Happy New Year

Thank you for reading my blog in 2011.
nilo
( I'll save the 'looking back' post for next year. )

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More Alan Turing

( Although I have read more Fearless Symmetry haven't done a new summary yet. )

Anyway, I have selected two Alan Turing videos. First a clip from Nottingham Trent University in the series FavScientist.



Then the great Derek Jacobi as Turing in a clip from the docu drama Breaking the Code.

What is the origin of mathematics ?

If radio astronomers would discover a signal containing a repeating sequence of prime numbers then they would claim to have found extra-terrestrial intelligent life. Why? Because they consider mathematics as universal throughout the entire universe to which only intelligent life forms have access.

Mathematics is universal. That basically means that mathematics is discovered and not created. We use, for example, $\pi$ as the ratio between circumference and diameter of a circle, that ratio is the same everywhere in the universe. Not the symbol $pi$, the decimal number system and so forth.

We take the existence of mathematics for granted, we don't question when and how it was created. Where does that vast body of mathematics come from? Was it created with the Big Bang? If so, than the ( mathematical ) models of physicists that explain their Big Bang theory look naively simple.

It's rather vague to discuss if there was mathematics before the Big Bang. But -if- there was a point from which everything was created than that creation must have included -all- of mathematics. If we don't accept that than we are saying that we are the most intelligent life form in the universe, because mathematics is created by us and not discovered by us.

What -is- the origin of mathematics?

Peter Neumann speaks about Galois.

Found this video more or less by accident on YouTube.

Merry Christmas !

*★Merry★* 。 • ˚ ˚ ˛ ˚ ˛ •
•。★Christmas★ 。* 。
° 。 ° ˛˚˛ * _Π_____*。*˚
˚ ˛ •˛•˚ */______/~＼。˚ ˚ ˛
*° •˛• ☃｜ 田田 ｜門｜ ☃˚╰☆╮
...To ALL readers :) ♥♥♥

Calculating Galois Groups in Mathematica

If you are ( like me ):
- ( relatively ) new to Galois Theory
- looking for software to support your study of Galois Theory
- prefer software written in Mathematica because you know your way around in it,
then you should continue reading this post.

This year I became quite a fan of the sites of StackExchange. It is a priceless source of readily available know-how. I did not know where to begin looking for Galois Theory Software ( although I knew it existed ) so I posted a question in Mathematics StackExchange here: http://math.stackexchange.com/questions/93689/software-for-galois-theory As you can see I got answers fairly quick. It seemed that Sage, Magma have built-in support for Galois Theory. Both Mathematica and GAP have add-on package solutions. Needless to say these solutions will differ in capabilities, speed and so forth. But I wanted to focus on studying Galois Theorym and not wander off in software land. I accepted the Mathematica answer and pursued that route.

The package did not work!

Written more than a decade ( make that a century or more in software time ) ago or FIVE major releases of Mathematica ago. It got terminally deprecated. Function names used in the package were used in later releases of Mathematica with other, new functions. Other used functions got deprecated and were finally terminated. Software written with an older release can only be opened through the compatibility manager in Mathematica which is quite good at fixing issues. Not this time, which I ascribe to the sheer age of the package. In my confusion I posted the following question in Stack Overflow : http://stackoverflow.com/questions/8624000/how-to-handle-tag-arrow-is-protected-message-in-mathematica

With some help I was able to correct the issues. I don't know how or where to post it because the download came from here: http://library.wolfram.com/infocenter/Articles/2872/

So if you don't want to go the same issues just follow the posts above. I am of course willing to share my version 8 compatible version of the package.

Fearless Symmetry 6/23: Equations and varieties

I read the sixth chapter of Fearless Symmetry.

Part 1: Algebraic Preliminaries

Chapter 6: Equations and varieties

Logic of Equality

An equation is a statement, or assertion, that one thing is identical to another. In mathematics we replace is by = and use symbols that stand for the terms.

History of equations

Long before algebra as we know it, ancient peoples were working with equations.
A triangle whose three sides have lengths 3, 4, and 5 is a right triangle which is an example of a Diophantic equation because the unknowns are restricted to integers. Around the late 1500s Descartes added the connection between algebra and geometry now known as analytic geometry. Descartes, as a philosopher believed that the physical universe was governed entirely by the laws of geometry. Newton ( and Leibniz ) discovered that this wasn't true, they had to invent calculus to solve their scientific problems mathematically.

