The Mathematics Blog

The integers and the natural numbers have the same cardinality

2012-02-02T14:11:00.002+01:00

One could argue that there are twice as many integers as there are natural numbers since for every natural number there are two integers: $1 \mapsto (1,-1)$, $2 \mapsto (2,-2)$.

You can imagine that there was quite some opposition from within the mathematics community when Georg Cantor (1845-1918) proposed the following theorem:

Two sets A and B have the same cardinality if there exists a bijection, that is, an injective and surjective function, from A to B.

A sequence for $\mathbf{N}$ is $$s(n) = \sum_{k=1}^n 1 \ ,$$
$$ \begin{array}{cc}
n & s(n) \\
1 & 1 \\
2 & 2 \\
3 & 3 \\
4 & 4 \\
\cdots & \cdots \end{array} $$

Likewise a sequence for $\mathbf{Z}$ is: $$s(n) = \sum_{k=1}^n (-1)^{k+1} \cdot k \ ,$$
$$ \begin{array}{cc}
n & s(n) \\
1 & 1 \\
2 & -1 \\
3 & 2 \\
4 & -2 \\
\cdots & \cdots \end{array} \, $$ We can thus establish a bijective (one-to-one) map between $\mathbf{N}$ and $\mathbf{Z}$.

By the theorem above we can conclude that $\mathbf{N}$ and $\mathbf{Z}$ have the same cardinality ( 'number of elements' ).

Open University Library Services launches 2012 Student Survey

2012-01-31T09:50:00.000+01:00

If you are a student at the Open University now is the time to give your opinion about the Library Services. What is your opinion about the fact that subscriptions on important journals can only be accessed with a delay of a year? I bet that this restriction is not applicable to academic staff. I know that the Technical University in Delft, Netherlands is subscribed to all Springer e-books. The OU does not. - Wasn't one of the motives for tripling the fees that an education at the OU is tantamount to one at any other university?

The Open University is interested in -your- opinion.

Mathematics starts with observation

2012-01-29T10:40:00.000+01:00

No matter the complexity or size of the mathematical model or the power of the computers it runs on: if the model has been built from an ill-perceived problem, the results are worthless. Garbage in = garbage out. Have you ever said to yourself after you ( finally ) solved that difficult pure mathematics problem : "If I only looked at the problem in that way from the start I would have solved it immediately." Note the "looked at ... in that way". Even the purest of mathematics must be perceived by the 'mind's eye'.

How ( well ) do you perceive? The picture below is NOT a trick picture, nor has it been 'Photoshopped' in any way. What do you see? Look. Just look.

Symmetry ?

2012-01-29T00:19:00.001+01:00

Browsing photo albums on facebook often reveals interesting and beautiful pictures.

I find patterns like this very interesting because most people would call this a symmetric pattern. Yet, I know of no mathematical symmetry that accurately describes the symmetry we see here. Yet, mathematics claims to have categorized all symmetries of the plane.

[Video] Deriving Binet's formula for the Fibonacci numbers

2012-01-28T12:05:00.001+01:00

One of those formulas every mathematician loves ( I think ):

$$F_n = \frac{1}{\sqrt{5}}( \phi^n - (1-\phi)^n )$$

Here's how MathDoctorBob explains it.

Although I like The Doctor's videos ( I wished the real doctor would show up accusing me for abusing his name but taking me for a ride in his phone box anyway ), I wouldn't like to have Doctor Bob as a tutor in class, I simply wouldn't be able to catch up and I am not the audible type anyway. I like to read a bit, play and think a bit, read a bit, and so on. Every person has its own unique style of learning that works for him. Part of studying is discovering your own learning style.

Oh, and I think this formula beats the one of Binet ( although strictly speaking not in closed form ), because it fascinates me that the Fibonacci numbers are actually -in- the triangle of Pascal.

$$F_{n+1} = \sum_{k=0}^{n} {n-k \choose k}$$

Study tip: Kill procrastination with the Pomodoro Technique

2012-01-27T16:25:00.000+01:00

If you procrastrinate a lot while studying then you need to apply the Pomodoro Technique. Read what an experienced has to say about it: Marco Giannone about studying and the Pomodoro Technique.

Mathematica Stack Exchange in Public Beta

2012-01-27T16:07:00.000+01:00

If you are into Mathematica then this site is worth a visit. They are now in Public Beta.

You will find expert advise on that site. It is absolutely amazing. Entire issues are solved. I dealt a lot with commercial support organizations charging millions (!) in license fees every year for which they gave crap support in return ( not referring to Wolfram ). Sites like Stack Exchange are -in part- an answer to that, I suppose. They are like a wake-up-call for the very expensive low quality commercial support sites.

Affine transformation rules - Revisited

2012-01-26T21:14:00.000+01:00

Following yesterday's post here are the 'five rules' which aren't rules in Mathematica. Basically there is only one rule where the affine transformation consisting of invertible matrix $A$ and vector $t$ are mapped to a 3-by-3 matrix after which composition of affine transformations ( including translations only ) can be done by multiplying matrices.

f[A_, t_] := ArrayFlatten[{{A, Transpose[{t}]}, {0, 1}}]

Click to enlarge size.

An alternative definition of an affine transformation.

2012-01-25T15:16:00.002+01:00

If I am not careful enough in doing everything in the inefficient M336 way I might be heading for some really bad marks. Let me explain.

Affine transformation as in M336

An affine transformation is a transformation of the form $$\mathbf{x} \rightarrow A\mathbf{x} + \mathbf{p},$$ where $A$ is an invertible linear transformation and $\mathbf{p}$ some constant vector.

Why not:

An affine transformation is of the form $$\mathbf{x} \rightarrow A\mathbf{x} + \mathbf{p},$$ where $A$ is an invertible matrix and $\mathbf{p}$ a vector.

Details matter in mathematics.

Not important, to the point: for calculation purposes the notation $f=t\left[ \mathbf{p} \right] \circ \lambda\left[ \mathbf{A} \right] $ is used which requires five additional rules to remember:
R1 $t\left[ \mathbf{p} \right] \circ t\left[ \mathbf{q} \right] = t\left[ \mathbf{p+q} \right]$
R2 $\lambda \left[ A \right] \circ \lambda \left[ B \right] = \lambda \left[ AB \right]$
R3 $\lambda \left[ A \right] \circ t\left[ \mathbf{p} \right] = t\left[ A \mathbf{p} \right] \circ \lambda \left[ A \right]$
R4 $( t\left[ \mathbf{p} \right] \circ \lambda \left[ A \right] ) \circ ( t\left[ \mathbf{q} \right] \circ \lambda \left[ B \right] ) = t\left[ \mathbf{p}+A\mathbf{q} \right] \circ \lambda \left[ AB \right]$
R5 $(t\left[ \mathbf{p} \right] \circ \lambda \left[ A \right])^{-1} = t\left[ -A^{-1}\mathbf{p} \right] \circ \lambda \left[ A^{-1} \right]$

What an ugly and never seen before notation. Br! This hurts my eyes.

Alternative

For calculation purposes we define the block matrix
$$R = \left( \begin{array}{cc}
A & \mathbf{t} \\
0 & 1 \end{array} \right) $$
Example: if $A=I$ and $\mathbf{t}=(t_1,t_2)^T$ and $\mathbf{x}=(x,y)^T$, then
$ R\mathbf{x} = \left( \begin{array}{ccc}
1 & 0 & t_1 \\
0 & 1 & t_2 \\
0 & 0 & 1 \end{array} \right) \cdot \left( \begin{array}{c}
x \\
y \\
1 \end{array} \right)= \left( \begin{array}{c}
x+t_1 \\
y+t_2 \\
1 \end{array} \right)$

No rules to remember, only elementary matrix algebra. We have used the fact that a translation in $R^n$ is basically a rotation in $R^{n+1}$. So, the same idea works for affine transformations in $R^3$ which can be modeled by a rotation-matrix in $R^4$.

Open University TMAs

2012-01-24T12:20:00.001+01:00

Although the 2012 course year has not even started I bet that experienced OU students are already working on their first TMA. The best ( if not only ) advice I can give to ( beginning ) students is that you can't start soon enough on your TMAs. You don't have to ship them until the cut-off date of course. Until then you can always improve on your answers.

Students need good role models for writing mathematics. This is a reason for the complete write-ups of solutions to many examples, since most additional situations do not provide students with any models for solutions to the standard problems. This is bad. Even worse, lacking full solutions written by a practiced hand, inferior and regressive solutions may propagate. I do not always insist that students give solutions in the style I wish, but it is very desirable to provide beginners with good examples. - Paul Garrett.

In the setting of the Open University this means that you should copy your TMA answers as much as you can from the 'Solutions to the exercises' section in the booklets. That's how the model solutions look like and that's what your tutor likes to see. - I have given perfect answers -not- in the style of a booklet which made the tutor rather nervous because it was not what she -expected-. Play along.

( Taking an advanced course like M336 requires -revision-. It is adviced to do this with the M208 materials, which is probably the best from the viewpoint of the M336 course. But if you are really interested in Algebra you should read the books as well. I suggest reading the following answers on Stack Exchange if you want advice on algebra books:
- Good abstract algebra books for self study
- Requesting abstract algebra book recommendations
Visit the course forums but there is much more:
- Social media for mathematicians )

Maths and ballet

2012-01-24T09:58:00.001+01:00

Dance looks beautiful when arm and leg movements are coordinated.

Did you know that counting is a very important part of ballet. Dancers must constantly be counting their steps in order to keep time with the music.

New York City Ballet Soloist Robert Fairchild talks about his role in Douglas Lee's "Lifecasting". Notice that he mentions the complex counting schemes he has to use.

What is the topology of the universe ?

2012-01-23T17:33:00.004+01:00

Do we know what the topology of the universe is? Is it possible to determine? By us, I mean. - Some thoughts, nothing more.

