Transcript Week 7 lectureAx

MATH10001 Mathematical
Workshop
Mathematical Modelling and Problem
Solving
Traditional view of maths?
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Maths is useless
The only jobs maths can be used for are
accountancy and teaching
Maths has no link with the real world
The truth is that mathematicians have
changed the world,
Radio, digital revolution
Engineering
Computers
Security
Health
Telecommunications Fourier
Fourier Transforms
Joseph Fourier, 1768-1830
Mobile phones, digital radios, MP3 players
etc. all use these ideas.
, MP3 players etc all use the
sameideas
Internet shopping
Internet transfer protocols based on
mathematics
Credit cards and online shopping
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Modern encryption algorithms like RSA use
prime number theorems
Scanners
Security screening at the airport
MRI scanners in hospitals
Mathematical modelling in engineering:
eg. Finite element modelling to reduce vibration
Divide car into small
cuboids/tetrahedra.
Treat it a bit like masses
and springs in a network.
Vibrations modelled using eigenvalues of matrices.
Traffic Management
Variable speed signs
 “shock waves”
Many models of traffic flow
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Road design
Traffic control
Better throughput
Better safety
modelling
Real World
testing
Mathematical
Model
Modelling Cycle
Explanations
problem
solving
Solution
& Predictions
interpretation
Steps in the modelling cycle
1. Identify the problem
2. Define the variables and parameters
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Variables are quantities that can change in a problem
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Parameters are quantities that remain constant
3. Make assumptions
4. Write down a model - relationships between variables
5. Solve the model
6. Test the solution against the real life problem
7. Refine the model
Example – modelling the growth of bacteria
We start with 500 bacteria in a Petri dish. After one day we
have 525 bacteria, after two days we have 551 bacteria.
1. The problem is to find a formula for the number of bacteria
after n days.
2. Variables include the population, growth rate, time,
temperature, amount of food, amount of space left in dish
etc. Parameters include initial population, size of dish, initial
amount of food.
3. To simplify the problem we make certain assumptions –
ignore the amount of food, temperature and space in dish
and assume that the growth rate is constant.
4. Define the model: first introduce some notation:
n  time in days from start of experiment
k  growth rate
an  population after n days
a0  initial population
We write down the relationship
an  an1  kan1  (1  k )an1 for any n  1.
5. We have
an  (1  k )an 1  (1  k ) an  2  ...  (1  k ) a0 .
2
n
Assuming a growth rate of 0.05 (from our observations) we get
an  (1  k ) n a0  (1.05) n  500 for all n  0.
6. Test the solution: we test our solution against observations.
Day
Actual population
Predicted population
0
500
500
1
525
525
2
551
551
3
575
579
4
598
608
5
610
638
6
620
670
We can see that the model works well at the start but after 6
days the model is not accurately predicting the population. This
tells us that we need to modify our model.
7. The growth rate appears to be decreasing over time. This
could be due to a change in food available or room to grow.
We chose to ignore those variables in the original model. As
the growth rate is changing we could replace our constant
rate by one which is a function of n.
Why do we need mathematical models? Why don’t we simply
make lots of observations?