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Thursday, April 28, 2011

A surprising vector norm

A vector norm of an n-dimensional vector $\mathbf{x}$ denoted by $\parallel\mathbf{x}\parallel$, is a real-valued function of $\mathbf{x}$ such that:
-$\parallel\mathbf{x}\parallel > 0$
-$\parallel k\mathbf{x}\parallel = k \parallel\mathbf{x}\parallel$
-$\parallel \mathbf{x+y}\parallel \le \parallel\mathbf{x}\parallel+ \parallel\mathbf{y}\parallel$

It can simply be verified that the Euclidean length is a vector norm:
$$\parallel\mathbf{x}\parallel = \sqrt{x_1^2 + x_2^2 + \cdots + x_n^x}$$

There is an entire class of norms called the $l_p$-norms:
$$\parallel\mathbf{x}\parallel_p = (x_1^p + x_2^p + \cdots + x_n^p)^{\frac{1}{p}},$$
of which the Euclidean length or Euclidean norm is a member for $p=2$.

A very interesting ( and surprising ) norm is the $l_\infty$-norm:
$$\parallel\mathbf{x}\parallel_{\infty} = \max{(|x_1|, |x_2|, \cdots, |x_n|)},$$
this behavior can be explained from the fact that the higher the $p$, the more $l_p$ is dominated by the element of the largest magnitude.

Sunday, April 24, 2011

Are mathematicians likely to have Asperger Syndrome?

When the sun shines, we should drop our mathematics and go out 'to enjoy the Sun'. Those who don't are considered odd. They must 'have a personality disorder' of some kind. If you spend time on something 'you don't need anyway', like mathematics, something must be wrong with you.

Maybe there -is- something wrong with me after all? Gary Numan, a musician I like, has Asperger syndrome. Maybe I have the same disorder? It would be an explanation for my 'strange behaviour' since Numan is usually considered somewhat strange too. Considering that, according to some statistics, over half of the population has some form of personality disorder, this is a normal question to ask. I looked up a test for Asperger Syndrome, completed it, and here is the result:

 Click to enlarge

24. 'An average math contest winner' - Exactly right, if you change winner to contestant of course. Programmers score 21, which I am and explains that even in that context I am considered somewhat as a Nerd, sometimes anyway. The average man scores 18, while a male biologist scores 15. A score of 32 strongly indicates Asperger. Asperger is a form of autism. - I am in the clear. If you are interested in your own score on the Nerd scale you'll find it at this site.

Some people who have (had) Asperger.
- Albert Einstein
- Isaac Newton
- Leonardo da Vinci
- Bill Gates
- Robin Williams and
- Shakespeare.
who all did, "sort-of" well.

From: source

(*) Even if the 'real' answer is Yes, the answer is still NO. Because Asperger Syndrome is merely an invention by psychiatrists. A real disease can be determined during an autopsy. The zillions of diseases, syndromes ( supposedly caused by chemical imbalances ) are inventions to justify prescriptions of psychiatric drugs which are as addictive to the patient as profitable to the manufacturer. ( See: CCHR )

Thursday, April 21, 2011

I don't know what people who go to watch a performance of the New York ballet actually experience. In the Black Swan Thomas Leroy ( played by Vincent Cassel ) says:

To beauty! - Thomas Leroy

I can't agree more. Beauty, it is probably the best word to describe what happens when we get touched by a piece of mathematics, or even a line of Mathematica code.

I found a free, Creative Commons licensed, book about Mathematica programming. I must emphasize that it is a book about programming the core Mathematica language. It is possible to solve and communicate about mathematical problems without actually programming Mathematica but there will come a time when your problems require ( some ) programming. Although Mathematica ships with extensive on-line documentation several ( excellent ) books have been written about the subject most of which I have reviewed in this blog. Among them, and not reviewed yet, is Mathematica programming: an advanced introduction.

