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Friday, December 31, 2010

Happy 2011

.
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                | | | |( (_| || (_) )| (_) )| (_) |
                (_) (_)`\__,_)| ,__/'| ,__/'`\__, |
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     | |`\ |(  ___/| \_/ \_/ |         | |(  ___/( (_| || |
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                                            -=*  POP! *=-
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Looking back on 2010 - 1


The number of readers of this blog has been more or less constant during the first two years of collecting stats. Somewhere in October both the number of unique visitors and pages viewed doubled. ( Visits were high in the exam-period. ) Visitors stabilized on this new level in November and December. I don't think there is one explanation for the doubling but attributing factors might be regular 'on-topic' posting and tweets about new posts. It is nice to know that someone is reading your scribblings.

Somewhere during 2010 I implemented MathJax ( LaTeX for HTML ). Because Blogger does not officially support MathJax ( yet, I hope ), I am leeching resources elsewhere. This slows down MathJax somewhat. Recently I installed the AMS extensions which further slowed down MathJax performance. One way or the other I'll fix this issue.

A conjecture about perfect numbers

Perfect number

In number theory the sum of the divisors is denoted as $\sigma$: $$\sigma(n) = \Sigma_{d/n} d$$ and $s(n)=\sigma(n) - n$ is the sum of the proper divisors. A perfect number is equal to the sum of its proper divisors. All known perfect numbers are even, it is unknown if odd perfect numbers exist. The number $2^{p-1}(2^p-1)$ is perfect if and only if $(2^p-1)$ is prime.

Conjecture

Show that: if $p$ is odd then
$$ 2^{p-1}(2^p-1) = \sum_{k=1}^{\frac{p+1}{2}-1} (2k-1)^3$$
( Notice that $2^{p-1}(2^p-1)$ yields a perfect number if $(2^p-1)$ is prime. )

Example

$6$ is perfect, since $6 = 1 + 2 + 3.$
$28$ is perfect, since $28= 1 + 2 + 4 + 7 + 14.$

Any perfect number ( except 6 ) can be represented as a sum of cubes.
$\begin{array}{ccc}
\underline{p} & \underline{Pf} &\underline{s}\\
3 & 28 & 1^3 + 3^3 \\
5 & 496 & 1^3 + 3^3 + 5^3 + 7^3 \\
7 & 8128 & 1^3 + 3^3 + ... + 15^3 \\
13 & 33550336 & 1^3 + 3^3 + ... + 127^3
\end{array}$

Proof

My exercise for New Year's Day. ( You may have noticed that I like doing 'sums'. ) Later...

( Source:
- A primer of analytic number theory, From Pythagoras to Riemann by Jeffrey Stopple
)

Thursday, December 30, 2010

Stirling numbers in Discrete Calculus

Definition

Stirling numbers of the second kind represent the number of k-partitions of an n-set and are recursively defined as $\left\{ 0,0 \right\} = 1$, and $\left\{ n,k \right\} = \left\{ n-1,k-1 \right\} + k \cdot \left\{ n-1,k \right\}$. They are used in the Discrete Calculus to convert powers to factorial powers, i.e. $n^2 = n^{\underline{1}} + n^{\underline{2}}$.

The matrix below shows the Stirling numbers for $n=0$, to $n=5$.
$\left(
\begin{array}{cccccc}
1 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 \\
0 & 1 & 1 & 0 & 0 & 0 \\
0 & 1 & 3 & 1 & 0 & 0 \\
0 & 1 & 7 & 6 & 1 & 0 \\
0 & 1 & 15 & 25 & 10 & 1
\end{array}
\right)$

Example

Calculate $\sum_{k=1}^{n} k^5$.

$\Sigma \Delta k^5$
$=\Sigma ((n+1)^{\underline{1}} + 15(n+1)^{\underline{2}} + 25(n+1)^{\underline{3}} + 10(n+1)^{\underline{4}} + (n+1)^{\underline{5}} )$
$=\frac{1}{2}(n+1)^{\underline{2}} + 5(n+1)^{\underline{3}} + \frac{25}{4}(n+1)^{\underline{4}} + 2(n+1)^{\underline{5}} + \frac{1}{6}(n+1)^{\underline{6}}$
$=-\frac{n^2}{12}+\frac{5 n^4}{12}+\frac{n^5}{2}+\frac{n^6}{6}$
$=\frac{1}{12} n^2 (1+n)^2 \left(-1+2 n+2 n^2\right)$

\begin{array}{lll}
\underline{n} & \underline{n^5} & \underline{\frac{1}{12} n^2 (1+n)^2 \left(-1+2 n+2 n^2\right)} \\
1 & 1 & 1 \\
2 & 32 & 33 \\
3 & 243 & 276 \\
4 & 1024 & 1300 \\
5 & 3125 & 4425
\end{array}

Wednesday, December 29, 2010

Continued Fractions

I started studying continued fractions..., a vast subject. ( 'new land!' )

Here I calculate the Extended GCD of $9976$ and $6961$ using Blankinship's matrix method. A by-product of the calculation ( column 1 ) are the numbers for expressing $\frac{9976}{6961}$ as a finite continued fraction.
$\left(
\begin{array}{cccc}
- & - & 9976 & 6961 \\
- & 9976 & 1 & 0 \\
1 & 6961 & 0 & 1 \\
2 & 3015 & 1 & -1 \\
3 & 931 & -2 & 3 \\
4 & 222 & 7 & -10 \\
5 & 43 & -30 & 43 \\
6 & 7 & 157 & -225 \\
7 & 1 & -972 & 1393\\
& 0 & 6961 & -9976 \\
\end{array}
\right)$

$\frac{9976}{6961}=1+\frac{1}{2+\frac{1}{3+\frac{1}{4+\frac{1}{5+\frac{1}{6+\frac{1}{7}}}}}}$

Monday, December 27, 2010

Five proofs for the sum-formula of 1+2+3+ ... +n

The running totals of 1,2,3 ... are called the triangular numbers. We will show that $1 + 2 + 3 + \cdots + n = \frac{n(n+1)}{2}$

Proof-1 Gauss's proof

( Gauss supposedly came up with this proof when he was 8 years old. On this page you will find more than 100 different tellings of this story. )
$s = 1 + 2 + 3 + \cdots + n$
$\underline{s = n + (n-1) + (n-2) + \cdots + 1}$
$2s = (n + 1) + ((n-1)+2) + ((n-2)+3) + \cdots + (1+n) \Leftrightarrow $
$2s = n \cdot (n + 1) \Leftrightarrow $
$s = \frac{n(n+1)}{2}$

Proof-2 By induction

Let $S=\left\{ n \in \mathbf{N} | \sum_{k=1}^{n} k = \frac{n(n+1)}{2} \right\}$
Clearly $1 \in S$
Assume, $n \in S$:
$\sum_{k=1}^{n+1} k = \frac{n(n+1)}{2} + (n+1) = \frac{(n+1)(n+2)}{2}$, or $n \in S \Rightarrow n+1 \in S$
Now, since $(S \subset \mathbf{N} \wedge 1 \in S \wedge n \in S \Rightarrow n+1 \in S ) \Rightarrow S=\mathbf{N}.$

Proof-3 With the Pascal Triangle

Because $n^k$ is in the PT for any $k \in \mathbf{N}$, sums of polynomials with integer coefficients can be read from the PT. ( $n={n \choose 1}$, $n^2={n \choose 1} + 2{n \choose 2}$, and so forth. )
$\underline{n}$
0: 1
1: 1 - 1
2: 1 - 2 - 1
3: 1 - 3 - 3 - 1
4: 1 - 4 - 6 - 4 - 1
5: 1 - 5 - 10-10 - 5 - 1
From $n$ we seek the second column, one row down or ${n+1 \choose 2}= \frac{n(n+1)}{2}$

Proof-4 Geometric

Look at the pattern
X
and
X---Y

X
X-X
and
X---Y-Y
X-X---Y

X
X-X
X-X-X
and
X---Y-Y-Y
X-X---Y-Y
X-X-X---Y

The number of X's and Y's are equal. The triangle X-pattern with base of n X's is replaced by a rectangular shape of n+1 by n X's OR Y's.

Proof-5 With Discrete Calculus

The discrete analog of solving a differential equation.
$\Delta f(n) = n+1$
$\Sigma \Delta f(n) = \Sigma (n+1) $
$f(n) = \frac{1}{2}(n+1)^{\underline{2}} + C$
$f(n) = \frac{1}{2}(n+1)n + C$
Since $f(1) = 1, C=0$
$f(n) = \frac{n(n+1)}{2}$

Sunday, December 26, 2010

Divisibility by 7

Let n be an integer with last digit d.
Repeat until divisibility of n by 7 has been decided:
- Set m to n with last digit removed.
- Set n to m - 2d
- Determine if n is divisible by 7.

