Well the debate has been going on the science and physics OU Fora about the meaning or not of quantum physics as all such debates we do seem to going around in circles those of my fellow OU students reading this blog may find part 1 of the discussion here
http://learn.open.ac.uk/mod/forumng/discuss.php?d=1016527
and part 2 here
http://learn.open.ac.uk/mod/forumng/discuss.php?d=1029089
You will see that I have come close to losing my temper some times with a certain person, Still I'm not the only one who finds his attitude slightly infuriating which is reassuring.
Any way as promised here is a brief summary of the Pilot wave approach. This was intially thought of by De-Broglie and the basic idea is that each particle is accompanied by a Pilot wave. (Not that a particle is a wave or vice-versa), this Pilot wave guides the particles in situations such as the two slit experiment giving rise to the characteristic interference pattern on the screen, but a particle only passes through one slit at time. It was proposed at a conference at Solvay in 1927 but wasn't really taken up as most physicists were more impressed with the views of Bohr/Heisenberg and Schrodinger. The rest is history as they say, physicists got on with the business of using quantum physics to calculate the properties of molecules, atoms and solids etc and the pilot wave theory got quietly dropped.
That was until David Bohm rediscovered it and published two papers in the early 1950's these weren't taken seriously most notoriously because David Bohm claimed that the theory required the use of hidden variables to explain things like paticle states. So it was ignored until Bell came on the scene, as is well known he devised an experimental test which could distinguish between hidden variables and the standard predictions of quantum mechanics, it was shown that the predictions of quantum were vindicated and hidden variables were ruled out. You might have thought that would be the end of the story physicists could get on with the real business of developing the applciations of quantum mechanics to ever more and more complex problems. However there was a get out clause (there always is) In deriving his contradiction Bell made two assumptions
i) There were no hidden variables dictating spin components
ii) There were no non local interactions affecting the measurement of 1 particle a long distance away from another one.
As both were required it was perfectly possible for Bohmians to reject ii) and keep i). Most physicists were until quite recently prepared to accept i) and reject ii). However i) has its own problems if taken literally it implies that properties are created by an act of measurement against our notions of common sense.
Of course all that is a bit of an exaggeration, if on the statistical interpretation the wave function is simply a means to generate probabilities then measuring something does not create a property of a particle. All that happens as say when one throws a dice is that one of the possiblities is realised. But then that means that quantum mechanics is no more than an algorithm for correlating the mathematics of quantum mechanics with probabilities and doesn't really explain anything. Well I think I still hold that view, but nevertheless I'm slowly being persuaded, that there is more to the De-Broglie Pilot wave theory than I first would expect.
There has been quite an industry actually using the De-Broglie Bohm theory to perform calculations. Most impressive is it's explanation of the two slit experiment, It shows how the two slit experiment can be explained with each particle taking a definite trajectory and passing through a single slit, there is none of the usual problems associated with believing that an electron splits in two then magically reforms when it is detected. Neither is there any notion of wave packet collapse occuring as a result of measurement.
A really good introduction to the De-Broglie Bohm pilot wave theory is given here by these lectures by Mike Towler.
http://www.tcm.phy.cam.ac.uk/~mdt26/pilot_waves.html
Best to start with the popular lecture
http://www.tcm.phy.cam.ac.uk/~mdt26/PWT/towler_pilot_waves.pdf
I still think there are major problems with it which I will expound on in another post. Anyway I;ll leave you to enjoy the lectures and make your own minds up. The fact that something has moved from being a speculative tool to one where real calculations can be made has to be something.
On another note to do with real waves I was slightly disappointed with the results of my last TMA from MS324 in the low 80's. I missed a key point in one of the questions about the boundary conditions and in my favourite question number 2 I dropped a few marks because I missed out the first term in the series still a reasonable score but not as high as I hoped.
