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 KĂ¶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’HĂ´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. )
Monday, November 30, 2009
Watched lectures 15,16 and 17 of Abstract Algebra E222
Sunday, November 29, 2009
Extending C2 x C2 to Q8
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 Q8abstract 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 Q8abstract and Q8:
f: Q8abstract > Q8
by
{ 1 > 1,
a > i,
b > j,
ab > k,
c > 1
ac > i,
bc > j,
abc > k }.
Q8 is not something like ( C2 X C2 ) : C2, where X stands for direct product and : stands for semidirect product, but it is very likely something similar. I read briefly that there are ways to construct groups other than using the direct or semidirect product. Will / must take some time to check this out.
The fact that Q8 can be constructed and has a fairly simple presentation predicts that there must be similar methods for all other finite groups.
Tuesday, November 24, 2009
Watched lecture 14 of Abstract Algebra E222
Gross: "The most important theorem in calculus is the Intermediate Value Theorem."
What is the error?
Sunday, November 22, 2009
Watched lecture 13 of Abstract Algebra E222
One question was as follows.
Q. Show that if a group has a unique element of order 2 then it is part of the center.
A. Order of element is equal to order of conjugate and because there is only one element of order 2 the following is true.
a = g a g^1 for all g in G, or
a g = g a for all g in G,
and thus a belongs to the centre of G.
The lecture introduced subgroups of GL(n,F):
 the orthonogal group O(n,F) and
 the special linear group SL(n,F).
Where GL(n,F) consists of invertible matrices in O(n,F) this is further reduced to matrices with the property that the transposed matrix is equal to the inverse matrix. These matrices turn out to have determinants 1 or 1. ( Not true that all matrices with determinant 1 are orthogonal ). Matrices with +/ 1's on the diagonal are orthogonal as well as permutation matrices.
The elements of the Special Linear group are further reduced to those with a determinant value of 1.
A concrete orthogonal group is O(2,R) as subgroup of GL(2,R). This group consists of 2 by 2 matrices with elements from the real numbers. These matrices are linear transformations of vectors in R2.
More groups and geometry to follow.
Friday, November 20, 2009
Group Explorer 2.2
Watched lecture 12 of Abstract Algebra E222
What is six billion ?
Personal Brain 5.5 released
Wednesday, November 18, 2009
Rotman
As Rotman put it...
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.
I think that's the trick to understanding his book.
As a matter of fact I already started. As Rotman wrote ( many ) are not difficult. The following exercise is an easy one if you studied Stirling numbers, if you didn't I wouldn't call it simple.
Exercise 1.5 ( in Rotman ). If 1 < r < n, then there are (1/r) [n(n — 1)... (n — r + 1)] rcycles in Sn.
Why doesn't he use Stirling numbers ? I am going to need to review Stirling numbers.
Now I remember, I encountered this before and decided I had to study some Discrete Math first. Well, I did. So that won't hold me up studying this book. I think all the prerequisites are 'in' now for attacking this book.
Tuesday, November 17, 2009
Professors
Watched lecture 11 of Abstract Algebra E222
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.
Watched lecture 10 of Abstract Algebra E222
Anyway, a very abstract and fast talk starting with the definition of a vector space all leading up to the change of basis problem. The change of basis problem is in itself not easy and takes a lot of practive to get comfortable with. A lot of material to go through in just one lecture. It took me months to fully understand these topics. Just to compare: Strang took about 10 lectures to cover this material.
See also: http://ocw.mit.edu/OcwWeb/Mathematics/1806Spring2005/VideoLectures/index.htm
Sunday, November 15, 2009
Improving the study process
P.S.
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
Thursday, November 12, 2009
Watched lecture 9 of Abstract Algebra E222
Nothing much really in this lecture.
Vector spaces.
Spans.
Linear combinations.
Dimensions.
Maybe a hint to a very important notion in mathematics: all vectorspaces of dimension n are isomorphic.
Peter on Monday.
Tuesday, November 10, 2009
Watched lecture 8 of Abstract Algebra E222
Isomorphism Theorem
Vector spaces over an arbitrary field
 Definition field
 Examples of finite fields
Proof that Z/pZ is a field
 Added to what we know from Z/nZ as an additive subgroup of Z we must prove that each a in Z/pZ has a multiplicative inverse, so we must show that if a is not a multiple of p then there is an integer b such that a*b congruent 1 mod p.
