A week ago searching for alternatives to the Open University courses on logic a search led to this wonderful institution
http://www.saylor.org/
This offers a range of Free Online courses which are equivalent to courses studied at American Universities. Unlike coursera or Future learn which are examples of MOCC's the courses available can be accessed at anytime. Also they can be put together to earn the equivalent of a full degree provided you stay the course.
What is really impressive are the maths courses
http://www.saylor.org/majors/mathematics/
Having looked at the course content I think it's fair to say that they are at least equivalent to the OU and some such Abstract Algebra II and Real Analysis II go further than the OU do.
I have always wanted to understand politics and economics better and so am aiming to do at least the core sections in these disciplines.
http://www.saylor.org/majors/economics/
I've registered for microeconomics and also calculus of a single variable I which has a really good explanation of the epsilon delta definition of continuity. The great thing is that you can take your time there is no scrambling to complete a TMA Anyway hopefully I can zip through a lot of the maths courses in the next two years I'maiming for 1 module in maths every three months or so.
The great thing is that all this is free and the courses are being recognised as equivalent to having studied equivalent to that in an American universities. It can only be a matter of time before this becomes more and more widespread. When the OU is becoming more and more inflexible and restricted in the options they offer and the prices they charge unless it get's it act together institutions like Saylor will rightly supersede them.
As a bit of light relief I'm also doing the Future learn course on the Higg's boson
https://www.futurelearn.com/courses/higg
The lectures seem a bit more advanced than the assessment. the second lecture on the derivation of conservation laws such as momentum from imposing translational invariance on Newton's law is really neat. However the assessment is confined to a few multiple choice questions on general principles.
Anyway it's hello Saylor the way to go.
Monday, 17 February 2014
Sunday, 19 January 2014
ABRSM Grade 5 Theory Transposition by Numbers
One of the most difficult parts of the grade 5 theory exam is to get the intervals correct OK we can probably tell that it's a fifth but is it a perfect fifth or a diminished fifth or what? Similarly with transposition again yes transposing a piece up or down by a minor 3rd say means that the notes are displaced up or down a line or space but what about the accidentals, even worse what if it asks for a key signature change. A while ago when I was first studying music with the OU I hit on the idea that a lot of the confusion could be clarified by using numbers instead of letters. This works really well for intervals and transposition
First number the notes of the Chromatic scale starting with middle C as zero
So we have the table
Db Eb Gb Ab Bb
C C# D D# E F F# G G# A A# B
0 1 2 3 4 5 6 7 8 9 10 11
Each number is the number of semitones above middle C that the corresponding note is.
Then for transposition for grade 5 the main ones are
Up or down a major 2nd (2 semitones) simply add or subract 2 modulo 12 to the numbers then use the first line to work out the notes
Up or down a minor 3rd (3 semitones) simply add or subtract 3 modulo 12 to the numbers then use the first line to work out the notes
Finally up or down a perfect 5th (7 semitones)
So for example suppose we want to transpose a given melody down a perfect 5th we subtract 7 then add
12 if the new number is less than zero or subtract 12 if the new number is greater than 12.
Db Eb Gb Ab Bb
C C# D D# E F F# G G# A A# B
0 1 2 3 4 5 6 7 8 9 10 11
Performing the arithmetic gives for C 0 -> -7 then add 12 to get 5 (you only have to do this once)
then just start from 5 to give
Db Eb Gb Ab Bb
C C# D D# E F F# G G# A A# B
0 1 2 3 4 5 6 7 8 9 10 11
5 6 7 8 9 10 11 0 1 2 3 4
So that we see eg that C goes down to F (5) or Ab goes to D (2) and so forth.
So writing this table out and doing the appropriate arithmetic will give you a fail safe method of
transposing accurately especially for those tricky accidentals.