I think that most people think about the physical universe as one huge cube or sphere, filled with space-time, dark matter and dark energy, like the air in a balloon. The air of that 'balloon' however is polluted with billions and billions of galaxies. Somewhere in that balloon is our tiny earth from which we are observing everything and making smart conclusions so that we finally became able to understand the universe from its inception: the Big Bang. We even know what will happen to the universe in a few billion years. It's hard to check whether there has ever been a Big Bang. ( Physicists working in branches where results are verifiable are very modest. Think about weather predictions, and engineering another moon-mission: it would take decades. )

In popular science documentaries they bring us to 'the end of the universe', show us pictures of stars in faraway galaxies in extreme quality. How do they do that? It's not that the pictures are sent to us from that remote galaxy, they are taken from Earth('s orbit). Basically, data is captured, analyzed and processed. From that data a picture is generated: call it "probabilistic photography". ( It helps funding research if you can show some beautiful animations or pictures. What they really have though is a database with numbers. )

Physicists speak with authority and certainty about the Big Bang and the topology of the universe. That certainty is a pose, absolutely nothing is certain in physics.

I got kicked out of class once. Because I kept asking the physics teacher about gravity. "What -is- gravity?", ( I honestly didn't know that he couldn't have a clue at that time. ) I asked, when he started again about the g. constant. "Sir, -why- do objects fall to the ground?" Finally, he literally kicked me out of class.

To the point. I made two images.

Imagine that this is an abstraction of what we really know about the universe. The following image shows the same objects but from another viewpoint.

What is our viewpoint in relation to the vastness of what is around us? Does the topology of the universe permits us to see it all? Is the topology of the universe like that balloon or is it, perhaps, some topology not even discovered in the realm of mathematics? Would we ever be able to determine that?

Compare us with little intelligent tiny fish on the bottom of the ocean 'studying' the(ir) universe. What would their conclusion be?

Tiling Constructors in Mathematica

2012-01-21T13:51:00.001+01:00

Go to the Wolfram Demonstrations site or write your own, in Mathematica. ( Shamelessly plugging Mathematica, I am just a devoted fan of the product. )

http://demonstrations.wolfram.com/TilingConstructor/

Animated Penrose Tiling

2012-01-20T23:57:00.001+01:00

I am strolling around in the world of 2D Euclidean geometry. What can be so interesting about something -that- "simple"? This is an area of mathematics which is deeply connected to human imagination and art. Think Escher. I came across a YouTube video I would like to share. Worth the watch.

An animation of the celebrated Penrose non periodic tiling made with Povray, realized at the Department of Mathematics and Physics, Catholic University, Brescia (Italy). By Maurizio Paolini and Alessandro Musesti.

Get the most out of the Open University

2012-01-20T11:24:00.000+01:00

A reminder and a tip. - While you are a student at the Open University you have access to the Open University ( on-line ) Library Services. To get the most out of this valuable resource it helps to take a few training sessions in how to use the Library Services. - Training starts today! - http://www8.open.ac.uk/library/training-and-events/online-training-sessions

Guessing the Frieze Group

2012-01-19T11:22:00.000+01:00

There are probably as many Frieze Group ( and Wallpaper Group ) 'guesser programs' as there are programming languages. I chose a Frieze Group guesser that has been written in Mathematica.

http://demonstrations.wolfram.com/GuessingTheFriezeGroup/

If you don't have Mathematica, there is a free Mathematica Player. There are also Student and Home Editions ( with all functionality ) at a much reduced price.

Open University upgrades student websites

2012-01-17T12:06:00.000+01:00

Today the Open University will implement several changes on the student websites. The look and feel of the course websites have already been changed. I definitely like the changes. Anyway, M336 opened. M336 is an advanced group theory course about tilings, frieze and wallpaper groups, the classification of finite abelian groups and the sylow theorems.

This is a mini-talk about Frieze groups I found on YouTube:

Independent site for Mathematica in Stack Exchange close to entering beta

2012-01-12T12:48:00.001+01:00

Only a few more commits are required...

Stack Exchange is the place to go if you need help. ( ... from experts. )

2012, at last.

2012-01-11T15:26:00.000+01:00

Fair is fair. The Open University gave me a call, registration is done. - It is now 2012. For me, anyway: that is how I feel about it.

About the Open University

2012-01-10T21:05:00.001+01:00

Last year's problems seem simple with what I am going through now. What is going on? I have to withdray money from my student account which I hold with the Open University to pay for this year's courses, the procedure is simple and fast, really. First you call the course registration line, then you simply ask the student advisor to arrange it for you. But...

... I can't get through.

Not that I haven't been trying. Last week I gave up and wrote an e-mail asking for advice on how to establish contact. They say that it can take 'up to three working days before you get a reply'. It's past three working days already. But...

... no reply.

What's going on? Does it have to do with budget cuts? Queries about the new fees? I never had a problem to get through. The Open University is not just any university. They are =huge=. On a piece on the OU site I read that the OU is the biggest university in the UK with
- 250,000 students
- 7000 tutors ( Associate Lecturers )
- 1200 academic staff, and
- 3500 support staff
It is also an -international- university, they have 3500 students in Ireland, 9000 in the EU and another 7500 outside the EU.

In the meantime I am still waiting. I'll have to figure out something because I can't wait to start studying again. When everything is OK I do the reflection post first.

I haven't dropped the Fearless Symmetry series, on the contrary: I am styding algebra again. I have posted a question about Galois Theory here. Basically a course in Galois Theory makes you understand -why- polynomials with rational coefficients and degree five or higher can't be solved by radicals. At least not in general, if the corresponding Galois group of the polynomial is soluble however then there is a solutiuon. - Which you won't find in the textbooks. And that's what I find disappointing, to say the least. Maybe this is why a study in mathematics never seems to stop. A course answers a few of your questions but you'll have more questions after the course than you had before.

If Galois Theory is a blank for you, then this article ( pdf) might fill it, a bit. ;-)

( Mathematics PDF ) books formatted for e-readers...

2012-01-08T23:09:00.000+01:00

Formatting, typesetting books specifically for e-readers. I am a bit late to notice, but J.S. Milne is probably the first who formatted a mathematics book for an e-reader. Look at his Group Theory book ( PDF e-reader format ) here.

There was a time when moving to a new flat ( house ) meant carrying boxes of books and even more boxes of vinyl records. All the vinyl the average music collector had fits nicely on one USB stick with more than enough space left to store all his books, in e-reader format.

Milne did not have to rewrite his books ( maybe he made a few cosmetic changes, I don't know ) to reformat his PDF books for e-readers. Simply because he writes his books in LaTeX. He can publish to PDF in any format, he can have them nicely printed or he can merge his content in a wiki. His mathematical source stays untouched.

One of the things computers can't do yet is scanning a mathematics book, one of Euler's publications for example to LaTeX.

It makes me a bit sad that Windows 8 will be a lot like iOS from the iPhone ( Microsoft wasn't able to leap-frog Apple after Windows 95 ), I mean Windows 8 will be fingers based instead of pen-based. In the 1980s computers were for male geeks, today computers have become social media tools which are mainly used by women. I mean which male doesn't get bored by hanging out on facebook day-in day-out.

Writing mathematics is done with a pencil after which it is typeset.

What makes me sad is that just when pen-computers started to recognize handwritten math ( in Mathematica for example ), the world is flooded with an interface that is essentially geared for Facebook. Lots of pictures and every text has the intelligence of the two thumbs it has been typed with. Writing is just like reading. Very few people ( like to ) do it well. With or without e-reader.

Graph functions with Google Search

2012-01-05T12:16:00.000+01:00

Try this:
- https://www.google.com/search?q=x/2,+(x/2)%5E2,+ln(x),+cos(pi*x/5)

If you don't think that's cool, then try 'good old' WolframAlpha:
- http://www.wolframalpha.com/input/?i=x%2F2%2C+(x%2F2)%5E2%2C+ln(x)%2C+cos(pi*x%2F5)

Enjoy!

Example of a Galois Group of order 8 ( Introducing Math Doctor Bob )

2012-01-03T19:05:00.001+01:00

Regular readers must have noticed my interest in Abstract Algebra, of which I am currently studying, in different ways, the topic of Galois Theory. If you have chosen a different route in mathematics ( computation, statistics, and so forth ) or if you are at the early undergraduate level you may have difficulty picturing what Galois Theory is -all about-. I am trying to communicate that idea by summarizing the popular introduction to the field 'Fearless Symmetry' which basically introduces Galois Theory to the general ( but educated ) public. ( Currently working on part 7 out of 23). But as they say, one picture says more than a thousand words. For those that want to get an idea, fast and easy, and *now*, I recommend the following video ( mini lecture ). Don't expect you can master the subject by watching a ten minute video but the ten minutes are well worth it. The video lecturer is 'Math Doctor Bob', who uploaded about 600 mini lectures on various mathematical topics.

See also:
- Fearless Symmetry

Happy New Year

2011-12-31T15:00:00.017+01:00

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

2011-12-31T10:58:00.001+01:00

( 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 ?

2011-12-27T10:58:00.000+01:00

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.

2011-12-26T19:22:00.000+01:00

Found this video more or less by accident on YouTube.

Merry Christmas !

2011-12-24T19:23:00.000+01:00

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

Calculating Galois Groups in Mathematica

2011-12-24T19:22:00.001+01:00

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

2011-12-24T16:59:00.000+01:00

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

2011-12-18T16:22:00.006+01:00

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

2011-12-16T22:13:00.000+01:00

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

2011-12-16T21:19:00.007+01:00

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.

2011-12-15T17:32:00.000+01:00

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

2011-12-14T13:50:00.000+01:00

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

2011-12-13T13:19:00.002+01:00

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

2011-12-13T12:33:00.000+01:00

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

2011-12-11T15:49:00.003+01:00

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

2011-12-11T12:35:00.002+01:00

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.

2011-12-11T09:40:00.002+01:00

I -love- creative people.