( Actual proof is worked out in the video )
What are the finite fields beyond Z/pZ ?
 the finite fields are of order p^n where p is a prime and n>=1, so there are finite fields of order 2, 3, 4, 5, 7, 8, 9, 11, etc. ( note 6 = 2*3, 10=2*5 not of type p^n )
Definition of a vector space ( V )
 Additive abelian group
 With a map f: VxF > V which is called the scalar multiplication
 ( All rules are written down on board. )
Examples of vector spaces
 V={0}
 V=F
 V=F2
 V=Fn
 V=F[X], vector space of all polynomials p(x) with coefficients in F
Vector subspace
 A subgroup 'stable' under scalar multiplication
Vector space homomorphisms
 Linear transformations ( as we knew it ) are explained as group homomorphisms stable under scalar multiplication
 So for T: V> W we can define the Kernel of T as a subspace of V and the image of T as a subspace of W. We can also define the quotient space V/W analog to the quotient group
Book of beauty: Visual Symmetry
Saturday, November 7, 2009
Watched lecture 7 of Abstract Algebra E222
At the end, briefly for students with an interest in Algebraic Topology, Gross mentioned sequences like 1 > H > G > G' > 1. With examples 1 > Z3 > Z6 > Z2 > 1 and 1 > A3 > S3 > {1,1} > 1 which show that by knowing Z3 ~ A3 and Z2 ~ {1,1} does not say anything about the resultgroup.
Basicly a long abstract theoretical discussion about factorgroups, with basicly zero examples. How will group theory develop in people exposed to such lectures? I am not sure if I want to think about that.
Friday, November 6, 2009
The prototypical mathematician ( is NOT ).
A picture of the prototypical Linux type of person. As one gains experience in Linux one tends to start looking like the Ultimate Nerd. I wish I could understand why. Thank G_d, there are all sorts of mathematicians, maybe it's because mathematics in itself is so rich in subjects that there is no such thing as the prototypical mathematician.  I read somewhere that it is rather not done to say " I am a mathematician", the proper thing is " I studied math ", or something like that. One becomes a mathematician not before others ( in the field ) are calling you one.  I agree there is a difference between simply having done some math courses ( even if they add up to a B.Sc. or M.Sc ) and practicing mathematics at the research level.
Wednesday, November 4, 2009
Galois Theory
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^21=0 is C2 and of x^42=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.
Watched lecture 6 of Abstract Algebra E222
Arithmetic congruent mod n.
Addition
Multiplication
How a congruent b mod n is in fact an equivalence relation.
And thus induces a partition of the integers.
Cosets are nZ, 1+nZ, 2+nZ, ... (n1)+nZ
Addition can be defined on these cosets and then they have a group structure.
The map Z > nZ is then a homomorphism with 0 as kernel.
( Around min 35 or so I lost interest... I fast forwarded watching minutes here and there, just to make sure there was not introduced anything I did not know already. I hope I am not losing interest in the series all together. We'll see. )
Tuesday, November 3, 2009
Watched lecture 5 of Abstract Algebra E222
Properties of an equivalence relation:
 reflexive: a~a
 symmetric: a~b <=> b~a
 transitive: a~b and b~c => a~c.
A homomorphism f: G>H with kernel K which is a normal subgroup of G implies an equivalence relation on G where K is one of the equivalence classes. The other equivalence classes have the form aK = { ak; k in K, for some a in G}. aK is also called a left coset of K. ( Gross writes complete proof of this proposition on board. )
A bit of mathematical history about Lagrange ( born in Italy! ) who writes a letter to Euler at age 17 containing some very sophisticated mathematics. Euler immediately recognizes the genius of Lagrange and arranges further education for Lagrange who until that time learned his math through selfstudy.
(The famous) Theorem of Lagrange.
If G is a finite group and H is a subgroup of G then the order of H divides the order ( size ) of G.
More propositions are discussed.
 Groups of order p are simple.
 Groups of order p^2 are abelian.
 An is simple for n>=5.
 Any finite, nonabelian group has even order.
( Next lecture Peter. )
Monday, November 2, 2009
Watched lecture 4 of Abstract Algebra E222
Proof that e is mapped to e by any homomorphism.
Proof that inverses are mapped to inverses by any homomorphism.
Definition of Image.
Definition of Kernel.
Properties of the Kernel.
 subgroup;
 normal subgroup.
Any normal subgroup is the kernel of a homomorphism.
Example homomorphism.
f: GL(n,R) > R_x
f(A) = det(A)
f has as kernel the matrices with det=1, also called SL(n,R). ( Special linear group )
Example homomorphism.
f: Sn>GL(n,R)
f(p)=Ap ( permutation matrix associated with p )
f( (1,2,3) ) = {{0,0,1}, {1,0,0}, {0,1,0} }
Definition center of G.
