Study plan for 2011 is shaping up

The Study Plan for 2011 is shaping up.

With M336 Group Theory forthcoming in 2012 I will keep my Abstract Algebra 'warm' by reading:
- Goodman; Algebra Abstract and Concrete. ( Free e-book );
- Rose, Harvey E; A Course on Finite Groups;
- Cox, David; Galois Theory;

I will probably have some difficulty ( 'uneasyness', if you like ) studying MST209 because the materials will not be presented in mathematical but in scientific format. Since I am not very good at long reading sessions ( I get distracted too easily ) I prefer the Theorem / Proof / Example presentation because it leaves a lot to the reader, i.e.: read a bit, then DO a lot. Mathematica experiments for example. For that purpose I selected the books
- Kelley, Peterson; The Theory of Differential Equations - Classical and Qualitative. From the preface: "...Previous experience with differential equations is helpful but not required. Consequently, this book can be used either for a second course in ordinary differential equations or as an introductory course for well-prepared students. ...)", and
- Differential Equations DeMystified as an antidote to the previous book and as a guide to the HOW-TO's of solving DE's.

Top priority will of course have MST209 because it adds 60 points to my balance. I'll have to decide if I want and / or can add either M337 Complex Analysis or M381 Number Theory to the workload.

The Mobius strip

I blogged many times about the Mobius strip, a fascinating object and superb to introduce mathematics to the uninitiated. Algebraic Geometry is the field of mathematics where they try ( more often do ) to catch these objects in polynomial equations. A quick browse ( or read ) of this PDF may give you an idea of the field.

http://data.imaginary-exhibition.com/IMAGINARY-Moebiusband-Stephan-Klaus.pdf

Found while surfing IMAGINARY.

Analytical function representing the Fibonacci series

The MST209 exam has a different format than MS221 and M208 have. In 2006, for example, the format was as follows:
Part A. 15 multiple-choice questions, 2 marks each = 30 points. ( 1 hour )
Part B. 8 questions, 5 marks each = 40 points. ( 1 hour 15 min )
Part C. 3 out of 7, 15 marks each = 45 points. ( 45 min )
Yes. Maximum score is 115. Scores above 100 are set to 100.
The exam looks doable. And again questions on eigenvalues and eigenvectors. That would be three in a row: MS221, M208 and MST209. Considering the fact that one can prove Binet's formula for the Fibonacci numbers with them it's worthwhile having it firm under your math-belt.

There is also an analytical function for the Fibonacci numbers which rounded, gives an exact Fibonacci number if the input variable is an integer. Here is the related math.

Let $F_n = F_{n-1} + F_{n-2}, F_0=0, F_1=1$, show that $Fa_n=\frac{1}{\sqrt{5}}e^{n \cdot \log{\phi}}$, where $\phi$ is the Golden Ratio or $\frac{1+\sqrt{5}}{2}$. ( Round $Fa_n$ to get $F_n$. )

If we define the elements $F_{n}$ and $F_{n+1}$ as the vector $s_n= \left(
\begin{array}{c}
F_{n+1}\\
F_{n}
\end{array}
\right)$
then $F_n$ simply becomes
$F_n= \left(
\begin{array}{cc}
1 & 1\\
1 & 0
\end{array}
\right)^n
\cdot s_{0}$.

We can calculate the power of a matrix by diagonalizing the matrix. And this is where eigenvalues and vectors come in. If $\lambda_1, \lambda_2$ are eigenvectors with respective eigenvectors $E= \left( e_1, e_2 \right)$ we get $F_n= E^{-1}
\cdot
\left(
\begin{array}{cc}
\lambda_1^n & 0\\
0 & \lambda_2^n
\end{array}
\right)
\cdot
E
\cdot s_{0}$

The eigenvectors are the roots of the characteristic equation $\left|
\begin{array}{cc}
1-\lambda & 1\\
1 & -\lambda
\end{array} \right| = 0$ and are thus $\frac{1}{2} + \frac{1+\sqrt{5}}{2}$ and $\frac{1}{2} - \frac{1+\sqrt{5}}{2}$.

