| LIST OF THIRD YEAR PROJECTS FOR THE
ACADEMIC YEAR 2006/07 |
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Project
1: Is there a fair voting system?
There are a wide variety of different voting systems in use around the world. Each has been designed with a different purpose in mind, but we would like to believe that most of them are probably fair. But is it possible to prove that this is the case? This project will give an introduction to some of the various different voting systems that exist, consider what we might require from a fair voting system, and explain why our quest for a fair voting system may be doomed to failure. Part of this will be based on work of Kenneth Arrow (a Nobel prizewinner in Economics). |
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Project
2: A Mathematical model of the Severn
Barrage
Tidal estuaries offer a
means of extracting clean energy by using the movement of water to
power large turbines. Such a scheme was originally proposed for the
Severn estuary as long ago as 1925 and is now once again under serious
consideration. Tidal power is already used successfully to generate
electricity in many other parts of the world. The aim of this project
is to develop a simple mathematical model of such a system. This will
involve some inviscid fluid dynamics leading to a mathematical problem
requiring the solution of the relevant governing equation and boundary
conditions. The solution is likely to involve some analytical work
together with some algebraic computations. The main objectives are to
determine how the flow depends on the geometry of the system and to
discuss the results in the context of practical applications.
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Project
3: The Science of the Ethics of Finance:
Islamic Finance and other Ethical issues treated Mathematically Most financial instruments arising in
'western style' finance may be understood mathematically. Many
such instruments are designed
to work without ethical
constraints, or consideration of their social implications. On the other hand some financiers choose
to apply ethical and related constraints to their financial
instruments. A prime
example is the instruments of
Islamic finance (although there are plenty of others). These
instruments still operate in a framework which is prescribed by
Mathematical considerations, but they are also constrained by (say)
consideration of Sharia law. The purpose of this project is to provide
a mathematical, logical, semantic analysis of such instruments.
The first aim is simply to
explain the instruments mathematically, and analyse their relationship with the
ethical constraints which motivated them. The second aim is to use
this machinery to discuss
their fitness for purpose, and
perhaps to consider alternatives (and the reasons why
conventional instruments may not be acceptable). This is a rigorous project - following a
strict definition based construction of components, with logic
based arguments. It is anticipated that some of these
arguments will use mathematical language, and others logical argument in
English. As such a good mastery of English is (unfortunately) a
prerequisite.
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Project
4: The mathematics of
juggling
What kind of mathematics
can be used to describe juggling? And what kinds of questions can it
answer? This project will be an introduction to Combinatorics -- a branch
of mathematics used in many areas in Computers Science,
Statistics, and general Mathematics -- concentrating on juggling as a
source of particularly simple and easy to understand
examples.
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Project
5: Simple models of random processes
A core model in the
simulation of processes such as the evolution of stock market prices is
that of a random walk. The random walk on a line
(measuring, say, the cost of a stock over time) is closely related to the
combinatorics of the binomial theorem and Pascal's triangle. There
are a number of ways of generalising this to widen the potential for applications.
For example we might replace the walk
on the line by a walk on a more interesting shape. The idea of this project is
to investigate this problem by
applying and modifying a
number of techniques already broadly familiar from the undergraduate
syllabus.
The project is suitable for up to 3 or 4 students, who might collaborate to some extent in the scientific investigation, but who would be required to produce independent project reports. |
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Project 6: Non-linear stock
market price dynamics
Stock market prices are
strongly influenced by two classes of investors, so-called fundamental
value investors and so-called trend followers. Recently (2001) a simple
two-dimensional model has been proposed which simulates the influence of
such type of investors on prices in the stock market in a non-linear
fashion. One of the two dynamical variables is taken to be the logarithm of
the ratio of the observed price over the fundamental price. The other
variable is the market trend. The system is known to possess an unstable
fixed point when the observed price equals the fundamental one and the
market trend is zero, that is when the stock "does not know" in which
direction to develop. The
purpose of the project is to study this model in some detail depending on various market
parameters.
