AM1 ---
Orthogonal Polynomials
The Classical
Orthogonal Polynomials are a set of polynomials which arose in a number of
different situations and which share a number of similar properties. The object
of this set of projects is to examine a number of these different properties
for each of the functions of the set.
GROUP PROJECT
We are going
to examine these properties:
·
Differential Equation: Each of these polynomials satisfies a differential equation. The
polynomials arise out of series solutions of a differential equation.
·
Orthogonality Relation: If 1/Jn(x) is
the nth order polynomial of the particular type: the orthogonality relation
takes the form
where m does not equal to n and w(x) is a non
negative weighting function. This relation can be used to define the
polynomials.
where g (x) is a zeroth,
first or second order polynomial.
00
antn
(x) = g(x, t).
n=O
This relation comes in useful in proving a
number of other properties of the polynomials. Recurrence Relations Each
set of polynomials satisfies a relation of the form
and
(x) has n real distinct zeros in
the interval a < xi < b, where a and b are
the limits on the integral for the orthogonality relation. Also each zero
of 
(x) lies between a consecutive
pair of zeros of (x). INDIVIDUAL
PROJECTS
The aim of
each project is to digest the material in the text, completing the proofs where
necessary, to work out a few of the polynomials and draw their graphs.
I.
Laguerre Polynomials
II.
Associated Laguerre
Polynomials
III.
Legendre Polynomials
IV.
Chebyshev Polynomials of the
First Kind
V.
Chebyshev Polynomials of the
Second Kind
VI.
Chebyshev Polynomials of the
Thrd Kind
VII.
Chebyshev Polynomials of the
Fourth Kind
VIII.
Gegenbauer Polynomials
IX.
Hermite Polynomials
Notes will
be distributed on each topic.
AM2 --- Integral
Theorems of Vector Calculus
This project area
is based on MATH225 and includes integral theorems which follow from the
Divergence Theorem and Stokes' Theorem, integral expressions for scalar and
vector potentials for solenoidal (divF=0)
and irrotational (curlF=0)
vector fields, Helmholtz's Theorem, derivation of
Maxwell's equations from relevant integral theorems of electromagnetism,
derivation of heat equation, and finally, discussion of how one can establish
whether a solution to a governing partial differential equation is unique. All
these topics can be considered as an introduction to the theory of partial
differential equations and their applications in mathematical physics.
In “further
applications of vector calculus and related topics”, this project area can include the concept of the
delta function and conservation laws for fields with sources and/or sinks, the
Green's function method which is used to solve inhomogeneous boundary value
problems for linear partial differential equations (PDE' s),
and the method of images which is used to construct Green's functions for the
Laplace operator. We shall also consider Cartesian tensors (and show that the
stress tensor introduced in MATH225 is indeed a Cartesian tensor), discuss some
elements of the analytic function theory and show how this theory can be used
to study fluid flows. We shall also consider second order linear PDE's, describe the method of characteristics and how this
method can be used to solve the one-dimensional wave equation, and finally, we
shall discuss some elements of the potential theory, prove the mean value
theorem and the maximum principle for the Laplace equation, and find out
whether it is always possible to find a solution to the Laplace equation which
satisfies given boundary conditions.
The notes
will guide you through each topic step by step. You should read them carefully
and discuss any difficult points with your tutor until you understand them as
well as possible. In some cases you are referred to a book which explains
certain concepts in more detail. If you have any problems finding the book
please contact your tutor.
GROUP PROJECT
Weeks 1-4. Review relevant ideas from :MATH225:
Grad, div, curl; line, surface and volume integrals, flux integral; Divergence
Theorem; Stokes' Theorem; continuity equation for fluids, stress tensor and
equation of motion, incompressible fluid flows and associated stream functions, irrotational flows and associated velocity potentials.