Z-Equations

A rational number is any number that can be expressed as the ratio of two integers. Real numbers are rational iff it is a terminating decimal or a repeating decimal. The set of all rational numbers is usually denoted as Q. We will deal mostly with equations where all the constants are integers. Or equations "defined over the integers". A Z-Equation is an equality of polynomials with integer coefficients. One of the main problems in number theory is finding and understanding all solutions of Z-equations.

Varieties

Fix the attention on a particular Z-equation. Write S(Z) for the set of all integral solutions of that equation, S(Q) for the set of all rational solutions of it, and so on. We call S an "algebraic variety". The variety S defined by a Z-equation ( or a system of Z-equations ) is the function that assigns to any number system the set of solutions S(A) of the equation or system of the equations.

For example define the Variety S as x^2 + Y^2 = 1
Then
S(Z) = {{1,0),(0,1),(-1,0),(0,-1)}.
S(Q) = {t in Q | 1-t^2 / 1+t^2, 2t/1+t^2}.

We can reformulate Fermat's Last Theorem using varieties as follows.
For any positive integer n, let V_n be the variety defined by x^n + y_n = z^n. Then if n > 2, V_n(Z) contains only solutions where one or more of the variables is 0.

Systems of equations

The system
x^2+y^2=1
x>0
is valid and has solutions, but it does NOT define an algebraic variety because inequalities are not defined in C nor in any of the finite fields.

Take the system
x^2+y^2+z^2=w
w^4=1
x+y=z
then S(R) is the ellipse x^2+y^2+x y = 1/2.

Finding roots of polynomials

The easiest general class of varieties to look at would be those defined by a single Z-equation in a single variable, for instance, x^3 + x - 2 = 0. The study of this type of variety is dominated by the concept of the Galois group. ( More in 8 and 13 ). If f(x) is a polynomial the roots of f(x) are the numbers c such that f(c)=0.

Are There General Methods for Finding Solutions to Systems of Polynomial Equations?

On a purely number-theoretical level, leaving philosophy and logic behind, we also have the famous theorem of Abel and Ruffini: Unlike quadratic polynomials, for which we can use the quadratic formula, for polynomials f (x) of degree 5 or greater, there is no formula involving just addition, subtraction, multiplication, division, and nth roots (n = 2, 3, 4, . . .) that can solve f (x) = 0 in general.

Deeper understanding is desirable

The amazing discovery of Galois is that there is more structure to S(A). As we shall see, S(A) is not just a set; it is the basis for defining a representation of a certain group, called the Galois group. We will look at another series of very interesting and very important, though not so very simple, Z-varieties: elliptic curves. These two kinds of varieties will give us some of our main examples to help us understand Galois groups and their representations.

To be continued with 7. Quadratic reciprocity

Fearless Symmetry 5/23: Complex Numbers

I read the fifth chapter of Fearless Symmetry.

Part 1: Algebraic Preliminaries

Chapter 5: Complex Numbers

Well, I think it is safe to assume that readers of this blog know the complex numbers. This chapter is in fact about a subset of the Complex Numbers called the Algebraic Numbers. In FS they use $\mathbf{Q}^{Alg}$ as notation, whereas I have seen mostly the notation $A$ for the Algebraic Numbers.

Every algebraic number can be expressed as the root of of a polynomial equation with integer coefficients. So $\pi$ is not a member of $\mathbf{Q}^{Alg}$, but $\sqrt{2}$ is because $\sqrt{2}$ is a solution of $x^2 - 2 = 0$.

Visualisation of the (countable) field of algebraic numbers in the complex plane.
( From Wikipedia )

To be continued with 6. Equations and varieties

Thinking about mathematics

Mathematics is the study of quantity, structure, space and change. Mathematicians seek out patterns and formulate new conjectures. Conjectures are either proven by a formal mathematical proof or falsified by some counter-example.

Wikipedia on the definition of mathematics

Programmers developed a language which enables them to communicate effectively and at a high level about software and their profession: creating software. It is a small language but very powerful, it is the language of design. - Mathematics, as a language, enables scientists ( astronomers, biologists, chemists, ... ) to talk and think about the patterns in the universe.

