Tuesday, December 9, 2008
Mathematica 7
One of the new features is Finite Group Theory. I'll have a look how it compares to GAP. More later.
Monday, December 8, 2008
Scores MST121
Sunday, November 30, 2008
Sunday, November 9, 2008
Mathematics and traffic an application of cellular automata
There is a wealth of information on the net about math applied to solving problems in traffic like this one: http://www.trafficsimulation.de/
Saturday, November 1, 2008
Feedback MST121 TMA 01  1
You got 33 out of the 40 for TMA01 part 1. Well done.
Some remarks,
Try to give your answers as a conclusion.
Q1 (c) Look for an easy factorisation.
Make sure that the answers satisfy that x is not 0 and 2x1 is not 0 so that both are solutions.
Mathcad,
The course supports the use of the floating "symbolic toolbar"
The solution sould look like:
42 x^3 + 301x^2 + 154 x 840 factor ___ (7(2x+5)(3x4)(x+6)
Q2
Make sure that you get used to the decimal point and notthe comma.
Always give the range,.
Check the difference between decimal points and significant figures,
Wednesday, October 15, 2008
Cayley graph of D8
G = { a,b  a3=b2=1, ab=ba3 }
Set of elements / vertices {1, a, a2, a3, b, ba, ba2, ba3}
Generating set / colors {a, b} ( a yellow, b blue )
Two elements (x, y) are connected by an arc if y=ax (yellow arc), y=bx (blue arc).
Now you know the Cayley Graphs of all Dihedral Groups...
Monday, October 13, 2008
Tuesday, October 7, 2008
Journal of Group Theory
Wednesday, September 17, 2008
MST121
Saturday, August 9, 2008
Online Calculator
In Roman numerals too?! I wonder. Give it a try at: http://www.tusanga.com/
Wednesday, July 16, 2008
Mathematics as Body Art
Sunday, July 13, 2008
PI encoded in a crop circle
Linda Moulton Howe wrote on her website EarthFiles
“The fact that the Pi decimal point is included (in the 2008 Barbury Castle barley pattern) and there is rounding up to 10 decimal places is to me a little mind boggling!”Michael Reed, Astrophysicist
The Barbury Castle Glyph. A crop circle representing the number Pi.
The first ten digits of Pi are 3.141592654 with a decimal point after the first digit. Read the crop circle from the center on outwards and follow the lines in clock order. Then you see a line, a dot, a line and onwards. There are ten line segments. Next line segments start further away from the center of the crop circle. The relative lenghts of the individual line segments are then respectively 3 1 4 1 5 9 2 6 5 4. And these are the first ten digits of Pi with the last digit correctly rounded.
Pi is in that circle alright. But what does that mean? In my opinion this can only mean two things:
 this particular crop circle is manmade;
 if this crop circle is not manmade then it has been made by an intelligent 'being'.
I used to believe that at least some crop circles appeared instantaneously without human intervention. Nature is full of beautiful patterns, so why not in crop fields? For me it is very hard to believe that Pi encoded circles can suddenly appear.
The crop community is rather excited about this 'formation'. I am not because for me it is like the end of an era. Santa Claus is not real after all. The more I look at it the more I recognize it as manmade.
The intelligent beings who created this crop circle must have been math students. Must be. If they only were brave enough to come forward.  Producing a similar circle with the first ten digits of another mathematical constant would be more hilarious though.
Choose life. ;)
Sunday, July 6, 2008
An interesting tool
Sunday, June 22, 2008
Mathematics in the Movies
 A Beautiful Mind
 Flatland
 PI
 Proof.
All these movies are fairly recent. The oldest is PI, from 1998. So that's encouraging. There is a movie about Ramanujan / Hardy in the works. Looking forward to that one.
Saturday, June 21, 2008
Programming Mathematica: Finding Perfect Numbers
Finding a solution to this problem can be done with the following Mathematica program.
PerfectQ [n_]:=Apply[Plus,Divisors[n]]== 2n
PerfectSearch[n_]:=Select[Range [n],PerfectQ]
PerfectSearch[10000]
{6,28,496,8128}
See also: An Introduction to Programming with Mathematica
Tuesday, June 17, 2008
Third Isomorphism Theorem: Visual Example
For example in
D4 (1)
D4 / { e, a2 } = C2 X C2 (2)
{ e, a2, a, a3 } / { a, a2 } = C2 (3)
( D4 / { e, a2 } ) / ( { e, a2, a, a3 } / { a, a2 } ) = C2 X C2 / C2 = C2 (4)
but also:
D4 / { e, a2, a, a3 } = C2 (5)
(1)
(2)
(3)
(4)
(5)
Wolfram Workbench
Perfect for scientists, engineers, mathematicians, financial analysts, and educators who want to build applications for technical computing problems, Wolfram Workbench is an integrated development environment (IDE) for Wolfram products such as Mathematica, gridMathematica, and webMathematica.