An ideal reader for it would be a person who has some Mathematica experience as a user, needs to write programs more substantial than a few one-liners, and wants to understand the logic of the language and ways to program idiomatically, minimize programming effort and maximize program's efficiency. - Leonid Shifrin

I don't know why people write 'free' books. In general I am not in favor of free books simply because the author spent ( a lot of ) time creating the book and if it is of value to a reader some form of exchange needs to take place. Free does not mean 'of less quality' although it has to be said that an author of a free book is not helped by an editor. I can only say that I like Leonid Shifrin's Mathematica book and that I suggest you have a look at it. It can be read online, or downloaded as a PDF at his site http://www.mathprogramming-intro.org/

Wednesday, April 20, 2011

Do not call 911 for math problems

This had to happen some day. It just happened to be today.

Tuesday, April 19, 2011

Magic squares of type n-by-n

I wrote a small program that can generate a magic square for any size of a matrix. The idea is that this program can support me in my quest to generate some particular normal magic squares. The ideas I got thus far were too computationally intensive though. It remains an interesting enough playground for me, a challenging enough problem that I might solve one day. - I have read that you need some sort of portfolio of problems on which you can work alternately depending on how you feel, think, etc. on a given moment.

$$\left( \begin{array}{ccc} 9 & 6 & 3 \\ 0 & 6 & 12 \\ 9 & 6 & 3 \end{array} \right)$$
$$\left( \begin{array}{cccc} 9 & 6 & -1 & -4 \\ 5 & -6 & 3 & 8 \\ -8 & 7 & 6 & 5 \\ 4 & 3 & 2 & 1 \end{array} \right)$$
$$\left( \begin{array}{ccccc} 40 & 30 & -46 & 8 & -2 \\ 40 & -43 & 30 & -11 & 14 \\ -52 & 26 & 24 & 22 & 10 \\ -8 & 9 & 16 & 7 & 6 \\ 10 & 8 & 6 & 4 & 2 \end{array} \right)$$
$$\left( \begin{array}{cccccc} 165 & 118 & -86 & -79 & -59 & -38 \\ 84 & -187 & 48 & 46 & 8 & 22 \\ -113 & 42 & 20 & 19 & 36 & 17 \\ -78 & 32 & 15 & 14 & 26 & 12 \\ -43 & 11 & 20 & 18 & 8 & 7 \\ 6 & 5 & 4 & 3 & 2 & 1 \end{array} \right)$$
$$\left( \begin{array}{ccccccc} 344 & 272 & -144 & -180 & -124 & -66 & -46 \\ 228 & -391 & 70 & 68 & 66 & -17 & 32 \\ -202 & 62 & 30 & 58 & 28 & 54 & 26 \\ -194 & 50 & 48 & 46 & 44 & 42 & 20 \\ -94 & 38 & 18 & 34 & 16 & 30 & 14 \\ -40 & 13 & 24 & 22 & 20 & 9 & 8 \\ 14 & 12 & 10 & 8 & 6 & 4 & 2 \end{array} \right)$$
$$\left( \begin{array}{cccccccc} 795 & 644 & -283 & -272 & -261 & -250 & -213 & -124 \\ 493 & -890 & 96 & 94 & 92 & 90 & 17 & 44 \\ -406 & 86 & 42 & 82 & 80 & 39 & 76 & 37 \\ -329 & 72 & 70 & 34 & 33 & 64 & 62 & 30 \\ -252 & 58 & 56 & 27 & 26 & 50 & 48 & 23 \\ -175 & 44 & 21 & 40 & 38 & 18 & 34 & 16 \\ -98 & 15 & 28 & 26 & 24 & 22 & 10 & 9 \\ 8 & 7 & 6 & 5 & 4 & 3 & 2 & 1 \end{array} \right)$$

