Questions tagged [fixed-point-theorems]
A fixed-point theorem is a result saying that a function $F$ will have at least one fixed point (a point $x$ for which $F(x) = x$), under some conditions on $F$ that can be stated in general terms.
229
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Automorphisms of rationally connected varieties
Let $X$ be a smooth, rationally connected variety over an algebraically closed field of characteristic zero. Denote by $\mathrm{Aut}(X)$ the space of automorphisms of $X$ and for a given $\phi \in \...
2
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0
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Existence and uniqueness of fixed point in generalized condition of triangular norm
Definition 1) A Menger space is defined as a triple $\left( S,F,T \right)$ where $S$ is a set , $F$ is a collection of distribution functions and $T$ is a triangular norm function
$T:[ 0,1 ]\...
2
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2
answers
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On fixed point probability in discrete logarithm
Fix an integer $n>2$.
Question. What is the probability that, for a given $h\in\Bbb Z_n,$ there is no $$x\in[0,\varphi(n)-1]\cap\Bbb Z$$ such that
$h^{x\bmod\varphi(n)}\equiv x\bmod n$?
4
votes
1
answer
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Fixed point of a group action
Let $\mathbb{R}^\infty$ be the product of countably many real lines.
Assume that a finitely generated group $\Gamma$ acts on $\mathbb{R}^\infty$ (linearly and continuously) and there is a nonempty ...
8
votes
2
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Continuity of mapping sending a function to its (brouwer) fixed point
Let $f:[0,1]^n \rightarrow [0,1]^n$ be a continuous mapping. Brouwer's fixed point theorem says that $f$ has a fixed point, i.e., some $x$ such that $f(x) = x$.
Suppose we have a continuous family, i....
3
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0
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Formal way to prove existence and continuity in an integral equation
In the paper "A Boundary Value Problem Associated with the Second Painlevé Transcendent and the Korteweg-de-Vries Equation" by Hastings and McLeod, the authors study the ODE
$$\frac{\mathrm{d}^2 y}{\...
10
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1
answer
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Equivariant Riemann-Hurwitz
The Riemann-Hurwitz formula starts with a genus $g$ algebraic curve $Y$ and a ramified cover $\pi\colon X\to Y$ of degree $N$, with ramification indices $e_P$ and computes invariants of $X$, such as ...
3
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Is it possible to find a $\mathbb{C}$-algebra endomorphism of $\mathbb{C}[x,y]$ that fixes a given $w \in \mathbb{C}[x,y]-\mathbb{C}$?
Let $w \in \mathbb{C}[x,y]-\mathbb{C}$ and let $u \in \mathbb{C}[x,y]-\mathbb{C}[w]$.
Is it possible to find a $\mathbb{C}$-algebra endomorphism $f$ of
$\mathbb{C}[x,y]$ such that $f(w)=w$ and $...
1
vote
1
answer
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Random iteration of a set of monotone maps until fixed point
Let $P$ be a poset with a least element $\bot$ ($\forall x \in P.\ \bot \le x$).
Let $M$ be a set of monotone maps $P \to P$.
Call $x \in P$ reachable if $x = f_1(f_2(...f_n(\bot)...))$ for some ...
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3
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Closed manifolds with the fixed point property
The real projective plane ${\bf P}^2({\bf R})$ is the only closed surface with the fixed point property: all continuous maps from the plane to itself has a fixed point. The projective spaces ${\bf P}^{...
1
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1
answer
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How to compute the joint spectrum?
Let $(A_{1},A_{2}, \ldots,A_{k})$ be $k$ matrices in $M_{n}(\mathbb{R})$.
Is there an algebraic formula, as a generalization of "Determinant" for $k=1$, to compute the joint spectrum of ...
9
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1
answer
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Is $\{x_n\}$ a Cauchy sequence?
Let $(X,d)$ be a complete metric space and $f$ a mapping of $X$ into itself. Let $\{f^n(x)\}=\{x_n\}$ be the sequence of iterated transforms.
Suppose $f$ satisfies that for each $\varepsilon >0$,...
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3
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Does a certain contractive mapping have a fixed point?
Let $f:X\rightarrow X$ be a contractive mapping of a complete metric space satisfying
$$d(f(x),f(y))\leq\alpha(d(x,y))d(x,y)$$
where $\alpha:\mathbf{R}^+\rightarrow [0,1)$, and $\alpha(t_n)\rightarrow ...
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4
answers
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What are the major differences between real and complex Banach space?
Most theorems under real Banach space settings have their twin brothers for complex ones, say, the Hahn-Banach theorem. However, some theorems are not valid in complex Banach spaces, and vice versa.
...
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Reference request: an example of Bott residue formula's usage
Could you give me an example of a clear and beautiful application of Bott residue formula in torus-equivariant cohomology (see below)?
I found an example calculating a product of Chern classes on ...
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0
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spiral forward orbits of analytic functions near repelling fixed points
An anonymous referee informs me that forward orbits near fixed points of analytic functions, such that the members of the forward orbits lie on spirals, are well-known. His citation for this (p. 31 of ...