Mathematica provides a highlevel environment that uses a flexible programming language to integrate highperformance computing, a vast collection of algorithms, and tools for visualization, data processing, and document preparation. Programmers who want to develop code written in the Mathematica language can use Workbench to:
* Work with code in a specialized editor
* Debug programs at the source level
* Profile the code's execution
* Develop and run tests
* Work with Wolfram technologies
* Build and deploy Mathematica packages
* Code better in an integrated workgroup environment
Saturday, June 14, 2008
Open University UK
 200,000 students worldwide;
 +/ 560 students in NL;
 mathematics is a popular study among Dutch students ;
 most students are in the age group 3140;
 credits earned don't expire;
 there are no entrylevel requirements;
 earning 60 points a year may cost upto 2000 punds a year ( if not more ).
Anyway, I decided that I will register as a student and take my first course in September.
Sunday, June 8, 2008
Group Theory Applets
Examples:
(c) Sheffield University Open University Java Applets
Monday, June 2, 2008
Algebraic Graph Theory
"For every finitely generated group G there exists a graph X such that the automorphism group of X, Aut(X) is isomorphic to G."
An interesting theorem. Well, I thought so anyway. Just had to be. All based on my intuition.
The branch of mathematics which studies this area is called Algebraic Graph Theory.
Thursday, May 29, 2008
When you are not being able to study math
Sunday, May 25, 2008
Transforming ideas to results...
Cayley table of the group ( C3 X C3 ) : C2
Tuesday, May 20, 2008
Sunday, May 11, 2008
Theorem about normal subgroups
Let N be a normal subgroup of a group G and H be any subgroup of G. Then the intersection of H and N is a normal subgroup of H.
For the proof we use the following theorem.
A subgroup H is a normal subgroup in G if gH=Hg for all elements g in G.
In easy to remember math: "The intersection of a normal subgroup with another subgroup is normal in that subgroup." ( H&N is N(ormal)in H )
If this seems difficult: this theorem becomes trivial real fast.
Sunday, May 4, 2008
Morphisms
A homomorphism is a mapping
f: G> H
such that
f(1) = 1, and
f(xy) = f(x)f(y).
morphism  when f is 
Mono  injective 
Epi  surjective 
Iso  bijective 
Endo  any and when H=G 
Auto  bijective and when H=G 
Saturday, May 3, 2008
Math Video
Blogger has been notified, according to the terms of the Digital
Millennium Copyright Act (DMCA), that content in your blog mathematicsdiary.blogspot.com allegedly infringes upon the copyrights of others.
( This post contained a link to a torrent of the video Joy of Mathematics. )
Thursday, May 1, 2008
Math Speak
"How many consecutive digits of pi (3.1415 . . . ) can you display with a deck of cards?"
But it is the sort of question Charlie Eppes might have asked when he was eight years old or so.
Groups of Finite Order By Robert D. Carmichael
Link to Google Books
It is over 440 pages and contains many interesting exercises. I am going to try to solve the following question entirely with the Mathematica Abstract Algebra addon package.
( But more on this interesting question later. )
Wednesday, April 30, 2008
Displaying Cayley Tables in Mathematica
( The structure of the Ring Z[5]: addition and multiplication mod 5. )
0.x = 0 is not an axiom
The axioms for the integers are, for addition
(A1) a + (b + c) = (a + b) + c
(A2) a + 0 = a
(A3) a + (a) = 0
(A4) a + b = b + a
for multiplication
(M1) a(bc) = (ab)c
(M2) 1a = a
(M3) ab = ba
and for addition and multiplication ('distributive laws')
(D1) a(b+c)=ab+ac
(D2) (a+b)c=ac+ab.
So how do mathematicians prove that 0.a=0?
They do something like this.
a = a
a = (1+0).a (by A2 and M2)
a = 1.a + 0.a (by D2)
a = a + 0.a (by M2)
0.a = 0 by (A2).
Tuesday, April 29, 2008
The field of fractions of an integral domain
Create a set V of ordered pairs from {..., 2, 1, 0, 1, 2, ...} (integers) and {..., 2, 1, 1, 2, ...} (integers excluding 0). Elements of V are for example (3,1), (5,1) and (4,2).
Create an equivalence relation on elements of V. Two elements (a,b) and (c,d) are 'equivalent', 'belong to the same equivalence class' if ad=bc. For example (4,2) and (8,4) are equivalent while (4,1) and (8,4) are not.
Define addition '+' as (a,b) + (c,d) = (ad + bc, bd).
Define multiplication '.' as (a,b) . (c,d) = (ac, bd).
