TMA01 results in (Groups and Geometry).

I just received back my results today for M336.  I scored 82% on this particular piece, which considering I messed up an entire question, probably was a bit of a let off!

In brief, my geometry scored almost full marks apart from a few accuracy slips.  My group theory was less certain and I got very muddled between a cyclic group and a multiplicative group over the integers.

My tutor was kind enough to remark on some insights that I managed to use, when answering a groups question. I gathered from his comments,  that I answered it in a shorter and more compact way than had been intended by the question author,  taking a shortcut byconverting a group into the kernel of another group, in order to make the result fit the question.

I can't complain.

I have been at work for the last 4 days and have not touched the books.  I have been so tired that I have come home, eaten some supper and then fell into bed, asleep in mid-air.  I have two days off now, so it is full on studying to try and catch up.

I am noticing that I am forgetting a lot of the earlier stuff really quickly, so I need to start some consolidation work, sooner than I expected.  It looks like I may just need to settle for some lower TMA marks, so that I can study and revise the units, without falling behind.

Relentless Studying - Fatigue - Mission Creep

I feel absolutely exhausted this week.

My work schedule combined with a study schedule that sees me studying for 3hrs before work in the morning, three hours after work and then 8hrs on all of my days off, is the lengths I am having to go to, just to keep up with understanding the new material that appears on every page of my Unit books for number theory and group theory.

I know that I should probably rejoice in the fact that I obtained a distinction for my first bit of course work for number theory.  But I can't.  You see, I worked out that each of those 95 marks, took me around 1/2hr of work per mark, to obtain.  Thats over 40hrs of work for a piece of coursework that should take <10hrs.

The reasons are clear; this stuff is okay to grasp on first reading, but then to try and internalize and synthesize answers to novel questions, which may take a slightly different form to those examples given in the books, is a real challenge for such a complex area of knowledge.

The stuff on recursive functions is a nightmare and I have reached a chapter in the book that, for the first time in my mathematical career, I really don't understand.

There is some light at the end of the tunnel though.  As with Pure Mathematics which I took last year, content that I found impossible to grasp and nearly had a meltdown over, at the begining of that course, was internalized and slickly applied in the exam, to gain full marks in that section.  However, this course is much more difficult.  I'm not sure that I will be so lucky this time around.

What doesn't help is that I am experiencing some 'mission creep' to coin a U.S military term.  I am finding that I settle down to study a set amount of pages in a study session, and end up managing only half of that, because I have this obsession with doing all of the example questions set, so that I at least understand as many different applications of the preceding knowledge, as possible.  My methods feel tangental, at the moment, and they desperately need to be refined.

It's not as mad as it sounds - I missed out on full marks in last years exam, because there was a particular section of two questions, that I struggled to fully complete, yet when I looked back at the unit books after the exam, there they were smiling back at me, in black and white.  Had I attempted those two example questions, I would have been able to tackle the exam question, with confidence.

TMA01 Results (Number Theory and Logic)

Well, I received my results today for my first piece of coursework on this fiendishly difficult mathematics module.

I am pleased to say that I scored 95% for the whole paper.  I managed to pick up 100% of the marks for the NUmber Theory section, but dropped 5 marks on the Logic section.  I was surprised at this, as I felt that I had managed to sort out my URM functions and tables, quite well.  However, my tutor identified that all of my lost marks could have been recovered through some more thorough checking of the programs that I wrote.

I can't really argue with that, although I really did struggle to work out a suitable function that a particular program produced.  Having read the answer, I still can't fathom it!  Oh well.

So, I have been struggling through my second TMA for both Number Theory and Group Theory; but I am about one week behind on the unit books.  It looks like I won't be getting a week of rest at Christmas, at this rate.

I am working solidly, many more hours than I should be; but I am needing to, rather than choosing to, as I don't understand half of the material on the first or even second reading.

To celebrate this small first victory score, I will put down my logic books for today, and head down the pub for a pint of Bishops finger.

Ooo Matron!

A Welcome Group Theory Interlude

I'm posting rather than studying at the moment.  Procrastination rules...

I have to say that the fourth book of the Mathematics module Groups and Geometry with the OU, is a lovely rehash of some safe, comforting basic group theory.  I never thought I would be so glad to meet this material again, after M208 (Pure Mathematics).  I had struggled, at times, with the concepts on Cosets and Normal subgroups; so it wasn't exactly a passion of mine, at the time.

But, coming back to it with fresh and slightly more capable eyes, I am actually enjoying this unit.  It is basic, but it is abstract and axiomatic, two aspects I very much enjoy.  But mainly, this is a welcome interlude from the relentless TMA onslaught of this course and the Number Theory course M381.

However, I see from Duncan's posts, that this is the calm before the storm, for this particular module.

M381 TMA, away...

Just a quicky!