Distribute notes on core extension material (for group study over next 2
weeks): introduction of the delta function: properties of the delta function;
delta function in more dimensions. In “further applications of vector calculus and related topics”, we also
review Green's identities; multiply connected regions, Green's Theorem for
multiply connected regions.
Week 3 Discuss
progress with extension material and help with difficulties. Week
4 Group presentation of core material. Assign individual topics and
distribute materials.
INDIVIDUAL PROJECTS
Possible topics:
X.
Delta Function and Conservation Laws
XI.
Green's Functions
XII.
Cartesian Tensors
XIII.
Analytic Functions and Fluid Flows
XIV.
Method
of Characteristics and Wave Equation
XV.
Introduction
to Potential Theory
Alternative topics are:
I. Divergence Theorem and some related theorems
II. Stokes' Theorem and some related theorems
III. Expressions for scalar and vector potentials. Helmholtz's Theorem
IV. Integral theorems and Maxwell's equations
V. Heat equation
VI. Uniqueness of solution
Weeks 5-8 Help students with individual topics.
Week 9 Discussion of structure and content of group project, and division of responsibilities.
Weeks 10-11 Help students with project writing.
Week 12 Group presentation.
AM3 --- Population
Dynamics Using Maple
This topic is about the use of
mathematics, a calculator and/or the computer algebra system Maple, to analyze
a number of different models in population dynamics. To make the most of this
topic you will need to be comfortable about using Maple and be prepared to
extend your knowledge of its capabilities: to define functions, solve both
algebraic and differential equations, manipulate
matrices and plot graphs.
These notes guide you through the
topic step by step. You should read them carefully and discuss any points of
difficulty with your tutor until you understand them as well as possible. When
you write your chapter of the group project, you should demonstrate your
understanding by using examples which are different from those treated here,
and, wherever possible, by presenting the material in your own words.
GROUP PROJECT
1.
Single Species Models - Continuous Case
2.
Single Species Models – Equilibra
3.
Single Species Models – Harvesting and restocking
4.
Single Species Models – Continuous class-structured case
5.
Single Species Models – Discrete case
6.
Single Species Models – Discrete class-structured case
7.
Two Species Models – Continuous case
INDIVIDUAL PROJECTS
Project 1- The SIR model
The SIR model
is concerned with the spread of an infectious disease. The full model involves
the inter-related time-development of three classes of individuals:
Susceptible, infected and recovered/removed - see section 4 of the core notes.
In this project you will investigate two simpler versions of the SIR model
- in the spirit of mathematical modeling - and then an application of the full
model to discuss the effect of the Great Plague of 1666 on the population of
the
Project 2 - Discrete systems in a tropical paradise
Vegetable
Ivory from
Project 3 - Predator-prey models
Predator-Prey
(PP) models have two populations, X the predator, which uses y the prey, as a food source. In the simplest model, the development of the
two population densities is governed by a coupled system of two differential
equations - see section 7 of the core notes for generalities of the mathematics
of continuous 2-species systems.
In this project you will consider the basic theory of PP models for which
each population has a Malthusian growth law in the absence of the other. You
will then look at two enhancements of the basic model: the prey population has a
carrying capacity, and this will be applied to the real data for a population
of reindeer on a South Atlantic island; a model in which a proportion of the
prey can hide from the predator.
Project 4: Competition Models
Competition
models have two populations, X and Y, which compete for a food
source or living space. In the simplest model, the development of the two
population densities is governed by a coupled system of two differential
equations of the form: x’ = ax- bxy, y’=gy-dxy, where the parameters are assumed to be positive. These
are These are sometime called Lotka-Volterra
equations, named after the mathematicians who first studied them and other
related equations. Comment on the terms in these equations, and describe what
would happen to each population if the other were not present. We tend to think
of the species involved in such models as being from the animal kingdom - two
species of herbivores, for example. Can you think of a situation where one or
both of the species could be other than animal? This system of equations is one
of the few that can be integrated.