Video lectures on Theory Of Automata, Formal Languages and Computation

2011-12-10T18:35:00.000+01:00

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

2011-12-10T15:45:00.007+01:00

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

2011-12-09T13:44:00.001+01:00

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

2011-12-08T20:55:00.000+01:00

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 )

2011-12-06T11:35:00.000+01:00

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.

M381 'Challenge Exercise'

2011-11-26T19:36:00.000+01:00

Each Open University M381 Number Theory booklet has a number of 'challenge exercises'. This is one of them.

Half of the number we are looking for is a square, a third of the number is a cubic and lastly a fifth is a fifth power. Find a number satisfying these properties.

Solution on request.

Britain´s greatest code breaker / Alan Turing

2011-11-26T15:13:00.001+01:00

Alan Turing ( represents the team that ) decisively changed the course of World War II. He is among the greatest scientists of the 20th century, if not all times. Alan Turing is the inventor of computers, programming and artificial intelligence. All the computers in operation today, including the billions of smartphones are in fact Turing Machines, the computer Turing invented conceptuallly. His thoughts were revolutionary. Computers in his days, were people. People, computing.

The British of today are working hard on clearing their conscience on how they treated Alan Turing. In 2009 Gordon Brown said that "he is sorry for the "appalling" way World War II code-breaker Alan Turing was treated for being gay." And now there is this documentary called "Britain's greatest codebreaker" based on the biography of Turing and sessions Turing had with a psychiatrist in Manchester, Franz Greenbaum.

In my opinion Alan Turing could only have flourished in Britain because of his eccentricity. What would have happened with Turing if he had to work at a patent-office like Einstein? Asa Briggs, one of the codebreakers at Bletchley Park mentioned that Turing often came to work with his pajamas under his jacket. That described the culture in that group, I suppose. One of the most important things in Turings private life was the loss of his best friend, the love of his. He never got over it although in Bletchley Park he became friends with a woman, the only woman codebreaker on the team. He even proposed to marry her but Turing called it off, because he wanted to live an honest life. That was his first mistake...

His second mistake was going to the police and accusing a male prostitute of stealing 50 pounds. The police did not care about the 50 pound robbery at all, they could book a professor for gross indecency. Turing now lived in a world where, it seems, nothing out of the ordinary was accepted. Society really wanted him. His sentence was that he could choose prison or enforced body change. He chose the body change, which was an experimental chemical castration. His security privileges were removed as well, meaning he could not continue to work for the UK Government Communications Headquarters. The drug, a synthetic version of female hormones, had a catastrophic effect on his body.

Alan Turing sadly ended his life in 1954.

Alan Turing Documentary / Drama

2011-11-24T20:12:00.001+01:00

Alan Turing is among the mathematicians who are truly an inspiration to me and I am sure to many, many others. Find the latest documentary on Alan Turing here: Turing Documentary . A review follows soon, when I watched the movie.

Enjoy!

If you don't have access to Channel 4 ( like me ), then download ( and seed! ) the torrent.

Compiler development 2/2

2011-11-20T18:54:00.000+01:00

OK, what is the status It's in the technical design phase. Graphica is a 3D scene graph language, ( the first version of ) the Graphica compiler generates Mathematica source code that will be handled by Mathematica ( another target to compile to could be Java/jReality for example ). I am at the point of studying the theory behind compilers and the selection of tools, candidates I have pre-selected are:
- ANTLR,
- javaCC,
- jparsec and
- an all handcrafted option.
More options may pop-up while studying though. I selected four books to study:
- (1) Modern compiler implementation in Java, A.Appel, CUP 2004
- (2) Programming Language Processors in Java, D.Watt, D.Brown, Prentice Hall 2000
- (3) ANT LR Reference, Terence Parr, Pragmatic Bookshelf 2007
- (4) Compilers, PT&T, Aho, et al, Pearson 2007
Title (4) is aka the Dragon book and supposedly is the best book on the subject, I'll use that as a glossary. Title (1) supports a javaCC approach, and develops MiniJava, a subset of Java, (2) builds a compiler and interpreter for a language called Triangle ( with Java ). I already downloaded the source, created a project for it in NetBeans and tested a few compiles and runs of a Triangle program which is a Pascal type of language. Several bugs have been reported over the years which I have collected and should fix. Title (3) is clearly pro ANTLR. And (4) is the authoritative reference. Some of the books may seem old but I used a list from a 2010 university syllabus.

I'll have to solve one other issue on the Mathematica side of the project, and that is how to remotely create a Notebook using a kernel I am connected to in Java. Once that is solved I can continue programming, using a stub compiler if necessary.

The purpose of Graphica is to stimulate my personal development and knowledge of computer science, so any pro-s and con-s that I might mention are clearly mine and have to be seen in that light only.

To be continued

Compiler development 1/2

2011-11-19T23:06:00.001+01:00

Although programming domain-specific languages ( with tools like ANTLR ) is part of the standard toolkit (*) of an application programmer, programming compilers is still considered to be a different ballgame. ( Is it? ) I worked for a company once that developed a 4GL. A small group of experts worked on the kernel ( compilers and run-time ). They enjoyed an almost cult-like status, they were literally seen as guru's. The skill to write a compiler was considered God-like. Java was only just released and tools like ANTLR were years ahead of us. - The cruelty is, it always is, that developing compilers is not difficult, it is simple compared to developing satisfactory end-user applications.

More next post, in which I will explain how you can learn to write your own compiler fast and easy.

(*) If you expect a programmer to be carrying an actual physical "kit" on his gadget belt you are wrong. "Standard Toolkit" is programmers jargon for a minimal set of technical skills expected of any programmer. Every programmer knows how to query a database with SQL, creating and populating tables to optimize performance is however specialized knowledge expected only of a database administrator or DBA ( although many programmers have DBA skills ).

Mathematics of Mariner 9

2011-11-18T23:07:00.001+01:00

In 1971 (!), Mariner 9 transmitted pictures of the planet Mars over a distance of 135 million km, with a transmitter that had a power of only 20 watts (!). With such low power and taking into to account the radio noise of space it is remarkable that data was received at all. Yet, the pictures were near-perfect. How is this possible?

The ﬁrst pictures of Mars were taken at a resolution of 700 × 832 pixels. The images used 64 different shades of grey. The pictures were transmitted back to Earth by sending one pixel at a time, so we can think of a message as a single number between 0 and 63. The channel could send the two binary digits 0 and 1. Encoding each pixel as a string of 6 bits would not have worked well. A survey gave a probability of 0.05 that a bit would be ﬂipped by the channel, implying that about 26% of the image would be wrong.

It was acceptable for each pixel to be encoded by up to 32 bits, so Mariner sent a volume of more than five times of the original data. But by doing so error-detection and correction techniques could be used so that finally a picture had less than 100 incorrect pixels. Mariner transmitted over 7200 pictures of Mars like this.

Coding Theory is a branch of mathematics that uses Galois fields, polynomials, vector spaces and matrices.

135,000,000 km far, sent with a 20 watt transmitter

Study Tip - 4 ( Plan, plan and plan )

2011-11-15T21:55:00.000+01:00

It has been a while since my last study tip.

Learning style

I am not a type that can sit still for hours to read, think deep mathematics and make an occasional note in my virtual notepad in my memory. Although I would like to be like that it is not going to happen. They say Euler was like that. That's why he had no problem to continue with his mathematics after he became blind. Stephen Hawking also has this ability to do theoretical physics in his mind. In the Netherlands we have a former checkers World Champion, Ton Sijbrands, who can play and win or draw close to 30 games blind. It proves what the human brain is capable of, in principle. - I can't do it though because I have to be active, I must do something, better: create something. Otherwise I can literally doze off. A friend once said that I have a kinematic learning style. Different people have different learning styles. I can't sit in a lecture and listen to a lecturer either. It literally goes in one ear, leaves the other, without any storage or processing in between. I get extremely bored in lectures ( or courses ), days seem ages because I want to go home to -do- something.

Check-sheet

Over time I organized my style of studying into a series of to-do items. The first to do is always 'Make a detailed plan.' For example if I plan studying a booklet ( Open University study materials are delivered in booklets which often take two weeks to study, or something between 16 and 32 hours ) I create a check-sheet for it. ( If you have ever done a course in scientology, you know the concept. ) If you study from a check-sheet you won't have those blockages. They really work. The most important thing is that you create a plan that works for you. A plan helps you go through the materials and you know when to do what and no more. That way you won't fall in the 'I never have time-off trap'.

Statistic

Maybe you study to make a step on the career ladder, expect to make more money or maybe you simply want to be able to do your current job better. Or you study to get a(nother) job, or you finally want that degree. In any case you have a deep interest in the subject you study. Final study results come in after many years of study. They are hard to measure on a daily basis, even success on a module which usually takes a year is not a good measurement. TMA results? Yes, somewhat. But what if you got that 'good' result by cramming the TMA out in 48 hours of almost continuously cramming? You will have forgotten the materials on the exam. You have to set a study statistic you can measure on a daily basis. I make ( = do ) flash cards with the open source program Anki. For example when I study a definition, of a graph for example. I type in Anki Q: What is a graph. A. A graph G is a tuple [V,E] where V is a set of vertices. etc. Anki makes sure that I will never forget the definition by monitoring my recall on the question in a very efficient manner. But most of all, I am doing something while styding because I am alert and awake as to which questions I have to put in Anki. Naturally I try to formulate the answers in my own words. Anki then provides me a with a range of graphs and statistics about my progress.

Feedback

Have you ever made a plan which worked well for onze day but you threw away the next day because all the time unexpected things happen? I often made plans to study on times when I was not physically up to studying. I felt tired, spent too much time in traffic, got hold and so forth. That is very valuable information you can use when you make your next plan! And if you record the time spent on each topic your time-estimates will improve and improve.

That is what I mean by "plan, plan and plan".

Oh, I no longer think in time, my unit of doable work is the Pomodoro. More about pomos another time.