Bye for now
Tuesday, 29 May 2012
Sunday, 20 May 2012
Interpretation of quantum mechanics part I (Again)
Hi been having a heated discussion on the OU science fora about the meaning or not of quantum mechanics
Those who follow this blog will know that I tend to be quite sceptical about any attempts to go beyond the current formalism as
a) No new physics will come out of it (Or none that we can distinuguish between experimentally)
b) Any attempt to see the wavefunction as somehow real leads to all sorts of problems
i) The idea of superposition being something physical until observed seems to imply that we create reality
by an act of measurement
ii) In the two slit experiment the idea that an electron or large particle somehow splits in two and then magically reforms (if taken literally) at the detector seems totally incredible (where does the energy come from etc) why bother with CERN if we can split electrons in two simply by passing them through slits.
iii) If a particle really is a wave how come the pattern only emerges after several impacts on the screen rather than all at once. It is only after a statisitically significant number of events have occurred that anything like a pattern interpreable as a wave function can occur. So that the 'wave aspects' are esssentially statistical the usual fuss about the pattern still occurring even though there is only one particle in the intererometer being irrelevant (or just as relevant as the throw of a single dice).
For these reasons I prefer the statistical interpretation of quantum mechanics, which says that the 'wavefunction' is essentially a probability amplitude whose modulus squared gives the probability of certain events happening. This implies that the wave function is not a property of a single system but more a mathematical device for generating probabilities, it differs from that of classical probability in the sense that to account for the quantum mechanical viewpoint we have to use complex numbers. I then went on to show how you could account for the sinusoidal dependence of the probabilites on the phase factor for a two state system. Also how it was quite striking how classical probabilty could be recast in the language of quantum mechanics specifically the Dirac formalism. For a recap see these two posts
http://chrisfmathsphysicsmusic.blogspot.co.uk/2011/05/quantum-mechanics-of-two-state-systems.html
http://chrisfmathsphysicsmusic.blogspot.co.uk/2011/05/mathematics-of-two-state-systems-2.html
I also gave a reference to a paper by Marcella which gave an explanation of how the typical form of the two slit interference pattern can be interpreted as a single particle build up of many events, where the particle does pass through a single slit, which acts as a measuring device the uncertainty in the particles position being due to not being able to know precisely the position of the particle and being responsible for the wave like appearance.
http://arxiv.org/ftp/quant-ph/papers/0703/0703126.pdf
In all fairness I should point out that a subsequent paper has been written criticising the above paper
http://arxiv.org/abs/1009.2408
And this was presented as a falsification of the Marcella paper, by one of the contributers to the forum. I beg to differ, all it shows is that Marcella has hidden some of his assumptions and that the use of free particle eigenstates is equivalent to a classical wave theory. It would be possible to adapt the Marcella paper to make it more accurate eg by the use of a superposition of free particle eigenstates (often referred to as a Gaussian wavepacket) and removing the assumption that it was equally likely for the particle to emerge from one of the slits. All this would be a distraction from the main point, namely that it is possible to give a particle like interpretation of the two slit experiment using the formalism of quantum mechanics. Indeed as that is what physically happens namely particles really do appear individually and only eventually is a wave like pattern revealed it would seem bizzare to attribute wave like properties to individual electrons, neutrons or buckyballs. Obviously collectively the wave like properties are manifest so wave particle duality is simply when considered as a single entity quantons (for want of a better word) behave like particles but when considered collectively they behave like waves.
The statistical interpretation to my mind is the bottom line, it makes the least number of metaphysical interpretations, one can avoid all the usual problems, real wave collapse, and so forth. It by definition is consistent with the formalism so physicists can get on with the real job, namely developing and applying the formalism to predict and understand the properties of solids, stars, quantum fluids, elementary particles, lasers etc. Or in a word physicists can 'shut up and calculate' and leave the 'interpretational stuff' to other people.
For more information on the statistical interpretation this web site gives a good introduction and overview
http://statintquant.net/siq/siq.html
However as a consequence of the dialogue on the OU science forum, I have become a bit more interested in the so called Pilot wave theory initiated by De-Broglie and subsequently developed by David Bohm. I'll discuss more about it in another blog, giving my reasons as to why out of all the myriad interpretations of quantum mechanics which seek to go beyond the statistics, this is the one I consider most promising. One of the striking things is that the motivations behind the pilot wave theory, seem very similar to the motivations behind the statistical ensemble theory. I will point out the comparisons between the two in a later post. Also there has recently been an experiment showing that classical systems can show wave particle duality.Something that hitherto has never happened before. I'll leave you to ponder about the significance of this experiment for now.
