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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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 $...
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+50
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 ...
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48
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Converging integer function [closed]
How many integer functions f(x) are there such that:
for all elements in the set, applying f(x) finitely many times reaches an endpoint Q
Q is a fixed point of the function (applying it to Q returns ...
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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 ...
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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 $"\...
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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 ...
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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 ...
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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<\...
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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) - ...
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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 $...
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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 ...
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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 ...
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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 ...
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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].
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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 ...
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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 $(...
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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 ...
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$(M,d)$ 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, $f$ ...
1
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1
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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\...
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1
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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 $...
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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 ...
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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 ...
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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 ...
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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^* ; ...
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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), \...
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1
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How to find the function from fixed points?
A fixed point of a function $f$ is a number $t$ where $f(t)=t$. Geometrically, this means that the graph of $f$ crosses the line $y=x$ at the point $x=t$. Similarly, the graphs of $f\circ f$ and $f\...
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Fixed point of a function whose image is different to its domain.
Take a unit square in $\mathbb{R}^2$ and let $\gamma \colon [0,1] \to [0,1]^2$ be a continuous path from $[a,0]$ to $[0,b]$ where $0\le a,b < 1$ are constants. So we can think of $\gamma$ as ...
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Composite functions and fixed point free property
Given 2 functions: $f: X \to X$ and $g: X \to X$,
$f$ and $g$ are one-to-one
$f$ and $g$ are fixed point free
composite functions $f \circ g$, $\; g \circ f$,
$\; f^2 \equiv f \circ f\;$ and $\;g^2 \...
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Fixed point problem involving a minimization
Let $X \subset {\Bbb R}^n$ be a compact and convex set, and let $f : X \times X \to {\Bbb R}$ be a continuous and differentiable scalar field defined by
$$
f(x,x^*)=g(x) + \sum_{i=1}^n x_ih_i(x^*).
$$
...
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Do fixed lines of projective transformations contain a fixed point?
This post for linear transformations provides counterexample $(x,y)\mapsto(x+y,y)$, but the projective transformation $[x,y,z]\mapsto[x+y,y,z]$ does have a fixed point $[1,0,0]$ on any fixed line. Is ...
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Fixed point analysis at the origin that appears to be a source, but is undefined.
I have a 2D system of ODEs where $(x,y) \in [0,1]\times(0,1]$ as follows:
$$ \dot{x} = a\frac{x(1-x)}{y} - bxy $$
$$ \dot{y} = cx(1-y) - dy$$
where $a,b,c,d \in \mathbb{R}$. The system is clearly is ...
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1
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Generalization of Kakutani-Ky Fan Theorem without convexity assumptions
I'm wondering if there exists some extension of the Kakutani-Ky Fan Theorem
Theorem. Let $K$ be a nonempty, compact and convex subset of a locally convex space $X$. Let $f:K\longrightarrow 2^K$ be ...
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Disprove existence of fixed-point of endofunction constructor
This might be quite a naïve question, but I seem to be unable to find a good answer.
The set-valued functional
$$
F(X) = (X \to X) \cup \{ \bot \}
$$
is not monotonic: $F(\{t\}) = \{\bot,\mathrm{id}_{ ...
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A Problem on Compositions of Rotations [closed]
Let $A$, $B$, and $C$ be three different points such that compositions of the following three rotations
$$
R(C, 256^\circ) \circ R(B, 244^\circ) \circ R(A, 220\circ)
$$
has a fixed point. Determine ...
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Does a contractible locally connected continuum have an fixed point property?
I'm surprised that I can't find any research on this topic. Maybe it's too obvious? Kinoshita proved that contractible continuum do not have FPP, but his example is not locally connected. Maybe if we ...
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1
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Unique fixed point in a group action
I have been doing algebra exercises recently and I stumbled across this problem that I struggled to solve.
Suppose a finite group $G$ acts on a finite set $A$ so that for every nontrivial $g \in G$ ...
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1
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12 people throw their hats into a pile. They randomly pick a hat later. What is expected number of people who leave with their own hat?
Here is a similar question:
Question on the 'Hat check' problem
But the follow-up is, what is the standard deviation of number of people who leave with their own hat?
I understand that ...
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1
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Conjecture:A fixed point subset of real numbers are precisely closed and bounded interval.