I have completed and sent off my first TMA for Number Theory and logic.  It was very tough; but, already, on reviewing the answers, I am starting to feel a familiarity for some of the tools and techniques used in them.

So, it's bitter-sweet really.   I have put in far to many difficult hours on this TMA; but I am feeling an enormous sense of satisfaction in its completion.  I have no idea how I have done, but the challenge was good.

Tricky Number Theory

Oh Lord.

I am managing to understand number theory - (none of it is so difficult, that you are unable to follow the proofs with very careful reading) - but every time I come across another example problem in the book, It is clear that I would never have thought of answering it in the way described within the text.

What's more, having read the answer, I still can't understand how a number theory novice is supposed to know the answers to these things, from the sparse collection of theorems that are provided in the set books.

I reckon that with many extra hours of study and utilizing a rather handy technique that I have recently picked up for learning new material, I should be able to rote learn the minor theorem proofs, some key example solutions and a few other bits, in time for the exam.  But, I don't actually think that this is going to get me through the damn exam; however, it might just keep the blood pressure low enough to avoid going on Statins before June.

As my friend Chris would say.

K.B.O.

Reflecting on Number Theory

I think that both Chris and Duncan, were spot on, when they identified that the level of thought and abstraction involved in level 3 number theory, is much more thought provoking and much less formulaic, when it comes to answering problems based on the material.

On first examination, many of the number theory theorems presented in the course material are, ostensibly, elementary; and one can be fooled into thinking that they are, almost, trivial.  But this would be underestimating the power of such elementary building blocks on which all of mathematics is essentially laid.

Most of the worlds most beautiful architectural buildings are built using the basic materials that you can find in any common builders yard; but the ways that they are combined, built upon, measured and used as an expression of human thought, are essentially infinite.

Number theory is such a construction.  It is based on simple, axiomatic, undeniable truths, which lead to some very complex and thought provoking conclusions.  Glancing ahead, the strength in such constructions, is probably going to be both confirmed and blown apart at the same time, as I eventually discover what Herr Gödel had to say about these matters.

I can't wait.  The work will have been so worthwhile.

Walking Through Custard

Oh my!  I have spent the last three weeks, using up every available moment of my time, studying my geometry and number theory courses.  I hadn't realized just how much new stuff there is to learn; and not only learn, but manipulate and mould, into something that can be used in my TMA questions.

It has sunk in that number theory, is really, really hard.  I mean, not just hard, but insanely hard.  I'll try and explain a little.  I am currently staring at my TMA sheet, in which it has taken me 20hrs, to answer 3.5 questions out of 10.  I have looked at two of the questions and not only can I not work out an answer for them; I can't even recognize which bit of the course, they are supposed to be testing.

This isn't just some changing of symbols used, that is throwing me;  this is a case of me tackling this stuff without a full and comprehensive background in mathematics, at A level and 1st year university level maths.  These questions are calling on stock pieces of mathematical knowledge, that I am shaky on, combined with high level abstraction.

All I have been able to do, is work, work and work some more.  I have kept an accurate note of my time spent studying this month; and I am about on track with my studies and TMA work; but it has taken me an average of 30hrs per week, to get there.

And the worrying thing is, I still feel like I don't really know the material all that well.

All I can do is maintain this level of study for the entire course; and I won't give up.  I can't really say more than that, at this point.

This is the hardest that I have ever worked, in my life;  and I am going to make it count for something.

Division

Last year, I completed a T.M.A for my pure mathematics course with the Open University, in which one of the questions asked for a simple proof of a number theory example.  I forget exactly what the example was, but it involved proving that a certain expression was divisible by 12, or some other integer.

At the time, I had a bit of a brain freeze and struggled with the question for some days.  It consumed my days and nights until I finally gave in and rang my tutor, begging for salvation.  I though, at that point, that I wasn't cut out for number theory.

It wasn't until he asked some very probing questions, that we finally got to the bottom of where my difficulties lay.  It was the simple fact, that I didn't quite understand what was meant by 'divisible by'.

What?  Nonsense, I hear you say!  Every six year old knows how to divide two numbers.  But the simplicity hid a rather elegant idea, in that, to divide two integers, and get another integer, is quite difficult to prove or understand, from axiomatic principles.

This was born out when I delved into my first book of number theory, last week, and came across... you guessed it... a proof for the Division Algorithm (...how to divide two numbers).

Its not a short proof, but it is elegant, none-the-less.  If I had my LaTex editor working, I would write it below.  I might do that next month.

The division algorithm is really quite lovely and it is one of the few gems in this new course, including my favourite so far, the proof of general mathematical induction; and my least favourite, the rather puzzling, second principle of mathematical induction.  I say it's puzzling, as I haven't quite managed to figure out why, and when, you need to apply it to a problem, rather than just using the first principle.

I'm working on that one.

