Sticky Post – Read this first. Categories and Links in Mathemafrica

The navigability of Mathemafrica isn’t ideal, so I have created this post which might guide you to what you are looking for. Here are a number of different categories of post which you might like to take a look at:

Always write in a comment if there is anything you would like to see us write about, or you would like to write about.…

By | January 17th, 2018|Uncategorized|0 Comments

p-values: an introduction (Part 1)

The starting point

This is the first of (at least) 3 posts on p-values. p-values are everywhere in statistics- especially in fields that require experimental design.

They are also pretty tricky to get your head around at first. This is because of the nature of classical (frequentist) statistics. So to motivate this I am going to talk about a non-statistical situation that will hopefully give some intuition about how to think when interpreting p-values and doing hypothesis testing.

My New Car

I want to buy a car. So I go down to the second hand car dealership to get one. I walk around a bit until I find one that I like.

I think to myself: ‘this is a good car’. 

Now because I am at a second-hand car dealership I find it appropriate to gather some data. So I chat to the lady there (looks like a bit of a scammer, but I am here for a deal) about the car.…

By | August 21st, 2019|English, Level: Simple, Undergraduate|0 Comments

R-squared values for linear regression

What we are talking about

Linear regression is a common and useful statistical tool. You will have almost certainly come across it if your studies have presented you with any sort of statistical problems.

The pros of regression are that it is relatively easy to implement and that the relationship between inputs and outputs is linear (it’s in the name, but this simplifies the interpretation of the relationship significantly). On the downside, it relies fairly heavily on frequentist interpretation of probability (which is a little counterintuitive) and it’s very easy to draw erroneous conclusions from different models.

This post will deal with a measure of how good a model is: R^2. First, I will go through what this value means and what it measures. Then, I will discuss an example of how reliance on  R^2  is a dangerous game when it comes to linear models.

What you should know

Firstly, let’s establish a bit of context.…

By | August 18th, 2019|English, Undergraduate|1 Comment

Cantor–Schröder–Bernstein Theorem

Knowledge this posts assumes: What is a set, set cardinality, a function, an image of a function and an injective (one-to-one) function.

David Hilbert imagines a hotel with an infinite number of rooms. In this hotel, each room can only be occupied by one guest, and each room is indeed occupied by exactly one guest. What happens if more guests show up? Can they be accommodated for?

PAUSE: WHAT DO YOU THINK AND WHY?

Suppose we propose they cannot be accommodated for, since all the rooms are occupied. Hilbert then claims that he can define the functions f:A \mapsto B, and g:B \mapsto C, where A is a set containing all current guests, and f simply maps each guest to a room in the set B, and g maps each room in B to a new one in C. Notice that these functions must be injective, since if a room contains two different guests, those two different guests must be the same guest; recall f(a) = f(b) \rightarrow a = b.…

By | August 16th, 2019|Uncategorized|0 Comments

1.6 Partitions

Recall the  relation \equiv \text{ mod} (4) on the set \mathbb{ N}.

One of the equivalence classes is [0] = \{ ..., -8, -4, 0, 4, 8, ...\} which is equivalent to writing [0] = [4] = [-4] = [8] = [-8] ...

We could do this because the equivalence class collects all the natural numbers that are related to zero under the relation \equiv \text{ mod} (4)

 

The following theorem generalises this idea for any relation \equiv \text{ mod} (n) on the set \mathbb{ N}: for the integer n.

Let R be an equivalence relation on set A. If a, b \in A,  then [a] = [b] \iff  aRb.

Essentially, equivalence classes  [a] = [b] are equal if the elements  a, b \in A, are related under the relation R. And simultaneously, knowing that elements a, b \in A, are related under R means their equivalence classes  [a] = [b] are equal.

An equivalence class  \equiv \text{ mod} (n) divides set a A into n equivalence classes. We call this situation a partition of set A.

A partition of a set A is defined as a set of non-empty subsets of A, such that both these conditions are simultaneously satisfied:

 (i) the union of all these subsets equals A.

(ii) the intersection of any two different subsets is

 

Let’s return to our example: \equiv \text{ mod} (4) on the set \mathbb{ N}. We could represent this set as:

Modulus 4, General

  • NOTE: Each equivalence class above represents an infinite set and despite the drawing suggesting [0] is larger than [3] for instance, this is not true.
By | August 9th, 2019|Uncategorized|0 Comments

Review: Calculus Reordered

Book title: Calculus Reordered: A History of the Big Ideas
Author : David M. Bressoud

9780691181318

Princeton University Press
Link to the book: Calculus Reordered: A History of the Big Ideas

Discussions on the history of different fields are usually dry, wordy and generally, when you are studying the field, hard to read. This is because they are usually geared towards the general audience, and in doing so most authors tend to strip away the very exciting technical details. I expected the same treatment from the author, but I was pleasantly surprised.

