Category Archives: Maths

Statistics

End of Book 1 - Review of Understanding

A quick look at the mind map I produced for the whole of Book 1, very much a foundation for the rest of the materials on the module.

The complexity has increased necessarily, but I'd like to think the clutter has been reduced to a minimum. I was rearranging nodes as I was adding them in the hope of reducing clutter. You'll notice there are two white floating boxes not connected to anything externally. This was just to avoid clutter.

Some things to note about it that help me refer back to it:

It's split roughly into thirds, vertically. Each third is more or less a sub topic. More like themes, perhaps. Far left is foundational principles. Middle relates to basics of distributions. Far right is concepts surrounding the C.D.F. and P.D.F. (the definitions of which are in the white box in the middle of the far-right section).

Colouring helps a lot when needing to refer back to it, I found. Axioms are pink, properties are green, and definitions are blue. I found that in referring back to it, I needed some kind of differentiation between the definition of a certain distribution, and a normal definition. So all the yellow nodes you see are definitions of distributions.

In summary, to assist my understanding, I now have a graph that leads me from the most basics concept (like what an "event" is), to the definition of the standard normal distribution. I'll be referring back to this as I go!

Calculus, Statistics

Integration Revision

Putting these in a safe place for later. Found these to be a really good revision questions for integration.

\int^{\infty}_{0}x^{2}\lambda e^{-\lambda x}\:dx=\frac{2}{\lambda^{2}}

and

\int^{x}_{0}4xe^{-2x^{2}}\:dx=1-e^{-2x^{2}}

For context, the first one formed part of a question that required me to find the variance of a random variable, and the second one involved having to find the c.d.f. from a p.d.f.

(for my own reference, this was Book 1, Activity 5.2, p. 54 and Activity 6.1, p.62)

General, Maths, Statistics

Current Application of Mind Maps

I've finally completed the second section of the first book, and it's served as a good revision/introduction session. I'm glad I went to the trouble of creating a mind map for the main concepts that were covered. Here's how my mind map grew as I added to it:

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I've used this new mind map to refer back to several times during the exercises I've been set, and even the first assignment question. Admittedly it takes longer to get through all the learning material, as you have to take time out to contribute to the mind map as you go. Though I would say the time spent is worth it.

I also felt that this last image was the maximum size a mind map could take without becoming too cluttered. For the next section of the book, I'll be starting a brand new mind map. I'll try to keep going with this method for as long as I can, but it seems it's very useful.

Statistics

Module Number 7!

Summer is over!

Autumn is here!

That can only mean the new academic year is about to begin! (and the first module of the third and final stage of my degree!)

Up next? Applications of Probability (M343)!

I'm excited about starting this. When I started on this degree, I saw that there weren't many options for stats modules in the final stage unless you were specifically aiming toward a BSc in Statistics. I've read a number of articles discussing how there isn't enough stats in education at the moment and I'm inclined to agree. I'm hoping this will round my skills out nicely.

For all the info on this next module, check out the official page here.

The module website for students is due to open tomorrow, so I'll get access to all my materials and be allocated a tutor. What makes this module a little bit different is that it's a single credit module. The last two modules I've had (mathematical modelling and pure mathematics) were double-credit. So I'm wondering whether the workload will feel really light or not. I'm hoping it will be lighter, so I can contribute on here more frequently than for the last two!

General, Maths

The London Mathematical Society

In November '18, I decided to join the London Mathematical Society (LMS). Last month I attended a meeting there in Russell Square in Central London which involved a couple of lectures and a "swearing in" of new members.

As a part of the swearing in, new members have the opportunity to sign the LMS Member's Book. Interesting thing about this is it dates back to 1865(!). So after signing it, I flicked back the pages to find the signatures of both Arthur Cayley on June 19th 1865:

and James Clerk Maxwell on April 25th 1867:

Maths, Mechanics

A Lack of Updates

Study has been so busy this year, I've failed in keeping an up-to-date account of how it's been going. Here's a very quick run-though:

Assignment 2:

  • First and second-order differential equations.
  • Vector algebra and statics. (There are two blocks connected with taut string on a ramp and nothing is moving, what are the forces?)
  • Dynamics. (A block is sliding down a ramp with a certain friction, what are the forces?)

Assignment 3:

  • Matrices and determinants.
  • Eigenvalues and eigenvectors.
  • Systems of linear differential equations. (solving simultaneous differential equations).
  • Functions of several variables. (

        \[f(x,y)\]

    and

        \[f^{\prime}(x,y)\]

    instead of

        \[f(x)\]

    and

        \[f^{\prime}(x)\]

    ).

