M381 Number theory and logic First TMA

It’s been a while since my last post for which apologies. I have just sent off what I have managed to complete, for the first TMA for M381, Number theory and logic. Daniel and Duncan on their blogs have given a good overview of the topics so far and also the fact that at this level one is expected to think for one self a lot more. Duncan is way ahead as usual but then he did start early last year whilst I was busy on Fluids, Music and Quantum Mechanics. I started looking at the units about 5 weeks ago and  I have just about completed the units for TMA01, two number theory units and an introductory one on computability. Not much to add to what Duncan and Daniel have already said although I do find it strange that the course has decided to put computablity ahead of logic, a relatively gentle break in, would have been to tackle at least the basic elements of logic first, truth tables etc which most people are probabily aware of, before embarking on what is quite a challenging introduction to computability.
Again rather than introduce computability via Turing machines, the course has decided to use URM (Unlimited Register Machines) which were introduced in the 1960’s I think. As Duncan has pointed out URM machines are based around four simple commands
        Z(n) make the contents of the given register zero
        C(n,m) copy the contents of register n to m (Not the other way around which it is very easy to confuse)
        S(n) increment the contents of register n by 1
        J(m,n,q) jump to instruction q if the contents of register m are the same as register n.
URM machines are meant to simulate computer programming, but as a caveat I would add that the Jump instruction seems very similar to the now frowned upon GOTO statement, that has lead to piles and piles of spaghetti code and provides quite a great deal of diificulty in interpreting a URM code. Still it can be useful in providing loops until two registers are made equal simply by including an instruction of the form J(n,n,q) where q is a command earlier in the list. I dare say however a new formulation of computability, reflecting current programming practice, could be devised.

Anyway onto the TMA, it fell into two parts 6 questions on Number theory and 4 questions on computability
Question 1 A relatively straightforward one on the Eulidean Algorithm familiar from the last part of MS221

Question 2 A question on induction which had actually changed from the initial one, a bit tricky, but once one had seen how to proceed it was relatively straightforward

Question 3. A question on proving  a condition on the last three digits of any number  to be satisfied if the number that is to be divisible by 8. This was quite tricky at first but once you realised what was going on it became obvious. This question is really clever, the sort of thing that doesn’t involve any complicated maths, but requires some thought. I hope there are more questions like this as the course proceeds

Question 4. Another question on divisibility involving factorials divided into three parts first calculating the possible remainders of n^2 when expressed as 1Ok+r, Then proving what factorials are divisible by 10. Finally putting parts one and two together to find out what factorials can be expressed as squares Think I got most of this out.

Question 5 was on Fibonacci Numbers and as I hadn’t had time to look at this topic in any great detail I left it

Question 6 was a question on showing that there are infiinitely many primes of a certain form. As this was very similar to a theorem in the unit it didn’t require that much adaptation to generate a similar proof.

So Apart from question 5 feel reasonably confident that I have got most of this correct.

Second Part Computability part 1
Question 7 Asked us to demonstrate what would happen to a given URM program draw a flow diagram and work out what the code was trying to do. I think I got it out but it would have been easier if one of the copy instructions had their registers swapped ie C(n,m) instead of C(m,n) not sure whether this was a misprint or not.

Question 8 asked us to provide URM codes for two functions Think I got this out although for more complicated codes one has to make sure all the inputs are covered especially if one or more of the inputs are zero.
Question 9 Involved generating a primitive recursive function from two others. The definition of primitive recursion is a bit of a mouthful to say the least. Still after a few examples it begins to make sense. In general if a function is defined by recursion it means that a sequence can be generated such that the next term in the sequence can be generated from the others. That is quite straightforward but the definition appears like gobbldeygook and bears at first sight little relation to our intuitive ideas of recursion. However once one realises that the definition of primitive recursion as given in the text is a recipe for generating a primitively recursive function h from two other functions f and g then it makes a lot more sense.

Finally Question 10 was a question asking us to generate the URM code for a function h defined by primitive recursion from two other codes for f and g. There is a somewhat complicated recipe for doing this however if one follows the steps then one eventually gets there. The final part asked us to work out what the function h was Fingers crossed I think I got there.

Overall then should get a grade two, but having left out question 5, unlikely to get distinction.