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Project 7:
Graphs and maps
Given a map of an area divided into regions, how many
colours are needed so that no two adjacent regions have the same
colour? Can you redraw the map so that each region is replaced by a
square, without altering which regions are adjacent? Can you visit
every region exactly once without retracing your steps?
These are examples of problems
that can be solved using Graph Theory, an area of mathematics with many
practical applications (for example in computing and
telecommunications). This project will provide an introduction to the
methods used to solve such problems.
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Project
8: Applications of
physics to economics and finance:
Money, wealth and the stock market Many problems arising in
the context of economics and finance can be analysed by using methods and
concepts from physics. For instance, by analogy with energy, the
equilibrium probability distribution of money must follow the exponential
Boltzmann-Gibbs law characterized by a temperature equal to the average amount
of money per economic agent. Based on this observation one can construct
a "thermal machine" which
extracts a monetary profit between two economic systems with
different temperatures. With such analogies one can investigate stock market
fluctuations, probability distributions of income and many other problems in
economics and finance. The purpose of the project is to test such ideas which
were presented in a recent (2003) essay.
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Project 9:
Symmetry and Groups
The notion of symmetry is fundamental
in Sciences, for instance in Chemistry the symmetries of a molecule
determine some of its properties. The aim of these projects is to study
the symmetries of 2 dimensional figures, like tilings or wallpapers
(think for example of some of the patterns designed by Escher) and some
3-dimensional figures (for example some molecules). The mathematical
concept underlying these symmetries is the notion of a "group". This
will be used to classify symmetry types of figures.
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Project
10: Quantum
finance
Most financial models
which simulate market behaviour are of stochastic type. In such models coherent
effects, such as the influence of the trade of securities on the price
distribution of the next trade are not captured. Recently (2003) it has been
proposed to use a quantum theoretical description, which can take
such effects naturally into account. The idea of the project is to describe
such a quantum theoretical model. Key ideas are for instance that all
possible realizations of investors holding securities are taken as a basis for a
Hilbert space. Linear operators acting in this space describe basic
financial transactions, such as cash transfer and buying and selling of
securities. Simple Hamiltonians can be used to describe the temporal
evolution of the market. This
project can accommodate 1-3 students.
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Project 11: The
geometry of animation
Shrek is an oscar winning
animated film which was "shot" by creating a virtual world in a computer
and then rendering scenes from this world entirely in software. The
rules for this world were: 1. obey the director; 2. obey a few rules
of ordinary physics, and 3. obey the rules of 3D geometry (such as
those studied in the Geometry and
Vectors course). The idea of this project is to exercise
or reinforce our
understanding of this applicable geometry by making a "mini-Shrek" (perhaps 5
seconds of animation). Step 1 is to create a virtual geometrical world
consisting, say, of a single human hand with water (perhaps rain) running down
it and collecting in a droplet at one fingertip. Here "create"
means to give a geometrical description of the shapes and positions of
objects at each of a number of moments in time. This might be done
using one of the standard computer languages for describing geometry
(these languages are easy to learn, and no prior knowledge would be
required). Step 2 is to "film" this action (i.e. to choose where to view
it from, how to move the "camera"
etc.). Step 3 is to render this scene into a format
suitable for viewing on
your online CV/web page.
THIS PROJECT IS SUITABLE FOR UP TO 4 PEOPLE WHO WOULD PRODUCE SEPARATE SOLUTIONS TO STEPS 1 AND 2, BUT COULD COOPERATE ON THE UNDERLYING THEORY AND ON STEP 3. |
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Project
12: Quantum computing
Computer chips are made
smaller and smaller and one naturally imagines that eventually a limit is reached
in this process once the atomic scale is reached and quantum
mechanical effects become relevant. However, at this scale new possibilities
emerge due to the occurrence of a phenomenon called entanglement, which is
specific to quantum mechanics and shall be explored in this project. In fact P.
