Mathematica Workbench 2

Wolfram released a new version of their workbench named Workbench 2. It has all the features one can expect from a professional development workbench. Based on Eclipse 3.5 it comes with a Mathematica Debugger, facilities for Unit Testing, Deployment, Refactoring ( quick-fixes as they call it ) and tools for Documenting source. They have published a set of video tutorials here. With J/Link you can use Mathematica in Java and Java in Mathemica whatever you prefer.

Distance from Earth to its nearest copy.

To Infity to Beyond is one of those superb Horizon programmes. The UK fascinates me for many reasons but the quality of British television is one of them. ( I recently found a lot of Avengers episodes, the Spi Sci series from the sixties with John Steed, Mrs. -Emma- Peel and later with Tara King and 'Mother', what a pleasure to be able to watch them all again. But that's off-topic. )

To the Horizon programme they show how to calculate the distance between our planet Earth and the nearest exact copy. Yes. IF, the Universe is indeed infinite than there must be a planet Earth somewhere which is an exact copy of ours. This is not only some mental experiment like Hilbert's Hotel but it is possible to calculate the distance to the first copy ( there are many, of course ) in meters.

In meters: \[ 2^{10^118} * 10^26 m\] where \[ 10^118 \] is the number of particles in our universe, and \[ 2^{10^118}\] is thus the number of possible universes and \[ 10^26 m \] is simply the diameter of our own universe.

UPDATE:
I thought about this, watched the programme once more. I think the actual distance is significantly larger than the number above. The example in the program is an extrapolition of a universe consisting of 4 particles of which there are two types. The argument remains valid though. If the universe is infinite then there must be a copy of our Earth somewhere.

M208 - TMA01/2 draft completed

All questions done, in draft status. An advantage of typesetting TMA's is that you can always make them better. It's part of the fun of making TMA's. What were the questions like?
TMA01 consists of seven questions of which 2 have already been sent in and marked ( viciously unfair but that's another matter entirely ).
Question 3 was about graphing sets in the form of 2D regions.
Question 4 was about 2D transformations ( way simpler than MS221 which I don't understand because M208 is positioned as a successor to MS221 and the first 'real' pure mathematics course in M31 ), a translation, a rotation and a composition. Question 5 was about mathematical induction. Question 6 was about complex numbers. Question 7 was about equivalence relations. I liked questions 6 and 7 most. Part of question 6 was solving the following equation.
\[
z^4 = -8 + 8 \sqrt{3}i
\]
There is still some time to edit and review ( reviewing saves tons of marks ) before the cut-off. Then it is full steam ahead to the next TMA's.

And also: I want to do some long term ' thinking ' about my 'research interests' . Which part of mathematics do I really, really like? Do I want to know much more about? I should be able to answer that question by now. More about this topic soon.

Another Pi record

I have a customized home page on google.com/news with, of course, a section on news aboutt 'mathematics'. I find it odd that I missed this news item about Pi.

A computer scientist claims to have computed the mathematical constant pi to nearly 2.7 trillion digits, some 123 billion more than the previous record.

Odd, the item did contain the word 'mathematics' and there hasn't been any other major news pushing the item down on the list. ( Items can stay on the list for weeks ). Maybe I trusted Google a bit too much in providing me with -all- the news throught their excellent google news service.

M208 - Alert

Today I received my assessment for TMA 1 / Part 1: Q1 19/25, Q2 8/10. Or 27/35. Which means that I have to score 58 / 65 for the remaining questions to at least stay on 85. Considering how this tutor marks a clear Mission Impossible. - The marks were viciously unfair written in the ( red ) handwriting of an obsessed personality. I am going to ( already started ) take every possible legal step the OU regulations offer to solve this issue. I really have a very strong case. Proving the unfairness of the markings will be easy. - I don't want to go to an exam with an average TMA score below 85. I just won't.

Well, I have been warned. I don't know how I should call this. An enemy on the road. A guard at the gate. So M208 will be difficult after all ( ... ) Who would have thought that?