So first impressions on the whole quite interesting, the number theory part (so far) seems simpler than the computablilty part. But I do wonder whether number theory really counts as a theory, rather than it being a set of miscellenous facts about numbers. It’s not like group theory or analysis, or even quantum mechanics, where given a few axioms, everything follows from them.

As for computability, so far have just scratched the surface, the meat will come in the next two parts. Fortunately I have the Christmas holiday to get to grips with it. I suspect it will be the hardest part conceptually of this course. But it’s good to be challenged.

Bye for now Chris

SM358 exam debrief

As promised here is my summary (off the top of my head) of the questions and my responses to them
I think I'm on target for the top end of Grade 3 or if I'm lucky Grade 2 but it will only just get there.

The first part was 10 out of 12 questions of which I answered 9 but some were only sketches

Question 1 was about the degeneracies of energy levels in an infinite square well and the calculation of the wave length of light emitted when the electron fell from the higher level to the other.
Got most of this out 4  marks say

Question 2 Concerned the normalisation of of a given wave function and a calculation of the energy levels  again got most of this out 4 marks say

Question 3 I ignored think it involved calculation of quantum oscillator properties using annhilation and creation operators

Question 4 On the Born interpretation this had a different twist to the normal ones and threw me so I wrote down the born intepretation 1 mark out of 5

Question 6 Involved calculating the Energy eigenvalues of Lz for a given wave function then a question using the generalised uncertainty principle for Ly and Lx most out say 4 out of 5

Question 7 Involved calculation of the maximum separation of proton distance from a given radial wavefunction got most of this out again say 4 out of 5

Question 8 was fairly straightforward whether or not statements about the Bell Inequalities were true or false I think I got most of theses so 4 marks

I can't really remember most of the other questions in part 1 but they involved calculation of the expectation value of various quantites such as x or x^2. The calculation of bond order for a given molecular configuration and so forth. If my memory serves me correctly I got about 4 marks for two and say 2 marks for the other one
So pessimistically total marks for part 1 
so overall for part 1 I got 4+4+1+4+4+4+8+2 = 31 marks (Some may include full marks so possibly 35 marks )

Then parts 2,3 and 4 long questions

Part 2 I chose a question on eigenfunctions of a square well I think I got most of this so say 15/17

Part 3 was a question on spin functions the first part was really strange so I left it.losing 3 marks went onto the question in part 4 before coming to this question at the end..
  The rest should have been quite straightforward as it involved  calculating the coefficients of a given spinor in another basis and then the expectation value of Sy. I got the  first part out but chose the wrong matrix to calculate the expectation value. I might just scrape 8 or nine marks here

Part 4 was a question on eigenfunctions of the angular momentum operator in spherical coordinates
Got most of the first part out but then there were two part questions on of which involved showing a given wave function was an eigenfunction of J = Lz + Sz I couldn't quite get this. The last part should have been straightforward involving transition selection rules but my mind went blank so possibly 12 out of 17 here.

So part two is about 35 marks as well

35 + 35 is 70 so might just get grade 2 but a more realistic assessment would be for grade 3.

So overall not bad (certainly better than last june's car crash of Fluid mechanics) I decided not to do the resit as it clashed with my revision for quantum mechanics.

Number theory and logic awaits and I want to consolidate my music by doing the Grade theory exams I should be able to do Grade 5 in March and also piano lessons again I want to do grade 1 piano again in March. I will also get back to my own physics calculations. My first project is to examine the X ray diffraction theory behind the discovery of DNA. Then back to general relativity and also the physics of deep inelastic scattering which lead to the discovery of quarks. I will also do an Edinburgh dept continuing education course on Plato come January Enough to keep me busy for a while yet :)

Bye for now
 

Exam Time again

Well it's that time of year again for me (twice in one year) I have my quantum mechanics exam on Monday
I did my first practice past paper under exam conditions today and got borderline grade3/grade2 part of the problem is there is just not enough time available. For part 1 you have to answer 10 so called short questions in 1 hour and 30 minutes and then there are three long questions which you are supposed to allocate 30 mins to. This means that you almost have to download the already prepared answers in your head if you are to stand any chance of a grade 2.