( TO BE CONTINUED ... )

Mathematical Deceptions - ( Coast to Coast AM 21-oct-2010 )

C2CAM had a 3 hour during interview with Prof. C. Seife on mathematics

Prof. Charles Seife discussed the art of using pure mathematics for impure ends. He used the term "proofiness," (an analogue of Stephen Colbert's term "truthiness") to describe how people use numbers to prove things that they believe in their hearts are true, even if they're not factually true. People often lie in polls to make themselves feel better, he reported. For instance, in surveys men will typically overestimate the number of sexual partners they have had, while women will underestimate the number.

Advertising often uses numbers fabricated out of whole cloth, like saying a moisturizer delivers 70% more moisture in every drop. Ads are just "larded with...completely meaningless numbers that are supposedly backed by some sort of research which is either dubious or non-existent," he declared. Politicians sometimes manipulate the scale of graphs to make it appear like a dramatic effect or change is occurring, such as with tax cuts or budgets. Al Gore, in his film An Inconvenient Truth, exaggerated climate projections, as a way to further his agenda, Seife added.

Numbers are often inflated to make a person or organization seem more important, such as the number of people in attendance at a rally, or the circulation of readers for newspapers and magazines, he noted. Seife also discussed various developments in science, such as the Large Hadron Collider experiments at CERN. If "a certain pattern of missing energy turns up, it could be a very good sign that we have found another dimension beyond our own," he reflected.

For subscribers: http://www.coasttocoastam.com/show/2010/10/21

Otherwise: http://www.demonoid.com/files/details/2427492/2340388/ ( Opera Browser has a built-in BitTorrent Client )

Bernouilli equations

I watched approximately half of video lecture 4 of 18.03 today. Topic of lecture 4 is solving differential equations by direct or indirect substitution.

Direct and indirect substitution originates from Calculus Integration, I suppose. Take for example, the integral:
$$\int{x \sin(x^2)\ dx}$$
requires the direct substitutions $y=x^2, dy=2x \ dx$ to solve. However, the integral
$$\int{\frac{1}{\sqrt{1-x^2}}\ dx}$$
requires the indirect substitution $x=sin(u), dx=cos(u) \ du$ to solve.

A Bernoulli equation is a DE of type:
$$y' = p(x) \cdot y + q(x) \cdot y^{n}$$
Rearrange as follows:
$$\frac{y'}{y^{n}} = p(x) \cdot \frac{y}{y^{n}} + q(x) \cdot \frac{y^{n}}{y^{n}}$$
$$\frac{y'}{y^{n}} = p(x) \cdot \frac{1}{y^{n-1}} + q(x) $$
Now substitute $v=\frac{1}{y^{n-1}}$, and thus $v' = (1-n) \frac{y'}{y^{n}}$:
$$\frac{v'}{1-n} = p(x) \cdot v + q(x)$$
$$v' + (n-1)p(x) \cdot v = (1-n)q(x) $$
The last equation is a linear ODE in standard form.

Recipe for solving linear first order ODE

As I blogged before I am going to make more notes, summaries and stuff and store everything in a plex in a Personal Brain database. The following note basically summarizes 18.03 lecture 3.

$$a(x) \cdot y' + b(x) \cdot y = c(x)$$
$$y' + p(x) \cdot y = q(x)$$
$$e^{\int{p(x) \ dx}} \cdot y' + p(x) e^{\int{p(x) \ dx}} \cdot y = e^{\int{p(x) \ dx}} q(x) $$
$$( e^{\int{p(x) \ dx}} \cdot y )'= e^{\int{p(x) \ dx}} q(x) $$
$$e^{\int{p(x) \ dx}} \cdot y= \int{e^{\int{p(x) \ dx}} q(x) \ dx} $$
$$y= \frac{\int{e^{\int{p(x) \ dx}} q(x) \ dx}}{e^{\int{p(x) \ dx}}} + \frac{C}{e^{\int{p(x) \ dx}}}$$

( Would need some explanations but as a personal note it is clear to me. )

Watched MIT 18.03 videos 2 and 3.

