Sunday, 23 October 2011
Normal service will be resumed in two weeks
Hi I apologise for not been able to post anything of substance over the next two weeks but I'm afraid I'm having to devote a lot of my spare time to work at the minute. The crisis should be resolved in two weeks time so I'll resume normal service. I think this week is this blogs 1st birthday. I would like to thank all who find this blog interesting and for all the new friends I've made. Who says mathematicians/physicists are anti social. It's only that it's difficult to find like minded people I wish more people were interested in maths as opposed to Soap Operas or Football the world would be a much more interesting place.
Friday, 14 October 2011
Wednesday, 12 October 2011
M208 and M337 Exams Debriefing
Well what a day I started with M337 I have to say it was a fair exam even though I probably wont get higher than grade 3.
Part 1 has 8 questions the topics were
1) Allegedly simple questions on algebraic manipulation of complex numbers. However the last part was a fiendish raising a number to a complex power. In general for a complex number z
$$z^{\alpha} = exp(\alpha Log z) $$
where Log z = log|z| + iArg z
So it should have been simple to work out but I got bogged down
2) Sketching some sets and their difference and deciding whether or not the sets are regions, compact etc
Straightforward but I froze and for the life of me couldn't after about 2 attempts sketch the difference between the two sets.
It took me about 1/2 an hour to do these questions which should have been a simple warm up was not feeling very happy.
Then the core 3 questions on Complex Integration my favourite topic think I managed to do justice to all three parts and finished these in the next half hour so calmed down and felt a bit better,
Question 6 involved Rouche's theorem which is a method of finding zero's of a complex function within a given interval managed to get the first part out which is quite straightforward but unable to answer the last part as you had to work out how many of the zero's lay in the upper half and I couldn't see how to do it.
Question 7 was a standard one on fluid flows and their complex potentials and sketching the stream lines
I got the complex potential but couldn't sketch the stream line as it involved a complicated circle and I couldn't remember where the centre of a circle of the form
$$x^2 + y^2 + y = c$$ was
Question 8 was a question on Complex Iteration and determination of whether or not a point lay in the Mandelbrot set got about 3/4 of this question out
So reckon probably about 3/4 of part 1 answered correctly
Part II there were 4 questions
9 Part 1 was on the Cauchy Riemann conditions which I think I managed to answer more or less correctly
Part II was on curves and the effects of a transformation. Normally the curves are quite straightforward but this time they weren't and so only managed to answer about 1/2 of the second part
Then my last question 10 should have been a straightforward derivation of a Laurent Series for the function
$$\frac{z}{1-cosz} $$
however I couldn't get it out all the usual tricks didn't work and I couldn't see how to proceed
I abandoned this and was able to answer a question on the Complex Integration of the series as they gave the answer. The second part was about the singularities of the function but by this time my brain was frazzled so I gave up.
So about 1/2 of part 2 answered. So not a satisfactory paper. I've done enough to pass probably grade 3
but certainly no more.
M208 This went much better
Part 1 had twelve questions
1 Sketch of a graph.
2 Exercise on proof involving the converse of a statement
3 Question on whether or not two sets were groups
4 Another question on groups and their cosets
5 A question on row reduction
6 A question on basis for a linear transformation I did not answer this as I hadn't revised this so left it
7 A question on the solution to an inequality
8 A question on whether or not two sequences were convergent or not
9 A question on the symmetry groups of a hexagon
10 A question on homomorphisms
11 A question on L'Hopital's Rule
12 A question on showing that an Integral was less than a certain value
Think I got most of this correct
So apart from question 6 managed to give reasonably full answers to most of the questions but may have lost one or two marks here and there
Part II had 5 questions of which I answered 2
13 A question on the diagonalisation of a 3x3 matrix. Seemed to come out so confident that I got most of the marks for this question
17 A question on the Taylor polynomial of a function which I got most of
A question on the epsilon delta definition of continuity. By this time it was the last 15 minutes and I was feeling quite tired. Also as I did not write out a sample problem of this type in my handbook unlike some people I didn't phrase the answer properly so will probably only get a few marks for this question.
