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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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a global fixed point [closed]

Let $G$ be a finite subgroup of $\text{Isom}(\mathbb{E}^2)$. Show that $G$ has a global fixed point $c$ element $\mathbb{E}^2$, meaning that $g(c) = c$ for every $g$ element $G$. I admit that I ...
Alex Nguyen's user avatar
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Fixed point iteration and convergence and contraction mapping

Let $p\ge 2$ be an integer, $a>0$ and $f$, $g$ be two functions defined for $x>0$ by $g(x)=ax^{1-p}$, $f(x)=x+\lambda(g(x)-x)$. Let $x_{0}$ be an appropriate initial guess, close enough to the ...
maths and chess's user avatar
6 votes
2 answers
354 views

Permutation groups without global fixed point where every element has a fixed point

I am looking for examples (or interesting statements) of finite permutation groups $G\leq \operatorname{Sym}(n)$ with the following two properties: every element $g$ of $G$ has a fixed point, i.e., ...
Florian's user avatar
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Fixed Point Iteration Limit Point [closed]

This might sound dumb. But I cannot find a plausible answer to this question: Can fixed point iteration converge to another point than the fixed point of the continuous function?
Enes Arda's user avatar
1 vote
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"Heteroclinic orbit" for non-autonomous system?

Apologies if I am using language incorrectly, but I want to consider a non-autonomous ODE for $u(x),v(x)$ that is approximately autonomous in the limits of both $x\rightarrow\pm \infty$. An example of ...
octonion's user avatar
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Reconstructing a closure operator from a set of fixed points

Let $L$ be a lattice, not necessarily complete. We define a closure operator as a function $f\colon L\to L$ which is: idempotent, $f(f(x)) = f(x)$, isotone, $x\leq y \Rightarrow f(x) \leq f(y)$, ...
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Stability of fixed points in a 2D discrete time dynamical system

I am studying the stability of fixed points in the Henon map for a project, and I have a question regarding it. The Henon map we are studying is given by: \begin{align*} x_{n+1}&=1+y_n-\alpha x_n^...
Sarah Hadaidi's user avatar
4 votes
2 answers
227 views

Fixed point for some continuous function.

Let $f$ be a continuous function on $[a,b]$ ( $f: [a,b]\to \mathbb R$, $a < b$) such that $\int_a^b f(x) \, dx = \frac{b^2-a^2}{2}$. How can we prove that $f$ has a fixed point in $(a,b)$ without ...
Noname's user avatar
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Prove that $x-\frac{1}{3}\sin(x)= y$ has a unique solution for any $y \in \mathbb{R}$. [duplicate]

I am trying to prove that $ x-\frac{1}{3}\sin(x)= y$ has a unique solution for any $y \in \mathbb{R}$ by a fixed point method. My aim is to apply somehow the Banach Contraction Principle. I started by ...
Pingu's user avatar
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Is there a closed-form expression for this iterated mean?

Here is a simple Python implementation of the arithmetic, geometric, and harmonic means of a (non-empty) list of numbers: ...
Dan's user avatar
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How to find function $G(v_i)$, such that $\mathbb{P}\big[{\psi}(G(v_i))\big]= \frac{1}{G(v_i)}$. Is this a fixed-point problem?

The problem comes from an economic and market scenario. We have a function $$\psi(v_i)=v_i-\frac{G(v_i)-F(v_i)}{f(v_i)},$$ where the random variable $v_i$ is any real number (e.g., person $i$'s money)...
iDhone's user avatar
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1 answer
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If $p$ is a fixed point of a $C^1$ map $f:\Bbb R\to\Bbb R$, and $|f'(p)| < 1$, then $p$ is an attracting fixed point

This is Exercise $1.5.4$ in Introduction to Dynamical Systems by Michael Brin and Garrett Stuck. Let $f: \Bbb R \to \Bbb R$ be a $C^1$ map, and $p$ be a fixed point. Show that if $|f'(p)|<1$, then ...
stoic-santiago's user avatar
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Why is a repelling fixed point also an isolated fixed point?

