Complex Analysis

What is shear impossible with linear algebra... transforming a rectangle to a circle is easy with the complex mapping z: -> Exp[z].

Generalising solutions to problems

When writing a program it is a challenge to make a program reusable. The same can be done to solutions of exercises. You can try to generalise the solution. I generalized a result of a number theory problem I solved.

Let n be a positive integer and D the set of divisors of n. We are interested in the sum of all quotients a/b, where a,b in D and GCD(a,b)=1.

Example
n=5
D={1,5}
Quotients: {1/1, 5/1, 1/5}
Sum 6 1/5

n=12
D={1,2,3,4,6,12}
Quotients: {1/1, 2/1, 3/1, 4/1, 6/1, 12/1, 1/2, 3/2, 1/3, 2/3, 4/3, 1/4, 3/4, 1/6, 1/12}
Sum 33 7/12

Formula:
g[{x_,y_}]:=(1-x^(2y+1))/((1-x)*x^y)
f[n_]:=Apply[Times,Map[g,FactorInteger[n]]]
( In Mathematica format )
The function is Arithmetic, i.e. f[x]*f[y] = f[x*y] if GCD[x,y]=1.

Explanation.
If p is a prime then
f[p^n] = p^n + p^(n-1) + ... + 1 + 1/p + 1/p^2 + ... + 1/p^n
now use the fact that f is arithmetic and map the primefactorization of n to this function.

MT365 - CMA41

Completed the first 10 questions of MT365 / CMA41, covering Introduction and Graph Theory 1, today. There are another 10 questions to be done. Five about networks and another five on designs. The cut-off date is next week. MT365 is do-able but certainly not easy, let alone a giveaway.

Learning mathematics the fast way.

The video course I found has been streamed on EDGE. Some sort of system for distance learners. It is a 10 week course split over two subjects: vector analysis ( 4 weeks ) and complex analysis ( 6 weeks ). For the moment I am primarily interested in the complex analysis lectures. If everything goes well I do M337 next year. It is still very, very early but I am thinking of MST209 ( Mathematical modelling ) + M381 ( Logic and elementary number theory ) + M337 ( Complex analysis ) for next year. To the point. I am rather surprised by the sheer speed they go through a topic like complex analysis. They cover basicly what's in the book A First Course in Complex Analysis which has 500 pages. Well, you would say that's roughly 85 pages a week. Or 20 pages per lecture. Or three minutes per page. Now suppose a page is a theorem + proof + example that would reduce the time to one minute for a proof. Reasoning like this doesn't work of course but there is this Eternal Truth: "Mathematics is hard and there is no royal way." Some King ( don't remember which one ) supposedly asked one of his mathematicians that there surely must be a faster way -for him- to learn mathematics. The answer he got was "There is no royal way to mathematics." Maybe they discovered one after all, in Seattle.

AMATH 401/501 Video Lectures

Because of my special interest in number theory I read ahead on the topic of complex analysis ( M337 ) which is a prerequisite for analytical number theory. I like to watch video lectures because it gives me an idea about how the topic is presented to students. Until a few days ago I wasn't able to find any lectures on the topic. Although the lectures aren't currently online at the site of the University of Washington they must have been in the past or they are available to enrolled students only. Anyway, some student has been so very kind to upload 38 lectures given in 2009 of +/- 50 min each in a total of 6GB compressed video data to the filesharing site Rapidshare. I do not know if the student violated copyright laws or anything. I don't think so. I think the files are offline because having the files online involves quite a lot of bandwidth for the university which is expensive. Mathematics video lectures have a global appeal and seem to be very popular.

Turing Machine

Someone built a Turing machine close to how Alan Turing described it in his paper. Have a look:

$ \kappa(G) \le \lambda(G) \le \delta (G) $

Music of the primes

I watched another BBC mathematics documentary presented by Marcus du Satoy, called the music of the primes. Du Satoy wrote a book with the same title. ( Such a documentary is unheard of in my country The Netherlands. Too expensive, not enough viewers, I suppose. Whatever. Like there is no mathematics department in the Dutch Open University. )  It is amazing how Du Satoy can explain extremely complicated stuff to a wide audience in an entertaining way. Music of the primes is basically about the Riemann hypothesis.