Proof:
m = (n - d)/10
m = (n - d)/10 - 2d
m = ( n - 21d ) / 10
If 7 / n then 7 / ( n - 21d ) / 10 since (7,10)=1 and 7/21.

Example:
n = 8 641 969
m = 864 196
n = 864 196 - 18 = 864 178
m = 86 417
n = 86 417- 16 = 86 401
m = 8 640
n = 8640 - 2 = 8638
m = 863
n = 863 - 16 = 847
m = 84
n = 84 - 14 = 70
Divisible by 7.

I suppose trivia like this are only interesting for math(s)(*) enthusiasts and number geeks. Computational number theorists working with numbers of several million digits may actually use methods like this.

P.S.
(*) Just learned from Math is Fun that "Mathematics is commonly called Math in the US and Maths in the UK and in many other countries.".

Saturday, December 25, 2010

Merry Christmas

*★Merry★* 。 • ˚ ˚ ˛ ˚ ˛ •
•。★Christmas★ 。* 。
° 。 ° ˛˚˛ * _Π_____*。*˚
˚ ˛ •˛•˚ */______/~\。˚ ˚ ˛
*° •˛• ☃| 田田 |門| ☃˚╰☆╮
...To ALL readers :) ♥♥♥

Friday, December 24, 2010

[Books] - Problem-Solving and Selected Topics in Number Theory



Provides a self-contained introduction to classical number theory;
Includes step-by-step proofs of theorems and solutions to exercises;
Designed for undergraduate students ( ... );
( Olympiad-caliber problems )

My Christmas Present from myself ( who else? )... Do you mind? ;-) - This boook symbolizes my commitment to Number Theory. Besides M373 / M381 ( 2011 ) and MST209 / M336 ( 2012 ) I will focus my self-study on Number Theory. Besides Elementary ( or Classical ) Number Theory which is covered by M381 I have a shortlist of books on Analytical-, Algebraic- and Computational Number Theory.

Oh, and I think I'll ask Caz ( who has a column on Platform ) to blog about why people think they are successful in their studies. It could lead to some interesting discussion and hopefully valuable ideas. Where do they come from: Study Results <- Progress <- Concentration <- Focus <- Discipline ??

Thursday, December 23, 2010

OU final registrration date extended until 6/Jan-'11

Received the following e-mail.

The final date for reserving a place on an undergraduate course starting in Feb 2011 has now been extended until 6th January 2011, due to the adverse weather conditions.

( I have quite a 'registration-story' forthcoming too. )

Tuesday, December 21, 2010

[Sign of the times] To: Richard Branson

Richard Branson has criticised the British education system for "overeducating" students and failing to prepare them for the business world, according to leaked diplomatic cables. Branson is by no means the only successful businessman to choose experience over education. - The Virgin tycoon, who is dyslexic and left school at the age of 15, has previously said that he gained valuable business experience while his peers were at university. (From The Telegraph)

Dear Mr. Richard Branson,
All the money you made can't buy you a single pass of a 10 point level 1 course. I suppose 90% of the 16-year olds are smarter than you because you were so stupid to leave school at the age of 15. You may need brainless morons to obey your orders in exchange for a few coins, society needs some very smart people to solve the current global problems which you are partly responsible for.
Kind regards.

Monday, December 20, 2010

M373

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

M381

Just registered for M381 - Number Theory and Logic per Feb 2011. ( More later... )

P.S. $$\sum_{p \in \mathbf{P}} \frac{1}{p}$$
... diverges.

Wednesday, December 15, 2010

Two days and counting ...

... before the M208 result comes in. I wish that we all get the results we have hoped for and deserve considering the work put in.

Anyway, I am definitely going to do M381 Number Theory and Logic next year / month. I'll decide tomorrow if I add M337, MST209 or leave it with "just" M381.

Tuesday, December 14, 2010

[Video] - Why Pi?

Why Pi? is the title of Don Knuth’s 16th Annual Christmas Tree Lecture. Don Knuth is a famous computer scientist and the designer and programmer of TeX. The first book I seriously self-studied was Concrete Mathematics by Knuth, Ron Graham and Patashnik with beautiful stuff on the Fibonacci series, the Pascal triangle, combinatorial identities, generating functions and numbers and divisibility. It's a beautiful book. I haven't watched the ( entire ) lecture yet, i parked it on my watch-queue.

Sunday, December 12, 2010

Thursday, December 9, 2010

Reminder 2011 course registration

Ref: 47911183 Reminder - changes to registration dates for undergraduate courses starting in February 2011 - 22 Dec 2010.

I still haven't decided yet... I really don't know at this moment. Tired, tomorrow...

[Exercise] - 2a: The coconut congruence equation

The coconut problem is what is called an algebra word problem. In a course in elementary number theory we could skip the words and ask to solve the following equation:

$$\frac{4}{5}(\frac{4}{5} (\frac{4}{5} (\frac{4}{5} (x-1) - 1) -1) -1) \equiv 1 \mod{5}$$

[Sign of the times] - Wikileaks: Yes, USA: No

Often heard: "What can I do about it, anyway?"

Well, you can close your MasterCard account for example. Or stop buying books from Amazon. Don't buy American if you have the option. You can also support Wikileaks directly of course. I think they need it and deserve it. - They belong to the good guys on this planet. Wikileaks is more than just Assange, although he is important.

Just my 0,02c.

Wednesday, December 8, 2010

[Exercise] - 1a : Hint

Let $n \in \mathbb{N}$, show that $$f(n) = \frac{(2+\sqrt{3})^{1+2n}+(2-\sqrt{3})^{1+2n}+2}{6}$$ is a square.

Hint: MST121 / MS221 math suffices to solve this one, we need to find a function $A(n)$ such that $${A(n)}^2 = \frac{(2+\sqrt{3})^{1+2n}+(2-\sqrt{3})^{1+2n}+2}{6}$$
and
$$f(n) = A(n) \in \mathbb{N}$$
( To be continued. )

Tuesday, December 7, 2010

[Exercise] - 2

This is a famous problem. I found it on the Internet by searching for "mathematics, monkey, coconut, problem". - My version is in a Dutch book called Algebra by M. Riemersma.

Five men and a monkey were shipwrecked on a desert island, and they spent the first day gathering coconuts for food. Piled them all up together and then went to sleep for the night. But when they were all asleep one man woke up, and he thought there might be a row about dividing the coconuts in the morning, so he decided to take his share. So he divided the coconuts into five piles. He had one coconut left over, and he gave that to the monkey, and he hid his pile and put the rest all back together.

By and by the next man woke up and did the same thing. And he had one left over, and he gave it to the monkey. And all five of the men did the same thing, one after the other; each one taking a fifth of the coconuts in the pile when he woke up, and each one having one left over for the monkey. And in the morning they divided what coconuts were left, and they came out in five equal shares. Of course each one must have known there were coconuts missing ; but each one was guilty as the others, so they did not say anything.

How many coconuts were there in the beginning?”

To be continued ( i.e. answer and comment )

Monday, December 6, 2010

[Exercise] - 1

Let $n \in \mathbb{N}$, show that $$f(n) = \frac{(2+\sqrt{3})^{1+2n}+(2-\sqrt{3})^{1+2n}+2}{6}$$ is a square.

For $n=1$ to $5$ we have ($n, \ \sqrt{f(n)}, \ f(n)$):
$\begin{array}{lll}
1. & 3. & 9. \\
2. & 11. & 121. \\
3. & 41. & 1681. \\
4. & 153. & 23409. \\
5. & 571. & 326041.
\end{array}$

[News] - Math and industry

The European Science Foundation published the ESF Forward Outlook "Mathematics and Industry". Download a PDF copy from this page.

Sunday, December 5, 2010

[Sign of the times] - New York /12-2010

From an article in the New York Times: LAPTOPISTAN

credit: Piotr Redlinski for The New York Times 

( A new series of posts [Sign of the times] with mainly pictures. Common theme is of course the theme of this blog: "mathematics and / or study". ) Let me know if you think this series doesn't belong here.

Comment:
Take your laptop with you and you can go out on your own. Have coffee, work, study and enjoy the company of people around you. In the Netherlands public libraries are ( or already have ) transformed themselves into embassies of Laptopistan with free power-points and WiFi.

Saturday, December 4, 2010

[Sign of the times] - Abacus still in use

The Corporation is optimistic about the potential of the abacus system of mental arithmetic, which it introduced in two of its primary schools on Friday. The system is aimed at improving the comfort of young students with numbers and the mathematical functions.

Source: express buzz

Link: Abacus ( see the comment )

Link: An introduction to the abacus

Link: Chinese Abacus + Manual

Friday, December 3, 2010

Video Lectures on Number Theory

Among lectures on Calculus I,II and III, ( Introduction to ) Linear Algebra and ( Introduction to ) Differential Equations from the UCCS ( University of Colorado and Colorado Springs ) Department of Mathematics you will find video lectures on Math 311 Number Theory by Professor Dr. Seung Son here. I have watched most of them earlier this year. This week I watched some of them again.