This is Exercise $1.5.2$ in Introduction to Dynamical Systems by Michael Brin and Garrett Stuck. Let $X$ be a locally compact metric space and $f:X\to X$ be a continuous map. We say that a fixed ...
stoic-santiago's user avatar
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A problem on fixed points in complex

Prove that every holomorphic function on the closed disk $\overline{\Delta}(0,1)$ with $|f(z)|<1$ when $z\in \overline{\Delta}(0,1)$ has at least one fixed point in $\Delta (0,1)$. I was thinking ...
Quang123's user avatar
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numerically solving for the fixed points of a system of nonlinear ODEs

I was looking at an excellent lecture series on Robotics by Russ Tedrake, and he discusses Linear Quadratic Control (LQR) for system of nonlinear differential equations. So as he suggests, robots are ...
krishnab's user avatar
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Grimmett and Stirzaker P57 (Letter Matching) updated

https://math.stackexchange.com/posts/2854522/edit I was stuck on step 4 of the derivation below. A secretary types n different letters together with matching envelopes, she then drops the pile down ...
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Does there exist a non-trivial power series that equals its own coefficients on the positive integers? [duplicate]

Consider the power series $p(x) = \sum_{n=0}^{\infty} a_n x^{n} $ Suppose this power series has infinite radius of convergence. ie; $limsup_{n\rightarrow\infty} (|a_n|^{1/n}) = 0$ https://en.wikipedia....
Awkward Deduction's user avatar
5 votes
1 answer
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Definition of attracting fixed points in dynamical systems [book: Brin and Stuck]

I am reading the book Introduction to Dynamical Systems by Brin and Stuck. Here is the definition of attracting fixed points. Let $X$ be a locally compact metric space and $f: X \to X$ a continuous ...
FactorY's user avatar
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How to deduct Newton's method heuristically

Let $\alpha$ be a fixed point of a contractive function $g$ defined in $[a, b] \subset \mathbb{R}$ such that $g'(\alpha) = 0$ and $g''(\alpha) \neq 0$. My first question is how to prove that $x_{k + 1}...
Cyclotomic Manolo's user avatar
2 votes
2 answers
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How can I show that the unique fixed point of a composition of continuous functions implies another unique fixed point?

Consider the continuous functions $f : \mathbb R \to \mathbb R$ and $g : \mathbb R \to \mathbb R$. Is it possible to show that, if there exists a unique $x_0 \in \mathbb R$ such that $x_0 = f(g(x_0))$,...
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4 votes
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Automorphisms of CW complexes and fixed points

Let $X$ be a CW complex, and let $F:X\to X$ be an homeomorphism that sends each cell onto some cell; notice that we could say that $F$ is an automorphism of the CW complex since it preserves its cell ...
mathplayer's user avatar
4 votes
1 answer
194 views

Fixed point property and compactness

A topological space $X$ has fixed point property if for every continuous map $f:X\to X$ there exists $x\in X$ such that $f(x) = x$. Its easy to see that fixed point property is a topological property, ...
Jakobian's user avatar
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Galois fixed part of modules. Why does $|(1-\sigma)A||A^G|=|A|$ hold?

Let $L/K$ be a quadratic extension of number field $K$. Let $\sigma$ be a generator of $G=Gal(L/K)$. Let $A$ be a finite $Gal(L/K)$ module. Then, Why does $|(1-\sigma)A||A^G|=|A|$ hold ? Here, $|A|$ ...
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Fixed Point Property of Rationals

Does the space $X = [0,1] \cap \mathbb{Q}$ have a fixed point property (FPP)? i.e. can we find a continuous function, $f: [0,1] \cap \mathbb{Q} \rightarrow [0,1] \cap \mathbb{Q}$ without a fixed point?...
NPD's user avatar
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Calculating coordinates of vertices, given dimensions in an architectural floorplan

So, one of my friend is trying to learn autocad. They were given a floorplan. The floorplan had the dimensions. And they were asked to find the coordinates of the all the vertices of the plan. So we ...
user3851878's user avatar
4 votes
1 answer
151 views

IMO 1987/P1 - Combinatoric approach

I was was solving IMO 1987, Problem 1 and also found the first solution. However, I also tried a combinatorics approach but couldn't find any valid argument. My argument was as follows: Consider each $...
LeoMinor's user avatar
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5 votes
1 answer
157 views

A system of polynomials has Galois group G, a subgroup of $P_n$. Why are invariant polynomials of the roots rational, can you calculate them?

