brain activity log
07.04.2006 Friday 14:54 - be afraid
You can't hold me down forever
What doesn't kill me makes me strong
"The edge" - Annihilator
06.04.2006 Thursday 09:00 - targeting london
Nice news from Iakko. Plan confirmed and finalized: we're tripping to London in late May.
The curious note is that the tube station near our hotel has been a target
of a recent terroristic attack. The probability of two consecutive terroristic attacks
on the same target is very low: we're in a statistically safe place :)
05.04.2006 Wendesday 23:10 - ehm...
(still with respect, obviously)
Raga... un posso. Tregua. :D
05.04.2006 Wendesday 11:00 - the things you see
When I look at the edge
Oh it scares me.
Though I know I'll go,
always go back.
If there was somewhere to go,
my boat wasn't holed on the reef.
And my lack of belief
in the truth I am seeing...
To you I'll tie
my inner eye
which never, ever
"The things you see (when you haven't got your gun)"
04.04.2006 Tuesday - from fixed points to butterfly effect
A fixed point of a function is a point that is mapped to itself by the function.
For example, if f is defined on the real numbers by f(x) = x^2 - 3x + 4,
then 2 is a fixed point of f, because f(2) = 2.
Not all functions have fixed points: for example, the function f(x) = x + 1 has no fixed point on the reals,
since x is never equal to x + 1 for any real number.
In many fields, equilibrium or stability are fundamental concepts that can be described
in terms of fixed points. For example, in economics, a Nash equilibrium of a game is a
fixed point of the game's best response correspondence.
In compilers, fixed point computations are used for whole program analysis,
which is often required to do code optimization. The vector of PageRank values of
all web pages is the fixed point of a linear transformation derived from the World Wide Web's link structure.
The solutions of f(x) = 0 can be found by an iterative fixed point research.
An attractive fixed point of a function f is a fixed point x0 of f such that
for any value of x in the domain that is close enough to x0, the iterated function sequence
x, f(x), f(f(x)), f(f(f(x))), ... converges to x0. How close is "close enough" is sometimes a subtle question.
Attractive fixed points are a special case of a wider mathematical concept of attractors.
In dynamical systems, an attractor is a set to which the system evolves after a long enough time.
For the set to be an attractor, trajectories that get close enough to the attractor must remain
close even if slightly disturbed. Geometrically, an attractor can be a point, a curve, a manifold,
or even a complicated set with fractal structures known as a strange attractor. Describing the
attractors of chaotic dynamical systems has been one of the achievements of chaos theory.
An attractor is informally described as strange if it has non-integer dimension or if the dynamics
on the attractor are chaotic. Strange attractors are often differentiable in a few directions and like a
Cantor dust (and therefore not differentiable) in others.
The Henon attractor and the Lorenz attractor are examples of strange attractors.
The Lorenz attractor, introduced by Edward Lorenz in 1963, is a non-linear three-dimensional
deterministic dynamical system derived from the simplified equations of convection rolls arising
in the dynamical equations of the atmosphere.
The butterfly-like shape of the Lorenz attractor may have inspired the name of
the butterfly effect in chaos theory.
The butterfly effect is a phrase that encapsulates the more technical notion of sensitive
dependence on initial conditions in chaos theory. Small variations of the initial condition
of a dynamical system may produce large variations in the long term behavior of the system.
This is sometimes presented as esoteric behavior, but can be exhibited by very simple systems:
for example, a ball placed at the crest of a hill might roll into any of several valleys depending
on slight differences in initial position.
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