This is how the field of Rationals is formally constructed from the Integers.
Monday, April 28, 2008
Thursday, April 24, 2008
The differentiation matrix for arithmetic polynomials
But it is simpler to use the differentiation matrix for arithmetic polynomials:
The 5x5 matrix above is suitable for polynomials up to degree 4. It is possible to create a (n+1)x(n+1) matrix capable of handling polynomials up to degree n.
Proof:
Exercise (hint: use falling powers).
Question: Is there a compact way ( recursive, perhaps ) of describing the matrix capable of handling polynomials up to degree n?
Wednesday, April 23, 2008
Golden Ratio
But did you know that the function for the Fibonacci numbers is much more elegant if we explicitly use the Golden Ratio?
External Links
 Mathworld
Sunday, April 20, 2008
Tribonacci numbers
If we set the first three numbers at 0,1 and 1, then the generating function is
,
and the first 15 Tribonacci numbers are
0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149, 274, 504, 927, 1705, ...
Source: MathWorld
The Tribonacci Numbers look a lot like the Fibonacci numbers, that's how they got their name anyway. Who was Fibonacci? Fibonacci, or Leonardo of Pisa (11701250) was an Italian mathematician who introduced the arabic numbers ( the numbers we use today ) in Europe in his book Liber Abaci.
Saturday, April 19, 2008
Generating Function
( More later. )
Friday, April 18, 2008
Thursday, April 17, 2008
Philosophy of Learning
Philosophy of Learning
1. Amount learned is proportional to time put in.
2. Best way to learn is to figure out ideas yourself or teach them to someone else.
3. Second best is to do so with hints from others like your friends or us.
4. Third best is to get the ideas from reading; but pause in your reading to think about them.
5. Fourth best: unacceptable: don't get them at all.
6. The object of a lecture is not so much to inform you of important facts, but rather to stimulate you to try to learn about some concept.
7. The object of the course is to empower you to use the concepts of calculus in any context.
I would like to make some comments on these points.
1. Of course. I nevertheless disagree.
 Finding the right time to study is very important. Study when you feel energetic, hungry to learn, wanting to know and agressive enough to tackle any hard problem thrown at you.
 Better study one hour each day than seven hours every Saturday. The brain somehow needs backup time to process new concepts learned.
2. Very true! The trick is to find 'things' to find out for yourself which add the knowledge required for the course you are taking. About the 'teaching', I guess he means that you can verify if you have mastered a subject by explaining it in your own words.
3, 4, 5. Yeah...
6. Lectures. Personally, I don't like lectures. They cost you a LOT of time. You either understand what's being told at a lecture ( and that's because you already mastered the subject ) or you simply don't understand what's being told which makes it all a big timewaster if not worse.
7. Yeah...
Abstract Algebra and Mathematica
What is Galois Theory anyway? Galois only wrote one ( unfinished ) paper before he died at the age of 19 in a duel, but the contribution he made to mathematics is very significant. Before Galois mathematicians were searching for a formula to find the roots of a fifth degree polynomial equation.
Galois proved that it is impossible to find such a formula. It does not exist. In his proof he introduced the concept of a Group.
Abstract Algebra and computers. I learned a lot of concepts in Abstract Algebra by using the package of Hibbard and Levasseur: Exploring Abstract Algebra with Mathematica. It is a book and a software package combined as one. The package though, can be downloaded freely. Mathematica (6.0) is a requirement as the package is written in the Mathematica language.
Sunday, April 13, 2008
Aurifeuillian factorization
David Wells wrote the book Prime Numbers, The most mysterious figures in math. Well, there are some most mysterious theorems in his book...
What an insanity! Read more about Aurifeuillian factorization on MathWorld.
Saturday, April 12, 2008
Sum of consecutive cubics
It is of course possible to prove this identity by induction but that doesn't make you understand why the above is true. Proofs by induction generally don't contribute to understanding a problem.
( From Getaltheorie voor beginners )
Groups of Order 16
I am looking forward to thoroughly study this 12 page article. It's an excellent opportunity to repeat some Group Theory stuff and hopefully pick up some new insights. Group Theory is still my favourite subject.
Thursday, April 10, 2008
Problem ( Number Theory )
Show that:
If 24 / ( m * n + 1 ) Then 24 / ( m + n ).
numb3rs, 215: Running Man
Benford's Law. Very counter intuitive, but true. And explainable.
(Or: Be careful when creating a 'random' set! What is random anyway? )
Friday, April 4, 2008
Tuesday, March 25, 2008
An alternative GCD algorithm
If ( a=even AND b=even) then GCD(a,b)=2*GCD(a/2,b/2)
If ( a=odd AND b=even) then GCD(a,b)=GCD(a,b/2)
If ( a=even AND b=odd) then GCD(a,b)=GCD(a/2,b)
If ( a=odd AND b=odd) then GCD(a,b)=GCD(ab,b)
Example
(36, 27) = (27, 36/2)
(27, 18) = (27, 18/2)
(27, 9) = (279, 9)
(18, 9) = (18/2, 9)
(9, 9) Halt.