Shor showed in 1995 that one can exploit this characteristic of quantum
mechanics and design computers which are far more efficient than classical
computers. The idea of this project is to give an account of the general
principles and concepts of quantum computing, in particular on Shor's
algorithm. By comparing the amount of additions and multiplications needed in a
classical computer with the number of quantum gates needed, the potential
power of a quantum computer will become apparent.
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Project
13: Solving unsolvable problems
For most problems in
theoretical physics or applied mathematics analytical solutions are not
known. Often this is only a matter of consistent investigations, but
many times the situation is hopeless as a proper mathematical framework
does not exist. Non-linear integral equations are examples for this, as
there is no systematic solution procedure developed for them. With the
advance of computer technology more and more problems allow for a
numerical treatment. The idea of this project is to write ashort
computing program to solve a set of coupled non-linear integral
equations which play an important role in the theory of quantum
integrable models. The problem can be treated without reference to its
background in physics, but according to preferences one can also learn
about the contextin physics where they occur. This project can
accommodate 1-3 students.
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Project 14:
Dimensions and Fractals: measuring complexity
The notion of dimension we are mostly
familiar with goes back to Euclid's definitions of point, line,
surface and solid and to Descartes coordinate geometry. This is
the notion of topological dimension, which tells us that points
have dimension 0, lines have dimension 1, the interior of a square
has dimension 2 and that of a sphere 3 etc. However this concept of
dimension can not always be used, for instance if we imagine a
curve such that it covers all the 2-dimensional plane (as for
example the famous Peano curves) and try to answer the question
of what its dimension is we would not know whether to say 1 or 2.
Precisely fractals are objects for which a similar problem
arises and a more suitable definition of dimension (fractal
dimension) need to be introduced. Examples of those are the
so-called Hausdorff dimension (1919) or the box-counting
dimension, which for most examples coincide. It turns out that we can find many
systems for which the fractal dimension and
topological dimension do not coincide. In fact, a definition of fractal
introduced by
B. Mandelbrot is that of a set whose dimension is strictly larger
than its topological dimension. Nonetheless, the essential
property of a fractal is that it has and infinitely complex
structure which extends over all length scales. It is also a
mathematical feature of
fractals that they are often generated by
recursive equations. That is the case of two of the most widely
studied groups of fractals known: the Julia sets and the Mandelbrot
sets.
Fractals are very interesting objects
from a mathematical point of view but they also resemble very often real structures or patterns we find in nature
(like tree leaves, snow flakes, shells or the color patterns of
certain animals). That makes them all the more fascinating!
In this project I would like the students to write an essay about fractals: how did they first appear?, how has their study developed?, what are their main characteristics from the mathematical point of view? etc. Several students could do this project, in which case each of them should concentrate (after a general introduction) on a different kind of fractal and describe the fractal of his/her choice in more detail. It would be very interesting if the students could actually generate some fractals numerically and therefore understand better how they come about. |
| Project 15:
The theory of special relativity The theory of special relativity was
proposed by Albert Einstein in 1905 in his paper entitled "On the
Electrodynamics of Moving Bodies". About 300 years
earlier, Galileo had already stated that all uniform motion
was relative, and that there was no absolute state of
rest. In other words whether an object moves or not depends on the
relative position of the observer with respect to the observed object.
Einstein's theory is based on two fundamental axioms:
As we have said, the first
axiom was already quite old, but the second one implied a completely
new understanding of the way space and time are tied together and, in
particular, contradicted the Newtonian notions of absolute space and
time. It had various a priori "surprising" consequences that have since
been verified experimentally: the equivalence of matter and energy
expressed in the famous equation E=mc², where c is the speed of
light, the fact that time and space become relative to the
observer and that on the contrary the speed of light becomes an
absolute quantity, independent of the velocity of the observer
himself.