Magic squares of type 5-by-5

Still in 're-discovery mode' generating magic squares has become trivial, although the computational resources increase fast depending on the size of the square.
$$\left( \begin{array}{ccccc} 14 & 32 & 6 & 8 & 22 \\ 14 & 7 & 30 & 17 & 14 \\ 0 & 26 & 24 & 22 & 10 \\ 44 & 9 & 16 & 7 & 6 \\ 10 & 8 & 6 & 28 & 30 \end{array} \right)$$
$$\left( \begin{array}{ccccc} 4 & 22 & 6 & 28 & 42 \\ 4 & 37 & 30 & 17 & 14 \\ 20 & 26 & 24 & 22 & 10 \\ 64 & 9 & 16 & 7 & 6 \\ 10 & 8 & 26 & 28 & 30 \end{array} \right)$$
$$\left( \begin{array}{ccccc} 4 & 6 & 22 & 26 & 64 \\ 6 & 55 & 30 & 17 & 14 \\ 16 & 34 & 24 & 42 & 6 \\ 86 & 7 & 16 & 7 & 6 \\ 10 & 20 & 30 & 30 & 32 \end{array} \right)$$

Magic square
A magic square is a square matrix with elements in $\mathbf{Z}$ such that the totals of rows, columns and both diagonals are equal.
Normal magic square
A normal magic square is a magic square with elements $1,2, \cdots, n^2$ where $n$ is the size of the matrix.
Latin square
A latin square is a $n$ by $n$ square matrix containing $n$ times the first $n$ elements of the alphabet such that each row and each column contains each letter only once. (i.e. Cayley Table)

Further developments

Clearly, normal magic squares are the most desirable objects in the realm of matrices. I am making detailed notes about this work in the ( still? ) unpublished personal mathematics wiki I am setting up.

Monday, April 18, 2011

Open University in the news

The UK's Open University is to receive funding from the United States to improve staying-on rates among poorer students in US colleges.

Saturday, April 16, 2011

Tau-ism: amputating Pi to Tau

Some of the best and beautiful mathematics has been created centuries ago. Open any 18th century mathematics book or journal ( or older if you can read Latin ), do the same with a book or journal of any science of your choice, or compare it to a newspaper of the same period if that's what you prefer. My point is that the vast body of mathematics produced by generations before us is still accessible and thus usable. This accessibility is a unique characteristic of the science of mathematics.

I am not at all for borrowing words from common language and redefining them in the context of mathematics. Words like ring, group or space got an entirely different meaning, words like programming, number or weight may even confuse mathematics students, the full initiation in the language takes years. Personally, I would rather have used a new word for the concept of group, i.e. symmetry transformations, ( at least something that includes symmetry ). Suppose we would implement that word -now-. Before we know it we would have changed the language of math completely. Making the past of mathematics inaccessible for future generations. Something nobody really wants.

Now, what if, we would redefine Pi?! Michael Hart ( wisely ) calls it an immodest proposal in his 'The Tau Manifesto'. Naturally, I thought it was a joke, some sort of parody, but he seems to be serious. Naturally, I am flabbergasted, appalled by such an idea. It is like amputating pi to tau ( See the video for a detailed explanation. )

Mathematics is beautiful, but not perfect. If our ancestors made choices we wouldn't have made today we have to live with them. The cost of change outweighs the benefits manifold. Science includes safekeeping the discoveries of the past.

The video below has been viewed almost 400,000 times in one month.

Thursday, April 14, 2011

Linear Algebra Thoroughly Explained

M373 ( Optimization ) renewed my interest in Linear Algebra which always has been one of my favorite branches in mathematics. Unfortunately I never got any further with it then the standard course up to Eigenvectors. Benedict Gross said in one of his lectures "You can't learn too much Linear Algebra." I suppose that he meant that any investment in learning more Linear Algebra always pays off. Quite a few books on advanced linear algebra have been published, understandably most of them by algebraists which doesn't make these books very accessible, let alone of any practical value. I recently discovered a book called Linear Algebra Thoroughly Explained by Milan Vujicic and published by Springer in 2008. From the foreword: "There are a zillion books on linear algebra, yet this one finds its own unique place among them." The book introduces Complex Inner-product Vector Spaces, new methods for solving linear systems, Dual Spaces, Tensor Products and a lot more.

