AC Mathematical Modelling

23/04/2018

AC Mathematical Modelling session on Game Theory

23/04/2018

Long time no see! Hope everyone had a great Spring Break!

Today, we introduced our members to the fields of Game Theory. Terminologies such as strategy spaces, payoffs, outcome matrix were explained and the methods of finding pure and mixed Nash Equilibrium were covered in details using the penalty kick game, tennis volley game and the roommate dilemma. We also applied our knowledge into one of the most well-known games ---- Rock, Paper, Scissors ---- to find the optimum strategy for both players.

Try to see if you are able to solve the Ultimate Warfare problem using these game theory knowledge. Answers will be out next Monday!

Please see the link below for the google slides and the problem:

Game Theoretic Models Mathematical Modeling @ AC

02/04/2018

The Inter-UWC April Mathematical Challenge starts today!

Link to the problems:

26/03/2018

26 March, 2018
Here is a link to the Geogebra animation we have created modelling the relative movement of the Sun, Planet Earth and the Moon with respect to time in an (X,Y,Z) coordinate system:
https://ggbm.at/dKXpmAZw

To further generalize this model to one with an elliptical orbit, we looked into the properties of the ellipse and the Mathematical Proof of Kepler's Second Law.

Video on the derivation of the equation of the ellipse:
https://www.youtube.com/watch?v=BHtIBLsGdNU

Video on the mathematical proof of Kepler's Second Law:
https://www.youtube.com/watch?v=Pa3Of_3vpRc

19/03/2018

19/03/2018

19 March, 2018
After re-organizing our thinking process, we introduced two ways to tackle the problem:
1. Using data to fit curve
2. Using vectors, rotation matrices to model the position of the sun relative to the earth in a coordinate system
Some significant progress on the "light tracking solar panel project" are made by members in the session today.

Below is a link to the powerpoint that we covered today in the session:https://docs.google.com/presentation/d/16grx_ajd3LbQ5gKsZm447DBrfh07lYbFYHIQpbMS45I/edit?usp=sharing

Today is the second day in March where the sun rises exactly from the East (90) and set off in the West (270)! What a coincidence!

19/03/2018

Picture 1:
Members working on the "Tracking Solar Panel Project"

Picture 2:
Today is the second day in 2018 where the sun rises exactly from the east (90) and sets off in the west (270).

13/03/2018

This time, a 2 by 2 rotation matrix was covered in the club activity. So, members had a look at a question from the Further Math mock exam. Yuhan, a second year, did a good job in explaining how to prove the rotation matrix. Unfortunately, the last half of the question could not be covered due to the time constraints, so it was assigned as homework. We will go through it quickly next week and make some progress in our solar panel projects hopefully.

05/03/2018

5 March 2018
Today, under the guidance of and introduction to the mathematics of tracking solar panel and barrage in lagoons to generate electricity by Dr. Martin, group members set off to apply their previous mathematical knowledge to address complicated real-life issues that are related to our campus.

Let's wait and see what solution our lovely mathematicians will provide in the following weeks! Hopefully, the answers to these modelling problems will actually be implemented to make our college more sustainable!

05/03/2018

AC Mathematical Modelling Update!

In an attempt to out-reach the whole UWC community to form a platform for students interested in Mathematics and initiate collaboration and knowledge sharing across different maths clubs within the UWC circle, we have contacted several UWC colleges including UWC Li Po Chun, Maastricht, Robert Bosch, Pearson, Mahindra and Changshu China.

After several discussions with people from UWC AC, LPC, Maastricht, Robert Bosch and etc. We have come up with an inter-UWC monthly Mathematical Challenge to start with that will hopefully initiate discussions that people will want to see happen.

The monthly mathematical challenge takes place on every Monday of the Month. It consists of 5 questions that are an extension to the IB syllabus and some a little bit difficult questions. Anyone who wants to join the challenge will have 2 weeks to work on the problem sets and then 2 weeks to discuss with your co-years from the UWC community to learn from each other on the Facebook group:
Inter-UWC Mathematical Challenge

If you have any questions regarding the problems or if you would like to contribute mathematical questions to the challenge, please reach us at [email protected].

Further details and the questions can be found in the PDF:
https://drive.google.com/open?id=1CiOtMHTNjLQJlUGo-8AkHkp1P94TaneD

All the best,
AC Mathematical Modelling

20/02/2018

19 Feb 2018
In this session, we talked about how to see matrix multiplication from a row and column point of view. For example, if you use matrices to solve simultaneous equations, the row picture tells us the function of lines in a coordinate system. The solution then is the intersection of these lines. However, if you see it from column perspective, each column represents a vector in the coordinate and the solution is the stretch of each vector so that they add up to a given vector.

Using column pictures and row pictures, we are able to prove that diagonalization happens iff the diagonal values of the diagonal matrix are eigenvalues and the matrix P consists of eigenvectors as its columns. Moreover, matrix P shouldn't be singular; otherwise, we will not be able to get the inverse of P.

We also went over the challenge question last week using diagonalization and De Moivre Theorem. As it's a rather long and laborious proof, I have attached the whole calculations below.

13/02/2018

12 Feb 2018
We completed the last part of the "Analytic Hierarchy Process" model, which is the method to check the consistency of evaluation matrices, today. The beauty of the AHP model lies in that we can incorporate our subjective opinions, which may not necessarily be consistent, or in other words, convey equivalent relationship (Equivalent relationship means A:B=a, B:C=b, then A:C=ab), while still constraining our subjective biases to ensure that our results are valid at the same time.
We also talked about matrix diagonalization and how to calculate powers of square matrices easily. Below is the link to a PDF on what we have done in yesterday's session:
https://drive.google.com/file/d/1KxOPetSpwQdOILNYYM5DyADqltB8Qh2s/view?usp=sharing