First Impressions

Well,  I think I like my new courses.  Number Theory isn't as heavy on content, as I first thought, and I have been able to manage to stay on track for my first 1.5 weeks of study.  The main thrust of the material, is the mastery of Mathematical Induction as a way of providing simple proofs for integer problems.

I struggled with induction, when I studied M208 Pure Maths; but I think this was because I didn't allow enough time to complete practice problems, on the subject.  However, I have taken head of the warning in the material for the number theory course, which states that Induction must be mastered, asap.

So, I have spent hours doing problems and redoing them, until the method started to stick.  Having said that, I have pretty much struggled with the algebraic step in the last part of all of these proofs.  Now, I considered myself to be at an advanced intermediate level, when it comes to algebraic manipulation; but these examples in the texts, make huge jumps in the algebraic reasoning, that I have struggled to follow or replicate (even by punching the expressions into Wolfram Alpha).

I definitely need to ask my tutor for some advice, about that.  I do hope that I have not reached a natural ability ceiling, that hinders my progress on the course.

The groups course has been quite heavy on material volume, in the first book.  You only get 1.5 weeks per book, where as, you get 2 weeks for a similar amount of material in number theory.  Most of the material introduced, was basic concepts and language, that will be needed for the course; however, there was a cheeky little chapter on Affine geometry, which I started off hating, but then when I got all the examples right, without referring to the answers; I started to love it.  How fickle!

Also, I have to say, that I find it soothing, rather than boring, to manipulate tiling's.  I know that people who previously took the course, felt that it was all a bit dull.  I find that they are all so geometrically neat and satisfyingly symmetrical; just the right amount of order needed, after a torturous and chaotic day at work!

I am adopting a new strategy for my studies this year.  That is, I have decided to complete all of the practice examples in the books, having a good go at them, before referring to the answers.  In my last few courses, I had started to get into the habit of skimming the examples, kidding myself that I was saving time, or leaving myself extra practise examples for exam time.  I think it was a false economy and harmed my understanding of some of the more difficult concepts in analysis and linear algebra.

Lets see if it does me any good, this year.

Go...

They have arrived.  I now have the first few month's worth of study materials for both Number theory and The O.U Groups course.

The Groups course is full of groovy plastic overlays and lots of different handbooks and equation sheets; yikes!

I have already started both courses, completing my first hour of study, on each of them yesterday.  It's always difficult to say which will be the more challenging course; however, going solely on first impressions - I would say that the Groups course looks a little easier than the Number theory course.  Time will tell, as to whether this first impression, lasts.

On Your Marks... Get Set...

After a lovely summer break which included a month off work and a trip to France and then Barcelona; the kids are back at school (my sanity is returning), and I am now sitting by the front door mat, like an expectant dog, waiting to chew the hand of the post man, as he delivers my fresh, new, Open University course materials, for this year.

The last time I blogged, I mentioned some of the prep that I would do over the summer, including some of M337, in preparation for next year.  I didn't manage to do that, as I made a decision to start giving myself a 'proper' break between periods of study, so that I can maintain a higher level of intensity, when the courses begins.

For the last few years, I have studied pretty-well none stop, in some form or another, which while admirable in keeping up a study habit, has probably harmed some of final TMA marks, as I find my attention waning, after non stop studying for 24+ months.

After my summer break, I have picked up my French and Spanish Assimil language courses, that I did a few years back; and I have resolved to spend an hour a day (in my work lunch break and during my commute), to listening to these audio tracks, with a hope of reaching a better level of conversational proficiency, in the next few years.

I had found previously, when I took my OU Art History module's in 2004-2005, that doing a short period of language study (at that time French with the O.U course L192) each day, somehow made it feel easier to learn the Art History from all of those dusty old books.  I don't know the reasons for this.  Whether it was just some novelty and a different part of the brain being exercised (auditory and language centres), I will never know.  I will report back as to whether I perceive any difference in my Maths study, as I listen to a small bit of languages, alongside my formal studies, this year.

Anyway, I am booked onto Number Theory and Logic, along with Groups and Geometry.  I've read Duncan's posts on these topics, and it all sounds very interesting.  I just hope that it isn't too stressful.  I've almost forgotten how many times, I sat up until 2am, during 12hr marathons to get my TMA's completed for M208.  Only now, I have two courses to contend with.

Exciting stuff.

M337 Complex Analysis

My last course M208 Pure Mathematics, lightly touched upon complex analysis, only as far as arithmetic and basic manipulation of complex numbers, is concerned.  My current plans include completing the Open University course M337, Complex Analysis, in 2014.  As such, I am planning a very slow study of the course, prior to starting it, next year.

When I say slow, I mean glacial.

I am in no rush whatever.  So, alongside my Groups, and number theory preparation over the next few months; I am planning on studying a page or two from the first couple of units in M337, every few days.

I am hoping that the basic concepts will then slowly simmer in my unconscious mind, over the next year, prior to starting the course, proper, next October.