Take for example symmetry. Everybody, children included, has some notion of symmetry, i.e. reflection symmetry. When we are confronted with the art of Escher we realize that there is more to symmetry.

Circle Limit II 1959 woodcut in red and black, printed from 2 blocks

Symmetry is real, it is a physical phenomenon we can observe from the structure of atoms to the spirals of galaxies. Symmetry is more than a property of space it is what defines the structure of everything around us.

But what is quantity, and how we describe it? Here we enter the realm of numbers and number theory. Although, I am seriously fascinated by ( prime ) numbers, we cannot -see- them anywhere in nature. They are not real and seem inventions by humans. If we ever pick-up a signal from outer space that somehow involves prime numbers that would mean that that signal is created by extraterrestrial life.

When you study group theory you will find that there is a deep connection between groups ( i.e. symmetries ) and... prime numbers! Prime numbers are thus part of the signature of the Creator no matter what image you have of that concept.

Fearless Symmetry 4/24: Modular Arithmetic

I read the fourth chapter of Fearless Symmetry.

Part 1: Algebraic Preliminaries

Chapter 4: Modular Arithmetic

Chapter 4 is all about modular arithmetic.

Considering the goal of the book somewhere fields have to be introduced and in this chapter we find the first definition of a field.

Definition: A field is a number system where we can divide by anything nonzero.

Anything more precise would scare off the laymen casual reader for who the book is intended. I kind of like the definition myself. '... where you can divide anything by nonzero'.

Modular arithmetic is introduced as clock arithmetic of course with examples like: "Today is Tue. What day is it in 25 days?" or "The analog clock shows 8. What time will it show in 33 hours?"

Also, the extremely important concept of an equivalence relation is defined. There is much more about modular arithmetic in the book, of course.

To be continued with 5. Complex Numbers

Stirling numbers of the first kind.

Definition. S1[n,k] (Stirling number of the first kind) is the number of permutations of length n containing k cycles, multiplied by the sign of these permutations.

Example: 
Permutation     Cycles
123             (1)(2)(3)
132             (1)(23)
213             (12)(3)
231             (123)
312             (132)
321             (13)(2)
Thus:
S1[3,1]=2
S1[3,2]=-3
S1[3,3]=1

Stirling Numbers of the first kind are implemented in Mathematica as StirlingS1.

Higgs boson and the Euro

Yesterday CERN announced in a press release that 'they almost found the Higgs boson particle'. To me that sounds like a programmer telling me that his code is almost finished. 'It's 99% done'. ( The worst thing I ever heard was 'I'll finish the design when I am done coding.'. ) Anyway, they must get really nervous at CERN for budget cuts in these terrible economic times. - People at CERN rather don't communicate with common people. Unfortunately they depend upon our tax money to fund their expensive toy, the LHC. That's why they are always close to finding something, or they -think- that something exceeded the speed of light. As long as we pay their toy while everywhere else people are bleeding.

Fearless Symmetry 3/23: Permutations

I read the third chapter of Fearless Symmetry.

Part 1: Algebraic Preliminaries

Chapter 3: Permutations

In chapter 3 the concept of a permutation is explained and how they form groups.

Definition: A permutation is a one-to-one map from a set to itself.

Example 1:
Given the set {1,2} the possible one-to-one maps ( permutations ) are:
1->1, 2->2 and
1->2, 2->1.

Example 2:
For sets of three elements there are 6 = 3! possible permutations.

If we put all the permutations of a set in a set by itself and add the composition of permutations as the operation then this set becomes a group. Permutation groups are among the most important objects in group theory because every finite group is a subgroup of some permutation group.

There are two possible ways of notation when it comes to permutations. Let's consider the set {1,2,3,4} which has 24 possible permutations.

The permutation 1->1, 2->2, 3->4 and 4->3 can be written as [1 2 4 3] and also as (1)(2)(3 4) or short (3 4) this is the so called cycle notation. Thus (1 2 3) and [2 3 1] represent the same permutation. Clearly the cycle notation is more efficient, especially when considering permutations of large sets.

The composition of permutations means permuting one after the other. Unfortunately in some books it is done from left to right, in others from right to left.