M208 - TMA01 / Part 2

Started working on it today. The exercises aren't really difficult, I suppose. Well, everything can be turned into a challenge. I am trying to do as much as I can with Mathematica, that way I learn more about Mathematica. I am trying to find a way to visualize the effect of a linear translation on a certain are of the 2D-plane at the moment. I sort of feel the load of the 90 points at the moment. Have to ship all questions by the end of next week.

A research problem which interests me is the Collatz 3x+1 function, which is defined as follows:

\[
\begin{equation*}
f(n)=
\begin{cases}
3n+1 & \text{if $n$ odd ,}
\\
\frac{n}{2} &\text{if $n$ even.}
\end{cases}
\end{equation*}
\]
Collatz conjectured that recursively applying this function to a number always ends in the loop {1,4,2,1}.
Take for example 9.
f(9) = 28 , f(28)=14 , f(14)=7,  f(7)=22 , f(22)=11 ,f(11)=34 , f(34)=17 , f(17)=52, f(52)=26, f(13)=40 ,f(40)=20, f(20)=10 , f(10)=5 , f(5)=16, f(16)=8, f(8)=4 , f(4)=2, f(2)=1, f(1)=4
et voila...

Try 27, if you dare.

( Latex test post )

\[
\begin{equation*}
f(n)=
\begin{cases}
3n+1 & \text{if $n$ odd ,}
\\
\frac{n}{2} &\text{if $n$ even.}
\end{cases}
\end{equation*}
\]

MT365 - Video 7

Video 7 is called "This little flower went to market". Locations: Amsterdam Flower Market, Bloemenveiling Aalsmeer ( Flower Auction Aalsmeer, Holland ) and the Scilly Islands in Great Britain. An interesting documentary about decisions regarding transportation and distribution a flower producer has to take. It is not always possible, even when using mathematical methods, to find a guaranteed optimal solution although in many cases such a solution can be found. Problems shown involved the selection of  type of transport: land, sea or air and the difficult bin packing issue which involves loading several trucks with batches of flowers of different size.

Now that I have started MT365 I am learning various new features of Mathematica as well. Particularly those in the Graph Database and the Combinatorica package.

Videos Discrete Mathematics

(... Long ago I watched a programme ( I thought it was in the Twilight Zone series ) about a rich man and a poor man. Some catastrophy happened and in their neighbourhood so they had to leave. The rich man took gold with him, as much as he could carry. Gold keeps its value and buys everything at all times. The poor man carried water as it was the first thing he might need. - The walk out of town took longer than expected and took them right into the desert. The rich got thirsty and decided to buy some water from his neighbor who still had enough for days, perhaps weeks. But he didn't sell. - Gold turned out to be worthless. ...) - I recalled this while I found another very interesting series of video lectures on discrete mathematics. Last year or so, complete lecture series on video were scarce on the internet. There were the MIT lectures, nothing more. Now ever more turn up. That's great but it's impossible to watch them all.

The one I found today is special though. It is a lecture series by Steven Skienna about Concrete Mathematics, a famous book by Ron Graham, Don Knuth and Oren Patashnik. A difficult book with many challenging problems. Worth reading over and over. A classic. The first lecture is about the Josephus problem. I might watch a few lectures, why not? It is cetainly related to MT365.

Doing Graph Theory

What is Graph Theory like? ( After a few weeks of study anyway. ) Let's take the The Konigsberg Bridges problem as an example. This problem marks the start of Graph Theory and was solved by Leonhard Euler. The question was to find a route through the city crossing each bridge only once. Then the Graph Theorist enters the scene.

He translates the relevant parts of the city to a graph such that the seven bridges become seven edges and the various city areas shrink to mere vertices. He translates the original question to a question about the graph: is this graph Eulerian? If so, then the answer to the original question is yes.

But what exactly is an Eulerian graph? An Eulerian graph is a connected graph which contains an Eulerian trail. A graph theorists knows a theorem which helps him to decide if the graph contains an Eulerian trail. We however, have to explain first the words connected graph and Eulerian trail. An Eulerian trail is a closed trail that includes every edge. Again. Two new words! Closed trail and edge ( = connection between two vertices. ) A closed trail is a trail with start and finish at the same vertex ( = a dot in a graph. ) A trail is a walk in which all edges, but not necessarily all vertices, are different. A walk of length k is a succession of k edges of the form uv, vw, wx, ..., yz. This walk is denoted by uvwx...yz, and is referred to as a walk between u and z. Remains the word connected graph. A graph is connected if there is a path between each pair of vertices.