While watching a video on mathematical induction something amazing happened, not sure if I would call it a cognition, but it's close. Since I was able to do proofs by mathematical induction and thus understood it, I thought I was done studying mathematical induction. ( I mean both the MS221 and M208 exams included questions on induction). Well, I close-to-cognited that I didn't understand proofs by mathematical induction -at all-.

Do you? If so:
- state the first principle ( of mathematical induction ) using symbols only,
- state the second principle using symbols only,
- re-formulate: "Show that: ... $$\sum_{k=1}^{n}k = \frac{n(n+1)}{2}$$ ..." using Set Terminology,
- can you explain the difference between the first and second principle?
- give an example of a proof using the first principle,
- give an example of a statement which can only be proved with the second principle.

I failed ( note: past tense ) all answers to the questions above. Post is To Be Continued ...

Tuesday, November 30, 2010

MST209 Kick Off

Finished and approved the plan for MST209 today. Status: in-progress.

I have been revising some physics: classical ( Newtonian ) mechanics. Amazing what we can do with three simple laws and a few calculus tools: the foundation of modern technology.

Saturday, November 27, 2010

MST209 - Overview Blocks Units

I'll be doing MST209 next year. It is the most logical choice after MST121, MS221 and M208 the courses I have done sofar. The course looks quite interesting if you ask me. I already studied some topics based on the OpenLearn version of MST209 supported by corresponding MIT 18.02 / 18.03 video lectures. Although I don't study mathematics because of the stuff offered in MST209 I accept that MST209 is part of the mathematical language shared by all scientists ( and engineers ). Besides that MST209 is packed with examples I can play with in Mathematica, do some Mathematica programming as well.

Since I don't have the course materials yet ( haven't even registered yet ) I made a simple planning in a spreadsheet. Blocks / Units as rows, weeks as columns. The TMA cut-offs are estimated based on M208 data of this year. My study 'capacity' in net study hours is roughly as follows on a weekly basis.

NET STUDY(=RESOURCE) CAPACITY
Mon Tue Wed Thu Fri Sat Sun Total
2 6 2 2 - 6 6 = 24

Total 24hrs or 90P / year. ( Read: Work + Study and do nothing else. )

I scheduled the MST209 data in the schedule and it fits very well. I usually work two weeks on a TMA. Because I use LaTeX, do a lot of quality-checks, and start asap on a TMA, which means I might need to study stuff while working on the TMA. Anyway this means that I have two weeks to study a block, 5 Units or roughly 20 sections, do exercises, etc. You all know the drill.

Working on a degree is basically an excuse for spending time on math. In most circles it is considered anything between nerdy, nutty and plain vanilla crazy. ( Ignorance is NOT bliss. ) I am just explaining that my plan objective is not completing the project at the earliest possible date.

I have the following options.
- Blank MST209
- MST209 + M381 Number theory and logic ( As in my original overall plan )
- MST209 + M337 Complex analysis
- MST209 + MST326 Mathematical methods and fluid mechanics

My next step is, based on my study and OU experiences sofar, trying to 'fit in' another 30 point course ( any course for that matter ). If ( and only if ! ) that works I have to choose which module I will do next to MST209.

What's in MST209 :

BLOCK-1
Unit 1 Getting Started
This unit focuses mainly on mathematical techniques, but also covers some examples involving skills in the application of mathematics.

Unit 2 First-order Differential Equations
This unit considers in detail how a differential equation arises in a mathematical model with basic definitions and terminology associated with differential equations and their solutions.

Unit 3 Second-order Differential Equations
This unit considers second-order differential equations, that is, differential equations that involve a second (but no higher) derivative.

Unit 4 Vector Algebra
We often need to represent physical quantities such as mass, force, velocity, acceleration, time, etc., mathematically. Most of the physical quantities that we need can be classified into two types: scalars and vectors. This unit defines a vector and discusses ways of representing vectors in two and three (or more) dimensions. Also considered are ways of operating on and combining vectors - that is, they provide the fundamentals of vector algebra.


BLOCK-2
Unit 5 Statics
This unit and unit 6 lay the foundations of the subject of mechanics. Mechanics is concerned with how and why objects stay put, and how and why they move. This unit considers how and why they move. This unit assumes a good working knowledge of vectors.

Unit 6 Dynamics
Continuing on from Unit 5 this unit considers how and why objects move and outlines the procedure for solving dynamics problems. There is a video sequence associated with section 2 of this unit and is available on the DVD (order code MST209/DVDR01).

Unit 7 Oscillations
All around you there are mechanical systems that vibrate or oscillate. Each day you probable experience oscillations in a wide variety of forms: the buzzing of an alarm clock, the vibrations of an electric hair-drier or razor, the sideways movements of a train or boat, and so on. This unit describes a simple experiment involving an oscillating system, introduces Hooke's law as a model for the force exerted by a spring, and goes on to consider how this law applies in various situations where no movement takes place. It also shows how to use Newton's second law to model the oscillations of the simplest oscillating system, which consists of a single particle attached to a single horizontal spring.

Unit 8 Energy and Consolidation
This unit consolidates the mechanics covered in the previous units and introduces the topic of energy in mechanical systems.


BLOCK-3
Unit 9 Matrices and Determinants
This unit examines some of the properties and applications of matrices. It introduces the matrix method of solving large systems of linear equations, called the Gaussian elimination method, and explains the conditions required for this method to work.

Unit 10 Eigenvalues and Eigenvectors
This unit introduces eigenvectors showing simplified problems. It considers the eigenvectors and eigenvalues associated with various linear transformations of the plane and outlines situations where eigenvectors and eigenvalues are useful.

Unit 11 Systems of Differential Equations
This unit focuses on systems of linear differential equations relating two more functions and their derivatives. It shows how various situations can be modelled by a system of linear differential equations, how such a system can be written in matrix form, and how eigenvalues and eigenvectors can be used to solve it when the equations are homogeneous with constant coefficients.

Unit 12 Functions of Several Variables
This unit extends the calculus of functions of one variable to functions of several variables. It also discusses the application of functions of several variables to mechanics.


BLOCK-4
Unit 13 Modelling with Non-linear Differential Equations
In this unit we study the mathematical models associated with two physical systems: the growth of two interacting populations, one a predator and the other its prey and the motion of a rigid pendulum.
This is demonstrated through the use of Lotka-Volterra equations, which apply to a pair of interacting populations. How these equations can be linearized near an equilibrium state and the graphical representation of the solutions are discussed.

Unit 14 Modelling Motion in Two and Three Dimensions
This unit turns the attention to motion and forces in more than one dimension. It draws on ideas about vectors, forces and component forces and fundamental ideas about mechanics of particles, in particular Newton's second law. There is also mention of kinetic energy and potential energy. The video sequences associated with this unit are available on the DVD (order code MST/209/DVDR01), however it is not essential for you to view these.

Unit 15 Modelling Heat Transfer
This unit makes use of ideas relating to energy and first-order differential equations. It begins by developing models that could be used to answer questions such as the following.

How much does it cost to heat up a tank full of hot water?
What thickness of insulation should be applied to a hot-water tank?
What thickness of insulation should I use in my loft, and what savings would I make over a year?
What should the gap be in double-glazing?

Is it better to insulate the roof, insulate the walls or double-glaze the windows of my house?
The common factor in answering all these questions is the need to consider the transfer of heat energy between different regions of space. This unit introduces the basic ideas of heat energy and temperature. It discusses conduction and convection as well as a third mode of heat energy transfer, radiation.

Unit 16 Interpretation of Mathematical Models
This unit introduces the ideas of mathematical modelling. It discusses the five key stages of the mathematical modelling process in detail and looks at dimensions and units of physical quantities to see how they can be used to predict and check the outcomes of the modelling process.


BLOCK-5
Unit 17 Damping, Forcing and Resonance
This unit refers back to the contents of several earlier units. In particular it builds upon and extends the model of simple harmonic motion and uses this approach in analysing one-dimensional motion. It also returns to the concept of a resistance force proportional to the velocity of a particle.

Unit 18 Normal Modes
This unit continues with the theme of mechanics, in particular, it builds on earlier units that dealt with oscillations. In order to solve the equations of motion derived for the mechanical systems studied, it uses the methods used for solving systems of differential equations. This unit also draws heavily on the discussions regarding eigenvalues and eigenvectors.

Unit 19 Systems of Particles
The main objective of this unit is to show how to obtain useful information on the complicated motion of an object or system, and demonstrates that the concept of centre of mass is crucial to this process.