http://phys.org/news78650511.html
Those who follow this blog will know that I tend to be quite sceptical about any attempts to go beyond the current formalism as
a) No new physics will come out of it (Or none that we can distinuguish between experimentally)
b) Any attempt to see the wavefunction as somehow real leads to all sorts of problems
i) The idea of superposition being something physical until observed seems to imply that we create reality
by an act of measurement
ii) In the two slit experiment the idea that an electron or large particle somehow splits in two and then magically reforms (if taken literally) at the detector seems totally incredible (where does the energy come from etc) why bother with CERN if we can split electrons in two simply by passing them through slits.
iii) If a particle really is a wave how come the pattern only emerges after several impacts on the screen rather than all at once. It is only after a statisitically significant number of events have occurred that anything like a pattern interpreable as a wave function can occur. So that the 'wave aspects' are esssentially statistical the usual fuss about the pattern still occurring even though there is only one particle in the intererometer being irrelevant (or just as relevant as the throw of a single dice).
For these reasons I prefer the statistical interpretation of quantum mechanics, which says that the 'wavefunction' is essentially a probability amplitude whose modulus squared gives the probability of certain events happening. This implies that the wave function is not a property of a single system but more a mathematical device for generating probabilities, it differs from that of classical probability in the sense that to account for the quantum mechanical viewpoint we have to use complex numbers. I then went on to show how you could account for the sinusoidal dependence of the probabilites on the phase factor for a two state system. Also how it was quite striking how classical probabilty could be recast in the language of quantum mechanics specifically the Dirac formalism. For a recap see these two posts
http://chrisfmathsphysicsmusic.blogspot.co.uk/2011/05/quantum-mechanics-of-two-state-systems.html
http://chrisfmathsphysicsmusic.blogspot.co.uk/2011/05/mathematics-of-two-state-systems-2.html
I also gave a reference to a paper by Marcella which gave an explanation of how the typical form of the two slit interference pattern can be interpreted as a single particle build up of many events, where the particle does pass through a single slit, which acts as a measuring device the uncertainty in the particles position being due to not being able to know precisely the position of the particle and being responsible for the wave like appearance.
http://arxiv.org/ftp/quant-ph/papers/0703/0703126.pdf
In all fairness I should point out that a subsequent paper has been written criticising the above paper
http://arxiv.org/abs/1009.2408
And this was presented as a falsification of the Marcella paper, by one of the contributers to the forum. I beg to differ, all it shows is that Marcella has hidden some of his assumptions and that the use of free particle eigenstates is equivalent to a classical wave theory. It would be possible to adapt the Marcella paper to make it more accurate eg by the use of a superposition of free particle eigenstates (often referred to as a Gaussian wavepacket) and removing the assumption that it was equally likely for the particle to emerge from one of the slits. All this would be a distraction from the main point, namely that it is possible to give a particle like interpretation of the two slit experiment using the formalism of quantum mechanics. Indeed as that is what physically happens namely particles really do appear individually and only eventually is a wave like pattern revealed it would seem bizzare to attribute wave like properties to individual electrons, neutrons or buckyballs. Obviously collectively the wave like properties are manifest so wave particle duality is simply when considered as a single entity quantons (for want of a better word) behave like particles but when considered collectively they behave like waves.
The statistical interpretation to my mind is the bottom line, it makes the least number of metaphysical interpretations, one can avoid all the usual problems, real wave collapse, and so forth. It by definition is consistent with the formalism so physicists can get on with the real job, namely developing and applying the formalism to predict and understand the properties of solids, stars, quantum fluids, elementary particles, lasers etc. Or in a word physicists can 'shut up and calculate' and leave the 'interpretational stuff' to other people.
For more information on the statistical interpretation this web site gives a good introduction and overview
http://statintquant.net/siq/siq.html
However as a consequence of the dialogue on the OU science forum, I have become a bit more interested in the so called Pilot wave theory initiated by De-Broglie and subsequently developed by David Bohm. I'll discuss more about it in another blog, giving my reasons as to why out of all the myriad interpretations of quantum mechanics which seek to go beyond the statistics, this is the one I consider most promising. One of the striking things is that the motivations behind the pilot wave theory, seem very similar to the motivations behind the statistical ensemble theory. I will point out the comparisons between the two in a later post. Also there has recently been an experiment showing that classical systems can show wave particle duality.Something that hitherto has never happened before. I'll leave you to ponder about the significance of this experiment for now.
