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Cathedral RESIDENTS
..again: lot's of interesting mail today...

21022006

From The Nonlinear Workbook by Willi-Hans Steeb ( ISBN 981-256-278-8 ) :
Chapter 1

One-dimensional maps

1.1 Exact and Numerical Trajectories

map f: N-->N defined by f(x)=x/2 if x is even, 3x+1 if x is odd
In this file there are two illustrations of this formula. One directly below, made in Flash, the other one after the intro, a plain number crunching version using a Java Server Pages spaghetti of java and html code.

[Warning: the 'music' you are about to hear is rather irritating - mind your neighbours]

MINENONEMANEMU

This flash animation moves to one of twelve frames dependent on the value of X%12 until the period orbit ...421421... is reached. While moving towards its period orbit it also stores the values in an array.

Upon reaching the period orbit, this array is used to consecutively launch one of twelve samples according to the value stored. Since the length of each sample exceeds the time elapsing between each launch, the samples overlap, making for the weirdish polyphony you'll hear playing for a while after the orbit has been reached. The samples i made with the MBROLA speech synthesizer.

The composer in the pictures is Dimitri Shostakovich (1906-1975) , it should have been Schoenberg perhaps, although the dodecastuff is phony ( i use a system of melodies instead of notes).
It had to be Sjostakovich because my sister wrote poetry on him, not on Arnold.

MBROLA .pho file of the MINENONEMANEMU samples
Their only differentiation is in the pitch factor:12x an increase of 0,2 from 1 to 3,4
_ 169 0 95 89 70 98 114
m 53 85 125
i 134 11 124 28 127 38 128 65 124
n 99 12 123 25 129 62 130 75 135 91 136
e 139 22 136
n 89 4 145 40 146 63 146 84 151
o 167 8 164 21 165 38 168 90 167
n 111 26 163 53 151 70 145 86 137
e 169 2 122 17 125 32 125 50 130 100 130
m 100 16 125 35 125 48 129 75 130
a 174 9 137 20 137 34 142 55 142
n 63 46 140
e 163 8 122 34 125 69 125 77 122 89 122 98 118
n 103 48 115 67 122
u 393 4 125 11 129 16 126 20 128 24 130 28 126 58 127 66 130 70 127 92 127 96 71
_ 250 86 70


<UNFOLD>

Nonlinearity

From Wikipedia, the free encyclopedia
(Redirected from Nonlinear)
Jump to: navigation, search

This article describes the use of the term nonlinearity in mathematics. For other meanings, see nonlinearity (disambiguation).

In mathematics, nonlinear systems represent systems whose behavior is not expressible as a sum of the behaviors of its descriptors. In particular, the behavior of nonlinear systems is not subject to the principle of superposition, as linear systems are. Crudely, a nonlinear system is one whose behavior is not simply the sum of its parts.

Linearity of a system allows investigators to make certain mathematical assumptions and approximations, allowing for easier computation of results. In nonlinear systems these assumptions cannot be made. Since nonlinear systems are not equal to the sum of their parts, they are often difficult (or impossible) to model, and their behavior with respect to a given variable (for example, time) is extremely difficult to predict.

In nonlinear systems one encounters such phenomena as chaos effects, strange attractors, and freak waves. Whilst some nonlinear systems and equations of general interest have been extensively studied, the vast majority are at best poorly understood.

 

The function in grey above is a simple linear function, utterly predictable. Given a certain input, a real number, it will create a map of real numbers leading to its period orbit ...421421... by taking its output as new input for the same equation. This happens in fully predicatable manner. Nothing weird or nonlinear about it.



In the JSP example below i use this function 12 times,
repeating the iteration to create a rather silly image of time passing by as a virtual discussion takes place on what is more important: Code or Art (inspired by a rhetorical question raised in a recent discussion on the Rhizome Raw list).You get all these numbers cruched out and at the end there's a print command line printing a statement going either way.

Now that's a weird thing to do, mathematical people don't do this, because you can't learn anything from it, mathematically. It doesn't make it art either, perhaps it doesn't even make any sense, but i like to think it's a start of familiarising ourselves with what math can do in an arty context.

So there's no answers here and hardly any art, but i took it as a starting point for yet another string of things inside the Cathedral. Somehow i feel i have to learn about things like the Ruelle-Takens-Newhouse Scenario, Hopf Bifurcations and Heigway's Dragon, so if you're interested too, i'd be glad to take you along my dummie trail. Like most of you, i know next to nothing on these subjects, but I do know a tiny bit of coding so i 'll be very explicit about how to translate all the Mathematical Beauties to 'real world' java examples.