The aim of this project is that the student or students write down an introduction to the theory of special relativity, emphasizing the mathematical aspects of the latter and explaining how the two main axioms above imply the relativity of space and time. The project should also describe the main differences with respect to Galileo's theory and show how the theory of special relativity is consistent with the latter for velocities much smaller than the velocity of light. |
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Project 16:
Neural networks
The description of nature
is usually carried out by means of some model or theory based on various
mathematical equations. However, often such model is not at hand or the situation
one would like to describe is simply too complex. Neural networks are
designed to overcome this deficiency and have the ability to derive meaning
from very complicated or even imprecise data. They can be used to extract
patterns and detect trends that are too complex to be noticed by either
humans or other computer techniques. There are different kinds of neural
networks, such as biological neural networks, as for instance the human brain
or parts of it and also artificial neural networks originally referred
to electrical, mechanical or computational simulations or models of
biological neural networks. Meanwhile the field has expanded so much that some
applications do not clearly resemble any longer an existing biological
counterpart. The adaptability of neural networks is so large that their
applications nowadays range from fundamental subjects as biology and physics even to
banking, finance, insurance, marketing, manufacturing, etc. The
purpose of the project is to provide an introduction to neural networks, give
their brief history, investigate simple network architecture and consider
some realistic applications. Students are encouraged to write a simple
program for some neural network, such as for instance for pattern
recognition or similar. The programs could written in MATLAB, Mathematica or Maple
or similar languages. This
project can accommodate 1-4 students.
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Project 17:
Knot theory
Take two pieces of strings
and tie a knot in each. Although they
may look very different, is there a way to determine whether the two knots are in fact the
same? This is one of the fundamental problems in Knot Theory, and
these projects will investigate the various methods that have been
introduced to try to solve
it. The study of knots is
a relatively new subject, with applications in various areas of Biology,
Chemistry and Physics.
In these projects the students will each explore a different aspect of the subject. The goal is to write a clear but rigorous introduction to the chosen topic, and to apply the results obtained to a variety of examples. |
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Project 18:
Cryptology
Cryptology is the science
concerned with communications in secure and usually secret form. The term
cryptology is derived from the Greek kryptós (hidden) and
lógos (word). Security is "guaranteed" to legitimate users, the
transmitter and the receiver,
being able to transform information into a cipher by virtue of a key, i.e. a piece of
information known only to them. Although the cipher is inscrutable and often
unforgeable to anyone without this secret key, the authorized receiver can
either decrypt the cipher to recover the hidden information or verify that it
was sent in all likelihood by someone possessing the key. Until
recently this was mainly important to military circles, but in the computer
age the applications of cryptology become more and more widespread, as for
instance in on-line business transactions. Meanwhile cryptology has
developed into an important branch of mathematics and the idea of the project
is to write an account on cryptology and explain some of its main
concepts and techniques. This
project can accommodate 1-4 students.
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Project 19:
The Goldbach conjecture
In 1742 Christian Goldbach
proposed in a letter to Leonard Euler the following conjecture: "Every
integer greater than 2 can be written as the sum of three primes",
where he considers one as a prime number. Nowadays the conjecture is
usually formulated as: "Every even
number greater than 2 can be written as the sum of two primes". Even though this conjecture has
been checked by computers up to 2 x 10^17, a rigorous proof is still
not known. Between 2000 and 2002 a prize of 1.000.000 $ was offered for
the proof, but it remained unclaimed. Goldbach's conjecture remains one of the
most challenging unsolved mathematical problems. The idea of this
project is to give an account of some of the attempts made so far and
understand their limitations and to design a computer program which
verifies the conjecture up to X, where X is the challenge. This project can accommodate
1-3 students.
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Project 20:
The butterfly effect: what is chaos?