Pattern recognition

2011-11-14T19:41:00.000+01:00

6666666666666669
9999999999999999
9999999999999999
6699966666999666
6699966666999666
6699966666999666
6699966666999666
6666666666666666

Select all numbers in the quote above with your mouse. Press find: Ctrl+F, type 99. If you are not surprised then you have high pattern recognition skills.

Dreaming of ( massive parallel ) computing power

2011-11-14T17:48:00.000+01:00

Or: Numbers divisible by seventeen (17)

I am looking at properties of decimal numbers consisting of only 1s and 0s. Here is a sample of them which are divisible by 17.

\begin{array}{l}
100000001100000000001000100000100000010010 \\
100000001000001000000100110000000000100010 \\
1000000000010001000000010000000100000100000010010 \\
1000000000010001000001000000100100000001000000010 \\
10000001001000000010000000110000000000100010 \\
10000001001100000000001000100000001000000010
\end{array}

Again, these are just ordinary decimal numbers consisting of only ones and zeros. They are all divisible by seventeen (17). Divided by 17 they are:

\begin{array}{l}
5882353005882352941235300000005882353530 \\
5882353000000058823535300588235294123530 \\
58823529412353000000000588235300000005882353530 \\
58823529412353000000058823535300000000058823530 \\
588235353000000000588235300588235294123530 \\
588235353005882352941235300000000058823530
\end{array}

I really wonder where this track might bring me. Do I ( we ) have control over it, or is there some invisible hand leading us?

I do most of my calculations on computers a geek would frown at today but were dreaming of only five years ago. What is computing power? - Since I am not a gamer I thought that I never needed a high-end PC. Now, I think I need a whole wall of them. Serious. It's kind of crazy, isn't it. More of this madness in future posts, I promise.

Numbers divisible by seven

2011-11-12T22:30:00.003+01:00

Sometimes, doing an exercise runs out of control, slurps time and leads to posts like this:

\begin{array}{l}
111111101101010101001001 \\
111111101101001001010101 \\
111111010101101101001001 \\
111111010101001001101101 \\
111111001001101101010101 \\
111111001001010101101101 \\
101101111111010101001001 \\
101101111111001001010101 \\
101101010101111111001001 \\
101101010101001001111111 \\
101101001001111111010101 \\
101101001001010101111111 \\
10101111111101101001001 \\
10101111111001001101101 \\
10101101101111111001001 \\
10101101101001001111111 \\
10101001001111111101101 \\
10101001001101101111111 \\
1001111111101101010101 \\
1001111111010101101101 \\
1001101101111111010101 \\
1001101101010101111111 \\
1001010101111111101101 \\
1001010101101101111111
\end{array}

you might assume they are binary numbers. They might be, but they are just ordinary decimal numbers consisting of only ones and zeros. They are all divisible by seven. For your convenience I'll show them again, divided by seven this time.

\begin{array}{l}
15873014443001443000143 \\
15873014443000143001443 \\
15873001443014443000143 \\
15873001443000143014443 \\
15873000143014443001443 \\
15873000143001443014443 \\
14443015873001443000143 \\
14443015873000143001443 \\
14443001443015873000143 \\
14443001443000143015873 \\
14443000143015873001443 \\
14443000143001443015873 \\
1443015873014443000143 \\
1443015873000143014443 \\
1443014443015873000143 \\
1443014443000143015873 \\
1443000143015873014443 \\
1443000143014443015873 \\
143015873014443001443 \\
143015873001443014443 \\
143014443015873001443 \\
143014443001443015873 \\
143001443015873014443 \\
143001443014443015873
\end{array}

Even mathematics tries to steal your time, sometimes.

Fabric of the Cosmos

2011-11-12T08:53:00.000+01:00

I think I finally 'get it'. That space-time is not just a word, but a fabric. The fabric of the cosmos.

PBS did it again. They created a documentary you'll still be talking about in ten years time. If that sounds strange, think of the documentaries of Jacob Bronowski, Carl Sagan and Marcus du Sautoy.

Anyway, fabric of the cosmos is running now on PBS. Don't miss it.

"The Fabric of the Cosmos," a four-hour series based on the book by renowned physicist and author Brian Greene, takes us to the frontiers of physics to see how scientists are piecing together the most complete picture yet of space, time, and the universe. With each step, audiences will discover that just beneath the surface of our everyday experience lies a world we’d hardly recognize—a startling world far stranger and more wondrous than anyone expected.

100.000th blog view expected on 11/11/11

2011-11-10T19:51:00.001+01:00

This blog has had close to 100,000 page views since August 2008 and it looks like that the 100,000th page will be viewed on 11/11/11. This is a coincidence, of course. But with all the hype surrounding 11/11/11 I thought I should mention it. - Another example of the NumerologyIsNotMathematics principle. ;-)

EDIT:
( Although not 11/11-'11 yet on all parts of the world. )

[News] - London student protests revisited

2011-11-09T21:16:00.001+01:00

Students from all over the UK ( i.e. Scotland ) came to London today to protest.

Why?
- Fees are trebling from GBP 3,000 to GBP 9,000.
- The privatization of education will start.
- Entire faculties will be closed.
- And more.

Maybe the people behind these measures think that people will do anything to obtain a descent education including taking loans to finance university. From the USA we know that this means lifelong enslavement to the bank...

And that's just what they are protesting against at Occupy London ( and in zillions of other cities around the world ). So maybe the two will meet and join forces.

See also:
- [News] - London: More student protests

Graphica at Google Code

2011-11-08T14:14:00.000+01:00

Graphica is now a real project. I don't know how many projects get started, I do know a lot of them die unfinished. I can only say that I hope this won't happen to Graphica. I will try to push it forward at a steady pace. It is not that I thought about it yesterday and well, started a project to 'see what happens'.

- I have been thinking about it for years.
- I silently programmed a prototype in Mathematica,
- made an extensive study of Mathematica graphics,
- studied how to develop a language-to-language compiler in Java,
- spec'ed the language,
- selected the Java frameworks, and tools to start with,
- chose a license

so: it's time to start hacking! At last. All I need now is a home for my code to live:

Graphica at Google Code.

Du Sautoy on CERN's speed of light result

2011-11-08T13:57:00.002+01:00

I missed a Marcus du Sautoy ( pronounce: desoto ) documentary on CERN's issues on the speed of light. I watched it yesterday though. Another quality documentary by the BBC. I think there was a one-minute babble by the home expert on everything on Dutch tv. I have heard that the BBC receives hundreds if not thousands of complaints every week, well maybe that's what keeps them sharp.

To the point.

Faster than the speed of light could be made so fast due to the rich catalog of the BBC. A lot of material in this episode came from the catalog, no doubt. The struggles scientists had with light are explained up to the point that Einstein entered the scene and explained it all. Einstein said that the speed of light is the same everywhere and independent of how it is measured so time became variable which is hard to understand but true. Later Einstein said that time is the limit of speed in the universe. Nothing can travel faster than light. Voila $E=mc^2$.

Well, a neutrino traveled faster than light There are 16 types of fundamental particles, three of which are a neutrino. Pauli predicted neutrinos in 1930 but he thought it wouldn't be possible to ever find one. We are crossed by billions of neutrinos every second which is possible because everything, including us, is built up from atoms which are mostly empty. Neutrinos are extremely small and have no charge. But they still have a tiny mass so their speed is limited to the speed of light according to Einstein's laws. The amazing result from CERN is about neutrinos traveling faster than light.

For me, as a non-physicist, I find it unbelievable that it is possible to make such precise measurements. If the neutrinos and light were athletes running the 100m they would have beaten light only by a few millimeters.

Scientists are skeptical because time travel would become possible and it contradicts with previous results in the late 80s when it was measured that light and neutrinos emitted by a supernova reached us at almost the same time. But in favor of CERN is a previous result measured in Chicago which at the time was considered an error.

I lost it when Du Sautoy began about tachyons. Theoretical particles with imaginary or negative mass which could travel faster than light, mathematically of course, but so did anti-matter which was predicted through mathematics. There are also circumstances where an absolute speed limit doesn't make sense like black holes and the first second after the big-bang. And of course Einstein's theory and quantum-mechanics are incompatible.

String theory might have an explanation which satisfies everyone though. In their multi-dimensional bulk we live on a 3D membrane. It could be that the neutrinos left our membrane into the 4th dimension and so appeared faster. I bet a lot of SF fans could have come up with a similar explanation. String theorists are actually paid to talk about the fourth up to the 10th or eleventh dimension.

If you haven't seen the documentary yet, you'll be able to find it somewhere, I did.

Graphica preview

2011-11-07T17:32:00.000+01:00

I don't think there are much limits, if any, to the graphics you can create with Mathematica, besides your own imagination of course.

My study project involves mathematics, computer science and art. It is called Graphica, it is basically an IDE to develop applications using Mathematica Graphics. For this I have designed a new simplified language to create -and animate- graphics. The most basic application is of course creating graphics. I have included some pics. Of course this is a project 'in progress'. I intend to release it as an open source app. More as the project develops.

The following images are candidates for the standard object library. They are created in Mathematica with SphericalPlot3D and are called 'butterfly' in Graphica.

Various polyhedra which are easy to create in Mathematica by accessing the PolyhedronData database functions.

More examples in one of my facebook albums, if you are interested.

Thoughts about number theory (1)

2011-11-04T09:15:00.002+01:00

Has it ever happened to you that you skipped a proof because no matter how hard you tried you simply "didn't get it"? - Or worse, that you had to learn an algorithm but the 'why' wasn't given, let alone a proof. - Mathematics may be hard and difficult but at the end of the day you should 'own' the theorems. Where ownership stands for the notion that you could have created the theorem, in principle, yourself. Or look at ownership at this way: browse through a mathematics book that you studied two or three years ago, or even longer. Everything in it looks really simple because slowly over the years, you took ownership of that particular subject of mathematics.