Unit 20 Circular Motion
The theme of this unit is rotational motion. It concentrates mainly on analysing the circular motion of a particle. This can be used to model a wide range of situations, such as a child on a swing, the pendulum of a clock and a chair-o-plane roundabout at a fairground. This unit builds on many of the ideas from earlier units mainly: polar coordinates, vectors, torque and Newton's second law.


BLOCK-6
Unit 21 Fourier Series
This unit is concerned with the technique of expressing a periodic function as a sum of terms, where each term is a constant, a sine function or a cosine function. This unit assumes you have a background knowledge of the definition of the period (unit 7), forced oscillations and resonance (Unit 17, and integration by parts (Unit 1).

Unit 22 Partial Differential Equations
This unit builds on ideas previously introduced in unit 12 regarding The diffusion equation and the wave equation, in the context of modelling the vibrations of a taut string (such as guitar string).

Unit 23 Scalar and Vector Fields
The main focus of this unit is the differential calculus of scalar and vector fields, i.e. the study of how scalar and vector fields vary from one point to another. A brief introduction to the properties of orthogonal matrices and pictorial representations of scalar and vector fields is given along with an extended discussion of the gradient function of a scalar field. Cylindrical and spherical polar coordinate systems for specifying points in three dimensions is also introduced in this unit.

Unit 24 Vector Calculus
This unit discusses the divergence of a vector field, the curl of a vector field, the scalar line integral and linking line integrals curl and gradient. This unit also builds on the concepts of kinetic energy and potential energy.


BLOCK-7
Unit 25 Multiple Integrals
This unit generalises the idea of an integral still further to deal with two and three dimensions by introducing two new kinds of integrals, called area integrals and volume integrals. How area integrals can be evaluated as combinations of two ordinary integrals, is shown and applications of area integrals, including the evaluation of centres of mass of planar (i.e. two-dimensional) objects are described. How volume integrals can be expressed as combinations of three ordinary integrals and how area integrals can be used to compute the area of a curved surface are also demonstrated.

Unit 26 Numerical methods of Differential Equations
This unit introduces the study of numerical methods for differential equations. It covers the Taylor's theorem with exercises, recaps Euler's method for solving initial-value problems involving first-order differential equations and goes on to explain that more efficient methods exist. Three new methods known as Runge-Kutta methods, are derived and a way of determining how small the step size h would need to be in order to achieve a given accuracy for a given initial-value problem is established.

Unit 27 Rotating Bodies and Angular Momentum
This unit deals with the motion of extended bodies, and in particular with their rotational motion. Rotating bodies, Angular momentum, Rigid-body rotation about a fixed axis and rotation about a moving axis are all covered in this unit.

Unit 28 Planetary Orbits
This unit shows how Newton's laws of motion and Newton's law of universal gravitation can be used to predict the orbits of planets around the Sun. In particular, it shows that Kepler's laws of planetary motion can be derived using Newtonian mechanics.

Watched MIT 18.02 - lecture 11

In 18.02 lecture 11 Prof. Denis Aroux talks about differentials and the Chain Rule. Two of the examples used to illustrate the main topic are of particularly interesting: a new proof for the differentiation of products and quotients, and the conversion between rectangular and polar coordinates.

A main result of this lecture is $$df = f_u \frac{du}{dx} + f_v \frac{dv}{dx},$$ where $f$ is a function of two variables $u,v$ which are both dependent on $x$, and $f_u, f_v$ are partial derivatives. The quotient rule can be derived from this result as follows. Let $g(x) = \frac{u}{v}$, with $u,v$ both dependent on $x$ :
$\begin{align*}

df &=f_u \frac{du}{dx}+f_v \frac{dv}{dx} \\
&=  \frac{1}{v}\frac{du}{dx}-\frac{u}{v^2}\frac{dv}{dx} \\
&= \frac{ v \frac{du}{dx}-u \frac{dv}{dx} }{v^2}
\end{align*} $
The last expression is the quotient rule for differentiation.

This lecture inspired me to some experimentation ( play ) with Mathematica's PolarPlot function. A polar coordinate is in fact a function of two variables $x,y$ which are both dependent on $r$ and $\theta$ with $x=r \cos(\theta)$, $y=r \sin(\theta)$. By applying the theory above one suddenly gets control over geometric objects like this:

Click to enlarge

Finally the concept of a gradient was mentioned which is merely a vector of partial derivatives. Gradients are the topic of lecture 12. I designed some problems and exercises ( and other experiments ) for functions in polar coordinates. I am delighted I feel more able in that regard.

Thursday, November 25, 2010

Watched MIT 18.02 - lectures 9,10

The Second Derivative Test is a procedure for determining if a critical point is a maximum, a minimum, a saddle or a degenerate.

PROCEDURE:
- Calculate $f_x, f_y, f_{xx}, f_{xy}, f_{yy}$
- Calculate the critical points ;

Then for each critical point:
- Calculate $A = f_{xx}(x_0, y_0)$
- Calculate $B = f_{xy}(x_0, y_0)$
- Calculate $C = f_{yy}(x_0, y_0)$
- $AC-B^2$.

Apply the Second Derivative Test
If $AC-B^2 > 0$ and $A > 0$ minimum
If $AC-B^2 > 0$ and $A < 0$ maximum
If $AC-B^2 < 0$ saddle
If $AC-B^2 = 0$ degenerate

Click to enlarge


EXAMPLE:
( See Mathematica print )
$f(x,y)=e^{x^2-\frac{x^4}{4}-y^2}$

At $(0,0): \left\{AC-B^2, A\right\} = \left\{-4,2\right\}$
Saddle.

At $(-\sqrt{2},0): \left\{AC-B^2, A\right\} = \left\{8e^2,-4e\right\}$
Local maximum.

At $(\sqrt{2},0): \left\{AC-B^2, A\right\} = \left\{8e^2,-4e\right\}$
Local maximum.

.

Marcus du Sautoy on Symmetry ( Explained )

If you watched du Sautoy's talk on symmetry I posted yesterday and your background in Group Theory matches MS221 or M208 you might have lost him when he was talking about the symmetries found in the Alhambra. Let me explain.

These symmetry groups are called the Wallpaper Groups and they are part of the Open University Course M336 Groups and Geometry. Although there are only 17 wallpaper groups ( mathematical proof in M336 ) recognizing them in nature or in a piece of art is difficult. Well, it takes practice. This beautiful applet ( Java required on your PC ) lets you experiment with the wallpaper groups. Some theoretical background can be found here.

Enjoy!

Wednesday, November 24, 2010

Watched MIT 18.02 - Lecture 8

A lecture in which Prof. Denis Auroux says: "Applied mathematics is physics with a different set of symbols." To the math.

Lecture 8 starts a new unit in the 18.02 series and is about functions of several variables, the graph of a function of two variables, contour plots of functions of two variables and partial derivatives.


The image above shows the graph and the contour plot of $f(x,y)=1-(x^2+y^2)$ followed by the partial derivatives as calculated by Mathematica.

Partial derivatives are defined as follows:
$$\frac{\partial f}{\partial x}(x,y) = \displaystyle\lim_{\Delta x \to 0}\frac{f(x+\Delta x, y) - f(x,y)}{\Delta x}$$
$$\frac{\partial f}{\partial y}(x,y) = \displaystyle\lim_{\Delta y \to 0}\frac{f(x, y+\Delta y) - f(x,y)}{\Delta y}$$

.

[News] - London: More student protests

London, 24 nov 2010
Students protest against fees trebling from GBP 3000 to GBP 9000.

[Video] - Marcus du Sautoy on Symmetry

A talk on Symmetry by Marcus du Sautoy.



Marcus du Sautoy is also the narrator of the beautiful BBC - Open University co-production "The story of maths" which is a 4 hour documentary ( in 4 parts ) accompanying the OU course TM190 The story of maths. ( Level 1 / 10 points / 1 CMA only ).

Tuesday, November 23, 2010

[TIP] - Join a Study Group

I joined an MIT 18.02 Online Study Group since I think 18.02 and MST209 have many similar topics. There are several other Study Groups as well. If you have a Facebook account you can login with your Facebook Id.

http://openstudy.com/

MIT 18.02 Study Group

Monday, November 22, 2010

More on MST209 in relation to MIT video lectures

MIT has a video lecture series on multivariable calculus: 18.02, which is in fact a prerequisite for 18.03. I looked further into the topics of MST209 and I now think that 18.02 is a much better preparation for MST209 than 18.03.

It's more or less like this:
MST209 = 18.02 + ( part of ) 18.03
MST209 + MST326 = 18.02 + 18.03

18.02 has lectures on
Lecture 15: Partial Differential Equations
Lecture 16: Double Integrals
Lecture 19: Vector Fields
Lecture 21: Gradient Fields
Lecture 25: Triple Integrals
Lecture 27: Vector Fields in 3D
Lecture 30: Line Integrals
which are topics in MST209.