Video lectures

Here you will find the VIDEO ARCHIVE of the UCCS Department of Mathematics Math Courses. Lectures  relevant to MST121, MS221, M208, MST209 and M381. New video lectures come available regularly. Although I have seen quite a few Number Theory lectures already I might go through them once more but now with M381 in mind. ( I like the Chinese professor who lectures Number Theory. ) Anyway, I don't watch lectures to learn new stuff but I use them as a test to verify if I understand everything that is being lectured. If so, I learned the material in the book well enough.

Something is wrong

One million pages of new math every year.

I was in the library today where I read the book 'Letters to a young mathematician' from professor Ian Stewart. Stewart answers questions of his ( hypothetical ) niece Gem who wants to become a mathematician. If you have a question about a career in mathematics it may very well be that you will find the answer in Stewart's book. In the preface he writes that he considers the book an update to Hardy's famous book 'A mathematician's Apology'. Things changed a bit compared to Hardy's days. Where Hardy spent four hours a day on math and the rest of the day in leisure ( i.e. watching cricket ) the research mathematicians of today have 10 to 12 hour workdays doing a lot more than basic research.

Anyway, in one letter Stewart, very much part of the current mathematical elite, explains how the young mathematician should pursue for a career. First he almost apologizes for the fact that her thesis advisor will probably be someone who just got his Ph.D him/her-self ( the number of female mathematicians is growing fast as Stewart explains elsewhere ). This remark alone should be red flagged. He then explains in the same apologetical fashion that you can consider yourself lucky if you have a good teacher. Most teachers only teach because it comes with the job of researching. And a good researcher is the one with the most publications. And since there are approximately 500,000 research mathematicians worldwide this explains the creation of one million pages of new math every year. Just imagine the industry behind it.

Attacking a serious mathematical problem like the Poincare conjecture or the Riemann hypothesis may take decades of  work with only a small chance of successful completion. I think I am beginning to understand why Grigori Perelman doesn't like the establishment and vice versa.

Something is wrong.

Proof that harmonic series diverges

Let $H_n=\sum_{k=1}^n{\frac{1}{k}}$
We investigate $H_{2^k}$
$H_{2^0}=H_1=\sum_{k=1}^1{\frac{1}{k}}=\frac{1}{1}=1+0(\frac{1}{2})$
$H_{2^1}=H_2=\sum_{k=1}^2{\frac{1}{k}}=\frac{1}{1}+\frac{1}{2}=1+1(\frac{1}{2})$
$\begin{align*}
H_{2^2}
&=H_4\\
&=\sum_{k=1}^4{\frac{1}{k}}\\
&=\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}\\
&\gt \frac{1}{1}+\frac{1}{2}+(\frac{1}{4}+\frac{1}{4})\\
&\gt \frac{1}{1}+\frac{1}{2}+\frac{1}{2}\\
&\gt1+2(\frac{1}{2})
\end{align*}$

$\begin{align*}
H_{2^3}
&=H_8\\
&=\sum_{k=1}^8{\frac{1}{k}}\\
&=\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}\\
&\gt\frac{1}{1}+\frac{1}{2}+(\frac{1}{4}+\frac{1}{4})+(\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8})\\
&\gt \frac{1}{1}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}\\
&\gt1+3(\frac{1}{2})
\end{align*}$
By induction it can be shown that
$H_{2^k}\gt1+k(\frac{1}{2})$
Since we can make $k(\frac{1}{2})$ as large as we want by choosing a value for $k$ this implies that we can also make $H_{2^k}$ as large as want, i.e. $H$ diverges.

Started on MT365-TMA01/CMA41. Started on question 1 and got lost in the mechanics of GraphPlot and it's options. Learned some Mathematica again. I have to increase my efforts on MT365. Yellow Alert.

Exercise - continued

A positive integer is written on each face of a cube. Each vertex is then assigned the product of the numbers written on the three faces intersecting the vertex. The sum of the numbers assigned to all the vertices is equal to 1001. Find the sum of the numbers written on the faces of the cube.

I found this exercise in a problem book for Number Theory. The type of book which is used for training young students who participate in Math Contests. I spend a few hours on this problem today and managed to solve it in my third attempt or so. The risk of problems of a more than elementary nature is that they may cost a lot of time. Although I managed to solve it I can't say I did it in a standard way, using some cookbook recipe. All sorts of ideas kept coming up. The gateway to the solution was a link I noticed between this problem and the Instant Insanity problem.

I'll post the answer if someone requests for it, of course. Although I more or less gave it away above.

How am I doing ?

