# Questions tagged [fixed-points]

In mathematics, a fixed point (sometimes shortened to fixpoint, also known as an invariant point) of a function is an element of the function's domain that is mapped to itself by the function. That is to say, c is a fixed point of the function f(x) if and only if f(c) = c. A set of fixed points is sometimes called a fixed set.

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### What are the fixed points of the arithmetic derivative over the non-negative integers?

I just watched When the derivative of a number is not zero -- The arithmetic derivative by Michael Penn. I like to explore things visually and computationally, so I found this recursive implementation ...
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### Calculate Intersection point of 2 Lines with angle [closed]

I would like to calculate an intersection point of two lines in a 2D area. I think it should be really simple but i cannot figure it out. I have two points P1(x,y), P2(x,y) and 2 angles alpha and beta....
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### Proof for fixed point under specific circumstances

Prove that every $f \in C(I, \mathbb{R})$ with $I := [-1, 1] \subset \mathbb{R}$ and $f(I) \subseteq I$ has a fixed point. This would be true if $f$ is a contraction on $I$, since then Banach's fixed ...
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### Stability of fixed points

Hi I was exploring the bifurcation diagram of the logistic map when I came across the concept of fixed points and their stability. However, I don't really understand how it is stable when |f′(x)|<...
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### When is the minimum equal to the fixed point for this function?

Consider a smooth function $f:R^n\times R^n\rightarrow R$ such that $g(y) = \text{argmin}_x f(x, y)$ is a contraction with the unique fixed point $x^*=g(x^*)=\text{argmin}_x f(x, x^*)$. My question is,...
1 vote
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### Prove the $k$-th power of the logistic map with parameter $\mu = 4$ has $2^k$ fixed points

I'm trying to solve an exercise in which I need to prove that the logistic map with parameter $\mu = 4$, $F_4:[0,1]\to[0,1]$, $F_4(x) = 4x(1-x)$, satisfies that for every positive integer $k$, $F_4^k$ ...
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### Continuous function on the unit interval with commuting compositions

Let $I$ be the unit interval and $f, g:I\to I$ be continuous functions. Assume that $f\circ g = g\circ f$. This post shows that if $f$ and $g$ in addition are assumed to be increasing then $f$ and $g$ ...
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### Given a random permutation $\sigma$ of a set $\{1,...,n\}$. Assume we want to compute $X$=# of integers $i$ with $\sigma(i)=i$. What is $E[X]$?

Given a random permutation $\sigma$ of a set $\{1,...,n\}$. Assume we want to compute $X$=# of integers $i$ with $\sigma(i)=i$. What is the expected value of $X$? Potential Solution Part 1 I had an ...
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### Tarski's Fixed Point but with an Order Reversing Function

I have read the Tarski's fixed point theorem for powersets. It requires an order-preserving/monotonic function. I wonder if there are results (with strengthened conditions) for order-reversing ...
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### Prove that if $x^*$ is an equilibrium point of $y(n+1)=f^2(y(n))$, then it need not be equilibrium of $x(n+1)=f(x(n))$

This question is related to Problem 13 in section 1.5 in Elaydi's "An Introduction to Difference Equations": Prove that if $x^*$ is an equilibrium point of $x(n+1)=f(x(n))$, then it is an ...
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### Prove points close to attracting fixed points converge to the attracting fixed point

I am trying to prove that if $x$ is an attracting fixed point of $f(x)$, then if $x_{n+1}$ is "close enough" to $x$. Then $x_n = f(x_{n+1})$ is even "closer" to x and as n goes to ...
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### In lambda calculus, how many fixed-point combinators are there? [closed]

In lambda calculus, how many fixed-point combinators are there? I am familiar with Curry’s paradoxical combinator a.k.a. the $Y$-combinator and Turing’s fixed-point combinator, $\Theta$, which are ...
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### Lambda Calculus functions that are fixed-points of themselves

In general, a fixed-point of a function, $f$, is a point $x$ in the domain of $f$ such that $f(x) = x$. In lambda calculus, a function can take a function, including itself, as input, so we can even ...
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My idea to show this is to notice that a projective transformation is in general given by the form $\phi(z)=\frac{az+b}{cz+d}$, then $\phi(z)=z$ gives a second order equation with at most two ...