http://phys.org/news78650511.html
Sunday, 6 May 2012
The Importance of Pedantry
Hi as a break from Topology and Waves I have been working on my long term ambition to understand the Big Bang line by line. I'm currently trying to understand the derivation of the Friedmann equations from General relativity. This exercise is somewhat tedious and heartbreaking to say the least. However reading some text books eg Lasenby I have noticed at best a misleading error in the calculations of the connection coefficients and this has led me to all sorts of confusion. A connection coefficient is a three indexed quanitity which can be calculated from the metric tensor g as follows
$$\Gamma^{i}_{jk}= \frac{1}{2}g^{il}(\partial_{k}g_{li}+\partial_{j}g_{lk}-\partial_{l}g_{jk})$$
Where the indices run from 0-3 0 corresponding to t and 1,2,3 x,y,z or whatever coodinate system is required. As there are potentially 128 different terms to caclulate one can see the temptation to take short cuts. However if we interchange the indices we get
$$\Gamma^{i}_{kj}= \frac{1}{2}g^{il}(\partial_{j}g_{li}+\partial_{k}g_{lj}-\partial_{l}g_{kj})$$
Which for cases where g is a diagonal metric ie the only non zero terms are
$$g_{00},g_{11},g_{22},g_{33}$$ or $$g_{ii}$$
as in the case of the Robertson Walker, the first equation reduces to
$$\Gamma^{i}_{jk}= \frac{1}{2}g^{ii}(\partial_{j}g_{ii}+\partial_{k}g_{ii}-\partial_{i}g_{kj})$$
and the second equation reduces to
$$\Gamma^{i}_{kj}= \frac{1}{2}g^{ii}(\partial_{k}g_{ii}+\partial_{j}g_{ii}-\partial_{i}g_{jk})$$
But as addition is commutative this is just the first equation with the first two terms interchanged and the last term in both equations will be zero for the off diagonal terms (ie when j does not equal k) . So for diagonal metrics then we must have
$$\Gamma^{i}_{jk} = \Gamma^{i}_{kj}$$
However this symmetry is often missed by standard textbook accounts (eg Lasenby et al page 377) and the claim is made that the only non zero off diagonal connection coefficients for the Robertson Walker metric are
$$\Gamma^{1}_{01}, \Gamma^{2}_{02},\Gamma^{2}_{12},\Gamma^{3}_{03},
\Gamma^{3}_{13},\Gamma^{3}_{23}$$
When they should also include the other off diagonal terms:
$$\Gamma^{1}_{10}, \Gamma^{2}_{20},\Gamma^{2}_{21},\Gamma^{3}_{30},
\Gamma^{3}_{31},\Gamma^{3}_{32}$$
The reason why this is important is because a key quantity in general relativity is the Ricci Tensor which can be calculated from the connection coeffients as
$$R_{ij}=\partial_{j}\Gamma^{k}_{ik}-\partial_{k}\Gamma^{k}_{ij}+
\Gamma^{l}_{ik}\Gamma^{k}_{l}{j}-\Gamma^{l}_{ij}\Gamma^{k}_{lk}$$
So if some gullible reader like myself neglects the other off diagonal terms then you will not get the correct answer for the Ricci Tensor and will spend many hours of frustration wondering why you can't get the answers quoted in the text books. Of the many books I have which give results for the connection coefficients for the Robertson Walker metric at least three namely
Narlikar " Introduction to Cosmology" page 106
http://www.amazon.co.uk/An-Introduction-Cosmology-J-Narlikar/dp/0521793769/ref=sr_1_1?ie=UTF8&qid=1336329189&sr=8-1
Lasenby "General Relativity An Introduction for physicists" page 377
http://www.amazon.co.uk/General-Relativity-An-Introduction-Physicists/dp/0521829518/ref=sr_1_1?s=books&ie=UTF8&qid=1336329247&sr=1-1
and
Collins, Martin and Squires "Particle physics and Cosmology" page 373
http://www.amazon.co.uk/Particle-Physics-Cosmology-P-Collins/dp/0471600881/ref=sr_1_2?s=books&ie=UTF8&qid=1336329321&sr=1-2
All neglect the other off diagonal connection coefficients for the Robertson Walker metric in their tables of connection coefficients.
