The Open university in it's infinite wisdom has decided to merge three of the 4 pure maths courses currently offered into a third level 60 point course entitled Further Pure maths. This means that the current courses
M338 topology, M336 Geometry and groups and M381 logic and number theory are being phased out.
More alarmingly the new course is dropping the logic part from the number theory course, the applications of group theory to geometry from the Geomerty and groups course and discussion of continuity in terms of open sets, a major part of any topology course, will be replaced by a discussion in terms of convergence of sequences in metric spaces. In compensation there will be an introduction to rings and fields. There is also a move of courses to an October start.
However as far as I am concerned I shall be sticking to the Old courses. Given that fees will probably skyrocket over the next few years. My plans to do a degree in Maths followed by the MSc and a similar exercise in Philosophy over the next 5 years will probably have to be extended to seven or even eight years.
I have also been reassured by people on the first class forum that two key courses namely approximation theory and functional analysis in my plan for the maths MSc will not (despite the misleading rubric on the course site) be phased out so I can do the MSc courses in a more logical order. Also I'm swithering as far as philosphy is concerned whether or not to specialise in Analytical philosophy for which the infrastructure is not really there for part time study via distance learning at postgraduate level or Continental Philosophy for which there is a distance learning MA available via Lampeter University.
http://www.trinitysaintdavid.ac.uk/en/courses/postgraduatecourses/maeuropeanphilosophy/
I'll talk more about the difference between Analytical philosophy and Continental philosophy later but my mind is tending towards trying to understand both schools of thought and the Lampeter MA does seem to give a good grounding in the area of Continental Philosophy. Given that most people have heard of Kant, Nietszche, Heidegger and Foucault but not say Michael Dummett, Hilary Putnam, Kripke or Davidson (All leading lights in contemporary analytical philosophy) then it is clear which has had the more impact on society as a whole. Also the chance to read in depth such classics as The Critique of Pure Reason, Nietszche's Genealogy of morals and Heidegger's Being and time seems to good to miss.
The first priority is still the Maths degree followed by the MSc and also to complete my undergraduate philosphy eductation by doing the Philosophy of Mind OU course next year, followed by some of Geoffrey Klempners Philosophy Pathways modules especially Philosophy of language and Metaphysics.
http://www.philosophypathways.com/programs/pak2.html
So the timetable for the Maths courses over the next few years looks like
Feb 2012 - Oct 2012 Topology (M338), Group Theory M336, Philosophy of Mind
Oct 2012 - July 2013 Number theory and logic (M381) / Mathematical methods and fluid mechanics(MS326) also do Geoffrey Klempners pathways modules.
October 2013 Start Maths MSc with Caclulus of Variations
Also 2 modules from Lampeter University MA
Writing about Philosophy and Hermeneutics
2014 Approximation theory and Applied Complex variables
Year 2 of Lampeter MA Kant and Nietzsche
2015 Functional analysis and either Analytical Number theory 1 or Fractals
Year 3 of Lampeter MA Heidegger and Directed reading on one book probably
Adorno's Negative dialectics
2016 Dissertation For both Math's MSc and Lampeter MA
2017 onwards see what my options for doing a part time Phd in either maths or philosophy are.
This will take me upto the age of 60 by which time I should be able to move to part time work whilst concurrently doing a part time PhD, at the minute the demands of a full time job mean my options are fairly limited. This extended period also gives me time to save a bit more money for the course fees.