Exercise 1:
Show that (ab)(cde)*(ae)(bc)(d)=(ac)(bde).

( I would solve it as follows: )
Right hand side:
A B C D E
_ D _ E B Apply (bde)
C D A E B Apply (ac)

Left hand side:
A B C D E
_ _ _ D _ Apply (d)
_ C B D _ Apply (bc)
E C B D A Apply (ae)

E C B D A
C D _ E _ Apply (cde)
C D A E B Apply (ab)

And it shows that LHS = RHS

Sets of permution form a group under composition because:
- composition leads to a new permutation ( closure )
- the neutral element is the do-nothing permutation, i.e. (1)(2)(3).
- every permutation has an inverse because it can be permuted back to the original positions.
- composition of permutations is associative.

Note that the composition of permutations is NOT ( always ) commutative.

To be continued with 4. Modular Arithmetic

M381 'Challenge Exercise' - Revisited

Sorry, Paddy and Jobidaker for the late reply. ( Auto-accepting comments has its drawbacks too. )

My solution for the 1/2, 1/3, 1/5 and 1/7 case is the following. Follow the pattern for the solution to the general case. Click to enlarge pic.

Please let me know if you think a smaller number qualifies.

Fearless Symmetry 2/23: Groups

I read the second chapter of Fearless Symmetry.

Part 1: Algebraic Preliminaries

Chapter 2: Groups

Definition: A group G is a set with a composition defined on pairs of elements, as long as three axioms hold true:
1. For any three elements x,y,z in G: x*(y*z) = x*(y*z)
2. G contains an element e such that for all x in G: x*e = e*x = x.
3. For any element x in G, there is an element y in G such that x*y=e.

Example 1:
The group of rotations of the sphere in R3: SO(3) or the Special Orthogonal Group in 3 dimensions. -The set G is the collection of all rotational symmetries of the sphere, i.e. if we rotate the sphere by any angle, the sphere doesn't noticeably change. The group property basically means that if we rotate the sphere over any angle A, after this over an angle B, it is the same if we would have rotated it in one go, but over some different angle. Also any rotation has an inverse: rotating it over the opposite angle. This makes the rotations a group. SO(3) is in fact a Lie group because these rotations can be done arbitrary small which is not the case when considering the symmetry group of for example a cube. Lie groups capture the concept of "continuous symmetries".

For me personally, this is the time to review chapters 1,2 and 3 of Naive Lie Theory by John Stillwell, Springer 2008. There you will find that the ( 4 dimensional ) quaternions are intimately related to the group SO(3) and that the quaternions can be expressed as 'complex 2-dimensional rotations' or complex 2 by 2 matrices. - This explains why quaternions are frequently used in 3D-(game)-programming.

To be continued with 3. Permutations

Fearless Symmetries - Reviews

In the eye of the general public the protypical math student suffers a range of personality disorders, i.e. autism, Asperger of course. When they suffer from a Narcissistic Personality Disorder as well, they merely look dumb and arrogant, the complete opposite of what they are trying to achieve.

Perhaps the most accessible introduction for a "very naive layperson" is Ash and Gross's Fearless Symmetry.

by Anonymous on Mathematics StackExchange

Anyway, Ash, one of the authors of Fearless Symmetry listed several reviews of the book on his page here.

Mathemusician puts Pi to music.

I -love- creative people.

Video lectures on Theory Of Automata, Formal Languages and Computation

NPTEL released a new series of video lectures featuring Prof.Kamala Krithivasan from the Department of Computer Science and Engineering IIT in Madras on  the Theory of Automata, Formal Languages and Computation. The first lecture in the series is called GRAMMARS AND NATURAL LANGUAGE PROCESSING.

Fearless Symmetry 1/23: Representations

I read the first chapter of Fearless Symmetry. See: Reading ( conflicts time management ) for the history on this topic.

Part 1: Algebraic Preliminaries

Chapter 1: Representations

The goal of the book is 'Mod p linear representations of Galois groups' and how these representations help to clarify the general problem of solving systems of polynomial equations with integer coefficients.

It is very important to know that a mathematical definition can redefine a commonly word used elsewhere. ( A simple group is not 'simple' but complex. A tree is a graph, a 'tree' in the forest is not. ) Sometimes an object is defined by listing its properties and following that a proof is given of the existence of such an object.

