Mosaic view

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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.

Generating 4 x 4 magic squares

Inspired by Ramanujan, I am playing, experimenting, with 4 x 4 magic squares. I have ( independently ) found a way to generate them, infinite many of them if necessary. What's interesting about it is that theoretically the method should work for n x n magic squares.

15	0	9	8
5	5	4	18
5	13	10	4
7	14	9	2

15	10	10	7
6	13	5	18
4	13	11	14
17	6	16	3

16	106	160	79
156	85	6	114
1	88	182	90
188	82	13	78

Two follow-up projects come to mind. 1) A reading experiment. Since I discovered part about mathematics independently ( thanks to Ramanujan's 3 x 3 formula, of course ) reading about this topic in the literature should be quite different compared to reading about a subject you know nothing about, i.e. when the reading-protocol is 'discovery'. Anyway, that is the experiment. 2) A writing experiment. I'll try to write down the method I use with as much rigor as I possibly can. Will I be able to produce something readable? - Unfortunately I have more urgent tasks to handle.

Ramanujan's first paper on highly composite numbers

I am still reading The Indian Clerk. In between reads I look up relevant information from that period. Like when Ramanujan publishes his first paper in the book I tried to find what I could find of it through the Open University Library Services.

Early 1914 Ramanujan arrived in England. During the mornings he worked with Hardy and in the afternoons he followed lectures. One of the topics he was working on when he came to England were what Ramanujan called "highly composite numbers".

A highly composite number is a positive integer with more divisors than any positive integer smaller than itself.

The earliest paper of Ramanujan I could find was communicated by Hardy and published in the Proceedings of the London Mathematical Society; January 1915, Vol. 14 Issue: 1 p347-347, 1p.

If you have access this is a link: Highly Composite Numbers. ( pdf - 63 pages ).

page 2 of the paper

Update: 31/3-'11
The paper on highly composite numbers was not the first paper of Ramanujan. He published before he came to England in the Indian Journal of Mathematics. According to the "collected papers of Ramanujan", it was his 15th publication.

The Indian Clerk - ( Ramanujan Revisited )

I am reading The Indian Clerk by David Leavitt. Faction about the Hardy / Ramanujan collaboration. Reading a novel, through the appropriate reading protocol, gives an experience close to reality. Through reading The Indian Clerk you travel back in time to 1914 and meet Hardy, Littlewood and Ramanujan and a host of other very interesting people.

Hardy (l) and Ramanujan (r).

While in school Ramanujan spent more time on his own mathematics than on what he had to learn for his mathematics exams. As a result he was not drilled in doing proofs but doing mathematical research became natural to him.

Many schools include a research project as part of the graduation requirements for their mathematics majors. But most students are at a loss to create their own research questions, leaving this task to their advisors. It would be better if the student came up with their own research question that involved significant mathematical investigation and the creation of original mathematics. This is a daunting task: most graduate students are unable to do this, and rely on their advisors to frame a suitable area for investigation. The task is further complicated by the fact that many questions relating to undergraduate mathematics have “already been solved,” while many of the unsolved questions require so much specialized background to understand or so much existing research to review that the preparation needed to tackle the problem is itself a major project. - Jeff Suzuki in "But How Do I Do Mathematical Research?” on MAA website.

... the intensity of his interest in mathematics led him to pay scant attention to the other subjects in which he was obliged to show some facility. ... On each occasion, his interest in his own mathematical researches was so all consuming that he neglected his more quotidian studies, with the result that he failed his examinations and lost his scholarships. p118 ...no wonder he doesn't understand how to do a proof! p125 - The Indian Clerk

Ramanujan got -kicked out- of all schools he attended. Not fit for mathematics.

In the early 1900s, long before internet, good math books were rare in India or too expensive for Ramanujan. He learned his math from an encyclopedia type of collection of formulas and theorems. He must have re-invented the math that was not in that book because when he came to England he knew more or as much as Hardy, the leading Number Theorist of that time, but -in a different way-.

Long before Ramanujan arrived in England he was called the Indian genius. What is a 'genius' anyway? Newton was a 'genius'... Why didn't Newton share his invention of Calculus for such a long time? I suppose because ( knowledge ) power corrupts. Thank God Leibniz ( and others... ) independently invented Calculus around the same time. Science is for humanity not for the individual ( company ).