Still having answered most of part 1 and 3/4 of part 2 I think I've done enough to get grade 2 and if the examiners are feeling generous might get distinction.
Some people might think I'm being perverse in refusing to annotate the handbook. My argument is that the exam is a test of how much you know without any crutches. Not an ability to spot model answers to a question and then simply copying them out. There was much heated debate on this in the forum. In my opinion annotation should not be allowed however no doubt other people will think differently.
So a bit disappointed by my performance on M337 but reasonably satisfied with my performance on M208
Will try and review both courses in the next week or two and also how I plan to use down time productively.
Part 1 has 8 questions the topics were
1) Allegedly simple questions on algebraic manipulation of complex numbers. However the last part was a fiendish raising a number to a complex power. In general for a complex number z
$$z^{\alpha} = exp(\alpha Log z) $$
where Log z = log|z| + iArg z
So it should have been simple to work out but I got bogged down
2) Sketching some sets and their difference and deciding whether or not the sets are regions, compact etc
Straightforward but I froze and for the life of me couldn't after about 2 attempts sketch the difference between the two sets.
It took me about 1/2 an hour to do these questions which should have been a simple warm up was not feeling very happy.
Then the core 3 questions on Complex Integration my favourite topic think I managed to do justice to all three parts and finished these in the next half hour so calmed down and felt a bit better,
Question 6 involved Rouche's theorem which is a method of finding zero's of a complex function within a given interval managed to get the first part out which is quite straightforward but unable to answer the last part as you had to work out how many of the zero's lay in the upper half and I couldn't see how to do it.
Question 7 was a standard one on fluid flows and their complex potentials and sketching the stream lines
I got the complex potential but couldn't sketch the stream line as it involved a complicated circle and I couldn't remember where the centre of a circle of the form
$$x^2 + y^2 + y = c$$ was
Question 8 was a question on Complex Iteration and determination of whether or not a point lay in the Mandelbrot set got about 3/4 of this question out
So reckon probably about 3/4 of part 1 answered correctly
Part II there were 4 questions
9 Part 1 was on the Cauchy Riemann conditions which I think I managed to answer more or less correctly
Part II was on curves and the effects of a transformation. Normally the curves are quite straightforward but this time they weren't and so only managed to answer about 1/2 of the second part
Then my last question 10 should have been a straightforward derivation of a Laurent Series for the function
$$\frac{z}{1-cosz} $$
however I couldn't get it out all the usual tricks didn't work and I couldn't see how to proceed
I abandoned this and was able to answer a question on the Complex Integration of the series as they gave the answer. The second part was about the singularities of the function but by this time my brain was frazzled so I gave up.
So about 1/2 of part 2 answered. So not a satisfactory paper. I've done enough to pass probably grade 3
but certainly no more.
M208 This went much better
Part 1 had twelve questions
1 Sketch of a graph.
2 Exercise on proof involving the converse of a statement
3 Question on whether or not two sets were groups
4 Another question on groups and their cosets
5 A question on row reduction
6 A question on basis for a linear transformation I did not answer this as I hadn't revised this so left it
7 A question on the solution to an inequality
8 A question on whether or not two sequences were convergent or not
9 A question on the symmetry groups of a hexagon
10 A question on homomorphisms
11 A question on L'Hopital's Rule
12 A question on showing that an Integral was less than a certain value
Think I got most of this correct
So apart from question 6 managed to give reasonably full answers to most of the questions but may have lost one or two marks here and there
Part II had 5 questions of which I answered 2
13 A question on the diagonalisation of a 3x3 matrix. Seemed to come out so confident that I got most of the marks for this question
17 A question on the Taylor polynomial of a function which I got most of
A question on the epsilon delta definition of continuity. By this time it was the last 15 minutes and I was feeling quite tired. Also as I did not write out a sample problem of this type in my handbook unlike some people I didn't phrase the answer properly so will probably only get a few marks for this question.
Still having answered most of part 1 and 3/4 of part 2 I think I've done enough to get grade 2 and if the examiners are feeling generous might get distinction.
Some people might think I'm being perverse in refusing to annotate the handbook. My argument is that the exam is a test of how much you know without any crutches. Not an ability to spot model answers to a question and then simply copying them out. There was much heated debate on this in the forum. In my opinion annotation should not be allowed however no doubt other people will think differently.