When I am stalled on a particular topic or proof, I simply accept the proposition knowing that somehow my brain is working on it. It is a better alternative than remaining stalled. It is possible though that you are stalled because the author decided to leave out 'a few details'. Some topics in number theory, for example, are simple if you look at it from a group theory perspective. The author then has to decide if his proposed audience has already studied group theory or not. And even then his publisher may decide otherwise because from his perspective the audience should be made as large as possible.

Number Theory may be called the Queen of Mathematics the Queen needs a lot of help from the 'people'. There is analytical, algebraic, combinatorial and computational number theory and I wouldn't be surprised if there are a few more. The theory of prime numbers and group theory are strongly interconnected for example. Numbers are among the first mathematical objects that man studied but is a number an object? Unlike graphs, sets and geometric or topological shapes numbers don't really exist. Even groups exist, not just as sets, but they are part of nature itself in the form of symmetries everywhere. What -is- one? Like the one in 'one apple'? I argue that 'number' is a property like color, and the rest is just physics.

What about 'God invented the integers', 'Number theory is beautiful' and so on? The truth is that man invented a God that supposedly invented the integers. I have been studying number theory for a while now, and I haven't found its beauty yet. Just open problems, a lot of open problems everywhere. - Inspector Columbo would call that loose ends. And for him that is proof of human error. - That these open problems are a challenge is another issue.

Google Public Data Explorer

2011-11-01T11:00:00.000+01:00

Google Public Data Explorer is really a great tool of practice and exploration if you are doing a course in descriptive statistics. It is a sort of Google Maps - plus. Plus data. It does certain things that WolframAlpha does but interactive and with more data. You can even embed data that you created with this tool in your own webpages or reports.

The Google Public Data Explorer makes large datasets easy to explore, visualize and communicate. As the charts and maps animate over time, the changes in the world become easier to understand. You don't have to be a data expert to navigate between different views, make your own comparisons, and share your findings.

Try it yourself here: http://www.google.com/publicdata/home

Graphica - Scene Graph Language

2011-10-31T10:29:00.000+01:00

Like everyone else, I suppose, I have a need to express my creativity. To facilitate this I am coding a small programming language to create graphical art. Although, for me, Mathematica is -the- language to do that I want to have a language that kids of age 8 to 88 ( TinTin or Donald Duck readers ) could learn. By using it geometric transformations like rotations and translations are learned in a natural way.

The graphic like the one below, including the rotation, is an example what the language could create with a few lines, a Mathematica Kernel is required but will run in the background only.

To be continued.

M381 - Challenge Exercise

2011-10-29T21:04:00.000+02:00

In the Open University course Number Theory and Mathematical Logic the additional exercises sections of the workbooks are complemented with several 'challenge exercises'. This is one of them.

Let $n$ be an odd positive integer.
Prove that there are $\tau(n)$ ways of writing $n$ as a sum of consecutive positive integers.
For example, if $n=9$, $\tau(9)=3$ because $9$ has three divisors $1,3,9$ and the three sums are: $9$, $4+5$ and $2+3+4$.

To be continued.

Rene Descartes

2011-10-27T22:14:00.002+02:00

In the book 'Problem Solving and Number Theory' I read

The law of quadratic reciprocity was discovered for the ﬁrst time, in a complex form, by L. Euler who published it in his paper entitled “Novae demonstrationes circa divisores numerorum formae $xx + nyy$.

Was there no notation for exponents in the time of the great Euler? I did not know how to formulate a query for Google, so I asked a question at Math/StackExchange.

I found out that Descartes introduced the notation for $x^2$. Descartes is famous for his quote "Cogito ergo sum", "Je pense donc je suis" or in plain English "I think, therefore I am". Descartes was my favourite mathematician in secondary school because he invented analytic geometry, one of the milestones in the development of mathematics. Before Descartes geometry was done strictly in the Euclidean way, by compass and ruler. Thanks to Descartes' quadrant and coordinates, geometric shapes like lines and circles could be respresented by algebraic equations. They become objects to do calculations with.

So far about Descartes.

Also, thanks to kind repliers I discovered the bookset "A History of Mathematical Notations, vols 1 and 2., by Florian Cajori". A real gem if you like the history of mathematics. Because the copyright expired one is allowed to freely download the original version. See the comments below the original question at StackExchange for a link to the full version of the book.

Link:
- Question at StackExchange

Social Media for mathematicians

2011-10-23T13:52:00.000+02:00

The places 'to be' for mathematicians are -not- Twitter and facebook, although they are both great places to socialize, chit-chat you know what I mean, they basically steal your valuable time.

Mathematicians hang out at:
- http://math.stackexchange.com/
- http://www.reddit.com/r/math
- http://www.stumbleupon.com/to/stumble/topic:247
- http://mathoverflow.net/
- http://mathblogging.org/
- http://arxiv.org/

Thanks to Gaurav Tiwari, read his review at his own site My Digital Notebook

The difference between mathematics and numerology

2011-10-22T15:09:00.000+02:00

Numerologist is a four-letter word among mathematicians ( I can imagine ). Is numerology the same as number theory? Not really, the difference between numerology and number theory, as a branch of mathematics, can best be explained using an example.

Both numerologists and mathematicians study identities like $$2^2 + 3^2 + 5^2 + 7^2 + 11^2 + 13^2 + 17^2 = 666.$$ A mathematician would notice that this is the sum of the first seven primes and rewrite the sum as $$\sum_{k=1}^{7} p_k = 666.$$ The right-hand side sum is a number with equal digits, so the mathematician might look at other sums of consecutive primes and verify if these sums have similar patterns. Basically a mathematician is interested in anything that might lead to the formulation of a theorem, a proposition to prove mathematically.

A numerologist would immediately notice that $666$ is the number of "The Beast", the representation of evil in the Christian belief system. Then, a numerologist might consider prime numbers divine, since they are the building blocks of all integers, and might try to formulate some law of good and bad represented in the sum-formula. Numerologists also believe that future events can be predicted so they will be extra alert to that.

I believe that it was until the Middle Ages that there was no real distinction between the profession of numerologist and mathematician. Newton has been called the last Alchemist, perhaps he was the last mathematician / numerologist as well. ( History, I am afraid, is not my strongest point. )

Solving quadratic congruence equations in Mathematica

2011-10-22T10:23:00.000+02:00

This is as yet the last post in the LQR series. If you are only interested in -solving- ( quadratic ) congruence equations then this is the way to do it in Mathematica:

As a general Diophantic equation:

In[1]:= Reduce[x^2==123456+1299709 k,{x,k},Integers]

Out[1]= (C[1]\[Element]Integers&&x==427784-1299709 C[1]&&k==140800-855568 C[1]+1299709 C[1]^2)||(C[1]\[Element]Integers&&x==871925-1299709 C[1]&&k==584941-1743850 C[1]+1299709 C[1]^2)

Or slightly more elegant as a pure congruence equation ( thanks to: Mr. Wizard ):

In[2]:= Reduce[x^2 == 123456, x, Modulus -> 1299709]

Out[2]= x == 427784 || x == 871925

Shanks-Tonelli algorithm for solving quadratic modular equations

2011-10-21T16:19:00.003+02:00

M381 unit 6 is about the Law of Quadratic Reciprocity. An application of the LQR is solving quadratic modular equations like: $$x^2 \equiv 499 \ \text{mod(617)}.$$ M381 contains a method that can fast determine if that equation is solvable. It does not contain however a fast algorithm for finding the actual solutions. One such algorithm is the Shanks-Tonelli algorithm and can be found on Planet Math which as often gives a much clearer presentation than the messy Wikipedia-entry.

See also: Tutorial for Quadratic Equations

Quadratic reciprocity in a finite group.

2011-10-21T09:21:00.001+02:00

Law of Quadratic Reciprocity

Let $p$ and $q$ be distinct odd primes. Then $$\displaystyle \left({\frac p q}\right) \left({\frac q p}\right) = \left({-1}\right)^{\frac {\left({p-1}\right) \left({q-1}\right)} 4}$$ where $\displaystyle \left({\frac p q}\right)$ and $\displaystyle \left({\frac q p}\right)$ are defined as the Legendre Symbol $\displaystyle \left({\frac{a}{p}}\right) := a^{\frac{(p-1)}{2}} \pmod p$.

Gauss considered his work on the Quadratic Reciprocity Law among his major achievements. I don't 'get that', not now anyway, that's a call for more study on the topic.

Now and then, when I browse through papers, or otherwise, I find an interesting mathematical paper... ( that I can actually read ). Actually, I was browsing through a book called Reciprocity Laws, from Euler to Eisenstein by Franz Lemmermeyer, it contains more than 100 proofs of the Quadratic Reciprocity Law. I hoped to find a proof I could appreciate by it's beauty. Although most proofs are based on Gauss's Lemma ( as the proof in M381 ) but there are proofs in other realms of mathematics like Group Theory. Group Theory -as we know it today- did not exist in Gauss's time. That's why I am going to spend some time studying the following paper 'Quadratic reciprocity in a finite group.'

An easy problem

2011-10-18T13:44:00.000+02:00

A wise man rode into a desert village one evening as the sun was setting. Dismounting from his camel, he asked one of the villagers for a drink of water.‘Of course,’ said the villager and gave him a cup of water. The traveller drank the whole cupful. ‘Thank you,’ he said. ‘Can I help you at all before I travel on?’‘Yes,’ said the young man. ‘We have a dispute in our family. I am the youngest of three brothers. Our father died recently, God rest his soul, and all he possessed was a small herd of camels. Seventeen, to be exact. He decreed in his will that one half of the herd was to go to my oldest brother, one third to the middle brother and one ninth to me. But how can we divide a herd of 17? We do not want to chop up any camels, they are worth far more alive.’ ‘Take me to your house,’ said the sage. When he entered the house he saw the other two brothers and the man’s widow sitting around the fire arguing. The youngest brother interrupted them and introduced the traveller.