Of course you have to remember if the original melody has a given key signature to take into account
the key notes. Thus for Bb major the accidentals are Bb and Eb so every time you see a B or an E remember
these are Bb and Eb
This also works for key signature changes so suppose I start in Eb and I want to go up a major second
Eb is 3 according to the table add 2 to give me 5 the key signature is now that for F major ie simply with Bb
In the next post I will show how a similar technique can be used for intervals
First number the notes of the Chromatic scale starting with middle C as zero
So we have the table
Db Eb Gb Ab Bb
C C# D D# E F F# G G# A A# B
0 1 2 3 4 5 6 7 8 9 10 11
Each number is the number of semitones above middle C that the corresponding note is.
Then for transposition for grade 5 the main ones are
Up or down a major 2nd (2 semitones) simply add or subract 2 modulo 12 to the numbers then use the first line to work out the notes
Up or down a minor 3rd (3 semitones) simply add or subtract 3 modulo 12 to the numbers then use the first line to work out the notes
Finally up or down a perfect 5th (7 semitones)
So for example suppose we want to transpose a given melody down a perfect 5th we subtract 7 then add
12 if the new number is less than zero or subtract 12 if the new number is greater than 12.
Db Eb Gb Ab Bb
C C# D D# E F F# G G# A A# B
0 1 2 3 4 5 6 7 8 9 10 11
Performing the arithmetic gives for C 0 -> -7 then add 12 to get 5 (you only have to do this once)
then just start from 5 to give
Db Eb Gb Ab Bb
C C# D D# E F F# G G# A A# B
0 1 2 3 4 5 6 7 8 9 10 11
5 6 7 8 9 10 11 0 1 2 3 4
So that we see eg that C goes down to F (5) or Ab goes to D (2) and so forth.
So writing this table out and doing the appropriate arithmetic will give you a fail safe method of
transposing accurately especially for those tricky accidentals.
Of course you have to remember if the original melody has a given key signature to take into account
the key notes. Thus for Bb major the accidentals are Bb and Eb so every time you see a B or an E remember
these are Bb and Eb
This also works for key signature changes so suppose I start in Eb and I want to go up a major second
Eb is 3 according to the table add 2 to give me 5 the key signature is now that for F major ie simply with Bb
In the next post I will show how a similar technique can be used for intervals
Tuesday, 14 January 2014
On line piano tuition for Grade 1 and above
Well tomorrow I take my first official piano lesson with the aim
of doing grade 1 by June. I have been practicing on and off
for about 18 months now and I can't say it has been smooth sailing
to say the least. Inded I felt in a bit of a rut as I was playing
the Grade 1 pieces badly and never really completing them.
However over Christmas I found these amazing
on line videos by Alison Sparrow who has some really useful tips
http://www.youtube.com/user/theonlinepianotutor
Plus being very attractive to look at. Could be the Nigella
Lawson of piano and violin teaching.
Anyway her tip on practicing really helped me get to grips with
three of the Grade 1 pieces. When you try and set up a practice
session the temptation is to try to and play the piece as
a whole with the inevitable stumbling. Alison's method for which
I cannot thank her enough is to break it down.
At grade 1 the pieces are usually 16 bars what Alison suggests
is play the first 4 bar phrase 4 times. Then play the second
4 bar phrase 4 times then play the first and second phrases
together 4 times. Then move onto the third phrase play that
4 times, then the 4th phrase 4 times. Then put the 3rd and fourth
phrases together 4 times. Then finally play the piece 4 times
repeat this for about 2 weeks and you should have the piece
under your belt. Anyway it certainly seems to have helped me.
This is so obvious when pointed out to you but not at all obvious
when you are practicing on your own.
Another good website I have found is that of Shawn Cheeks
especially his 'boot camp' sight reading course. Shawn points
out that most musicians rely on their ear and memory. What
this misses is actually engaging with the music as it is written
as his career progressed he found it more and more difficult to
tackle the more difficult pieces. So he decided to go back to basics
and really learn the music. Have a look at his introduction
on You tube. Again I have been following his boot camp and
can see the improvement. Certainly spelling out the notes whilst
learning a new piece is really helpful
His philosophy is outlined here
http://www.youtube.com/watch?v=u3V-0iS8JMY
and you can follow the links to the other parts.