So a bit disappointed by my performance on M337 but reasonably satisfied with my performance on M208
Will try and review both courses in the next week or two and also how I plan to use down time productively.
Monday, 10 October 2011
Tick Tock E Day approaches
Well tomorrow is E day for me I have Complex Analysis in the morning followed by Pure Maths in the afternoon. Having been through two or three past papers for both courses. I seem able to do most of the
questions in Part 1 and about 1 and a half questions in Part 2 in the time available. So distinction is probably not likely and certainly not for Complex Analysis. Anyway I'll be happy with grade 2. I'm not going to do anymore revision. I'm just going to try to relax till tomorrow. It's a bit like a Kid waiting for Christmas Eve to be over so that he can open his presents.
Good luck to all my fellow course mates for tomorrow
I'll report back on Wednesday when I'll either be over the moon or down in the dumps
Till then Bye for now.
questions in Part 1 and about 1 and a half questions in Part 2 in the time available. So distinction is probably not likely and certainly not for Complex Analysis. Anyway I'll be happy with grade 2. I'm not going to do anymore revision. I'm just going to try to relax till tomorrow. It's a bit like a Kid waiting for Christmas Eve to be over so that he can open his presents.
Good luck to all my fellow course mates for tomorrow
I'll report back on Wednesday when I'll either be over the moon or down in the dumps
Till then Bye for now.
Friday, 30 September 2011
M337 TMA04 back and Hume
So got my last TMA for M337 back overall average in the high seventies so looking for grade 2 pass if the exam goes well. Have to say don't really feel I own the subject in the way I do for M208 where my average was in the mid nineties. Still we will see. Not long now D day or should it be E day is the 11th October Complex Analysis in the morning and Pure Maths in the afternoon. This weekend will be exclusively on Complex Analysis revision I hope to work through two papers thoroughly.
Anyway started my Hume course last night. The focus is on Hune's enquiry which is a smaller version of his magnum opus A Treatise on Human Nature. Hume attempts to apply the experimental method so successfully applied by Newton to the study of human nature, in a way it's a precursor to both psychology and sociology and it is arguable that from these perspectives that Hume has been superseded by developments in say cognitive psychology, or sociology. However Hume has an importance in establishing the limits of human knowledge, which Kant was able to develop further.
Hume as is well known is an empiricist, for him the rationalist speculation of Descartes or Leibniz over exagerates what can be achieved by reason alone. They sought to try to base human knowledge on what could be proved purely by reason in order to arrive at absolutely certain knowledge. Hume and other empiricists such as Locke argue that what cannot be based on human experience, is speculation and they would argue the dangers of exaggerating what can be achieved purely by the use of reason. A bit like the debate that still goes on today between Platonists in mathematicians for whom mathematics uncovers the hidden secrets of a world of which the empirical reality is a mere copy. In contrast to those who see us using mathematics as an aid to understanding, but would not make want to make the mistake of assuming that to every mathematical concept there must correspond an element of reality.
Hume begins his enquiry by contrasting what he calls the easy philosophy with more abstruse philosophy. The 'easy' philosophy purports to be an extension of common sense and attempts to describe the importance of say virtue in ways which can easily be understood. By the use of poetry or literature. In Hume's day novelists such as Addison or the poet Pope would be adherents to such philosophy, in our day think of Alain de Boton.
Hume does not dispute the popularity of such philosophy but 'edifying philosophy' has it's limits and indeed there is a tedency for those who prefer 'edifying' philosophy to denigrate the more abstract philosophy because it is obscure and can only be understood by a few people. Hume offers three main arguments against this
1) A knowledge of abstract knowledge can help poets and artists, for example a painter gains much from a knowledge of anatomy.
2) Abstract knowledge satisfies human curioisity and should be encouraged even if it does not benefit mankind directly. Somewhat optimisitically, Hume claims that politicians and lawyers would benefit from a study of abstract philosophy as they would have a better understanding of how government works. Here Hume approaches Plato's ideas that the guardians of the state should be trained in abstract philosophy.