Linear Algebra Thoroughly Explained @ Springer.

Tuesday, April 12, 2011

Antikythera

In the 20th century Alan Turing designed the first truly general purpose computer: The Turing Machine. Charles Babbage ( 1791-1871 ) created his working difference engine and Blaise Pascal ( 1623-1662 ) created the the first desktop calculator. Can it get more mind-blowing than that? I think so.

The Antikythera mechanism is an astronomical computer thought to have been built in 150BC. It was rediscovered on the Antikythera shipwreck in 1900 and has since astounded researchers by its mechanical complexity. It's the oldest known computer, a relic dating back 2000 years and rediscovered at the bottom of the ocean.

Watch this two part 20 min mini-documentary.

Sunday, April 10, 2011

Magic squares of type 3-by-3 ( continued )

A few details on 3-by-3 true magic squares:
2 | 9 | 4
7 | 5 | 3
6 | 1 | 8
has square symmetry so there are 8 magic squares with digits 1-9 and constant number 15, i.e.:
2 | 7 | 6
9 | 5 | 1
4 | 3 | 8
after a reflection in the main diagonal.

An example of an 'almost true magic' square is:
3 | 4 | 8
10 | 5 | 0
2 | 6 | 7
since it has nine different digits, if we call 10 a digit ( in base 16 for example ) and constant number 15.

A few other nice ones with constant number 15 are:
5 | 9 | 1
1 | 5 | 9
9 | 1 | 5

7 | 3 | 5
3 | 5 | 7
5 | 7 | 3
.

Some remarks following my previous post on the subject, ( in Coast or Horizon-style )

* We are dealing with maps 'up' to a higher dimension. This would mean that if we would ever be able to travel to higher dimensions we would appear to have all sorts of symmetric qualities in the eyes of higher dimensional beings

* Since we can code a (simple) color using three digits we could say that magic squares are the visible 3-by-3 matrices while the other matrices remain invisible to the human eye.

* Any point in 3-space has a corresponding magic square related to it.

So far. Inspiration for this mini project on 3-by-3 magic squares: thanks to David Leavitt / Ramanujan.

And... OU Course M373.

Saturday, April 9, 2011

Tips for Study Tech

I read two articles by Sato on the QED Insight blog.

How To Read A Mathematics Textbook
Posted on April 5, 2011 by Santo

and

Posted on April 8, 2011 by Santo

I have summarized the articles and added info where needed with the typical OU mathematics student in mind:

* Find alternative books, which approach the subject from a different perspective than the course textbook.

* Read with pencil and paper by your side. By the end of each reading session you should have written out a nice list of questions to work on; If you can't answer the questions by yourself ask your tutor or visit a mathematics site like Planet Math, or The Math Forum

* Work through a large number of exercises. Top students when pressed for time, fall back on just doing the assigned exercises.
Some specialized books that you might consult for training in problem-solving are:
- The Art and Craft of Problem Solving, by Paul Zeitz;
- Problem-Solving and Selected Topics in Number Theory, by Michael Th. Rassias
- How to Solve Problems, by Wayne Wickelgren
- How to Solve Problems New Methods and Ideas, by Spyros Kalomitsines

* Daily work is key. Do your best to budget your time so that you devote some time every day to each of your courses.
* Be systematic. Keep a notebook of questions (might be electronic), and note your answers, too. This will become a wonderful record of the evolution of your understanding. Also keep solutions to your exercises and problems in a neat notebook. Review the notebooks often, ideally daily.

Always remember that a desperation for marks is counterproductive to learning.

Ramanujan's magic square formula - Revisited

 Ramanujan's genius (r) was discovered by Hardy (l)
At a very young age Ramanujan designed the following formula for a 3 by 3 magic square:

C+Q | A+P | B+R
A+R | B+Q | C+P
B+P | C+R | A+Q

where A,B,C are integers in arithmetic progression and so are P,Q,R.

