Linear Programming ( M373 )

A series of 24 video lectures on Operations Research of which the first 12 are on Linear Programming. These 12 lectures cover more or less block 2 of course M373. The lectures are easy to follow since they go in a slow ( but steady! ) pace with lots and lots of examples. TIP!

More here: NPTEL

As usual Mathematica can be your virtual math coach through the commands:
- Maximize
- Minimize
- LinearProgramming
( See the Mathematica docs for more. )

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.

Generating 4 x 4 magic squares ( 2 ) - ( MathCad = MathCrap )

To be precise, my squares would have been magic if I would have only used the numbers 1 to 16 once each. Like in this famous magic square found in 'Melancholia' by Albrecht Dürer.

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

I am trying to find the parameters required by my method to generate this one. As a test.

I am not sure if this is just procrastination or that this qualifies as 'real mathematics'. I suppose it depends on the result and how I document it. Positive side-effects are that I am now much more interested in M373 and that I am still getting better in Mathematica. - Mathematica can be used as a super calculator of course, but I am beginning to use it as a tool to actually do some 'thinking-work' for me.

Am I still studying for B31? Of course!

Well, M373 entered -Yellow Alert- phase. A plus of M373 is that the topics are really, really interesting. But, I don't like MathCad. It is such a honky tonky load of crap, unbelievable. And then we get the 2001 version! I use the latest version, of course, crapwise certainly an improvement over 2001. - The Open University obviously wasn't by their right mind when they chose PTC ( the manufacturers of MathCad ). Perhaps they had no choice because PTC is the only supplier of mathematics software based on British soil. And Maple ( which seems to be almost as good as Mathematica ) courses are being dropped. Students do have limits as to what they will accept though.

Just about to ship a M381 TMA.

Complex analysis videos

I am dividing my (study-)time between M381, M373, David Leavitt's novel The Indian Clerk (TIC) and self-study activities related to analytical number theory and complex analysis. ( And when I finished reading The Indian Clerk I'll start a book about the life of Kurt Goedel, or I might reread the book on Alan Turing. ) Reading, always keep reading, is my motto.

Excellent mathematicians work harder, have more luck in chosing their subjects and belong to more influential networks than less excellent mathematicians. Is that true? If we can prove so many theorems couldn't we have created one or two as well? Of course. Unless it was a 'theorem' of Ramanujan. His work, providing one has access to it ( read: understands it ) is of the jaw dropping class.

How did Srinivasa Ramanujan perceive the mest world?

In my self-study project I am struggling with the proof of the Prime Number Theorem as well as with understanding the Riemann hypothesis (RH). - In TIC it is G.H. Hardy's wish to prove the RH. At that point in his life he lectures at Cambridge. People in his surroundings we meet in the book are ( amongst others ) John Littlewood, Betrand Russell, Ludwig Wittgenstein and John Maynard Keynes. One day Hardy receives a letter full of mathematical scribblings. The impact of the magnitude of these scribblings reaches him, but slowly. The letter came from no one less than Srinivasa Ramanujan, an until then unknown Indian mathematician. Hardy discusses the letter with his young collaborator Littlewood and the story unfolds... An excellent companion to this reading adventure is Number Theory in the spirit of Ramanujan.



The main goal I have set for myself this -mathematical- year is understanding the PNT. The best way to reach that point was in my opinion studying Apostol's Analytic Number Theory. I make progress, but slow, although I am not terribly behind on schedule. The single most important effect though is that I now feel naturally motivated and ready to attack complex analysis. Last but not least: another set of video lectures on Complex Analysis by Bernd Schröder from Louisiana Tech University.

Lecture 1: Introduction

A measure I took based on my experiences with M208 last year is implementing a TMA-(latex-)code-freeze-date. I have set that date for next Tuesday when I'll start the check-double-checks. As M381 is a level 3 course I am content with the 65-sure I am at right now. Although there is no such thing as a 'sure' before the result is 'in'.

My URM emulator now automatically concatenates two URM programs. I want to automate substitution as well as primitive recursion. Although I am still struggling a bit with the manual implementation of a primitive recursive function at the deeper URM level. Knowing that this all leads ( and I am sure it does ) to understanding Goedels incompleteness theorems makes this all a worthwile adventure.

M373 applied to computer games

The fact that computer games are computationally intensive forces game programmers to invent tricks to do their calculations in a minimal number of steps. A typical M373 optimization problem. It is much faster to calculate $\frac{1}{\sqrt{x}}$ using Newton approximation.

From the source code of Quake.

float InvSqrt(float x){
   float xhalf = 0.5f * x;
   int i = *(int*)&x; // store floating-point bits in integer
   i = 0x5f3759d5 - (i >> 1); // initial guess for Newton's method
   x = *(float*)&i; // convert new bits into float
   x = x*(1.5f - xhalf*x*x); // One round of Newton's method
   return x;
}

Link: Understanding Quake's fast inverse-square-root

M373 site opens

Confirmation of the first cut-off date: it is indeed as soon as 15 Feb for TMA01. I better start on it asap, as I have to go through quite some revisions and new stuff as well.

Looked briefly through course book of Block 1/Unit 2: ( again ) about solving systems of linear equations. But this time enough tools are supplied to solve systems of zillions equations ( if necessary ). Lots of linear algebra and matrix stuff. Looks cool to me.

No details yet on tutors and tutorials. As far as I am concerned Edinburgh is fine.

More about M373 as I go through this course.

M373 TMA01 is due soon

M373 of which the website opens on Monday published the TMA cut-off schedule. The course officially starts February 5th and has a cut-off date for TMA01 on 15 feb. I would think that this was a typo but since the first TMA only counts for 10 points it may very well be possible. I suppose this means that I have to start working on this TMA immediately.

[M373-1] status: started

Started ( unofficially ) on M373 today using the unit-1 course sample.

Unit 1 is called 'Introduction to Iterative Methods.' and is a repeat and continuation of MS221 B1 Iteration. The Newton Raphson method is also discussed which was part of MST121/MS221.

Proofs of theorems are given ( of course ) but questions on the exam will not be about proofs. The emphasis is on 'hands-on' calculation using computers where possible. Again, the use of MathCad is encouraged.

Haven't finished the unit yet, have to do more exercises.

Registration

Today I completed the final step in the registration process. Although it was rather difficult to contact the Open University by phone today ( already at my second try an excuse message was included in the answering system ) when I finally got through to the call-center in Manchester the registration was handled in less than five minutes. Tomorrow, or Thursday the confirmed reg ( of M373 and M381 ) will be visible on my Student Home page and the materials will be shipped by the warehouse somewhere next week.

M373

Just registered for M373 - Optimization per Feb 2011. ( More later... )