Category Archives: Maths

Chapter 4 of 24 - Fermat's and Wilson's Theorems

Okay. Another chapter down. Very much "congruence continued".

Sections I covered were Fermat's Little Theorem, representation of fractions by decimals, Wilson's Theorem, and polynomial congruences. On top of that, I've managed to complete my first assignment based on these first four chapters. Reviewing that feedback will be very interesting.

One thing I have noticed is that the materials in the book are far more challenging than the questions in the assignment. I wonder if I'm being lured in to a false sense of security here. I suppose I'll only know when I review the past exam papers for this module...

Can't quite believe I've reached this point as the next chapter is "Examples of groups". Hello group theory my old friend!

Chapter 3 of 24 - Congruence

I've worked with congruences before, but this chapter was a lot more fun than I thought it would be. Introduction to properties, classic divisibility tests and simultaneous linear congruences.

Questions in this chapter seemed a lot more achievable. Though looking at the assignment, the related question seemed completely out of my reach. It was that feeling of fear you normally get in an exam when you see a question you don't understand and your brain shuts down. I've got some work to do in this area, as I feel I've been experiencing that feeling fairly consistently in the last two exams I've had (stats and complex analysis). -both of which have been covid lockdown at-home exams.

What was good though was in this chapter there was some "highest common factor" questions thrown in there, so that was really useful. Good for building confidence.

It seems there's more congruence work in the next section, "Fermat's and Wilson's Theorems".

Chapter 2 of 24 - Prime Numbers

Right. Another very tough chapter complete.

Well I say complete. I did have to leave out a number of the exercises again. Of the answers to the exercises that I understand, the answers in most cases are inspired. There's simply no way I could ever make the logical bounds demonstrated in those answers.

I was so disturbed by the difficulty of the exercises in Chapter 1, I decided to consult my tutor about it. He replied explaining that a lot of the exercises are meant to teach you by you looking at the answers so you're kind of meant to get stuck. This made me feel a lot better, but now it's just down to the time I can spend reviewing and learning from each answer.

But yes. Prime numbers. Fun chapter! Covered an intro to The Primes, the prime decomposition of integers, the infinitude of primes, famous problems concerning primes, and Fibonacci numbers. Fibonacci numbers are far more interesting than I ever realised, it turns out.

Okay. Time to move on to Congruence...

Chapter 1 of 24 - Foundations

Well 2021 has gone quickly hasn't it?

The workload in complex analysis became so large that I couldn't contribute to this blog for the rest of the academic year. This is frustrating as I felt there were some really important things to learn from my marked assignments. If I ever find time to write them up, I will do.

Though needless to say, it's suddenly October. My new (and final) module begins! M303 Further Pure Mathematics. This will be a big one. It's a final-stage double-credit module and contains an absolutely enormous amount of material.

Last week I received an email from my tutor that could be summarised to two words: "GET AHEAD". Of course that's when the fear struck, so I decided to be tactical with the first chapter. There were two sets of exercises I didn't attempt, because I realised I knew enough about the material to answer the assignment questions. However practically all of the exercises I did try, I couldn't complete. At this early stage, I'm perhaps feeling it might've been a mistake to choose this module. Having said that, I have been able to answer the first two questions of the assignment.

The first chapter was about number theory. Proof by Mathematical Induction, highest common factors, lowest common multiples, the Euclidean Algorithm, and Diophantine equations. Not only all that, but all the theories, definitions, propositions and lemmas that go along with them.

I feel this was so difficult because you need to use all the theories and definitions like a palette of different paints. In the same way as the artist it feels like the mathematician needs to use the theories and definitions to paint a picture. Problem being, it felt like I was being taught what the primary colours were for the first time. Still, we've moved on, and I've marked the exercises I need to return to. I'm hoping that down the line I'll be able to return to them and they'll make a bit more sense to me.

Deep breaths.

Back in the saddle.

Next chapter? Prime numbers...

Complex Analysis Assignment 1

My first complex analysis assignment has been marked and returned. I don't think I've ever felt the urge quite so much to learn from my mistakes.