M208
- TMA01 85 ( part 1  77% , part 2 89% )
- ahead schedule
- TMA02 done, one review required before shipment; cut-off = 9/4
- feeling OK about Linear Algebra

MT365
- behind schedule
- Graphs 1 studied
- TBD: Networks 1, Designs 1
- CMA41 due on 7/4
- TMA01 cut-off 26/4

I use a new way of studying MT365, based on checksheets and Personal Brain. Will evaluate after TMA01. MT365 introduces truckloads of new concepts and theorems. It is obvious that my priorities should be on MT365 for now. At the end of the day M208 is more important though, it brings in 60p and is a required course. If I fail MT365 for some reason I could change to another topic, this is not possible with M208 since it covers the basics of mathematics.

Exercise

A positive integer is written on each face of a cube. Each vertex is then assigned the product of the numbers written on the three faces intersecting the vertex. The sum of the numbers assigned to all the vertices is equal to 1001. Find the sum of the numbers written on the faces of the cube.

Lecture on Infinity

Enrico Bombieri, IBM von Neumann Professor, School of Mathematics. This lecture, accessible to a wide audience, explores how mathematics has arrived at its present pragmatic view of infinity and some of the counterintuitive paradoxes, as well as some of the positive results, deriving from its acceptance. It concludes with a view of how computer science is leading today to a new precise concept, namely the impossibly large in the realm of the finite.

Supposedly accessible to a wide audience. Find the lecture at this link.

Update: Boring. Slow. Amazing people actually take the time to sit in and listen to this bloke. People like this are responsible for the fact that mathematics has such a low appeal to many. - Nothing shows he actually wanted the talk to succeed. His "slideshow" is a notepad file. He must have a big name. Only names get away with a performance like that.

M208 - TMA02 done

I completed the remaining questions 4 and 5. Question 4 is about permutations, permuationcycles and conjugate permations. Question 5 is about subgroups, cosets and quotient-groups. Not extremely difficult but I am not going to make an estimate for this one. Not for M208. - M208 has four books remaining, Linear Algebra, Analyis A, Group Theory B and Analysis B. Next topic Linear Algebra. They cover a lot of ground for Linear Algebra in a short time. Compared to Real Analysis anyway. - My StudentHome has been updated with the TMA01 score of 85/100. - I would be very disappointed if I scored lower than 85 on TMA02. I have to wait and see.

Result M208 - TMA01 is in

Only MST121 - TMA01 with a score of 82 was worse than this. I have to admit that I was rather happy with the score of 82/100 at that time, still am in a way. MST121, although 'Level 1' is hard. I managed to squeeze out an overall result just in the disctinction zone. For part 2 I scored 59/65 so added to the 27/35 from part 1 giving a total score of 85/100 for TMA01.

( My M208 tutor marks with the purpose of returning the TMA with the -lowest possible- score. I couldn't face the light of day with an attitude like that. The pages of the questions where I scored 100% are still full with deep, agressive RED V's. Not a particularly nice sight, considering the time I spent writing it. )

On to the next TMA's. I hope to finish TMA02 on Group Theory tomorrow, or at least this week, not soon after I'll start on Linear Algebra TMA03.

For M208 I have done most of the actual studying. That's not what I can say of MT365 which feels like a tsunami of definitions, definitions and definitions. I never said this study was easy or simple.

Numerous 10-45 code ones

If you think this video is somewhat off-topic, think twice, start with the alternatives.

Pi day

Without cheating and without special practice 3.14159265358979. Well, 15 digits, more than I know of any other irrational number. Yet, I can't say that Pi is my favourite number. Don't think I have one, although I can think of some candidates like i, the imaginary number or e the base of the natural logarithm, or the golden section \[\phi=\frac{1+\sqrt{5}}{2}.\].

Fermat's Room ( L'habitació de Fermat )

"Four mathematicians who do not know each other are invited by a mysterious host on the pretext of resolving a great enigma...." That's how Fermat's room starts, with a simple invitation. Agatha Christie like but very original nevertheless. A mathematical thriller indeed, but a thriller and a good one. It/s one of those movies that occupies you completely.

Photography with a touch of math: Caleb Charland

This picture somehow reminds me of the movie Pi... the best math movie I have seen to date.

Caleb Charland has a website.

Complex mapping

'Reading ahead' on Complex Analysis it suddenly daunted how I can easily draw graphs of complex mappings using Mathematica. - Where would I be without Mathematica?
Define f: C->C by f[z]=Tan[2z].
We investigate the image of the unit-circle in the complex plane under f.