Sunday, 27 March 2011
Friday, 25 March 2011
Change of Title
Hi I've decided to change the title of this blog hope you all find it amusing
The Failed physicist tag means that like many of my colleagues at work means that I failed to make into academia and short and fat stands for itself
The Failed physicist tag means that like many of my colleagues at work means that I failed to make into academia and short and fat stands for itself
Sunday, 20 March 2011
What type of mathematician are you
This is probably going to be an inchorent post but what the heck. I want to lift up the corner of a debate which has raged since the Ancient Greeks about the meaning of mathematics. This is partly motivated having watched (finally as Nilo would say) the film Pi. In it the claim is made that everything can be reduced to numbers and their patterns with the implication at least according to the protagonists view that numbers are the foundation of the universe and everything is constructed out of them. For those unaware of the plot the protagonist becomes obssessed with trying to find a pattern in the digits of pi to the extent that he ignores everything else and shuts himself away . Despite some wise advice by his tutor to give up the pursuit which he himself tried he persists but to no avail and eventually drives himself insane and it's only by taking a drastic measure that he is able to free himself of his obsession. All pretty bleak and I suppose the film was trying to make the point that such obsessions can be destructive. However without such obsessiveness a lot of maths that we know just would not have happened. Maurice Wiles locked himself away for seven years with minimal contact with the outside world whilst he worked on Fermat's last theorem. And speaking personally I couldn't juggle the demands of a full time job with a family life and pursue my interests in mathematics. Not that I'll ever contribute much to the field.
So obviously if people are going to make such sacrifices there must be something about mathematics which drives them. A lot of mathematicians (and I suspect Number theorists are probably most prone to this) are attracted by Plato's ideas. In a very brief nutshell Plato had the idea that to everything that existed there was an ideal form of which any concrete realisation was a poor copy. Thus whilst we can draw a triangle it will never be an ideal triangle but just a pale imitation of it's real form. The extension of this idea to mathematics as a whole is encapsulated in the view that when a mathematician proves a theorem he is discovering something about the nature of reality. Or to put it another way to every mathematical concept there is a corresponding element of reality. For these people if mathematics wasn't doing this then it would be a waste of time. Current Platonists include Roger Penrose, Max Tegmark and G H Hardy who expounded this view in his book a mathematicians Apology. On this viewpoint mathematicians are the new priesthood entrusted with finding the fundamental structure of reality and every advance in Maths brings us closer to the ultimate truth. There is a quasi religious zeal about all of this hence the reason I suppose why some mathematicians become obsessed with their work. Ok I've probably over simplified this viewpoint so I apologise in advance for what follows if I've oversimplified.
I'm afraid I don't share this view at all, for me mathematics is a construction which we impose on the world around us which more or less fits. The first philosopher to expound this view was Kant (or at least he gave the most coherent statement of this view). Kant building on ideas from Hume argued that it was impossible to step outside the way we perceive the world to describe the world as it is initself. Indeed one cannot separate the world as it appears to us from the perceptual apparatus that we use to perceive it with. Very crudely we can only see with a limited range of wavelength's of light, it is true we can extend this range by using different wavelengths, hence all the various branches of astronomy from Infra red right up to gamma ray astronomy. However the way the world appears in these different branches is still a function of the wavelength used to investigate the phenomenon. Nevertheless Kant argued that there were certain concepts (which he called categories ) which were necessary to make sense of the way the world appeared to us.
(This is further elucidation added 25th March) Amongst these categories was the world of mathematics. We are forced to use mathematics as a key means of structuring the world of our experience. That is mathematics forms part of what Kant called the syntthetic a priori. Kant made a distinction between synthetic and analytic
concepts. Synthetic concepts are those which add some empirical information to the concept whereas analytical concepts are those which are contained in the definition. Thus Chris is a bachelor automatically implies that Chris is unmarried this is an example of an analytical statement. Whereas Chris is a short fat failed physicist tells you empirical information about Chris which is more than anything that could be deduced from the fact that Chris is a bachelor this is an example of a synthetic statement . Mathematics according to Kant involves concepts which are more than just empirical information but on the other hand are not just analytical concepts.
In the early part of the 20th century boosted by developments in logic due to Frege and Russell, the hope was that mathematics could be reduced to purely analytical concepts. This was dealt a major blow by Godel's theorems which if I understand it correctly has shown that any axiomatic system sufficiently complex to include the Peano axioms of arithmetic (Notably Russell's Principia Mathematica) is either incomplete or inconsistent. By incomplete is meant that there are known mathematical truths which are outside the given axiomatic system. By inconsistent is meant that the system is obviously inconsistent. The sting in the tail as far as I understand it is that if you try to make the system complete it becomes inconsistent. So much for trying to reduce arithmetic to logic. Interestingly enough geometry and abelian groups do not suffer from this defect. Russell's early book the Principles of Mathematics is full of hubris about how he has defeated the Kantian idea that mathematics is some form of construction used to make sense of the world as it appears to us. In the light of Godel's theorem such hubris seems unjustified. (When I do M381 in a couple of years time I'll be in a stronger position to assess the validity of the above formulation any experts reading this please let me know if I'm incorrect). So given that mathematics is not purely analytic there is some constructive element to mathematics which isn't done justice to on a purely Platonist account.