Consequently there has been quite a lot of post-assignment learning... :/

This assignment featured a very brief introduction to complex numbers as a refresher, then broadly covered complex functions, the concept of continuity and complex differentiation.

So in no particular order, below are some notes on mistakes I made and how I could've avoided them! There's a lot to reflect on here...


Read questions carefully. One of the first very simple questions read "express z in polar form and determine all fourth roots". I did the second bit, but not the first.


I feel this is a bit "Complex Numbers 101", but the square root sign is defined as the principal square root (of a complex number), i.e. there's no need to calculate the second root.


If you're using the triangle inequality, state it specifically.


Again, this is fairly "Complex Numbers 101", but the polar form of a complex number isn't just a cosine function as the real part, and a sine function as the imaginary part. The arguments to both functions must be identical to qualify as "polar form". ie, you should be able to write the complex number as an exponential form.


Top tip: Be mindful about using identities. In complex analysis there are loads of them and they help a great deal.


When working out the inverse of a complex function, it's important to use your common sense. Part of one inverse I'd calculated had a square root in it. Just by looking at that, you know it could never produce a unique answer (it isn't a one-to-one function).


For another, I had to find the inverse of \text{Log}(3z) and the domain of that inverse. I got this spectacularly wrong. I'd written: given w=\text{Log}(3z), hence z=e^{3w}.

Trick here was to exponentiate each side, leading to e^{w}=3z. But the domain of the inverse isn't affected by the "3" above, the image set of the original function is still \{z: -\pi <\text{Im}z \leq \pi\}.


Some complex functions are very very different to their real equivalents. Case in point: \text{cosh}(x)\neq 0 , \forall x \in \mathbb{R}, but \exists\: z \in \mathbb{C}\: \text{s.t.}\: \text{cosh}(z)=0. Which leads to the next note:


If \text{cosh}(z) is the divisor in a complex quotient, you need to show that it's only 0 for values outside of the given range of the equation (eg |z|<1).


For one question, I had to prove that f(z)=z^{i},\:\: (\text{Re}\:z>0) was continuous. I thought this was easy.

z^{\alpha},\: \alpha \in \mathbb{C} is a basic continuous function on \mathbb{C}-\{x\in\mathbb{R} : x \leq 0\}. So if you let \alpha=i, then f(z) is continuous, right?

Not quite. I had entirely forgotten to state that the given set (\text{Re}\:z>0) is a subset of the set I gave: \mathbb{C}-\{x\in\mathbb{R} : x \leq 0\}.

The answer can appear obvious sometimes, but you have to keep your answer rigorous, otherwise you risk losing half marks or whole marks here and there.


Note:
z^{\alpha} = e^{\alpha Log(z)}
z^{\alpha} \neq e^{z Log(\alpha)}
🙁


For one question I had to prove whether a set was a region or not. For reference, a region is a non-empty, connected, open subset of \mathbb{C}. In the usual manner, if you can prove that any of those three properties don't hold then you've managed to prove that your set isn't a region. Easy.

I realised I could prove a set was closed, and hence not a region. Turns out this was incorrect. A set being "closed" and a set being "not open" hold two completely different definitions, and are seen as different things. I was meant to show it was "not open" as opposed to showing it was "closed".

In other words, mathematically:

Closed is not the same as not-open.
Closed is not the opposite of open.
Not-open is the opposite of open.


Again, here I needed to provide a proof based on the properties of various objects. Given a set that was compact (closed and bounded), I needed to prove that a function f was bounded on that set.

The Boundedness Theorem states that if a function is continuous on a compact set, then that function is bounded on that set.

The function was: f(z) = \frac{1}{7z^{7}-1}

I proved that the given function was continuous on it's domain, but I'd failed to prove it was continuous on the set. Here, I needed to show where the function was undefined, THEN show that those points at which it was undefined all lay outside of the set. So there was quite a lot of work I missed out from this answer.


My simultaneous requirement for the Cauchy-Riemann theorem, AND the Cauchy-Riemann Converse theorem within a proof ended up not flowing very well logically. Once again, I'd jumped ahead with my logic. As soon as I had seen something obvious, I felt the urge to state it immediately.