In mathematics and physics chaos theory
is understood as the study of
certain non-linear dynamical systems which under particular conditions exhibit the phenomenon known
as chaos. The latter is most
famously characterized by the extreme sensitivity of these systems (equations) to the initial
conditions. Many examples of
such systems appear in the context of physics: the atmosphere, the solar system, plate tectonics,
turbulence in fluids but also in other domains such as economics or
populations growth. Some of
the best known systems which
exhibit chaos are:
The
Lorenz model: this is one the most interesting and more widely studied systems of equations
which exhibit chaos. The
equations involved are three coupled first-order non-linear differential equations which describe in
a simplified way the
convection rolls arising in the atmosphere
ẋ(t)=a (y(t) - x(t)), ẏ(t)= x(t)(r-z(t))-y(t) and ż(t)= x(t)y(t)-b z(t), with x(t) representing the intensity of
the convection and y(t), z(t)
related to the horizontal and vertical temperature distributions. a, r, b are positive
constants. In his
original work, Lorenz took a=10 and b=8/3 and varied r. He found chaos for
r=28. For these values of the constants, a three dimensional plot of the variables
{x(t),y(t),z(t)} for
different values of t yields
the famous Lorenz strange
attractor, characterized by its
shape, similar to the wings of a butterfly.
The
logistic map: Another
system of equations which
sometimes exhibits chaotic behaviour is the so-called logistic equation. The latter describes
a discrete one-dimensional
dynamical system characterized by
the equation:
x(n+1)= a x(n) (1-x(n)), where n=0,1,2 ...,
a is a constant parameter and x(n) takes values between 0 and 1. This equation was originally used as a
simple demographic model, with
x(n) representing the amount of individuals of a certain population at year n. For values of a>3.57 the system
exhibits chaotic behaviour,
which in particular means the values of x(n) for n very large are extremely sensitive to the
initial value x(0). Before
reaching the chaos regime there is a region where another peculiar phenomenon occurs namely that
of bifurcation.
The aim of this project
will be that the students
familiarize themselves with the most basic aspects of the theory of chaos and write an introduction about
the subject. Then each
student should concentrate on one of the models described above (or other examples) and answer if
possible questions such as:
for which values of the parameters does chaos emerge? What are the characteristic of the solutions which
lead us to conclude the existence of chaos? Are there
regions were stable solutions exist (fixed points)? How can be determine
those? how do the solutions
of our equations approach these fixed points? What is an attractor? What is a
bifurcation? etc. It would be very instructive if the students could use
some computer program (such
as MATHEMATICA or EXCEL) in order to investigate the equations numerically.
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Project 21: The
shape of the world
How do we know the world is round?
Is it possible that we live on a big donut rather than a sphere? Of
course, if we travel in space then we can prove that the surface of the
Earth is a sphere but people knew it long before travelling into space.
What was the evidence?
These projects will investigate different mathematical methods to tell surfaces apart and study their properties. The Poincaré conjecture is concerned with the same problem in 3 dimensions. Last August, Grigori Perelmann was awarded (but declined) the Fields Medal for his proof of this conjecture. |
| Project 22: The millennium problems In
May of the year 2000, in a highly publicized meeting in Paris, the Clay
Mathematics Institute announced that seven $1 million prizes were being
offered for the solution of each of seven unsolved mathematical
problems. These problems had been chosen by an international comitee of
mathematicians to be the most difficult and most important in the field
today. The seven millennium problems are:
You can find more
information about them in the Clay Mathematics Institute website www.claymath.org. A very nice and
simple introducction to each of the problems is also given in the book:
The millennium problems by Keith
Devlin. The idea of this project would be that you write a report
about each (or some) of this problems trying to answer questions such
as: what is the problem about? why do you think it is so difficult to
solve it? do you know of any attempts to solve it? why were they not
successful?
