Thursday, December 31, 2009
Lecture 23 of Abstract Algebra E222
Summary.
 Sylow Theorems
 Groups of order p*q
 Groups of order p^2 * q
 S5
Topic of the lecture: A5.
 Prove the following proposition: " If G is a simple group of order 60 then G is isomorphic to A5."
And more A5 stuff.
Wednesday, December 30, 2009
Jokes about math and mathematicians
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 )
Tuesday, December 29, 2009
Monday, December 28, 2009
Symmetry is everywhere
Sunday, December 27, 2009
Mathematica now supports handwriting recognition
I had a copy of Windows 7 Ultimate. UK version. That did not work as an upgrade of a Dutch version of Vista. Yakh. That delayed the upgrade several weeks. When I finally got hold of a Dutch 7 Ultimate I continued. MAJOR PROBLEMS, this time. The Wacom Driver ( I have a tablet PC ) did not work anymore, screen rotation portrait / landscape did not work, the HP specific buttons did not work anymore. I had to download Windows 7 drivers for most of the hardware. There is however no Wacom tablet driver for Windows 7. Bye, bye tablet. I thought for a day anyway. While browsing some forum I read a story of someone who experienced the same problems and was so kind to document what he did. He just reinstalled the Wacom Vista drivers. And that worked. There were several other issues but I managed to solve all of them. The entire upgrade took me about 12 ( Twelve ) hours. That includes the waiting and searching for problem solutions.
The good thing is that I can now use handwriting to enter mathematics into Mathematica. And that is extremely cool. It takes practice though, as with everything. Still discovering the howto's. The tablet features of Windows 7 did got a major upgrade compared to Vista. ( And Vista was already much better than XP in that respect. )  For now I decided to leave my desktop AS IS: Vista. An upgrade from Vista to Windows 7 is nice but it is not a must. Unless there is this specific feature you are after, like me with math handwriting recognition.
One more thing: initially handwriting did not work as described in the Mathematica documentation. After reinstalling Mathematica it did however work.
Thursday, December 24, 2009
Oxford numbers ( math mystery movie )
Wednesday, December 23, 2009
Mathematics in Movies ( 2 )
I got two math movies today. Fermat's Room and the Oxford Numbers. Both from 2008, maybe that explains why neither are on the page I mentioned. Haven't seen them yet though. Something for Christmas, perhaps.
I am not sure, but I think a movie about Ramanujan will be released in 2010. A must see. Ramanujan and Hardy in Cambridge.  Haven't watched Numb3rs in a while. I saved two seasons, for 'someday'. Well, maybe that day should be soon. Although Charlie is completely surreal, I mean he is an expert in every subject and has time to lecture, write books and be a fulltime consultant to the FBI as well. Secret: I adopted one of his habits: I often use a noisereduction headphone just to create silence around me, not to listen to music. It's great, you should try it while studying.
Tuesday, December 22, 2009
Lecture 22 of Abstract Algebra E222
He demonstrated a short way of calculating the conjugate of a permutation and he talked a bit about S5, including the number of conjugacy classes it has. Each partition of n in Sn represents a conjugacy class. So for 5 it is:
5 : 24
41 : 30
32 : 20
311 : 20
221 : 15
211 : 10
11111 : 1
Total 120=5!
The Stirling numbers of the first kind are used to calculate these numbers, which incidentally produce a Pascal triangle typeof matrix.
Sunday, December 20, 2009
Lecture 21 of Abstract Algebra E222
Then finite groups of certain orders were classified.
Order p, p prime. Cyclic groups of order p.
Order p * q, p > 2, p < q. Cyclic of order p*q and if p / (q1): Cq : Cp.
Order 2p, cyclic of order 2p and the Dihedral groups Dp.
Order p^2*q. Started with order 12:
 C12
 C6 X C2
 A4
 D6
 C4 : C3.
I can add that
 A4 = ( C2 X C2 ) : C3.
Gross sofar never talked about GAP, Mathematica, Magma or Maple. Group theory can get very complicated without the aid of a mathematics package. The groups were classified solely on the basis of the Sylow theorems in this lecture. The semidirect product has not been lectured yet. For those new I would say these lectures were real hard. Unnecessarily hard in my opinion. The next lecture is about the Symmetry group by a famous number theory guy ( forgot his name ).
Anyway, I discovered an interesting fact about S6 today, the symmetry group on 6 letters. It is the only symmetric group whose automorphism group is not eqaul to the group itself.  Now, why would that be? Seems like some deep fact to me.  This is were mathematics gets a grip on you. You must know why.
