The stereotype mathematician in jokes is a male writing some incomprehensible formula on a blackboard and not being able to communicate the meaning of it to his audience. There are however jokes where the mathematician is viewed more favourable. I found two of them on the site from Simon Singh. ( Simon Singh is an English author, journalist and TV producer, specialising in science and mathematics. He wrote a book on Fermat's Last Theorem. As I mentioned Singh collects jokes, you'll find his jokes on his website. )
An assemblage of the most gifted minds in the world were all posed the following question:"What is 2 + 2 ?"
The engineer whips out his calculator, taps away at it for a while and finally announces "3.99".
The physicist consults his technical references, sets up the problem on his computer, and announces "it lies between 3.98 and 4.02".
The mathematician cogitates for a while, oblivious to the rest of the world, then announces: "I don't know what the answer is, but I can prove an answer exists!".
The philosopher strokes his chin for several days, finally asking: "But what do you mean by 2 + 2?"
Finally the accountant closes all the doors and windows, looks around carefully then asks "What do you want the answer to be?"
( By Helen Arney )
An astronomer, a physicist and a mathematician were holidaying in Scotland. Glancing from a train window, they observed a black sheep in the middle of a field.
"How interesting," observed the astronomer, "all Scottish sheep are black!"
To which the physicist responded, "No, no! Some Scottish sheep are black!"
The mathematician gazed heavenward in supplication, and then intoned, "In Scotland there exists at least one field, containing at least one sheep, at least one side of which is black."
( By Stephen Oman )
The course is divided into three related areas: graphs, networks and design. The Introduction introduces two themes of the course, combinatorics and mathematical modelling, and illustrates them with examples from the three areas.
Graphs 1: Graphs and digraphs discusses graphs and digraphs in general, and describes the use of graph theory in genetics, ecology and music, and of digraphs in the social sciences. We discuss Eulerian and Hamiltonian graphs and related problems; one of these is the well-known Königsberg bridges problem.
Networks 1: Network flows is concerned with the problem of finding the maximum amount of a commodity (gas, water, passengers) that can pass between two points of a network in a given time. We give an algorithm for solving this problem, and discuss important variations that frequently arise in practice.
Design 1: Geometric design, concerned with geometric configurations, discusses two-dimensional patterns such as tiling patterns, and the construction and properties of regular and semi-regular tilings, and of polyominoes and polyhedra.
Graphs 2: Trees Trees are graphs occurring in areas such as branching processes, decision procedures and the representation of molecules. After discussing their mathematical properties we look at their applications, such as the minimum connector problem and the travelling salesman problem.
Networks 2: Optimal paths How does an engineering manager plan a complex project encompassing many activities? This application of graph theory is called ‘critical path planning’. It is one of the class of problems in which the shortest or longest paths in a graph or digraph must be found.
Design 2: Kinematic design The mechanical design of table lamps, robot manipulators, car suspension systems, space-frame structures and other artefacts depends on the interconnection of systems of rigid bodies. This unit discusses the contribution of combinatorial ideas to this area of engineering design.
Graphs 3: Planarity and colouring When can a graph be drawn in the plane without crossings? Is it possible to colour the countries of any map with just four colours so that neighbouring countries have different colours? These are two of several apparently unrelated problems considered in this unit.
Networks 3: Assignment and transportation If there are ten applicants for ten jobs and each is suitable for only a few jobs, is it possible to fill all the jobs? If a manufacturer supplies several warehouses with a product made in several factories, how can the warehouses be supplied at the least cost? These problems of the system-management type are answered in this unit.
Design 3: Design of codes Redundant information in a communication system can be used to overcome problems of imperfect reception. This section discusses the properties of certain codes that arise in practice, in particular cyclic codes and Hamming codes, and some codes used in space probes.
Graphs 4: Graphs and computing describes some important uses of graphs in computer science, such as depth-first and breadth-first search, quad trees, and the knapsack and travelling salesman problems.
Networks 4: Physical networks Graph theory provides a unifying method for studying the current through an electrical network or water flow through pipes. This unit describes the graphical representation of such networks.
Design 4: Block designs If an agricultural research station wants to test different varieties of a crop, how can a field be designed to minimise bias due to variations in the soil? The answer lies in block designs. This unit explains the concepts of balanced and resolvable designs and gives methods for constructing block designs.
Conclusion In this unit, many of the ideas and problems encountered in the course are reviewed, showing how they can be generalised and extended, and the progress made in finding solutions is discussed.
Introduction Real Functions and Graphs is a reminder of the principles underlying the sketching of graphs of functions and other curves. Mathematical Language covers the writing of pure mathematics and some of the methods used to construct proofs. Number Systems looks at the systems of numbers most widely used in mathematics: the integers, rational numbers, real numbers, complex numbers and modular or ‘clock’ arithmetics.
Group Theory (A) Symmetry studies the symmetries of plane figures and solids, including the five ‘Platonic solids’, and leads to the definition of a group. Groups and Subgroups introduces the idea of a cyclic group, using a geometric viewpoint, as well as isomorphisms between groups. Permutations introduces permutations, the cycle decomposition of permutations, odd and even permutations, and the notion of conjugacy. Cosets and Lagrange’s Theorem is about ‘blocking’ a group table, and leads to the notions of normal subgroup and quotient group.