I'm not denying the effectiveness of mathematics in it's domain of applicability and the fact that many systems in the world can be described by linear mathematics is remarkable, but one mustn't be seduced into thinking that linear mathematics for example is the ultimate truth about reality. Try applying linear mathematics to biological systems and you will end up with a completely different ball game. Neither am I saying that because mathematics provides one way of looking at the world it's no different from astrology or voodoo. Thus I do not want to be accused of post modernist relativism which seems to be the fear of those who take something akin to the Platonist viewpoint.
For me there is a compromise which falls between the absolutist claims of the Platonists and the subjective relativism of the post modernists which is that if a theory purports to model some aspect of reality then provided it is empirically adequate that is all that matters. Progress in science or mathematics is found by finding the limits of the applicability of that theory but that doesn't mean the usefulness of that theory should be discarded if say for example another theory comes along which incorporates the successes of that theory but extends the range of applicability. For example despite all the advances of quantum mechanics engineers still design bridges using Newton's laws of motion and I still design antenna's using Maxwell's equations. Obvously we now know that a lot of the metaphysical baggage associated with Newton or Maxwell's theories such as absolute space and time and the aether are no longer tenable. But their equations still work in the domain they were intended to apply to. Thus they are empirically adequate even if their metaphysics is somewhat dubious. (I'll expound more in another post). Clearly Voodoo or astrology is not going to help you design bridges or antenna's thus they cannot be seen as empirically adequate for these particular problems but unlike voodoo or astrology we know the limitations of the applicability of Newtonian physics or classical electromagnetisn whereas by definition the limitations of voodoo and astrology are just not defined.
The implication is that it just is not true that theories such as Newtonian mechanics or classical electromagnetism have been falsified as someone like Popper would claim. What is true is that the metaphysical baggage associated with these theories has been falsified, but in their domain of applicablilty they provide perfectly adequate representations of the world as it appears to us and are perfectly adequate for all intents and purposes to make concrete predictions about phenomenon which they were constructed to explain. Any attempt to lessen their significance in explaining the world around us because they are seen as approximations to the ultimate truth misses the point. I do not know for example how to model rigid body motion or design an antenna using superstring theory or whatever is alleged to be the current ultimate theory of reality. It really is not helpful to see Newtonian physics or classical electromagnetism as having been superseded by quantum physics. Obviously it is helpful to have found their limitations clearly the metaphysical baggage is no longer valid and if that is the case then that places a severe question mark against the metaphysical baggage associated with more fundamental theories such as say superstrings or M theory or their claims to provide access to a hidden reality behind the world as it appears to us. So the concept of empirical adequacy provides a reasonable compromise between the idea that theories in mathematics and science are some how providing the absolute truth about reality and that there is no difference between Astrology or Voodoo and science or mathematics. It also does justice to how science and engineering is practiced today.
Ok so where does this leave us. Mathematics is essentially a game of rules and axioms some of which may apply to the real world and others may not. It should not need something like Platonism to justify its existence as a human activity. It is essentially a puzzle solving activity (those who are into Wittgenstein will know where I'm coming from) and I would argue that getting hung up on how mathematics relates to the world involves all sorts of unresolvable problems.
