A Linear algebra problem

I have this linear algebra problem in the context of quantum mechanics. Let $latex \mathbf{f}_\lambda$ be a family of linear operators so to each $latex \lambda \in \mathbf{R}$ we have a linear operator $latex \mathbf{f}_\lambda : \mathcal{H} \to \mathcal{H}$ where $latex \mathcal{H}$ is a complex vector space if one is [...]

By | October 12th, 2016|0 Comments

Dependent Types

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 $latex x, y : A$ then the statement "$latex x$ is [...]

By | October 11th, 2016|0 Comments

Checking direction fields

I was recently asked about how to spot which direction field corresponds to which differential equation. I hope that by working through a few examples here we will get a reasonable intuition as to how to do this. Remember that a direction field is a method for getting the general [...]

By | October 11th, 2016|0 Comments

Cellular Automaton

By | October 4th, 2016|0 Comments

Group Theory in a Nutshell for Physicists, by Tony Zee – A review

I studied group theory for the first time around 15 years ago at the beginning of my PhD. There were six of us in the class, and I found it both a magical, as well as a mysterious subject. We had a great lecturer, but the way that the course [...]

By | September 11th, 2016|0 Comments

Radius of convergence of a series, and approximating polynomials

I hinted today that there were sometimes issues when you did a polynomial approximation, that if you tried to find the value of a function a long way from the region about which you're approximating, that sometimes you wouldn't be able to do it. This is related to an idea [...]

By | August 19th, 2016|0 Comments

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. 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 [...]

By | August 17th, 2016|0 Comments

Convex functions

What I learnt in class today: A convex function f:\mathbb R\to\mathbb R is defined as satisfying 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 \smallsmile . An [...]

By | August 15th, 2016|0 Comments

Integration by Parts – Lightbulb Education

Integration by Parts

By | August 10th, 2016|0 Comments

Integration By Substitution – Lightbulb Education

Integration By Substitution

By | August 8th, 2016|0 Comments
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