Saturday, December 19, 2009
Linear Algebra ( Books for 2010 )
 1. Vector spaces
 2. Linear transformations
 3. Isomorphism theorems
 4. Modules I
 5. Modules II
 6. Modules over a PID
 7. The structure of a linear operator
 8. Eigenvalues and eigenvectors
 9. Real and complex inner product spaces
 10. Structure theory for normal operators
Part 2: topics, contains another 9 chapters.
I think I am ready for part 1, since M208 contains linear algebra as well, this book definitely comes on my 2010 list. I just decided I am going to make a list for the math books I want to study in 2010 besides M208, MT365.
I was actually studying a book on Group Representation Theory. That's a topic which relies heavily on linear algebra. When I was studying Maschke's Theorem I realized I had to review my linear algebra, especially inner product spaces. And then I found Roman's book. My understanding of vector spaces (1) is ok, linear transformations (2) as well, and if not, M208 has lots of stuff on that. I studied the isomorphism theorems (3) in group theory, I think they are more or less the same in linear algebra. I have some notion about modules (4,5,6): like vector spaces but with scalars from a ring instead of from a field. Have to study them deeper, I suppose. Maybe the CG Modules ( vector spaces where the vectors can be multiplied with group elements as well as scalars ) I studied are a sort of modules, have to check it out. Looking forward to learn more about structure of linear operators (7) and (10) as well. The stuff in Eigenvalues and eigenvectors
(8) and Real and complex inner product spaces (9) is familiar but most likely goes much deeper here.
From a first browse through the book I can say that I like the style. Clearly written and enough examples. I hate books that don't have examples. I think that authors who don't include examples in their books are  A) too lazy, or B) not really willing to communicate their knowledge, or C) sadistic. In all cases bad people.  The book is packed with exercises but alas for the selfstudy student: no answers. I haven't made my mind up about authors who do provide exercises but keep the answers to themselves. Fortunately I found some problem books on advanced linear algebra as well. More on that another time, perhaps.
Thursday, December 17, 2009
Assessment strategy M208, MT365.
For M208: one exam, and the OCAS consists of 7 (not 4) TMA's.
Had a brief look at some past exams from both courses. Both exams seem doable, passable. Most of the MT365 is entirely new to me, so I must be careful not to underestimate this one. Rating the course at level 3 must have had a reason.
For the moment I'll stop preparing, prereading for M208, MT365, will do other math stuff.
Wednesday, December 16, 2009
Registered for MT365
What's in MT365? ( Although it's a level 3 course there aren't much hard prerequisites, just a certain mathematical maturity whatever that is. )
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 wellknown KÃ¶nigsberg 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 twodimensional patterns such as tiling patterns, and the construction and properties of regular and semiregular 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, spaceframe 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 systemmanagement 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 depthfirst and breadthfirst 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.
Registered for M208
What's in M208?
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 threedimensional 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 epsilondelta approach to limits and continuity, and relates these to the sequential approach to limits of functions. Differentiation studies differentiable functions and gives l’HÃ´pital’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.
Saturday, December 12, 2009
Real matrix representations of quaternions
i * j = k, j * i = k
j * k = i, k * j = i
k * i = j, i * k = j
i^2 = j^2 = k^2 = 1.
The quaternion group has eight members { 1, 1, i, j, k, i, j, k } and is nonabelian.
The most common presentation of Q8 is < a,b  a^4 = 1, a^2 = b^2, b^1*a*b = a^1 >.
Using only real numbers the quaternions can be represented as 4by4 matrices as follows.
Thursday, December 10, 2009
MS221: Grade 2 pass ( Cont. )
( From my Open University home page )
All in all, it has been a few hours since I know the result. I am happy with it, of course I am. I am fully motivated for the courses next year, that's what counts now. I can't wait to make another TMA actually.
Should register asap. Registration for the 2010 courses closes the 16th.
MS221: Grade 2 pass
2nd reaction: relief, not the feared grade 4 pass
3rd reaction: disappointment only 3pnts below distinction
4th reaction: doesn't mean a thing because 90% passed at eirher d, 2, 3 or 4
5th reaction: jealousy, 20% or so did get a distinction
Just recording what I felt when I read the pass.
I MUST DO BETTER NEXT TIME.
Time to sign up for M208, MT365.
( I had 82/100 on the examinable component, that's really bad. Yakh. )
Wednesday, December 9, 2009
Playing games with groups
 finding all groups ( of a specific order );
 finding the presentation formulas;
 finding the automorphism group;
 finding the character table;
 drawing the Cayley graph;
Once you start on this you get a sort of natural need for the use of either the Mathematica Abstract Algebra package, or even GAP ( or similar ), that is a mindset which makes learning the language rather easy.