So obviously if people are going to make such sacrifices there must be something about mathematics which drives them. A lot of mathematicians (and I suspect Number theorists are probably most prone to this) are attracted by Plato's ideas. In a very brief nutshell Plato had the idea that to everything that existed there was an ideal form of which any concrete realisation was a poor copy. Thus whilst we can draw a triangle it will never be an ideal triangle but just a pale imitation of it's real form. The extension of this idea to mathematics as a whole is encapsulated in the view that when a mathematician proves a theorem he is discovering something about the nature of reality. Or to put it another way to every mathematical concept there is a corresponding element of reality. For these people if mathematics wasn't doing this then it would be a waste of time. Current Platonists include Roger Penrose, Max Tegmark and G H Hardy who expounded this view in his book a mathematicians Apology. On this viewpoint mathematicians are the new priesthood entrusted with finding the fundamental structure of reality and every advance in Maths brings us closer to the ultimate truth. There is a quasi religious zeal about all of this hence the reason I suppose why some mathematicians become obsessed with their work. Ok I've probably over simplified this viewpoint so I apologise in advance for what follows if I've oversimplified.
I'm afraid I don't share this view at all, for me mathematics is a construction which we impose on the world around us which more or less fits. The first philosopher to expound this view was Kant (or at least he gave the most coherent statement of this view). Kant building on ideas from Hume argued that it was impossible to step outside the way we perceive the world to describe the world as it is initself. Indeed one cannot separate the world as it appears to us from the perceptual apparatus that we use to perceive it with. Very crudely we can only see with a limited range of wavelength's of light, it is true we can extend this range by using different wavelengths, hence all the various branches of astronomy from Infra red right up to gamma ray astronomy. However the way the world appears in these different branches is still a function of the wavelength used to investigate the phenomenon. Nevertheless Kant argued that there were certain concepts (which he called categories ) which were necessary to make sense of the way the world appeared to us.
(This is further elucidation added 25th March) Amongst these categories was the world of mathematics. We are forced to use mathematics as a key means of structuring the world of our experience. That is mathematics forms part of what Kant called the syntthetic a priori. Kant made a distinction between synthetic and analytic
concepts. Synthetic concepts are those which add some empirical information to the concept whereas analytical concepts are those which are contained in the definition. Thus Chris is a bachelor automatically implies that Chris is unmarried this is an example of an analytical statement. Whereas Chris is a short fat failed physicist tells you empirical information about Chris which is more than anything that could be deduced from the fact that Chris is a bachelor this is an example of a synthetic statement . Mathematics according to Kant involves concepts which are more than just empirical information but on the other hand are not just analytical concepts.
In the early part of the 20th century boosted by developments in logic due to Frege and Russell, the hope was that mathematics could be reduced to purely analytical concepts. This was dealt a major blow by Godel's theorems which if I understand it correctly has shown that any axiomatic system sufficiently complex to include the Peano axioms of arithmetic (Notably Russell's Principia Mathematica) is either incomplete or inconsistent. By incomplete is meant that there are known mathematical truths which are outside the given axiomatic system. By inconsistent is meant that the system is obviously inconsistent. The sting in the tail as far as I understand it is that if you try to make the system complete it becomes inconsistent. So much for trying to reduce arithmetic to logic. Interestingly enough geometry and abelian groups do not suffer from this defect. Russell's early book the Principles of Mathematics is full of hubris about how he has defeated the Kantian idea that mathematics is some form of construction used to make sense of the world as it appears to us. In the light of Godel's theorem such hubris seems unjustified. (When I do M381 in a couple of years time I'll be in a stronger position to assess the validity of the above formulation any experts reading this please let me know if I'm incorrect). So given that mathematics is not purely analytic there is some constructive element to mathematics which isn't done justice to on a purely Platonist account.
I'm not denying the effectiveness of mathematics in it's domain of applicability and the fact that many systems in the world can be described by linear mathematics is remarkable, but one mustn't be seduced into thinking that linear mathematics for example is the ultimate truth about reality. Try applying linear mathematics to biological systems and you will end up with a completely different ball game. Neither am I saying that because mathematics provides one way of looking at the world it's no different from astrology or voodoo. Thus I do not want to be accused of post modernist relativism which seems to be the fear of those who take something akin to the Platonist viewpoint.
