Mathematical Modelling: Part 1

25 Jun

I’m giving a talk to year 9/10 students (~15-16 year olds) this evening on mathematical modelling. So  I thought I would write a summary here so my thoughts are in order before I give this evening’s talk.

What is mathematical modelling?

A description of a system that uses mathematical concepts and language (Thanks wiki). The keys things in this definition is:

  • ..description.. a model is just a description. The difference here is that the description isn’t just words, we have the whole language of maths available to us
  • system… the description has to be of something that has some sort of order in it. Which is, admittedly most things. But not all. Think  xkcd055 (ueseless):
  • uses mathematical concepts and language: As I pointed out initially, the description is in maths. And not just the maths you see in high school . We like using graph theory, group theory and pictures to get out models across as well. And all the other maths

What are some examples mathematical models?

  • e = mc^2 : this models the relationship between energy and mass.
  • Modelling the vibrations of a molecule. Compounds have their molecules ordered in a stuctured way. Due to the structure, there is certain ways the molecules can move. We use group theory to model the symmetries of the molecules in order to determine the vibrations
  • Road networks. The roads become the edges of a graph, the locations become the nodes. Now that we have a nice graph we can use well known algorithms like Dijkstra’s to find the shorted route between two locations.

What type of models are there?

  • Empirical vs. mechanical models. For an empirical model we collect data and try to fit a model to it. Mechanical  models we take something we already know, and try to adjust it to fit some other problem.
  • Discrete vs. continuous models: Discrete models only take in given quantities, continuous models take all values. Population growth is normally a discrete model: we don’t want to model the population down to the very second. On the other hand, modeling the leak in a tap is normally continuous.
  • … There are really heaps more. I could go on forever if I had to list them all
  • I’m going to focus on discrete models

What do we use models for?

  • To help explain a system. Mathematical models are often very concise, so it is much easier to see what is really happening.
  • To make predictions. Understanding how a model works means that we have a better idea on what will happen in the future, meaning we can make more accurate predictions.
  • To assist in making decisions. Since we can now  make predictions, we can make decisions. Big business loves making accurate predictions

Next post! Actually doing some modelling.

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One Response to “Mathematical Modelling: Part 1”

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  1. Mathematical Modelling: Part 2 | I like stuff - 26 June, 2013

    […] Continued on from Mathematical Modelling: Part 1 […]

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