Linear Algebra Vectors and Conics is an introduction to vectors and to the properties of conic sections. Linear Equations and Matrices explains why simultaneous equations may have different numbers of solutions, and also explains the use of matrices. Vector Spaces generalises the plane and three-dimensional space, providing a common structure for studying seemingly different problems. Linear Transformations is about mappings between vector spaces that preserve many geometric and algebraic properties. Eigenvectors leads to the diagonal representation of a linear transformation, and applications to conics and quadric surfaces.
Analysis (A) Numbers deals with real numbers as decimals, rational and irrational numbers, and goes on to show how to manipulate inequalities between real numbers. Sequences explains the ‘null sequence’ approach, used to make rigorous the idea of convergence of sequences, leading to the definitions of pi and e. Series covers the convergence of series of real numbers and the use of series to define the exponential function. Continuity describes the sequential definition of continuity, some key properties of continuous functions, and their applications.
Group Theory (B) Conjugacy looks at conjugate elements and conjugate subgroups, and returns to the idea of normal subgroups in this context. Homomorphisms is a generalisation of isomorphisms, which leads to a greater understanding of normal subgroups. Group Actions is a way of relating groups to geometry, which can be used to count the number of different ways a symmetric object can be coloured.
Analysis (B) Limits introduces the epsilon-delta approach to limits and continuity, and relates these to the sequential approach to limits of functions. Differentiation studies differentiable functions and gives l’Hôpital’s rule for evaluating limits. Integration explains the fundamental theorem of calculus, the Maclaurin integral test and Stirling’s formula. Power Series is about finding power series representations of functions, their properties and applications.
What I never understood is that in books on Group
Theory Q8 is shown as a concrete group, i.e. the
group of quaternions
{i, j, k | i^2 = j^2 = k^2 = -1, i*j=k, j*k=i, k*i=j }
and not as an abstract group. Well, I just
discovered that it is fairly easy to construct Q8
from C2 x C2 ( which is often shown in abstract
form and concrete form: the Klein4 group ).
The group C2 x C2 has the following presentation:
<a,b | a^2 = b^2 = 1, a*b = b*a >.
The group Q8-abstract has the following presentation:
<a,b,c | a^2*c = b^2*c = 1, a*b*c = b*a >,
members of this group are:
{ 1, a, b, a*b, c, a*c, b*c, a*b*c }.
The following isomorphism can be established
between Q8-abstract and Q8:
f: Q8-abstract -> Q8
by
{ 1 |-> 1,
a |-> i,
b |-> j,
ab |-> k,
c |-> -1
ac |-> -i,
bc |-> -j,
abc |-> -k }.
Abstract Algebra
Theory and Applications
Thomas W. Judson
Stephen F. Austin State University
February 14, 2009
To the Reader
Exercises in a text generally have two functions: to reinforce the reader's grasp of the material and to provide puzzles whose solutions give a certain pleasure. Here, the exercises have a third function: to enable the reader to discover important facts, examples, and counterexamples. The serious reader should attempt all the exercises (many are not difficult), for subsequent proofs may depend on them; the casual reader should regard the exercises as part of the text proper.
GL(2,2) is the general linear group of dimension 2 over GF(2).
Gross asked which group is isomorphic to GL(2,2) ?
( I stopped the video and gave it a try. )
GF(2) has the following tables for addition and multiplication.
+ 0 1 x 0 1
0 0 1 0 0 0
1 1 0 1 0 1
The possible maps from F2 -> F2 have the following matrices :
0 0 1 0 0 1 1 1
0 0 0 0 0 0 0 0
0 0 1 0 0 1 1 1
0 1 0 1 0 1 0 1
0 0 1 0 0 1 1 1
1 0 1 0 1 0 1 0
0 0 1 0 0 1 1 1
1 1 1 1 1 1 1 1
Elements of GL(2,2) are the matrices which have determinant 1.
1 0 1 1 0 1 1 1 1 0 0 1
0 1 0 1 1 0 1 0 1 1 1 1
Order 1 2 2 3 2 3
We now see that GL(2,2) is generated by
0 1 1 1
1 0 1 0
and is isomorphic to S3.
Gross however had a different ( smarter ) approach as follows.
F2 is the following set:
{ (0,0), (1,0), (0,1), (1,1) }
A linear transformation from F2 to F2 must fix (0,0)
so the elements of GL(2,2) are the permutations of
(1,0), (0,1) and (1,1) with group S3.
I had my life to live over again, I would have made a rule to read some poetry and listen to some music at least once a week; for perhaps the parts of my brain now atrophied would have thus been kept active through use. The loss of these tastes is a loss of happiness, and may possibly be injurious to the intellect, and more probably to the moral character, by enfeebling the emotional part of our nature.