For me there is a compromise which falls between the absolutist claims of the Platonists and the subjective relativism of the post modernists which is that if a theory purports to model some aspect of reality then provided it is empirically adequate that is all that matters. Progress in science or mathematics is found by finding the limits of the applicability of that theory but that doesn't mean the usefulness of that theory should be discarded if say for example another theory comes along which incorporates the successes of that theory but extends the range of applicability. For example despite all the advances of quantum mechanics engineers still design bridges using Newton's laws of motion and I still design antenna's using Maxwell's equations. Obvously we now know that a lot of the metaphysical baggage associated with Newton or Maxwell's theories such as absolute space and time and the aether are no longer tenable. But their equations still work in the domain they were intended to apply to. Thus they are empirically adequate even if their metaphysics is somewhat dubious. (I'll expound more in another post). Clearly Voodoo or astrology is not going to help you design bridges or antenna's thus they cannot be seen as empirically adequate for these particular problems but unlike voodoo or astrology we know the limitations of the applicability of Newtonian physics or classical electromagnetisn whereas by definition the limitations of voodoo and astrology are just not defined.
The implication is that it just is not true that theories such as Newtonian mechanics or classical electromagnetism have been falsified as someone like Popper would claim. What is true is that the metaphysical baggage associated with these theories has been falsified, but in their domain of applicablilty they provide perfectly adequate representations of the world as it appears to us and are perfectly adequate for all intents and purposes to make concrete predictions about phenomenon which they were constructed to explain. Any attempt to lessen their significance in explaining the world around us because they are seen as approximations to the ultimate truth misses the point. I do not know for example how to model rigid body motion or design an antenna using superstring theory or whatever is alleged to be the current ultimate theory of reality. It really is not helpful to see Newtonian physics or classical electromagnetism as having been superseded by quantum physics. Obviously it is helpful to have found their limitations clearly the metaphysical baggage is no longer valid and if that is the case then that places a severe question mark against the metaphysical baggage associated with more fundamental theories such as say superstrings or M theory or their claims to provide access to a hidden reality behind the world as it appears to us. So the concept of empirical adequacy provides a reasonable compromise between the idea that theories in mathematics and science are some how providing the absolute truth about reality and that there is no difference between Astrology or Voodoo and science or mathematics. It also does justice to how science and engineering is practiced today.
Ok so where does this leave us. Mathematics is essentially a game of rules and axioms some of which may apply to the real world and others may not. It should not need something like Platonism to justify its existence as a human activity. It is essentially a puzzle solving activity (those who are into Wittgenstein will know where I'm coming from) and I would argue that getting hung up on how mathematics relates to the world involves all sorts of unresolvable problems.
M337 progress slow but steady
This blog topic covers my progress on Complex Analysis, I'm about 3/4 a way through the first TMA
The questions cover the following topics
1) First part Warm up on Complex numbers and nth roots of unity
Sketch of sets and some basic inequalities Initially confusing especially as it involved the backwards form
of the triangle Inequality which is
|a+b| >= ||a|-|b||
It wasn't obvious at first what to do but I think I got there in the end
2) Transformations of a complex functions analogous to similar questions in M208
Inverse functions and their domains. I Got really bogged down here. There is an important distinction to be made between a function specified over all the domain of Complex Numbers and a restriction to a specific part of the the domain. For example take the function cosh(z) where z = a+ib because cos(iz) = cosh(z) there is a periodicity associated with cosh(z) that isn't present in the corresponding real function. As a consequence cosh(z) only has a well defined inverse if the domain of cosh(z) is restricted. Anyway my confusion arose as I failed to distinguish between a function and it's restriction.
3) Questions on Limits of Complex Functions and specification of regions. There are some intriguing links with topological concepts in this unit.
4) Differentiability of Complex Functions At last something I recognise namely the Cauchy Reimann Conditions. In most mathematical methods courses on Complex Variables these are introduced almost
immediately. This course has take 4 units to get here. However here again there is a lot of stuff being covered here about the domain of complex functions. Yes you can eventually use the same formula for differentiating real functions as for complex variables but there are a lot of subtleties which I confess to not fully understanding yet may be second time round.
