TITLE
Ptolemy's Inequality
SOURCE
Dr John Pulham, Department of Mathematical Sciences
University of Aberdeen
e-mail pulham@maths.abdn.ac.uk
AREA
history, metric spaces
KEYS
Ptolemy, inequalities, metric spaces
LEVEL
final year
LENGTH
1/8 final year (but short). 30? pages
PREREQ
a basic knowledge of metric spaces would be an advantage.
HISTORY
used once
DESCRIPTION
This short project really starts off from Apostol's paper [1].
This should be studied and reported on.
A general description of Ptolemy's work and of Alexandrian
mathematics in general should be included.
References:
[1] T.M. Apostol "Ptolemy's inequality and the chordal metric"
Math Mag. 40 (1967)
[2] C.B. Boyer "A History of Mathematics" Wiley 1948
[3] D.J. Schattschneider "A multiplicative metric"
Math Mag 49 (1976)
[4] G.J. Toomer "Ptolemy's Almagest" Duckworth (1984)
TITLE
Square functions and curves
SOURCE
T.P.McDonough, Department of Mathematics, University of Wales,
Aberystwyth.
e-mail:tpd@aber.ac.uk
AREA
Analysis
KEYS
periodic functions, approximating functions, Fourier series.
LEVEL
elementary undergraduate.
LENGTH
One of three investigations undertaken by students in a group
of about four. The course is of one semester duration and is
1/12 of a student's degree commitment for the year.
Each student presents a report on each of the topics
investigated by the group. This report is the basis of the
student's assessment.
PREREQ
A-level Mathematics.
HISTORY
The project has actually been used in this form.
DESCRIPTION
The graphs of two simple periodic functions are sketched --
one is a step function, the other is formed with line
segments at $pi/4$ to the axes.
The periodic functions themselves may be described using
functions such as sgn($x$), abs($x$) ( = $|x|$) and int($x$)
( = $\lfloor x \rfloor$) and the elementary functions.
Find such descriptions for these and similar functions.
(Perhaps, you might choose some simpler ones to start with).
These functions may also be approximated by continuous
functions such as
$$a_1 \sin kx + a_2 \sin 2kx + a_3 \sin 3kx$$
by choosing the constants $a_1$, $a_2$, $a_3$ and $k$
appropriately.
Make a number of choices and quantify how well your
approximations fit the functions.
The locus of the point ($\cos t$, $\sin t$), with $t$ real,
is a circle centred at the origin.
This may be considered as an approximation to a square curve
obtained by inscribing a square in the circle.
Can you find a better approximation of the square curve
using the trigonometric functions sine and cosine?
Can you describe the square curve parametrically in the form
($f(t),\ g(t)$) where $f(t)$ and $g(t)$ are expressions
involving the simple functions sgn, abs and int?
Generalise to the case of a regular hexagon and a regular
octagon.
References:
"Square functions"; J.Thomas; Math.Gaz.vol.72; no.462
"Square waves"; S.Morley; Math.Gaz.vol.71; no.456
"The Mathematical Experience"; P.J.Davis \& R.Hersh
(Fourier Analysis)
TITLE
Constructibility of the Regular Polygons
SOURCE
Dr John Pulham, Department of Mathematical Sciences
University of Aberdeen
e-mail pulham@maths.abdn.ac.uk
AREA
Geometry, Algebra, number theory
KEYS
geometric constructions, field extensions
LEVEL
final year
LENGTH
1/8 final year. 30-40 pages
PREREQ
a first course in rings and fields. Galois theory is not
needed.
HISTORY
used
DESCRIPTION
A regular polygon is one which has all its sides of the same
length and all its vertex angles the same size
(and is convex?). A problem that was considered by the Greeks
and which remained of interest for a long time afterwards
was to determine which of the regular polygons can be
constructed using a ruler and compass. It was finally solved
by Gauss. The required result is that an $n$-gon can be so
constructed iff $n=2^kp_1p_2\ldotsp_r$ where the $p$s are
distinct primes of the form $2^{2^a}+1$.