Ramanujan told ( to Eric and Alice Neville while they were interviewing him in Madras ) that his inventions were sent to him by the Goddess Namagiri in his sleep. Goddess?!, Red Alert! Stop.

There are two options here. Ramanujan's story about the Goddess Namagiri is true, or Ramanujan must have found a powerful method to do mathematical research of which he had no intention of sharing, like Newton, who used his then secret, Calculus to come with all sorts of amazing results. To declare someone a genius is giving up the search for his methods. How did Ramanujan do it? It's not in Ramanujan's notebooks let alone in Leavitt's book. Yet I tend to believe that if we want to come any closer to the Source of mathematics it is through Ramanujan.

Generating a list of permutation matrices with Mathematica

The following function defined by

f[n_] := Map[# // MatrixForm &, 
  Map[SparseArray[{i_, i_} \Rightarrow 1, {n, n}][[#]] &, 
   Permutations[Array[# &, n]]]]

maps $n$ to $S_n$, the symmetric group of $n$ elements, and displays its elements as a list of permutation matrices.

Mathematica keeps surprising me.

The function above is basically a one line function in Mathematica. ( Mathematica is a functional programming language. Every Mathematica instruction is a function, including IF and WHILE. Compare Microsoft Excel. All data in Mathematica is essentially a list. )

M381 Mathematical Logic is about the Universal Register Machine, the URM. It is a theoretical computer with only four instructions. Z(n): zero Register n, S(n) add 1 to Register n, C(m,n) copy register m to n and J(m,n,p) if the contents of Registers m and n are equal then jump to the instruction at line p. - The function above is computable and thus can be implemented in the form of an URM program. I don't know how many registers and instructionlines would be involved but it will be in the hundreds if not thousands. Just think of programming the one line function in a ( higher ) programming language like Java without using any third party imports.

P.S.
All Mathematica output can be copied as LaTeX and pasted in your LaTeX IDE, unfortunately MathJax leaves the handling of tables and matrices to HTML ( not entirely logical because you are publishing to the web and not to a PDF document ). That is the reason I included a printscreen image of a Mathematica notebook in this post.

LaTeXDraw

Although typesetting mathematics has somewhat of a learning curve you will get used to it and after a while it will become your only option for writing mathematics. If you aren't using LaTeX yet, there will come a day when it is hard to understand you could do ever without it. Like the Internet, for example.

Typesetting mathematics is one thing. Typesetting pictures, or drawings is an entirely different ballgame. Where LaTeX is built upon Don Knuth's original TeX, the community created tons of add-on packages for LaTeX. For the typesetting of graphics there are two major options: PSTricks and TikZ/PGF. Although the possibilities of creating professional graphics in your documents seems limitless, creating them by hand, i.e. through code, is hard. Again, the community came with a solution by creating IDEs for these packages.

( As I blogged before ) M381 TMA01 requires the inclusion of a flowchart. What is simpler to draw than a flowchart? Well, it is the lines, arrows, text insertions, fonts and all that which makes it a challenge. I decided to install the IDE for PSTricks called LaTeXDraw and I must say that the first results are hopeful. LaTeXDraw is a graphical drawing editor for LaTeX and can be used to generate PSTricks code. LaTeXDraw is developed in Java and thus runs on top of Linux, Windows, and Mac OS X. The LaTeXDraw GUI consists of two panels, a left panel drawbord and a right panel code editor. The panels remain in-sync while either graphically modifying the drawing or via code. Using the mouse you can drag and drop template figures and draw them, basically like as in any drawing tool. The PSTricks commands for LaTeX are instantly generated in the code-editor. When you are done a simple copy and paste into your document completes the process.

It looks as though LaTeXDraw becomes part of my basic software toolkit. I'll keep you posted.

LaTeXDraw

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.

Primitive recursion

I really appreciate the books on Mathematical Logic from M381. Take for example the very concise definition of primitive recursion.