Tuesday, December 8, 2009
Aut( Q8 ) = S4
The elements of Q8 are:
1:1, 2:1, 3:i, 4:j, 5:k, 6:i, 7:j, 8:k.
The generators are:
3:i, 4:j.
Then there are 24 isomorphisms, since there are 6 possible generators i, j, k, i, j, and k but after 1 has been chosen, 4 remain ( choose i then i can't be chosen anymore ).
So up to here I have shown that Aut(Q8)=24, not yet that it is isomorphic to S4. ( That's for later. )
Monday, December 7, 2009
Extensions of a group
The groups of order 16 are:
C16
C8 x C2
C4 x C4
C4 x C2 x C2
C2 x C2 x C2 x C2
D16
Q16
QD16
D8 x C2
Q8 x C2
C8 : C2
C4 : C4
(C4 x C2) : C2 (a)
(C4 x C2) : C2 (b)
At this moment I can construct
C16
C8 x C2
C4 x C4
C4 x C2 x C2
C2 x C2 x C2 x C2
D16
D8 x C2
Q8 x C2
C8 : C2
C4 : C4
Work to do on:
Q16
QD16
(C4 x C2) : C2 (a)
(C4 x C2) : C2 (b)
Saturday, December 5, 2009
C13 symmetry.
The presence of the number 13, a Fibonacci number, is not a coincidence in this picture.
(Picture from the book Visual Symmetry.)  Btw. that book could be a terrific Christmas present. Although interest in mathematics is not required, appreciation of beauty is.
Breaking news ( I missed ): TeXnicCenter 2.0 alpha released
To the point: the new version. It is really great news that the developers took the time to upgrade TXC. Although near perfect already new technologies in IDEs became standard in recent years which aren't in TXC 1. Cold folding is one of them and it has been implemented in TXC 2 according to the website. Coldfolding is really a musthave nowadays. I recently blew TXC and started to use an Eclipse based TeX IDE. After only one TMA I returned to TXC which simply works best for me.
I will install and test the version of TeXniCCenter 2.0 alpha and report my findings here.
TexNicCenter
Friday, December 4, 2009
Watched lecture 20 of Abstract Algebra E222
Thursday, December 3, 2009
Change of plan ?
Since I have to register for the 2010 courses before the 14th this month I have to make up my mind fast. I thought to be sure about 2010 but watching the E222 videos changed my mind. It was my plan to do MST209 and M336 next year, thus leaving M208 for 2011. ( Ideal would be M208 + MST209 but is just not advised to take on a load like that. If you slip a few weeks for whatever reason it is impossible to catch up. So M208 + MST209 is out of the question. )
Both M208 and M336 contain Group Theory. M208 introduces the basic concepts: groups, subgroups, cosets, Lagrange's theorem, normal subgroups, and quotient groups in one module. A second module covers conjugacy classes, homomorphisms and group actions. All stuff I understand fairly well. That's why I though M336 was an option although M208 is a prerequisite. M336 reviews the M208 stuff and then covers counting with the aid of group actions ( necklace problem, I suppose ). The theory of abelian groups is covered fairly deeply just as the Sylow Theorems. These topics are half of the course. The other half is about geometry using group theory. The solids in two and three dimensions ( what I have just seen in the E222 videos ), tilings, frieze patterns, lattices and the wallpaper patterns.
My conclusion is that M336 in 2010 and M208 in 2011 is not an option. M208 + M336 is an option though. M208 in 2010 and M336 in 2011 is not an option because M336 doesn't run in 2011, so M208/2010, M336/2012 is the second option.
Recapping...
MST209 + M336  NOT
1. M208 + M336  Option
2. M208 ( only )  Option
3. M208 + MT365  Option
Both M208, MT365 have MST121, MS221 as prerequisite, so that fits. ( That is if I have a pass for MS221 ).
Just MST209 is an option as well, I suppose, although in that case I might jeopardize the overall plan.
Wednesday, December 2, 2009
Watched lecture 19 of Abstract Algebra E222
Tuesday, December 1, 2009
Watched lecture 18 of Abstract Algebra E222
 Group actions.
 Counting formula.
 Conjugation action.
 Conjugacy classes.
 Class equation.
 Symmetry group of the Tetrahedron (A4),Icosahedron (A5).
 Nice proof of the fact that a group of order p^n always has a nontrivial centre.
 At the time this video was recorded mathematicians conjectured that shape of the universe was SO(3)/A5.
( For me not a particularly simple lecture, but am still following. )
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(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.)