A Whimsical Introduction to Graph Theory (1)

Part 1 – What are Graphs?


Mathematics is full of fascinating ideas and concepts. These can, however, be very challenging to tackle and make sense of, especially when you are put under pressure to answer questions about them! In this post, and those to come, I hope to share some insight into these concepts without getting too formal. Where some definitions and more technical bits are introduced, they will be explained at end of the post: look out for the dagger \dagger symbols!

To begin, let’s ask what we do in mathematics. The first step in any area of maths is almost always to abstract things. We take some concept we want to be able to work with and pull out the essential ideas. From a bunch of maps we may take out just destinations and the routes between them; from 3D objects we may only need to know what ways we can rotate them and still see the same thing; from a collection of algorithms we may only care about how long they take to run on a computer, and so on.…

By | February 12th, 2018|Level: Simple|0 Comments

MAM1000W 2017 semester 2, lecture 1 (part ii)

The distance problem

If I want to know how far I walked during an hour, I can ask how far I walked in the first five minutes, and how far I walked in the second five minutes, and how far I walked in the third five minutes, etc. and add them all together. ie. I could write:

 

d=d_1+d_2+d_3+d_4+...d_{12}

 

Where d_i is the distance walked in the i^{th} five minutes. To calculate a distance, we need to know how fast we are going, and for how long. In fact:

 

distance=velocity \times time

 

where you can think of velocity as the same thing as speed (though there are subtle differences which you will find out about later). This formula works if the velocity is constant, but what if it is changing. Well, if we have a graph of velocity against time, then we can think about splitting the graph into intervals (like the five minute intervals above), and approximating that during a small interval of time, the velocity is roughly constant.…

MAM1000W 2017 semester 2, lecture 1 (part i)

I wanted to put up a little summary of some of the most important things to remember from the end of last semester. There was a sudden input of new concepts, so let’s put some of them down here to get a clear reminder of what we need to know. A few things in this post:

  • The antiderivative
  • Sigma notation
  • Areas under curves

Antiderivatives

An antiderivative of a function f on an open interval I is a function F such that:

 

F'(x)=f(x) for every x\in I

 

Note that we say an antiderivative, not the antiderivative. There can be many functions whose derivatives give the same thing. While we know that:

 

\frac{d}{dx}\sin x=\cos x

 

And therefore we can say that \sin x is an antiderivative of \cos x. However, we can also say that:

 

\frac{d}{dx}(\sin x+3)=\cos x

 

So \sin x+3 is also an antiderivative of \cos x. In fact for any constant c it is true that \sin x+c is an antiderivative of \cos x. We write this statement as:

 

\int\cos x dx=\sin x+c

 

This is called the indefinite integral of \cos x with respect to x.…

Unsolved!: The History and Mystery of the World’s Greatest Ciphers from Ancient Egypt to Online Secret Societies by Craig P. Bauer – A review

This book was sent to me by the publisher as a review copy.

This is a book of some impressive magnitude, both in terms of the time span that it covers (being millennia), as well as the ways in which it discusses the context and content of the ciphers, most of which, as the title suggests, are unsolved. The book starts with perhaps the most mysterious of all unbroken ciphers: The Voynich Manuscript (the entirety of which can be found here). This story in itself is perhaps the most fascinating in the history of all encrypted documents, and that we still don’t know if it truly contains anything of interest, or is just a cleverly constructed (though several hundred year old) hoax makes it all the more intriguing.

The writing rather effortlessly weaves between the potential origin stories, the history of the ownership of the manuscript and the attempts to decode it.…

By | July 29th, 2017|Book reviews, English, Level: Simple, Reviews|1 Comment

Quantum Computing (Part 0): A Brief Introduction

These days, the vogue in science and technology is all things quantum, especially in the continuously-advancing frontiers of computation and data analysis. Applications such as AI and crunching through Big Data require ever-faster processing of ever-larger datasets. The amount of data generated by companies such as Google and Facebook is already mind-boggling and is only set to increase at an exponential rate. However, it is clear that in many contexts, simply adding datacenters and processors is not going to be enough: a complete paradigm shift is required if these kinds of technologies are to become effective in the lives of billions of people around the world.
This is where the quantum realm comes in. At the smallest scales of our universe, the behaviours of the “classical” mechanics governing our everyday lives is replaced by something entirely different; the laws of quantum mechanics are unintuitive and confusing, hiding layers of complexity in ways that simply cannot exist at our macroscopic scales.

By | July 10th, 2017|Level: Simple|0 Comments

You’re (probably) a Bayesian – whether you like it or not!

