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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.



03.04.2006 Monday - windy stop (with a smile)

:)



02.04.2006 Sunday - diary and some goodies

Yesterday we have been in Florence and returned really late. Couldn't sleep this morning, so woken up early. In the first minutes thought that because of the missing hours of sleep and because of recent developments it would be a bad day. ...but after opening the window discovered that there was a fantastic Sun outside: been waiting for this since several months.

Never undestimate the energy emitted by the Sun: a ball of plasma with a diameter of 1.392 million km and a mass of about 2.0x10^30 kg. Approx 5500 K at surface... ... man ... that's HOT :D

Etherea sent me an english translation of my CV. Thnx Ethy :******

In her blog I have also found this funny image of Iakko :D

(Hum.. Cle... there is no way to link your blog directly without loosing the external frame)

While getting my coffee I've added a couple of comparison operators to the SSMatrix class. The scalar ops are still missing, but the sun is calling me thus I'll add them the next time :). I've also upped the first SSComplex template that can be used with SSMatrix. Enjoy :)

It seems that there is some new movement around KVirc.

For chronicle purposes I also to need to record the following (important) note.

Segmentation fault. Core dumped.

"Hum... still need some debugging"



01.04.2006 Saturday - April Fools

1st April. In many countries this is the day of "Jokes". "Prima Aprilis: uwazaj bo sie pomylisz".. which is in Polish and means "1 April, be careful because you might be wrong". "April Fools" and "Pesce d'Aprile". The (net-)jokes already started.

Heh.. after you read some of them, all the of the web starts to look like a joke :D

An Insightful post on slashdot stated:

Try date -u and you will see that it is in fact April Fool's Day.

Note that April Fool's Day, as defined by the International April Pranksters Association, goes by UTC, not by local time zones. Because IAPA is not widely recognized as an international standards body and many people's problems to understand time zones, this has led to some problems since its introduction in 2002. It is especially uncertain for regional publications, which are reluctant to adopt the new standard because they fear to irritate their audience. One example is the Hubsborough Gazette, which famously spread confusion on the evening of March 31st, 2004 (EST) when an article claiming that aliens have attacked the Whitehouse appeared on their website. Despite the seemingly obvious nature of the hoax, many believed it and called the authorities or local clergery for guidance. One family even is reported to have spent two weeks in their backyard bunker. Since then, the editor has announced that they will only publish April Fool's articles during the hours when April 1st of their local time zone and UTC overlap, and take down articles afterwards. Many publications have followed their example in the following years.

The guys from Centrica fooled me about bex2 emitting all zero size output... that would be a problem. It is a 1st April joke..isn't it ? :D



31.03.2006 Friday - diary and other goodies

Hum.. a lot of images (in random order):

  • The Diophantine Equation
  • A nearly black cat
  • The Youla-Kucera theorem (kinda hard proof for the multivariable case)
  • A pair of sweet lips :)
  • The missed Frantz sister's party: mea culpa :(
  • Francesca in the lecture room
  • Silvia pressing for messaging
  • Samuel spending up to the last cent for a drum set
  • The two girls near Skile tripping three times around the block
  • A sensation close to deja-vu
  • Iakko laughing hard after drinking 4 Hoegarden glasses :D
  • ...
Nice four days :)

The promised LU decomposition code is here. Pretty straightforward once the formula is known. The file contains also the algorithms for matrix determinant (computed either by LU or by a recursive adaptation of the Liebnitz fomula) and for matrix inversion. Enjoy :)

"I have a monster in me and if I wake him up he is probably going to kill me"



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