All this goes to show is that one should not just naively take on trust results quoted in text books I can only agree with my colleague Duncan in his last post about the importance of pedantry when trying to understand maths or physics books
http://matrices-reloaded.blogspot.co.uk/2012/04/last-book.html
On another note I have decided not to embark on the composition course immediately as my finances couldn't really cope with it and also I want to spend more time on my extra curricular physics.
$$\Gamma^{i}_{jk}= \frac{1}{2}g^{il}(\partial_{k}g_{li}+\partial_{j}g_{lk}-\partial_{l}g_{jk})$$
Where the indices run from 0-3 0 corresponding to t and 1,2,3 x,y,z or whatever coodinate system is required. As there are potentially 128 different terms to caclulate one can see the temptation to take short cuts. However if we interchange the indices we get
$$\Gamma^{i}_{kj}= \frac{1}{2}g^{il}(\partial_{j}g_{li}+\partial_{k}g_{lj}-\partial_{l}g_{kj})$$
Which for cases where g is a diagonal metric ie the only non zero terms are
$$g_{00},g_{11},g_{22},g_{33}$$ or $$g_{ii}$$
as in the case of the Robertson Walker, the first equation reduces to
$$\Gamma^{i}_{jk}= \frac{1}{2}g^{ii}(\partial_{j}g_{ii}+\partial_{k}g_{ii}-\partial_{i}g_{kj})$$
and the second equation reduces to
$$\Gamma^{i}_{kj}= \frac{1}{2}g^{ii}(\partial_{k}g_{ii}+\partial_{j}g_{ii}-\partial_{i}g_{jk})$$
But as addition is commutative this is just the first equation with the first two terms interchanged and the last term in both equations will be zero for the off diagonal terms (ie when j does not equal k) . So for diagonal metrics then we must have
$$\Gamma^{i}_{jk} = \Gamma^{i}_{kj}$$
However this symmetry is often missed by standard textbook accounts (eg Lasenby et al page 377) and the claim is made that the only non zero off diagonal connection coefficients for the Robertson Walker metric are
$$\Gamma^{1}_{01}, \Gamma^{2}_{02},\Gamma^{2}_{12},\Gamma^{3}_{03},
\Gamma^{3}_{13},\Gamma^{3}_{23}$$
When they should also include the other off diagonal terms:
$$\Gamma^{1}_{10}, \Gamma^{2}_{20},\Gamma^{2}_{21},\Gamma^{3}_{30},
\Gamma^{3}_{31},\Gamma^{3}_{32}$$
The reason why this is important is because a key quantity in general relativity is the Ricci Tensor which can be calculated from the connection coeffients as
$$R_{ij}=\partial_{j}\Gamma^{k}_{ik}-\partial_{k}\Gamma^{k}_{ij}+
\Gamma^{l}_{ik}\Gamma^{k}_{l}{j}-\Gamma^{l}_{ij}\Gamma^{k}_{lk}$$
So if some gullible reader like myself neglects the other off diagonal terms then you will not get the correct answer for the Ricci Tensor and will spend many hours of frustration wondering why you can't get the answers quoted in the text books. Of the many books I have which give results for the connection coefficients for the Robertson Walker metric at least three namely
Narlikar " Introduction to Cosmology" page 106
http://www.amazon.co.uk/An-Introduction-Cosmology-J-Narlikar/dp/0521793769/ref=sr_1_1?ie=UTF8&qid=1336329189&sr=8-1
Lasenby "General Relativity An Introduction for physicists" page 377
http://www.amazon.co.uk/General-Relativity-An-Introduction-Physicists/dp/0521829518/ref=sr_1_1?s=books&ie=UTF8&qid=1336329247&sr=1-1
and
Collins, Martin and Squires "Particle physics and Cosmology" page 373
http://www.amazon.co.uk/Particle-Physics-Cosmology-P-Collins/dp/0471600881/ref=sr_1_2?s=books&ie=UTF8&qid=1336329321&sr=1-2
All neglect the other off diagonal connection coefficients for the Robertson Walker metric in their tables of connection coefficients.
All this goes to show is that one should not just naively take on trust results quoted in text books I can only agree with my colleague Duncan in his last post about the importance of pedantry when trying to understand maths or physics books
http://matrices-reloaded.blogspot.co.uk/2012/04/last-book.html
On another note I have decided not to embark on the composition course immediately as my finances couldn't really cope with it and also I want to spend more time on my extra curricular physics.
Tuesday, 24 April 2012
M338 TMA02 Back
So got this back reasonably pleased top end of grade 2. Made a silly mistake in the last question and as predicted didn't do all that well in question 2. My tutor says I should be pleased as it was a hard TMA anyway going to take a week off from OU maths till Monday of next week hopefully giving it a bit of a rest will help my subconscious digest it all.
