Ramanujan's magic square formula

Today I started to read the Ramanujan biography ( The e-book version, of course. ) The book looks promising. What was it like to communicate with someone gifted with such powerful mathematical insights? I am hoping ( a bit less after every failed try ) for a book that pulls me in that world of it's own. I have been there before on many occassions, while reading other books of course. Scientologists have a special word for it: exteriorizing.

The Man Who Knew Infinity.

At a very young age he 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.

Try it, it works!

MS221 - TMA04: Done

I reviewed MS221 - TMA04, made some minor corrections and decided that the A is ready for shipment to T. I nevertheless wait with shipping the TMA ( Cut-off date is 30 sep anyway ) because I haven't received TMA03 back yet. If I get a high enough mark on that one I might even get away with a low score ion TMA04. We'll see.

MS221 exam close by

I copied this from a post by "Tma Machine" :

Here’s how a rough outline of what’s in the 2008 paper (hopefully this won’t constitute copyright infringement!):

Part 1
Question 1: Finding a closed form for a recurrence system
Question 2: Identifying and sketching a conic
Question 3: Stating the rule for some isometries, and for a composite isometry, and then using the double-angle and half-angle formulas to show a result
Question 4: Classifying fixed points of a curve, then sketching the graph of the curve and using graphic iteration construction
Question 5: Identifying basic linear transformations, applying them to a vector, and stating an invariant line for each one
Question 6: Finding eigenvalues, eigenlines and eigenvectors.
Question 7: Differentiation
Question 8: Integration
Question 9: Finding and manipulating Taylor series about 0, and classifying stationary points
Question 10: Finding the modulus and argument of a complex number, converting them from Cartesian to polar form and vice versa, and using the formula for powers of complex numbers.
Question 11: Using Euclid’s Algorithm and working with exponential ciphers.
Question 12: Combining variable propositions, finding a case for which a given proposition is false, and finding the converse of a proposition.

Part 2
Question 13: Looks like it’s about conics, but I haven’t done this one yet.
Question 14: Linear transformations
Question 15: I haven’t done this one yet either, but it looks like it involves differentiation, integration and stationary points.
Question 16: Groups

I sort of expected this, so I am not surprised. But I am not entirely ready for the exam yet. I still have several weaknesses but I am confident I can handle these in time. To be honest I wished the exam was over and done with. It is obviously my objective to score in the 90-100 because that is what I score for TMAs. But I check, doublecheck, re-check my TMAs at least two times. That makes a lot of difference.

Logic puzzle

In lecture 2 Dr Kamala describes a murder mystery which can be solved entirely using logic in a Sherlock Holmes or Hercule Poirot like manner! Very nice.

Discrete Mathematical Structures

Lectures on YouTube by Prof. Kamala Krithivasan, Department of Computer Science and Engineering, IIT Madras.

Beautiful lectures for the whole planet to watch.

Excercise

I think this is a nice exercise at the MST121 / MS221 level.

Show that : Sin ( 18 degrees ) = 1/4 * ( SquareRoot(5) - 1 ).

Will use it to practice the trig formulas for the MS221 exam.

Sage or Mathematica ?

Sage and Mathematica are both very powerful mathematics tools. For an absolute beginner both packages will do, I suppose.

Mathematica is closed source and for the price of a full edition Mathematica package you can also buy a state-of-the-art Sony VAIO or other sexy netbook or tablet you are thinking of buying. The Mathematica programming language is a so called propriety language. That means that it is only used in Mathematica and the owners of Mathematica can change the language whenever they want. Mathematica has a steep learning curve. Now that I master the basics of it I am very satisfied with what Mathematica can do for me.

Because Sage says it is possible to integrate Mathematica into Sage ( haven't figured yet out how, I suppose I need the Mathematica for Linux version ) I am having another look at Sage. Sage is Linux only but you can run it entirely from within a browser. So you can install Sage on a Linux server ( for example in a VMWare virtual machine ) and run it on Windows in your browser.

( More later about my Sage findings. )

http://www.sagemath.org/

Characteristic of a ring

The characteristic of a ring is the number of times you must add the multiplicative identity element in order to get the additive identity element. If adding the multiplicative identity element to itself, no matter how many times, never gives us the additive identity element, we say the characteristic is 0.

Video: Solving Cubic Equations

Video: Solving Cubing Equations

And more: Pythagoros theorem ( a^2 + b^2 = c^2 ) is not really by Pythagoras. Pythagoros is the probable leader of a cult called the Pythagoreans. There is an interesting formula for enumerating all the Pythogarian triples like (3,4,5) (5,12,13), etc.
In the next clip we learn that Fermat bought a translation from Greek to Latin of Diophantus' work and started his work on Number Theory from there on. It was also in Fermat's lifetime that Calculus was born. ( Fermat already worked on the currently very important Elliptic Curves which Andrew Wiles used to prove Fermat's famous last theorem. ) A method is shown to find rational points on elliptic curves. An elliptic curve is the graph of an equation of type y^2 = x^3 + px + q.
Then the video becomes less interesting. It seems that in academic talks it is custom to speak with great admiration of certain other scientists, mathematicians especially if you know them or are related to them. The speaker, professor Gross had John Tate as his thesis advisor. John Tate is a celebrity in a the Elliptic Curve branch of mathematics. So Gross shows a picture of him and Tate and all the other graduate students at that time. Then he mentiones all these students by name including the universities where they are now teaching as a mathematics professor.
( I suppose they had their euphoric moments too when they simply passed an exam. After a while more impressive results are needed to feel the same high, like being appointed as professor. )
See the rest for yourself if you are still interested...

MS221 TMA03 - Result

MS221 TMA03 - Result is in... 99.

Lost a point due to a typo. This once again proves that you can literally save points by reviewing, QA-ing, checking whatever you call it. And check your results as much as possible by using a mathematics package like MathCad or your own preferred software, as long as it can do the math.

MST121 - Overall Course Result ( Final )

I have received the marked MST121 TMA04 and now also all data have been updated in my student record.

Assignment Score
CMA 41 83
CMA 42 84
TMA 01 82
TMA 02 88
TMA 03 95
TMA 04 90

TMA 01 82 has been substituted to 87.7 giving a final score of 88.84 rounded to 89.

An 89 for this is course is much more than I deserve. Especially in the beginning of MST121 I did not study enough ( november and december last year ). During this period I also had a last minute approach towards TMA's. I definitely lost points there.

Anyway, I learned that in order to get high marks much more is expected of the student. TMA01 and TMA02 were quick and dirty jobs using pen and pencil. As from TMA03 I changed to LaTeX including several QA reviews using Mathematica ( if possible ).                                          

Chinese Remainder Theorem

Solve systems of linear congruence equations using the Chinese Remainder Theorem.

Find a video lecture here ( Lecture #12 )

I was solving a problem from ' 104 Number Theory problems ' which required an application of the CRT. I had forgotten the algorithm. Instead of looking it up in a book I watched Song's lecture.