In this
project, we shall consider determining the equilibria, and examining the
behaviour near to these, for several systems
Project 5: Harvesting and Drug Administration
In the project,
consider the following scenario: A population n(t) which satisfies the Malthus
law with relaxation time r, having initial population density n0 = n(0) , is
reduced by N after each interval of time r, What is the population density at t = r, , immediately before and immediately after the reduction has taken
place? Find the relationship between the various parameters which ensures that
the population has not become extinct. Assuming that condition holds, allow the
population to develop in a Malthusian way.
Several
tasks will be carried out.
Project 6: Population Growth and Earth’ Human
Carrying Capacity
It is very easy to turn the predictions of a model into reality in one's
mind and to forget that the model is only a model, and as such, depends on a
number of assumptions, which may be only partially true (or not true at all!)
The chief assumption in the Malthus model is that the growth rate constant is
actually a constant. This means that no changes in birth rates due to birth
control programmes or economic circumstances, no changes in life expectancy and
child mortality, and so on. Quite clearly, such assumptions are unreasonable to
a greater or lesser degree. More accurate models, as used by the UN, try to
take account of these complexities. Nevertheless, the very crude Malthus model
is still useful in at least three important ways: It provides a starting point
for more sophisticated models; It provides a framework
of understanding; Its predictions provide a yardstick for comparison of
empirical data and the results from more complex models.
This project investigates the Malthus model and the Verhulst
model.
Project 7: The Verhulst Model
and Immigration/Harvesting
In the project, your first task is to investigate the Malthus model in
its basic form. Given a table of data for the population of the
Whether or not you are satisfied with any of the fitted curves, do any
of them give sensible predictions for the population of the
Your next tasks will to modify the model to allow immigration and harvesting.
Project 8: The Verhulst Model
and Immigration/Harvesting
Given a specific 2-species system of
two coupled ODEs: x’ = f(x)+
axy, y’=g(y)+bxy, where f,
g are quadratic, show that there can be at most four equilibria. Give a
mathematical example where there are no equilibria. Is your example
biologically feasible? Feasibility and Stability will be our main tasks in this
project.
Project 9: Equilibria and Symbiosis
First consider the
equations for two species with populations X and Y
which
interact with one another in a symbiotic way so that each species improves the
conditions for growth of the other species. In the simplest model, the
development of the two population densities is governed by a coupled system of
two differential equations ad the form: x’ = -ax+bxy, y’=-gy+dxy. Compare this equation with those in projects 3 and 4.
Solve this equation to give a relationship between y and x. If
you tackle this by hand, don't expect to obtain y as an explicit
function of x, and in particular it is not easy to draw the trajectories
by hand in the phase-plane. However the Maple solution is explicit, involving
the Lambert W function - find out about it using the Maple help system.
Further investigate
a more realistic model when carrying capacity is allowed.
Project 10: Predator-prey models with Harvesting
The PP model for population densities of two-species is a system of two coupled differential
equations. Harvesting can be modeled by adding a new term to one of the
equations. For different scenarios, through Maple solutions, describe the general
properties of the solutions and think of harvesting strategies which might
prevent extinction.
AM4 --- Topics
In Mathematical Economics
This project is concerned with three topics:
using Maple to further investigate topics covered in the MATH227 lectures;
extending the Theory of Demand as covered in the MATH227 lectures; and an
initial investigation of linear models in microeconomics. The use of Maple may
well be helpful in the latter two topics as well, to streamline the calculations.
GROUP PROJECT
1.
Introduction
The idea of a utility function to quantify a
consumer's preferences was introduced in the lectures. As usual we consider two
commodities and define U(x,y)
to be the utility function where x and y are the amounts of the two
commodities. What properties are required of a utility function? The group
should consider various possibilities for a utility functions and use Maple to
check that the properties are satisfied. (Start with some known examples from
the lectures.)
