Mathematical Modelling for Infectious Diseases – a course at UCT (19th-30th September 2016)
For anybody interested in the mathematics of infectious disease modelling, the following should be very interesting.
Convex functions
What I learnt in class today:
A convex function 
![f(\lambda x + (1-\lambda )y)\leq \lambda f(x)+(1-\lambda )f(y) \quad \forall x,y\in \mathbb R,\ \forall \lambda \in [0, 1].](http://s0.wp.com/latex.php?latex=f%28%5Clambda+x+%2B+%281-%5Clambda+%29y%29%5Cleq+%5Clambda+f%28x%29%2B%281-%5Clambda+%29f%28y%29+%5Cquad+%5Cforall+x%2Cy%5Cin+%5Cmathbb+R%2C%5C+%5Cforall+%5Clambda+%5Cin+%5B0%2C+1%5D.&bg=ffffff&fg=000&s=0 "f(\lambda x + (1-\lambda )y)\leq \lambda f(x)+(1-\lambda )f(y) \quad \forall x,y\in \mathbb R,\ \forall \lambda \in [0, 1].")
Thus, the shape of a convex function is like 
…
Integration by Parts – Lightbulb Education
Integration by Parts…
Integration By Substitution – Lightbulb Education
Integration By Substitution…
The mathematical equation that caused the banks to crash
The Black-Scholes model is a mathematical equation invented by Fischer Black and Myron Scholes that first appeared in their seminal paper of 1973 opening a new wave of selling and buying financial contracts. This economic formulation was well received and recognized to be effective by the financial community to the extent that it won Black and Scholes a noble price in 1997. However, on the 19th October, 1987 – The Black Monday, the world experienced a severe shock when the markets suddenly crushed bringing to light the flaws in the mighty celebrated Black-Scholes model. The number one mistake in the model was the assumption that a given contact could be priced at the same volatility level irrespective of the strike price – the price a contract owner had to pay at the expiry date of the contract. A more economic perspective is discussed here.…
Does knowledge of past years affect the Keynesian Beauty Contest result?
Last year I played the 2/3 numbers game, also called the Keynesian Beauty Contest with my first year maths class. The discussion can be found here: http://www.mathemafrica.org/?p=11143 I wanted to know if, telling my class the results from last year (including sketching for them the histogram of results), would change how they chose their numbers this year. Of course I can’t tell if it changed them, but what is fascinating is that either:
- Their guesses (if I didn’t tell them about the results from last year) would have been very different from those last year, or:
- They were completely unaffected by knowing what people did last year, which really means that they believed that the rest of the class would have been unaffected.
I plot here the results from the last three years and you can see how similar the results are, year on year.
You can see that the distributions are relatively similar, and the means are extremely close.…
Picturing volumes of revolution
One of the homework questions this week was the following:
Let
a) Sketch the region R and find its area.
b) Find the volume of the solid obtained by rotating the region R around the y-axis.
The first thing to do is to sketch the graphs of 
Now we have to imagine bringing out a third axis, perpendicular to the picture above, ie. coming out towards us. We then want to rotate the red form here about the vertical axis. This we can imagine doing in the following animation:
Given this form we can then think about either taking horizontal cross-sections through it, which will give us thin annuli, or we can take vertical, circular slices to give us thin shells. Adding these together and integrating should give us the same answer whichever way we choose to slice it, but one way will be considerably easier.…
An integral expression for n!
I gave a challenge question at the end of class a week or so ago. Here I will give the solution and show that it gives us something rather strange and surprisingly useful.
I wrote down the following, and asked you to prove it:
For 
And the number of ways of pulling no objects out of a bag is 1, because you just don’t pull anything out.…
A general formula for partial fractions
Out of the blue I wrote down a rather confusing mass of indices and summations on the board a few days ago. Writing this down at the last minute was perhaps a bad idea, but I wanted to show what the general form for expanding a fraction into partial fractions was. Here I’m just motivating it a little more. It’s not something that you will need to use, but it’s often good to write things down in as general a form as you can.
Let’s say that we have an expression of the form:
Where P(x) is some polynomial of degree less than 3 (because the denominator is degree 3). We can write this as:
To find A, B and C, you cross-multiply, and then match coefficients of powers of x with those in P(x). If you have an irreducible quadratic in the denominator you will have terms of the form:
in your partial fraction, and of course if it’s an irreducible quadratic to an integer power greater than one, you will have multiple terms, just as you do for the (x-2) expression in the example above.…
The 17 equations that changed the world
I was so excited the first time I read Ian Stewart’s book entitled “The 17 equations that changed the world“. The book is written in simple and easy to understand language with interesting practical examples for applications. I immediately wrote to Ian Stewart requesting if I could reproduce his work in form of posters in both English and French to be used at AIMS-IMAGINARY in Senegal in 2015 (see here). I remember his only reply was “please proceed but I won’t be able to attend since I have prior committments”. Business Insider published a list of these equations emphasing further how intuitive they are (see here). I do strongly believe every school in the world be it elementary, college, secondary, technical, university, you name it, should have these posted up or painted on the walls of their science departments/offices, classrooms, laboratories etc; in all langauges applicable.…