3) Perhaps his strongest argument is that quite often rational principles are misused as a cloak to justify all sorts of superstition. Indeed it is quite surprising that so called analytic philosophers of religion such as eg Plantiga or Peter Geach, see their main task to use modern techniques of analytical philosophy to maintain standard Christian doctrine no matter how obnoxious. For example the specious justifications often given for why God if he is all powerful and knowing allows evils such as Auszchwitz or Tsunami's to happen. (I might sound off about this in another post). For Hume the only way such specious reasoning can be countered is to master the abstruse metaphysics so that it's weaknesses can be exposed for all to see. Hume offers the hope that by enquiring seriously into the nature of Human Understanding it can be shown by an exact analysis of the powers and capacities of the Human mind that it is not fitted for such remote and abstruse subjects. He hopes to undermine the foundations of obscure metaphysics which has in the past served only to shelter superstition and absurdity.
In the next section Hume discusses the Origin of Ideas of which I will give a basic summary of next week.
If I get time this weekend I shall as promised review Richard II. I'll be watching Henry IV part 1 this weekend.
Anyway started my Hume course last night. The focus is on Hune's enquiry which is a smaller version of his magnum opus A Treatise on Human Nature. Hume attempts to apply the experimental method so successfully applied by Newton to the study of human nature, in a way it's a precursor to both psychology and sociology and it is arguable that from these perspectives that Hume has been superseded by developments in say cognitive psychology, or sociology. However Hume has an importance in establishing the limits of human knowledge, which Kant was able to develop further.
Hume as is well known is an empiricist, for him the rationalist speculation of Descartes or Leibniz over exagerates what can be achieved by reason alone. They sought to try to base human knowledge on what could be proved purely by reason in order to arrive at absolutely certain knowledge. Hume and other empiricists such as Locke argue that what cannot be based on human experience, is speculation and they would argue the dangers of exaggerating what can be achieved purely by the use of reason. A bit like the debate that still goes on today between Platonists in mathematicians for whom mathematics uncovers the hidden secrets of a world of which the empirical reality is a mere copy. In contrast to those who see us using mathematics as an aid to understanding, but would not make want to make the mistake of assuming that to every mathematical concept there must correspond an element of reality.
Hume begins his enquiry by contrasting what he calls the easy philosophy with more abstruse philosophy. The 'easy' philosophy purports to be an extension of common sense and attempts to describe the importance of say virtue in ways which can easily be understood. By the use of poetry or literature. In Hume's day novelists such as Addison or the poet Pope would be adherents to such philosophy, in our day think of Alain de Boton.
Hume does not dispute the popularity of such philosophy but 'edifying philosophy' has it's limits and indeed there is a tedency for those who prefer 'edifying' philosophy to denigrate the more abstract philosophy because it is obscure and can only be understood by a few people. Hume offers three main arguments against this
1) A knowledge of abstract knowledge can help poets and artists, for example a painter gains much from a knowledge of anatomy.
2) Abstract knowledge satisfies human curioisity and should be encouraged even if it does not benefit mankind directly. Somewhat optimisitically, Hume claims that politicians and lawyers would benefit from a study of abstract philosophy as they would have a better understanding of how government works. Here Hume approaches Plato's ideas that the guardians of the state should be trained in abstract philosophy.
3) Perhaps his strongest argument is that quite often rational principles are misused as a cloak to justify all sorts of superstition. Indeed it is quite surprising that so called analytic philosophers of religion such as eg Plantiga or Peter Geach, see their main task to use modern techniques of analytical philosophy to maintain standard Christian doctrine no matter how obnoxious. For example the specious justifications often given for why God if he is all powerful and knowing allows evils such as Auszchwitz or Tsunami's to happen. (I might sound off about this in another post). For Hume the only way such specious reasoning can be countered is to master the abstruse metaphysics so that it's weaknesses can be exposed for all to see. Hume offers the hope that by enquiring seriously into the nature of Human Understanding it can be shown by an exact analysis of the powers and capacities of the Human mind that it is not fitted for such remote and abstruse subjects. He hopes to undermine the foundations of obscure metaphysics which has in the past served only to shelter superstition and absurdity.