‘Wait,’ said the wise man, ‘I think I can help you. Here, I give you my camel as a gift. Now you have 18 camels. One half goes to the eldest, that’s nine camels. One third goes to the middle son, that’s six camels. And one ninth goes to my friend here, the youngest son. That’s two.’ ‘That’s only 17 altogether,’ said the youngest son. ‘Yes. By a happy coincidence, the camel left over is the one I gave to you. If you could possibly give it back to me, I will continue on my journey.’ And he did.

What went wrong ?

A difficult problem

2011-10-18T13:40:00.001+02:00

Prove or disprove the existence of God.

Answering this question requires a definition of God, which we don't have of course. We could propose a temporarily definition like 'God is the entity that created life on planet Earth, a modest definition because it leaves the question of who created the universe unanswered.

Although still debated, Gödel's incompleteness theorems imply that we won't be able to build computers with a conscious mind capable of creating new mathematics or writing a program that solves all open problems in mathematics. Besides cloning and modifying what we have ( DNA ) we won't be able to create a 'machine' smarter than ourselves.

Our own existence is a paradox we don't understand. History proves that in cases like that humans are creative in inventing a God capable of fixing any problem. Despite centuries of scientific advances we still need a God to explain our existence.

Off-topic: 15 October 2011 - World-wide protest day

2011-10-15T13:12:00.002+02:00

Was it Cairo or New York, - Occupy Wall Street - where it started? It sure is spreading. I want to express my solidarity with the people out on the streets today in almost 1600 cities world-wide: Occupy Together.

Since the revolution started in Egypt I watched and compared the news from BBC, CNN, Infowars, Al Jazeera, RT ( Russia ), and Press TV ( Iran ). Now Press TV has been taken off-air in the UK. Draw your own conclusions. Here is what Infowars ( US Libertarian ) had to say about it: Press TV taken off air in UK – war for freedom of speech now on.

Now that the alternative media are flourishing, it turns out that we haven't had a free press for decades, Press TV made that painfully clear.

During the last world war, scientists created the atom bomb, period. 'Because they were afraid the Germans would be first.' We have to assure that they don't have an easy excuse like that next time, a free Internet ensures that.

M381 exam

2011-10-14T11:22:00.001+02:00

Did M381 exam. All the questions were doable, easy as a matter of fact. Honestly. I am sure I have full marks for the first question I made. But when I was done and looked at the clock it was already past three o'clock. "Should I have been able to do that question in 10 minutes?", I thought. Maybe. I have a painful ear-infection at the moment, and I was drugged of course, prescription painkillers, doctor's order. And antibiotics, of course. I felt like walking on the moon, in an astronaut's suit. Whatever the outcome may be, I can set it right. More later on the exam, when I feel better.

The book will never die.

2011-10-10T16:22:00.000+02:00

A lot has been written about real paper books versus ebooks. Both have their distinct advantages and disadvantages. I have to admit that I read mostly ebooks. They are cheaper and easier to get, store and carry. Mathematics ebooks often have the PDF format. So you need a reader that can handle PDFs. If you use your PC or laptop than a PDF reader is all you need. Most people know Adobe Reader but there are much better programs than Adobe around. With free readers, just like Adobe. My PDF reader of choice is Foxit Reader 5. What I particularly like about Foxit is that it is lightweight, i.e. loads and acts fast. Foxit uses tabbed reading, like internet browsers. If you were in the middle of five books, close Foxit, the program nicely remembers which books you were reading and on what page you left. Most of all, I like the feature that I can highlight what I read. It is almost as if I was reading in a paper book.

Link:
- Foxit Reader 5.

Hypercomputation

2011-10-09T12:32:00.000+02:00

Only recently lightspeed as the ultimate limit of speed got challenged. It seems they have been challenging the Turing Machine for a while too.

If you start with studying mathematics you are only three or four centuries behind on contemporary mathematics. That's quite a lot of catching up to do. - Some fields started their development in the previous century though. Like mathematical logic, a field I have been studying this year, and have written about so now and then in this blog. Part of mathematical logic is the theory of computation which showed us what can be computed and what can't. That what can be computed is what can be computed on a Turing Machine, period. - That idea is challenged however in the theory of hypercomputation. A new field in mathematics which is trying to go beyond the limits of the Turing Machine.

A book with an overview of the theory is the following.

The mathematics of revolution

2011-10-08T11:32:00.000+02:00

The Occupy Wall Street movement is spreading. It reminds me of Conway's Life game. It could spread enormously and still die out soon, or it could cause permanent change without real massive demonstrations. Nobody knows, nobody can predict this. Still, I think it will be very interesting to find mathematical patterns in global, Internet connected, demonstrations like this. I am sure mathematicians of government agencies are working on it.

If a government does not educate even one generation it is lost. It is in their own interest, it is in the interest of the ruling elite to give all citizens a good education. I think that rule fits all political systems. In the US however they use education to enslave people for the rest of their life to the bankers. Paying back student loans turns out to be very difficult. On top of that teachers get fired or are underpaid. - That does not seem right to me.

Primitive recursive function

2011-10-07T23:53:00.005+02:00

Normally you calculate n factorial with Factorial[n] or short n!. Mathematica handles the details of the function for you and prints the result.

In the course M381 you have to prove that functions like factorial are a primitive recursive function. This basically means that the function can be defined only in terms of itself, add one, or set to zero. A primitive recursive definition of factorial would look as follows in Mathematica.

suc[n1_] := n1 + 1
add[n1_, 0] := n1
add[n1_, n2_] := suc[add[n1, n2 - 1]]
mul[n1_, 0] := 0
mul[n1_, n2_] := add[mul[n1, n2 - 1], n1]
fac[0] := suc[0]
fac[n_] := mul[n, fac[n - 1]]

As you can see no other Mathematica functions than "+ 1" and "= 0" are used. The functions suc, add, mul, fac are defined for the first time.

For example:

In[67]:= Factorial[6]
fac[6]

Out[67]= 720

Out[68]= 720

Comment on "From analog to brain computing ".

2011-10-04T13:19:00.000+02:00

Mathematicians have a tendency to regard texts which are not written using 'protocol' as irrelevant. Long ago I wrote a note to a mathematician and his reply was that I should formulate my thoughts in 'standard mathematics'. I did my very best 'to make myself clear'. It was not enough. The thing is mathematicians lose their authority when they leave familiar territory. Well, at least I received a reply. ( Although that was all he did. And I haven't given up on the problem I was working on... )

The blog received a comment, containing what I, for the moment, call 'out-of-the-box' thinking. Non 'standard mathematics' at least. I have moved the comment to this post in an attempt to share this with as much as possible readers. I will reply. But later, I have to let it work on me first.

Regarding our active internal analog math...

I'm a civil/environmental engineer by education but I've been working off and on on a theory which takes the tact that all abstract math symbols and expressions are secondary and arise from a handful of internal analog "math" artifacts and processes. This may not be a very polite thing to say to a mathematician, but I am wondering if you have impressions along the same line?

It turns out that we all get energy to think and do math and other things from the respiration reaction (organics + oxygen -> water + carbon dioxide +energy). And basically, what that means, if you remember your biology or organic chemistry, is, body-wide, within our cells is a ~steady creative flow of about 10^20 water molecules per second -- coming from the 160 kg of O2 we each respire each year. Generally, each water molecule is sort of tetrahedral in shape with two positive and two negative vertices and so, it turns out that there are at least six ways each water molecule can orient within an enfolding field when it first comes into being at a respiration site. That also means that a chain of n-molecules can form in 6^n different ways. Thus a sequence of 12 molecules could form in 6^12, or about 2 billion different ways. A chain of eighteen molecules could associate with 6^18 or 10^14 different impressions. Now, in this analog math theory, I am assuming that repeating vibrations in the environment ought to result in formation of similar stacks and chains of structurally coded water molecules being formed. This gets us a rather crude image of the vibrations of our internal and external environment forming an internal echo or representation within this active internal analog "math", or "language".

I say it's active because the 6^n stacks of water molecules are really also structurally coded hydrogen-bonding packets and such things, when they unfurl, are connected with and influential in protein-formation and protein-folding, which is to say, memory formation and muscle movement, which is to say, in our case, ALL human expression, perhaps beginning with our nearly universal actions and impressions of counting each of our ten fingers and ten toes, and the like.

Bizarre stuff, huh? Lots of little internal Turin devices writing out structural coded signals.

I'm wondering if mathematicians are taught this type of internal analog math as the basis of the abstract math symbols and expressions, or if they are given different associations or impressions, perhaps leaving it that there is just an uncanny (and unknown) relationship between much or all of nature and math?

Also, I vaguely see the similarity between 2^n binary or boolean math and the 6^n "multiple-state structural coding" that I've made up or stumbled onto. I expect the trend continues with starting with other polyhedra which have limited orientations "within enfolding fields" -- when a containing structure is added. My general hunch is the initial condition IS actually significant for us and we can immediately get to multiple states (relevant to ~quantum mechanics/quantum gravity) by starting with tetrahedron and adding the enfolding cube container, rather than the way it's done presently of beginning with the xyz-cubic framework and adding variants.

Initial conditions do matter in mathematics, don't they?

Best regards,
Ralph Frost

@frostscientific
http://magtet.com/images/phpshow.php

Thanks you, Ralph Frost.

Who am I ?

2011-10-04T12:51:00.000+02:00

I am the square root of -1. Who am i?

and of course from GEB:

This sentence contains ten words, eighteen syllables, and sixty-four letters.

From Mathworld - Self-Recursion

From analog to brain computing

2011-10-02T10:52:00.001+02:00

Before digital computing took over completely, analog computing was dominant for a short while. An analog computer is based on the creation of a model which represents the problem to be solved. But mathematical models of problems can be created of ( almost ) any problem and these models can be implemented on a digital computer. A digital computer is nothing more than a convenient, fast, Turing Machine or equivalent thereof, i.e. the URM or Abacus. And from Mathematical Logic ( Goedel ) we know that these systems have its limitations. It is theoretically impossible to create a program that solves all mathematical problems. - But physicists and biologists say ( and why should we disagree? ) that we -are- computer ( brain ) controlled machines.