Example homomorphism G> Aut(G) i.e. Klein4 > S3
Watched lecture 3 of Abstract Algebra E222
Watched lecture 3 of Abstract Algebra E222. ( A lecture by Peter, Gross's assistant, if he isn't a professor yet, he will be soon, I suppose. )
Review of lectures 1 and 2.
 Groups. And examples of groups GL(n,R), Sn, Z+.
 Subgroups. Cyclic subgroups.
 ( Hom(Rn, Rn) has the structure of a vectorspace. )
 All subgroups of Zn are of the form bZ. ( Emphasis on importance of proof of this proposition.)
 ( Studying the subgroup structure of a group is in general very difficult. )
 Example of a cyclic subgroup of GL(2,R). The group generated by {{1 1}, {0,1}} is {{1 n}, {0,1}} n in Z.
Example of an isomorphism.
G1 = {i, 1, i, 1}
G2 = {(1,2,3,4}, (1,3),(2,4), (1,4,3,2), ()}
G1 and G2 are isomorphic by i > (1,2,3,4)
( Permutations are here in cyclic notation which are not introduced in the course yet. )
Example of an isomorphism.
G1 = {R,+}
G2 = {R\{0},*}
G1 and G2 are isomorphic by f: G1>G2; x > e^x
Proof:
f(x+y)=e^(x+y)=e^x * e^y = f(x)*f(y).
Klein4 group.
V={() , (1,2)(3,4), (1,3)(2,4), (1,4)(2,3)} as a subgroup of S4.
V={ {{1 0}, {0,1}}, {{1 0}, {0,1}}, {{1 0}, {0,1}}, {{1 0}, {0,1}} as a subgroup of GL(2,R).
Definitions.
Automorphism.
Homomorphism.
Image ( of a homomorphism)
Next lecture Gross on images of homomorphisms ( and more ).
( Thank you, Peter. )
Sunday, November 1, 2009
MS221 course result ...
Watched lecture 2 of Abstract Algebra E222
Example of group: GL(n,R), invertible nXn matrices with elements a_ij taken from R.
Definition of a group
 operation is closed
 operation is associative
 group has identity element
 elements have inverses
Definition of S(n), group of all bijective maps f: S>S, the symmetry group of n elements, with as operation the composition of maps
Definition of a subgroup
 closed
 group has identity element
 elements have inverses
Examples
 S1
 S2
 S3. This example was very messy. Instead of correctly naming the elements e, s, s2, t, st, s2t he named them e, t, t', s, s' and s'', not clearly emphasizing that s' was in fact st and so on. Here he could have nicely drawn a Cayley Table but he didn't.
Definition of transposition as the exchange of only two elements. ( Very important concept )
Example of all the subgroups of Z.
 bZ, all multiples of an integer including {0}, so {+/2, +/4, ...}, {+/3, +/6,... } are all subgroups.
Proof that all subgroups of Z are of form bZ.
 bZ is a subgroup
 any subgroup is of type bZ
This last part was really excellent, he used the Euclidian division algorithm to complete this part of the proof.
Definition of cyclic subgroup. The smallest subgroup containing an element g. This is the collection {g,g^2,g^3,...}. This subgroup can be either finite or infinite.
Definition of the order of an element g of a group. The smallest positive integer e such that g^m=e.
Next time his colleague / assistent will be lecturing and he will start with Lagranges theorem about that the order of a subgroup is a divisor of the order of a group. So I'll expect cosets will be introduced as well.
Note:
Two surprising, remarkable comments from Benedict Gross of which I am not sure I agree:
 " You cannot learn too much Linear Algebra "
( I agree Linear Algebra is important and fun but imho it will never be able to grasp deep theorems in say Number Theory. I am aware of the importance of Linear Algebra in Group Theory especially Representation Theory )
 " I do not recommend writing out multiplication tables "
( Playing with Cayley Tables gave me definitely more insight in the structure of many groups, if you have Mathematica or GAP producing a Cayley table is not difficult. I doubt if Gross has actual experience with either of the two, or he hides it carefully until later. )
It is Artin...
( While browsing through some old blog entries and I stumbled upon this entry about Lang or Artin. )
Saturday, October 31, 2009
Education in the US
Today, I watched the first videolecture of Harvard's E222 course on Abstract Algebra. The lectures are about the book Algebra by M. Artin. ( Well known to me, I selfstudied it in 2007 ). Anyway Benedict Gross, the lecturer, was talking about other Harvard courses in numbers. I tried to look up the topics of these courses and then I found out that a login is required for that.
That's it! There is secrecy about the actual content of the mathematics programmes of the various universities. That is called capitalism. Competition among universities. It is also a way to hoaxup the socalled 'quality' of the educational material. They are asking ludicrous fees for reading out loud Artin's books. It's a system to introduce classes in society.