Sure, this takes place within the Cathedral, so you may expect some irregularities along the path, and as you should know by now, nothing here gets completed or substantial without it proving it's worth to Our Lady Cathedral-Mother Dear. In other words, as usual, i start these things on stupid impulse, & no one can tell what may come of it. Now i know there do exist some good introductions out there, but i can't bring myself to going through all that, just reading it. I need to be doing something with it, otherwise i just clean forget everything i read. That, i think, about covers the Why of things.

There's a boring start to all subjects, so here's the 'Hello World' of (Non)linearity: the one-dimensional maps below take the instant of their creation as an input (request time of this file + the time it takes to compute the previous maps), effectively showing their computation takes time. Wow, spec - ta - cu - lar!

Well, at least it clearly shows that linearity in functions doesn't lead to predictable outcomes when applied in the real world. Time is a nasty bugger and we need to keep a clear picture of things as they happen. Math is math, code is code, running code is running code and people discussing the real world all need to be addressed on their own plane of consistency. As you can see, a lot of things can happen between those spaces in a split second.

Finally, visualisations like these may not give you a better understanding of the mathematics that underlie them, and math itself doesn't need it much or not at all, but they may be a welcome alternative to the tempting images of mystification that come to (my) mind (at least) when i think of the names mentioned above or hear someone mentioning them in real world language that he or she claims matters to me as well. If i don't understand, at least i can see what i don't understand. So perhaps this coding business is about a very private form of ownership or lack thereoff, after all.

</UNFOLD>

© dv 2006 - Free Art License - L.M. December 31, 2016 11:32 PM

<%@ page contentType="text/html;charset=utf-8" language="java" import="java.math.*"%>

<%

// first use a boolean to alternate tables

boolean mod=true;

StringBuffer sb=new StringBuffer();

for (int k=6;k>0;k--){

if(mod){

mod=false;

%>

//in one html table give results for f(x) with original input the time of the users request in Milliseconds sinds january 1 1970 descending to when the function reaches the period orbit ...421421...

Date d=new Date();

String tijd=""+d.getTime();

BigInteger X= new BigInteger(tijd);

BigInteger TWO= new BigInteger("2");

BigInteger THREE= new BigInteger("3");

int T=350;

for (int t=0;t<T;t++){

 if((X.mod(TWO)).equals(BigInteger.ZERO)){

X=X.divide(TWO);

}else{

X=X.multiply(THREE);

X=X.add(BigInteger.ONE);

}

sb.append(X+"::");

out.print(X+"<br>");

}

<%

//in another table give the same results ascending

for (int i=349;i>0;i--){

out.print(sb.toString().split("::")[i]+"<br>");

}

%>

<%
}else{
mod=true;
%>

<%

//turn around the sequence of tables -first use what's still in the StringBuffer for descending output at the end of this table be sure to empty the StringBuffer thus

sb.delete(0,sb.length());

//insert the art-code stuff anywhere between the tables

//because the computation takes quite some time, you get different dates to refer to (ususally one second will elaps, if not, someone upgraded my server). Do not do this (a lot) on your site, the application server does not like all that work.

//if you refresh the page a few times, you'll see that for some amount of milliseconds (the input), the function reaches its period orbid, for others it doesn't. It would, eventually if you change the iteration from 350 to sth higher, say 700.

//in the next episode we'll start using Processing to avoid all of this Server Stress. Surprisingly, the user's computer can count too, so we don't need Big Mama. & JSP spaghetti is nice because it looks more complicated, but it's also unnecessarily confusing...

}//you opened a for loop up there, remember?
%>

CODE|ART
is not a question

Relevant Math articles in Wikipedia

http://en.wikipedia.org/wiki/Nonlinear

http://en.wikipedia.org/wiki/Periodic_function

<%boolean mod=true; StringBuffer sb=new StringBuffer(); for (int k=6;k>0;k--){ if(mod){ mod=false;%>

 

 

<% Date d=new Date(); String tijd=""+d.getTime(); BigInteger X= new BigInteger(tijd); BigInteger TWO= new BigInteger("2"); BigInteger THREE= new BigInteger("3"); int T=350; for (int t=0;t"); } %>
<%for (int i=349;i>0;i--){ out.print(sb.toString().split("::")[i]+"
"); }%>
<%}else{ mod=true;%>

<%for (int i=349;i>0;i--){ out.print(sb.toString().split("::")[i]+"
"); }%>
now
is
the
time
is
now

<% Date d=new Date(); String tijd=""+d.getTime(); out.print(" "+d+" & code is more important than art
"); //StringBuffer sb=new StringBuffer(); BigInteger X= new BigInteger(tijd); BigInteger TWO= new BigInteger("2"); BigInteger THREE= new BigInteger("3"); int T=350; for (int t=0;t"); } sb.delete(0,sb.length()); out.print(" "+d+" & art is more important than code
"); %>

 

<%} }%>

If you have the time, please place comments on the no comments place please
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