So so far this course is really stretching me I was in despair a couple of weeks ago as nothing seemed to gell but slowly I'm getting there. I suspect that as far as this course goes I will not be really forging ahead or have time to explore parallel paths. Still having got out my initial quagmire I can get on with other courses, I hope to crack TMA02 for M208 next week
Have fun
The questions cover the following topics
1) First part Warm up on Complex numbers and nth roots of unity
Sketch of sets and some basic inequalities Initially confusing especially as it involved the backwards form
of the triangle Inequality which is
|a+b| >= ||a|-|b||
It wasn't obvious at first what to do but I think I got there in the end
2) Transformations of a complex functions analogous to similar questions in M208
Inverse functions and their domains. I Got really bogged down here. There is an important distinction to be made between a function specified over all the domain of Complex Numbers and a restriction to a specific part of the the domain. For example take the function cosh(z) where z = a+ib because cos(iz) = cosh(z) there is a periodicity associated with cosh(z) that isn't present in the corresponding real function. As a consequence cosh(z) only has a well defined inverse if the domain of cosh(z) is restricted. Anyway my confusion arose as I failed to distinguish between a function and it's restriction.
3) Questions on Limits of Complex Functions and specification of regions. There are some intriguing links with topological concepts in this unit.
4) Differentiability of Complex Functions At last something I recognise namely the Cauchy Reimann Conditions. In most mathematical methods courses on Complex Variables these are introduced almost
immediately. This course has take 4 units to get here. However here again there is a lot of stuff being covered here about the domain of complex functions. Yes you can eventually use the same formula for differentiating real functions as for complex variables but there are a lot of subtleties which I confess to not fully understanding yet may be second time round.
So so far this course is really stretching me I was in despair a couple of weeks ago as nothing seemed to gell but slowly I'm getting there. I suspect that as far as this course goes I will not be really forging ahead or have time to explore parallel paths. Still having got out my initial quagmire I can get on with other courses, I hope to crack TMA02 for M208 next week
Have fun
Wednesday, 9 March 2011
Just a quick one
Hi just a quick one to say I finally got my GENSTAT licence key should anyone do M346 and you run into problems then contact the M346 course team directly. So I spent an exciting evening learning how to read in files, generate statistics and plot histograms and normal plots. Hopefully things will get better I'll start the TMA over the weekend.
Also got my first TMA for M208 back pleased with the result but it's early days yet. Dropped 1 mark in the induction question as I used a short cut in writing out mathematical expressions in terms of P(k) and P(k+1) which Alan told me was meaningless as P(k) is a proposition not a mathematical expression (ouch). Apart from that most of the rest of the TMA went well. The questions were on
a) Drawing Sets and applying transformations
b) Composite Functions
c) Proof by Induction
d) Complex Numbers
e) Equivalence Classes
There has been quite a discussion on equivalence classes on the M208 forum partly because I suspect that their significance is not really bought out. In group theory they play a major part in partitioning the group by it's various symmetry relations. For dihedral groups in particular the conjugacy classes correspond to the varous symmetry operations for a triangle for example we have the identity, three rotations and two reflections. Each of these belong to a separate conjugacy class. The classes play an important role in representation theory which I hope to expand on in more posts. Essentially if we represent each symmetry operation by a matrix the sum of the diagonal elements of each type of matrix (the trace ) is the same for each type of symmetry operation so instead of using a group table to represent all the invariant operations we can still retain a lot of the essential information of the group by concentrating purely on the traces of each type of matrix. These are called characters for more watch this space .....
Also got my first TMA for M208 back pleased with the result but it's early days yet. Dropped 1 mark in the induction question as I used a short cut in writing out mathematical expressions in terms of P(k) and P(k+1) which Alan told me was meaningless as P(k) is a proposition not a mathematical expression (ouch). Apart from that most of the rest of the TMA went well. The questions were on
a) Drawing Sets and applying transformations
b) Composite Functions
c) Proof by Induction
d) Complex Numbers
e) Equivalence Classes
There has been quite a discussion on equivalence classes on the M208 forum partly because I suspect that their significance is not really bought out. In group theory they play a major part in partitioning the group by it's various symmetry relations. For dihedral groups in particular the conjugacy classes correspond to the varous symmetry operations for a triangle for example we have the identity, three rotations and two reflections. Each of these belong to a separate conjugacy class. The classes play an important role in representation theory which I hope to expand on in more posts. Essentially if we represent each symmetry operation by a matrix the sum of the diagonal elements of each type of matrix (the trace ) is the same for each type of symmetry operation so instead of using a group table to represent all the invariant operations we can still retain a lot of the essential information of the group by concentrating purely on the traces of each type of matrix. These are called characters for more watch this space .....