Why not show all these definitions, relations and corresponding theorems in a graph?! Exactly. ( There is an aspect of self-reference here but I lack the knowledge of terms to exactly describe it. Maybe after M381. )

I created this graph using Personal Brain. It is possible to share and publish brains on the internet in order to collaborate on some project. MT365 is the first course I am using Personal Brain for. It is too soon to evaluate though.

MathJax

Yesterday I found MathJax. Thanks to Twitter, btw. Twitter can be very helpful in the process of collecting valuable information from the web.

MathJax in brief:
High-quality display of LaTeX and MathML math notation in HTML pages
Supported in most browsers with no plug-ins, extra fonts or special setup for the reader
Easy for authors, flexible for publishers, extensible for developers
Supports math accessibility, cut and paste interoperability and other advanced functionality
Powerful API for integration with other web applications

We are now able to write math on the web.
\[\sum_{k=0}^{n} {n \choose k} = 2^n \]

No plugins needed. Use simple Latex. If you don't know Latex let Mathematica write it for you.

( testing ... LaTeX )

The probability of getting \(k\) heads when flipping \(n\) coins is:

\[\sum_{k=0}^{n} {n \choose k} = 2^n \]
\[P(E) = {n \choose k} p^k (1-p)^{ n-k} \]

An Identity of Ramanujan

\[ \frac{1}{(\sqrt{\phi \sqrt{5}}-\phi) e^{\frac25 \pi}} =

1+\frac{e^{-2\pi}} {1+\frac{e^{-4\pi}} {1+\frac{e^{-6\pi}}

{1+\frac{e^{-8\pi}} {1+\ldots} } } } \]

( ... more later !!! )

MT365 - More video programmes

MT365 - Video 2.
Video 2 is about tilings at the Alhambra. One of the options I considered for 2010 was studying m336 next to m208. I decided not to because of the geometry part of M336: it's about tiling patterns which I find a rather difficult subject. Designs 1 of MT365 is about the same subject. But where in M336 one must be able to prove that the only rotational patterns possible are of orders 2,3,4 and 6 that fact is accepted as is without proof in MT365. Designs 1 could very well be the step I need to start confident on M336 I have in my plan for 2012.

MT365 - Video 5.
This video is about the proof of the four colour theorem. No more than four colors are needed to color any map. Thomas Kempe, a 19th century mathematician ( played by an actor of course ) explains his proof in detail based on unavoidable sets and reducibility. I knew it couldn't be the real proof because I knew the real proof wasn't found until 1976 but I didn't see why Kempe's proof was flawed. Neither did the mathematicians he explained it to at the time. After ten years or so it became apparent that his proof wasn't entirely correct. In 1976 the theorem was proved but by a computer. The proof consisted of hundreds of computer-generated pages. - I wonder if there are still people around who hope for a compacter proof. The 1976 proof was largely based on Kempe's proof by the way. What if there is an entirely different approach possible to tackle this problem?

Received MT365 course materials

As you can see on the pic below today I received the first shipment of MT365 course materials.


I already watched one video on the DVD, 'the location problem'. A programme about the Fire Department in Rotterdam, my home town. How the Fire Department decided on locations for the various departments spread over the city. Most of the mathematics was done by the University Twente.

The shimpment contains besides various forms,
- course handbook ( very important ! )
- course guide
- graphs 1
- networks 1
- design 1
- graphs 2
- networks 2
- design 2
- 3 audio CDs
- notes for the audio programs
- 1 DVD containing 7 half hour programs
- notes for the video programs
- assignments ( CMA41, CMA42, TMA1, TMA2 )
- graph theory software
A standard booklet like 'graphs 1' has often five chapters, lots of exercises and answers and has on average +/- 75 pages with clear illustrations. All the materials I studied sofar were excellent. If you are on 8h/week for MT365 ( 30 points ), one booklet is studied in 16 hours or over a period of two weeks. Most of the time is ( should be ) spent on exercises anyway, which is fun time.