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0
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Fixed-point iteration depending on a parameter
Let $f\colon X\times \mathbb{R}\to X, (x,\varepsilon)\mapsto y$, with $X$ open, be a continuous function in both arguments. Consider the following fixed-point iteration
\begin{align}
x_{k+1} = f(x_k,\...
2
votes
1
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Fixed point theorem for a nonconvex set in a Banach space
Generally speaking, I am looking for a generalization of the Schauder fixed point theorem, which applies to the situation described briefly below.
All references I read (e.g. E. Zeider 'Nonlinear ...
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1
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Number of fixed points in Zagier's involution (Fermat's Theorem) [closed]
Zagier's has found a famous one sentence proof for Fermat's theorem on sums of two squares. It centers on the following involution of the set $S= \lbrace (x,y,z) \in N^3: x^2+4yz=p \rbrace $ having ...
2
votes
1
answer
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characterization of normality by selection theorem
The Urysohn's extension theorem states that a space $X$ is normal iff every continuous function $f:A \rightarrow \mathbb{R}$, with $A$ a closed subset of $X$, can be extended to a continuous function $...
0
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Fixed point shape property
Question: Provide (or prove that it's not possible) a metric compact space which has the fixed point property but not the fixed point shape property.
Here is the definition of f.p.s.p.("map" means ...
3
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1
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leray schauder fixed point and schauder fixed point
I have seen these 2 fixed point theorem and I think the condition of Leray Schauder fixed point theorem is very strong and we require to consider the fixed point of $u=\sigma Tu$ $\forall \sigma \in[...
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Proof of Lefschetz-Hopf Fixpoint Theorem with de Rham cohomology?
Looking for a proof of the Lefschetz-Hopf Fixpoint Theorem with the de Rham Cohomology.
(I´m more interestet in the Formula then just the simple statement that if the Lefschetz number is not zero ...
3
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0
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203
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Can the following system of equations be solved analytically/in a closed form?
From a constrained non-linear maximization problem I obtained the following system of equations:
$a_1=\frac{1+a_3-\sqrt{a_2a_3}\sqrt{v_1}}{1+\sqrt{\frac{a_3}{a_2}}\sqrt{v_1}}$
$a_2=\frac{2-a_3-\sqrt{...
3
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0
answers
550
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On a matrix algorithm involving rank-one projections
Let $\{v_i\}_{i=1}^N$ be a set of $n$-dimensional real vectors spanning $\mathbb{R}^n$. Let $p\in [0,1]$ be a rational number and consider the iteration
\begin{equation}
X_{k+1}=\frac{1}{N}\sum_{i=1}^...
1
vote
1
answer
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Fixed point of quantum operations
A quantum operation is defined as
\begin{equation}
\varepsilon(\rho)=\sum_{k}M_k\rho M_k^{\dagger}
\end{equation}
where $\varepsilon(\rho)$ takes an initial state $\rho$ to some final state $\rho'$ ...
18
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1
answer
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Does every automorphism of a separably rationally connected variety have a fixed point?
Let $k$ be an algebraically closed field. Let $X$ be a smooth, projective variety over $k$ that is separably rationally connected, i.e., there exists a $k$-morphism $u:\mathbb{P}^1_k \to X$ such that ...
8
votes
2
answers
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A property stronger than the fixed point property
Assume that $X$ is a topological space. We say that $X$ satisfies the strong fixed point property if the graph of every surjective continuous self-map intersect the graph of every continuous self-...
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1
answer
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A weak fixed point property
The usual fixed point property can be interpreted in terms of non empty intersection of the graph of all maps with the graph of the identity map.
This motivates us to consider the following "weak ...
7
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2
answers
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Intermediate value for a vector-valued function
Consider a vector-valued function $f: [0,1]^n\rightarrow[0,1]^n$. Write $f(x)=\{f_1(x), ..., f_n(x)\}$ with $x\in[0,1]^n$, where the $f_i: [0,1]^n\rightarrow[0,1]$ are continuous functions with the ...
6
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A tangential fixed point property for manifolds embedded in Euclidean spaces
Assume that $M$ is a compact orientable manifold which is embedded in some Euclidean space $\mathbb{R}^{N}$
We say that $M\subset \mathbb{R}^{N}$ has the tangential fixed point property if for ...
3
votes
2
answers
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Example of measure of non-compactness
I can't understand the following example of measure of non-compactness, which was given in this article.
Definition: A non-negative function $\phi$ defined on the bounded subsets of $X$ will be ...
3
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2
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Reference request: using integral equations to study asymptotics of ODEs
I was told by my supervisor that one way to study the asymptotic behaviour of solutions to ODEs is "to reformulate them as integral equations, and use fixed-point kind theorems on the resulting ...
7
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2
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Does a continuous map $f$ from the $n$-ball $B$ into $R^n$ such that $B\subset f(B)$ have a fixed point?
