57
votes
Accepted
Does iterating the derivative infinitely many times give a smooth function whenever it converges?
I was able to adapt the accepted answer to this MathOverflow post to positively answer the question. The point is that one can squeeze more out of Petrov's Baire category argument if one applies it ...
- 91k
28
votes
Does the Brouwer fixed point theorem admit a constructive proof?
It's easier to think about the Intermediate Value Theorem, which is equivalent to the Brouwer Fixed-Point Theorem for the unit interval.
The main issue is that dichotomy for (Cauchy) real numbers is ...
- 42.6k
21
votes
Accepted
Does the Brouwer fixed point theorem admit a constructive proof?
You are correct in observing the flaw in the claims for BFPT to be constructive: There is no algorithm that takes a sequence in the unit hypercube and outputs some accumulation point of it. This task ...
- 3,366
18
votes
Is the fixed point property for posets preserved by products?
This an open problem, as far as I know. The finite case was solved by Roddy in 1994.
There were some more general results proved by other people later:
see this paper by Bernd S. W. Schröder for a ...
- 827
18
votes
Accepted
How is this fixed point theorem related to the axiom of choice?
I'll deduce Zorn's Lemma from your fixed-point theorem. Suppose $P$ is a poset violating Zorn's Lemma; so all chains in $P$ have upper bounds, but there's no maximal element. Consider the poset $Q=P\...
- 68.4k
17
votes
What are the major differences between real and complex Banach space?
There are some differences. For example Bishop-Phelps theorem, which holds only in real Banach spaces. In my opinion, this qualifies as a "major theorem".
MR1749671
Lomonosov, Victor
A counterexample ...
- 78.1k
16
votes
Is it possible to find a $\mathbb{C}$-algebra endomorphism of $\mathbb{C}[x,y]$ that fixes a given $w \in \mathbb{C}[x,y]-\mathbb{C}$?
If we view $w$ as a map $\mathbb A^2 \to \mathbb A^1$ and the geometric generic fiber has a trivial automorphism group, then there will be no nontrivial automorphisms of $\mathbb C[x,y]$ fixing $w$.
...
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16
votes
Does iterating the derivative infinitely many times give a smooth function whenever it converges?
(Edit) A simplified shorter version of this answer is in Fedor Petrov's comments (as Iosif Pinellis pointed out) :
Assume the sequence $f^{(n)}$ is dominated, that is, there exists $g$ locally ...
- 2,162
15
votes
Is Schauder's Conjecture Resolved?
I have taken the following from the review of the following paper "Schauder's conjecture on convex metric spaces" written in 2010 :
One of the most resistant open problems in the theory of ...
- 30.2k
14
votes
Applications of Lawvere's fixed point theorem
It follows from Lawvere's theorem that for most spaces $X$ there is no space-filling curve for its path space, $\alpha: I \to X^I$, working here in the category of $k$-spaces. (Yes, that would also ...
- 50.6k
13
votes
Accepted
Homeo-Fixed point property
consider the connected sum of the klein bottle with the projective plane
there is a map with no fixed point
collapse to the klein bottle and rotate
any homeo induces a map on mod two homology which ...
- 421
13
votes
Accepted
Z/p action on finite contractible complex
If a finite $p$-group $P$ acts on a finite-dimensional $CW$-complex $X$ which is acyclic mod $p$, then the fixed point set $X^P$ is also acyclic mod $p$. This is a special case of "Smith theory" (see ...
- 29.9k
12
votes
Is Schauder's Conjecture Resolved?
In Points fixes des applications compactes dans les espaces ULC published in in the arXiv in 2010 Robert Cauty wrote
il y a d’ailleurs une erreur dans la demonstration du
lemme 3 de [2], qu’il n’y a ...
- 13.4k
12
votes
Is the fixed point property for posets preserved by products?
The question you asked is one of the main long-open problems in fixed point theory of posets.
Commenting on Paul Taylor's interesting answer: the property he describes is called strong fixed point ...
- 2,054
12
votes
Is there a square with all corner points on the spiral $r=k\theta$, $0 \leq \theta \leq \infty$?
The points on the spiral with $\theta=$
$$
3.4535999354657\\
15.1248305526170\\
22.0370015553781\\
16.3950081067565
$$ form a square.
The numbers can be generated from this Mathematica code, if ...
