### About Siphelele Danisa

I am a Mathematics student at the University of Cape Town.

## The Res-Net-NODE Narrative

This is second post in the blog series, and it is meant to give a broad narrative of the content for next two blog posts. Like the previous post, it will be more of an overview, but the two posts that will follow it will unpack and discuss deeply whatever appears in this one and slightly more.

Humble Beginnings: Ordinary Differential Equations

The story begins with differential equations. Consider $f$ such that $f:[0,T]\times \mathbb{R}^n\to \mathbb{R}^n$ is a continuous function. We can construct a rather simple differential equation given this in the following way. We let

$\begin{cases} {y'(t)}=f(t,y(t))\\ y(0)=y_0\in \mathbb{R}^n \end{cases}$

A solution to this system is a continuous map that is defined in the neighbourhood of $t=0$ such that this map satisfies the differential equation.

Ordinary differential equations are well-studied, and we know that, for example, a solution to the given differential equation will exist whenever the function $f$ satisfies the following:

$(\forall x,y\in \mathbb{R}^n)(\exists C>0 (\in \mathbb{R}))(||f(t,y)-f(t,x)||\leq C||y-x||)$

This property is known as Lipschitz continuity. A function that satisfies this condition is said to be Lipschitz.…

By | March 18th, 2020|Uncategorized|Comments Off on The Res-Net-NODE Narrative

## Deep Q Learning, Briefly!

This blog post is a direct translation of a talk that was given by the author on the 17th of February 2020. The ideas was to very briefly introduce Deep Q-Learning to an audience that was familiar with the fundamental concepts of reinforcement learning. If the person reading this is not familiar with these basics, then a very great introduction can be found here: An Introduction to Reinforcement Learning. Without the additional details from the talk, one will note that this post is rather brief, and should really be used as a tool to gain an overview for the method or a gateway to relevant resources. This will not the case for posts later in the series, because the intention is to deal more with the mathematical aspect of reinforcement learning.

Basic Reinforcement Learning Notions

The idea behind reinforcement learning is that there is an agent that interacts with the environment in order to achieve a certain task.…

## Review: Calculus Reordered

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

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.…

## Linear Algebra for the Memes

I recently saw a post on Quora asking what people generally find exciting about Linear Algebra, and it really took me back, since Linear Algebra was the first thing in the more modern part of mathematics that I fell in love with, thanks to Dr Erwin. I decided to write a Mathemafrica post on concepts that I believe are foundational in Linear Algebra, or at least concepts whose beauty almost gets me in tears (of course this is only a really small part of what you would expect to see in a proper first Linear Algebra course). I did my best to keep it as fluffy as I saw necessary. I hope you will find some beauty as well in the content. If not, then maybe it will be useful for the memes. The post is incomplete as it stands. It has been suggested that this can be made more accessible to a wider audience than as it stands by possibly building up on it, so I shall work on that, but for now, enjoy this!

## Basics of Vector Representation

Not so long ago, I started reading some linear algebra, just out of interest. I was uncertain about whether or not I would understand the concepts, or if it would be worth it to go through all the trouble. I can now say that it was worth it. Honestly, it was the most frustrating, but at the same time rewarding, experience. I have come to realise that there are things that we often have to accept without knowing the beauty of the logic behind their existence, and the idea presented here is one of them. This post answers a simple question about vector notation.

You might have asked yourself at some point in your life (… or maybe you haven’t, but you should): Why is it “legal” to write a vector,$A$, as ${ A=(a_{1},a_{2},\ldots,a_{n}) }$, and why can we switch between different notations without finding trouble (for example, we can represent the vector in the form: ${A = \sum\ a_ia^*_i}$ ) ?…

## Counting to Infinity

I haven’t  really been active, so please accept my apologies since my first post seemed to promise that I would write quite often than I have done. But it was all for a good cause, so I have decided to share something which I found interesting while reading up some Mathematics this holiday (any corrections, additions of definitions, etc. will be appreciated).

Now, the idea I am going to talk about is the Cardinality of a set. In simple terms:
Definition 1: Cardinality is a “measure  of the size” of a set.

Suppose that we have a set, $A$, such that $A=\{a,b,c,d,e\}$. The cardinality of the set, denoted by $|A|$, is $5$, because there are $5$ elements.
It is indeed worth noting that unlike ‘lists’, in Mathematics, order and number of elements doesn’t determine much concerning the identity  of a set, so $\{a,b,c\}=\{a,a,a,b,c\}=\{a,b,b,b,c,c\}=\{a,...,b,...,c, ...\}$ as long as you use the same elements, they are all equal, and of course, cardinalities would be the same, because all of them are a representation of the same mathematical entity.