I don't have any number theory books to pre-study, prior to October this year, other than my trusty copy of 'How to Prove it' by Velleman.  It contains some nice set theory along with some logic and practise at working out basic proofs.  I would recommend that book to anyone who wants extra practice at creating proofs from first principles, in a systematic, logical way.

Conjugate Groups, Abelian Groups and Equivalence Relations.

Whilst conducting some revision of my material from the Open University course M208 Pure Mathematics, today; I noticed something quite satisfying, within one of the proofs of that book,  (P.16, Unit GTB1, Conjugacy).

When I studied this material, last year ( I think it was around May/June time?), I had glossed over many of the proofs in the book, and leaned some of the material on face value.  This strategy served me well, such that I had enough time to practise questions and gain an overview of each topic, to allow me to pass the course.

However, I am now going through some of those longer proofs, in some very fine detail.  Why? Well, I want to enrich the material that I learnt last year; I want to gain more experience of reading proofs, so that I can begin to write some of my own; and It is providing good, previously missed connections, between the material that I have already studied.

This leads me onto the chosen proof of today's post:

'In any group G, the relation 'is conjugate to, denoted by ~, is an equivalence relation on the set of elements of G'.

I just want to highlight one small part of that proof.  For those interested in seeing the entire proof; let me know, and I will happily reproduce it, once I sort out my LaTex editor, next week.  Alternatively, any search on the web, will probably locate it somewhere.

The part that caught my attention, was in the section of the proof ' E2 SYMMETRIC'.

This is where we are trying to prove that for all elements x,y of G; then if x~y , it follows that y~x.

The strategy used is one where the equivalence relation x~y is denoted as  y = gxg^-1.  And the proof uses some basic manipulations to show that this is equivalent to x = g^-1 y (g^-1)^-1; or simply put, y~x.

The first step seems simple enough:

If y = gxg^-1, then  g^-1 y g = g^-1(gxg^-1)g

[All we have done here is multiply both sides of the expression by g^-1 on the left and g on the right, of each monomial part].

It then follows that through the associative property for groups; we can further manipulate the expression to give:

 g^-1 y g = g^-1(gxg^-1)g
              = (g^-1 g) x (g^-1 g)
              = exe                                [e is the identity element of Group G]
              = x.

Now, this particular proof takes one final step, to show that this group is symmetric.

Why? you might ask.

Well, what we have been left with, is  x = g^-1 y g.  The problem with this expression, is that although it looks very similar to the expression that we need to find; you will see that the g^-1 and the g are actually reversed (they are the wrong way around).

This matters, because the proof actually makes no mention of whether the group G,  is an Abelien group, or not.

An Abelian group is one where the group has the additional - commutative property - such that:

For all g1, g2 elements of G;    g1 o g2 = g2 o g1.

So, the order of the elements, before they are operated on, does not matter in an Abelian group.  In an Abelian group, the expression x = g y g^-1 and x = g^-1 y g, would be equivalent.

Hence, because our proof makes no mention of the group G, being Abelian, we can assume that we need a final stage to this proof, in order to complete it.  We need to 'swap the two conjugating elements back around.

This act of 'swapping around'  the conjugating elements  g and g^-1, makes the whole proof work for a non- Abelian group, as well as an Abelian group.

Thus,  we need this final step:

Since

 (g^-1)^-1 = g,   (this is basic manipulation of indices);

substituting this into our expression, we have

g^-1 y (g^-1)^-1 = x.

 Here, all we have done is restated the expression, in a slightly different way, but it is still essentially the same form of expression.

And finally, we have shown that g^-1 conjugates y to x, so y is conjugate to x.  It doesn't matter that the element conjugating is actually g^-1, since as long as we have the inverse element, (g^-1)^-1, on the right of the conjugated element, it all still makes sense.  All we ever  need, is at least one element of G, to conjugate x~y and y~x, in order for the group to be symmetric, in this way.  It matters not whether that element is g or g^-1 or  or if you want to go crazy, ((g^-1)^-1)^-1; or any other random element of G, for that matter.

Therefore, since y is conjugate to x and x is conjugate to y; the Group G has the symmetric property.

I'm sorry if my lack of LaTex editor made this post difficult to follow.  I will endeavour to fix that little problem, by next week.

So, by delving into this particular proof, in detail, I have been able to subtly shift my own understanding of Groups; by naturally combining an earlier concept (Abelian Groups), with the concepts of equivalence relations and conjugacy.

Neat!


  [Caveat: I am a student, rather than an authority on Group Theory; so if you notice an error in my reasoning  / preceeding proof; please let me know, so that I may correct it.]

A Group Theory diversion

I was re-reading unit GTB1 tonight, of the Open University course M208 Pure Mathematics.

I just thought I would note down something that I have noticed, but which I have only just realized, on reviewing the material, this year.

Just a bit of background; M208 was the first pure mathematics course that I completed, and part of it contained sections on Group Theory.  As any diligent group theorist would do, I began by learning the Axoims that lead to the definition of a 'Group', from a set of any given elements.