The Cauchy-Riemann theorem proves that a function is not differentiable at certain points. The Converse theorem then proves that a function IS differentiable on certain points. After using the Cauchy-Riemann theorem, it was extremely obvious where the function was differentiable, so I stated it. Then, as a matter of course, plodded through the Converse theorem to prove it. Complete lack of discipline! 🙂

Complex Functions: Domains, Image Sets and Inverses

I can imagine having to refer to these notes regularly, so I'm putting them here!

Image Sets

  1. State the domain A of f(z)=w.
  2. Rearrange so w is a function of z (to discover the condition under which w remains valid.

e.g., for f(z)=\frac{1}{z-1}:

    \begin{align*} f(A) =&\: \left\{ \frac{1}{z-1}\::\:z\in\mathbb{C}-\{1\}\right\} \\ f(A) =&\: \left\{w=\frac{1}{z-1}\::\:z\neq 1\right\} \\ f(A) =&\: \left\{w\::\: z=\frac{1}{w}+1\::\:z\neq 1\right\} \\ f(A) =&\: \{w\::\: w\neq 0\} \\ f(A) =&\: \mathbb{C}-\{0\} \\ \end{align*}


Domain of Combined Functions

Domain of combined functions are the intersection (A\cap B) of the domains of all component functions and that of the combined function. e.g:

f(z)=\frac{z-1}{z}\:,\:\: z\in\mathbb{Z}-\{0\}

g(z)=\frac{z}{z-1}\:,\:\: z\in\mathbb{Z}-\{1\}

\frac{f\left(z\right)}{g\left(z\right)} = \frac{z^{2}-2z+1}{z^{2}} \:,\:\: z\in\mathbb{Z}-\{0,\:1\}


Domain of Composite Functions

For f and g with domains A and B respectively, the domain of g\circ f is:

A-\{z\::\: f(z)\: \notin\: B\}

e.g., for:

f(z)=\frac{z-1}{z}\:,\:\: z\in\mathbb{C}-\{0\}

g(z)=\frac{z}{z-1}\:,\:\: z\in\mathbb{C}-\{1\}

\text{domain of }f\circ g = \text{domain of }g - \{z\::\:\frac{z}{z-1} \:\notin\: \mathbb{C}-\{0\}\}

\text{domain of }f\circ g = (\mathbb{C}-\{1\} ) - \{z\::\:\frac{z}{z-1} =0\}

\text{domain of }f\circ g = (\mathbb{C}-\{1\} ) - \{0\}

\text{domain of }f\circ g = (\mathbb{C}-\{0,\:1\} )


Inverses

  1. Determine image set of f(z)=w.
  2. Invert f(z) to find a unique z in the domain of f.

For f(z)=\frac{1}{z-1}

f(A) = \{\frac{1}{z-1}\::\:z\in\mathbb{C}-\{1\}\}

f(A) = \{w=\frac{1}{z-1}\::\:z\:\neq\: 1\}

f(A) = \{w\::\: z=\frac{1}{w}+1\:\neq\: 1\}

f(A) = \{w\::\: w\:\neq\: 0\}

f(A) = \mathbb{C}-\{0\}

(all same as above for finding an image set)

z=\frac{1}{w}+1 gives a unique soluition in \mathbb{C}-\{0\}, hence f has a unique inverse rule:

f^{-1}(z)=\frac{1}{w}+1\:,\:\:\:z\in\mathbb{C}-\{0\}

A New Year of Study Begins!

September has come around already!

My virology work placement came and went so quickly. Great experience. I was certainly not expecting it to be such a creative process. Or should I say "necessarily creative process". It seems with research like this, you really do need to be mindful of other results popping out of your work. If the tangent appears to be more important/meaningful than the original work, then it's best to follow it!

Of course this creative approach gives rise to a bit of a problem. The project has the potential to meander. I suppose this would be fine on a longer time scale, but for my all-too-brief 8 weeks it meant I wasn't able to neatly draw a line under it by the end of the placement.

Though having said that, the work that I was looking at wasn't time-dependant so I still have however long I want to complete the project in my spare time. After this next year of study I'd love to return to it.

