segunda-feira, 13 de dezembro de 2010

Diversification vs. Unification

Fortunately, if a condition of vocabulary balance does exist in a given sample of speech, we shall have little difficulty in detecting it because of the very nature and direction of the two Forces involved. On the one hand, the Force of Unification will act in the direction of decreasing the number of different words to 1, while increasing the frequency of that 1 word to 100%. Conversely, the Force of Diversification will act in the opposite direction of increasing the number of different words, while decreasing their average frequency of occurrence towards 1. Therefore number and frequency will be the parameters of vocabulary balance.

Since the number of different words in a sample of speech together with their respective frequencies of occurrences can be determined empirically, it is clear that our next step is to seek relevant empiric information about the number and frequency of occurrences of words in some actual samples of speech.

(George K. Zipf, Human Behavior and the Principle of Least Effort, 1949)

Double superlatives

Yet when we offer a prize to the submarine commander who sinks the greatest number of ships in the shortest possible time, we have a double superlative -- a maximum number and a minimum time -- which renders the problem completely meaningless and indeterminate, as becomes apparent upon reflection. Double superlatives of this sort, which are by no means uncommon in present-day statements, can lead to a mental confusion with disastrous results.

As pointed out years ago, the frequent statement, "in a democracy we believe in the greatest good for the greatest number," contains a double superlative and therefore is meaningless and indeterminate. (In Part Two we shall see that the distribution of goods and services are in fact governed by a single superlative.) Intimately connected with the "singleness of the superlative" is what might be called the singleness of the objective whose implications are often overlooked (i.e., the pursuit of one objective may preclude or frustrate the pursuit of the scoond objective). These two concepts apply to all studies in ecology.

(George K. Zipf, Human Behavior and the Principle of Least Effort, 1949)

segunda-feira, 6 de dezembro de 2010

Mandelbrot Set

The Mandelbrot set M is defined by a family of complex quadratic polynomials
$P_c:\mathbb C\to\mathbb C$
given by
$P_c: z\mapsto z^2 + c$, where c is a complex parameter.

If the sequence $(0, P_c(0), P_c(P_c(0)), P_c(P_c(P_c(0))), \ldots)$ (starting with $z=0$) does not escapes to infinity, the complex number c is said to belong to the set.

A simple Octave/Matlab code was created to plot the complex numbers on the Mandelbrot set.

for k1=1:length(x),
for k2=1:length(y),
for it=1:100,
z = z^2+c;
if(z < Inf), k++; mset(k)=c; end;
figure; plot(mset,'k.');