Now, one of those axioms is the property of associativity.  That is:

For all g1, g2, g3 elements of G,

g1 o (g2 o g3) = (g1 o g2) o g3.

(the numbers are supposed to be subscript, but my Latex editor is currently 'up the Shoot'.)

Whilst I was studying the course, I had assumed that a group probably needed three elements in a set, in order to meet this axiom's criteria.

I say I assumed this, however, to be honest, I had never actually given it any real thought.

So, on revision of this material in 'slow time'; this is the sort of question, that seems to be popping into my head.  A promising sign, in my eyes.

So, I glanced at the issue tonight and, of course, I quickly concluded that the axioms do not state that you need three elements in a set, in order to meet their requirements.

For example, if I have a set {1}; this could be a group under multiplication.  Why?  Well, it meets the axioms of Closure, Identity, Inverses and Associativity; where g1, g2 and g3 are all the set element {1}.  There are no axioms that say that the elements in a set must be different, or that the elements must be distinct.

Why?  Well, the axioms do not go into numbers of elements, or other topics, for that matter.  The way that they are written, means that they do not assume that there is more than one element or even that the elements actually exist.

Having said that the Axioms certainly do not state, that there should be a none existent element, or that there should be a group that exists, which contains less than three elements.

So, the axioms certainly don't prove that there is a set of less than three elements that can form a group.

Logical, but confusing, I think.

Pure Mathematics

Good lord, has it really been 8 weeks since my last post?  I've had a bit of a break from all things studying for the last two months, as I had to defer my O.U module, Astrophysics, whilst I recovered from some health issues.

Before I left, I managed a couple of TMA's and also some of the research elements; and whilst I am sad to leave this interesting subject, I can't help but feel that it is probably actually for the best; as I can now, truly concentrate, on my new found love affair with pure mathematics.  My degree profile will also probably look a a little more focussed, since it will contain all level 3 maths modules.

Also, in a strange twist of fate;  whilst I have just effectively added another year to my level 3 study schedule; I hadn't realized that by doing Astrophysics,  I would have been unable to complete Complex analysis, before moving onto the MSc.  This would have been quite foolhardy, in my opinion.  So, this could be a blessing in disguise.

It now means that I will take two pure maths modules in October, as planned (number theory and logic, with groups and geometry), but because I will now need to study more level 3 modules in 2014, I can now add complex analysis, to my quiver.  I had planned to self study the complex analysis books, that were kindly sent to my by Chris F; but I think that Chris was quite right when he suggested, some time ago, that any future sponsor for post-graduate maths research / work, would question why one did not have such an important module.

Anyway, the study break has given me a good opportunity to start slowly re-reading some of my O.U module Units from M208, Pure Mathematics, which I completed last year.  I have forgotten a fair bit, and there was some of it that I never entirely understood, the first time around (epsilon-delta,  I'm looking at you!)

So, I have a rather lazy summer ahead, with some hopefully enjoyable revision of pure maths.  I also plan to dust off my copies of  Hardy's - Course in Pure Mathematics; Brannan's - Geometry and Spivak's - Calculus (Which should really be called Real Analysis, judging by the content of the book).

A Question of Pure Mathematics

My mentor, Chris F, made a comment back on January 3rd, which I didn't get  around to answering, but it is an important question and one that helps to explain what has been driving me in my study experiment that 'kicked-off' this blog, a couple of years ago.

Chris's comment is below:

"Changes of direction are always likely as you find out more about the subject and what it can involve. You haven't really explained what it was that has given such an antipathy to Applied maths or physics especially as you seemed so enthusiastic at the start."

I don't think I am feeling antipathy, as such, towards science or applied maths.

It is much more a case of self-examination /understanding and a little bit of discovering what makes me tick.  And while it may sound a little trite, I really do, now, understand myself a bit better than when I first started.

From a pragmatic perspective, I look back at my premise for this experiment when I decided to pursue a PhD in physics.  It appeared to me, to be the most difficult of human pursuits and I wanted to see if I could stay the course;  and if I couldn't? I wanted to know at what point I dropped off.

The experiment progressed and I began to find that I wasn't enjoying the applied areas of mathematics, as much as areas such as number theory or analysis.  I don't know if it was the way that the Open University presented applied maths that started my dislike of the subject, or whether it is just the way that I am wired up? I find that if I can understand something axiomatically; from first principles, then I seem to better understand and enjoy the subject on a very fundamentally level.

I do think that there might be a little bit of being a 'control freak', that is causing my problems.  Meaning, that I generally struggle to accept something, and by definition, I tend to wrestle with it, if I am told just to 'accept' that the foundations that something is built on, are correct.