18.02 Multivariable Calculus

M203/M208 Exam Results 2006-2009

M203/M208
YearTook examPass
200669%92%
200771%90%
200869%93%
200967%92%

Took exam: Exam candidates / Final Registrations
Pass: Candidates passed / Exam candidates

Source: Open University

Sunday, November 21, 2010

[Review] - Atom 1/3 : The Clash of Titans

The Clash of Titans ( Atom 1/3 ) by Professor Jim Al-Khalili is a documentary about the birth of quantum mechanics. - According to Al-Khalili the discovery of the Atom is the most important achievement in science, ever ( amazing how every scientist says something like that about his or her field ). I wished he hadn't said that in the documentary itself because besides that Atom is one of those documentaries the BBC is world famous for ( starting with The Ascent of Man with Jacob Bronowski ). The program is educational, the photography is in one word beautiful and Al-Khalili is a real story teller, he turns the birth of quantum mechanics into an action movie with heroes, winners and losers. Atom was first showed on TV in early 2008 when I missed it but I have been able to pick up a copy of the entire three part series recently. With MST209 scheduled for next year more posts about physics will be inevitable, I suppose.

Watched MIT 18.03 - lecture 14

In this lecture we hear that Prof. Mattuck plays the cello. Yet another example of a mathematician with a love ( or talent ) for classical music. To the math.

This lecture is about the following differential equation that represents a system with a resonance frequency $\omega$ and the input to the system with frequency $\omega_1.$

$y'' + \omega^2y = \cos{(\omega_1x)}$

If $\omega_1 \neq \omega$ then the solution is: $y=C1\cos(\omega x)+C2\sin(\omega x)+\frac{\cos(\omega_1 x)}{\omega^2-\omega_1^2}$

If $\omega_1 = \omega$ then the solution is $y=C1\cos(\omega x)+C2\sin(\omega x)+\frac{x\sin(\omega x)}{2\omega }$

Saturday, November 20, 2010

MST209 + MST326 as a 90 point option ?

It seems that MST209 is merely a ( level 2 ) introductory course on differential equations and that MST325 Mathematical Methods and fluid mechanics is the real deal on differential equations as far as the Open University is concerned.

Let me repeat the names of the courses once more:
MST209 - Mathematical methods and models
MST325 - Mathematical Methods and fluid mechanics
Just so that we agree that the word differential equations does not appear in either of the course names. - There is nothing wrong with changing a course description so that it will have appeal to a wider audience as long as the topic remains clear. I.e. "ODEs and PDEs" ( a name I can definitely imagine if the an occasional nerd slipped into the math community ) to "Ordinary and Partial Differential Equations", or: "Differential Equations" to "Mathematical modeling with Differential Equations". I do not understand the strange naming conventions of the Open University.

Names are just names lets have an in-depth look at the course descriptions.

MST209 - What you will study

This course will be of particular interest to you if you use mathematics or mathematical reasoning in your work and feel that you need a firmer grounding in it, or if you think you might find it useful to extend your application of mathematics to a wider range of problems. The course should also be suitable if you are teaching A-level applied mathematics, or if you intend to do so; the material on mechanics, in particular, gives a very careful treatment of the basic concepts of this subject. The teaching is supported and enhanced by the computer algebra package Mathcad.

Around half of this course is about using mathematical models to represent suitable aspects of the real world; the other half is about mathematical methods that are useful in working with such models. The work on models is devoted mainly to the study of classical mechanics, although non-mechanical models – such as those used in heat transfer and population dynamics – are also studied. The work on methods comprises topics chosen for their usefulness in dealing with the models; the main emphasis is on solving the problems arising in the real world, rather than on axiom systems or rigorous proofs. These methods include differential equations, linear algebra, advanced calculus and numerical methods. Many are implemented in Mathcad, so you can use the computer to solve more difficult problems and to investigate case studies.

The mechanics part of the course begins with statics, where there are forces but no motion, and then introduces the fundamental laws governing the motions of bodies acted on by forces – Newton's laws of motion. These are first applied to model the motion of a particle moving in a straight line under the influence of known forces. Undamped oscillations are discussed next. Newton's laws are then extended to the motion of a particle in space. The motions of systems of particles are modelled. Next we look at the damped and forced vibrations of a single particle. Then we look at the motion (and vibrations) of several particles. Finally, we investigate the motion of rigid bodies.

The methods part of the course covers both analytic and numerical methods. The analytical (as opposed to numerical) solution of first-order and of linear, constant-coefficient, second-order ordinary differential equations is discussed, followed by systems of linear and non-linear differential equations and an introduction to methods for solving partial differential equations. The topics in algebra are vector algebra, the theory of matrices and determinants, and eigenvalues and eigenvectors. We develop the elements of the calculus of functions of several variables, including vector calculus and multiple integrals, and make a start on the study of Fourier analysis. Finally, the study of numerical techniques covers the solution of systems of linear algebraic equations, methods for finding eigenvalues and eigenvectors of matrices, and methods for approximating the solution of differential equations.

MST326 - What you will study

In simple terms, we think of a fluid as a substance that flows. Familiar examples are air (a gas) and water (a liquid). All fluids are liquids or gases. The analysis of the forces in and motion of liquids and gases is called fluid mechanics. This course introduces the fundamentals of fluid mechanics and discusses the solutions of fluid-flow problems that are modelled by differential equations. The mathematical methods arise from (and are interpreted in) the context of fluid-flow problems, although they can also be applied in other areas such as electromagnetism and the mechanics of solids.

Because of its many applications, fluid mechanics is important for applied mathematicians, scientists and engineers. The flow of air over objects is of fundamental importance to the aerodynamicist in the design of aeroplanes and to the motor industry in the design of cars with drag-reducing profiles. The flow of fluids through pipes and channels is also important to engineers. Fluid mechanics is essential to the meteorologist in studying the complicated flow patterns in the atmosphere.

The course is arranged in 13 units within four blocks.

Block 1 is the foundation on which the rest of the course is built.

Unit 1 Properties of a fluid introduces the continuum model and many of the properties of a fluid, such as density, pressure and viscosity. The basic equation of fluid statics is formulated and used to find the pressure distribution in a liquid and to provide a model for the atmosphere.

Unit 2 Ordinary differential equations starts by showing how changes of variables (involving use of the Chain Rule) can be applied to solve certain non-constant-coefficient differential equations, and leads on to the topics of boundary-value and eigenvalue problems. It concludes with an introduction to the method of power-series for solving initial-value problems.

Unit 3 First-order partial differential equations extends the earlier version of the Chain Rule to cover a change of variables for functions of two variables, and shows how this leads to the method of characteristics for solving first-order partial differential equations.

Unit 4 Vector field theory relates line, surface and volume integrals through two important theorems – Gauss’ theorem and Stokes’ theorem – and formulates the equation of mass continuity for a fluid in motion.

Block 2 starts by investigating the motion of a fluid that is assumed to be incompressible (its volume cannot be reduced) and inviscid (there is no internal friction).

Unit 5 Kinematics of fluids introduces the equations of streamlines and pathlines, develops the concept of a stream function as a method of describing fluid flows, and formulates Euler’s equation of motion for an inviscid fluid.

Unit 6 Bernoulli’s equation analyses an important equation arising from integrals of Euler’s equation for the flow of an inviscid fluid. It relates pressure, speed and potential energy, and is presented in various forms. Bernoulli’s equation is used to investigate phenomena such as flows through pipes and apertures, through channels and over weirs.

Unit 7 Vorticity discusses two important mathematical tools for modelling fluid flow, the vorticity vector (describing local angular velocity) and circulation. The effects of viscosity on the flow of a real (viscous) fluid past an obstacle are described.

Unit 8 The flow of a viscous fluid establishes the Navier-Stokes equations of motion for a viscous fluid, and investigates some of their exact solutions and some of the simplifications that can be made by applying dimensional arguments.

Block 3 looks at a class of differential equations typified by the wave equation, the diffusion equation and Laplace’s equation, which arise frequently in fluid mechanics and in other branches of applied mathematics.

Unit 9 Second-order partial differential equations shows how a second-order partial differential equation can be classified as one of three standard types, and how to reduce an equation to its standard form. Some general solutions (including d’Alembert’s solution to the wave equation) are found.

Unit 10 Fourier series reviews and develops an important method of approximating a function. The early sections refer to trigonometric Fourier series, and it is shown how these series, together with separation of variables, can be used to represent the solutions of initial-boundary value problems involving the diffusion equation and the wave equation. Later sections generalise to the Fourier series that arise from Sturm-Liouville problems (eigenvalue problems with the differential equation put into a certain standard format), including Legendre series.

Unit 11 Laplace’s equation is a particular second-order partial differential equation that can be used to model the flow of an irrotational, inviscid fluid past a rigid boundary. Solutions to Laplace’s equation are found and interpreted in the context of fluid flow problems, for example, the flow of a fluid past a cylinder and past a sphere.

