I recently received a message from a friend and the heading of the post perfectly describes what was said to me. Thereafter, an interesting integration question was sent to me. It read as follows:

I must admit, it does look quiet scary. My immediate thought was that some sort of substitution was required but I really had no idea as to where and how this should be done. Two pages and a headache later, I thought to myself why don’t I get a rough idea of what the answer should look like. Once again, let’s start approximating things (as it turns out, my approximation gets me the absolutely correct answer).

I looked at the integration bounds and noted that the point x = 3 was in fact the midpoint. I then decided to construct a Taylor polynomial for the above function around the point x = 3. It was done as follows:

You might be wondering why I didn’t bother taking anymore derivatives.…

Why ? I assume this is the question on everyone’s mind. (Whether you’re a Math lover or not)

The simple answer would be that we all love pie, now don’t we?

Before I begin discussing any technicalities, I’d like to acknowledge that it is possible for some of us to find the concepts easy whilst others might struggle with them. This is the reason why I’m choosing to speak in a very simple and understandable manner. (I’m baby proofing my post!)

Firstly, let us have a look at the Maclaurin series of

Aside: A Maclaurin series is a polynomial which approximates a function around the point x = 0. The level of accuracy decreases as you move further away from x = 0. The only way to get an exact answer and not an approximate is to let the sum go to infinity.

Below is a table of the first few derivatives of .…

Essentially, the entire theory of electromagnetism can be found in the following four equations:

These are Maxwell’s Equations in differential form, not in integral form, which is the way they are often introduced. I will discuss them in this form, however, as I believe the differential equations convey more elegantly their physical meaning straight from the mathematics. Let’s get started.

Fields

If you ever did high school physics, you should have some idea of what electric and magnetic fields are. Below is an example of each (depicted using field lines):

What these field lines show is the direction of the respective fields at each location (indicated by arrows) as well as their relative strengths (indicated by density of field lines). So what are these fields actually representing? Well, for electric fields, this shows you the direction an object with a positive electric charge would be forced, while a negatively charged object would feel a force in the opposite direction.…

I have this linear algebra problem in the context of quantum mechanics. Let be a family of linear operators so to each we have a linear operator where is a complex vector space if one is unfamiliar with functional analysis (like I am) or is a Hilbert space if one is. Let’s suppose that this family is differentiable.

Suppose further that is always a Hermitian operator. Suppose that has a discrete spectrum of eigenvalues . I need to show the following:

Theorem

Now here is a “proof,” it is not quite rigorous since there are probably a lot of technical details regarding functional analysis that I’m missing out on but:

Proof We begin by differentiating the eigenvalue equation with respect to using the product rule:

After multiplying by and rearranging terms we have the following:

Now we can take the adjoint of both sides of the eigenvalue equation to get that since because the eigenvalues of a normal operator are real.…

This blog post will carry on from the previous one, and introduce dependent types. So what is a dependent type? To motivate the idea let’s talk about equality. Remember that we interpret propositions as types, so if we have then the statement “ is equal to ” corresponds to some type, let’s call it . This type depends on its values, for example we expect to be able to prove (i.e. construct) , but not to be able to prove and so we will have an equality type that depends on its values. This idea is also being explored in various programming languages. These languages have a type like , where means that is a list of elements from the type . Since the length of the list is part of its type which is known ahead of time, it is impossible to ask questions like, “What is the first element of this empty list?” Indeed dependent types are so powerful that one can write a compiler and be sure that the compiler preserves the meaning of a program.…

For anybody interested in the mathematics of infectious disease modelling, the following should be very interesting.

A course on the application of mathematical modelling and computer simulation to predict the dynamics of infectious diseases to evaluate the potential impact of policy in reducing morbidity and mortality. (click to go to poster).

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.