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## AIMS-IMAGINARY – Maths and Science Exhibition and Workshop in Mbour, Senegal

I was lucky enough to attend the first of these workshops in AIMS, Muizenberg and indeed that is where Mathemafrica was born. This should be a great event! Photo is taken from the first program:

Text below taken from the workshop/roadshow/exhibition website

This second edition of the AIMSIMAGINARY Maths and Science Roadshow, Workshop and Exhibition will showcase interactive visual and hands-on tools used to stimulate interest in maths and sciences among diverse groups of people. The event targets primary, secondary, high school and university learners and teachers/lecturers. It will consist of (1) an AIMSIMAGINARY Maths and Science Roadshow packed with hands-on activities and discussions, (2) an AIMSIMAGINARY Maths and Science Exhibition, (3) discussions by interested participants willing to be part of the AIMSIMAGINARY network to share ideas and plan for future events, and (4) a Science Slam event. The event is being organized by the African Institute for Mathematical Sciences (AIMS) and supported by IMAGINARY and the Mathematisches Forschungsinstitut Oberwolfach, and the Government of Senegal.

## Game Theory can make you money…

I plan on doing some game theory posts in the near future, but for now, see how some very clever tactical thinking can do you wonders…

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## UCT MAM1000 lecture notes part 40 – 3D geometry and vectors part iii

The scalar, or dot product

We have seen now how to add together vectors and how to multiply them by scalars, but we haven’t seen how to multiply two vectors together. In fact it’s not all that obvious what it means to multiply two vectors together. A vector has a magnitude and a direction, how do you multiply directions? The answer is that there are two different ways to multiply together vectors. The first way which we will explore now is the scalar, or dot product. This will take two vectors and the product of them using this rule will give us a scalar. We definitely want something that is linear in both of the magnitudes of the vectors. That is to say that we want some way of multiplying together vectors so that when we double the magnitude of one of the vectors, we double the product. We will express the scalar or dot product of two vectors as: $\vec{v}.\vec{w}$.…

## UCT MAM1000 lecture notes part 39 – 3D geometry and vectors part ii

Vectors

Vectors are quantities which have both magnitude (ie. size) and direction. The most common examples of these are velocity ($3ms^{-1}$ to the right) and force (10 Newtons pointing vertically down). The easiest way to describe such a quantity is an arrow, where the magnitude gives the length and the direction is given by, well, the direction of the arrow. The important point about this is that the position of the vector itself doesn’t matter. In the figure below we place the same arrow in several different places and they are all the same vector.

A vector, with magnitude given by its length and direction given by the direction of the arrow, placed at different points in the plane. Note that the position of the start of the arrow is now important, just the relationship between the start and the end of the arrow.

We could define a vector by the length and the angle that it makes with the horizontal axis, but in general we define it by how much it goes in the horizontal direction and how much it goes in the vertical direction, that is, how much it goes in the $x$-direction and how much it goes in the $y$-direction.…

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## 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.…

## Galileoscope in action

Solomon Malesa, who won the translation competition has sent through photos of him in action with the Galileoscope. Congratulations again to Solomon, and thank you for the great pictures!

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## UCT MAM1000 lecture notes part 37 – differential equations part vi – second order differential equations

Second Order differential equations

We are only going to look at a particular subset of all possible second order differential equations (that is, equations which contain at most second derivatives) but these particular equations are absolutely ubiquitous across every field of science. The particular subset we are going to look at are linear, homogenous second order differential equations with constant coefficients. These can be written in general as:

$\frac{d^2y}{dx^2}+b\frac{dy}{dx}+c y=0$

It is linear because it contains at most (and in this case at least) a single power of $y$ in each term. It is homogenous because there is no term which has no powers of $y$ (ie. the right hand side is not a constant), and the coefficients $b$ and $c$ are any real numbers (though you can extend this to having complex numbers very easily). We will see that depending on the relationship between these numbers ($b$ and $c$) we can have very different behaviour of the equation.…

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## UCT MAM1000 lecture notes part 36 – differential equations part v – first order differential equations

First order linear differential equations

We are now going to deal with another subset of first order differential equations which in some ways are easier than the previous and in other ways more complicated. These are linear first order differential equations. The general form of a first order linear differential equation is:

$\frac{dy}{dx}+P(x)y=Q(x)$

where $P(x)$ and $Q(x)$ are any functions of $x$.

Very importantly, I’m leaving off the fact that $y$ is dependent on $x$ in the notation, but you should remember that this is really $y(x)$ and that is the function you are trying to solve for.

Sometimes you will be given an equation which is not obviously in this form but it can be transformed to this form. For instance:

$\frac{1}{y}\frac{dy}{dx}=x^2+\frac{\sin x}{y}$

This can easily be transformed into the canonical form for a linear first order DE. We are going to try and rewrite the left hand side of the equation in a form which will mean that we can solve the differential equation very easily.…

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## UCT MAM1000 lecture notes part 35- differential equations part iv – separable differential equations

Separable differential equations
In some ways these are the easiest differential equations to solve in theory, though in practice the final step (that of integrating) may be difficult or impossible. A separable differential equation is one of the form:

$\frac{dy}{dx}=\frac{f(x)}{g(y)}$

where $f(x)$ and $g(y)$ are any functions of $x$ and $y$ respectively. For instance:

$\frac{dy}{dx}=x y$

is of this form where $f(x)=x$ and $f(y)=\frac{1}{y}$. The reason that these equations are simple in theory is because we can rearrange them to be:

$g(y)dy=f(x)dx$

ie. we have all the $x$ stuff on one side and all the $y$ stuff on the other and then we can integrate both sides:

$\int g(y)dy=\int f(x)dx$

and that’s it. As long as you can do the integrals, you can get a function $y$ in terms of $x$. let’s look at some examples:

$\frac{dy}{dx}=x y$

gives the following integral:

$\int\frac{1}{y}dy=\int x dx$

and so:

$\ln |y|+c_1=\frac{x^2}{2}+c_2$

here we have one constant of integration from each side of the interval, but because they are just constants, we can put them into one constant and call it $c$:

$\ln |y|=\frac{x^2}{2}+c$

we can then rearrange this to give:

$|y|=e^{\frac{x^2}{2}+c}=e^ce^{\frac{x^2}{2}}$

We can then call $e^c$ just a constant, and let’s call it $y_0$ because we can see that when $x$ is zero, $|y|$ is just going to be given by this constant:

$|y|=y_0e^{\frac{x^2}{2}}$

This has two solutions (one where $y$ is positive, and one where it is negative), so we can choose one of them, depending on the initial condition for $y$.…

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## UCT MAM1000 lecture notes part 34 – differential equations part iii – Direction flows and Euler’s method

We haven’t yet studied any general ways to solve differential equations. In the first case of exponential growth we found an easy way to solve the equation, but for the logistic equation we just gave the solution and showed that it indeed satisfied the equation. Here we are going to look at some methods for finding not the exact solution, but approximations of the solutions. The first method is the method of Direction Fields and it will give us a good idea of what the solutions are going to look like. The second method, Euler’s method will give us an approximation to a single solution and we will be able to improve it to get arbitrarily good solutions to any differential equation (so long as there aren’t particularly nasty pathologies in the differential equation).

Direction Fields

Let’s take a differential equation:

$\frac{dy}{dx}=x+y$

Note that sometimes we will say explicitly that $y(x)$ and sometimes we will leave it implicit, because the equation has a derivative of $y$ with respect to $x$.…