Assignment 4:

  • Mathematical modelling. (Introduction to writing a mathematical modelling paper).
  • Oscillations and energy. (Forces in a spring system, and potential energy).
  • Forcing, damping an resonance. (Forces in a system of springs and dampers).
  • Normal modes. (Oscillations of particles in a spring system).

Assignment 5:

  • Systems of differential equations. (Equilibrium points of two differential equations and use of the Jacobian matrix).
  • Fourier series.
  • Partial differential equations.

Assignment 6:

  • Vector calculus. (Scalar fields and gradients).
  • Further vector calculus. (Conservative vector fields, curl, and divergence).
  • Multiple integrals. (Area and volume integrals).

Assignment 7:

  • Writing a 3000-word mathematical modelling paper.

All the above are now complete, and I'm currently working through the last few units. Namely:

  • Systems of particles.
  • Circular motion.
  • Rotating bodies and angular momentum.

There's a requirement for me to submit one assignment per month, from October to May. These double-credit modules are hectic...

Out of all of the above, Assignment 7 had to be the most nerve-racking. Most assignments normally read like exam papers, though for Assignment 7 I had to write a report. So not only had I never written a report like this on this course before, I'd never written a report like this ever! Results are due in a couple of weeks, so based on the outcome I'll be keen to break down where I went wrong.

Favourite bits from the above have to include my first ever hand-calculation of a Fourier series in Assignment 5, and a question where I had to prove a result of Archimedes (287-212 BCE) regarding relative volumes, using modern calculus in Assignment 6. I'd also always wanted to know more about the construction of a Jacobian, so Assignment 5 was good for applying my new knowledge.

Automata Theory, Proofs, Pure Maths

Pumping Lemma

Way way off the beaten path here, but this is the best example of usage of the pumping lemma I've seen. Just need somewhere to put it... The below is taken from here. Theorem: Let L be a regular language, and w be a string. Then there exists a constant c s.t. \forall w \in L, |w| \geq c. We can break w into three strings, w=xyz, s.t.:
  • |y| > 0
  • |xy| \leq c
  • \forall k \geq 0, xy^{k}z \in L
Method to prove that a language L is not regular:
  • At first, we have to assume that L is regular.
  • So, the pumping lemma should hold for L.
  • Use the pumping lemma to obtain a contradiction:
  • Select w s.t. |w| \geq c.
  • Select y s.t. |y| \geq 1.
  • Select x s.t. |xy| \leq c
  • Assign the remaining string to z.
  • Select k s.t. the resulting string is not in L.
  Problem: Prove that L=\{a^{i}b^{i} | i \geq 0\} is not regular. Solution:
  • At first, we assume that L is regular and n is the number of states.
  • Let w=a^{n}b^{n}. Thus |w|=2n\geq n.
  • By the pumping lemma, let w=xyz, where |xy| \leq n.
  • Let x=a^{p}, y=a^{q}, and z=a^{r}b^{n}, where p+q+r=n, p \neq 0, q \neq 0, r \neq 0. Thusly |y| \neq 0.
  • Let k=2. Then xy^{2}z=a^{p}a^{2q}a^{r}b^{n}.
  • Number of a\text{'s}=(p+2q+r) = (p+q+r)+q=n+q.
  • Hence, xy^{2}z=a^{n+q}b^{n}. Since q\neq 0, xy^{2}z is not of the form a^{n}b^{n}!!!!!!!!!!!!!!!
  • Thus, xy^{2}z \notin L. Hence L is not regular.
Maths, Mechanics

Unit 1 COMPLETE!

My first unit is done! I managed to find time to complete the first assignment too. This is timely, as the course officially starts tomorrow.

This unit was interesting. Looking at the title "differential equations" I thought "Oh, okay, fine. I've done this before". Then opening up the book, I realised "Oh wait. I haven't quite done THIS before". That's when the fear crept in... how much of a learning curve would I be facing in this new unit?!

Forging through, I had to concede defeat on one or two examples, but I came out the other end confident about all the material I'd covered. Only now I've completed the assignment do I feed confident about my ability on the subject matter, but it's also one of those areas where I need to keep up the practise. 

All too often, it's the first unit that you return to at revision time and think "what IS all this?!". It's been hinted at that themes of second order differential equations reoccur throughout the module so I'm hoping I'll stay in practise. 

So I'll give the assignment a final proof-read and then submit it tomorrow on the first official day of the module. Good start.

Maths, Mechanics

Book Arrival

My books finally arrived.

I've got nine months to get all of that inside my brain. -but then I will also need to be able to write answers to questions on the subject matter at close to the speed of sound by the time of the exam...