Everyone who read Ramanujan's life story probably agrees with me that Mathematics should be free and accessible to everyone on the planet wether studying in Cambridge, through the Open University or at home selfstudying a copy of Artin's book downloaded from internet.
What I like about the US is, that it is home of Mathematica AND Sage, of Bush/Obama AND Ron Paul, and a zillion other contradictions.
When closure is sufficient for a subset to be a subgroup.
If G is a finite group and S is a subset of G then closure in S suffices for S to be a subgroup.
Proof:
S is a subgroup if for all a,b,c in S
( i ) ab in S  closure
( ii ) a(bc)= (ab)c  associativity
( iii ) e in S  has identity
( iv ) a^(1) in S  has inverse
let's prove them one by one:
( i) is proposed to be true ;
( ii ) is true for G and thus true for S ;
( iii ) since G is finite there is an integer n such that a^n = e thus e in S
( iv) since a a^(n1) = e all a have an inverse.
[]
When in exercises the word 'finite' is added to group like 'G is a finite group ' we know that for G the group axioms are true like (i) to (iv) above AND that there is an integer n such that for all g in G g^n = e.
Friday, October 30, 2009
Michio Suzuki
Links:
 In memoriam M. Suzuki
 Suzuki, Group Theory I
Sunday, October 25, 2009
Skiena's CSE 547
10 years old  1999 was clearly a small bandwidth area.
Nice to have anyway.
Video lectures Abstract Algebra
MATH E222 Abstract Algebra  http://www.extension.harvard.edu/openlearning/math222/Enjoy!
( Update 1feb/'11: )
You might also like:
 Video lectures number theory
 [News]  Video Lectures Algebraic Topology ( for Undergraduates )
 video lectures complex analysis
Saturday, October 24, 2009
M336  Groups and Geometry
Axioms and examples
Subgroups
Generating subgroups
Cyclic groups
Group actions
Group axioms
Subgroups and cosets
Normal subgroups and quotient groups
Isomorphisms and homomorphisms
Generators and relations
Equivalent colourings
Group actions
The counting lemma
The cycle index
Polya's enumeration formula.
Direct products
Abelian groups and groups of small orders
Cyclic groups
Subgroups and quotient groups of cyclic groups
Direct products of cyclic groups
Finitely presented abelian groups
The reduction algorithm
Existence and uniqueness of torsion coefficients and rank
Finitely generated abelian groups
Finite abelian groups
Subgroups of abelian groups
Permutation groups
Conjugacy  pgroups
Sylow psubgroups
Sylow's first and second theorems
Sylow's third theorem
Applications of the Sylow theorems
Subgroups of prime power order
Review
Groups of order 2p
Groups of order 12
Where now?
Thursday, October 22, 2009
Plan for 2010.
Just decided to do MST 209 first instead of M208. As far as possible from the Open University website I compared both courses. My main conclusion is that the pay off in "skills" is much higher from MST 209. Why start on the theoretic Real Analysis if your can't solve the basic differential equations? And MST 209 has to be done anyway. I can still add Groups and Symmetry to my schedule for next year.
Yep. As far as MST 209 is concerned I made up my mind.
Tuesday, October 20, 2009
About MS221 exam (3)
Well, yes and no.
Your final result is the lowest of the two!
So a 100 for TMA's and a 40 for exam = 40.
so a 40 for TMA's and a 100 for exam = 40.
At the end of the day only the exam counts, that is if your TMA's were 85+.
Intermezzo
About the MS221 exam ( 2 )
I
2nd order recurrence system (6)
conics (6)
matrix calculation (6)
composite isometry (4)
linear transformation rotation / reflection (4)
differentiation (3)
integration(3)
taylor series (1)
function iteration (1)
logic (1)
complex numbers (4)
number theory (4)
group theory (6)
49
IIconic with xy component ( rotated )
iteration( 4 )
logic ( 8 )
12
So that is 61 or a grade 3 pass.
In a TMA @home situation I would have a 99/100 no doubt about it whatsover.
About the MS221 exam
The exam itself. Well, it was hard, hard, HARD! Everybody was still writing after three hours. I just finished the last question at 17.30. Wow, did time fly! I didn't know time could go that fast. The questions were as expected just as I predicted in an earlier post. 12 + 2 questions in three hours is a lot. No time to check, doublecheck, evaluate and play around with the exercises as I was used to when doing TMA's.
A distinction is impossible. I underestimated the timepressure component. There is no time to think all you can do is robotically do the exercise as if on autopilot. But you can train for this so I think distinctions will be possible in future exams.