Statisticians have long been separated into two camps as to how they philosophically interpret their trade. These schools of thought are usually called Frequentists and Bayesians.

Frequentists believe that a probability, p\in[0~ 1] , associated with a specific possible outcome of an observable occurrence or process, is simply telling you that, could you observe this occurrence (or process) infinitely many times, the fraction of such observations that would yield that specific outcome is p . Using the age-old coin toss example: tossing the coin is the occurrence or process and recording a Heads or Tails are the two observation. The number 0.5 \left(P(\text{Tails})=0.5=P(\text{Heads})\right) tells a Frequentist that, in the pursuit of infinitely many coin tosses, the ratio of Heads recorded to the number of tosses performed asymptotically approaches 0.5. And that’s all! The value should not be interpreted as the most likely outcome for the next observation or sample taken from the process (though I’ve always wondered how a Frequentist would gamble…).…

By | February 3rd, 2016|English, Fun, Level: Simple, Uncategorized|0 Comments

AIMS-Senegal – picture story of a maths communication adventure

Dear all,

this is a picture story, so be prepared to see many pictures! And it is an adventure too, since we (I will explain later who is “we”) tried something quite new: an interactive highly technical mathematical exhibition in Senegal, including a road show with high school students and Master maths students, talks, conferences, workshops and discussions, and games! And: all in three days!

Let’s start:

Day 0:

The day before the opening of the exhibition and the start of the roadshow. And my first day in Senegal. I arrive around 1:30 am in the morning and spend the night in Dakar. At 11:00 I am picked up by a car from AIMS-Senegal, who is organising the event together with the Next Einstein Initiative and IMAGINARY. In the car, I meet two other international participants: Niane Kode from Senegal, who works at AIMS-Cameroon, and Marcos Cherinda from Mocambique, an ethno-matematician, who also attended the AIMS-IMAGINARY workshop in 2014.…

By | November 16th, 2015|English, Event, Level: Simple, Uncategorized|5 Comments

A little medical statistics

Originally written by John Webb

tw: fictionalised statistics of disease rates.

———

Today there are many tests that are widely used to detect life-threatening diseases early. How effective are they? Should they be believed?

At a routine checkup, your doctor tells you that there is a simple and inexpensive blood test that can detect a rare but particularly
nasty form of cancer. You agree to have the test done, and the doctor takes a blood sample and sends it off to the pathology laboratory.

Two days later the doctor calls to tell you that the test has come up positive. The good news is that the cancer can be cured since it
has been caught at an early stage. The bad news is that the treatment, though effective, is very expensive and has a number of unpleasant side-effects.

Before agreeing to treatment you need to do a little bit of basic arithmetic.…

By | November 2nd, 2015|English, Level: Simple|0 Comments

Mathematically speaking, what is a contradiction?

The world of predicate logic interests me, especially how it provides a foundation for understanding the logic behind many mathematical proofs. It is interesting to know how the negation, contrapositive and inverse are defined with respect to some implication  A \Rightarrow B   ( A \wedge \neg B, \neg B \Rightarrow \neg A  and   B \Rightarrow A respectively). What got me thinking about predicate logic again was when I asked myself, “What is a contradiction?”

My big Collins Dictionary and Thesaurus defines ‘contradict’ as “to declare the opposite of (a statement) to be true” (verbatim). But, this leaves some room for debate as the meaning of the word “opposite” is not logically clear. Is the negation true? Is the inverse true? My reasoning says that a contradiction of the above implication is defined as  A \Rightarrow \neg B . In a less formal way (and also less strongly),  A \not \Rightarrow B .

Let me briefly pause here for the sake of those unfamiliar with the symbols I have already used. \neg denotes ‘not’, \Rightarrow denotes ‘implies’, \wedge denotes ‘and’, and A and B are symbols which represent a statement, such as “dogs are black” or “black animals are dogs”.…

By | November 1st, 2015|Level: Simple, Uncategorized|1 Comment

UCT MAM1000 lecture notes part 38 – 3D geometry and vectors part i

A lot of the following is going to be rather intuitively clear, but we need to build up a framework where we are all speaking the same language to develop the powerful tools that we are going to find over the coming sections. We will be dealing here specifically with three dimensional space but we will discuss along the way the extension of these concepts to higher dimensional spaces. The higher dimensional stuff is not examinable but I think that sometimes it helps to understand the things which are special about three dimensions, and the things which are not.

In particular, I can recommend having a look at the web page of John Baez who discusses the regular polytopes in different numbers of dimensions here.

It’s clear that to define where you are in three dimensional space you need to set up a few key ingredients first. What you need is first of all an origin – a place to call home from which you will relatively describe your position.…