Block 4 returns to applications of the mathematics to fluid flows.

Unit 12 Water waves uses some of the theory developed in Block 3 to investigate various types of water wave, and discusses several practical examples of these waves.

Unit 13 Boundary layers and turbulence looks at the effects of turbulence (chaotic fluid flow) and at the nature of boundary layers within a flow, introducing models to describe these phenomena.

Now let's have a look at MIT 18.03, the introductory undergraduate course on differential equations at MIT:

18.03 - Description

This course is a study of Ordinary Differential Equations (ODE's), including modeling physical systems.

Topics include:
Solution of First-order ODE's by Analytical, Graphical and Numerical Methods;
Linear ODE's, Especially Second Order with Constant Coefficients;
Undetermined Coefficients and Variation of Parameters;
Sinusoidal and Exponential Signals: Oscillations, Damping, Resonance;
Complex Numbers and Exponentials;
Fourier Series, Periodic Solutions;
Delta Functions, Convolution, and Laplace Transform Methods;
Matrix and First-order Linear Systems: Eigenvalues and Eigenvectors; and
Non-linear Autonomous Systems: Critical Point Analysis and Phase Plane Diagrams.

I would say that, roughly, MST209 + MST326 = 18.03, but this means that I must add MST209 + MST326 to my options for 2011. I would not have seen this option if I hadn't been studying MST209 ( OpenLearn ) and the 18.03 lectures.

Difference 18.03 - MST209 ( OpenLearn edition )

Taken from MST209 (OpenLearn edition) Book 2: Second order differential equations:





The 18.03 method however is as follows:

$2\frac{d^2y}{dx^2}-2\frac{dy}{dx}+y=2e^{-x}$

$(2D^2-2D+1)y=2e^{-x}$,
( where $\alpha=-1$ and $p(D)=2D^2-2D+1$ )

$y_p = \frac{2}{p(\alpha)}e^{-x} = \frac{2}{5}e^{-x}$

I don't know if the linear differential operator ($D$) is discussed in any part of MST209, if not the conclusion is simple: MIT 18.03 does a better job at teaching differential equations than MST209 since Matuck said that he $D$ operator plays an important role in the rest of 18.03 ( lectures 14-33 ).

Friday, November 19, 2010

Watched MIT 18.03 - lecture 13

I watched another lecture on differential equations by Prof. Arthur Matuck.



Lecture 13 is about finding solutions to the following ODE:
$y'' + Ay' + By = e^{\alpha x}$, with $\alpha$ a complex number.

The general solution is
$y = C_1 y_1 + C_2 y_2 + \frac{e^{\alpha x}}
{\alpha^2 +A\alpha + B }$

where
$C_1 y_1 + C_2 y_2 = y_h$ is a solution the homogeneous part of the ODE and

$\frac{e^{\alpha x}}{\alpha^2 +A\alpha +B} = y_p$ is a particular solution of the ODE.


( I have not blogged about lectures 10,11 and 12 although I have seen them. Lectures 11 and 12 in particular were highly theoretical but did not add much in terms of new definitions or theorems.)

Tuesday, November 16, 2010

[New bookmark] - OU on iTunes (2)

The Open University on iTunes. Just had a brief look, although I think it's a good thing that the OU is present on iTunes considering its appeal to so many. I don't like it personally, though. I don't like the iTunes program -at all-, and I don't want to be controlled by Apple. To the point, as I said, had a brief look and I found a lecture by Mark Ronan about his book Symmetry and the Monster. It's a book about Group Theory.

Mathematics => A blog - Bookmarks

M208 result expected at 17 dec 2010

Your module result should be available by Friday 17 December 2010.
(By e-mail from the Open University)

Monday, November 15, 2010

Before Pythagoras

An interesting exhibition in New York - "Before Pythagoras: The Culture of Old Babylonian Mathematics November 12 - December 17, 2010"

Visit the website of Before Pythagoras

Sunday, November 14, 2010

Orbit-Stabilizer Theorem

Let $G$ be a group, $X$ be a set, with $g \in G$, $x \in X$.

The number of elements in the orbit of $x$ is equal to the index of the stabilizer of $x$ in $G$:
$|\text{Orb}(x)| = [G: \text{Stab}(x)]$ ( Orbit-Stabilizer Theorem )

The total number of orbits is equal to the number of elements in $x$ fixed under an action of $g$, summed for all elements in G and finally divided by the size of $G$:
$|\text{Orb}| = \frac{1}{G} \sum_{g \in G} \text{Fix}(g)$ ( Counting Theorem )

Two theorems in which $\text{Orb}$ occurs but with a distinct different meaning.

Making study notes (2)

What I have in mind is a set of notes similar to ( but from memory and in my own words ) this set from prof. Kiyoshi Igusa on Group Theory. This is a clear 60-page summary of an undergraduate Group Theory course. It can't replace a book but it will help when you are reviewing. It gives you yet another view on the topic.

Producing PDF documents like this is very simple when you use LaTeX with a good LaTeX IDE. The work on presentation and lay-out issues is zero. This is why most mathematics books look more or less the same. Which is a good thing imo, just think of future generations accessing the vast amounts of new mathematics created in our time.

Saturday, November 13, 2010

Making study notes

I have been puzzling for quite a while on how to best make notes in a way that they are productive, i.e. help in getting a better understanding of the topic, as well as reusable.

A while back I came to the idea of building a linked repository of definitions and theorems in a Personal Brain database ( plex ). Although I will be using PB whenever I can, building a repository is not what it will be. As an activity of understanding the topic it may be effective but not efficient. - I have read, seen and heard it all over the place: what really makes you understand mathematics is 1: doing exercises, 2: teaching mathematics. Although I am not a teacher, thus have no students, I can pretend as such and prepare lectures. The lecture notes and slides become the end product, they are reusable and the math presented is well understood by me. It's an idea, worth trying.

I have been a tiny bit, actually. It suddenly daunts that you can choose the format of the theorem, fonts, Greek letters, symbols used, etc. But most of all, I can present it in the most beautiful and clear way I think is possible. - ( I am currently writing a technical note on Group Actions ). M208 stuff, I know, but it will return in M336.

[Fact] - How the Enigma cryptography machine works

A-hum... if you were expecting a short and simple explanation of the inner workings of the notorious Enigma then I have to disappoint you. If you insist in knowing though, there is a good cryptography book which does all the explaining. And the author decided to publish an on-line version of the book for free. The author is Nigel Smart and you will find his book on this page. You will need a postscript reader, you'll find the Enigma stuff in Chapter 4. The book is self-contained, which more or less means that it is a book at the undergraduate level. Download viewers and stuff at this site.

[TIP] - LaTeX for OpenOffice

I copied this pic from the site of OOLatex, which is a LaTeX extension for the Open Office Suite.

Click to enlarge
To be frank I have only just begun with exploring the possibilities to produce texts with ( LaTeX -) mathematics on Powerpoint ( or Impress ) format. - I have learned many times over that I shouldn't write with the presentation in mind. I have only recently implemented this simple rule here on this blog and the effort slowly begins to pay off. But that's another issue.

Friday, November 12, 2010

[Fact] - Wroskian of two functions

The Wroskian of two functions $f(x),g(x)$ is
$\left| \begin{array}{cc}
f(x) & g(x)\\
f'(x) & g'(x)
\end{array} \right| = f(x) \cdot g'(x) - f'(x) \cdot g(x)$.

(From MIT 18.03 lecture 11.)

[News] - London student protests and Open University fees

Students were protesting ( "rioting" according to mainstream media ) this week in London against a proposed trebling of student fees from 3000 to 9000 pounds.



P.S.
I wonder if the Open University fees will be increased equally. As a foreign student I pay 800 pounds per 30 points ( seriously more than UK citizens, which I think is fair but that doesn't mean I am happy with it ) or roughly 2400 pounds per year. I am not sure if I want to pay 7200 pounds a year because my mathematics study is a hobby, period. I am sure the same is true for many other students abroad. At the end of the day the increase won't add up to the loss in students. From my point of view I hope the Open University only slightly increases overall fees and drops the subsidies to locals.

Thursday, November 11, 2010

Poetry in mathematics.

While I was looking for 'poetry in mathematics', all I could find was this article in the Journal of on-line Mathematics and its Applications (volume 6 October 2006) by JoAnne Growney called Mathematics in Poetry.


As I find it hard enough to describe what I mean by Poetry in mathematics, in my native language Dutch, I won't even give it a try in English. But I would like to show this snippet of math as an example of what I consider deeply poetic. Although the context is math, to me this is poetry, period.

\[\begin{aligned}
1^2 &= 1^3\\
(1+2)^2&=1^3+2^3\\
(1+2+3)^2&=1^3+2^3+3^3\\
(1+2+3+4)^2&=1^3+2^3+3^3+4^3\\
(1+2+3+4+5)^2&=1^3+2^3+3^3+4^3+5^3\\
(1+2+3+4+5+6)^2&=1^3+2^3+3^3+4^3+5^3+6^3\\
\cdots &= \cdots
\end{aligned} \]

$(1,9,36,100,225,441,\cdots)$.