The short questions seem to range over a variety of topics whereas the long questions can be grouped into 3 quite distinct groups

PArt 1 A question on a particle in a potential usually square but sometimes they will throw a wobbly and ask about a non square potential or just some general type questions. This is a banker question

Then there appears to be a choice between a question involving Transmission or reflection at a boundary or one on Harmonic oscillator raising and lowering operators. Will hope the question on potential is reasonable

Part 2 A question on spin states in a magnetic field again a banker question the alternative is usually a question on the Bell inequalities

Part 3 A question on hydrogen atom wave functions which could get messy however it's a topic I feel reasonably confident in although in my trial I couldn't for the life of me see how to get the normalisation of a given hydrogen atom wave function correct.

The second question is usually on something involving molecules, solid state physics, perturbation theory or Transistions induced by radiation. I've not paid much attention to this part which comes at the end of the course so I'll go for the question on Hydrogen atom wavefunctions. If it looks really tough then if a question on perturbation theory comes up I'm reasonably confident I could tackle that one.

In part 1 typical questions include normalising a wave function via the coefficient rule. Calculating a probability density function from Born's rule. A bit on hermitian operators, one or two questions on the Bell Inequalities and spin states, Then a whole bunch of questions on either molecular or atomic orbitals
Selection rules between states induced by radiation one or two waffly questions on solid state physics and so forth

Ok prediction top end of grade 3 or if I'm lucky grade 2

On another note I've got the material for Number Theory and Logic which has officially started  but so far apart from a quick glance I've not had any chance to look at it. The first unit looks reasonably straightforward covering Triangular numbers, Induction and Divisibilty some of which was covered in the last unit of MST221 so hopefully I will be able to get up to speed,

Post match analysis on Tuesday till then that's all folks

SM358 quantum physics TMA04

So final TMA submitted for SM358 but still have 3 on line assessments to complete one nearly finished and of course I missed out on TMA02 as was bogged down in completing the EMA for the music course and desperately attempting to do revision for Fluids

TMA04 concentrates on the first part of the 3rd block quantum mechanics of matter this involves an overview of the hydrogen atom and it's wave functions, and it's extension to many electron atoms plus an introduction to pertirbation theory. The rest of the unit is (IMHO) a rather inadequate introdution to molecular physics, Solid state physics and quantum optics. Each of these deserve at least a 30 point unit to themselves and again I feel the OU is selling it's physics graduates short. Anyone thinking that by completion of this course, electromagnetism and say the relativistic universe has a degree equivalent to that of a standard course in undergraduate physics would be kidding themselves. Still I don't write the rules and it is not the point of this post to beef (yet again ) about the gulf between OU courses and those of brick univesities.

So the questions on this TMA divided into 3

1) A question based on one particular wave function of the hydrogen atom, calculating various quantities such as the expection value of 1/r and 1/r^2 and their uncertainty. As they say in the books this was tedious but straightforward

2) Again more questions on the hydrogen atom wave functions involving a hypothesised modification to the coulombic interaction. In the context of the hydrogen atom this is obviously unphysical however it may correspond to the force between two nucleons such as the potential between two deuterons. Any way the point was to test our ability to calculate the first order corrections to a change in the potential. Again this was straightforward but tedious and it took a couple of goes before all the signs and factors of pi cancelled out to get the correct answer.

One of the skills one needs in solving problems concerning hydrogen atom wavefunctions is patience and the ability to obtain the correct numerical factor. This is somewhat tedious but a necessary skill just hope the exam questions aren't so pernickity and fiddly.

3) The final question probably the most straightforward involved calculation of the various atomic terms associated with a multi electron atom and their degenaricies. Just one point that wasn't really mentioned in the text namely how to correlate the total number of States given with a given L and S nuimber with the individual l and s numbers of the valence electrons. Fortunately an example in the additional exercise book gave a clue

So all in all I think I've done reasonably well on this TMA. I omitted an essay type question at the start of question 3.

So that apart from completing the exam and the online assessments is that as far as quantum mechanics goes. It really doesn't seem all that long ago since I started and now it's more or less finished the feeling does seem a bit weird to say the least but the same is true of any OU course. It is just rather a pity to say the least that there are no follow up courses as far as the OU is concerned.

Statisical and Solid State Physics

Well it's nearly the end of SM358 unfortunately the open university has no where else to go as far as quantum physics is concerned. A next logical step is Statistical Physics and Solid State Physics A search through the internet has yielded these two courses which seem to give a good over view of the subject and are based on courses given to those in their third year at Oxford and Cambridge whats more they have problem sheets associated with them


Statistical Physics

http://www.damtp.cam.ac.uk/user/tong/statphys.html

Solid State Physics

http://www-thphys.physics.ox.ac.uk/people/SteveSimon/condmat2013/condmat.html

If any one is interested in forming a study group to work through these courses after the exams could they please let me know. Obviously we would have to share any solutions amongst our selves as if they were published on the internet then the authors might well restrict access to the lecture notes.