I am confident I'll pass but on a lower grade I had in mind. I am 200% motivated to continue my mathematics study. Either Pure Mathematics (60) + Groups and Symmetry (30) or maybe just Pure Mathematics (60) depending on other commitments.
The question on group theory was really easy, I saw all answers immediately in a flash, but writing all the answers down still took me 10 minutes ( or more ).
Anyway, my WIN of today is that I:
 am confident I'll pass
 am full of motivation to continue ( will register very soon )
 have learned from the experience in order to do better in future exams.
2h30 before MS221 exam
Staying calm is definitely a quality.
Preparing at Uni.
My goal was ( is ) a distinction.
Had another look at the specimen exam. In an at home TMA like situation with Mathematica a 90+ score is very well possible.
But with only three hours at the worst possible moment of the day (14.30  17.30) and with only a calculator, I am not sure at all.
Thank G_d, we are allowed to use the MST121/MS221 handbook.
Speculating...
To get a grade 4 pass...
33 + 7.
11 / 12 questions @ 3/6 + 1/2 @ 7/14.
That is still a lot.
OR
5/12 questions @ 6/6 + 2/2 @ 5/14.
To get a grade 2 pass...
12 /12 questions @ 4/6 2/2 @ 11/14
More realistically, questions are expected on the following subjects ( expected result )
I
2nd order recurrence system (6)
conics (3)
composite isometry (3)
function iteration (3)
linear transformation rotation / reflection (3)
matrix calculation (3)
differentiation (6)
integrationb (6)
taylor series (0)
complex numbers (3)
number theory (6)
group theory (6)
logic (6)
II
conic with xy component ( rotated )
eigenvalue problem ( 11 )
volume of 3D body
proof by induction ( 17 )
54 + 17 = 71 grade 2 pass
More tomorrow or after the exam.
Saturday, October 17, 2009
Result MS221  TMA04 is in
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.
Wednesday, October 14, 2009
The story of mathematics (4)
Make an infinite square of all fractions such that the first row contains all fractions with numerator 1, the second row with numerator 2, etc. Do the same for the denominators but by colomn. The square should look like:
1/1 1/2 1/3 1/4 1/5 1/6 1/7 1/8 ...
2/1 2/2 2/3 2/4 2/5 2/6 2/7 ...
3/1 3/2 3/3 3/4 3/5 3/6 ...
4/1 4/2 4/3 4/4 4/5 ...
5/1 5/2 5/3 5/4 ...
...
It is then possible to traverse this square like
1/1
2/1 1/2
3/1 2/2 1/3
4/1 3/2 2/3 1/4
The sum of the numerator and denominator is constant by row. Because this traversal includes every fraction it is possible to map the fractions in a 1to1 relation with the natural numbers and thus the set of fractions has the same number of elements as N. ( It is easy to include the negative fractions in the traversal, the result is the same. ) It turned out that a similar reasoning is not possible for the reals R and thus R has an infinity which is more infinite than the natural numbers. ( It wasn't in the documentary but I know that there is a third level of infinity which is the same as the total number of possible curves in a 2Dspace ( plane ). Theoretically there is an infinity of possible infinities ( I think ).
The mathematician Hilbert and his famous problems for the 20th century was introduced. The first problem was called the Continuum Hypothesis which asked if there was a set in between Q and R. Around that time Kurt Godel proved that there are statements in every possible system of mathematics which cannot be proved. Until the 30's of the last century Europe had the world centers of mathematics in cities like Gottingen and Paris. Then the Nazis came but many scientists and artists left. In the US Princeton was established to become the next world center of science and mathematics. Among others Godel and Einstein lectured there. Needless to say both were Jewish.
Many famous mathematicians had some sort of psychological issue. Godel suffered paranoia and Cantor had a bipolar disorder. In the 30's there were no antidepressants or ADHD type of drugs so they must have had a dark life. Yet, I don't think having a disorder is a prerequisite for being able to produce excellent math because there are many more mathematicians without a disorder. I am not even sure if the mathematical community has more psychological issues than other groups of scientists.
At Princeton there was a young mathematician Cohen who took on Hilbert's first problem. His answer wasn't what the community expected but because Godel approved his paper the community accepted that there are two types of mathematics: one where the Hypothesis is false and one where the hypothesis is false.
Monday, October 12, 2009
Base conversion
What is 777 in base 3 ?