As an example, I know that a lot of differential equations just 'work' rather beautifully  and can be used to describe some of the most elegant of scientific ideas.  But there is a grumbling part of me, that doesn't like accepting some of these equations, because I haven't understood them from first principles.  For example, as I read through some of the MST209 maths units, I felt like I was being taught the odd tool to tackle certain types of differential equations; but only if they were of a certain type.

However, when I studied group theory, despite it being utterly frustrating when being asked to apply this maths to wallpaper patterns or polyhedra etc... I knew that I could just follow back through the unit's axioms and always come out the back of it, with a very clear understanding of why, and exactly how, it worked.

Please don't misunderstand me; this post is not about me having a go at applied maths, or in some way stating that it is inferior.  In many ways applied maths and science is clearly breathtaking.

My problem is, that I don't have the time (or the brains!) to go back to first principles in learning applied maths, that will provide me with a sufficient understanding of the mathematical background to allow my brain to accept and understand some of the tools and techniques that are taught.

Just to be clear; to try and go back to first principles in much of the applied maths and science that is needed to tackle real world problems, would be wholly unproductive, unless you were trying to understand the subject, for its own sake; rather than use it in a real and practical way.

I have to say, that I just don't think that I have the special type of abilities or intelligence that allows one to tackle, use and expand on applied maths and scientific principles.

Pure maths, I can do, I can understand (mostly) and I crave it, when I am not studying it.  It may not be much of an explanation, but it just 'feels' right, somehow.

It is this craving, that is probably going to be the biggest and most important factor in keeping me studying into and beyond postgraduate work.  Without it, I am surely doomed to failure?

So, as a reality check:

Do I now believe that I will be able to successfully work towards (and enjoy) postgraduate Physics studies?

Regrettably, no.

Do I believe that I will be able to successfully work towards (and enjoy) postgraduate studies in Pure Mathematics?

Absolutely, yes.

So then, comes the question about my blog.  I am not into revisionist practices that wipe away previous paths and dreams; and part of the experiment embodied within this blog, is in keeping a diary of my path regardless of which direction it takes.  So the blog will remain in its current format, and I will continue to contribute posts without much change in style and content.

I am a little worried that my blog précis and personal statement may confuse readers, as I begin to lean more towards pure mathematics, from October this year.

So I may remove a few words in the 'about me' section, to make my current goals a little clearer.  But I shan't be adding any new ones.

And lets not forget, that I still have seven months of astrophysics ahead of me for which I will still need those Jedi powers, to keep me on track!

First Taste of Real Scientific Research

Okay, TMA02 for my current course Astrophysics, was successfully completed and pinged to my tutor via the enigma, that is, the electronic TMA service with the Open University.

Actually, I like the fact that I can just upload my coursework as an electronic file and send it, without having the last-minute scramble to the post office, several days before the cut off date.  Life is difficult enough, without having to give up 48-72hrs of prep time to the damn, inefficient and utterly unreliable Royal Mail service; so sending my work as 'naughts and ones', via the phone-line is much easier and a lot less stressful.

I think that one of my better decisions in the recent-past, was to anticipate having to send in TMA's as an electronic document, by starting to learn and use Latex in my coursework.  I started doing so as early as my first level 1 maths course MST121, Using Mathematics.  It wasn't necessarily needed; and I believe that I was in a minority, doing so.  However, taking so much time to write out my answers using LaTex, at such an early stages, allowed me to practice and become adept at using the system; thus, writing level 3 physics TMA's and research or post graduate  mathematics scripts, has not become prohibitively slow and cumbersome.

It's worked fairly well, as I've now committed many of the keyboard short-cuts to long term memory, and I can 'knock out' an in-line formula in seconds, rather than tens of minutes or hours.  Any advantage at this level, needs to be grasped firmly, as the study material is difficult enough!

I say LaTex; but actually, I cheat by using Mathtype which is a 'what you see is what you get' equation editor; but it is fantastic and does a really professional job.  I can't recommend it highly enough.

Anyway, having completed TMA02, I now have my sights rising to meet the next important step in my scientific journey this year.  I am due to complete my first piece of real research within astrophysics.

I don't know all of the details yet; but I do know that I will be crunching large amounts of astrophysical data that have been collected from the Sloan Digital Sky Survey (SDSS) archive.  I will then use this data and proceed to devise a project concerning the optical spectroscopy of previously unstudied quasars; a prospect that will make my recent hard slog and study  of stellar evolution and nucleosynthesis, well worth the effort.

Been Sick, But Back on Track

I'm a month or two older, since my last blog post and my daughter tells me that I have 6 new grey hairs too.  Well, I've had one of those times in the last couple of months, where a lingering winter virus has caused me to spend several weeks with a headache and not much motivation.

Having said that, I have continued with my S382 Astrophysics studies with the OU.

Chapter one was a challenge for me because I haven't done any University level 'sciency-stuff' for a long time and because my recent background has been firmly planted in pure mathematics, I have struggled with silly things, such as working out what units go with what quantities.  Is it banana-lengths per second, or Joules per gnats-whisker?  Who knows?

