Mathematica 7

Wolfram released a new release of Mathematica: 7. More here.

One of the new features is Finite Group Theory. I'll have a look how it compares to GAP. More later.

Scores MST121

Received my work with the tutor's remarks today. I scored 82/100. This assignment counts for 20% of the exam so I now have 16/100 with 80 to go in five assignments.

A trigonometric identity

Isn't that beautiful? ( Can you prove it? )

Mathematics and traffic an application of cellular automata

As a commuter living in a densely populated area with few public transport options I know too much about the daily traffic jams here. At work it is the #1 subject to discuss with colleagues. The Dutch speak about the traffic jams, weather and football ( but only if their team is winning ). Like eskimos have a zillion words for snow the Dutch have quite a vocabulary on traffic jams.

There is a wealth of information on the net about math applied to solving problems in traffic like this one: http://www.traffic-simulation.de/

Feedback MST121 TMA 01 - 1

Yesterday I received an e-mail from my math tutor with the score for a tutor marked assignment.

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 2x-1 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)(3x-4)(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,

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...

test

 

Hello world

Journal of Group Theory

The Journal of Group Theory is devoted to the publication of original research articles in all aspects of group theory. Articles concerning applications of group theory and articles from research areas which have a significant impact on group theory will also be considered.

MST121

Today I received the first and second shipment of MST121 ( Using Mathematics ) study materials. Looks good, have a lot of reading and planning to do. ( For the last two months or so I focused on learning ADF/Struts that's a web framework (partially) from Oracle. But that's off-topic here! ) I feel relieved that I am now following a well tested study path.

Online Calculator

"... This free online symbolic calculator enables you to define variables and functions as well as to evaluate expressions containing numbers in any number system from 2 (binary) over 8 (octal), 10 (decimal) and 16 (hexadecimal) to 36, roman numerals, complex numbers, intervals, variables, matrices, function calls, Boolean values (true and false) and operators (and, or, not ...), relations (e.g. greater than) and the if-then-else control structure. Comments are C-style /* */ or //. ..." 

In Roman numerals too?! I wonder. Give it a try at: http://www.tusanga.com/

Mathematics as Body Art

Found this picture somewhere on internet. The equation -is- beautiful, no doubt about that, so why not paint it on your body if you are into that stuff anyway?

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:

1. this particular crop circle is man-made;
2. if this crop circle is not man-made 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 man-made.

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. ;-)

An interesting tool

While browsing the OpenLearn LearningSpace site I found a new software tool for Visual Thinking: Compendium. I currently use Mind Manager for 'visual thinking' but imho Mind Manager is more suited for supporting project planning and designing documents and presentations than stimulating deep thought about a particular topic you don't yet fully understand. A tool that supports the thinking from not completely understanding to a full cognition of the subject would be very welcome in my collection of software. I'll give it a try.

Mathematics in the Movies

Not counting Numb3rs there aren't many movies truly about math. As far as I know anyway. I know of the following movies ( in alphabetical order ):
- 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.

Programming Mathematica: Finding Perfect Numbers

Find all numbers less than or equal to 10000 which are equal to the sum of their proper positive divisors. These are the so called Perfect Numbers. For example 6 has proper positive divisors 1,2 and 3 ( 1+2+3=6 ).

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

Third Isomorphism Theorem: Visual Example

Let G, H and K be groups such that K is a normal subgroup of H and H is a normal subgroup of G. Then ( G/K ) / ( H/K ) = G / H. This is the Third Isomorphism Theorem or the Chain Theorem.

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

I almost gave up on finding a copy of Wolfram Workbench. If you are serious about coding in Mathematica it's a must have.

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 high-level environment that uses a flexible programming language to integrate high-performance 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

Open University UK

I was at a presentation about studying ( mathematics ) at the Open University in the UK. The presentation was held in The Hague, Lange Houtstraat 11. I arrived early which gave me the opportunity to talk to a course consultant prior to the presentation. She answered most of the questions I had. The presentation took about 45 minutes and was very informative. Some interesting facts are:
- 200,000 students world-wide;
- +/- 560 students in NL;
- mathematics is a popular study among Dutch students ;
- most students are in the age group 31-40;
- credits earned don't expire;
- there are no entry-level 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.

Group Theory Applets

I have found several Java Applets that can aid in learning ( teaching ) some concepts in Group Theory. Find them on ShOp Java Applets.

Examples:

Powers of permutations
Orbits and stabilizers
Fixed sets for orbit counting
Permutations: the symmetric group S4
Cosets, Lagrange and factor groups

(c) Sheffield University Open University Java Applets

Algebraic Graph Theory

I am interested in graphical representations of finite groups. Today I discovered that...

"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.

When you are not being able to study math

I am in England for a BPEL course. Haven't done much math for the last three days. It makes me feel empty. Math is like playing an instrument. You must practice every day. It is bearable because just before I left I had this idea which I can freely think about anytime, anywhere.

Transforming ideas to results...

A drawing of an idea. How much time would it cost me to convert a group presentation like 'a4+1' to an Excel sheet containing the corresponding Cayley Table? - I certainly would like to have such a program ( coded by myself ). But will I hold on to the idea long enough?

Cayley table of the group ( C3 X C3 ) : C2

I constructed the Cayley table of the non-abelian group ( C3 X C3 ) : C2 which is the semi-direct product of the abelian groups C3 X C3 and C2.