We repeatedly divide by 3 and store the remainder until 0 is left:
777 = 3 * 259 + 0
259 = 3 * 86 + 1
86 = 3 * 28 + 2
28 = 3 * 9 + 1
9 = 3 * 3 + 0
3 = 3 * 1 + 0
1 = 0 * 3 + 1
Now the number is equal to the stored remainders in reverse order:
1001210
Let's check if the value in base 10 is indeed 777:
1 * 729 = 729
0 * 243
0 * 81
1 * 27 = 27
2 * 9 = 18
1 * 3 = 3
0 * 1
Finally we add 729 + 27 + 18 + 3 = 777.
Don works for basically any coniguration,
Sunday, October 11, 2009
Pythagorian triples were known in 1900BC
Just read In A primer of analytic number theory by Jeffrey Stopple, 2003 that there is a Babylonian cuneiform tablet (designated Plimpton 322 in the archives of Columbia University) from the nineteenth century b.c. that lists fifteen very large Pythagorean triples; for example, 127092^2 + 135002^2 = 185412^2.
This means that they must have known how to generate these numbers with
x = 2*s*t, y = s^2 − t^2, z = s^2 + t^2
For s=2,.t=1 we get the well known x=4, y=3, z=5 and
for s=3, t=1 we get 6, 8, 10 and
for s=4, t=2 we get 16, 12, 20.
1900 BC.
So 1900BC is much closer than I thought. The current blazing speed of scientic development has only been reacned since the last few hundred years or so.
Thursday, October 8, 2009
Blog or diary?
With the MS221 in less than two weeks I'll be free to study whatever I like until at least february 2010 when I will be doing 'Pure Mathematics' and 'Groups and Geometry'. I think I am going to selfstudy Introduction to Analytic Number Theory by Apostol. That will prepare me for three OU courses: Number Theory (M381) and Analytic Number Theory I ( M823 ) and II ( M829 ).
I would very much like to be able to prove, or at least understand the proof of, The Prime Number Theorem.
Friday, October 2, 2009
Thinking of the future
We'll see.
Tuesday, September 22, 2009
Ramanujan's magic square formula
At a very young age he designed the following formula for a 3 by 3 magic square:
C+Q  A+P  B+R
A+R  B+Q  C+P
B+P  C+R  A+Q
where A,B,C are integers in arithmetic progression and so are P,Q,R.
Try it, it works!
Wednesday, September 16, 2009
MS221  TMA04: Done
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 ( Cutoff 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.
Tuesday, September 15, 2009
MS221 exam close by
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 doubleangle and halfangle 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
I sort of expected this, so I am not surprised. But I am not entirely ready for the exam yet. I still have several weaknesses but I am confident I can handle these in time. To be honest I wished the exam was over and done with. It is obviously my objective to score in the 90100 because that is what I score for TMAs. But I check, doublecheck, recheck my TMAs at least two times. That makes a lot of difference.
Sunday, September 13, 2009
Logic puzzle
Saturday, September 12, 2009
Discrete Mathematical Structures
Beautiful lectures for the whole planet to watch.
Sunday, September 6, 2009
Excercise
Show that : Sin ( 18 degrees ) = 1/4 * ( SquareRoot(5)  1 ).
Will use it to practice the trig formulas for the MS221 exam.
Sage or Mathematica ?
Mathematica is closed source and for the price of a full edition Mathematica package you can also buy a stateoftheart Sony VAIO or other sexy netbook or tablet you are thinking of buying. The Mathematica programming language is a so called propriety language. That means that it is only used in Mathematica and the owners of Mathematica can change the language whenever they want. Mathematica has a steep learning curve. Now that I master the basics of it I am very satisfied with what Mathematica can do for me.
Because Sage says it is possible to integrate Mathematica into Sage ( haven't figured yet out how, I suppose I need the Mathematica for Linux version ) I am having another look at Sage. Sage is Linux only but you can run it entirely from within a browser. So you can install Sage on a Linux server ( for example in a VMWare virtual machine ) and run it on Windows in your browser.
( More later about my Sage findings. )
http://www.sagemath.org/
Saturday, September 5, 2009
Characteristic of a ring
Friday, August 28, 2009
Video: Solving Cubic Equations
And more: Pythagoros theorem ( a^2 + b^2 = c^2 ) is not really by Pythagoras. Pythagoros is the probable leader of a cult called the Pythagoreans. There is an interesting formula for enumerating all the Pythogarian triples like (3,4,5) (5,12,13), etc.
In the next clip we learn that Fermat bought a translation from Greek to Latin of Diophantus' work and started his work on Number Theory from there on. It was also in Fermat's lifetime that Calculus was born. ( Fermat already worked on the currently very important Elliptic Curves which Andrew Wiles used to prove Fermat's famous last theorem. ) A method is shown to find rational points on elliptic curves. An elliptic curve is the graph of an equation of type y^2 = x^3 + px + q.