I have been studying systems of equations based upon iterating a polynomial. The complete Galois group of these systems is only a subgroup of the permutation group. There are many invariant ...
tippy2tina's user avatar
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0 answers
92 views

What are the generators of polynomials symmetric under a subgroup of the permutation group

I have been studying iterated polynomials, specifically let $P(x)=x^2+c$ and consider the equation $P(P(P(x)))=x$. After dividing out the solutions of $P(x)=x$, we have six solutions, which obey the ...
tippy2tina's user avatar
0 votes
1 answer
143 views

Composition of linear mapping and fixed point.

Question: $E= \mathbb{R}$ or $\mathbb{C}$, and $E$ is a normed vector space$(E; ||.||)$. Let $a$ be a linear continuous mapping and we writte $a^p = a \circ a \circ ... \circ a$ $p$ times (with $"\...
X0-user-0X's user avatar
0 votes
1 answer
28 views

A problem involving a simple discrete-time system

A discrete-time system is described by the state equation $$ V(k+1) = A V(k) + B u(k) $$ with $V(k) = [x(k), y(k)]^T$ being the state vector, and $$ A = \begin{bmatrix} 2 && - 3 \\ 0.5 &&...
Quadrics's user avatar
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1 vote
1 answer
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How to find a function with several bifurcations of specified type

Suppose I want to construct a function $h(x,r)$ such that it exhibits a bifurcation of type $a$ at the point $(x_{1},r_{1})$ and bifurcation of type $b$ at the point $(x_{2},r_{2})$. One alogirhtm to ...
Mani's user avatar
  • 402
1 vote
1 answer
64 views

How many permutations of set are there with 4 fixed points?

Given the set of numbers $A = \left \{ 1, 2, 3, 4, 5, 6, 7, 8 \right \}$. How many permutations leave exactly four numbers fixed? I've considered this as a solution... but then I stumbled upon a ...
runtotherescue's user avatar
2 votes
0 answers
36 views

Convergence of Two interrelated Sequences

Consider two sequences described below: $$\alpha_{t+1} = (1-\beta_t^2)\alpha_t,$$ $$\beta_{t+1} = (1-C \alpha_t\alpha_{t+1})\beta_t,$$ where $C>0$ and we know $0 <\beta_1<1$ and $0<\...
abolfazl's user avatar
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1 vote
0 answers
55 views

There exists an expansion function from $S_1 \to S_1$ that is continuous and has no fixed points?

If $S_1 = \{ x \in \mathbb{R}^2 : \|x\| = 1 \}$ is it possible to construct a function $f: S_1 \to S_1$ that is continuous and for all $x, y \in S_1$ we have $|f(x)-x| \ge \dfrac{1}{10}$ and $|f(y) - ...
Átila Luna's user avatar
9 votes
0 answers
145 views

Integer series related to the functional differential equation $f'(x) = f(f(x))$

There was an interesting discussion on the functional differential equation $$f'(x) = f(f(x)) \tag{1a}$$ The essence of which was to find a solution of $(1a)$ by Taylor expansion of $f$ about a point $...
Dr. Wolfgang Hintze's user avatar
3 votes
1 answer
48 views

The function $f(n) = \sum_{d \mid p_n\#} \mu(d)\sum_{r^2 = 1\mod d}\lfloor\frac{p_{n+1}^2 - 2 - r}{d}\rfloor$ has no fixed point $f(n) = n$?

Definition. $$ f(n) := \sum_{d \mid p_n\#} \mu(d)\sum_{r^2 = 1\mod d}\lfloor\frac{p_{n+1}^2 - 2 - r}{d}\rfloor $$ where $p_{n+1}$ is the $(n+1)$th prime number. And where it is understood that each ...
SeekingAMathGeekGirlfriend's user avatar
1 vote
2 answers
97 views

Why the fixed elements under translation transformation $f(x) \mapsto f(x+1)$ of a field of rational functions must be constants?

Let $F$ be a field, and let $E=F(x)$ be the field of rational functions over $F$. We can see that the mapping $f(x) \mapsto f(x+1)$ is an automorphism of the field $E$. My question is, how to prove ...
David Lee's user avatar
1 vote
2 answers
34 views

Fixed points of sum of factorial of digits function

Let $f:\mathbb{Z}\rightarrow \mathbb{Z}$, $f:x=\sum_k x_k 10^k \mapsto \sum_k x_k!$ where $x_k$ is the $k$-th digit of $x$ in base ten. This function came up in a Project Euler problem. The question ...
Jackson Walters's user avatar
-1 votes
2 answers
105 views

Does $f:\mathbb{R} \rightarrow \mathbb{R}$ satisfies $f(f(x)) = x$ have a fixed point? [closed]

Given continuous mapping $f:\mathbb{R} \rightarrow \mathbb{R}$ satisfies $f(f(x)) = x$. Does this function has a fixed point? I know it has fixed point on domain [0,1].
Alice's user avatar
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0 answers
60 views

Is there a way to compute oscillating iterated functions?