- 16.7k
12
votes
Accepted
A variation of the Ryll-Nardzewski fixed point theorem
The claim does not hold. Let $F$ be the free group on $\{a,b\}$ and $X⊂F$ be the words whose last letter is $a$ or $b$. Let $\xi=\delta_a+\delta_b−\delta_{a^{-1}}−\delta_{b^{-1}}\in\ell_1(F)$. Then ...
- 8,836
11
votes
Is the fixed point property for posets preserved by products?
Strengthening the hypotheses on $P$ and $Q$ from a fixed point property (every endofunction has some fixed point) to the existence of a fixed point operator $\mathsf{fix}$ with
$$ \text{for any } f:P\...
- 6,638
11
votes
Does the Brouwer fixed point theorem admit a constructive proof?
I have thought about this recently, and here is I think the best constructively valid statement one can extract from Brouwer fixe point theorem (framework : internal logic of an elementary topos, real ...
- 32.5k
11
votes
Accepted
Does Peano's existence theorem admits a constructive proof?
I think that the heart of the question is "Can one give a rigorous meaning to 'there is no constructive proof to the Peano's theorem'?" The answer to this is yes, but the answer is not as simple as ...
- 9,442
11
votes
Closed manifolds with the fixed point property
According to this paper, there are various such manifolds constructed as cartesian products and connected sums of real, complex and quaternion projective spaces.
For example, it is shown that $\...
- 6,314
11
votes
Accepted
Has Goedel's Second Incompleteness Theorem been proven using Lawvere's Fixed Point Theorem?
As suggested by the OP, I'm turning my comment into an affirmative answer. WARNING: The presentation of Joyal's proof in the paper I cited contains an incorrect conclusion about Joyal's sentence (that ...
- 4,847
10
votes
fixed point property for maps of compacts
If $X$ is a connected polyhedron, a point $x\in X$ is said to be a global separating point if $X-\{x\}$ is not connected.
Theorem ([1], Theorem 7.1)
In the category of compact connected polyhedra ...
10
votes
Can the Lawvere fixed point theorem be used to prove the Brouwer fixed point theorem?
Edit: Will Sawin has pointed out some difficulties with this answer. I'm going to leave this up for at least a while, in case anyone has any ideas about repairing it. Or perhaps this could be a ...
- 50.6k
10
votes
Does the Brouwer fixed point theorem admit a constructive proof?
One constructive version of the Brouwer fixed-point theorem which I find very elegant and illustrating runs like this:
Theorem (constructive Brouwer fixed-point theorem)
Let $B$ be the closed unit ...
- 1,596
10
votes
Does a continuous map $f$ from the $n$-ball $B$ into $R^n$ such that $B\subset f(B)$ have a fixed point?
Edit: I'm retaining my original answer below, but here is a simpler formula which builds on Goldstern's modification of my answer.
Work in the complex plane. Let
$$A = \{z \in \mathbb{C}: 1 \leq |z| \...
- 37.4k
10
votes
Accepted
leray schauder fixed point and schauder fixed point
Note that Leray-Schauder is usually proven by using the hypotheses to construct a mapping that satisfies the conditions of the Schauder fixed point theorem, and then appealing to the Schauder fixed ...
- 31.6k
9
votes
A variation of the Ryll-Nardzewski fixed point theorem
The claim does not hold, as the nice example by Naratuka Ozawa shows.
The purpose of this answer (or rather, extended comment) is to share a related fixed point theorem.
Theorem:
Let $X$ be a non-...
- 10k
9
votes
Accepted
Connected vertex-transitive graph with the fixed-point property
Let me convert my comments into an answer.
There is no such [EDIT:] finite graph $G$. Indeed, something stronger can be said. Suppose that a group $\Gamma$ acts transitively by permutations on a ...
- 19.1k
8
votes
Fixed point theorems
I forgot who proved it, but the statement is nice and very easy to prove: A function $f:X\to X$ is fixed point free if and only if there is a partition of $X$ into three subsets s.t. $f$ maps each of ...
8
votes
Can the Lawvere fixed point theorem be used to prove the Brouwer fixed point theorem?
Here is a sort of partial solution. I doubt it will be very helpful, but anyone who wants to read it is free to do so.
Let $I$ be any space with the fixed point property. We will construct a space $X$...
- 116k
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