiative corrections to processes  from the Standard Model to model the scattering  of such reactions  as electron neutrino scattering,  electron positron annhilation or investigate the predicted properties of the Higgs boson.

My aim as far as my Open University studies are concerned is to get the MSc in mathematics  concentrating on the pure side. Mainly because modern physics uses a lot of pure maths concepts such as functional analysis, group theory and differential geometry for which the average theoretical physicist is ill prepared. Indeed, despite the fact that I can solve Maxwell's equations, apply quantum mechanics to solve problems in scattering theory and atomic and molecular spectroscopy, apply General Relativity to investigate the properties of stars and cosmological models and so forth. There are two significant books in the foundations of physics which I've had for thirty years which may have been written by people from a different planet and it is my long term ambition to study these.

These are

1) Hawking and Ellis 'The Large Scale structure of space time'
This book was the first coherent account of the singularity theorems of classical General Relativity showing that apparent singularities which occur in say the Schwarzschild solution or the Big Bang model were not accidents of the coordinate system but inherent in the basic equations of General Relativity. The background needed to understand this is topology and co-ordinate free differential geometry.

2) Von Neumann "The Mathematical Foundations of Quantum Mechanics"
This book showed that the underlying mathematical structure of quantum mechanics was that of a complex Hilbert Space. To understand this the mathematical background needed is a grasp of real analysis, complex analysis, Linear analysis, and last but not least Functional analysis.

Unfortunately the mathematical background of the average theoretical physicist is not adequate to this especially in the fields of analysis . Hence the Open University maths courses. Next year I embark on

M208 Introduction to Pure Maths

And taking a bit of a risk as it presumes a knowledge of real analysis got from M208.

M337 Complex Analysis