Category Archives: Maths

Chapter 24 of 24 - Fractals

Finally reached the end of the materials.

First section introduced some examples of fractals. Second section was "Using the Contraction Mapping Theorem" which introduced the notion of distances between compact sets, ultimately the "Hausdorff Distance", and the notion of contraction mappings that kind of translate your base shape down to the next iteration of the fractal. Last section was on dimensions of fractals, which enforced a lot of logarithm revision.

Again, this section was merciful when compared to the average difficulty of some of these chapters.

Suddenly I'm putting the finishing touches on my last ever written assignment. Nearing the summit of Maths Mountain.

From here on out it's mostly revision. Going over past assignments and preparing for my last ever maths exam. Let's see what useful little bits of information I can find and document over the next month or so...

Chapter 23 of 24 - Completeness

Loving the irony that the penultimate chapter is on "completeness".

First two sections covered completeness of metrics as defined by sequences, including examples of basic well-known metrics that are complete.

Third section was on the contraction mapping theorem, which seemed fairly profound. It allows you to show that generalised differentials or integrals have unique solutions. I needed to speed through this section, and I wonder whether a review of past papers will reveal that I need to spend more time on this. Like a lot of this, as some of the answers are so lengthy, I do wonder how feasible it is that any of it will be in the exam.

The last section covered methods by which you can find a completion of a metric. This presented a framework that was really interesting, but offered no examples. There was even an admission that the process they'd presented was lengthy and with examples and not mention of it in the assignment, I can only guess it too is not covered in the exam.

So. One quick review of the ongoing assignment and I'll finally be on to my last chapter. Ever.

Chapter 22 of 24 - Compactness

The first section here was a stern-sounding "Two Important Theorems". It covered the boundedness theorem and the extreme value theorem, both I've covered before in previous pure maths modules, but this related them directly to metric spaces.

Second section decided to introduce compactness through the concept of sequential compactness, which essentially describes the "d-closed and bounded"-ness of a set. Though is inevitably difficult to learn quickly because of strange concepts involving sequences in metric spaces.

Third was "functions and sequential compactness", which was even more strange as it extended section two to also cover all functions in C[0,1], and various hard-to-understand \epsilon-\delta theorems.

Section four was "compact metric spaces" and the notion of covers and sub covers. Far easier to understand, but depending on the set it can be tricky (for example) to find a finite sub cover from an infinite cover. With a bit of practise you get the trick though.

Section five was how sequential compactness and compactness are equivalent for metric spaces, and the last section covered unions and intersections of compact spaces. Again, these sections I sped through as section 5 was largely theorems, corollaries and lemmas, and section 6 was largely unexamined.

Chapter 21 of 24 - Connectedness

First section here was on homeomorphisms, which is essentially a brief intro in the foundations of topology. I've been wanting to study topology for years, but as is the trend, I didn't have time to dwell. Again, we don't get examined on topology, so this will be something I'd want to come back to after my final exam.

Next up was "closed and open sets revisited", which is practically exactly what you expect.

The third section introduced the concept to connectedness properly, and the fourth went on to explore connectedness in Euclidean space.

At this point, I'd learned enough to tackle the assignment questions. I didn't read through the last two chapters, "path-connected spaces" and "the topologists cosine", but I didn't feel too guilty about this. I'm already vaguely familiar with path-connected spaces, and the last section seemed to be unlikely to be examined. I'll review both of these sections during my revision period though.

All in all, this chapter was mercifully easy to understand.

Chapter 20 of 24 - Public-Key Cryptography

Well that's a pretty sexy title isn't it. Shame I'm not being examined on any of it.

I got half-way through the first section on "cryptosystems based on modular arithmetic" before I decided to pick my battles wisely and return to chapter 19. As a consequence I never got to the meaty bit of the material. -namely encryption and decryption in the RSA cryptosystem.

The second section that I got nowhere near was "cryptosystems based on elliptic curves". This section looked difficult and lengthy, so I'm glad I discovered it wasn't examined before I reached it.

I'll keep this chapter in the back of my mind in case I want to revisit it at any point in the future (after my exam).