Rewriting Ramanujan's scheme somewhat to
2Q+R | 2P+2R | P+Q
2P | P+Q+R | 2Q+2R
P+Q+2R | 2Q | 2P+R

where P,Q,R are in the Rationals, it is clear that every (P,Q,R) yields a magic square with constant number 3 (P + Q + R).

I conjecture that for any 3 by 3 magic square a triple (P,Q,R) can be found in the Rationals such that they fit the above scheme. Finding a proof for this is one of my 'problems'. Naturally, I would be very interested in any counter-example.

Friday, April 8, 2011

Decomposition of the magic square on Drurer's Melancholia

 Durer's Melancholia

Durer's Melancholia is world famous for its magic square on the top right wall.

16 3 2 13
5 10 11 8
9 6 7 12
4 15 14 1

It is a real magic square because it contains the integers 1,2, ..., 16 on 4x4 places. The constant of this magic square is 34. I have been thinking if there are any 'beautiful' ways in decomposing it into two or more (magical) squares. Best I have come up sofar are two squares with constant number 17.

1 3 1 12
3 10 0 4
9 1 6 1
4 3 10 0

15 0 1 1
2 0 11 4
0 5 1 11
0 12 4 1

Add them and the result is Drurer's magic square.

Memorizing digits of PI - revisited

A quality shared by all great mathematicians of the past is that they did not try to make an impression with their knowledge but instead shared their knowledge to the benefit of all. Someone who keeps his knowledge to himself demotes knowledge to a set of tricks and himself to a mere magician.

I don't know the author of the comment below but in the tradition of the best, he wrote the following comment to the post 'How to remember 1000 digits of Pi' which I received earlier this week but only read today. Not often do I receive a comment that I promote to a full post. As you will see it deserves to be.

Remembering the first 1,000 digits of Pi is a very simple task. It sounds daunting at first, but I assure you it is much easier than you think. The method you are using only complicates the process. You need to use the Dominic Method with the Journey Method. The Dominic Method uses 100 characters, each representing a number and letter code.

For instance: 11 on my list is Andre Agassi; his action is playing tennis, naturally. To remember 4 digits at one time, you pair 1 character with the 2nd characters action. So for the first 4 digits of Pi, we have: 3.1415 which corresponds to: Andy Dick writing on a blackboard (1415). AD=14, AE=15. AE is Albert Einstein and his action is writing on a blackboard. So pairing the first character and giving him the second character's action gives you a sequence of 4 digits.

The NEXT step, after memorizing your 100 person list (which gives you 10,000 memory storage locations) is to put them on a Journey. Take a familiar Journey around your house, neighborhood, etc. For instance, start in your bathroom: You have Andy Dick writing on a blackboard IN YOUR BATHTUB. Then you have Norman Bates (92) playing with toys (65)IN YOUR TOILET...then at your sink....you get the idea.

If you want clarification on this or help with your 100 person list, let me know. People have used this technique to memorize 75,000 digits of Pi. I have a journey with 1,000. A simple 50 mile round-trip stretch of highway with landmarks to "Put" these characters on will suffice for 250 storage locations of 4 digits (1 person with 2nd action) at each, which equals 1,000 digits. Email me at: 1fastbmw328@gmail.com if you want more info/help.

The author clearly knows what he is talking about as he refers to the Dominic method and the Journey method, both seem well known methods in the realm of memorizing. ( I must admit that I was not aware of either of them. Another reason why I enjoy learning new stuff. ) It seems that the key is Memorizing the 100 Person List. That is do-able, I suppose.

Clearly, I will start learning more about the methods and start experimenting with them. More soon.

Fast factoring using quantum computers.