Let $a$ be a natural number and let $g: \mathbf{N} \times \mathbf{N} \rightarrow \mathbf{N}$ be a function. The function $h: \mathbf{N} \rightarrow \mathbf{N}$ is said to be defined by primitive recursion from the constant $a$ and the function $g$ if
- $h(0) = a$,
- $h(n+1) = g(n, h(n))$.

Example:
- $h(0) = 1$
- $h(n+1) = g(n, h(n)) = (n+1) \times h(n)$
is the well-know faculty function.

( Open University, M381 ML-1 )

Beautiful, isn't it?

The Mathematical Tripos - ( Read: mathematics is easy )

In its classical nineteenth-century form, the tripos was a distinctive written examination of undergraduate students of the University of Cambridge. Prior to 1824, the Mathematical Tripos was formally known as the "Senate House Examination".From about 1780 to 1909, the "Old Tripos" was distinguished by a number of features, including the publication of an order of merit of successful candidates, and the difficulty of the mathematical problems set for solution. By way of example, in 1854, the Tripos consisted of 16 papers spread over 8 days, totaling 44.5 hours. The total number of questions was 211. The actual marks for the exams were never published, but there is reference to an exam in the 1860s where, out of a total possible mark of 17,000, the senior wrangler achieved 7634, the second wrangler 4123 and the lowest scoring candidate 237. ( W'pedia )

Click to enlarge

Click to enlarge

Mathematics is easy... Nowadays.

Investigating LaTeX graphics ( M381 TMA due soon )

M381 TMA01 consists of nine questions, 5 about number theory and 4 about mathematical logic. Question 6b asks us to draw a flowchart of a URM program. Since I typeset my TMA's ( instead of using a pen or worse: using a word processor ) I need to insert the flowchart as a graphic. Although almost any graphic can be inserted in a document ( including scans from manually drawn stuff ) I want to give the TMA a professional look and feel ( no: it won't add a single point, but are you transforming yourself into a knowledgeable mathematician or are you only interested in the grades? ). I investigated what the possibilities are for producing high-quality graphics with LaTeX.

It was as if I entered a new world entirely. I looked at PS-Tricks before but got overwhelmed with its possibilities ( and possible steep learning curve ), this time I had a better look. And I looked at PGF/TikZ as well. Both look very impressive and have the support of third-party editors. Both seem to have an established and satisfied user-base. I might have overlooked something but it seems that PS Tricks and PGF/TikZ are the competing 'standards' for native LaTeX graphics.

My next step is installing and as a test trying to create the flow-chart graphic for the M381 TMA.

PS Tricks graphic example

Links:
- PS Tricks
- PGF/TikZ (TikZ ist kein Zeichenprogramm)

HapPi PiDay

March, 14th PI day.

$$\pi = \sqrt{6 \sum_{k=0}^\infty \frac{1}{k^2}}$$

( PDF document )

The Euler spiral

Students of literature read Shakespeare; students of music listen to Bach. In mathematics such a tradition is, if not entirely absent, at least fairly uncommon. This book is meant to address that situation. Although it is not intended as a history of the calculus, I have come to regard it as a gallery of the calculus. - (William Dunham)

I am reading Dunham's book. Calculus Gallery. While reading I realized that I couldn't do the integral $$\int \cos{(x^2)}\ dx ,$$ anyway to make a long story short I discovered yet another beautiful curve ( most beautiful ever? maybe ): the Euler spiral, it is parametric plot of two Fresner integrals.

An Euler spiral has the property that its curvature at any point is proportional to the distance along the spiral, measured from the origin.

Exercise ( read: question ) How can we rotate the spiral?

Peek preview of Dunham's book.

Gödel, Escher, Bach video lectures

Since the logic part of M381 Number Theory and Mathematical Logic leads to understanding Gödels Incompleteness Theorems, I am alert at stuff on Godel. I happened to come across a video series ( note: Real format, if you can't already play Real videos I would think twice about installing Real, I think they invented the concept of adware. ) from MIT of which I am sure most of you will like. It's about the book Gödel, Escher and Bach ( video ). More than six hours about that mind-blowing book.