Presentation of the group ( C3 X C3 ) : C2

{a,b,c | ab=ba, ac=c^2a, bc=c^2b} = (C3XC3):C2

Theorem about normal subgroups

Normal subgroups are very important objects in Group Theory. One of the 'must-never-forget'-theorems is the following.

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.

Morphisms

Let G,H be groups and x,y elements of G.

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

Math Video

Blogger has been notified, according to the terms of the Digital
Millennium Copyright Act (DMCA), that content in your blog mathematics-diary.blogspot.com allegedly infringes upon the copyrights of others.

( This post contained a link to a torrent of the video Joy of Mathematics. )

Math Speak

There are circles where this is common speak, really:

"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

I have found a beautiful book on Group Theory which was first published in 1937. I am not at all surprised that it was reprinted in 2000.


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 add-on package.



( But more on this interesting question later. )

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 statement 0.x = 0 is not an axiom and can thus be proved.

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).

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.

Math is more than this...

( How very cliche to put a Mandelbrot fractal in a Math blog. )

The differentiation matrix for arithmetic polynomials

Calculating a difference function is a straightforward process:





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?

Golden Ratio

The Fibonacci numbers are well known.



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

Tribonacci numbers

The Tribonacci numbers are defined as



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 (1170-1250) was an Italian mathematician who introduced the arabic numbers ( the numbers we use today ) in Europe in his book Liber Abaci.

Generating Function

( In mathematics ) a generating function is a formal power series whose coefficients encode information about a sequence a(n) that is indexed by the natural numbers.

( More later. )

More Mathematica Graphics

Example of a Mathematica Graphic

" Polyhedral Microcosm "

Philosophy of Learning

I found this on a webpage from the MIT OpenCourseWare Calculus course.

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 time-waster if not worse.
7. Yeah...

Abstract Algebra and Mathematica

I found a promising e-book on ( Explatory ) Galois Theory. The book is written at the undergraduate level and it has 'Explatory' in the title because the text uses and refers to the Mathematica ( and Maple ) packages.

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.

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.

Sum of consecutive cubics

Show that ...


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 found this article The Groups of Order Sixteen Made Easy, by Marcel Wild, from the January 2005 American Mathematical Monthly.

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.

Problem ( Number Theory )

Let m, n be positive integers.

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? )

One of Riemann's notes

Latex for Blogger

Try it at Equation Editor

An alternative GCD algorithm

The greatest common divisor ( GCD ) of two integers a and b is usually calculated with the ( well-known ) Euclidean Algorithm. There is however an alternative algorithm which is based on an entirely different idea. Let's illustrate this idea with an example. Let a, b be integers, a >:b and (a, b) = GCD(a,b). Then the following rules can be applied recursively until a=b=GCD(a,b):
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(a-b,b)

Example
(36, 27) = (27, 36/2)
(27, 18) = (27, 18/2)
(27, 9) = (27-9, 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.

Calculating squares

Try ( without a calculator )

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

Create a triangle from the sequence of odd numbers s[n]=2n-1 by writing s[1] on the first row, s[2] and s[3] on the second row, the next three numbers from the sequence on the third row, ... the next k numbers on the k-th row. For example:

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.

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)

Roman numerals

Just noticed that the BBC uses Roman numerals for copyright dating. (c) MMVIII.

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

My goal

I decided that I should have a goal related to my math hobby. I don't know if that is good or bad. It is my goal to get this degree in mathematics. It has been in the back of my mind for a while. It's time to come out of the closet. Having a secret goal is a sort of fear of failure I guess. Working towards a goal is more fun than having achieved a goal because once a goal has been achieved new goals turn up. Talking ( writing ) about what I am doing is the purpose of this blog anyway. ( To be continued. )

Inner product of two matrices

Let M(m,n)[R] be the vector space of m by n matrices with elements in R and scalar field R.

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.

The psychology of Charlie Eppes.

He is often shown "stimming". "Stimming" is basically self-stimulation, a repetitive sound or motion made by people who have difficulty integrating all the sensory input around them as a buffer between themselves and the sensory stimulus.

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 single-minded 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". )

Projection Matrix

A matrix P is a projection matrix IFF P=P*P.

Henry Pitcher

92-year-old to graduate from UH with a bachelor degree in math Crossing the stage to graduate from the University of Houston, summer 2007 semester culminated a quest that Henry Pitcher, 92, began 75 years ago. Full Story

Collatz' original drawing

Math and art

Although I seriously doubt if mathematics and art are even close I added http://math-art.net/ to the 'Cool Sites' list. I do agree that math pics can be beautiful. "... Where Mathematics and Art blends into a zen-like state of peace ...".

Venn diagram

Great Britain, The United Kingdom, or is it England: confused? Charlie Eppes ( Numb3rs ) would probably give a mini-lecture on Set Theory before drawing a Venn diagram to explain the difference. We all use math every day.

Collatz conjecture

The 3x + 1 problem or Collatz conjecture is simple in its formulation but until now no proof is known that it is in fact true for any n. A conjecture is generally believed true but a formal proof is still to be found.

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?

Quadratic residu

A number q is called a quadratic residue modulo n if there exists an integer x such that x^2 = q mod n.

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

Number theory exercise

Determine the integers n for which there exist integers x and y such that n divides ( x + y - 2 ) and ( 2x - 3y -3 ).

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.