Your job is to write an essay centred on this result. Space
and personal choice will determine exactly what you put in
and what you leave out, but some side issues that you might
like to consider are (1) does it make a difference if the
compasses hold their shape (modern ones) or collapse (Greek
ones) when lifted off the paper? (2) Constructions with
compasses alone. (3) Constructions using a ruler and a fixed
circle. (4) Constructions where the rules are relaxed
slightly in various ways.
The main elements of the proof of the main theorem are
fields and groups. So the essay will be a mixture of
geometry and algebra with a bit of history thrown in to
set the scene.
References:
[1] Jones, Morris and Pearson : "Abstract Algebra and Famous
impossibilities"
[2] C.R. Hadlock : "Field Theory and its Classical Problems"
[3] H. Dorrie : "100 Great problems of Elementary Mathematics"
[4] Dan Pedoe : "Geometry and the Liberal Arts"
[5] F. Richman : "Number Theory: An Introduction to Algebra"
TITLE
Chord envelopes
SOURCE
T.P.McDonough, Department of Mathematics, University of Wales,
Aberystwyth.
e-mail:tpd@aber.ac.uk
AREA
Geometry
KEYS
Geometry, straight lines. envelopes, cardioid, epicycloid.
LEVEL
elementary undergraduate.
LENGTH
One of three investigations undertaken by students in a group
of about four. The course is of one semester duration and is
1/12 of a student's degree commitment for the year.
Each student presents a report on each of the topics
investigated by the group. This report is the basis of the
student's assessment.
PREREQ
A-level Mathematics.
HISTORY
The project has actually been used in this form.
DESCRIPTION
The circle can be described parametrically by
$x(t) = \cos t$,$y(t) = \sin t$ and the corresponding point
may be loosely referred to as the point $'t'$.
If $f(t)$ and $g(t)$ are two functions of $t$, the chord
$C(t)$ joining the points $'f(t)'$ and $'g(t)'$ is also a
function of $t$. As $t$ varies, the chords $C(t)$ are seen
to be tangents to another curve which is known as their
envelope.
Plot a selection of such chords when the functions $f(t)$
and $g(t)$ are of the following simple forms
\halign{\tabskip=20pt#&\hfill#&&${#}$\hfill\cr
&(i)&f(t)=at+b&g(t)=ct+d\cr
&(ii)&f(t)=at+b&g(t)=t^2 \cr
}
Where possible, find parametric descriptions of these
envelopes and plot them directly.
Make some other simple choices for the functions for $f(t)$
and $g(t)$ involving, for example, trigonometric functions,
exponential functions, etc.
In the case of envelopes which are closed curves, find the
area enclosed by them -- approximately, if necessary.
By taking sufficiently many values of the parameters and
finding the areas approximately, can you suggest how the
area depends on the parameters?
References
[1] "Circles,chords and epicycloids"; A.F.\& L.A.Beardon;
Math.Gaz.vol.73; no.465
[2] "The area of a cardioid"; W.W.Wilson;
Math.Gaz.vol.67; no.441
TITLE
Factorization of Polynomials
SOURCE
Dr John Pulham, Department of Mathematical Sciences
University of Aberdeen
e-mail pulham@maths.abdn.ac.uk
AREA
algebra, computer algebra
KEYS
computer algebra, polynomials, factorization
LEVEL
final year
LENGTH
1/8 final year. 30-40 pages
PREREQ
Basic knowledge of polynomials, modular arithmetic and
Chinese remainder theorem
HISTORY
used once
DESCRIPTION
Computer algebra systems are now very efficient at
factorizing large polynomials over the integers (and more
general rings). Surprisingly, this process is essentially
algorithmic whereas factorizing over the reals is certainly
not. Factorizing
$x^{15}+8x^{14}+18x^{13}+28x^{12}+23x^{11}+9x^{10}-25x^9
-49x^8-61x^7-23x^6+15x^5+41x^4+22x^3+3x^2-7x-3$$
as
$$ (x-1)(x^2+5x-3)(x^6+x^5+x^4+x^3+x^2-2x-1)(x^2+x+1)^3$$
is almost trivial.