Speaking which... my next year of study has started! I won't be informed of my assigned tutor for about another month, but I have my learning materials. So learning has begun! This year it's complex analysis. Very excited to be looking at this subject, and to be writing up the areas I have difficulty with on here.

(Side note: Wow, I've been writing on this blog for 5 years?!)

A Brief Change

Normally I'd take a well-earned break from mathematics during the Summer. Recharge for my next module that starts up in September.

Not this year!

This year, I've managed to take a short career break from my normal job to work as a work placement student in mathematics research!

So for eight weeks I'll be getting a taste of real life mathematics research! I've been lucky enough to be accepted into the mathematical biology research group at the University of York. Specifically, I'll be looking at mathematical virology, but the relevance to the current times is purely by chance: I first started arranging this placement about a year ago.

In my placement I'll be using group theory and linear algebra to produce predictions of virus structure.

It seems that all viruses appear to have the same symmetry as an icosahedron. But it turns out that you can find more icosahedral symmetry by translating an icosahedron along its axes of symmetry to create a larger non-crystallographic structure. When you do this according to strict rules, it turns out that you can start to predict overall virus structure. You can predict not just what it looks like on the outside, but what it may look like inside too.

I'm very early on in the position, but it's already fascinating. I'll be updating here when I can about how I get on with the experience.

Exams

I had my statistics exam at the beginning of the week.

This exam was weirder than most. Because of the pandemic, the OU had decided to turn the usual three-hour sit down exam into an "end-of-module" assignment that could be done at home within a period of 24 hours.

As soon as I heard this news, I had mostly negative feelings. The exam at the end of a 9-month module is a chance to really show off what you've learned. In three hours you have to recall, at speed, a very large assortment of problem solving skills and take full advantage of nurtured intuition. Some people can just stroll into an exam and do well, but I need to work very hard to walk into that exam hall with any confidence.  Preparing for these exams for me is like training for a marathon, or a mountain climb. It's exhausting.

I start attempting past papers under exam conditions on Saturdays and Sundays four to five weeks before exam day. Then the week before the exam I take a whole week off work to spend practically ten days straight doing past papers, marking them harshly, then reviewing them and revising further.

By the time I arrive at the exam in the exam hall in June, those three hours feel exactly like I'm running that marathon or climbing that mountain I've been training for.

Walking out of an exam realising like you were prepared and knowing it's all over is an enormous feeling. The final punctuation of nine months hard work.

Hearing that that wasn't happening this year was a let down. I'd be denied completing my marathon.

Though despite the fact that the "exam" was to be completed at home, I trained just the same. To the point where I felt I couldn't have been more prepared. I was comfortable and determined to complete the at-home exam in three hours regardless of how long I was given.

However.

On the day, I downloaded the exam pdf. I scrolled through it. And I realised that they had changed the distribution of the questions in the sections just enough that I was not prepared for it in the way that I was hoping. For the past seven years of past papers, you could guarantee certain topics would appear.

Not here.

For the past seven years, you should guarantee that within each topic, you'd be given a certain set of sub-topics.

Not here.

Immediately I was glad that I was not running my marathon. If I had been, I would have had to be stretchered away from the starting line by medics.

That day was a battle. Over the entire course of the exam (which took way way longer than three hours) I thought "how was I not prepared for this?". It shook me, and it was the only thing I could think about.

Here, in the third and final stage of this degree, I may have found that there is something fundamentally wrong with the way I learn.

Being kind to myself, this was generally a hard exam. It was statistics, which by its nature is non-intuitive (a lot of people find it so anyway). I did think that this module was aimed at students studying an actual degree in (just) statistics, and that I probably didn't have the background knowledge that other stats students did. And as my mathematician friend has pointed out, it's unlikely it was hard for just me. If an exam is hard, it's generally hard for everyone.

So where do I go from here? It's difficult isn't it. Amongst those 500 or so pages I learn from, should I pay attention and make notes on 'the fleeting comments on page 274 that I never got tested on once and seemed insignificant'? ......Regardless, it seems my revision technique as it stands isn't sufficient.

Assuming I will be in an actual physical exam hall for three hours in June 2021 for my complex analysis exam, I need a better revision strategy.