GCD(36,27)=9.
Compare using the Euclidean Algorithm
36 = 1 * 27 + 9
27 = 3 * 9 + 0 Halt.
GCD(36,27)=9.
However, this doesn't mean that the Euclidean Algorithm is always faster.
Monday, March 17, 2008
Calculating squares
21^2 ?
37^2 ?
If you need a calculator to calculate simple squares then you may need the following simple rule.
21^2 = 441
37^2 = 1369.
Or using ( x  k ) * ( x + k ) + k^2 = x^2
21^2 = 20 * 22 + 1^2 = 440 + 1 = 441
37^2 = 34 * 40 + 3^2 = 1200 + 160 + 9 = 1369.
Cubic numbers
The vertical column contains the sums by row of the numbers in the triangle. It is easy to see that this column contains the cubic numbers.
Sunday, March 16, 2008
Primes
Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the mind will never penetrate. (Leonard Euler)
Monday, March 10, 2008
Roman numerals
I  1
V  5
X  10
L  50
C  100
D  500
M  1000
1 I
2 II
3 III
4 IV
5 V
6 VI
7 VII
8 VIII
9 IX
10 X
11 XI
12 XII
13 XIII
14 XIV
15 XV
16 XVI
17 XVII
18 XVIII
19 XIX
20 XX
Sunday, March 2, 2008
My goal
Sunday, February 24, 2008
Inner product of two matrices
Let A, B, C elements in M(m,n)[R] and x,y elements in R. Define the map
f: V x V > R by (A,B) > Tr(B^T * A).
Since
 (xA+yB,C) = x(A,C) + y(B,C)
 (A,B) = (B,A)
 (A,A) >= 0
(proofs are trivial)
f is an inner product.
Saturday, February 23, 2008
The psychology of Charlie Eppes.
Charlie has sensory integration issues. This means he has difficulty integrating sensory input around him into a cohesive whole, something that most of us do automatically but is sometimes an exhausting task for individuals in the autism spectrum. Charlie wears enormous headphones with no music playing while figuring out a math problem.
Charlie fails to see the big picture at almost every turn and has extreme difficulty anticipating spontaneous behavior. Charlie is most comfortable with inductive reasoning, and despite his high intelligence levels, finds deductive reasoning difficult at best.
Charlie struggles to interpret social cues from others, when he picks up on them at all. Charlie rarely intuits when he annoys everyone around him, either with his singleminded preoccupation with mathematics or his inability to accurately read the nuances of social situations. He is very literal minded, often rambles past the point where anyone is listening, and in fact, sometimes DOES NOT NOTICE everyone has stopped listening.
Charlie fails to connect his concrete theorizing to abstract human reality, and in fact, doing so is deeply disturbing to him.
Charlie displays an extremely limiting and persistent social naivete that also can translate as extreme social immaturity. He is 27, doesn't date, doesn't notice that his very pretty friend is probably interested in him.
There is the compulsive and completely preoccupying nature of Charlie's obsession with discrete mathematics.
( This entry is a summary of this article on "tv makes you stupid". )
Thursday, February 21, 2008
Sunday, February 17, 2008
Henry Pitcher
Wednesday, February 13, 2008
Saturday, February 9, 2008
Math and art
Venn diagram
Friday, February 8, 2008
Collatz conjecture
The conjecture says that if you repeatedly apply the following rule to a number the end result will always be one (1):
 if even then divide by 2;
 if odd then multiply by 3 and add 1.
Example:
9
28
14
7
22
11
34
17
52
26
13
40
20
10
5
16
8
4
2
1
Take any number and try it.
What about 27?
Saturday, February 2, 2008
Quadratic residu
For example
a a^2 a^2 mod 7
1 1 1
2 4 4
3 9 2
4 16 2
5 25 4
6 36 1
7 49 0
8 64 1
9 81 4
etc.
The quadratic residues mod 7 are 1,2 and 4. The set {1,2,4} is a group under multiplication mod 7:
* mod 7 1 2 4
1 1 2 4
2 2 4 1
4 4 1 2
Monday, January 21, 2008
Number theory exercise
Friday, January 4, 2008
numb3rs
We all use math every day; to predict weather, to tell time, to handle money. Math is more than formulas or equations; it’s logic, it’s rationality, it’s using your mind to solve the biggest mysteries we know.
I like numb3rs. I suppose one of the ideas behind the series is to promote mathematics as one of the cool professions. Well they should, because mathematics is definitely cool.
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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.)