In the next section Hume discusses the Origin of Ideas of which I will give a basic summary of next week.
If I get time this weekend I shall as promised review Richard II. I'll be watching Henry IV part 1 this weekend.
Tuesday, 27 September 2011
M208 TMA07 back and Gotterdammerung
Got My last TMA for M208 back yesterday just made 90 due to rushing some group theory questions. Think I definitely prefer analysis to group theory or at least to OU group theory. As I've said before the parts that are relevant to physics is representation theory basically concentrating on representing group operations by matrices but this does not seem to play any part in the OU treatment.
Saturday was spent on a last tutorial for M208 Alan my tutor has been wonderfully supportive and I'm lucky to have him. He will be tutoring me for Topology and Group theory next year. Revision for M208 is under control but have to admit M337 needs some attention. I shall focus on this till Monday hopefully I can do two or even three papers by then.
On Sunday as a complete contrast I went to a concert performance of my favourite Wagner Opera
Gotterdammerung played by the Edinburgh Players Opera group.
http://www.operascotland.org/news/81/Edinburgh+Players%27+Opera+Group
Of which one of my work colleagues is the Leader. This has been an annual concert in the late summer the players meet for a whole weekend and friends are invited to come along on the Sunday.
It provides an opportunity for both amateur orchestral players and up and coming singers to perform in roles which they would not otherwise have the opportunity to do so. Ok so there were a few squeeks and squawks but there were no hitches. It is an amazing experience and I still have the music ringing in my head.
Saturday was spent on a last tutorial for M208 Alan my tutor has been wonderfully supportive and I'm lucky to have him. He will be tutoring me for Topology and Group theory next year. Revision for M208 is under control but have to admit M337 needs some attention. I shall focus on this till Monday hopefully I can do two or even three papers by then.
On Sunday as a complete contrast I went to a concert performance of my favourite Wagner Opera
Gotterdammerung played by the Edinburgh Players Opera group.
http://www.operascotland.org/news/81/Edinburgh+Players%27+Opera+Group
Of which one of my work colleagues is the Leader. This has been an annual concert in the late summer the players meet for a whole weekend and friends are invited to come along on the Sunday.
It provides an opportunity for both amateur orchestral players and up and coming singers to perform in roles which they would not otherwise have the opportunity to do so. Ok so there were a few squeeks and squawks but there were no hitches. It is an amazing experience and I still have the music ringing in my head.
Friday, 16 September 2011
TMA06 M208 Back
Well got the 6th TMA for M208 back not bad in the 90's but the elusive 100% escapes me it highlighted one point which I had totally missed. Also it just goes to show how counterintuitive analysis can be.
The series
$$\sum_{0}^{\infty}\frac{1}{n}$$ is divergent but the sum
$$\sum_{0}^{\infty}\frac{-1^n}{n}$$ is convergent
to some people on the forums this seems obvious but to me given that the second series has alternating terms I would think it diverges to two different series. Anyway at least I know now.
Started revising my plan is to do 1 paper per week per course under exam conditions at the start of each weekend and then to consolidate my answers revise topics I'm not sure about until, the last few days before the exams. Which I will take off and try and do two papers for each course on the last four days. That will make a total of five past papers per course which should be enought. At least the exams are on the same day so I can divide my time equally on each course before each exam.
Anyway I'll review Richard 2nd later on this weekend and also summarise the first section of Hume's Enquiry.
The series
$$\sum_{0}^{\infty}\frac{1}{n}$$ is divergent but the sum
$$\sum_{0}^{\infty}\frac{-1^n}{n}$$ is convergent
to some people on the forums this seems obvious but to me given that the second series has alternating terms I would think it diverges to two different series. Anyway at least I know now.
Started revising my plan is to do 1 paper per week per course under exam conditions at the start of each weekend and then to consolidate my answers revise topics I'm not sure about until, the last few days before the exams. Which I will take off and try and do two papers for each course on the last four days. That will make a total of five past papers per course which should be enought. At least the exams are on the same day so I can divide my time equally on each course before each exam.
Anyway I'll review Richard 2nd later on this weekend and also summarise the first section of Hume's Enquiry.
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