“If somebody can build a quantum computer with all that it promises to be, prime number decryption could then occur in real time and that would mean all of the encryption that’s used by banks, governments and the military would be crackable.”
- Andrew Cleland, Ph.D., Physicist, UC-Santa Barbara"

Last Thursday, (31/3-'11) Linda Moulton Howe was on Coast with her monthly report. Linda covers environmental topics for Coast ( Japan quake, Gulf oil spill, mystery of the missing bee populations and so forth ), so I was surprised to hear her talking about prime number factorization.

All encryption nowadays is based on the RSA-encryption method which is a so called public-key cryptography method. The key required for enciphering is publicly known so that everyone can encipher data. Deciphering the data is only possible by the person who issued the key. The enciphering key is ( based upon ) the product of two very large primes while deciphering requires the individual primes. Cracking the code involves the factorization of a large prime number which we aren't very good at not even using brute force on networks of supercomputers. We simply can't do it.

Announcing that you are doing research that will eventually solve the prime factorization issue is of course mind-blowing. As I understood it they are going to build a machine that somehow mimics the quantum behavior of the particle world up-to a quantum computer. The analog of quantum mechanics is then that the computer computes everything until you ask it a question. So when you ask it to factor a large prime it comes immediately with an answer out of the scope of all answers. This answer is then immediately verifiable, something which we can do very fast.

Well, that's the idea. It won't work in all cases of course. Imagine the following question: Is the Riemann Hypothesis true? Yes (or No), will the quantum computer then answer. Remains the task for us to verify, and prove the RH.

This professor either does not understand the concept of Computability, or: he is very smart in using scare tactics to talk some money out of the pockets of some banks, or: his genius is multiple times greater than that of Alan Turing.

Coast to Coast is known for ( and owes its popularity to ) bringing the news 'unscreened' because 'it could be true'.

Thursday, April 7, 2011

Geometry of a table

A mathematician is an organism that transforms coffee into theorems.
Paul Erdös

True. But I bet that he wasn't drinking his coffee at an ordinary coffee table.

Wednesday, April 6, 2011

My advice about studying mathematics is to go deep, to slowly let it sink in and revise, revise and revise. I.e. imagine a student who just reached the mandatory 360 (540) points: if he would not pass a subset of all his previous exams -today- wouldn't the degree be a joke? Anyway, I am all for a final exam covering a subset of all previous exams. - Yet, I don't think students will like the idea. But why not? Once the math is 'in' you would only have to maintain it during the lifetime of the course.

 Hardy (l), Littlewood (r)
 Ingham

An advice given by Littlewood to Ingham ( both were giants in Number Theory while Littlewood, together with Hardy of course, got Ramanujan to Cambridge ) is the following:

Albert Ingham was educated at Stafford Grammar School, and from there he won a scholarship to Trinity College, Cambridge, in December 1917. After spending a few months in the army towards the end of World War I, he began his studies in January 1919. An outstanding undergraduate career saw him awarded distinction in the Mathematical Tripos and win a Smith's prize and the highest honours. In 1922 he was elected to a fellowship at Trinity for a dissertation on the zeta function and his next four years were occupied only with research, a few months of which were spent at Göttingen. During this time Ingham was greatly influenced by Littlewood who gave him the advice to:-

... work at a hard problem: you may not solve it but you'll solve another one.

An advice I have taken long ago. When you generally work on harder problems than those presented to you in TMA's and/or exams, your TMA questions tend to become trivial. This is one possible method to score high marks for your TMA's. Other known proven recipe's which I do not advise are:
- target a student in the forums and offer him help ( not knowing he is going to do all the helping himself );
- post portions of your questions in Dr Math type of forums, mathematicians are very helpful;

If you have a blog ( in mathematics, or any other field ) and people start acting friendly then they are working towards asking you
- to 'exchange' some OU course books;
- to 'compare' answers on a TMA.
Make sure you always say NO, you don't owe them anything. Friendly visitors come in many disguises.

Sunday, April 3, 2011

Dimensions - Documentary movie about mathematics

Dimension Two - Hipparchus shows us how to describe the position of any point on Earth with two numbers… and explains the stereographic projection: how to draw a map of the world.