Video 3 is about a straightforward recipe for solving a linear ODE of the first order:


Video 2 is on numerically solving DE's with Euler's method:

Getting started on MST209 Differential Equations

In order to get a feeling for MST209 I will be watching the video's of MIT 18.03 Differential Equations by Prof. Arthur Mattuck. I watched lecture 1 today. Since there are 33 video lectures in total I doubt if I'll get through all of them by february though. The purpose is getting back into differential equation stuff. OU OpenLearn has several booklets from the MST209 course available which give a reasonable idea on what to expect of MST209. Both MIT 18.03 and MST209 are less formal than M208, they are mainstream courses teaching mainly how-to's, i.e. how-to solve a particular type of differential equation. - My choice for MST209 next year is more or less certain by now. Three options remain. 1) do just MST209 and concentrate and spend time on self-study of topics that really interest me (i.e. Abstract Algebra stuff ). 2) Add M381 Number Theory + Logic. 3) Add M337 Complex Analysis. Well, I haven't decided yet. MST209 involves 7 TMA's, 2CMA's and probably a MS221, M208 'type-of-exam', i.e. 12 questions for 70, plus 2 out of 5 for 30.

Announcing two new websites.

I received this message earlier today.

Hi Nilo Just to let you know I've gone live the link is http://chrisfmathsphysicsmusic.blogspot.com/ Hope you find it interesting Best wishes Chris

Go here: http://chrisfmathsphysicsmusic.blogspot.com/.

This might be the moment to 'open-up' a website in its very early beginnings. The site will change often and is going to contain my ( mathematical ) notes. Anything I ever might publish will go through there. At least, that's the idea. I don't have a domain yet, let alone a webserver so I used Opera Unite's out-of-the-box Webserver. The address is http://home.ndroock1.operaunite.com/webserver/

See also the Bookmarks page

Nicolas Bourbaki

From Preface to the Third Edition of Galois Theory by Ian Stewart

Nicolas Bourbaki is the pseudonym of a group of mathematicians — mostly French, mostly young — who tidied up the mathematics of the mid-20th century in a lengthy series of books. Their guiding principle was never to prove a theorem if it could be deduced as a special case of a more general theorem. To study planar geometry, work in n dimensions and then let n = 2.

Note-taking

I found a way to enter LaTeX in Personal Brain thoughts. Exporting a plex to a website is simple. I hope to demo it soon. I publish my work via Opera Unite which means that I don't have to upload files to a server because the web server runs as an Opera process / thread on my computer. I don't leave it on 24/7 though. The exciting part is that you can use Personal Brain to:
- manage your mathematical notes in LaTeX ( plexes of 300,000 thoughts exist );
- query thoughts;
- reorganize database structure instantly;
- publish thoughts to a website;
- export thoughts to other programs;
- import thoughts ( Mathematica, for example can export notebooks to LaTex ).

Personal Brain is a sort of Mind Manager without the constraint that everything has to fit in a tree structure. They have a Microsoft sort of version scheme: home, business, enterprise. The home version is free, it's not an expiring trial.

Personal Brain website

Books on Galois Theory

Starting a self-study project involves finding the right books. I found the following books on Galois Theory which are aimed at beginners in the topic. For my purpose the following books are the most useful.

Galois Theory by David Cox, Wiley 2004
This is a very beautiful book of close to 600 pages, every page shows that the author loves the subject and really tries to explain the subject. It has sections on Galois Theory in Mathematica. ( In study-tech terminology: it effectively handles the first barrier to study, i.e. lack of mass, by making the subject tangible in the form of Mathematica functions. The student can explore the subject in a concrete fashion. ) Essential for self-study it has hints to selected exercises.

Exploratory Galois Theory by John Swallow, Cambridge University Press 2004
The author wrote software of his own in the Mathematica language which is available for download.

Galois Theory (3rd ed) by Ian Stewart, Chapman & Hall, 2004
Ian Stewart (*) is well know in England ( and beyond ) and this was his first book. I have added this book to my list because there are four different proofs of the Fundamental Theorem of Galois Theory in it.

Most good introductory books on Abstract Algebra have a few chapters on Fields and Galois Theory but it seems they are merely included to create an appetite for more.

Choosing books is a critical phase in self-study. I started with the book of Weintraub which he probably started very enthusiastically but it got denser and denser almost by page. ( Good book but needs extensive lecture notes by a teacher. ) Via Google Books -and other sources- you can find and browse any book on the subject. The reviews on Amazon can be helpful too.

The books are decided upon. On to phase 2: planning.