I've also put myself forward for grade 5 music theory this march
What i haven't done is any number theory or logic. I have
given myself 4 days this weekend to complete the assignment
if i don't then I will probably quit I can't really say I'm
enjoying the course OK I haven't really put much effort into
learning it. Right now though my musical interests are
dominanting if I can achieve grade 5 and 6 theory and grade
1 and 2 piano by the end of the year and get back into general
relativity then I wont feel so bad about abandoning M381.
I guess the non stop deadline of assignments exams, followed
by going straight back to other assignments over the past two
yeara has taken it's toll there is a small chance I'll continue
but I can't see it.
Friday, 3 January 2014
Instrumentalism or Why we should Shut up and calculate.
Prior to the festive season, I got myself in one of my perenial debates on the physics forum about the meaning or not of quantum mechanics. My main protaganist will be well known to those who follow the debates. He claims to endorse the Copenhagen Interpretation but also denies that he is an instrumentalist. A somewhat inconsistent opinion in my view. Anyone it’s not my job to help him see the contradictions in his position.
I do want to make a defence of instrumentalism however, that is some what missed by self styled philosophers of physics, but is in fact the current practice of most physicists.
OK what is instrumentalism ? essentially it is the view that the main aim of science is to provide empirically adequate models of nature without bothering to much about how the underlying concepts used to make the predictions correspond to reality.
What do I mean by empirically adequate, it is essentiallly the condition that the predictions of the theory when instantiated in a concrete model give reasonable agreement with experiment. For the quantative sciences such as physics this makes the theory testable or at least a given model of a given phenomenon predicitable. If the model does not give reasonable agreement with experiment then one can try and make the model more accurate by including more terms in the model or trying another approach.
It is important to make a distinction between theories and models this distinction is often blurred. A theory is a set of general principles in physics, there are about 8 sets of general principles which have been discovered
Classical physics
Newton’s Laws of motion.
The macroscopic laws of thermodynamics.
Maxwell’s equations.
Modem Physics
Special relativity.
General relativiry.
Non relativistic quantum mechanics.
Statistical physics.
Quantum field theory.
Note I do not include speculative theories such as superstrings because so far there is not one concrete prediction that has come out of it. At this stage it is ‘Not even wrong’
In order to describe natural phenomenon, one takes one of the above set of principles appropriate to the phenomenon in question. Then with the aid of mathematics and empirical information, such as the masses of particles involved sets up the appropriate equations and solves them either analytically or for complicated problems one has to resort to computers.
So for example to model the properties of stars as they collapse, one needs a combination of Statistical physics and General relativity along with an appropriate equation of state. To model planetary motion one would use either Newton’s laws of motion or for say mercury one has to resort to General relativity. A simple model would neglect the interactions between the planets a more complicated model would include these. Deciding on what approximations are appropriate is a necessary skill of a good physicist.
As we apply the above general principles to more and more phenomenon we gradually begin to understand how nature works and whats more with the aid of mathematics can predict how systems will behave. We can predict the energy levels of a molecule or solid. We can predict the orbits of planets, we can predict the decay rates or scattering cross sections of particles. We can even predict the rate of expansion of the universe from it’s early stages by a combination of Einstein’s general theory of relativity. relativistic statistical physics and a knowledge of the basic particles involved.
All of this is so obvious to a practicing physicist, but so called self styled philosophers of science aren’t happy with this. For them the aim of science is not to make concrete predictions of phenomenon but to describe reality as it is initself. They want to concentrate on the meaning of the general principles but therein lies a problem because some of the concepts used can be quite obscure.
For example in classical physics the nature of gravitation remained obscure all one could say about it was that it obeyed an inverse square law. In thermodynamics the concept of entropy also remained obscure although it had a perfectly precise meaning in terms of a measure of heat transfer . Also whilst Maxwell’s equations involved electrical and magetic fields their real nature remained obscure and all sorts of weird and wonderful ideas about the ‘real nature’ of an electric field involving vortices and eddy currents in the Aether were prevalent at the time. However more concrete these models appeared rather than Maxwell’s equations they didn’t really add much to the understanding of electromagnetism and these were ultimately shown to be wrong as the Aether was proven not to exist. The situation in intepreting the nature of an electric field so exasperated Helmholtz that when asked what Maxwell’s theory was he replied that Maxwell’s theory was Maxwell’s equations. Quite similar to the attitude which I favour of ‘Shut up and calculate’ when it comes to interpreting quantum mechanics.