A topological space $(X, \tau) $ is called a fixed point space iff
$ f\in C(X, X) $ implies $\textrm{Fix}_f(X) \neq \emptyset$
Theorem: $(X, \tau) $ FPS implies $(X, \tau) $ connected.
Proof: Suppose ...
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nonlinear odes: physical intuition behind nonlinear terms like $x^2$ and $x^3$.
I have been reading Strogatz's book on nonlinear differential equations, as well as watching Robert Ghrist's videos on nonlinear ODEs and discrete dynamical systems. I am just doing this as a self ...
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1
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1
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Identify the Bifurcation of a map
This question is taken from Glendinning's textbook "Stability, instability and chaos": Given the map $x_{n+1} = \mu - x_n^2$, determine the type of bifurcation which occurs at $\mu = -\frac{...
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Differentiable self-map of finite order fixing open set is the identity
Suppose $U \subseteq \mathbb{R}^m$ is some connected open subset and let $f: U \to U$ be a differentiable map with $f^n = \text{id}$ for some $n\in\mathbb{N}$. Let $\text{Fix}(f)=\{x\in U: f(x) = x\}$....
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Evaluate $\frac{d^{n-1}}{dw^{n-1}}\csc^{-1}(w)^n$ or $\frac{d^{n-1}}{dw^{n-1}}\left(\frac{\sqrt{w^2-1}+i}w-1\right)^m$
With Lagrange reversion:
$$x\sin(x)=1\iff x=\{2\pi k+\csc^{-1}(x),(2k+1)\pi-\csc^{-1}(x)\}$$
Therefore:
$$x_{2k}=2k\pi+\sum_{n=1}^\infty\frac1{n!} \left.\frac{d^{n-1}}{dw^{n-1}}\csc^{-1}(w)^n\right|_{...
- 10.6k
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0
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47
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Finding the fixed point of a 1 dimensional map.
I am struggling with finding the fixed point x* as a function of r for the one dimensional map:
$$x_{n+1}=f(x)=x_n+log(r(2-x_n))$$
The solution is apparently:
$$x_n^*=2-1/r$$
I understand a fixed ...
- 1
0
votes
0
answers
48
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Tips for finding equilibria of high-dimensional nonlinear dynamical systems
In order to perform stability analysis, I seek to find fixed points in a couple of relatively high-dimension (10-30 state variables) nonlinear dynamical systems composed of ordinary time-derivatives. ...
- 1
2
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0
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85
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The miraculous fish problem: sums of cubes of digits halts to 153 for multiples of 3
The fun name is a nod to the verse:
So Simon Peter climbed back into the boat and dragged the net ashore. It was full of large fish, $153$, but even with so many the net was not torn. - John 21:11
...
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0
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Properties of the bifurcation diagram for the logistic function
Once the bifurcation diagram has been plotted ($x_{n+1}=rx_n(1-x_n)$), there are 3 elements or properties that I don't know haw to explain, and I have not found any article where they are explored. ...
1
vote
1
answer
84
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How many fixed points of $\tan x$ are there if $x>0$, $x\neq k \pi + \frac{\pi}{2}$?
Is the following statement is true or false
Consider the function $\tan x $ on the set $S=\{ x \in \mathbb{R} : x \ge 0 , x \neq k \pi + \frac{\pi}{2} \text{ for any } k \in \mathbb{N} \...
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3
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Sum of sign of permutation * number of fixed points over $S_{n}$
Let $n\ge3$ be a positive integer and $S_{n}$ the set of all permutations of {1, 2, 3, ..., n}. For every permutation $\sigma \in S_{n}$, we denote by $\epsilon (\sigma) $ the sign of $\sigma$ and by ...
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Consider the function $f(x)=\sin(\cos(x))$. What are the attractive fixpoints for $f(x)$. [closed]
Consider the function $f(x)=\sin(\cos(x))$. What are the attractive fixpoints for $f(x)$, ie. the set of values of $x$ for which $f \circ f \circ f \circ \cdots \circ f(x) = x$ in the limit?
Let
$$F_n(...
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1
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Proof that $T^nx$ converges.
Let $B$ be a real two times two matrix with eigenvalues $\lambda\in (1,\infty)$ and $\mu\in (0,1)$ and we define $$T:S^1\rightarrow S^1;~~x\mapsto \frac{Bx}{\|Bx\|}$$
I want to show that for all $x\in ...
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