Tagged Questions
1
vote
1answer
35 views
Limiting value of iteration $x(k+1) = A x(k) + B u(k)$ for summable $u(k)$
A matrix $A$ is known to converge such that $\lim_{k\rightarrow \infty} A^k = \bar{A} \neq 0$. We have an iteration defined as
$$x(k+1) = A x(k) + B u(k), \ \ k\in \mathbb{Z}_+.$$
$\{u(k), ...
0
votes
2answers
65 views
Definition of metastability
I was reading Terence Tao's blog post on analogies between soft and hard analysis when I saw that the soft analysis statement "$x_n$ is convergent" corresponds to the hard analysis statement "$x_n$ is ...
6
votes
1answer
190 views
The Mandelbrot Set Membership
To define the Mandelbrot Set we consider a sequence of complex numbers $z_0$, $z_1$, $z_2$, $z_3$, with the following conditions:
$$
\begin{cases}
z_{n+1} &= &z_n^2 + c &\text{ for }n\geq ...
9
votes
1answer
202 views
Number of limit points of a continued exponential
Inspired by the work of C. Bender, I recently played with continued exponentials (like continued fractions but with exponential functions ;) ). Given all prefactors are equal to 1, the continued ...
5
votes
3answers
498 views
The golden ratio and the logistic equation
I have this bonus question in an assignment: How is the golden ratio related to the logistic equation in discrete time?
The logistic equation is $x_{n+1}=rx_n(1-x_n)$. The professor suggested ...
5
votes
5answers
190 views
Periodic sequences on finite alphabet
Let $\Sigma=\{A,B,C\}$ be an alphabet, and let $\Sigma^{\mathbb{N}}$ be the set of infinite sequences on $\Sigma$ (ie $ABCBCCCBABC...$). By outside conditions, I have several subsequences that are ...
6
votes
1answer
99 views
converging to cosine by iteration
In what sense (if at all) does the iteration $x \mapsto 2x^2 - 1$ converge to $\cos 2^n x$ in the unit interval [-1,1]?
One might try to plot in Mathematica:
...