I've looked into iterated functions for a bit more than a year (especially thanks to tetration), but there's still things I do not quite know about them, especially when online searching isn't really ...
Pierre Carlier's user avatar
1 vote
0 answers
44 views

How many inversions in a permutation contain a fixed point?

An inversion occurs in permutation $X$ when $i < j$ and $X_i > X_j$, or when $i > j$ and $X_i < X_j$. A fixed point occurs when $X_i = i$, i.e., it is a cycle of length 1. If the pair $(...
virtuolie's user avatar
  • 171
0 votes
1 answer
79 views

How do I find the fixed points of one isometry, from the fixed points of another isometry?

I have an isometry group $G$ which acts on units of a ring $R$ (not neessarily the full isometry group). I know the set $P_f$ of fixed points in $R$ of one member $f$ of $G$. Can I use the group ...
it's a hire car baby's user avatar
1 vote
1 answer
51 views

$(M,d)$ complete metric space and $f : M \longrightarrow M$ such that $f^p$ is a contraction. Then, $lim f^n(x) = a$, for any $x \in M$.

I try to solve this problem: "Show that if $(M,d)$ is a complete metric space and $f: M \longrightarrow M$ is a map such that exists $p \in \mathbb{N}$ for which $f^p$ is a contraction, then, $f$ ...
Caio Leonardo Duarte Bargas's user avatar
1 vote
1 answer
59 views

If $Fix(f^2)$ is connected, then $Fix(f)$ is also connected.

Let $f: [0,1] \to [0, 1]$ be a continuous function. Let $Fix(f^2)=\{x \in [0,1]: f^2(x)=x\}$ (where $f^2=f \circ f $),$\hspace{0.1cm}$ $Fix(f)=\{x \in [0,1]: f(x)=x\}$ and $Per_2(f)=\{x \in [0,1]: mín\...
Blue's user avatar
  • 293
2 votes
1 answer
146 views

Warsaw circle has the fixed point property

I'm looking for any hint on proving that the Warsaw circle has the fixed point property. By this I mean that for every continuous function $f$ from the Warsaw circle onto itself there exists a point $...
emilio j's user avatar
0 votes
0 answers
23 views

Existence of recurrence wrt an equivalence relation implies existence of recurrence wrt equality

Assume I have $f : A \to B$, $g_a : A \to A$, $g_b : B \to B$ and $R$ an equivalence relation on $B$. If I know $f(a)$ is $R$-related to $g_b (f (g_a (a)))$ for all $a\in A$, is it the case that there ...
Michael Norrish's user avatar
1 vote
0 answers
65 views

This function has a fixed point, $F^4(x)=x$. Why?

Consider two points on the unit sphere, $C_1$ and $C_2$. Let these points be close enough such that circles of radius $r$ drawn around each point intersect at two points, $N_1$ and $N_2$. The vectors ...
cms's user avatar
  • 127
3 votes
0 answers
75 views

Probabilistic Kleene's recursión theorem

Kleene's recursion theorem states that for every total computable function $f$ there is an index $e$ such that $$\phi_{f(e)} = \phi_e,$$ where $\phi_n$ is a valid enumeration of the partial computable ...
Keplerto's user avatar
  • 463
0 votes
1 answer
34 views

Prove equivalence between properties of relations using fixpoint calculus

The problem Let $a$ be a binary endorelation of some countable set $S$, i.e. $a \subseteq S \times S$ . I need to show that the following properties are equivalent: $ (a^*)^{-1} ; a^* \subseteq a^* ; ...
mell_o_tron's user avatar
0 votes
0 answers
58 views

Finding fixed points of a specific constrained nonlinear PDE

I'm looking for attracting fixed points of the following differential equation in a vector $F$ of length $n$, and $M$ is a known $n \times n$ square matrix: \begin{align} \frac{dF}{dt} = I(F), \...
Minty's user avatar
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