Dimension Three – M.C. Escher talks about the adventures of two-dimensional creatures trying to imagine what three-dimensional objects look like.

The Fourth Dimension – Mathematician Ludwig Schläfli talks about objects that live in the fourth dimension… and shows a parade of four-dimensional polytopes, strange objects with 24, 120 and even 600 faces

Buy The DVD! If you like the movie.

M381 - TMA02 (1)

Looking ahead at M381 TMA02 Number Theory part.

TMA02 has 9 nine questions, 4 for Number Theory for a total of 50 points and 5 for Mathematical Logic also for a total of 50 points.

The Number Theory questions are about:
1. Chinese Remainder Theorem (10)
2. Congruences ( applied to ISBN numbers ) (11)
3. Fermat's Little Theorem (16)
4. Quadratic congruences / residues (13)

Doable, it seems (now).

More later on the ML part.

I noticed that TMAs 03 and 04 have been posted, so now the set is complete.

Saturday, April 2, 2011

Paul Erdös - documentary

Some people consider Paul Erdös as the most prolific mathematician who ever lived. Born in Hungary in 1913, Erdös wrote and co-authored over 1,500 papers and pioneered several fields in theoretical mathematics. "N is a Number: A Portrait of Paul Erdös" is a documentary about his life and work. - It is now available for on line viewing on Internet. - N is a Number. Portrait of Paul Erdös.

 Paul Erdös 1913 - 1996

Don't expect anything about his (other) addiction in this documentary. In an article by Paul Hoffman published in November 1987, Atlantic Monthly profiled Erdös and discussed his Benzedrine habit. Erdös liked the article, "...except for one thing...You shouldn't have mentioned the stuff about Benzedrine. It's not that you got it wrong. It's just that I don't want kids who are thinking about going into mathematics to think that they have to take drugs to succeed."

Friday, April 1, 2011

M381 TMA01 shipped

Just physically shipped M381 TMA01 to tutor.

This gives you a rough idea about the topics and available points per question. The qualifications 'easy, normal, hard and very hard' are of course my personal opinion.
1. Euclidean algorithm (12) - Normal
2. Mathematical induction (10) - Easy
3. Division (11) - Normal
4. Greatest common divisor (20) - Hard
5. Division (12) - Very hard
Number Theory (65)

6. URM trace and flow-chart (10) - Normal
7. URM programming (10) - Hard
8. Primitive recursion(5) - Hard
9. URM programming and primitive recursion (10) - Very hard
Mathematical Logic (35)

Counting very easy as -2 ( none this TMA ), easy as -1, normal as 0, hard as 1 and very hard as 2 yields a total which divided by the number of questions 9, added 10 and multiplied by 5 gives a hardness indicator ranging from 0 ( very easy ) to 100 ( very hard ). For M381 TMA01 (( 0 - 1 + 0 + 1 + 2 + 0 + 1 + 1 + 2 ) = 6 * 5 / 9 + 10 ) * 5 = 75.

So when I say that M381 TMA01 was hard you know exactly what I mean. ;-)

You may have noticed that the ratio ( Number Theory points / Mathematical Logic points ) is 2. This is because for this TMA we had to study 2 Number Theory OU booklets and 1 Mathematical Logic OU booklet. For TMA02 this is 2/2.

Welcome to The Bridge

Mathematics: is it the fabric of MEST?
This is my voyage
My continuous mission
To uncover hidden structures
To create new theorems and proofs
To boldly go where no man has gone before

(Raumpatrouille – Die phantastischen Abenteuer des Raumschiffes Orion, colloquially aka Raumpatrouille Orion was the first German science fiction television series. Its seven episodes were broadcast by ARD beginning September 17, 1966. The series has since acquired cult status in Germany. Broadcast six years before Star Trek first aired in West Germany (in 1972), it became a huge success.)