Then the video becomes less interesting. It seems that in academic talks it is custom to speak with great admiration of certain other scientists, mathematicians especially if you know them or are related to them. The speaker, professor Gross had John Tate as his thesis advisor. John Tate is a celebrity in a the Elliptic Curve branch of mathematics. So Gross shows a picture of him and Tate and all the other graduate students at that time. Then he mentiones all these students by name including the universities where they are now teaching as a mathematics professor.
( I suppose they had their euphoric moments too when they simply passed an exam. After a while more impressive results are needed to feel the same high, like being appointed as professor. )
See the rest for yourself if you are still interested...
Thursday, August 27, 2009
MS221 TMA03  Result
Lost a point due to a typo. This once again proves that you can literally save points by reviewing, QAing, checking whatever you call it. And check your results as much as possible by using a mathematics package like MathCad or your own preferred software, as long as it can do the math.
Friday, August 21, 2009
MST121  Overall Course Result ( Final )
Assignment Score
CMA 41 83
CMA 42 84
TMA 01 82
TMA 02 88
TMA 03 95
TMA 04 90
TMA 01 82 has been substituted to 87.7 giving a final score of 88.84 rounded to 89.
An 89 for this is course is much more than I deserve. Especially in the beginning of MST121 I did not study enough ( november and december last year ). During this period I also had a last minute approach towards TMA's. I definitely lost points there.
Anyway, I learned that in order to get high marks much more is expected of the student. TMA01 and TMA02 were quick and dirty jobs using pen and pencil. As from TMA03 I changed to LaTeX including several QA reviews using Mathematica ( if possible ).
Wednesday, August 19, 2009
Chinese Remainder Theorem
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.
Tuesday, August 18, 2009
Sample MS221  Exam
I would say that anything is possible with regards to this exam. I could fail but I could just as well get a high score. It all depends on proper training from now till the exam. I am sure that a lot of exam training will pay off.
But first I have to finish MS221  TMA04.
How I spent some spare study time
I completely understand and am thus able to prove, ( now and in the future, I am sure ) the formula for the Sigma numbertheoric function. The sum of the dividers of a natural number. For example:
12 = 2^2 * 3
sigma(12)
= 1 + 2 + 3 + 4 + 6 + 12
= ( 1 + 2 + 4 ) + ( 3 + 6 + 12 )
= ( 1 + 2 + 4 ) * ( 1 + 3 )
= 7 * 4
= ( 2^3  1 ) / ( 2  1 ) * ( 3^2  1 ) / ( 3  1 )
= 28.
Furthermore, I am beginning to ger a practical understanding of MĂ¶bius inversion.
And I firmly consolidated knowlegde of and skills regarding
 Properties of integer division
 Euclidean algorithm for finding the GCD
 Linear diophantine equations
 Congruences
 etc.
Tuesday, August 11, 2009
How am I doing relatively ?
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 80100 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.
Saturday, August 8, 2009
Studying mathematics with SuperMemo
I used to do a lot of selfstudying so a I used a lot of study time on tasks which aren't necessary when you do an OU course. I nevertheless got used to allocating time to studyimprovement on a regular basis. If I could only use SuperMemo then studying becomes a process which automatically produces a lot of statistical data about the studying process itself. But producing learning materials for SuperMemo can be timeconsuming in itself. That fact kept me from using SM thus far. But with the other tools I currently have like Mathematica, TeXnicCenter and so on I must be able to organize my way of studying, doing exercises etc. so that it will be fairly easy to document my work as a SM item/answer pair.
It is worth another try...
Thursday, August 6, 2009
MST121 overall course result
Overall course result
PassOverall continuous assessment score (OCAS) : 89
Overall assessment score (OAS) : 89
Substitution applied at : 87.7
Wednesday, August 5, 2009
GĂ¶del, Escher, Bach
And now MIT made a series of six video lectures about the book. Haven't seen it yet but I expect them to be interesting. GĂ¶del's incompleteness theorem is a topic in the course Number Theory and Logic which is part of my course plan.
Sunday, August 2, 2009
OU Courses for 2010
After MST121 and MS221, both M208 and MST209 are good options to do next. They can be done both but in that case you have to do one TMA per two weeks. That is a lot. Too much I can handle. Unfortunately.
Saturday, August 1, 2009
Graphs, networks and design
Formal grammars and languages
Sunday, July 26, 2009
Math writing recognition in Mathematica and Windows 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.
Tuesday, July 21, 2009
MS221  TMA03 Completed
TMA03 questions
1. Derivatives
2. Graph sketching
3. Indefinite integrals
4. Area and volume calculations
5. Taylor series
6. ( more ) Taylor series
Monday, July 13, 2009
PI is female.