If $f$ is a continuous map from the $n$-ball $B$ into itself, the Brouwer fixed point theorem guarantees a fixed point. What if we assume that $f$ maps $B$ into all $R^n$, and $f(B)$ contains $B$? For ...
6
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2
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Convergence of Fixed-Point Iteration of a dependent map
Suppose that we have two mappings $T_1(\cdot): Y \mapsto Y$ and $T_2(\cdot,\cdot) : X \times Y \mapsto X$ where both $X$ and $Y$ are compact and convex subsets of the same Euclidean space. Furthermore ...
1
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0
answers
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Condition for maximizer of convex combination to be expansion mapping
I have $\Pi_n:\mathbb R^{n+1}\rightarrow \mathbb R$ and $F_n:\mathbb R^2\rightarrow \mathbb R$ with $$F_n(x,a)=\Pi_n(x,...,x,a)$$
$$f_n(x)=\operatorname{ArgMax}_{a\in\mathbb R}\{F_n(x,a)\} $$
such ...
7
votes
1
answer
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Z/p action on finite contractible complex
Let $p$ be a prime and $X$ a finite contractible CW-complex. Assume $\mathbb Z/p$ acts on $X$. Then it is easy to see that there has to be a fixed point. (E.g. use Lefschetz's fixed point theorem or ...
1
vote
1
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Row-stochasticity of the Jacobian matrix of a stationary distribution
Let $P_{\mathbf{p}}$ be a $n \times n$ row-stochastic matrix whose entries are a function of a probability vector $\mathbf{p} \in \mathbf{R}_{> 0}^n$, $\sum_i p_i = 1$ and define the following ...
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Does Peano's existence theorem admits a constructive proof?
$$y(t)=y_0+\int_0^t b(y(s))ds$$ $b\in C(R^d)\cap L^\infty(R^d)$
The classical proof for Peano's existence theorem in ODE need use the Ascoli's theorem, so it's not constructive. When $d=1$, in the ...
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0
answers
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Multi-podal points
Two points $x,y \in \mathbb{R}^n$ are called antipodal if $x = -y$.
Stated differently, $x,y$ are antipodal if:
They have the same absolute value in each of their $n$ coordinates;
Each of their non-...
3
votes
1
answer
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Fixed point property for intersection of spaces which are homeomorphic to a disk
The following question is question 9.8 from Miller's paper ``Some interesting problems
'':
Question Suppose $D_n$ a subset of the plane is homeomorphic to a disk and for every
$n\in \omega, D_{n+...
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votes
0
answers
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Fixed point theorem in ordered spaces
Can someone provide a proof or a source containing a proof of the following theorem
Theorem: Let $D$ be a subset of the cone $K$ of partially
ordered space $E,$ $F:D\rightarrow E$ be nondecreasing. ...
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votes
4
answers
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Does the Brouwer fixed point theorem admit a constructive proof?
Wikipedia and a few websites (and a few mathoverflow answers) say there is a constructive proof of the Brouwer fixed point theorem, some others say no. The argument for a constructive proof is always ...
0
votes
1
answer
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Fixed point property for the projectivization of manifold of fixed rank matrices
Let $M$ be the manifold of all matrices in $M_{n}(\mathbb{R})$ with fixed rank $0<k<n$. The projectivization of $M$ is denoted by $PM$.
Does $PM$ satisfy fixed point property?
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vote
1
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203
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Totally non fixed point property
Edit: According to the comment of Pietro Majer, I revise the question
Is there a non singleton compact connected Hausdorff topological space $X$ for which the following property hold?:
"Constant ...
2
votes
0
answers
272
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Fixed points of self maps
Given $m$ points on $S^n$, is there an explicit polynomial self $1-1$ map of minimum degree $f:S^n\rightarrow S^n$ that fixes only these $m$ points? Can we say something about symmetry group of $f$ if ...
1
vote
1
answer
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Existence of a fixed-point free map in a manifold [closed]
I'm having some to proof a question. I have to show that a compact manifold that admits a nowhere vanishing smooth vector field has a smooth map fixed-point free homotopic to the identity map.
I know ...
5
votes
1
answer
147
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Continuity of central point operation
Stanisław Mazur and Stanisław Ulam, in their joint paper, characterized the mid-point $\ \frac{a+b}2\ $ in a Banach space in pure metric terms (without algebra). This allowed them to show that any two ...
1
vote
1
answer
208
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Lefschetz fixed notation
If $f\colon X\to X$ is a self-map of a nice space with isolated fixed points, then the Lefschetz fixed point theorem relates a global number to local numbers. Some write: $L(f)=\sum_{x\in \mathrm{Fix}(...
8
votes
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Do commuting homeomorphisms of the $2$-disk have a common fixed point?
Problem 2.20 (attributed to Lima) in Kirby's list of unsolved problems in low-dimensional topology asks:
Are there commuting homeomorphisms of the $2$-ball $B^2$ without a
common fixed point?
...