Sunday, 6 March 2011
Progress (or lack of it)
Amended 10th April now that I've learnt how to use Math JAX
Sorry for not blogging recently I've been a bit bogged down at work. This has meant an alarming decay in my productivity vis a vis mathematics. Still I'm on target as far as the course calendars are concerned.
Finished and posted the 2nd part of the first TMA for M208. Am half way through the TMA on M337 (which doesn't need to be in until April) so so far so good. The only sticking point is the Linear Statistical modelling course I have been unable to get a licence key for the software. This sort of thing makes me paranoid I installed the software following the instructions. It then said you could register online but when I pushed the button claiming to be able to access the website nothing happened. So I was forced to use the offline procedure I sent an e-mail off to the guys who run GenStat but so far have heard nothing. I will wait till the middle of the week then start complaining. I really hate this beauracrcy and being made to feel like a criminal simply because I want to use a piece of software which I've paid for. Doesn't look like a great start to a course that I wasn't particularly interested in in the first place we'll see what happens.
On a brighter note next topic in M208 is group theory. Neil has opened up a shared activities website to share practical applications of group theory. Many many years ago as part of my physics degree I studied group theory (from a physicists) perspective as it applied to problems in quantum mechanics specifically
the application of representation theory to classify the normal vibrations of molecules and it;s application to particle physics. It seems a good opportunity to revise this interesting topic.
To start off I've looked at how to simplify the calculation of Cayley tables for dihedral groups I have poor visualisation skills so get really confused when a shape is rotated through N/2pi degrees then reflected about an obscure axis. Some people seem to be able to do this naturally I can't fortunately algebra comes to the rescue I'll post more in a Latex document. However (and apologies to those reading this who know this already).
Dihedral groups are the groups associated with the symmetry of regular polygons. It can be shown (as I will expand further on my Latex document) that there are only two symmetry operations namely rotations and reflections.
The Cayley table splits into 4 quadrants
rot refl
rot rot refl
refl refl rot
The elements of the group are rotations through $$\frac{2\pi}{N}$$ where N is the Number of sides of the polygon
and reflections about a given axis represented by s. A rotation through $$\frac{j * 2\pi}{N}$$ where j an integer running from 1 to N-1 is denoted by $$r^j$$.
The elements of the group are $$(e,r^1, r^2 ....... r^(N-1), s rs r^1s ....... r^(N-1)s)$$
the second half being rotations of the basic reflection. reflections obey the basic rule
$$s^2 = e$$ .
The multiplication rules are really quite straightforward
A rotation through $$\frac{k*2\pi}{N}$$ followed by another rotation $$\frac{j*2\pi}{N}$$ is simply another rotation
$$r^j * r^k = r^{(j+k)}$$
where j+k is evaluated to modulo N.
A reflection followed by a rotation is simply
$$r^j * r^{k } s = r^{(j+k)} s.$$
The other two operations are a bit more complicated but it is quite straightforward to verify this magic rule
$$r^k s = s (r^k)^{-1}$$
(Those who are into quantum mechanics will recognise this as an anticommutation relation)
This means that for products of the form
$$r^j * sr^k$$ we get $$r^j * (r^k)^{-1} s$$
But $$(r^k)^{-1}$$ is simply$$ r^{N-k}$$
(so for a 5 sided polygon the inverse of $$r^2$$ is simply$$ r^{5-2} = r^3$$)
Finally a product of the form
$$ r^j s * r^k s = r^j s s (r^k)^{-1} = r^{N-k+j}$$ as $$s^2 = e$$
Where the second stage has been gotten from the anticommutation relations.
The point is that now it is relatively straightforward to generate the cayley table for such a dihedral group
without having to use matrices, two line symbols or get yourself confused looking at figures,
Anyone care to have a go at say 5 or 6 sides.