Sunday, 22 April 2012
MST324 TMA02 (Nearly finished my pudding)
Well after finally eating my meat for M338 I've nearly finished my pudding for MST324. What a joy (says he optimistically) The questions were on the wave equation and Fourier transforms.
Question 1 Involved using D'Alembert's solution to the wave equation to solve a particular boundary problem.
Question 2 (The best) involved solving the inhomogeneous damped wave equation for a particular function using seperation of variables. This is a showcase question consolidating all the techniques we have covered so far. The steps are
1) Write the function u(x,t) as a product of two functions f(x)g(t)
2) Substitute these into the partial differential equation and this gives two separate ordinary differential equations in both f(x) and g(t) in terms of a constant .
3) Solve the homeogeneous differential equations for f(x) and g(t) using the appropriate boundary conditons
4) Use the inhomogeneous function on the RHS of the partial differential equation to obtain particular integrals for f(x) and g(t).
5) The resulting solution is then u(x,t) = f(x)g(t)
This is maths at it's best, some of the steps might be a bit tricky but I think it's really cool how it all hangs together.
Question 3 Another question on a solution of a partial differential equation by separation of variables. This was another nice question and a bit simpler than question 2
Question 4 Testing your knowledge of the properties of Fourier transforms based around a question involving the recurrence relationships of Hermite Polynomials. This was OK but a bit tricky in parts and I still have to do the last bit.
So overall I'm quite pleased and this TMA has been a joy to do, there was none of the head scratching associated with M338, but I can see the last bit of ice cream in the tub and it will soon be back to struggling with the many definitions of continuity for Topological spaces and so forth. Anyway it looks like I'm not the only one to have struggled with the TMA for M338. According to my friend Duncan who is doing M208 this year, Alan my tutor for M338 has said that it is taking him 2 hours per question per person to mark the scripts. So if the tutors find it difficult it's not surprising that we do.
Question 1 Involved using D'Alembert's solution to the wave equation to solve a particular boundary problem.
Question 2 (The best) involved solving the inhomogeneous damped wave equation for a particular function using seperation of variables. This is a showcase question consolidating all the techniques we have covered so far. The steps are
1) Write the function u(x,t) as a product of two functions f(x)g(t)
2) Substitute these into the partial differential equation and this gives two separate ordinary differential equations in both f(x) and g(t) in terms of a constant .
3) Solve the homeogeneous differential equations for f(x) and g(t) using the appropriate boundary conditons
4) Use the inhomogeneous function on the RHS of the partial differential equation to obtain particular integrals for f(x) and g(t).
5) The resulting solution is then u(x,t) = f(x)g(t)
This is maths at it's best, some of the steps might be a bit tricky but I think it's really cool how it all hangs together.
Question 3 Another question on a solution of a partial differential equation by separation of variables. This was another nice question and a bit simpler than question 2
Question 4 Testing your knowledge of the properties of Fourier transforms based around a question involving the recurrence relationships of Hermite Polynomials. This was OK but a bit tricky in parts and I still have to do the last bit.
So overall I'm quite pleased and this TMA has been a joy to do, there was none of the head scratching associated with M338, but I can see the last bit of ice cream in the tub and it will soon be back to struggling with the many definitions of continuity for Topological spaces and so forth. Anyway it looks like I'm not the only one to have struggled with the TMA for M338. According to my friend Duncan who is doing M208 this year, Alan my tutor for M338 has said that it is taking him 2 hours per question per person to mark the scripts. So if the tutors find it difficult it's not surprising that we do.
Tuesday, 10 April 2012
M338 TMA2 away
Well for better or worse (probably worse) TMA02 is away. I have no real confidence that I understand what is going on. However for interest here is a summary of the questions and my response. Part 1 had to calculate d(x,y) a so cakked distance function for various x y
Question 1 a distance function is defined and we have to show that it is a metric ie we have to show
M1 d(x,y) >= 0 with equality only if x = y
M2 d(x,y) = d(y,x) ie d(x,y) is symmetric
M3 Finally show that d(x,z) >= d(x,y) + d(y,x) (ie, the triangle inquality)
The metric had 2 forms depending on whether or not given two points in R^2 (x1,y1) (x2,y2) y2 = x2 or not
First part just asked you to show that you can apply the definition correctly think I did OK on this.