—Charles Darwin
Galois Theory is a topic which is, at least in the algebra books I have, covered in the last chapter as the most beautiful result of algebra. I know that Galois introduced group theory and proved that it was impossible to solve an equation of type f(x)=0, where f(x) has a term of x in the 5th degree or higher, by means of a formula. ( Solving the quintic by radicals is how it is described. ) What bothers me is that I still can't follow the proof, or worse: I simply don't get it.
I found a hint though. The Galois Group of x^2-1=0 is C2 and of x^4-2=0 the Galois Group is the Dihedral Group of order 8 ( symmetry group of the square ). Will play a bit with these examples, I hope it will break some ice.
Update: the field we work in is Q.
All TMA's are done now. - Due to the substitution rule which is used at the OU the 87 for TMA02 has been replaced by 91,25.
What's it worth? Well, it means I am placed for distinction but that has to be proved at the exam next tuesday. I haven't got a clue how I'll perform during the exam. It's in the middle of my afternoon dip: 14.30 - 17.30. There isn't a single topic I don't understand that's why I was able to score high TMA's. But an exam has a time limit.
I am not afraid of the exam because even at a score of 15 I am entitled to a resit and 40 is a grade 4 pass. But, but, but: I need 85 for a distinction. Still three days to go to the exam.
I reviewed MS221 - TMA04, made some minor corrections and decided that the A is ready for shipment to T. I nevertheless wait with shipping the TMA ( Cut-off date is 30 sep anyway ) because I haven't received TMA03 back yet. If I get a high enough mark on that one I might even get away with a low score ion TMA04. We'll see.
Here’s how a rough outline of what’s in the 2008 paper (hopefully this won’t constitute copyright infringement!):
Part 1
Question 1: Finding a closed form for a recurrence system
Question 2: Identifying and sketching a conic
Question 3: Stating the rule for some isometries, and for a composite isometry, and then using the double-angle and half-angle formulas to show a result
Question 4: Classifying fixed points of a curve, then sketching the graph of the curve and using graphic iteration construction
Question 5: Identifying basic linear transformations, applying them to a vector, and stating an invariant line for each one
Question 6: Finding eigenvalues, eigenlines and eigenvectors.
Question 7: Differentiation
Question 8: Integration
Question 9: Finding and manipulating Taylor series about 0, and classifying stationary points
Question 10: Finding the modulus and argument of a complex number, converting them from Cartesian to polar form and vice versa, and using the formula for powers of complex numbers.
Question 11: Using Euclid’s Algorithm and working with exponential ciphers.
Question 12: Combining variable propositions, finding a case for which a given proposition is false, and finding the converse of a proposition.
Part 2
Question 13: Looks like it’s about conics, but I haven’t done this one yet.
Question 14: Linear transformations
Question 15: I haven’t done this one yet either, but it looks like it involves differentiation, integration and stationary points.
Question 16: Groups
Solve systems of linear congruence equations using the Chinese Remainder Theorem.
Find a video lecture here ( Lecture #12 )
I was solving a problem from ' 104 Number Theory problems ' which required an application of the CRT. I had forgotten the algorithm. Instead of looking it up in a book I watched Song's lecture.
I mean how do my study results compare to other students? Do I actually care??! I read about people receiving TMA results of 99, 100 and so on. Than my average of 90 sofar seams low, very low as a matter of fact. On the other hand those who 'blog' 'brag' including me, I suppose. I wouldn't shout it around if I passed on 41/100, although it has exactly the same value to the overall degree result. - If a lot of students are scoring in the 80-100 range then the TMA's are just too simple imho. Since the student can make the assignments at home that means they can be quite challenging.
Substitution applied at : 87.7
Have been waiting for this a long time. Already have the tablet PC with Vista on which handwriting recognition works perfect. So all I need now is Windows 7. That's at least a valid and worthwile reason to upgrade to Windows 7.
PolarForm[z_] := {Abs[z], Arg[z]}
Example:
PolarForm[1+i]={ SQRT[2] ,PI/4 }
MS221 TMA02 - Result is in... 87. Very disappointing at first. I hoped for 90+. Looking back at the effort I put in book B I can only be satisfied with the 87 points. Had to do a lot of MathCad work. ( A program I can't get used to. I suppose that I am spoiled by Mathematica. )
VERY OFF-TOPIC: MJ
( His biographer made the right prediction.
I don't think he has it in him to take his own life. I don't see him putting a gun to his head. It'll be an accidental overdose - something like that. ( Stacy Brown, July 2005 )When MJ announced the London concerts I had the feeling it wasn't Michael but a double. )
Just got this promising book. A history of Group Theory from the beginning until the discovery of the Monster Group. Will publish a review here when I have read it.
Mathematics: is it the fabric of MEST?
This is my voyage
My continuous mission
To uncover hidden structures
To create new theorems and proofs
To boldly go where no man has gone before
(Raumpatrouille – Die phantastischen Abenteuer des Raumschiffes Orion, colloquially aka Raumpatrouille Orion was the first German science fiction television series. Its seven episodes were broadcast by ARD beginning September 17, 1966. The series has since acquired cult status in Germany. Broadcast six years before Star Trek first aired in West Germany (in 1972), it became a huge success.)