PNT is the most difficult theorem I encountered thus far

$$\sum_{d/n}\frac{\mu(d)}{d} = 1 -\sum \frac{1}{p_i} + \sum \frac{1}{p_i p_j} - \sum \frac{1}{p_i p_j p_k} + \cdots$$
Apostol, Chapter 2: Arithmetical Functions

Although I am doing M381 and M373 this year, my real goal for this year is ( read: was ) self-studying Apostol's Analytic Number Theory. I came to realize that a goal like that must be wrong if I want to go anywhere. It may look like a SMART goal but it is not. ( SMART goals are S(pecific), M(easurable), A(chievable), R(ealistic) and T(rackable). ) Since self-study projects are not finished with an exam, it is hard to measure if the goal has been achieved, the goal is not specific enough. After about 1/4th of the study year I changed my goal to: "understanding the proof of the prime number theorem (PNT)". Although the main purpose of Apostol's textbook is just that. I am now free of the strict planning I imposed on myself, i.e. March: do AANT Chapters 2 and 3, etc. I can focus entirely on understanding the proof no matter where the knowledge comes from. Once I understand the theorem I should have a new outlook on the Number Theory field and decide where to go from there.

I chose Apostol's book because it is used in OU Courses M823 and M829, of which I sincerely hope are still around when I am eligible to do them. I thought it was more or less the Analytical Number Theory Bible, well it is not, it seems. The classic work in Analytical Number Theory is the following book by Iwaniec and Kowalski.

Terence Tao explains the music of the primes.

A talk on primes by Terence Tao. ( Tao won all the prizes there are to give. Charlie Eppes for real? A living Ramanujan? Decide for yourself. )

We start with a "sound wave" that is "noisy" at the prime numbers and silent at other numbers; this is the von Mangoldt function. Then we analyze its notes or frequencies by subjecting it to a process akin to Fourier transform; this is the Mellin transform. Then we prove, and this is the hard part, that certain "notes" cannot occur in this music. This exclusion of certain notes leads to the statement of the prime number theorem. - Tao

What is the inverse of a cube?

Schatz discovered in 1929 that the Platonic solids could be inverted, and one of the products of the inversion of the cube was the oloid.

Link: Oloid at Mathematica mathworld

The mysteries of kinematic inversion: the old man and the oloid

( 30 min. sort of documentary follows, where Don Cruse talks about Paul Schatz, in three parts, more interesting than I expected! )

About proofs in ( studying ) mathematics

Exercise, show that for $n>1$: $$\sum_{d/n} \mu(d)=0,$$ where $\mu$ is the Möbius function.

There has been a time when I did not like proofs. I suppose that was when I got first introduced to the theorem - proof - protocol. My goal was to study ( just ) enough so that I could do the exercises, or better ( read: worse ) questions from exams of the previous years. I must have complained about a lack of examples, I am sure. Going through all the compulsory proofs must have been hard because I never studied doing proofs as such. The idea that proofs can be beautiful was known to me but I thought I would never understand its meaning.

I liked mathematics a lot because I liked doing exercises. Think of the satisfaction the completion of a Sudoku puzzle can give you. Every completed exercise gave me a similar rush. It is as with Sudoku's: you know you will complete the game because, in essence, they are all the same ( at the same level anyway ). At a certain moment you begin to realize that doing exercises doesn't add anything to your mathematical abilities. You begin to search for ever harder exercises, preferably those that require a lot of study before you can complete them. And thus you can be sure of increasing math skills.

Reading ( read: thinking ) about new mathematics is interesting but does not let the subject sink in, digest into your system, make it part of your toolbox. - Doing exercises does. I suppose this is the principle of the books that present the theory in the form of ( almost just ) exercises. The theory is summarized, for example, in the form of a short list of theorems and then slowly, exercise by exercise you construct all the proofs yourself. If the solutions are included then books like that are priceless. It is, in fact, the method the Open University uses to present mathematics. The mathematics books of the Open University are not just like any other mathematics book.

Anyway, I made it full circle today... I know it is a wide open door: Doing mathematics is the best way to learn mathematics. But... use a consistent exercise set. If you notice you are cherry picking exercises that seem interesting you have fallen into the trap of an inconsistent ( or not hard enough ) set. In the protocol of mathematics a sequence of exercises for the reader contains a single message.

