March 2015 – Mathemafrica

AIMS-IMAGINARY Workshop 2014

A little blog about the conference in which Mathemafrica was conceived! This first appeared on the ICTP math blog on the 18th of November 2014.

One of the perks of working in mathematics is that I get to travel the world as part of my job. This time it was Cape Town, South Africa; I flew down there to participate in a workshop on maths `outreach’ activities in Africa. The workshop was held at the African Institute for Mathematical Sciences (AIMS) and was organised in partnership with IMAGINARY. Will introduce the organisers in a bit but let me focus on the idea of mathematics outreach for now.

For me, mathematics outreach is an attempt to share mathematical ideas with non-experts in the field. The term non-expert includes a broad spectrum of people stretching from school children to senior citizens. Math-outreach is usually done through exhibitions and the written media but the definition allows for many more forms of communication.…

By Tarig| March 15th, 2015|English, Event|1 Comment

An explanation for the multiplier effect in a Keynesian macroeconomic model

In this post I will provide a mathematical basis for the multiplier effect which is found when changing an autonomous variable in the aggregate expenditure function (AE). AE is a function which represents the total amount of money that is spent by all consumers in an economy, and is made up of two components: autonomous expenditure (which is exogenous in relation to income) and induced expenditure (endogenous to income). In other words, autonomous expenditure is the y-cut of the AE funtion, and any increases in autonomous expenditure will shift AE vertically. Induced consumption refers to any expenditure that is over and above the level of expenditure at the y-cut, and is directly related to the gradient of the AE function (which includes the marginal propensity to consume¹). Referring to Figure 1 below, let the increase in autonomous expenditure be y₁. y₁ thus shifts AE₁ to AE₂.…

By Aidan Horn| March 15th, 2015|English, Level: intermediate|4 Comments

The domain of a composite function

In this article I outline a systematic way of finding the domain of a composite function. A definition that can be used for this purpose follows:

$D(f \circ g) = \{x|x\in D(g) \wedge g(x) \in D(f)\}$

(Vaught, 1995:18)

Where $D(\lambda) =$ $\text{the domain of }\lambda$

The explanatory method which follows is to show how to use this definition in different examples.

Example 1

Solve $D(\ln(\ln(\ln x)))$ .

Solution:

Let $\ln(\ln(\ln x)) = f(g(h(x)))$

$\begin{minipage}{3in} \begin{align*} \text{Firstly, }& D(g\circ h) = \{x|x\in (0,\infty)\wedge \ln x\in (0,\infty)\} \\ & \ln x\in (0,\infty) \Leftrightarrow x\in (1,\infty) \\ \therefore \; \; & D(g\circ h) = x \in (1, \infty) \\ ~\\ \text{Now, }& D(f \circ (g\circ h)) = \{x|x\in (1,\infty)\wedge \ln(\ln x)\in (0,\infty)\} \\ & \ln(\ln x)\in (0,\infty) \Leftrightarrow \ln x\in (1,\infty) \Leftrightarrow x \in (e,\infty) \\ \therefore \; \; & D(f \circ g\circ h) = x \in (e, \infty) \end{align*} \end{minipage}$

$\square$

Example 2.1

Let $f(x)=x+1$ and $g(x)=x^2$ where $D(g)=[-2,2]$ .

Find $D(f\circ g)$

Solution:

$\begin{minipage}{2in} \begin{align*} f\circ g(x) &= x^2+1 \\ D(f\circ g) &= \{x|x \in [-2,2] \wedge x^2 \in \mathbb{R} \} \\ & x^2 \in \mathbb{R} \Leftrightarrow x \in \mathbb{R} \\ \therefore \; \; & x \in [-2,2] \end{align*} \end{minipage}$

$\square$

Example 2.2

Consider the same constraints as in Example 2.1, but with $D(f)=[-2,1]$

Solution:

$\begin{minipage}{3in} \begin{align*} D(f\circ g) &= \{x|x \in [-2,2] \wedge x^2 \in [-2,1] \} \\ & x^2 \in [-2,1] \Leftrightarrow x^2 \in [0,1] \Leftrightarrow x \in [-1,1] \\ \therefore \;\; & x \in [-1,1] \end{align*} \end{minipage}$

$\square$

References

Vaught, RL. 1995. Set theory: An introduction. 2nd edition. Boston: Birkhäuser.

LaTeX and PDF format here

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By Aidan Horn| March 4th, 2015|English, Level: intermediate|0 Comments

AIMS-IMAGINARY Workshop 2014

An explanation for the multiplier effect in a Keynesian macroeconomic model

Women in Mathematics for Social Change & Sustainable Livelihoods

The domain of a composite function

3rd International Workshop on Nonlinear and Modern Mathematical Physics 9-11 April 2015, Cape Town, South Africa

3rd International Workshop on Nonlinear and Modern Mathematical Physics 9-11 April 2015, Cape Town, South Africa