Is that a paradox? Humans can do more than computers, we can solve mathematical problems, in fact we -created- the concept of a 'Turing Machine'. This leads us to Roger Penrose. In The Emperor's New Mind, 1999 he claims that artificial intelligence in computers is impossible. He argued that the human brain must exploit a type of physics that he described as 'non-computable'. By this he means beyond algorithmic computing, and thus digital computing.

A picture that keeps fascinating me is that of a predator bird flying high over its prey before, at a carefully -chosen- moment, it makes the dive and following kill. And this is all done with a tiny bird brain. The best comparable thing made by humans thus far is the drone. A huge flying case loaded with bombs operated by a battery of digital computers assisted by human -computers-. Although humans have created a model of a flying bird, it is operated by a human computer on the ground.

Analog computers were special purpose computers, designed to solve one specific problem. A predator bird will never be able to learn new behavior, it cannot be trained to live with chickens. Not immediately anyaway, if ´evolution´ made the bird.

Let me summarize before this turns into a rant.
- There are other models of computing than the Turing machine, i.e. analog computing, brain computing.
- Digital computing is superior over analog computing, brain computing is superior over digital computing.
- Analog and digital computing are human creations we fully understand.
- We don't understand brain computing (yet?).
- Mathematical logic and computability theory study algorithmic ( digital ) computing.

Goedels theorems are somewhat like Russell's paradox in set theory. Goedel's incompleteness theorems are statements about logic and number theory deduced in and with the rules of logic.

Heatwave : day off study.

2011-10-01T23:15:00.000+02:00

Took a day off study today. Have been putting a lot of time in studying lately. Felt like work instead of fun. We have a mini-heatwave in The Netherlands. For the 1st of October it was the hottest day ever ( since recorded weather anyway ). I don't like summers, especially when they turn up in my favorite season autumn. I mean, I think everybody has been off-schedule today. - Tomorrow, I take a day off as well: I really missed working with Mathematica. I got really interested in the foundations of computer science lately. Will read about formal languages, grammars and parsers tomorrow. And of course will have a look at the Mathematica built Lisp interpreter. - I am working on a program myself, SceneGraphica, I need to spend time on that as well. I think I will start all over. Nothing will be lost though. I wouldn't have had the ideas I have now without the effort put in the early versions.

The Limits of Mathematics ( or: a Lisp interpreter in Mathematica )

2011-10-01T13:40:00.000+02:00

( ... ) mathematics because it is an extremely difficult road to traverse. The terrain is extremely demanding. The amount of work and concentration required to build the foundation necessary to continue extending the framework is immense. ( ... ) - David Andrews

Mathematics, as if you have never seen a skyscraper and are traversing the streets of Manhattan. With that mindset, you can only think that people -walk- to the 60th floor... Anyway, feeling overwhelmed by the sheer size and complexity of maths is not going to help. Only people willing to teach, without ulterior selfish motives, can help. One can write a book about mathematics to impress peers, as a way to meet publication quotas or to -teach-. Like the book The Limits of Mathematics does for example. It is a clear taste of the best mathematics has to offer, an invitation to go on to the next level.

The table of contents says it all:
- Randomness in arithmetic and the decline and fall of reductionism in pure mathematics
- Elegant LISP programs
- An invitation to algorithmic information theory
- The limits of mathematics
- Appendix. LISP interpreter in Mathematica

The appendix contains the source code of a Lisp interpreter coded in Mathematica. I love that. But the book starts with a clear description of the massive changes taking place in ( the thinking about ) mathematics during the first half of the twentieth century, from Hilbert to Turing.

Goedel, Escher, Bach - Lecture 6

2011-09-30T21:07:00.000+02:00

In the first six minutes or so Curry gives a fairly good summary of Goedel's Theorem. Unfortunately this is the summary of the previous lecture which was not recorded. It seems nothing is free, not even free video lectures because it turns out the best ( not implying the rest is good ) is missing.

After rushing through formal stuff he wastes five minutes about a three-layer stupid joke about a book he had not read.

I quick-scanned through the rest of the video. Not worth watching, really. Too bad. I looked forward to this.

The take home message of the course. All provable things are true but not necessarily al true things are provable.
Justin Curry

Now that I am mostly through all M381 stuff I am glad it included mathematical logic. I would -not- have done it as a stand-alone course. Logic is hard in the beginning, like most new subjects. It needs time to work on you. I will get back to this in the next M381 post.

Previous posts on the series:
- Lecture 1
- Lecture 2
- Lecture 3
- Lecture 4
- Lecture 5

About M381 (1)

2011-09-26T23:28:00.000+02:00

Regular readers of this blog know that I have a sort-of rage-button like the Hulk: it is called MathCad. Thank God, I have the anti-dote almost always open and ready: Mathematica. I am not going to repeat why MathCad is a danger to your mental health, but I have to press the MathCad button at least once.

In almost all mathematics courses you can do at the Open University there is software involved. They either deliver a standard package, or ship custom software especially developed for the course ( i.e. MT365 ). The house-package of the Mathematics Department of the Open University is MathCad, version 2001. I have argued that MathCad alone is a reason -not- to choose for the Open University. What a disgrace...

( Calming down. )

They do however recognize that software, computers, tools are relevant in mathematics. Especially in Number Theory computers are used in active research. Another area where they use software in active research is: mathematical logic. Stronger: research in Number Theory is impossible without computers.

These facts are not even mentioned in M381. There are many open source tools available for Number Theory, even more for Mathematical Logic. Not a word about it in M381. One, if not -the- reason is the fact that course development in the Open University is done in a project organization. A project is created with the objective to create course X which will then be used for the next 10 or so years. It is exactly the opposite of what one would expect of a university education. It is not reasonable to expect the Open University to be at the forefront of mathematical research. Simply because other universities in the UK have that role. But it is reasonable to expect more than a static expose of 19th century Gauss number theory and early 20th century logic from Church, Turing and Goedel. In fact, the field is presented as abstract and of theoretical importance only. But Number Theory and Mathematical Logic are extremely relevant and applicable in many industries! But I did not learn that from the course and that is sad.

It took me a lot of work but I found some relevant learning tools in the fields of number theory and mathematical logic. More about those later in this blog.

(*) - I may have misunderstood the concept of 'University' in the UK. I think many universities in the UK are what we call in the Netherlands 'schools'. They deliver professionals with a degree in all fields through excellent education but they don't do research and so on. They don't add to the body of knowledge. They process and transfer knowledge. That description fits the Open University as well. - A marketing issue is that students like to have a 'university' education. And marketing people love empty ~~heads~~ boxes, they have a fancy word for it too: the 'packaging'. Does that make sense?

Goedel, Escher, Bach - Lecture 5

2011-09-24T18:56:00.003+02:00

A few months ago I started to watch the MIT video lecture series on Goedel, Escher, Bach. Due to time constraints I wasn't able to complete watching the entire series. Today I continued with watching lecture 5. I have learned quite a lot on the subject through M381 and I am about to really 'get it' as far as the Goedel Incompleteness Theorems are concerned. My first reading of GEB took months and now parts of the book begin to look simple. If you don't know what I mean browse through a mathematics book you thought was hard, a few years ago. It often seems if there is 'nothing in the book'. The odd thing with Goedel ( and with all mathematics, I suppose ) is that in your mind you think you can explain it to a laymen in one or two sentences. ( It is -that- simple, I am afraid. ) The power of mathematics is that it can capture an entire knowledge tree in a single word. That word remains meaningless without understanding of all the words in the knowledge tree.

A bit about lecture 5.

Dress shows arrogance

I wonder if Justin Curry would go to a job interview in that Club Med outfit. Students are paying customers ( and a pool of cheap labor for lucrative research deals the university makes ) deserving respect from teaching staff.

Formal number theory ( as in M381 )

Justin talks about Typographic Number Theory, ( formal number theory in M381 ). For example $$\forall x ( \neg x = \mathbf{0} ( \exists y x = y') )$$ can be interpreted as

"Every x that is not equal to 0 is the successor of some y."

Leibniz was the first to propose a formal language for number theory. He asked whether it was true that an algorithm could decide if a statement in number theory was true. - Although in M381 this question is answered negatively that does not mean computers can not play a role in proving mathematical propositions. There is an abundance of ( open source ) software for proving theorems.

( Not in video: ) Isabelle a formal proof theory assistant has been used in testing an operating system kernel written in C and assembler. It not only verified that the spec was implemented correctly but it also discovered hundreds (...) of programming and design (...) errors which were not found by traditional testing methods.

Previous posts on the series:
- Lecture 1
- Lecture 2
- Lecture 3
- Lecture 4

Neutrino's supposedly faster than light.

2011-09-22T22:15:00.001+02:00

Off-topic:

Neutrinos Travel Faster Than Light, According to One Experiment

I got really excited when I read that "... particles had been found that travel faster than light ...". ( local teletext ). Wouldn't that be amazing? I mean, all the major scientific discoveries seem to have been made in the distant past. ( Physics and mathematics, I mean ). That would change overnight if Einstein's law of relativity got broken.

My guess? A measurement error. They are talking about a particle that traveled 60 nanoseconds faster than expected. One nanosecond is 1/10^9 part of a second. I am sure they have found some ingenuous way to measure speeds involving figures like that, but NOT when the figures depend on GPS! Come on, GPS.

