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.
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Must diffeomorphisms on $S^n$ have fixed points?
Question.
Let $N_1 \simeq S^k$ and $N_2 \simeq S^l$ be disjoint smoothly embedded spheres in $S^n$ with $k + l = n$.
Suppose a diffeomorphism $\psi: S^n \to S^n $ preserves $N_1$ and $N_2$, and that ...
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Proof of convergence of replicator type II dynamics
One of the evolution equations used in evolutionary game theory is the Replicator type II dynamics
$$
x_i(t+1)=x_i(t)\frac{(Ax(t))_i}{x^T(t)Ax(t)}
$$
where $A$ is an $M\times M$ payoff matrix with ...
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Higher-order self-coincidences for a timelike geodesic under $k$ compact time-shift diffeomorphisms
Let $(M^{n+1}, g)$ be a smooth, connected, globally hyperbolic, vacuum Einstein ($\operatorname{Ric}(g)=0$) Lorentz manifold with $n \ge 1$. Assume $M$ carries a complete, everywhere timelike Killing ...
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Fixed points of non-expansive mappings
Let $A$ be a non-empty finite subset of ${\bf R}^n$, $n\geq 2$.
Prove or disprove the following:
There exists a norm $\|\cdot\|$ on ${\bf R}^n$ and a map $T:{\bf R}^n\to {\bf R}^n$ with the following ...
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behavior of the moving average point in Brouwer theorem
Given a continuous operator from the unit disk $S := \{x \in R^2 : ||x||_2 \le 1 \} $ to itself like $ f: S \to S$ , we define the sequence $ x_0 , x_1 = f(x_1) , x_2 = f(x_1) , ..., f(x_n) = f(x_{n-...
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Explicit Data of Homotopy Fixed Points in Lurie's TFT
Lurie's classification theorem [1, Theorem 2.4.26] classifies fully extended
tfts for $G$-manifolds in terms of homotopy fixed points:
Let $C$ be a symmetric monoidal $(\infty, n)$-category with ...
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Lipschitz retraction constant of $B^+$ into $S^+$ in $L^2([0,1])$
In Hilbert space modeled by $L^2([0,1])$ we can define a set $B^+=\{x\in B(0,1): x(t)\geq0 \quad \forall t\in [0,1] \}$ and $S^+=\{x\in S(0,1): x(t)\geq0 \quad \forall t\in [0,1] \}$ where where $B(...
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Is the fixed point index bounded?
I am working with the notion of fixed point index presented in the book "The Lefschetz fixed point theorem" of Robert Brown (MR283793, Zbl 0216.19601) and I would like to know if given any ...
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Convergence of iterated average Bayesian posterior to high entropy distribution
Setup
Assume $p_Y \in \Delta^n$ is a probability vector obtained by $p_Y=L_{Y|X}p_X$, where $L_{Y|X} \in \mathbb{R}^{n \times m}$ is an arbitrary likelihood (i.e, a column stochastic matrix) and $p_X \...
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Is there a syntactic proof that first-order positive inductive definitions are conservative?
Every first-order positive inductive definition has a fixed point. It follows that, if the biconditional is thought of as an axiom in the language obtained from the background language by adding a new ...
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Transitive groups with fixed-point free elements of prime power order
A well-known result of Fein, Kantor and Schacher says that if $G$ is a finite group which acts transitively on a set $X$, then $G$ contains an element of prime power order without fixed letters. ...
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How to express a quadratic polynomial exactly as a power series [closed]
I claim, for $\operatorname{artanh}(\rho) = \frac{1}{2} \ln\left(\frac{1+\rho}{1-\rho}\right)$, i.e., the inverse hyperbolic tangent function, the following holds approximately under assumptions given ...
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Do these polynomials with a complex kind of ‘Vieta jumping’ exist for all $k$?
Inspired by a recent question about sequences defined by $s_{n+1}=s_n^2-s_{n-1}^2$, I started wondering whether non trivial real or complex cycles of any length $k\geqslant3$ fixed by such a sequence ...
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Generator of an analytic semigroup
Perhaps I have a naive question. My question is as follows:
When we consider a Cauchy proposition of the following form:
$$
\begin{cases}
x'(t)= -Ax(t)+ F(t,x(t)) &\text{for}\ t> 0 \\
x(0)=...
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Teaching suggestions for Kleene fixed point theorem
I will take over two lectures from a colleague in which we discuss fixed point theory in the context of complete partial orders, and culminates in showing the Kleene fixed point theorem (see f.e. ...
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