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  \sin x is an antiderivative of \cos x, 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 will come up with some clever notation for the antiderivative soon.…

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

Group Theory (lecture 2) by Robert de Mello Koch

As promised in the previous post, here is the second lecture by Prof Robert de Mello Koch on Group Theory.

 

 

Please comment if you have thoughts or questions from this video.

How clear is this post?
By | February 8th, 2017|English|0 Comments

An integral expression for n!

I gave a challenge question at the end of class a week or so ago. Here I will give the solution and show that it gives us something rather strange and surprisingly useful.

I wrote down the following, and asked you to prove it:

 

\int_0^\infty e^{-t} t^N dt=N!

 

For N\ge 0, N\in \mathbb{Z}. Now, N! can be thought of as the number of different orderings of pulling N objects out of a bag (without replacement) when they are all different. If you have N things in a bag, then there are N possible things that you can pull out first. There are then N-1 ways of pulling out the next object, N-2 ways of pulling out the next, etc. and finally, when you’ve pulled out N-1 objects there’s only a single possibility of pulling out the last. So:

 

N!=N(N-1)(N-2)(N-3)...3.2.1

 

And the number of ways of pulling no objects out of a bag is 1, because you just don’t pull anything out.…

Where did that substitution come from?

If you want to understand maths, you really have to do it. I recommend going through these examples and using the substitutions given here as hints. Get a blank piece of paper, put your notes away and try to do these examples and see if you get the same answers as in class. If you don’t, write in the comments, and we can see where things may have gone astray.

 

I’ve been teaching integration by substitution, including by trig substitutions over the last few days, and a frequent question which a newbie substituter will ask is “how did you know to make that substitution?”. It’s a very reasonable question, and one that takes practice to build the correct intuition, but I’ll do my best to give some motivation now as to why we made some of the substitutions we made. We won’t solve the integrals, but we will motivate here why we make particular choices for substitutions.…

By | July 22nd, 2016|Courses, English, First year, MAM1000, Undergraduate|2 Comments

Welcome to Reproducing Kernel Hilbert Space

In a series of posts I hope to introduce Mathemafrica readers to some useful data analysis methods which rely on operations in a little back-water of Hilbert space, namely Reproducing Kernel Hilbert Space (or RKHS).

We’ll start with the “classic” example. Consider the data plotted in figure 1. Each data point has 3 “properties”: an x_1 coordinate, an x_2 coordinate and a colour (red or blue). Suppose we want to be able to separate all data points into two groups: red points and blue points. Furthermore, we want to be able to do this linearly, i.e. we want to be able to draw a line (or plane or hyperplane) such that all points on one side are blue, all points on the other are red. This is called linear classification.

Figure 1: A scatter of data with three properties: an x_1 coordinate, an x_2 coordinate and a colour.

Figure 1: A scatter of data with three properties: an x_1 coordinate, an x_2 coordinate and a colour.

Suppose for each data point we generate a representation of the data point \phi(x)=[x_1, x_2, x_1x_2] .…

By | April 30th, 2016|English, Level: intermediate, Uncategorized|0 Comments

Greetings from Bibliotheca Alexandrina

I just received the info below on Bibliotheca Alexandrina and its new African Networks and I thought it makes sense to share it on mathemafrica. I really like the idea of historical places connecting to new technologies. Here is a link to a film (in French) about the historical library at Alexandria:

 

Greetings from the Bibliotheca Alexandrina (BA) in Egypt. We are pleased to announce the launch of “BA African Networks”. Below you will find a description of the networks. You are most welcome to explore our portal and our five networks through the following link. http://afn.bibalex.org/GeneralPortal.aspx

The Bibliotheca Alexandrina (BA) follows in the footsteps of the Ancient Library of Alexandria as a meeting point for cultures and civilizations. It aims to rise to the digital challenge in order to develop African innovation through the use of science and technology in networks that extend throughout Africa. The new goal set by the BA is to connect those with common interests and expertise to hasten the development of knowledge and enable immediate sharing of knowledge and contributions.

By | April 7th, 2016|English, News, Uncategorized|0 Comments

Do You Find Mathematics Scary?

A few weeks ago I attended a lecture by Johnathan Lewin, regarding the use of technology when teaching and it was brilliant, and I’m not even talking about his use of technology. The passion that Johnathan speaks with and the passion he has for Mathematics is explosive and practically contagious.

 
He uses a number of different programmes and applications to assist him in the classroom. He even records his lectures (he captures the audio and a visual of the learning materials and then makes them available to his students). He is in favour of designing the materials in front of the learners in order for them to see how the Mathematics is created rather than to arrive with some neatly prepared sides and show them what Mathematics looks like. He wants them to engage in it at all levels and not just see the perfect final product, if you wish.…

By | March 29th, 2016|English, Fun|2 Comments

Mathemafrica and the Next Einstein Forum (NEF) #1

Dear Mathemafrica readers,

I am sitting at the AIMS-Senegal institute in Mbour (about 1.5 hours drive South of Dakar) and together with my team member Sebastian and the AIMS-Senegal staff and students, we are preparing an interactive mathematics exhibition (as part of the IMAGINARY – open mathematics project).. It will be shown as of March 8 at the Next Einstein Forum (NEF), to be held at a huge conference venue just outside Dakar.

There will be many ministers, scientists, politicians, even presidents from (apparently all fifty-four) African countries – and also many international guests – joining the NEF, with the goal to discuss about scientific innovations, collaborations and solutions in Africa!

We have to plan to blog live from the NEF, with insights and views from participants. And of course, we will let you know about our exhibition, about a new competition, we will launch and everything happening around!

Prepare yourself for the NEF at:

gg2016.nef.org/

iameinstein.org

www.facebook.com/NextEinsteinForum

And the Twitter hashtag: #AFRICASEINSTEINS

Please find a picture from our first technical setup yesterday at the AIMS Institute.…

By | March 6th, 2016|English, Event, News, Uncategorized|0 Comments