I am afraid that this could turn into something very, very embarrassing for CERN. I hope their P/R people are as brilliant as the scientists they employ. ( Except the guy who was -?- responsible for measuring the speed of neutrino's, of course. )

Exercise ( algebra )

2011-09-22T00:16:00.000+02:00

Given that $$x^n-y^n = (x-y) (\sum_{k=1}^n x^{n-k}y^{k-1} )$$ with for example: $$x^4-y^4 = (x-y)(x^3 +x^2y +xy^2 + y^3)$$.
a) How would you factorize $x^5 + y^5$?
b) Generalize.
c) Prove the identity above for $x^n-y^n$ using mathematical induction.

Exercise ( logic ).

2011-09-18T17:40:00.000+02:00

Knowledge of mathematics is not required to solve the following exercise, but it will sure help ;-)

Dr. Who asked you for a ride in the Tardis. Naturally, you couldn't decline, it might be a matter of national, if not global, importance. The Tardis landed on the Planet of Truth which is is inhabited by people who always tell the truth. A minority however decided to lie, always. It is Doctor Who's mission to seek and destroy all liars. First you must get to the Capital of the Planet. Two roads fork out. Should you go left, or right? An inhabitant approaches, greets you and gives you the privilege of asking him one Yes/No question.

Ask him where the capital is, left or right on the fork, but beware he might be a liar!

Credit follows with the answer.

Take the challenge, test your ability to think logically.

Study Tip - 3 ( mathematics does not have to be difficult )

2011-09-17T20:56:00.001+02:00

Being a programmer who studies math (, a math studying programmer, whatever ) gives me access to the literature of the foundations of computer science. Now and then I browse the libraries. I thought to have a found a book that fits my level, so I read the preface to check if I was right. The author Alan Parkes made the following interesting statement.

For Students : I wrote this book partly because when I studied this material as part of my own Computing degree, I had to work really hard to understand the material, a situation which arose not because the material is too difficult, but because it was not well presented ...

from Preface of A Concise Introduction to Languages and Machines by Alan P. Parkes, Springer 2008.

I have seen video of a lecture on MSRI where the researcher / professor held 'a talk' ( there is a subtle difference between 'a talk', a presentation and a lecture, 'a talk' is a euphemism for a not well prepared presentation ) about a subject from scribbled notes. Most of the lecture he stood with his back to the audience mumbling while scribbling math. End of lecture. Goodbye. It was expected of the audience to write up notes from his mumblings and scribbles. - Folk like that, probably based on a brilliant thought they had decades ago, get the opportunity to write books too. Once in the hands of ( then ) students like Alan Parkes they make the learning experience troublesome to say the least.

Well written, illustrated mathematics followed by examples, more examples, examples of exercises, exercises with written solutions and so forth make studying mathematics easy. ( Think of an Open University booklet, a good one ). Open University booklets are written by course teams for students and not by professors who write to impress... who actually?

Tip 4: Change books

If you have difficulty with a ( mathematics ) subject then find another book about the subject. Check how the other author explains the topic. Try a third, fourth if necessary. This method is guaranteed to work. When you are close to revision for an exam it stimulates to read through the material written by someone else. Different exercises, and so on.

Finally: authors who write for students are read, respected and remembered.

Off-topic: 9/11 Revisited (4-4)

2011-09-11T18:54:00.000+02:00

What a totally insane day this is. Everybody knows that the official 9/11 story is a lie and the mainstream media are repeating that same lie. The US government is criminal, period.

People that have been very important to 9-11 truth are:
- Alex Jones,
- Dr. Stephen Jones,
- David Ray Griffin,
- Prof. James H. Fetzer,
- Dr. Judy Wood
- and many others
I deeply respect their courage to come forward and tell the truth.

I say no more.

Visit
- Scholars for 9/11 Truth
- Architects and Engineers for 9/11 Truth

Previous posts:
- Off-topic: 9/11 Revisited (1)
- Off-topic: 9/11 Revisited (2)
- Off-topic: 9/11 Revisited (3)

Reading fiction about mathematics

2011-09-10T12:00:00.000+02:00

This is a lighthearted, enjoyable novel about a bumbling but likable mathematics professor at an unnamed university who believes that he has discovered a solution to a famous mathematical puzzle known as "Beauregard's Wild Number Problem."

A novel set in the world of contemporary mathematics: The Wild Numbers by Philibert Schogt. Doesn't that sound great? I mean actually having the time to leisurely read novels, stroll through the city, drink coffee somewhere. Haven't read it yet, but I will, someday, I must really. :-)

Is M381 True or False ?

2011-09-09T10:56:00.003+02:00

In facebook I have a photo album with mathematics related pictures. Sometime ago I added the picture below. I thought it was a joke...

... but I could not have been further from the truth. The picture could have been taken at any Open University M381 tutorial. This student is not clueless, on the contrary: he contemplates about some deep mathematics.

Anyway, if you have always wanted to give an answer to the layman's question: "But can you -prove- that 1+1 = 2?" then M381 is your course.

Prime?

2011-09-08T23:59:00.000+02:00

A prime p is a number with two positive divisors: 1 and p. Note how this definition nicely excludes 1 which has only one positive divisor. Then the primes are: $$2,3,5,7,11,13,17,19,23,29, \cdots$$
Wait! What about:
$$ 5 = -i \cdot (1 + 2i) \cdot (2 + i)$$
This is an example of a factorization in the quadratic field of Gaussian Integers. It is therefore not enough to say that a number is prime. Primality is relative in relation to the number field.

The mathematician Lamé thought to have cracked Fermat's Last Theorem in 1847. Needless to say that his proof contained an error. He overlooked the fact that prime factorization was not unique in a number system he used in his proof.

In Mathematica factorization is done with:

FactorInteger[n],
factorization using Gaussian integers is done with:

FactorInteger[5, GaussianIntegers -> True].

Off-topic: 9/11 Revisited (3)

2011-09-07T22:24:00.001+02:00

What have -you- seen on 9/11? Are you sure that what you saw on TV really happened?

Ten years, what does that mean? A first year math student of 18 was only 8 in 2001. The story must be told and re-told because 9/11 is an important part of the history of -the world-. - Anyone older than, say 28, probably remembers 9/11 as if it was yesterday. I do, definitely, I can't forget 9/11. Especially in periods like this, with the 10th anniversary and so forth, it keeps going through my mind. Why?! I have asked that question many, many times. Because I have seen things on TV that are scientifically impossible but everybody accepted it as the truth. That brought me in a condition of doubt. Therefore the following video came as a relief to me. Things that I thought to have seen were finally confirmed.

This is the video that confirmed what I thought to have seen. ( Three parts, part 2 and part 3 on YouTube ).

Previous posts:
- Off-topic: 9/11 Revisited (1)
- Off-topic: 9/11 Revisited (2)

Perrin numbers

2011-09-06T22:35:00.000+02:00

Let $P(0)=3, P(1)=0, P(2)=2$ and $$P(k) = P(k-2) + P(k-3.)$$ These numbers are called the Perrin numbers. They have the interesting property that $\mod{[P(k), k]} = 0$ in almost all cases when k is prime. ( Otherwise we would have found a true prime generator! ). In any case the property holds until $k=271441.$ Interesting, isn't it? See the table below for the first 40 Perrin numbers.

k	P[k]	Mod[P[k],k]	PrimeQ
2	2	0	True
3	3	0	True
4	2	2	False
5	5	0	True
6	5	5	False
7	7	0	True
8	10	2	False
9	12	3	False
10	17	7	False
11	22	0	True
12	29	5	False
13	39	0	True
14	51	9	False
15	68	8	False
16	90	10	False
17	119	0	True
18	158	14	False
19	209	0	True
20	277	17	False
21	367	10	False
22	486	2	False
23	644	0	True
24	853	13	False
25	1130	5	False
26	1497	15	False
27	1983	12	False
28	2627	23	False
29	3480	0	True
30	4610	20	False
31	6107	0	True
32	8090	26	False
33	10717	25	False
34	14197	19	False
35	18807	12	False
36	24914	2	False
37	33004	0	True
38	43721	21	False
39	57918	3	False
40	76725	5	False

Off-topic: 9/11 Revisited (2)

2011-09-06T09:24:00.002+02:00

I am a big fan of the BBC because of its documentaries, Sherlock, Doctor Who, Torchwood and even EastEnders ( although I am 'clean' these days ). For many years I looked at the BBC as -the- independent source of world news, but 9/11 changed that, unfortunately. Reading the following below may shatter your trust in the BBC.

Links:
- The BBC’s Instrument of 9/11 Misinformation ( from: VetaransToday by prof. Jim Fetzer )
- Off-topic: 9/11 Revisited (1)

Fibonacci Sequence - Revisited

2011-09-03T14:30:00.000+02:00

Some thoughts about my progress in maths. - I browsed through some old blog posts to get a feeling for how much I progressed with my maths. That was the original purpose anyway. Six years ago I wrote about the Fibonacci sequence as I am nowadays. It often feels if I make no real progress. I see mountains ahead of me, when I look back I see a tiny road. I don't believe in working hard, working smart should do the trick. On the plus side I can say that today I can prove Binet's formula and give at least three alternative formulas for the series. I don't know I even knew the formula in 2006. If progress is thinking about the Fibonacci series as something trivial then that is not what I expected. When I was in secondary school the mathematics teachers seemed to have acquired infinite knowledge about mathematics. I am sure that I know at least as much about math as the average high school teacher. That should give me a sense of satisfaction, achievement. It does not. Link: - Max Cohen draws the golden spiral

Off-topic: 9/11 Revisited (1)

2011-09-02T12:44:00.000+02:00

My blog is about mathematics and in particular studying mathematics. I have kept the blog ad-free and I have avoided posts about politics and/or religion ( with one exception, a one-time post about 9/11 ). In little over a week it will be ten years ago since the 9/11 attacks. These attacks defined the major events in the last decade and shaped the world as we know it today. Counting down to the 9/11 memorial I'll make a few posts about 9/11. I hope you don't mind.

The first 9/11 post is a link to VT Veterans Today, the Military & Foreign Affairs Journal. The VT article is called 9/11: Video of Missile Hitting Pentagon Leaked.

It concerns the following video.