Anyway, In this last month, I have had one computer marked assignment and one TMA to write.  I managed 83% on the TMA, which was mainly based around working out energies involved in hydrogen burning stars, and the like.

The maths was very straight forward, but I did struggle with the descriptive essay style questions.  I stuck to what I knew and used as many formulas as I could find, to pad out my answers, which worked okay.  I didn't finish the last question due to time constraints, so I think I would have been on track for a +85%, had I finished it.

But, I didn't. However,I'm glad of a solid start.  I have finished 3/4 of the CMA, but I am struggling with it a little, as it is testing my non-existent basic science knowledge, which I am trying to dredge from my memory banks formed in the 1980's.  If only I had been more attentive in school!  But I wasn't.

Anyhow, this morning, I am working steadily on TMA02.  This is a liberal artists dream; lots of paragraphs, with the odd formula for decoration.  It is a very different TMA to the first one.  Verbose, yet devilishly difficult, in places.

I am about a week behind on reading the chapters, but I have a week to catch up over Easter, with ten days off work, so I am confident of keeping on track.

With regards to the debate that I was having with Duncan about 'why stars are hot'; I didn't manage to speak with a tutor about this matter; however, having studied the material, I can see that the stars get hot because they contract under gravity, which raises the pressure / kinetic energy of the stellar material, until it is hot enough to ignite hydrogen fusion.  Once the fusion begins, it acts to maintain that heat over the required lifetime of the star, until around 10% of the hydrogen has been burnt.  It then evolves into a different form, depending on its original starting mass.

So, I agree with Duncan, that the book was rather misleading as it didn't seem to put enough daylight between the terms 'hot enough to initiate fusion' and the subsequent maintenance of that heat/fusion over the lifetime of the star.

Astrophysics is difficult enough, without such supereminent elements of knowledge being shrouded in mild confusion by set texts.

I'll let you know how TMA2 goes.

Time Dilation

A quick update.  Due to some horrible time constraints caused by work; I have had to ditch S383 and just study S382 until October.

It's one of those risks that you tend to run, as a full time employee and a part-time student.  It is a transient situation, which I hope will be sorted out before October comes, as I am planning in two modules then.

I decided to stick with the Astrophysics over the Cosmology, simply because it appears to me, to hold the best hope of obtaining a grade 1 pass overall.  Also, it looks so damn interesting!

The only way that I could have held on to two modules this summer, would have been for me to find a way to make the rest of the world move at a velocity that was near to the speed of light, relative to my house.  I might then have stood a chance of finding the time.

If anyone reading this, doesn't understand the last paragraph; I now have a book for sale, that will bring you right up to speed.

Checking with the OU

In the interests of good scholarship, I will remove the previous post and ask an ou tutor to check through the points made, before re-posting it with any updates.

I think that as Duncan and I have begun to debate this subject, that we may have strayed from the original premise of 'why the sun is hot' and entered into a broader discussion of how that heat is sustained over a long period.

A discussion that has opened up complexities that depend on many different factors such as mass, temperature, luminosity, etc... and one in which I am not yet qualified to argue successfully.

Anyway, I will post any update, as soon as I have approached one of the course academics.

Why the Sun is Hot. Part II

Post under review.

Please check back soon for update.

Why the Sun is Hot (spoiler alert!)

Well, I never!

I have been left slightly flabbergasted this morning, since I have just had a lifelong belief, overturned by the course notes from my course S382, Astrophysics.

What do I mean?

Well, I had always assumed, nay, believed; that the reason why the sun and other stars, are so god-damn hot, was because of all that fusion energy going on within the star.  I mean, just get too close to a 1 megaton H-bomb as it detonates, and you would feel the effects of why that assumption might seem correct.

However, If one explores the equations that govern luminosity, and work out, from the proton-proton chain fusion reactions that occur at the core of the sun; you would see that for each square meter of nuclear material that is available to 'burn', it only produces approximately 300W of power per cubic metre.

The course notes use the example of imagining three 100W light bulbs in a cupboard, as an equivalent amount of energy release per unit volume.

So then, why is the sun so damn hot?

Well, it turns out that the fusion energy only really prevents the sun from collapsing in on itself uncontrollably, due to the energy released from this reaction which counters the gravitational energy from the mass of all that material.

The Sun is hot, simply because, it is so massive, that it has a mindbending amount of gravitational potential energy.  As all of this mass attracts itself and causes a contraction, the gravitational potential energy, is converted to kinetic energy.  And, fast moving particles, are very hot.

So, the main source of heat for all stars is caused by this conversion of energy from gravitational to kinetic.  The fusion reaction energy just seems to retard the gravitational collapse.

Well, it impressed me anyway....

From Newton to Lorentz, In Two Short Weeks...

