Monthly Archives: November 2010

The I-V-VI-IV chord sequence

It is a surprising fact how many pop songs are based on this ostinato (in the refrain, in not in the whole song). The same holds for the rotated sequence with regards to the minor scale: the sequence yields C,G,a,F in major and a,F,C,G in minor, i.e.  I-VI-III-VII. Some people have uploaded mixes in YT that compile many of —but, alas, not all— these songs in sequence or simultaneously!

Today I was listening to a pop CD with songs of an old friend of mine, and —what a surprise— one of the best songs in it is based on I-V-VI-IV at the refrain.

Why is this sequence so popular? Everything that is made on it seems to have a hit-potential. Is it something like the normal distribution? Does everything converge to this?

The black holes of the email communication

I would define an email black hole (EBH) as an email address or a website, who receives email about an issue but does not answer (to the surprise of the sender). In this case, an EBH is the opposite of a spam server: it absorbes unexpectedly instead of emitting unexpectedly (or in the language of vector fields, spam functions as a source and EBH as a sink).

Some examples from my personal experience (all in Berlin):

  • Once I wanted to go to the dentist of the university clinic. I used the email option of the clinic in order to make an appointment. They never answered.
  • Once I wanted to complain about the problematic delivery of a newspaper and used their email form. I was never informed if anybody read it.
  • Once I wanted to see a lawyer and used her email form. She never replied.

Which consequences has this behaviour? That almost everybody offers an email contact option (no matter if they actually support this option), because they don't want to give the impression that they don't move on. And one liar brings millions of others behind him.

A nice anonymity test

Many people use the Internet and supply numerous webpages with much private information in 2 ways: consciously and unconsciously. Former and latter are huge topics. Here I would like to say something about the latter.

The unconscious supply of information happens because the average user does not know

  1. what the applied software and hardware of his/her computer actually does;
  2. what the visited server actually does.

Of course, in order to know exactly what happens, one has to be an IT-specialist. But some basic things should be clear. Not only because the surfer can put himself in danger, but also because too many naive surfers can put in the long term the non naive in danger, or —at least— make their life harder.

A nice anonymity test is offered by JonDos GmbH here. Green is good, red is bad.

A Greek Paradox (out of many)

Many people are happy about the result of the local body elections in Thessaloniki. They hope that the new Mayor will make the city more human friendly. I don't hope, I am sure he will. More human friendly. But until Thessaloniki becomes really human friendly, we will have to wait some decades (I would estimate —at stable and continuous improvements— about 40-50 years), because the problems that caused the current situation are active for many years.

I will not write here anything about the gigantic problems of Thessaloniki, but about a fact that is totally absurd and functions as an obstacle to the democratic desicions in Greece.

In Greece, the place where one lives and the place where one votes do not have to be the same. E.g. a person who is raised up in place A and then moves to study or work to place B remains officialy inhabitant of A, unless he goes through the fight with the bureaucratic dragon and moves his electorial rights as well. The result is, that people travel on the election day back to their root-village or city, they see old relatives and friends, they vote, and the whole process gets a character of feast. Of course, if they decide to travel. Sometimes these journeys are quite long, and not all people have the desire to spend energy, time, security and money in order to have lunch with old buddies and vote for a person that has nothing to do with their lives. Thus: a huge number of people remain at home and just don't vote, or they vote at a place where they don't live!

On the other hand, Greeks that live abroad do not have the option to elect at the embassy.

Who can fight with Bureaucracy? Well, who wants to?

Project Euler

Project Euler is a project that offers mathematical problems that should be solved by computer programming. The problems are easy to understand and a theoretical solution is always obvious. But... a solutions is regarded as acceptable, if the algorithm that is applied lasts at most 1 minute. And of course the problems have such parameters that make brute force inefficient. So the ambitious solver has to invent good ideas or look for some in mathematical books, that solve the problem quickly.

Solving or just dealing with a problem makes much fun, because one experiences the power of mathematics and recalls old or learns knew knowledge.

Recommended to everyone who loves mathematics and programming!

Here is the webpage.