PI / 4 = 1  1/3 + 1/5  1/7 +1/9  1/11 + ....
I played a bit with this formula and it took me approx. 4000 terms to get the first four decimals stable. Beautiful but not practical.
I would say that PI is female.
Tuesday, June 30, 2009
MS221  TMA03
Sunday, June 28, 2009
Handbook of Mathematics
Saturday, June 27, 2009
Mathematica TIP(2): ComplexForm; PolarForm
PolarForm[z_] := {Abs[z], Arg[z]}
Example:
PolarForm[1+i]={ SQRT[2] ,PI/4 }
Open University Students
Thursday, June 25, 2009
MS221 TMA02  Result is in
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 OFFTOPIC: 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. )
Tuesday, June 23, 2009
The Story of Mathematics ( 3 )
Marcus du Sautoy clearly loves mathematics and he is an excellent storyteller. When I was young I saw a similar series by Jacob Bronowski which laid the foundation for my love of maths. I hope du Sautoy's program had a similar effect on young people watching it.
Still one episode to go on 20th century math.
Sunday, June 21, 2009
Symmetry and the Monster by Mark Ronan
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.
The Story of Mathematics (2)
Saturday, June 20, 2009
The Story of Mathematics
Thursday, June 18, 2009
Contemplating a Project
Contemplating... ... ...
( More later )
Feedback Delay on MST121 CMA42
Maybe some questions were too simple / difficult and are cancelled, leaving 24 questions which count just as in CMA41.
I wonder when the results do come in.
Wednesday, June 17, 2009
MS221  TMA02  ( 2 )
( Pffhhh. )
Time to study new things again! Book C is on calculus, while book D has introductions to number theory, group theory and logic.
Saturday, June 13, 2009
MS221  TMA02
Tuesday, June 9, 2009
MST121:DONE
By doing MST121 I primarily learned what it is like to study at the OU. I haven't seen any entirely new mathematics but that wasn't to be expected from a level 1 course. I am already busy with MS221 and I suppose it will be M208 and / or MST209 after that. All in all I have enjoyed the course.
Monday, June 8, 2009
Yellow Alert
Sunday, June 7, 2009
Protocol for reading a mathematics book
Once upon a time there was a programme on the radio for young children called Listen with Mother. (In those days it was assumed that it would be the mother who would be at home with the children.) In the first programme in 1950 the storyteller, Julia Lang, introduced the story she was about to tell by saying 'Are you sitting comfortably? Then we'll begin'. Apparently this introduction was not planned, but it caught on, and was used regularly until the programme ceased in 1982. When it comes to reading mathematics, however, this is not an appropriate beginning. A mathematics book cannot be read like a novel, sitting in a comfortable chair, with a glass by your side. Mathematics books need to be worked at. You need to be sitting at a table or a desk, with pencil and paper, both to work through the theory and to tackle the problems. A good guide is the amount of time it takes you to read the book. A novel can be read at a rate of about 60 pages an hour, whereas when it comes to many mathematics books you are doing well if you can read five pages an hour. (It follows that, even at 12 times the price, a mathematics book is good value for money!)
I would like to add that a good mathematics book can be read over and over. Some math books are companions for life.
Tuesday, May 26, 2009
MST121  TMA04 ( 3 )
Tuesday, May 19, 2009
MST121  TMA04 ( 2 )
Tuesday, May 12, 2009
MST121  TMA04
Saturday, May 9, 2009
Layed the foundation for new TMA's
Wednesday, May 6, 2009
MS221 second shipment arrived
I proposed a rather exceptional way of sending a shipment of books. The OU handled it as if it was a regular request. "... The quality of an organisation can be measured by their capacity of handling exceptions. ..." Isn't that true? Things usually go wrong when questions are asked.
Anyway, I received books C and D of course MS221 today. Book D is more or less an introduction to M208, I suppose. D has topics on Complex Numbers, Number Theory, Group Theory and Logic. Book C is calculus, again. I just to find calculus so booooring. I must say I find it less boring today and am actually looking forward to MST209 about model building which has LOTS of stuff on solving DE's.
MST209, M208 are both 60ers and are both next on my left. I don't think I can handle 120 in a year. 90 is doable for me, but 120? Don't think so. Havent ruled out the option of doing both MST209 and M208 in 2010. I havent planned courses for sep 09 / jan 10 yet so I could do the video based MIT selfstudy course on DE's in that period which would be an excellent technical preparation for MST209. But again, all options are still open.
As in football... concentrate on winning the next match instead of winning the competition.
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Welcome to The Bridge
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.)