You are to study at least one of the currently available
methods for performing such a factorization (e.g. Berlekamp's
method). Present the mathematical theory of the method and
show, by useful examples, how it can be applied.
You could also investigate the actual methods used by such
packages as REDUCE, Maple and Mathematica.
References
[1] D. E. Knuth The Art of Computer Programming : Vol 2
Addison-Wesley
[2] R.B.J.T. Allenby `Rings, Fields and Groups'
[3] J.F. Humphreys & M.Y. Prest `Number, Groups and codes'
TITLE
Floor polynomials
SOURCE
T.P.McDonough, Department of Mathematics, University of Wales,
Aberystwyth.
e-mail:tpd@aber.ac.uk
AREA
Analysis
KEYS
Step functions, polynomials, floor function, graphs, recursion.
LEVEL
elementary undergraduate.
LENGTH
One of three investigations undertaken by students in a group
of about four. The course is of one semester duration and is
1/12 of a student's degree commitment for the year.
Each student presents a report on each of the topics
investigated by the group. This report is the basis of the
student's assessment.
PREREQ
A-level Mathematics.
HISTORY
The project has actually been used in this form.
DESCRIPTION
Polynomial functions in one real variable and with real
coefficients
$$
a_nx^n+a_{n-1}x^{n-1}+\ldots+a_2x^2+a_1x+a_0
$$
are well-known.
Describe the principal properties of such functions.
In particular, how do the the roots (their number and
location) depend on the coefficients?
And how do the regions in which the function is increasing
or and decreasing,respectively, depend on the coefficients?
The {\em floor polynomial functions} in one real variable
and with real coefficients are defined as follows
$$
a_n\lfloor x^n\rfloor +a_{n-1}\lfloor x^{n-1}\rfloor
$$
Study a selection of such functions of degrees 1, 2, 3 and
4, comparing them with the corresponding polynomial functions.
Determine the roots of these floor polynomial functions.
How many roots do such functions have?
Or does the concept of root need to be modified for such a
question?
These functions will have a zero slope almost everywhere.
Can you provide a new and appropriate concept.
Can you develop formulas for integrating these functions?
Modify other familiar concepts concerning functions so that
you can make meaningful statements about the floor
polynomials regarding these concepts.
\vspace{7pt}
Families of polynomials may be described by the following
iterative process:
$$
f_n(x) = A(n,x)f_{n-1}(x) + B(n,x)f_{n-2}(x),
\hspace{.25cm}n\ge 2
$$
where $f_0(x)$ is a constant, $f_1(x)$ is a linear polynomial
in $x$ and $A(n,x)$ and $B(n,x)$ are simple polynomials in
$n$ and $x$.
Examine some of these families when the functions
$A(n,x)$ and $B(n,x)$ are particularly simple polynomials
in $n$ and $x$, e.g. constant and/or linear in $n$ and/or
$x$.
What statements can you make about their graphs, roots,
integrals, etc. and about the corresponding floor
polynomials?
References:
[1] "Legendre polynomials"; B.Symons; Math.Gaz.vol.66; no.436
TITLE
The Lorentz Group
SOURCE
Dr John Pulham, Department of Mathematical Sciences
University of Aberdeen
e-mail pulham@maths.abdn.ac.uk
AREA
Group theory, Relativity
KEYS
Lorentz group, relativity, matrix groups
LEVEL
final year
LENGTH
1/8 final year. 30-40 pages
PREREQ
basic group theory and some special relativity
HISTORY
used once
DESCRIPTION
TITLE
SOURCE
AREA
KEYS
LEVEL
LENGTH
PREREQ
HISTORY
DESCRIPTION
References:
TITLE
SOURCE
AREA
KEYS
LEVEL
LENGTH
PREREQ
HISTORY
DESCRIPTION
References:
TITLE
SOURCE
AREA
KEYS
LEVEL
LENGTH
PREREQ
HISTORY
DESCRIPTION
References
The aim of the project is to study fairly thoroughly the
structure of the Lorentz group and its action on Minkowski
space. The Lorentz group is the group of all 4 by 4 real
matrices $L$ satisfying $
Particular attention should be paid to those features in
which the Lorentz group differs in its behaviour from that
of the Orthogonal group. For example, null rotations.
