Questions tagged [iterated-function-system]

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-2
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0answers
44 views

Union of attractors (Fractals)

Is union of attractors of different IFSs (iterated function systems) attractors of another IFS? Is union of fractal sets always fractal? I wonder the answers for intersection and difference. I could ...
0
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0answers
24 views

Do iterative formula only solve for one positive root?

For all the questions I have done using an iterative formula, the quadratic equation only ever solves to one root, I am not sure is this always the case, or is there a method to find multiple roots? ...
2
votes
1answer
94 views

Iterations of function over rationals

Let $F = \mathbb{Q}\setminus\{-1,0,1\}$ and $f$ the function defined over F by $f(x) = \dfrac{x^2-1}{x}$. Show that : $$ \bigcap_{n\ge 1} f^n(F)= \emptyset$$ I don't have many ideas for this problem....
8
votes
1answer
113 views

Asymptotic behavior of $a_{n+1}=\frac{a_n^2+1}{2}$

Define a sequence as follows: $$a_0=0$$ $$a_{n+1}=\frac{a_n^2+1}{2}$$ I would like to know the asymptotic behavior of $a_n$. I already know (by roughly approximating $a_n$ with a differential equation)...
0
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1answer
30 views

Iterated function in differential equations

I am currently working on a problem that involves differential equations that contains iterated functions. The problem can be described as that one seeks the solution for the equation $$ \dot{x} = f^...
0
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2answers
34 views

iteractively solve power equation

I have an equation like that: $$ x = (x-A)^{2/3}+B $$ And I want to find the value of x. The problem is that during the iteration, x can became negative and then appears complex number. But that ...
1
vote
1answer
34 views

How many mappings have an IFS?

In most of paper about iterated function systems and fractals which I read, an iterated function system is taken as finite set of contraction mappings like $\{f_1,f_2,...,f_n\}$. In some paper, the ...
1
vote
2answers
79 views

understanding $\theta,\Theta, d\theta, \mu(d\theta)$

Let $(\mathcal S, d)$ be a metric space. let $\{f_\theta: \theta \in \Theta\} $ be a family of Lipschitz functions on $\mathcal S$ and $\mu$ be a probability distribution on $\Theta$. Suppose that $\...
1
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2answers
82 views

Collatz $2x + 1$ conjecture?

Do we know of any Collatz theorem involving similar functions. For example what do we know about iterations of: $$ f(x) = \begin{cases} \dfrac{x + 1}{2} \text{, if } x \text{ is odd}. \\ 2x + 1, \...
0
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0answers
24 views

selecting a number according to probability

Could anyone just tell me what does he, mean by at the point $2$, selecting a number according to the probabilities, $p(x_0)$? does he meant to chhose the largest one out of $p_i(x_0), i=1,2,\dots,N$?,...
1
vote
1answer
60 views

Is it possible to simplify this nested GCD?

Is it possible to simplify this nested GCD? $$\gcd\bigg(\gcd(m^2,\sigma(m^2)),\frac{m^2}{\gcd(m^2,\sigma(m^2))}\bigg)$$ Here, $\gcd(m^2,\sigma(m^2))>1$ and $\sigma(m^2)$ is the sum of divisors ...
0
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1answer
39 views

Limit involving iterated function $f_a(x)=x^2+a^2$

I have long ago give up trying to find a nice formula for the $n$th iteration of functions in the form $$f_a(x)=x^2+a^2$$ However, it would be interesting to consider the asymptotic growth of the ...
0
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0answers
53 views

Self-similarity dimension for IFS or attractor of IFS

We can have same attractor from different iterated function systems. So i wonder about self-similarity dimension concept is for IFS or its attractor. We know that when IFS satisfies the open set ...
0
votes
1answer
15 views

convert iterated function to continuous function

Let's say we have a function: $$v(t_n) = a\cdot(t-t_{n-1}) + v(t_{n-1}) \cdot d^{t-t_{n-1}}$$ Where $v$ = veolcity $t$ = time $a$ = acceleration $d$ = friction or damping What is basically does is ...
0
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1answer
34 views

Number of solutions for composite function

If $f(x)=4x(1-x)$ The function is defined over $\Bbb{R}$. Find the number of real solutions of $$f\circ f\circ f(x)=\frac x3.$$
0
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0answers
65 views

Behavior of 2-norm of k-th power of matrix

I got this problem from Greenbaum's book of iterative methods. In page 14 he mentions that the 2-norm of matrix $A^k$ is asymptotically behaves like $v \left( \begin{array} { c } { k } \\ { j - 1 } \...
-1
votes
1answer
41 views

understanding iterated function system by Markov chain [closed]

The Iterated Function System at node $i$ is a discrete time Markov chain on the state space ${\cal S}_i=\mathbb{R}^d$. The chain is specified by an integer $m$ and a collection of maps $f_j^{(i)}: ...