My protaganist in the OU debates on the fora doesn’t like the above view for him and many others like him such as Karl Popper this reduces physics to engineering (as if that were a bad thing). Well I’ve got news for him and Karl Popper and other anti-instrumentalists most of what is published in physics journals today is an application of one of the 8 above sets of general principles to model a given phenomenon. Indeed it is only by continuing the process of detailed modelling that we understand how nature works. If that’s ‘just’ engineering so what ?
The rest arguing about the nature of gravitation, the wave function, whether or not particles are real or waves are, really doesn’t move us forward and to some extent it doesn’t matter, as I can still use the general principles coupled with empirical information to make concrete predictions about natural phenomenon. What other means of understanding nature do we have ? It is only by shutting up and calculating that we will get any where. The attitude prevalent amongst certain people that one should always be looking for more new general principles is misguided, the need for new general principles will be forced upon us when we are able to probe nature at more and more higher energies or shorter length scales.. Until then we should all shut up and calculate.
Friday, 27 December 2013
Seasons Greetings and Plans for the New Year
Well another year has passed and it’s that time of the year when one starts to think about the future. This year promises to be a busy one although I shall be reducing my OU commitments to at the most one 60 point course per year.
I’m currently doing the Number Theory and logic course and completion of that will conclude my second open degree. If I get grade 2 it will put me on line for a 2:1. I then want a break of at least a year from OU work.
The courses will consist of
Level 1 MST121 Introducing Mathematics Pass
Leve 2 MST221 Exploring Mathematics Distinction
M208 Pure Maths Grade 2
A208 Philosophy and the Human Condition Distinction
A224 Music Grade 3
Level 3 M338 Topology Grade 4
MST324 Waves Diffusion etc Grade 3
SM358 Quantum Mechanics Grade 2
Then M381 number theory and logic
The only thing that would persuade me to change my mind is if A303 Philosophy of the Mind gets a reprieve and is presented next year. I wont know till about March. I could replace MST324 and M338 with it and would hopefully get a better grade we’ll see.
Not that I wont be busy I want to concentrate on music and my own physics and maths studies for a while. I intend to finally get a piano tutor with the aim of doing grade 1 piano in June and possibly Grade 2 by the end of the year. Also to practice the theory doing grade 5 and 6 theory this year alongside the OCA composition courses
http://www.oca-uk.com/subjects/music.html
I like the idea that eventually I’ll be able to compose a symphony (construct is probably a more accurate term). As I said before A224 is fine for an overview, but it is short on hammering the basics. grade 5 theory whilst quite simple harmonically is quite demanding in terms of getting transposition, the names of the keys and intervals at your finger tips. All necessary pre-requisites for a budding composer. One of the most reliable methods of workling out intervals and transpositon is to use modulo aritthmetic which I will expand on in another post
I also want to get on with my physics I left hanging in the air about 18 months ago the derivation of the Friedmann equations which govern the expansion of the universe from General relativitry and I’m taking the current break as an opportunity to get back into it. Any calculatiion in general relativity, even the most simplest such as those for the expansion of the universe or the Schwarzschild metric involves pages and pages of tedious algebra and it nearly broke my heart round about June of last year. Still having done the spade work I can now get on to the physics, watch this space.
Also I need to crack on with the second part of number theory and logic the computablity part of which awaits me. Given Duncan’s and Daniel’s experiences of it I’m not looking forward to it. Still as it is an important part of maths then it is necessary.