Second part show that the metric satisfies M1 and M2 of the definition of a metric
M1 is simply that d(x,y) >= to 0 with equality only if x = y. A bear trap here is to neglect to show that if
d(x,y) = 0 x = y think I managed to negotiate this sucessfully. Again this seemed relativiely sttaightforward
(In what follows e means element of )
Then one had to show that given a,b e R^2, that |a1-b1|+ |a2-b2| < d(a,b) again this seemed straightforward
But this was supposed to be a hint to be used in proving the triangle inequality ie to prove that d satisfied M3
I seemed to prove that M3 was satisified directly without using the hint so there is a nagging doubt that I missed some cases.
Then one had to sketch 2 open balls of d for various d(x,y) an open ball is such that d(a,r) < r for a e R
where a is the centre of the open ball, This is a generalisation of the definition of an open interval
This was relatively straightforward although conterintuitive for the second ball. I came to the conclusion that there was no open Ball centered on (1,1) with radius 1/2 which satisfied the second condition of the metric,
logic tells me I'm correct but my instinct doesn't. Antyway I couldn;t fault my logic so it stands for better or worse.
It was then desired to show that d(x,y) was not metrically equivalent to the Euclidean metricd2(x,y) The euclidean metric is simply the disance between two points Here I totally failed. The definition of two metrics d1 and d2 being equivalent is that it is impossible to find two real numbers m and M such that
md1(x,y) < d2(x,y) < M d1(x,y)
for all x,y e R^2 and we were supposed to use a solution to the first part as a hint to show this. Unfortunately all I could demonstrate was that given d2 and d1 it was indeed possible to find two number m and M such that the above inequality was satisfied. I'm looking forward to Alan's solution to put me right (watch this space).
So for question 1 reckon I've got about 75% of the full marks at least I've explained my confusion
The first part of Question 2 introduced a definition of some open sets and we had to show that they formed a topology on R the first part.
A topology T on a set X is a collection of subsets of X satisfying the following axioms
T1 the topology must include the empty set and the set X
T2 the topology T must be such that any two intersections of the subsets of T must also be an element of T
T3 the topology T must be such that any unions of subsets of T must also be an element of T
For finite toplogies it is sufficient to demonstrate this for any two subssets of X which are elements of T.
The elements of T are called open sets of X, their complements are closed sets of X.
At first sight the definition of an open set of X may seem totally arbitrary but T2 and T3 guarantee a form of closure.
I reckon I got this correct. The second part introduced a new definition of left continuity and we had to show that a specific function was left continous. I think I got most of these parts correct,
Finally we had to show that given a function is both left continuous and monotonically increasing that the funcition was indeed continuous on R.
The background to this is that, so far the purpose of M338 has been to generalise the M208 definition of continuity to reformulate it in terms of whether or not the inverse of f(a) gives rise to an open set of X. As my mastery of the various terminology is still a bit hazy I'm not convinced I got anything resembling a perfect answer to this part of the question.
Question 3 tested whether or not you understand basic definitions such as closure, boundary, exterior interorior or the 'density' or 'non density' of certain finite sets given a topology T defined on those sets
As there were no explicit examples (unlike M208) this involved a) understanding some basic definitions and b) how to apply them. I think I got there in the end but was a bit of a struggle to say the least. Also when defining a map from a set X to X if the function exclude a certain element a then the inverse of f(a) = 0 the empty set, But this is still part of the topology of T. So the ambiguity when testing for continuity is whether or not f(a) has an inverse mapping onto the topology, I decided that it did as 0 is still an open set of T but I can see arguments for the other point of view. So again confident that I got about 75% of this question correct.
Question 4 involved sketching certain sets finding their exterior and interior and boundary and deciding whether or not they were open or closed or neither. I think I got most of this correct.
So overall I might be pushing the boundary between grade 2 and grade 1 for this one but if I get above grade 2 that will only be due to the genoristiy or otherwise of Alan's marking. In terms of really understanding the subject I'm still only on block A2. I neeed over the next 4 weeks to get to grips with A3 and A4 before moving onto block B which admittedly looks a bit more straightforward. So yes I'm being pushed out of my comfort zone, but as my mate Neil said I wanted a 'brain fuck' and I think I've got this in spades. Obviously my confidence has been quite shattered and as a caution for those who think this is the ultimate in pure maths, it should be noted that a course in Topology, along the lines of M338 usally forms a second year course in most brick universities. Anyway I've eaten most of my main course so for the next two weeks, I'll be having my pudding of MST324 I can't wait.