[TIP] - Copy MathJax LaTeX source

If you see mathematical equations in this blog then they are created using MathJax. If you move your mouse cursor over the equation and then right-click a little MathJax menu appears. You can for example copy the LaTeX source of the equation and use it in another application.

Wednesday, November 10, 2010

[News] - Video Lectures Algebraic Topology ( for Undergraduates )

Video lectures Algebraic Topology recorded at the university of Sydney, Australia.

Algebraic topology is the study of shapes. In the first lecture you will learn that a mathematician calls his donut a torus:

[News] - Springer.com releases LaTeX Search

The Springer LaTeX search lets you search through over 2,660,787 LaTeX code snippets to find the equation you need.
Try it here: http://www.latexsearch.com/

Tuesday, November 9, 2010

Received the prospectus 2010 - 2011

Received the prospectus 2010 - 2011 today. I like browsing through the prospectus although I know what I want. And that is more than I can possibly handle. So, I don't know what I will -do- yet. Planning is finding a balance between want and can I suppose.

An exercise in arithmetic

This tomb holds Diophantus. Ah, what a marvel! And the tomb tells scientifically the measure of his life. God vouchsafed that he should be a boy for the sixth part of his life; when a twelfth was added, his cheeks acquired a beard; He kindled for him the light of marriage after a seventh, and in the fifth year after his marriage He granted him a son. Alas! late-begotten and miserable child, when he had reached the measure of half his father's life, the chill grave took him. After consoling his grief by this science of numbers for four years, he reached the end of his life.

No tricks. A (simple) exercise in arithmetic encoded in a story about Diophantus. What is your answer?

Monday, November 8, 2010

Watched MIT 18.03 - lecture 9

I skipped lectures 7,8 (watched 7 partially and had a brief look at 8 ) because during this first serious confrontation with differential equations I want to follow the route of MST209 ( OpenLearn version ).

In 18.03 lecture 9 Prof. Mattuck talks about differential equations of type $y'' + Ay' + By = 0$. They are of the 2nd order, have constant coefficients and are homogeneous. The procedure for solving them is surprisingly similar to solving second order recurrence equations. The DE has a characteristic equation $r^2 + Ar + B=0$. If both roots are real and distinct the general solution then looks like $y=c_1 \cdot e^{r_1x} + c_2 \cdot e^{r_2x}$. The other two cases ( i.e. a pair of complex conjugates, two real equal roots ) are discussed during the rest of the lecture.

Sunday, November 7, 2010

Case: solving linear ODE with trig RHS

In this case we solve the differential equation $y' + 3y = \sin{x} + \cos{x}$. Note the extensive usage of complex numbers.

\[\begin{aligned}
y' + 3y &= \sin{x} + \cos{x}\\
e^{3x} \cdot y' + 3 e^{3x} \cdot y &= e^{3x}(\sin{x} + \cos{x}) \\
(e^{3x} \cdot y)' &= e^{3x}(\sin{x} + \cos{x})\\
(e^{3x} \cdot y)' &= e^{3x}\sqrt{2}\cos{(x-\frac{\pi}{4})}\\
(e^{3x} \cdot y)' &= \Re{\left\{ \sqrt{2} e^{(x-\frac{\pi}{4})i+3x}\right\}}\\
\int{(e^{3x} \cdot y)' \ dx} &= \Re{\left\{ \int{ \sqrt{2} e^{(x-\frac{\pi}{4})i+3x} \ dx}\right\}}\\
\int{(e^{3x} \cdot y)' \ dx} &= \Re{\left\{ \int{ \sqrt{2} e^{(3+i)x-\frac{\pi}{4}i} \ dx}\right\}}\\
e^{3x} \cdot y &= \Re{\left\{ \frac{\sqrt{2}}{3+i} e^{(3+i)x-\frac{\pi}{4}i} \right\}} + C\\
y &= \Re{\left\{ \frac{\sqrt{2}}{3+i} e^{(x-\frac{\pi}{4})i} \right\}} + Ce^{-3x}\\
y &= \frac{1}{5}\cos{x} + \frac{2}{5}\sin{x} + Ce^{-3x}
\end{aligned} \]

Step 1:
$y' + 3y = \sin{x} + \cos{x} \Leftrightarrow$
$e^{3x} \cdot y' + 3 e^{3x} \cdot y = e^{3x}(\sin{x} + \cos{x}) $
We want to multiply the LHS and RHS with a factor $p(x)$ such that $p(x) \cdot y + 3 p(x) \cdot y' = (y\cdot p(x))'$. This implies that $p'(x) = 3p(x) \Leftrightarrow p(x) = \int{3p(x) \ dx} \Leftrightarrow p(x) = e^{3x} + C$. For our purpose $C=0$ suffices.


Step 2:
$e^{3x} \cdot y' + 3 e^{3x} \cdot y = e^{3x}(\sin{x} + \cos{x}) \Leftrightarrow$
$(e^{3x} \cdot y)' = e^{3x}(\sin{x} + \cos{x})$
We implement our objective from the previous step. This step is allowed due to the product rule of differentiation: $(f\cdot g)'(x) = f(x)g'(x) + g(x)f'(x)$.


Step 3:
$(e^{3x} \cdot y)' = e^{3x}(\sin{x} + \cos{x}) \Leftrightarrow$
$(e^{3x} \cdot y)' = e^{3x}\sqrt{2}\cos{(x-\frac{\pi}{4})}$
We want the RHS to be a single trigonometric function:

\[\begin{aligned}
\cos{x}+\sin{x} &= \Re{(e^{ix})} + \Re{(-ie^{ix})}\\
&=\Re{(e^{ix}-ie^{ix})}\\
&=\Re{((1-i)e^{ix})}\\
&=\Re{(\sqrt{2}e^{-\frac{\pi}{4}i} \cdot e^{ix})}\\
&=\Re{(\sqrt{2}e^{(x-\frac{\pi}{4})i})}\\
&=\Re{(\sqrt{2}(\cos{(x-\frac{\pi}{4})}+i\sin{(x-\frac{\pi}{4})}))}\\
&=\sqrt{2}\cos{(x-\frac{\pi}{4})}
\end{aligned} \]


Step 4:
$(e^{3x} \cdot y)' = e^{3x}\sqrt{2}\cos{(x-\frac{\pi}{4})} \Leftrightarrow$
$(e^{3x} \cdot y)' = \Re{\left\{ \sqrt{2} e^{(x-\frac{\pi}{4})i+3x}\right\}}$
If we write the RHS again as the real part of a complex expression we can integrate a single exponential function which is preferable due to its simplicity.


Step 5:
$(e^{3x} \cdot y)' = \Re{\left\{ \sqrt{2} e^{(x-\frac{\pi}{4})i+3x}\right\}} \Leftrightarrow$
$\int{(e^{3x} \cdot y)' \ dx} = \Re{\left\{ \int{ \sqrt{2} e^{(x-\frac{\pi}{4})i+3x} \ dx}\right\}}$
An intermediate step before integration, the RHS needs tyding up in the next step before integration over x.


Step 6:
$\int{(e^{3x} \cdot y)' \ dx} = \Re{\left\{ \int{ \sqrt{2} e^{(x-\frac{\pi}{4})i+3x} \ dx}\right\}} \Leftrightarrow$
$\int{(e^{3x} \cdot y)' \ dx} = \Re{\left\{ \int{ \sqrt{2} e^{(3+i)x-\frac{\pi}{4}i} \ dx}\right\}}$
In this step we have rewritten the RHS so that it is clear how the integration has to be done.


Step 7:
$\int{(e^{3x} \cdot y)' \ dx} = \Re{\left\{ \int{ \sqrt{2} e^{(3+i)x-\frac{\pi}{4}i} \ dx}\right\}} \Leftrightarrow$
$e^{3x} \cdot y = \Re{\left\{ \frac{\sqrt{2}}{3+i} e^{(3+i)x-\frac{\pi}{4}i} \right\}} + C$
In this step we have performed a straightforward integration of the LHS and RHS.


Step 8:
$e^{3x} \cdot y = \Re{\left\{ \frac{\sqrt{2}}{3+i} e^{(3+i)x-\frac{\pi}{4}i} \right\}} + C \Leftrightarrow$
$y = \Re{\left\{ \frac{\sqrt{2}}{3+i} e^{(x-\frac{\pi}{4})i} \right\}} + Ce^{-3x}$
We divide LHS and RHS by $e^{3x}$.