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| Project 23: How to become a millionaire
by solving a mathematics problem Various mathematical
problems are around for a rather long time without having been solved to this
day. In order to stimulate the interest of Mathematicians several prizes
have been announced for their successful solution. For instance, the
Clay Mathematics Institute has named seven prize problems which will be
awarded with 1000000$. One such problem is for instance the Poincaré
conjecture: If one stretches a rubber band around the surface of an apple, then
one can shrink it down to a point by moving it slowly, without tearing it
and without allowing it to leave the surface. On the other hand, if one
imagines that the same rubber band has somehow been stretched in the appropriate
direction around a doughnut, then there is no way of shrinking it to a
point without breaking either the rubber band or the doughnut. One says the
surface of the apple is "simply connected," but that the surface of the
doughnut is not. Poincaré, almost a hundred years ago, knew that a two
dimensional sphere is essentially characterized by this property of simple
connectivity, and asked the corresponding question for the three
dimensional sphere. Poincaré conjecture was recently solved by Grisha
Perelmann from St. Petersburg. He refused to accept the prize money as well as the fields
medal, which is the equivalent of the Nobel prize in Mathematics. The
idea of the project is to describe the Mathematics behind these problems and
possibly indicate to what extend progress has been made to solve them. You may
of course try to solve a problem and become a millionaire.
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Project 24: Number theory
applied to cryptography
Number theory is that
branch of mathematics concerned with the properties of integers.
It contains many
results and open problems that are easily understood, even by
non-mathematicians. More recently, the field has found
applications to practical
problems, the most popular of which is cryptography.
In this project we review the basic ideas of number theory that underpin several cryptographic protocols, with an emphasis on the Diffie-Hellman and RSA methods. |
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Project 25: Brownian motion in
Physics, Mathematics and Finance
Brownian motion is named
after the biologist Rober Brown (1773–1858). While examining
pollen grains and the
spores of mosses suspended in water under a microscope, Brown
observed
minute particles in the pollen grains executing a jittery motion.
He then
observed the same motion in particles of dust, and in other small
particles suspended
in liquids. Although he did not himself provide a theory to
explain the
motion, the phenomenon is now known as Brownian motion in his
honour. Albert
Einstein gave in 1905 a physical interpretation of Brownian motion
that helped
establish the reality of atoms, which until then had been regarded
as theoretical
speculation. More detailed theories followed Einstein's work, and
by now Brownian
motion is an important branch of the mathematical theory of stochastic
processes. In recent years, it has been found that Brownian
motion provides a
rather good description of the fluctuations of financial markets, and
has now become a fundamental object in all modern applications of
mathematics to Finance. The goal of this project is to present a
reasonably complete review of Brownian motion as a stochastic
process and of its various manifestations in the real world.
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Project 26:
Shadowgraphs in the ocean
One of the most important
mechanisms for the mixing of salt in the oceans is by ``salt
fingers''. These fingers consist of tall, thin (about 3cm) plumes
of water with a different heat and salinity to their
surroundings. One method that has been used to try and observe
these in the oceans is the "shadowgraph technique": a beam of light is
shone through a patch of water and the variations in the refractive
index of the water caused by the heat and salt variations causes the
parts of the beam to deviate. These deviations are observed by
projecting the beam onto a screen/camera. The purpose of this
project is to calculate the images that would be expected from the salt
fingers using
analytical and numerical methods and to compare these results with the images that were recorded in the oceans. |
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Project 27:
Wallpaper patterns and crystals
When you buy patterned
wall paper you will find that the pattern repeats itself, both along
the roll and across it. For big patterns the across roll pattern
may only be clear when it is hung. How many repeat patterns are
there? For example, in some patterns it will not matter which way up
you hang each strip (or even if you hang them horizontally!) while for
others it does. Some look the same in mirrors, others don't. In this
project you will determine rigorously the number of possible repeat
patterns for wall papers. This can then be extended to look at,
say, not repeating patterns in wall paper or the structures of crystals.
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Project
28: The use of Laplace transform in ordinary differential
equations
Many linear ordinary
differential equations do not have explicit solutions. One way that we
can examine the properties of the solutions is by using the method of
Laplace transforms. This technique will be covered in X3 Mathematical
Methods. It is a method that can sometimes reveal how a solution y(x)
behaves, for example, for small values of x or as x goes to
infinity.
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