Definition: A set is a collection of things which are the elements of the set.

Definition: A function f: A-> B from a set A to a set B is a rule that assigns to each element in A an element of B.

Definition: A morphism is a function from A to B that "captures at least part of the essential nature" of the set A in its image in B. ( Clearly "captures at least part of the essential nature" needs to be refined later. )

Definition: A representation is a morphism from a source object to a standard target object.

Example 1: Take A,B and the fact that B represents A. A may be a citizen, B her state rep and X the legal fact that B represents A by voting in the legislature on her behalf. A may be a(n abstract) group, B a group of matrices, and X a morphism from A to B. The ultimate in abstraction is representing A,B as dots and X as an arrow from A to B.

Example 2: In the context of counting, given any two finite sets A and B, a morphism is a one-to-one correspondence from A to B. A representation in this case is a morphism from a given finite set to one of the sets {1}, {1,2}, {1,2,3} and so on. So a flock of three sheep has the set {1,2,3} as its target.

To be continued with 2. Groups.

Solving the Königsberg Bridge Problem with Mathematica

One of the most famous problems in the history of mathematics is the Königsberg Bridge Problem because it clearly marks the beginning of Graph Theory.

"... In addition to that branch of geometry which is concerned with magnitudes, and which has always received the greatest attention, there is another branch, previously almost unknown, which Leibniz first mentioned, calling it the geometry of position. ..."
Soluti problematis ad geometriam situs pertinentis, Euler 1736

The Königsberg bridge problem asks if the seven bridges of the city of Konigsberg (*) over the river Preger can all be traversed in a single trip without doubling back, with the additional requirement that the trip ends in the same place it began. Euler proved in 1736 that there is no such traversal. (*) Königsberg = Kaliningrad in the Russian exclave between Poland and Lithuania.

Using Graph Theory the problem is equivalent to asking if the multigraph on four nodes and seven edges (see figure) has an Eulerian cycle. Using Mathematica we would model each bridge as an edge and the parts of the city as a vertex. There are different ways to create a graph in Mathematica, but this problem suits the method of creating a graph from an adjacency matrix. Since EulerianQ returns false we know that Mathematica confirms Euler's original answer.

Combinatorica

The following function in the Mathematica Combinatorica package ShowGraphArray[Partition[FiniteGraphs, 4]] produces the following graphic.

I found it in the book 'Computational Discrete Mathematics, Combinatorics and Graph Theory with Mathematica, by Sriram Pemmaraju and Steven Skiena, Cambridge 2003.' It's not the typical Proposition / Proof mathematics book but it does give you access to about 450 -practical- freely available ( if you have Mathematica ) tools to extend your problem solving toolkit.

Reading ( conflicts time management )

When I was in high school I ( and others of my generation ) considered reading a waste of time. Teachers took every effort to convince us of the opposite.

There is only one way you can effectively change yourself and that is through reading.

Some of the best memories I have are somehow related to reading. Good times! I have read quite a few time management books. These books are primarily written for people who want to do better in life, i.e. change themselves into more productive people. In none of these books I have found the advice: "Read more". Strange. Anyway, I managed to undo the habit of reading. And that is bad.

In "The New Student Hat" Hubbard explains why people don't read, he talks about the strange sensations people experience when they sit down and attempt to read a book.

Amazing reactions occur when conscious effort is made to do this. Dullness, perception trouble, fogginess, sleep and even pains, emotions and convulsions can occur when one knowingly sets out to BE THERE AND COMFORTABLY PERCEIVE with the various parts of a subject.
These reactions discharge and vanish as one perseveres ( continues ) and at last, sometimes soon, sometimes after a long while, once can be there and perceive the component.

When I sit down with a book ( or device ) I get these sensations which only go away when I CAN OCCUPY MY MIND ON A (MATH-)PROBLEM. I know exactly what Hubbard means, I have to -simply sit there and read-, confront it.

Many years ago Goedel, Escher, Bach helped me to through a reading-barrier. This time I chose "Fearless Symmetry, Exposing the Hidden Patterns of Numbers, by Avner Ash and Robert Gross published by Princeton 2006.

I'll keep you posted about my struggle to become a reader again.