# Tagged Questions

279 views

### Prove that a sequence defined by a recurrence relation converges

Consider the following recurrence relation: $$a_i = \frac{i+2}{2} \cdot \left(\frac{i}{i+1} - \sum_{j=1}^{i-1} \frac{2 a_j}{2i - j + 2}\right).$$ The first ten terms are: $0.75$ ...
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### Stability of rest points of a time-dependent sequence

Consider the following sequence: $$x_{t+1} = (x_t)^t$$ A fixed point $a$ satisfies the following: $$a = a^t$$ for all $t$ sufficiently large. $$a(1 - a^{t-1}) = 0$$ Clearly, $a=0$ and $a = 1$ ...
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### Why computing Fatou coordinate is so hard?

I'm trying to make images of Fatou coordinate for some polynomial maps. If I'm not wrong there is no explicit general formula/method for computing Fatou coordinate near parabolic fixed point. Is ...
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### Sequence of $r_n:=\frac{x_{n+1}}{x_n}$ where $x_n$ is a sequence

Let $f:X\rightarrow X$ be a function on some complete metric space $(X,d)$. Let, $x_n$ is a sequence in the metric space defined by $x_{n+1}=f(x_n)$ and starting from $x_0$. My questions are (1) ...
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### Dynamics of the repetitions of $f(z) = z^{2} +\frac{1}{4}$

This is probably a classic and most likely an easy question, but I was not able to find an answer. If we have the iteration $z_{n+1}=g(z_n)$, where $$g(z) = z + a z^{2},$$ with $a\ne 0$, then using a ...
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### Renormalization operator R

In a Robert Devaney's book ("an introduction to chaotic dynamical systems") is approached the quadratic map $F_\mu=\mu x (1-x)$. He introduce the renormalization operator R. R is a function of ...
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### What are Green's almost primes?

In a general-audience talk, Ben Green explains his famous proof with Terence Tao as an application of Szemerédi's theorem, but placing the primes within a smaller set of almost-primes in which they ...
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### Discuss the convergence of $\left \{ a_n \right\}$ where $a_{n+1}=\frac{a_0}{2}+\frac{a_n^2}{2},n\geq 1$

Let $$a_{n+1} = \dfrac{a_0}{2} + \dfrac{a_n^2}{2}$$ where $a_1 = \dfrac{a_0}{2}$ and $n\geq 1$ Discuss the convergence of $\left\{a_n\right\}$
The Collatz Conjecture is well known with the sequence $$f(n) = \begin{cases} n/2 &;\text{if } n \equiv 0 \pmod{2}\\ k\,n+1 &; \text{if } n\equiv 1 \pmod{2} \end{cases}$$ and $k=3$; the ...