As you can see this is a simple procedure with the Mathematica function ParametricPlot.

M208 - TMA02 - Kick off

Completed 3 out of 5 questions of M208 - TMA02 today. TMA02 is entirely on Group Theory.
The first question lists three algebraic structures which are possibly groups. Using the group axioms you have to determine if the structures are a group.
The second question is about a 2D object for which all geometric symmetries must be determined. The symmetries have to be written down in 2 line permutation notation. From these a Cayley Table for the group must be set up. Finally you have to determine with which well known group our group is isomorphic.
The third question is about cyclic groups, isomorphisms between them and subgroups of them.
Q4 and Q5 TBD at a later date / time. ( M208 is thus still on schedule. )

I often 'read ahead' in various mathematics books., those who are subscribed to my blog know that combinatorics, algebra and number theory are my favourite subjects ( not necessarily in that particular order ). Anyway, the subject that is 'holding me back' most and which I merely see as a 'supporting' subject is complex analysis. Sofar it occurred in Lie Theory and Analytic Number Theory. Subjects that ( I hope ) temporarily closed their doors on me. It's time to handle this issue. I found a series of 24 Mathematica Notebooks on 'Complex Analysis with Mathematica. Learning more practical applicationa of Mathematica -and- Complex Anallysis sounds great. - Because they are so obvious I forgot to mention fractals where complex functions play a major role.

Mathematical skills have a longer life than IT skills

Five years ago I invested quite some time in learning programming 3D graphical models using Java3D. I thought it was about time do some 3D programming once more. I planned to pick up Java3D where I left it. That turned out to be very simple indeed. Too simple, in fact. Java3D hadn't changed one release point. Is that bad? In IT it is. Java3D is not dead though. The API is frozen while they are working on a release for Java3D in JavaFX. The thing is that JavaFX isn't as successful as Sun hoped it would be. Adobe with Flash and Microsoft with SilverLight have their little plans too. In the meantime Flash kept its hold on the market-share. Things won't be the same at Sun since it is now an Oracle shop. So maybe all this 'work' on integrating Java3D in JavaFX was a waste of time.
And my personal time investment in learning Java3D? Lost. But not all together. While I am learning jReality at the moment ( another open source 3D geometry viewer written in Java )  I notice that it is not as hard as learning Java3D was. At that time everything, every concept, including the mathematics was new to me. I am leaning heavily on my Java3D knowledge while learning jReality. More on jReality soon, I hope.

3D Models of Polyhedra for M208: Virtual or Reality?

Included in the M208 course materials are kits to create physical models of the regular polyhedra. In the picture below you see the kit for the Icosahedron.

But it is always handy to play with a virtual model of the Icosahedron...

I think both models will help to burn these models in my brain circuits so that they become as natural as the cube or the sphere. I can already dream the tetrahedron and the octahedron ( and the cube of course ) it's time for the icosahedron and the dodecahedron.


Although I could have generated the Icosahedron with one command in Mathematica like this

I decided to use jReality because it is a professional 3D graphics engine ( capable of displaying Mathematica graphics files and a zillion other formats ) which I might need for some research I intend to do.

Mathematicians

More video's! Here. http://www.maths.bris.ac.uk/research/videos/ Video's about people who do mathematics. - Many thanks to the reader who pointed me to this site.

Estimate M208 / TMA01

1 19/25
2 8/10

3
5/5
2/5

4
2/2
2/3
3/5
3/5

5
4/5
3/5

6
2/2
11/13

7
11/13
2/2

Estimate: 77/100

I basically have all the answers right but I expect to lose quite a lot of marks due to the tutor. That's M208 I suppose.

M208 - TMA01/2 Done

Consider M208 TMA01 done. Question 7 about equivalence relations, while very easy at first sight, has been  designed as an error-mine. Other questions were about a proof using the contrapositive of the proposition, a proof by mathematical induction, solving a 4th degree equation yielding complex roots, graphically showing the difference of sets and one on isometries. That's about it. Seven more TMA's to go for M208. The next one is on Group Theory due by the end of -this- month. During a 60p course there is basically a TMA every month while there is one every two months for the 30p courses. Although in the case of MT365 a TMA includes a full CMA. - I still have to write a letter outlining why I don't agree with the marking of part 1 of the M208 TMA.