Exercises in a text generally have two functions: to reinforce the reader's grasp of the material and to provide puzzles whose solutions give a certain pleasure. Here, the exercises have a third function: to enable the reader to discover important facts, examples, and counterexamples. The serious reader should attempt all the exercises (many are not difficult), for subsequent proofs may depend on them; the casual reader should regard the exercises as part of the text proper.
(J. Rotman - An introduction to the theory of groups )

Exercise in geometry

This exercise is about observation and perception. The exercise is simple. ( Come on, geometry of triangles: piece of cake for MST121-ers and above. ) Observe your observation, perceive your perception. Draw your own conclusions.

Enjoy!

Princeton companion to mathematics

An overview of the entire field in 1000+ pages.

Different representations of sequences

The following five mathematical objects, i.e.: a sequence, a generating function, a recurrence equation, an arithmetical function and an asymptotic estimate, are all representations of the well-know Fibonacci sequence.

$$(1,1,2,3,5,8,13,21,34,55,...)$$
$$g[x]=\frac{x}{1-x-x^2}$$
$$a[1]=1; a[2]=1; a[n]=a[n-1]+a[n-2]$$
$$F_n = \frac{1}{\sqrt{5}} (\frac{1+\sqrt{5}}{2})^n - \frac{1}{\sqrt{5}} (\frac{1-\sqrt{5}}{2})^n$$
$$f(x)=\frac{1}{\sqrt{5}}e^{x\log{\phi}}$$

So, if we have to solve a problem involving sequences we have various options to choose from to represent the sequence we are working with. I had quite a cognition after my first confront with the consequences of this concept. Summation for example. Summation of a sequence is equal to multiplication with $\frac{x}{1-x}$ in the generating function realm. The GF of $(1,1,1, \cdots )$ is $\frac{1}{1-x}$ which means that the GF of $(1,2,3, \cdots )$ must be $\frac{x}{(1-x)^2}$.

The flexible inflexible university or The Open University

I will try to explain what I mean by the title of this post. "The flexible inflexible university or The Open University." In my perception a university should be a bridge to the state-of-the-art knowledge of a certain field. It maybe impossible to actually deliver the knowledge in the form of a B.Sc. or M.Sc. degree but then at least the student should leave with an up-to-date map to the frontiers of the field. Some sciences are so big, i.e. mathematics or expensive to research, i.e. ( particle ) physics that it would be unreasonable to expect of a single university to be that bridge. But in cases like that the university should be part of a cluster in any form as long as it is clear to their students.

Let me jump right into an example. Random matrix theory. It is not possible to study this topic at the Open University. It is not in the B.Sc. program nor in the M.Sc. program. Random matrix theory seems to be some inter-mathematical-discipline of probability and linear algebra. One could say, that it is -thus- covered in the Open University program. The issue is that it seems to be an area of interest and research. Actively used in engineering, finance and physics. Think of calculating the eigenvalues of 1000,000,000 x 1,000,000,000 matrices.

The flexible side of the Open University is well known, I don't have to deliver any arguments for that. The inflexible side has to do with the long lead time between course design, course development and development pay-back time which may be a cycle of at least 10 years. What is 10 years in mathematics you might ask? Ten years is always the same ten years. Engineering and finance develop at ever faster rates and they are extremely tool ( mathematics ) hungry. Universities where professors have direct links to society through contract research with large corporations for example can and will react sooner to those needs. And thus set an agenda for research. - There are more reasons like professors shopping for Ph.D. students for their own research that give brick universities an advantage over the OU.

A rather old document ( 2006 ) on random matrix theory which I found while 'stumbling'.

About the open university library services

I am still happy with studying polygonal numbers and representations of numbers as $n$ $n-$gonal numbers. I understand Langrange's proof of the four square theorem.

I did a Google search related to square representations and came across this JSTOR page. It is a 7 page PDF but downloading it would have cost me $12.--. The Open University Library Services came to the rescue. I searched within the category of mathematics for all documents with the title 'On the partition of numbers into squares'. This query returned zero results. A search for that same title in the JSTOR collection however did return a result. I could access the document for no charge at the OU Library.

I find it encouraging that I can read and understand this number theory article. ( From the American Mathematical Monthly, 1948 ).

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