I’ll be in two minds whether to continue with the Open University after June. Part of me wants to do the Maths MSc, the new third level course in music and philosophy and the new pure maths course. The other part wants to concentrate on my maths physics and music. Finish the big bang calculation, then get up to speed with modern cosmology, do the three Peskin and Schroeder research projects, understand the functional analysis formulation of quantum mechanics and also the Hawking and Penrose singularity theorems. I’m in my mid fifties and if I want to do all that and get to grade 8 piano and compose a symphony or two before I die then time is ticking on.
Anyway seasons greetings to you all and I hope 2014 brings you nearer to your goals.
Best wishes Chris
I’m currently doing the Number Theory and logic course and completion of that will conclude my second open degree. If I get grade 2 it will put me on line for a 2:1. I then want a break of at least a year from OU work.
The courses will consist of
Level 1 MST121 Introducing Mathematics Pass
Leve 2 MST221 Exploring Mathematics Distinction
M208 Pure Maths Grade 2
A208 Philosophy and the Human Condition Distinction
A224 Music Grade 3
Level 3 M338 Topology Grade 4
MST324 Waves Diffusion etc Grade 3
SM358 Quantum Mechanics Grade 2
Then M381 number theory and logic
The only thing that would persuade me to change my mind is if A303 Philosophy of the Mind gets a reprieve and is presented next year. I wont know till about March. I could replace MST324 and M338 with it and would hopefully get a better grade we’ll see.
Not that I wont be busy I want to concentrate on music and my own physics and maths studies for a while. I intend to finally get a piano tutor with the aim of doing grade 1 piano in June and possibly Grade 2 by the end of the year. Also to practice the theory doing grade 5 and 6 theory this year alongside the OCA composition courses
http://www.oca-uk.com/subjects/music.html
I like the idea that eventually I’ll be able to compose a symphony (construct is probably a more accurate term). As I said before A224 is fine for an overview, but it is short on hammering the basics. grade 5 theory whilst quite simple harmonically is quite demanding in terms of getting transposition, the names of the keys and intervals at your finger tips. All necessary pre-requisites for a budding composer. One of the most reliable methods of workling out intervals and transpositon is to use modulo aritthmetic which I will expand on in another post
I also want to get on with my physics I left hanging in the air about 18 months ago the derivation of the Friedmann equations which govern the expansion of the universe from General relativitry and I’m taking the current break as an opportunity to get back into it. Any calculatiion in general relativity, even the most simplest such as those for the expansion of the universe or the Schwarzschild metric involves pages and pages of tedious algebra and it nearly broke my heart round about June of last year. Still having done the spade work I can now get on to the physics, watch this space.
Also I need to crack on with the second part of number theory and logic the computablity part of which awaits me. Given Duncan’s and Daniel’s experiences of it I’m not looking forward to it. Still as it is an important part of maths then it is necessary.
I’ll be in two minds whether to continue with the Open University after June. Part of me wants to do the Maths MSc, the new third level course in music and philosophy and the new pure maths course. The other part wants to concentrate on my maths physics and music. Finish the big bang calculation, then get up to speed with modern cosmology, do the three Peskin and Schroeder research projects, understand the functional analysis formulation of quantum mechanics and also the Hawking and Penrose singularity theorems. I’m in my mid fifties and if I want to do all that and get to grade 8 piano and compose a symphony or two before I die then time is ticking on.
Anyway seasons greetings to you all and I hope 2014 brings you nearer to your goals.
Best wishes Chris
Tuesday, 26 November 2013
Result for M358 Quantum Mechanics
Well
Results are out for SM358 quantum mechanics I got a grade 2 (just) actually the
examiners this time were on the generous side
As I got
67% for the exam which technically should have been a grade 3 and 90% for the
Assessment.
It does seem a bit arbirtrary looking over the
past courses
M208 I
got 84% in the exam and averaged over 85% for the assessment and didn't not get
distinction
MS324 I got 67% in the exam and got over 80% for
the assessment but did not get grade 2
M338 Topology just scrapped a pass at 40%
although I think my marks were below that in the exam so passed
MST326 got 37% in the exam and above 80% for the
assessment but failed.
It's a bit worrying that depending on the whim
of an examiner you can miss out on a grade.
Tuesday, 19 November 2013
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
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
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