The physical significance of the results should also be
explored.
Evaluation of Pi
Dr John Pulham, Department of Mathematical Sciences
University of Aberdeen
e-mail pulham@maths.abdn.ac.uk
Analysis
Elliptic Functions, Analysis
final year
1/8 final year, 30-40 pages
elementary analysis, perhaps some complex analysis
used once
How to calculate the value of $\pi$ to a million decimal
places.
Most of the traditional methods for calculating $\pi$
converge rather slowly. This makes it time-consuming to
obtain even a few hundred decimal places of the value.
One important exception to this rule is the Gauss AM/GM
sequence method which provides good quadratic convergence
and hence the possibility of calculating a huge number
of decimal places.
The method relies on a link between the AM/GM process and
the properties of elliptic integrals. This will provide the
student with an excuse for studying the properties of
complete elliptic integrals as a prelude to the $\pi$
calculations. A more extensive project could start from the
theory of elliptic functions, though this would require a
significant background in complex analysis.
Multiprecision real arithmetic is now available in many
places (e.g. Maple and Mathematica) so there is no reason
why the student should not implement the method.
[1] Abramowitz and Stegun (eds) "Handbook of Mathematical
Functions" (17) Dover
Cubes and hypercubes
T.P.McDonough, Department of Mathematics, University of Wales,
Aberystwyth.
e-mail:tpd@aber.ac.uk
Geometry
Geometry, cubes, hypercubes, cross-sections.
elementary undergraduate.
One of three investigations undertaken by students in a group
of about four. The course is of one semester duration and is
1/12 of a student's degree commitment for the year.
Each student presents a report on each of the topics
investigated by the group. This report is the basis of the
student's assessment.
A-level Mathematics.
The project has actually been used in this form.
Determine the different types of plane cross-section of a
cube.
Are there other 3-dimensional solids with the same types of
cross-section exactly?
What are the different types of 1-dimensional cross-section
of the cube?
Describe the 4-dimensional analogue of the cube -- the
4-cube. Determine the different types of 1-, 2- and
3-dimensional cross-section of the 4-cube.
Continue this investigation for the $n$-cube where
$n = 5, 6, \dots$
[1] "A Steiner-type net for a cube"; J.Costello;
Math.Gaz.vol.75; no.474
[2] "How to iron a hypercube"; A.W.F.Edwards;
Math.Gaz.vol.75; no.474
[3] "The Mathematical Experience"; P.J.Davis \& R.Hersh
(Four-dimensional Intuition)
Integration in REDUCE - the Risch Algorithm
Dr John Pulham, Department of Mathematical Sciences
University of Aberdeen
e-mail pulham@maths.abdn.ac.uk
computer algebra
computer algebra, REDUCE, integration
final year.
1/8 final year. 30-40 pages
Some experience of field extensions is useful. Access to
REDUCE is important.
used
A computer algebra system like REDUCE is very effective at
the formal evaluation of integrals. The full internal
process being used, which should be studied, is complex but
has at its heart a methodology based on the Risch Algorithm.
Study this algorithm and its application.
[1] R.H. Risch "The problem of integration in finite terms"
Trans AMS 139
167-189
[2] Davenport, Siret and Tournier "Computer Algebra: systems
and algorithms for algebraic computation" Academic Press
[3] Buchberger, Collins, Loos and Albrecht "Computer Algebra:
symbolic and algebraic computation" Springer
[4] MacCallum and Wright "Algebraic computing with REDUCE"
OUP
[5] Geddes, Czapor & Labahn "Algorithms for Computer Algebra"
Kluwer