Having now completed two weeks worth of OU study on Relativity and  Astrophysics; I have found, to my surprise, that the astrophysics course seems to have far less reading in the early stages, than the relativity course.

If I were to gauge them in terms of study time taken; I would say that on average Astrophysics (S382) seems to take about 6hrs per week for each of the first two chapters.  There were a few things that caught me out initially.  Firstly, I had to spend some additional time on being thorough when calculating sums and using scientific notation.  I  also got tied up with a small bit about nucleosynthesis during the proton-proton chain.  I felt that the question on that topic, didn't fully correlate with the preceding diagrams.

As for the relativity course, there are a lot of conceptual difficulties when one starts to delve into the algebraic manipulation of some of the Lorentz calculations. But, as with Pure mathematics, I found that a strategy of 'just keep reading' held me in good stead.  This allowed me to gloss over any difficult passages and then come back to them at the end of the chapter, with a much better 'big-picture' view, from which to tackle them.

The relativity course (S383) did seem to take me about 10hrs of study per week for the first tranche.  More than I expected, but then we have just covered the history and mathematics of motion, courtesy of Messrs  Newton, Galileo and  Lorentz. 

No small feat in two weeks!

The Relativistic Universe

A quick post.  I have delved into the first book of The Relativistic Universe (S383), from the Open University; and, well, I just can't put it down!

It is one of the most interesting, easy to understand and yet detailed enough presentations of Special Relativity, that I have seen so far (and I've thumbed a few books on the subject, in the last two months).

The book is typical O.U fare, with less dense text, examples with clearly defined answers, leaving very few jumps in Algebraic manipulation to cope with; and a clean look that is sometimes missing from traditional textbooks.

I  like the way that the coordinate system for Lorentz transformations is also presented; for example, using
ct (the speed of light in a vacuum multiplied by time), to allow the first coordinate of the Lorentz equation to be stated as a distance (since distance is velocity multiplied by time). It certainly makes things easier to handle when lugging about all of those values.

Also, the use of S and S' as the two inertial time frames in standard configuration, rather than the sometimes confusing use of A and B, in some texts, is a breath of fresh air and certainly allows one to better visualize a Lorentz transformation problem.

So far, so good.

The Smell of Fresh Books!

They arrived this morning.  I now have 9 months of Astrophysics and Cosmology to look forward to.

First impressions on flicking through the books?  Very doable and, dare I say it, they look rather enjoyable.  The cosmology course (The Relativistic Universe), starts off by dipping straight into Lorentz transformations and tensors.  However, I feel suitably prepared having now self-studied the Special Relativity section of S207 (The Physical World) and also a book on tensors! (Fleisch).

I have also finalized my study of the prep material for these two courses, that the O.U provides as a bridge, to those who may not have followed a traditional study route (like moi)!

I found it, for the large part, more wordy than I am use to, coming from a year of pure mathematics; but a welcome diversion, none-the-less.

The Astrophysics books actually look the more taxing of the two courses, which surprises me, since I was expecting the opposite, based on my little knowledge of the previous presentations of these two courses.

Anyway, tomorrow, I am dipping my toe into both courses.  I have the books, I might as well just get on with it!

Great.

Astrophysics and Cosmology

As promised, I made my decision today, regarding whether to take two courses starting in Feb, or just the one.

Well, I chose two; both the courses Astrophysics (S382) and The Relativistic Universe (S383).

They both start on the 2nd of Feb and each have different challenges, in their own right.  Having self studied the 'Book 0', prep material that the O.U provides as a brush up on the prerequisite knowledge for both of these courses; I have already noted some teeny-tiny elements of these courses, that will probably drive me insane, by October.

I suspect that they wouldn't bother most people; but combine my recent drilling in pure-mathematical rigour, my dyslexia and a general bit of miserable old git; these little issues are surely going to send me 'round the bend'.

I guess that the issue for me, is the rather unclear and nonsensical way that astrophysicists and cosmologists appear to throw around mathematical and unit notations with wild abandonment, not dissimilar to the experience that I had this afternoon, pouring my Alphabetti Spaghetti onto two pieces of burnt toast.

For example, in mathematics, we could use the letter  r   for the radius of a circle.  But if you are a 'telescope-anorak', (as my wife has now taken to calling me! hmm), then you will have created your own symbols, just for the hell of it.

Hey, why not let  d  be the radius of an imaginary sphere when working out how bright a star appears in space?  Or, better still, lets use the symbol R on the same page, to describe the radius of that star.

This is a heady world where F is flux, not force; m is magnitude, not mass and  i  is an angle of inclination; which has nothing to do with complex numbers, I can assure you.

I think it's going to take some 'parrot-fashion' wrote learning, for me to grasp all of these different uses of the same symbols, within the same texts.

So, if you ever hear on the news, of a crazy man being committed to an institution after being found walking around Birmingham town centre shouting, 'Come back Group Theory, all is forgiven!'

Then please, think of me ;-)