M381 'Challenge Exercise'

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

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.

See also:
- Alan Turing documentary

Alan Turing Documentary / Drama

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

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

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

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 )

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

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

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

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

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

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

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

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

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

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)

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

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