# Tagged Questions

Fixed-point theorem is a result about existence of fixed points, i.e. points fulfilling F(x)=x, under some conditions on the function F. Results of this type appear in many areas of mathematics, e.g. functional analysis (Banach), algebraic topology (Brouwer), lattice theory (Knaster-Tarski, Kleene)...

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### Numerically solving a transport equation

I would like to solve this transport PDE numerically : $$\partial_t f + v(f) \partial_x f = 0$$ What I would want to do is "freeze" the velocity $v$ and solve a classical transport equation by ...
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### A generalization of Poincare-Birkhoff theorem

What could be the statment of a possible generalization of Poincare Birkhoff theorem for $M\times [0,\; 1]$ where $M$ is a compact orientable manifold?
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### Finite groups whose non-trivial elements have no fixed points

(I first asked this question on MathOverflow, but was recommended to ask here at Mathstackexchange instead.) I am interested in finite groups $G$ acting on a finite set $X$ with the following ...
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### Brouwer fixed-point theorem on non-convex sets [duplicate]

I read about the Brouwer fixed-point theorem in Wikipedia, and got confused about whether it holds when the domain of the function is non convex. On one hand, convexity is explicitly mentioned as a ...
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### Theorem about number of crossing-points between a function and a line

Assume $f(x)$, with $x \in [a,b]$. Take $u$ so that $f(a)<u<f(b)$. By the Intermediate value theorem, we know that $f(x)$ crosses $u$ at least once. My question is, given some extra information ...
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### Can somebody explain the notation $f \in C^4$

To give some context the full question is: Suppose $f \in C^4$ in a interval containing the root $\alpha$ and that Newton’s method gives a sequence of iterates $\{x_k\}$, $k = 0, 1, 2, \dots$ which ...
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### Brouwer's fixed-point theorem and the intermediate value theorem?

In one dimension, Brouwer's fixed-point theorem (BPFT) can be proved easily based on the Intermediate Value Theorem (IVT). Is the inverse also true? I.e, is it possible to prove the IVT directly from ...
By "comb space", I mean the space $X=([0,1] \times \{0\}) \cup (K \times [0,1])$, where $K=\{ \frac{1}{n} : n \in \mathbb{N}^+ \}$, without the leftmost vertical line segment. How to prove that this ...