The Adventure of π to the Normal distribution

Today, after giving the definition of the normal distribution a student asked me a very interesting question: how does the number π emerge in the Gaussian curve? The question is legitimate, because the Gaussian curve does not show any relation to the circle curve. I improvised an brief answer, and I will try to give here the path of this fascinating adventure of π. Firstly, the distribution function of a normal distributed random variable with expectation μ and variance σ2 is

\displaystyle\Phi(x)=\int_{-\infty}^{x}\frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(t-\mu)^2}{2\sigma^2}}dt

and its limit is

\displaystyle\lim_{x\rightarrow\infty}\Phi(x)=\int_{-\infty}^{\infty}\frac{1}{\sqrt{2\pi\sigma^{2}}}e^{-\frac{(t-\mu)^{2}}{2\sigma^{2}}}dt=1

This is a normalised form of the Poisson integral:

\displaystyle\int_{-\infty}^{\infty} e^{-x^2}dx = \sqrt{\pi}

Now the question is why this equals  \sqrt{\pi} . For this I would give 2 answers:

1. Path

Substitution x2=t gives

 \displaystyle\int_0^\infty e^{-x^2} dx =\frac1 2\Gamma \left(\frac 1 2\right) = \frac{\sqrt{\pi}}{2}

And how is \Gamma \left(\frac 1 2\right) = \sqrt{\pi}?

This can be explained over the Gauss form of the Gamma function:

 \displaystyle\Gamma (x) = \lim_{n\rightarrow\infty}\frac{n!n^{x-1}}{\prod_{i=0}^{n-1} x+i} = \lim_{n\rightarrow\infty}\frac{n!n^{x}}{\prod_{i=0}^{n} x+i}

These 2 different forms give at ½ when multiplied with each other the double of the Wallis product:

 \displaystyle 2\lim_{n\rightarrow\infty} \prod_{k=1}^n \frac{k^2}{k^2 - \frac 1 4}=\pi

Now, why does W(n):=\frac{k^2}{k^2 - \frac 1 4} \prod_{k=1}^n converge to \frac \pi 2 for n\rightarrow\infty (Wallis product)? This is because of the recursive relation

 \displaystyle \int_0^{\frac \pi 2} \sin^nxdx = \frac {n-1} n \int_0^{\frac \pi 2} \sin^{n-2}xdx

The recursive resolution of the terms of the following fraction yields:

\displaystyle1=\lim_{n\rightarrow\infty}\frac{\int_0^{\frac \pi 2}\sin^{2n+1}xdx}{\int_0^{\frac\pi 2}\sin^{2n}xdx }=\lim_{n\rightarrow\infty}W(n)\frac 2 \pi

and therefore the equation of the Wallis product.

We see here that π comes from the sine function to the approximation of the Wallis product, then as the value of Γ(½) and from there it evaluates the square of the Poisson integral. A long path!

2. Path

The second path that I will present is shorter and more direct. Because

\displaystyle\left(\int_{-\infty}^{\infty} e^{-x^2}dx\right)^2=\int_{R^2} e^{-x^2-y^2}dxdy

calculating the left side should give the expected result (π). This can be done by integrating over a circle with centre in 0 and radius a (B(a)) and then by transformation to polar coordinates:

\displaystyle\lim_{a\rightarrow\infty}\int_{B\left(a\right)}e^{-x^{2}-y^{2}}dxdy=\lim_{r\rightarrow\infty}\int_{0}^{2\pi}\int_{0}^{r}re^{-r^{2}}drd\phi=\pi


References:

Forster Otto: Analysis 1; Differential- und Integralrechnung einer Veränderlichen, 6,. verbesserte Auflage, o.O., vieweg, 2001.

Κουνιας Στρατής, χρόνης Μωυσιάδης: Θεωρία Πιθανοτήτων Ι, Κλασική πιθανότητα, μονοδιάστατες κατανομές, Θεσσαλονίκη, Εκδόσης Ζήτη, 1999.


PS. Wikipedia has an interesting article about the calculation of the Gaussian integral.

Long-term projects

Spare time is for me the time during which I do not earn money or look for jobs. Because this is extremely small, projects that I undertake in my spare time take months or years to end.

As I do not earn money from composing music or writing texts, such projects belong to my spare time. At some unusual days the productivity reaches peaks like yesterday: I wrote down the Tetris-text (in Greek) whose title was posing there for years and uploaded 3 posts (in English) in The Composing Adventure (Phase 1, Phase 2, Comment). I hope that the latter will be ready by the end of November...