Well, this will keep me busy besides M208.

Gems of Geometry by John Barnes

Gems of Geometry is a book, published by Springer, yet written by an amateur mathematician, John Barnes. Barnes is computer scientist who designed the language RTL/2 while working at ICI. He wrote several books on the ADA language.

Sofar I only browsed Gems of Geometry but I already love it. The chapters are:

1. The Golden Number
2. Shapes and Solids
3. The Fourth Dimension
4. Projective Geometry
5. Topology
6. Bubbles
7. Harmony of the Spheres
8. Chaos and Fractals
9. Relativity
10. Finale

More later on this book when I have read (parts) of it. I am curious what he has tell about these topics and what I will learn from it.

Four Cubes problem

After only one section on MT365 Graph Theory you will be able to fully understand and apply the techniques used in solving the Four Cubes problem, or Instant Insanity as it seem to be called. I found a YouTube video where some professor explains the problem.


I did not know the problem was so popular, yet another YouTube video on the subject ( warning: loud KrautRock music in background ).

About The Secret Life of Chaos

If you are studying mathematics then the BBC Four documentary 'The Secret Life of Chaos' is a must see. It starts off with the ideas of the brilliant Alan Turing and ends with a computer program that is able to simulate evolution itself.

In that particular program various 'brains' live with the purpose to 'walk'. The brains are programs that are able to fully rewrite themselves. Another program which acts as the enviroment kills of brains that don't perform well and selects programs to replicate. After several generations the brains are able to do things that human programmers would consider impossible to code. The idea is that we exist purely by chance based on very simple rules. The simulation proves that evolution works.

The behaviour of a flock of birds is briefly mentioned in the program. A few hundred pilots would not be able to mimick a flock of birds because in our perception of communication some leader has to give orders from top to bottom. The birds just follow some simple rules. I don't know if anywhere complex systems are being built based on that principle, if so I give them more chance to succeed than those who want to create complexity from scratch.

Leisure time

I haven't watched television in ages, I suppose it's at least ten years ago since I watched. Well, I occasionally watch a program. I have seen a couple of matches of Euro 2004. Netherlands - Portugal, if I remember correct. I read the news on a news ticker at the top of my desktop. I loaded the ticker with RSS feeds from all the major news sites. I subscribed to breaking news alerts via e-mail so the real news comes through. - At times I selectively crawl the internet to collect quality stuff to watch. I have tons of documentaries. This week I watched 'Touching the void' - about two climbers and their perilous journey up the west face of Siula Grande in the Peruvian Andes in 1985. As if I joined them. Mind blowing.

Tonight I'll be watching a BBC Four documentary ( 17 jan 2010 ) which is called The Secret Life of Chaos.

( One of the goals of my study is to be able to understand the mathematics of fractals. The OU has a course on them in the M.Sc. program. I suppose that would be level 4 then. But I am busy as it is with MT365 and M208. )

It looks promising. And otherwise I am going to listen to ( watch perhaps ) another Project Camelot interview.

Chaos theory has a bad name, conjuring up images of unpredictable weather, economic crashes and science gone wrong. But there is a fascinating and hidden side to Chaos, one that scientists are only now beginning to understand.

It turns out that chaos theory answers a question that mankind has asked for millennia - how did we get here?

In this documentary, Professor Jim Al-Khalili sets out to uncover one of the great mysteries of science - how does a universe that starts off as dust end up with intelligent life? How does order emerge from disorder?

It's a mindbending, counterintuitive and for many people a deeply troubling idea. But Professor Al-Khalili reveals the science behind much of beauty and structure in the natural world and discovers that far from it being magic or an act of God, it is in fact an intrinsic part of the laws of physics. Amazingly, it turns out that the mathematics of chaos can explain how and why the universe creates exquisite order and pattern.

And the best thing is that one doesn't need to be a scientist to understand it. The natural world is full of awe-inspiring examples of the way nature transforms simplicity into complexity. From trees to clouds to humans - after watching this film you'll never be able to look at the world in the same way again.