Step 9:
$y = \Re{\left\{ \frac{\sqrt{2}}{3+i} e^{(x-\frac{\pi}{4})i} \right\}} + Ce^{-3x} \Leftrightarrow$
$y = \frac{2}{5}\sin{x} + \frac{1}{5}\cos{x} + Ce^{-3x}$
We rearrange the RHS as follows:

\[\begin{aligned}
y
&= \Re{\left\{ \frac{\sqrt{2}}{3+i} e^{(x-\frac{\pi}{4})i} \right\}} + Ce^{-3x}\\
&= \Re{\left\{ \frac{\sqrt{2}}{3+i} e^{(x-\frac{\pi}{4})i} \right\}} + Ce^{-3x} \\
&= \Re{\left\{ \frac{1}{3+i} \frac{3-i}{3-i}(1-i)e^x \right\}} + Ce^{-3x} \\
&= \Re{\left\{ (\frac{1}{5}-\frac{2}{5}i)(\cos{x}+i\sin{x}) \right\}} + Ce^{-3x} \\
&= \Re{\left\{ \frac{1}{5}\cos{x} + \frac{2}{5}\sin{x} +i(-\frac{2}{5}\cos{x} + \frac{1}{5}\sin{x}) \right\}} + Ce^{-3x} \\
&= \frac{1}{5}\cos{x} + \frac{2}{5}\sin{x} + Ce^{-3x}
\end{aligned} \]

This completes the explanation.

Saturday, November 6, 2010

Touching complex analysis

Prof. Mattuck is in top form in lecture 6 when he talks about Euler and the beauty of complex numbers. Although it is a lecture in the DE series it can be watched as a stand-alone lecture. So if you are doing MST121, MS221 or M208 it is great fun to watch this lecture. He even touches the field of Complex Analysis when he explains differentiating $e^{i\theta}$. He notes ( jokes ) that time is always a real variable but he isn't so sure when the next Einstein comes around: he may very well decide we need complex time! Anyway, how would you integrate $\int{e^{x}}\cos{x}\ dx$ ? Prof. Mattuck says these integrals are easy if you switch to the complex domain. Finally he solves the beautiful equation $x^n+1=0$ ( as we have seen in MS221, M208 ).



Hopefully, after or during MST209, I will be able to analyze the synchronization of metronomes problem using differential equations some day.

Friday, November 5, 2010

Registration before 22/12

Received important e-mail from Open University:

As a continuing student you may be aware that in previous years the cut off date for course registration was later than for new students. However, this has sometimes led to course materials arriving late, late starts on courses, and some students having to travel a long distance for tutor groups. There has been increasing student dissatisfaction as a result and as such the final enrolment date for all students - whether new or continuing - will be 22nd December 2010 for courses that start in February 2011. Registration by any means, including online and over the telephone, will not be possible after this date, so you should register promptly to avoid disappointment. The courses affected by this change are all undergraduate 30 and 60 point courses, including short courses where there is a 2-stage registration process. Exceptions will be all Law courses and all remaining short courses.

If you intend to apply for financial support, and you live in England, Wales or Northern Ireland, you will need to ensure that you have reserved on your chosen course, and linked it to financial support by the 1st December 2010, and in order to guarantee that it is assessed in time we need to receive your application form back no later than 8th December 2010.

This is a necessary step in order to manage student numbers and expectations, and your co-operation and understanding is appreciated.

To reserve or register on your next course please log onto your StudentHome Page or visit open.ac.uk/study. Please note that all dates and exceptions are correct at the time of this email; please check individual course descriptions at www.open.ac.uk/study for any further updates or exceptions. If you need any further help or information, please contact the Student Registration and Enquiry Service on +44 (0) 845 300 60 90.


Please do not reply directly to this message as communications to this mailbox are not monitored.
We are happy to help you with any questions or concerns that you may have.
If any email address is specified in the body of this message then please use that.
Otherwise, we can be contacted via http://www.open.ac.uk/contact/

Khan Academy

I was listening to one of those great talks by Lew Rockwell. He talked about some 530 million dollar school building in Los Angeles built to house 4,000 (!) high school students and which looked like a prison. Probably because it is a prison for these kids, he added. Lew mentioned that teaching like that is not at all like it should be with the ( technological ) possibilities we have today. Or if you like, with our ( read: the US's ) financial situation. He pointed out that it can be done differently. That it in fact -is- done differently by The Khan Academy, a not-for-profit 501(c)(3) with the mission of providing a world-class education to anyone, anywhere.

For me too? I thought, immediately. And yes, he recently added video lectures for a course in Differential Equations to the approximately 1800 video's already stored on their website. http://www.khanacademy.org/ Besides differential equations they also have an impressive set of calculus video's.

Since I already started the 18.03 lecture program ( competition breeds innovation ! ) I probably continue to do so but I'll be making comparisons.

Thursday, November 4, 2010

Quaternion group as a Freegroup

The Quaternion group can be defined as follows $\{(a,b) : a^4=e,b^4=e,a^2=b^2,ab=ba^3\}$. Let's be practical and set a=i, b=j and let's implement this in Mathematica.

The following Mathematica code

$r:=\{ \text{iiii} \to \text{""} , \text{jjjj} \to \text{""} ,\text{ii} \to \text{jj},\text{ij} \to \text{jiii}\}$

$f[\text{x_}]:=\text{StringReplace}[x,r]$

$\text{NestList}[f,\text{"ijij"},5]$

yields:

$\{\text{ijij},\text{jiiijiii},\text{jjjjiiijji},\text{jjjiiiji},\text{jjjjjjiiii},\text{jj}\}$


Let me explain. The variable $r$ is a list which contains four ( production ) rules, i.e.:
$\text{iiii} \to \text{""}$ - Says that $i^4 = 1$.
$\text{jjjj} \to \text{""}$ - Says that $j^4 = 1$.
$\text{ii} \to \text{jj}$ - Says that $i^2 = j^2$. And $i^2=-1$ as we know.
$\text{ij} \to \text{jiii}$ - Says that $ij = jiii$. Or $ij=-ji$.

The command $f[\text{x_}]:=\text{StringReplace}[x,r]$ takes a string as input and applies the production rules once from left to right. This command can be repeated until the input string no longer changes. In the case of $"ijij"$ it took $5$ times and the input and resp. outputs were as follows.

$\{\text{ijij}$,
$\text{jiiijiii}$,
$\text{jjjjiiijji}$,
$\text{jjjiiiji}$,
$\text{jjjjjjiiii}$,
$\text{jj}\}$

As you see $(ij)^2=k^2$ is correctly evaluated to $j^2=-1$.

From here on it's fairly easy to generate all elements from the quaternion group by string concatenation and applying the production rules.

Wednesday, November 3, 2010

Logistic Equation

Watched lecture 5 in the MIT 18.03 series


In this lecture differential equations were discussed of type $\frac{dy}{dx}=f(y)$, i.e. with no independent variable in the RHS of the equation, aka autonomous differential equations. He showed how to get qualitative information about the solution without actually solving the Differential Equation. The logistic equation ( remember MST121? ) was discussed and used as a model in several problems. Without actually solving the equation much could be said about the solutions.

I read Lesson 1 of MST209/OpenLearn. The title is 'First Order Differential Equations' and contains procedures for Euler's Method, Direct Integration, Separation of variables and the Integrating Factor method. The lesson also contains several exercises with answers. All very much like a normal Open University lesson.

Monday, November 1, 2010

Change of plan

Since I am still in the planning phase I can change the plan whenever I think the overall sitiatopm nemefirs

I changed my mind on the books I will use next to the MST209 materials. If I don't understand something in book A I have a look how it is explained in book B. Comparing books also helps in determining what's really important. I am not ready yet for 'Kelley, Peterson' so I am going to drop their book.

- Elementary Differential Equations Sixth Edition C. Henry Edwards David E. Penney ( Used in MIT 18.03 )
- J. David Logan A First Course in Differential Equations ( Springer book, formal at the undergraduate level )
- Differential Equations with Mathematica Third Edition Martha L. Abell James P. Braselton ( Very practical with tons of do-able exercises in Mathematica )
- Differential Equations Demystified Steven G. Krantz ( Lots of Recipes, Exercises and Solutions )

All in all a balanced set of books to complement the core MST209 materials, I suppose.

Watched MIT 18.03 videos 4.

I watched video 4 of MIT 18.03 again today, this time in its entirety.


In this video it is shown how Bernoulli equations and Homogeneous DEs' can be solved by using the right form of variable substitution. Differential Equations is a lot like Calculus in the sense that you have to practice a lot.

Sunday, October 31, 2010

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.

Saturday, October 30, 2010

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 )

Friday, October 29, 2010

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. )

Thursday, October 28, 2010

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:

Wednesday, October 27, 2010

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.

Sunday, October 24, 2010

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

Saturday, October 23, 2010

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

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