2. Multimarket equilibrium and the Edgeworth
Box
Suppose we have two consumers A and B who start
out with certain amounts of two commodities. Will they wish to exchange some of
these commodities and if so how much?
3. Linear
Programming
There are many aspects of microeconomics which
behave linearly or approximately so. In multi-market systems, goods, e.g. steel
may be outputs of one process and inputs of another. With such systems it is
generally useful to describe quantities by a vector in commodity space, not distinguishing
between inputs and outputs. We consider n commodities and assemble them into an
n-dimensional vector. We also define a {\it primary} commodity to be one which
is only an input and never an output; a {\it secondary} commodity is one is an
output of some industry. Suppose there are m primary commodities and therefore
n-m secondary commodities. We shall consider the first (n-m) components of our
vector to represent the amounts of secondary commodities, and the remainder to
represent the amounts of primary commodities.
INDIVIDUAL PROJECTS
Project 1. Cost Functions
Suppose a producer uses amounts x and y of two
inputs, and that q(x,y) is
the amount of product made and c(x,y) is the cost of
using these inputs. We wish to minimise c(x,y). Write a Maple program to
evaluate the cost function for a given amount of product q(x,y) and c(x,y). (You may be
able to adapt your earlier program for the utility function.) Try various possibilities
for c(x,y) and q(x,y) and see what results you get for the cost functions.
Again, start with examples from lectures or homework to check you get the right
answers.
Project 2. Multimarket Equilibrium
Returning to the Edgeworth
Box for the consumers A and B, find the locus of the points for which their
indifference curves touch, i.e. the relation between y and x for these points. Plot this curve on your Edgeworth Box diagram. There are 8 small tasks to be done
in this project involving Maple solution of equations and various
investigations.
Project 3. Monetary Equilibrium
This project investigates the optimisation of utility functions under budget constraints. Next it considers some generalisations in which non specific values are used, either in the budget constraint or utility function. Finally it investigates any change in the original scenario that will ultimately affect the prices of the two commodities, as is the case in all real life situations.
Project 4. Wages and Leisure
We can regard income and leisure
as a pair of commodities which the worker wishes to exploit to his maximum
advantage. Suppose that x is his amount of leisure time and y is his income.
Also, suppose p is his rate of pay and T is his available time (e.g. T could be
1 day if he is being paid daily). Then he works for a time T-x and we have y=p(T-x), or px+y=pT. This is effectively a budget constraint. We suppose as
usual that there is a utility
function U(x,y) which the
worker will choose x and y to maximise subject to the
budget constraint. The analysis is as usual but with q=1 and also B=pT. We will ask whether the employer can make the worker
work longer by increasing the pay rate p. This may seem obvious but in fact
will depend on the utility function.
Project 5. Linear Programming and Input-Output Analysis
First consider an example from simple linear
programming: A printing company prints two weekly magazines, "Hello"
and "Eh up" using paper and ink. It takes 5 tons of paper and 1 pint
of ink to produce 1 batch of "Hello", and 4 tons of paper and 2 pints
of ink to produce 1 batch of "Eh up". Each week the company is
supplied with 17 tons of paper and 7 pints of ink. The profit on
"Hello" is $10 per batch and that on "Eh up" is $5 per
batch. How many batches of each magazine are produced per week and what is the profit?
If there are more than two variables, a
graphical method of solution is not possible. It is then necessary to work out
the size of the objective function at the vertices of the feasible region. To
calculate every vertex and the corresponding size of the objective function
would take a long time and is not necessary. The best way is to start at one
vertex, usually the origin if it belongs to the feasible region, and then find,
if possible, an adjacent vertex which increases the objective function. This is
continued until a vertex is obtained such that no adjacent vertex improves the
objective function. This vertex is then the optimal solution. Can you prove
that any vertex which is not the optimal solution always has an adjacent vertex
which improves the objective function?
Project 6. Further Study of Game theory
Some further tasks are waiting for your
solution.