The book contains 5 chapters, which are the following:

1) Accumulations
2) Ratios of Change
3) Sequences of Partial Sums
4) The Algebra of Inequalities
5) Analysis

Each of these chapters has a central theme that is being covered, but they are not at all disjoint. For instance, the last three contain the history of concepts that would normally be found in a first course for Real Analysis, while the first two are essentially the more applied spectrum to serve as some form of motivation for going through all this trouble, although they can certainly stand on their own.…

By | August 4th, 2019|Uncategorized|1 Comment

Investigating Practical Ordering of Grids

In Reinforcement Learning there is an environment known as Gridworld. In this environment you have a grid and there is an agent that learns how to find the shortest path from one cell to another. The theme of reinforcement learning is that you do not want to hard-code the rules, but you want the agent to explore until it can find a set of moves that are optimal for the problem at hand. Usually you can alter the grids to make the tasks tough–set ‘traps’, add obstacles, etc. We are considering grids with obstacles, and an interesting question that came up is the following,

Given two grids of size {N}, say {G, \,G'} which have respectively {k,l} obstacles where {k,l\in \mathbb{N},\,k,l\geq 0}, what are reasonable ways to put an order on the ‘complexity’ of the grids?

In other words, we want to be able to say that, for instance, in {G} the agent will find the optimal path more easily than in {G'} given any two grids {G,G'}.…

By | July 28th, 2019|Uncategorized|0 Comments

1.5 Equivalence classes (Infinite sets)

 

Let’s find the equivalence classes of the following finite set S:

Given S = \{ -1, 1, 2, 3, 4 \}, we can form the following relation R = \{ (-1, -1), (1,1), (2,2), (3,3), (4,4), (1,3), (3,1), (2,4), (4,2) \}.

Note: writing the relation R on set S in the following ways is equivalent:

-1R-1, 1R1, 2R2, 3R3, 4R4, 1R3, 3R1, 2R4, 4R2

or

-1\le -1, 1 \le1, 2 \le2, 3  \le3, 4 \le4, 1 \le 3, 3 \le 1, 2 \le 4, 4 \le 2

This relation, R has been given the symbol \le but it means “the same sign and parity” in this case. For instance, (1,3) or 1 \le 3 tells us that one and three are both odd and both have the same sign in set A (both positive).

The equivalence classes for this relation are the following sets:

\{ -1 \}, \{ 1, 3\} \text{ and } \{2, 4 \}

We obtained the above equivalence classes by asking ourselves:

  • How is the element -1 related to any other element in the set S under the definition of R?

Since R is defined as “the same sign and same parity,” then we’re really asking ourselves whether -1 has the same sign as any other element in S. Since all the other elements are positive, then -1 has the equivalence class containing only itself. Another question we would’ve asked ourselves is whether -1 is even or odd. …

By | July 25th, 2019|Uncategorized|1 Comment

The 2018 South African Mathematics Olympiad — Problem 6

The final round of the South African Mathematics Olympiad will be taking place on Thursday, 28 July 2019. I have been writing about some of the problems from the senior paper from 2018. A list of all of the problems can be found here.

Today we will look at the sixth and final problem from the 2018 South African Mathematics Olympiad:

Let n be a positive integer, and let x_1, x_2, \dots, x_n be distinct positive integers with x_1 = 1. Construct an n \times 3 table where the entries of the k-th row are x_k, 2x_k, 3x_k for k = 1, 2, \dots, n. Now follow a procedure where, in each step, two identical entries are removed from the table. This continues until there are no more identical entries in the table.

  1. Prove that at least three entries remain at the end of the procedure.
  2. Prove that there are infinitely many possible choices for n and x_1, x_2, \dots, x_n such that only three entries remain,

There are some heuristics that are often helpful when solving a problem, such as

  • Looking at small cases:

    This helps us to understand the problem and how the various pieces in the problem relate to each other.

By | July 23rd, 2019|Competition, English|1 Comment

On the invariant measure in special relativity

I’m writing this for my string theory class. We are basing our lectures on Zwiebach – A First Course in String Theory, and starting off with special relativity. Not everybody in the class has a physics background (pure and applied mathematics students), and so there are likely to be questions which come up which show where I have to fill in some knowledge. We had a question about the invariant measure in special relativity (SR) and why there was a different sign in front of the time term compared with the space terms. I’ll do my best to explain here. Note that I am not explaining it in the precise chronological order of discoveries.

We start the picture off with relativity before SR – that is, Galilean Relativity. This simply states that the laws of motion are the same in all inertial (non-accelerating frames). That may sound straightaway like SR, but there’s a crucial ingredient missing which we will see in a bit.…

By | July 23rd, 2019|Uncategorized|0 Comments

1.4 Equivalence classes

Let’s recall the definition of an equivalence relation:

 A relation R on a set A is termed an equivalence relation if it is simultaneously reflexive, symmetric and transitive.

Let’s look at more examples:

Example One: Let A = \{2, 11, 17, 20\} be a set with the following relation: R = \{ (2,2) (11,11) (17,17) (20,20) (2,20) (20,2) (11,17) (17,11) \}.

The relation described by R is termed “the same parity.” Elements x and y are said to have the same parity if they are both odd or both even. In our case, the elements 11 and 17 are both odd – hence have the same parity. Similarly, 20 and 2 have the same parity because they are both even. An element will always have the same parity as itself.

The elements that share the same parity as 11 can be grouped together to form a set: O = \{ 11, 17 \}. This is the set of all odd elements from A.

Similarly, the even elements can be grouped together to form the set: E = \{ 2, 20\}.

The new sets, O and E, form the equivalence classes of the relation R on set A.…

By | July 22nd, 2019|Uncategorized|0 Comments