MT365 - Status

I am on a 90-point schedule this year. Until now I have allocated saturday(-morning) and sunday(-morning) to MT365 and tuesday to M-208. I'll see how that goes. It's because I am fairly new to graph theory. I have studied book Graphs 1 / Section 1 this weekend. I collected 42 defintions, 4 theorems and 12 methods and / or How-To's into PB.

P.S.

The graph of K14 has exactly 91 edges.

Graph Browser

At the Wolfram site you can download a Graph Browser which requires either Mathematica 7 or a ( free ) Mathematica player. The graph browser connects to a graph database on the internet which contains quite a lot of graphs. In MT365 the only graph by name studied sofar is the Petersen Graph. A query on Petersen actually returned tha exact graph as we have studied.

Mathematica and Graph Theory

Mathematica can ( and probably will ) be a great help while studying Graph Theory MT365. The possibilities Mathematica offers are overwhelming. Being able to produce neat graphs from a list of edges is just the beginning.

The figure above is the well known Konigsberg graph.

The Handshaking Lemma ( First steps in Graph Theory )

Beginning Graph Theory is basically learning new words. I once heard that the Eskimo's have over one hundred  words for snow. It looks as though mathematicians have as many words for graphs. Graph, regular graph, null graph, complete graph, labelled graph, unlabelled graph, subgraph, cycle graph and this is only the beginning. The first theorem in Graph Theory is called 'The Handshaking Lemma'.

In any graph the sum of all the vertex degrees is equal to twice the number of edges.

MT365 - Course books online

Nigel, the course manager of mt365 was so kind to scan all the mt365 books and upload to the studenthome under course resources. Excellent job. The StudentHome page is more or less your interface with the university. Surfing to that page is actually 'going to the university'. All the materials are there, there is a library. A real one, online of course, but with access to resources you otherwise wouldn't have like NexisLexis databases. And there are of course forums to hangout so that you can talk to other students. It takes a while I suppose but after a while you feel part of that large OU community. To speak in Scientology terms: I found my 3D.

M208 - TMA01/1 done

Completed TMA01/1 of M208 today.
Question 1 ( 25 ): manual sketch of graph of rational polynomial function.
Question 2 ( 10 ): sketch of graph of function with different formulas on three different domain intervals.
A similar question was asked in TMA03 of MS221 so I thankfully used that TMA as a template of which I made a note in the answer, naturally.

Mathematics compared to Social Sciences

Comparing mathematics and social sciences? I wouldn't know where to begin. All I know is that they use math when they construct models of their problems. I came across the blog of an OU student in Social Sciences. I copied his comment on the reception of his marks on a first TMA:

Yesterday I got TMA01 back for DD101.
I’m not that pleased, to be honest. I got 70%, but I was expecting better than that. I was really pleased with what I wrote and I thought I would at least get into the 80s. Then again, this is my first assignment for my first course, so I had no idea what to expect. I have nothing to compare it to.
I’m also confused about the comments I got from my tutor.
They were all very positive, with praise for almost every element of the essay. The only negative comments were for minor technicalities, like having a full stop in the wrong place in a reference and not writing out a place name in full. If I’d read the comments and not looked at the grade, I would have expcted the grade to have been in the 90s!
Oh well, it was a solid pass, I guess. Onto the second assigment, which is due the first week of December!

That sort of frustration is unlikely in mathematics. When your score is 70 you know that you did not or incorrectly do problems worth 30/100. One of my ideals in life is the concept of a meritocracy. An organization form applicable to companies but also to countries I suppose where positions are granted based on 'the more you put in the more influence you get'. Some open source software projects are managed like that and are quite successful.

I wonder how close or how far away the world of mathematics is from being a meritocracy. Before one becomes eligible for any position a PhD is the bare minimum weight on your belt I suppose. But after that, I am not sure. I read a bio of a young, female professor. I don't recall her name. Asian but working in the US. I was impressed by the number of publications she had on her name in relation to her name. I then calculated the total number of hours she was given so far in her life. Presuming she was extremely efficient in everything she did, etc. I still found she could not have spend more then 40 hours ( or even less ) per paper. All the papers were collaborations. I then recalled a book I read once it was called 'The professor can't teach' or similar about how obsessed mathematicians are by publishing in volume. Have to read that book again some time.