Obviously I will have to try and consolidate my extremely limited understanding of the first part of M338 as well so it will be like eating porridge with salt as opposed to sugar. Still KBO (keep buggering on ) as Churchill is alleged to have said.
Saturday, 7 April 2012
Hand being Forced
Sometimes events just force your hand. As some of my followers know I had planned eventually to study composistion via the Open College of Arts.
http://www.oca-uk.com/distance-learning/music
However as I have been getting bogged down in Topology I had intended to postpone this till October, which would have given me time to pay all fees upfront and also get Topology out of the way. However they have just announced that their fees will be increasing by 60% for students who register after June 1st 2012. However they will keep a TA arrangement for those who register before then. So it looks like I'll be registering for the course in the next month or two, taking advantage of their installment plan. I have enough savings for their deposit, it also means less beer for the next 6 months but that's probably a good thing.
In a way I'm quite excited, but it means something is going to have to give over the next few years, I really can't study Maths, Philosophy and Music to anything like the depth I want to simultaneously. As my maths is important, especially as I have a long term ambiton to become an OU maths tutor, and I want to do some more undergraduate courses, especially M347 mathematical statistics and M336 groups, before I embark on the MSc. I think philosophy will have to take a back seat,for now. I will continue to do one or two courses via Edinburgh University every year and there are also Geoffrey Klempner's Pathway modules
http://www.philosophypathways.com/programs/pack.html
but it looks like I will have to abandon my plan of doing AA308 philosophy of the mind and wait till the replacement course comes in within the next couple of years.
So Provisional Plan is
Maths Music Philosophy
Current M338, MS324 Comp 1 (June) Philosophy of Arts (E/U)
Oct 2012 MST326 M381
June 2013 Comp 2 TBD
Oct 2013 M347 M336
June 2014 Composition of
Extended TBD
Piece (Comp 3)
Oct 2014 Start MSc OU 3rd level course
So if all goes well I could have composed my first symphony by June 2015. Not that it will ever get performed.
I had let my music Studies slide so it's good that I'm being forced to bring them back Still need to finish TMA02 for topology before doing anything else.
Added 2 pm
Have found a blog of someone who has been through the OCA music experience
http://www.fionacoulter.com/blog/
http://www.oca-uk.com/distance-learning/music
However as I have been getting bogged down in Topology I had intended to postpone this till October, which would have given me time to pay all fees upfront and also get Topology out of the way. However they have just announced that their fees will be increasing by 60% for students who register after June 1st 2012. However they will keep a TA arrangement for those who register before then. So it looks like I'll be registering for the course in the next month or two, taking advantage of their installment plan. I have enough savings for their deposit, it also means less beer for the next 6 months but that's probably a good thing.
In a way I'm quite excited, but it means something is going to have to give over the next few years, I really can't study Maths, Philosophy and Music to anything like the depth I want to simultaneously. As my maths is important, especially as I have a long term ambiton to become an OU maths tutor, and I want to do some more undergraduate courses, especially M347 mathematical statistics and M336 groups, before I embark on the MSc. I think philosophy will have to take a back seat,for now. I will continue to do one or two courses via Edinburgh University every year and there are also Geoffrey Klempner's Pathway modules
http://www.philosophypathways.com/programs/pack.html
but it looks like I will have to abandon my plan of doing AA308 philosophy of the mind and wait till the replacement course comes in within the next couple of years.
So Provisional Plan is
Maths Music Philosophy
Current M338, MS324 Comp 1 (June) Philosophy of Arts (E/U)
Oct 2012 MST326 M381
June 2013 Comp 2 TBD
Oct 2013 M347 M336
June 2014 Composition of
Extended TBD
Piece (Comp 3)
Oct 2014 Start MSc OU 3rd level course
So if all goes well I could have composed my first symphony by June 2015. Not that it will ever get performed.
I had let my music Studies slide so it's good that I'm being forced to bring them back Still need to finish TMA02 for topology before doing anything else.
Added 2 pm
Have found a blog of someone who has been through the OCA music experience
http://www.fionacoulter.com/blog/
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