So, this brings me to the end of Book E on rings and fields. Normally at this point, I would've finished the related assignment by now, but I'm still very much stuck on the question related to splitting fields. I'll spend the next day or so trying to make as much sense as I can of it to pull out a few extra marks.

Next up is the last book, Book F, and the last four chapters on metric spaces. Home stretch...

Chapter 19 of 24 - Fields and Geometry

Initially I had sped through this section, not paid much attention to it and moved straight onto chapter 20. See, my assignment seems to give me a choice of whether to answer a question on fields and geometry from chapter 19, or to answer a question on cryptography from chapter 20. Cryptography sounded far more interesting (and something you could easily apply in code!), so I kind of skipped chapter 19 entirely.

I got half-way thought chapter 20 and found out online that nothing from chapter 20 will actually be in the exam. The exam contributes to 80% of my final grade for this module, so picking my battles, I dropped chapter 20 and returned to chapter 19.

In an apparent change of pace, chapter 19 was laid out into just two sections. The first was revisiting field extensions, introducing some more nuanced facts about them. The second was an introduction to ruler and compass constructions, providing proofs of why you can't trisect the angle \frac{\pi}{3} and why you can't square the circle. It was interesting, but very involved. Especially when you're left to your own devices trying to prove why other geometric constructions might not be possible.

Now for a very quick write-up of chapter 20... :/

Chapter 18 of 24 - Fields and Polynomials

Chapter 18.

Oh, Chapter 18.

Easily the most challenging, exhausting, and the longest chapter yet.

Firstly preliminaries were covered, following by a very lengthy introduction to field extensions. This second section on field extensions was extremely frustrating as it's the only one so far that has required extra reading outside of the materials I was given. Next up was finite fields (very similar to the previous except not related to infinite fields). Then the fourth and last section was on splitting fields, which is again frustrating as the related question in the assignment seems to use language that isn't included in this section.

All in all a bit of a battle, and I still haven't been able to start the assignment question on splitting fields.

Once again, I'm running out of time, so I'll have to move onto chapter 19 and come back to this later...

Chapter 17 of 24 - Rings and Homomorphisms

We're hopping around a bit now. Book E is back to rings and fields.

I'm falling behind rather severely with the study at the moment. Needless to say, the panic plus the speed with which I needed to work through this chapter meant that a lot of the information in this chapter didn't settle in my brain in the way I needed it to.

First section was on fields of fractions (which I'm still not terribly clear on). Second section was introducing the concept of an ideal of a ring, followed by operations on ideals. Lastly, the fourth section was on quotients and was the one that introduced homomorphisms between rings, and explained how images and kernels relate to ideals of rings.

Okay. Calm the panic.

Very very quickly jumping on to the next section...

Chapter 16 of 24 - Open and Closed Sets

Sections in this chapter started with closed sets and open sets, which sound fairly straight-forward, but really they covered a formalisation of how sets are closed or open under a distance metric. eg, a set in two dimensions may be closed conventionally, but may not be closed under an unusual distance metric.

Next up were closures, interiors, boundaries, and the size of sets. Again, normally this would be fairly simple. -less so when looking at these in terms of the distance metric.

This chapter also included a little bonus seventh section introducing topology which looked great (and I've always wanted to learn more about) but unfortunately this section was not assessed, and I didn't have the time for extra reading. It seems this degree isn't just about learning maths, but specifically about learning maths fast.

So that brings Book D to a close!

Chapter 15 of 24 - Metric Spaces and Continuity 2

Another brutal chapter. Brain was pushed to the limits with this one.

First chapter was "new metrics from old", about how sets and distances can be combined to form new metrics. Second was "the Cantor metric" (Cantor space, just being what is essentially a binary tree). Next was "equivalent metrics", that involved a lot of inequalities. Fourth was "spaces of functions" which covered sequences of functions and their limits (different from sequences and limits). Lastly was "the max metric on C[0,1]". Conceptually, this was a jump as C[0,1] is actually the set of all continuous functions that map from [0,1] to \mathbb{R}. Had some really tricky questions in that last section. The majority of my brain power went into understanding how to prove that the function you apply to functions when you integrate is continuous. Exhausted.

On to the next chapter.