In the next post on this topic I'll show how these rules can also simplify the generation of conjugacy classes
(Apologies for the wide gaps still trying to master LATEX properly)
Best wishes Chris
Sorry for not blogging recently I've been a bit bogged down at work. This has meant an alarming decay in my productivity vis a vis mathematics. Still I'm on target as far as the course calendars are concerned.
Finished and posted the 2nd part of the first TMA for M208. Am half way through the TMA on M337 (which doesn't need to be in until April) so so far so good. The only sticking point is the Linear Statistical modelling course I have been unable to get a licence key for the software. This sort of thing makes me paranoid I installed the software following the instructions. It then said you could register online but when I pushed the button claiming to be able to access the website nothing happened. So I was forced to use the offline procedure I sent an e-mail off to the guys who run GenStat but so far have heard nothing. I will wait till the middle of the week then start complaining. I really hate this beauracrcy and being made to feel like a criminal simply because I want to use a piece of software which I've paid for. Doesn't look like a great start to a course that I wasn't particularly interested in in the first place we'll see what happens.
On a brighter note next topic in M208 is group theory. Neil has opened up a shared activities website to share practical applications of group theory. Many many years ago as part of my physics degree I studied group theory (from a physicists) perspective as it applied to problems in quantum mechanics specifically
the application of representation theory to classify the normal vibrations of molecules and it;s application to particle physics. It seems a good opportunity to revise this interesting topic.
To start off I've looked at how to simplify the calculation of Cayley tables for dihedral groups I have poor visualisation skills so get really confused when a shape is rotated through N/2pi degrees then reflected about an obscure axis. Some people seem to be able to do this naturally I can't fortunately algebra comes to the rescue I'll post more in a Latex document. However (and apologies to those reading this who know this already).
Dihedral groups are the groups associated with the symmetry of regular polygons. It can be shown (as I will expand further on my Latex document) that there are only two symmetry operations namely rotations and reflections.
The Cayley table splits into 4 quadrants
rot refl
rot rot refl
refl refl rot
The elements of the group are rotations through $$\frac{2\pi}{N}$$ where N is the Number of sides of the polygon
and reflections about a given axis represented by s. A rotation through $$\frac{j * 2\pi}{N}$$ where j an integer running from 1 to N-1 is denoted by $$r^j$$.
The elements of the group are $$(e,r^1, r^2 ....... r^(N-1), s rs r^1s ....... r^(N-1)s)$$
the second half being rotations of the basic reflection. reflections obey the basic rule
$$s^2 = e$$ .
The multiplication rules are really quite straightforward
A rotation through $$\frac{k*2\pi}{N}$$ followed by another rotation $$\frac{j*2\pi}{N}$$ is simply another rotation
$$r^j * r^k = r^{(j+k)}$$
where j+k is evaluated to modulo N.
A reflection followed by a rotation is simply
$$r^j * r^{k } s = r^{(j+k)} s.$$
The other two operations are a bit more complicated but it is quite straightforward to verify this magic rule
$$r^k s = s (r^k)^{-1}$$
(Those who are into quantum mechanics will recognise this as an anticommutation relation)
This means that for products of the form
$$r^j * sr^k$$ we get $$r^j * (r^k)^{-1} s$$
But $$(r^k)^{-1}$$ is simply$$ r^{N-k}$$
(so for a 5 sided polygon the inverse of $$r^2$$ is simply$$ r^{5-2} = r^3$$)
Finally a product of the form
$$ r^j s * r^k s = r^j s s (r^k)^{-1} = r^{N-k+j}$$ as $$s^2 = e$$
Where the second stage has been gotten from the anticommutation relations.
The point is that now it is relatively straightforward to generate the cayley table for such a dihedral group
without having to use matrices, two line symbols or get yourself confused looking at figures,
Anyone care to have a go at say 5 or 6 sides.
In the next post on this topic I'll show how these rules can also simplify the generation of conjugacy classes
(Apologies for the wide gaps still trying to master LATEX properly)
Best wishes Chris
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