Donnerstag, 23. März 2017

Summation formulas / Finite calculus

I made this discovery a couple of years ago. Today I recalled it and thought it might be appropriate to mention it in this blog.

From school we know this formula:

(I) 1 + 2 + 3 + ... + n = (n + 1) * n / 2

I discovered that apparently similar approaches work for the following summations:

(II) 1 * 2 + 2 * 3 + 3 * 4 + ... + (n - 1) * n = (n + 1) * n * (n - 1) / 3

(III) 1 * 2 * 3 + 2 * 3 * 4 + 3 * 4 * 5 + ... + (n - 2) * (n - 1) * n = (n + 1) * n * (n - 1) * (n - 2) / 4

And so on.

Every single equation can probably be proven by mathematical induction, but is there a way to prove it for the general case? That is:

(IV) Sum (i = 1 to n - k + 1) of Product (j = 0 to k - 1) of (i + j) = 1 / (k + 1) * Product (i = 1 to k + 1) of (n + 2 - i)

I posted about this to a forum and got informed about the tutorial on finite calculus available at:

https://www.cs.purdue.edu/homes/dgleich/publications/Gleich%202005%20-%20finite%20calculus.pdf

It shows that you can compute finite differences in a similar fashion as differentials, with the only difference being the use of falling powers.

So if f(x) = n * (n - 1) * (n - 2) = falling(n, 3), delta(f(x)) = falling(n + 1, 3) - falling(n, 3) = (n + 1) * n * (n - 1) - n * (n - 1) * (n - 2) = 3 * falling(n, 2). That means, falling(n + 1, 3) = falling(n, 3) + 3 * falling(n, 2) = falling(n - 1, 3) + 3 * falling(n - 1, 2) + 3 * falling(n, 2) and so on. Since falling(1, 2) = 0, we ultimately get falling(n + 1, 3) = 3 * (falling(2, 2) + falling(3, 2) + ... + falling(n, 2)). This is equivalent to formula II. Formula III can be proven in an analogous fashion, as well as - ultimately - formula IV.

Samstag, 11. März 2017

A note about Jungian functions

To explain further what I wrote about in my posting "Hugi bullshit", let me give you this example:

In a lecture I learned yesterday that 80 general practitioners were trained in Vienna in the past couple of years while there is a demand for 180 general practitioners. These are facts; just looking at the bare facts is what Jung calls "sensing". If you employ the function "thinking" according to Jung, you will come to the conclusion that the demand for general practitioners in Vienna is not met because too few general practitioners are trained. Now "intuition" is what gives significance and meaning to this observation: Apparently there is a shortage for general practitioners, which might lead to a change of the status of this profession. While general practitioners used to be considered "footmen of medicine", their increasing scarcity might increase the prestige of their profession and might ultimately even make them a new elite among medical doctors.

"Hugi bullshit"

I have bumped into an old debate in an online board about an article written by the infamous EP. A scener who read this article called it "Hugi bullshit".

Indeed, a lot of articles in Hugi are of philosophical nature, containing speculations about the nature of demos and trying to give demos a meaning. That does not appeal to everybody. This sort of articles exhibit the psychological function which Swiss pioneer of psychotherapy, Carl Gustav Jung, called "extroverted intuition". Since according to Jung and his followers only 20% of the population actively use intuition, it is possible that 80% of demosceners are not interested in this sort of articles.

The editor of the Zine diskmag, Axel of Brainstorm, said himself that Hugi articles contain "too much speculation" in his eyes and he has focused on presenting straight facts. It is a different style of diskmag, for a different style of personality. Some prefer the kinds of Hugi, others prefer the kinds of Zine. Both ways have their justification.

Freitag, 10. März 2017

Playing Video Games as an Adult

As a child, I played a lot of video games. As an adult in my thirties, there are still a couple of games I enjoy actively playing or watching videos of. I do not do that often any more, but when I try to think of what are my favourite games from today's perspective, these titles come to my mind:

Console:
- Shining Force 2 (Sega Mega Drive)
- Shining Force CD (Sega Mega CD)
- Top Gear (Super Nintendo)
- Terranigma (Super Nintendo)
- The Legend of Zelda: A Link to the Past (Super Nintendo)
- Streets of Rage 2 (Sega Mega Drive)
- Mighty Morphin Power Rangers: The Movie (Sega Mega Drive)
- Pocky & Rocky (Super Nintendo)
- Pocky & Rocky 2 (Super Nintendo)
- Mega Man: The Wily Wars (Sega Mega Drive)
- Sonic the Hedgehog (Sega Mega Drive)
- Sonic the Hedgehog 2 (Sega Mega Drive)

PC:
- Civilization VI
- Civilization IV: Beyond the Sword

Montag, 6. März 2017

Die Persönlichkeitstheorie von Uwe Rohr

Oft habe ich mit Dr. Uwe Rohr über die Unterschiede zwischen Normal- und Hochbegabten gesprochen. Das war eines seiner Lieblingsthemen, weil er sich als mutmaßlich Hochbegabter von den Normalbegabten in seinem Fach diskriminiert gefühlt hat. Dabei hat er aber immer nur von der Begabung gesprochen, auch wenn er eigentlich die Persönlichkeit meinte. Nach Uwe Rohr gäbe es nur zwei Typen von Menschen, eben Normal- und Hochbegabte. Diese würden in ihren jeweiligen Welten leben. Während sich die Welt der Normalbegabten um Dinge wie Einkommen, Prestige und Macht drehe, seien Hochbegabte eher an Ideen, Konzepten und Innovationen interessiert.

Ich finde, Uwe Rohr hat bei seinen Überlegungen zu wenig darauf geachtet, dass die Differentielle Psychologie nicht nur Intelligenzunterschiede zwischen den Menschen postuliert, sondern auch verschiedene Persönlichkeitstypen. Was Uwe Rohr den Hochbegabten zuschrieb, deckt sich weitgehend mit dem Rational-Temperament nach Keirsey. Die Normalbegabten nach Uwe Rohr sind hingegen nach Keirsey Guardian.

Es kann schon sein, dass der durchschnittliche Intelligenzquotient unter Rationals höher ist als unter Guardians. Schließlich stellen Guardians rund 80 Prozent der Bevölkerung, Rationals dagegen nur zehn, und die Mehrheit der Bevölkerung ist eben hinsichtlich ihrer kognitiven Begabung im Bereich der Normalbegabung (IQ zwischen 85 und 115). Aber es kann auch Guardians mit Intelligenzquotienten über 130 geben, ebenso Rationals mit durchschnittlichem IQ, und dennoch würde ein durchschnittlich intelligenter Rational eher dem entsprechen, wie Uwe Rohr sich einen Normalbegabten vorstellte, als ein hochintelligenter Guardian.

Wenn Uwe Rohr von der Benachteiligung der Hochbegabten durch die Normalbegabten spricht, dann mag es sich also in Wirklichkeit um eine Benachteilung von Rationals durch Guardians handeln. Diese könnte tatsächlich vorliegen, man denke nur daran, dass Guardians häufig die Neigung haben, andere Persönlichkeitstypen als pathologisch zu betrachten.