Two suitable textbooks to accompany both might be

http://www.amazon.co.uk/Concepts-Thermal-Physics-Stephen-Blundell/dp/0199562105

and by the author of the solid state course himself (and honour would suggest the least we could do is buy his book)

http://www.amazon.co.uk/Oxford-Solid-State-Basics/dp/0199680779/ref=sr_1_1?s=books&ie=UTF8&qid=1376838763&sr=1-1&keywords=Simon+Solid+State+physics

I have also been looking at the diffraction theory behind the discovery of DNA it really is quite clever and I hope to write up a paper which covers the theory and goes through the actual calculations. Watch this space

Coursera

Ok so there is an alternative to the OU which I’m going to use to supplement my current OU studies. I’m sure most people reading this blog will have heard of Coursera which is a free on-line set of courses provided by mainly American Universities. It’s like being a kid in a sweet shop. The courses are short.

I’m doing a) Mathematical Philosophy whiich started last week and I’m grateful for Gavin who pointed this out to me in a comment he made on the last post

https://class.coursera.org/mathphil-001/wiki/view?page=syllabus

b) Introduction to Logic which starts in September and should hopefully complement the Logic part of M381 https://www.coursera.org/#course/intrologic?utm_classid=971129&utm_nottype=class.welcome.before&utm_notid=-1&utm_linknum=1

c) Finally (for now) Analysis of a Complex Kind which starts end of October this hopefully will remind me about complex analysis and help me revise the topic

https://www.coursera.org/#course/complexanalysis?utm_classid=971056&utm_nottype=class.welcome.before&utm_notid=-1&utm_linknum=1

 It might also be a good introduction for those who are contemplating doing M337 next year

I really am begining to think the OU days are numbered for those who want to do courses mainly for interest given the cost of fees.Ok one gets more support but as Free internet provision of University level courses continues to grow it's difficult to see how the OU can justify it's expensive fee structure.

Incidentally I've just had a hell of a nightmare trying to publish this post I was using Internet Explorer 8 however blogger in their infinite wisdom have decided to withdraw support from Internet Explorer 8 so I've had to change to FireFox a case of if it's working we'll break it. Most inconvenient and tedious. Also in order to access the Coursera courses I had to download VLC media and access the videos that way but that came with a whole load of crap such as sweetsearch so I had to remove these manually. I hate computers.

Results out

Just a quick post to inform readers that my results are out

I got grade 3 for music with 61% OCAS (TMA's not bad since I only submitted 5 out of 6 ( and there is no substitution ) and 65% for the EMA (Extended Module Assessment) so reasonable there is as yet no feedback for the EMA but it's forthcoming

As for Fluids I failed with 37% for the exam and 71% for the continuous assessment I've been offered a resit but it will be capped most I can get is grade 4 .
Still as I would like the opportunity to hone my exam technique and reconsolidate the material  it will be worth doing. Also it's another excuse to keep up with partial differenctial equations and mathematical methods.

My course schedule for this year is

SM358 Quatum mechanics exam in October
M381 logic and number theory starting in October
Physics project on entanglement Feb 2014

Also for music
I will enrol for grade 4 theory exam in November with a theory exam every 6 months or so till I get to grade 8 start piano lessons in September and hopefully do grade 1 piano exams in March 2014

I decided it would be far to much for me to take on board AA308 the third level course in philosophy this year round. It's allegedly in it's last presentation this year with a replacement in Oct2014 but going on previous form I can't see this happening so hopefully it will be available in October   I really hope so as from a preliminary survey I don't really like the prospectus for the new course without being to harsh it fits into "waffle philosophy".That is  philosophy good for discussion at dinner parties but hardly cutting edge philosophy that one would expect for a third level course in the subject. On the other hand AA308 whilst not perfect does cover philosophy of the mind at a level comparable to that of other 3rd year undergraduate brick level courses.

I am seriously begining to think the only way in which to get  a rigorous training in philosophy is to bite the bullet and do the external London BA in philosophy even though I wont be able to start for another two years.  Watch this space.