Intereting Posts

isomorphism, integers of mod $n$.
On the existence of finitely generated injective modules (Bruns and Herzog, Exercise 3.1.23)
looking for materials on Martin Axiom
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Show that $\int_0^\pi \log^2\left(\tan\frac{ x}{4}\right)dx=\frac{\pi^3}{4}.$
Good way to learn Ramsey Theory
Why are maximal consistent sets essential to Henkin-proofs of Completeness?
Finding the first digit of $2015^{2015}$
How to integrate $\int\frac{\ln x\,dx}{x^2+2x+4}$
What is a counterexample to the converse of this corollary related to the Dominated Convergence Theorem?
Proof about cubic $t$-transitive graphs
Integral of the Von karman equation
Explain a surprisingly simple optimization result
Proving The Extension Lemma For Vector Fields On Submanifolds
Diophantine equations solved using algebraic numbers?

The stable manifold theoremtell us:

A local stable manifold $W^{s}_{loc}(x^{*})$, which is a differentiable manifold of class $\mathcal{C}^{r}$ and dimension $n_{-}, $ tangent to the stable subspace $E^{-}$ at $x^{*}$, and which can be representable as graph.

A local unstable manifold $W^{u}_{loc}(x^{*})$, which is a differentiable manifold of class $\mathcal{C}^{r}$ and dimension $n_{+}, $ tangent to the stable subspace $E^{+}$ at $x^{*}$, and which can be representable as graph.

- Smoothness of the boundary is the only obstruction for being a submanifold with boundary?
- Vanishing of the first Chern class of a complex vector bundle
- Number of Differentiable Structures on a Smooth Manifold
- parallelizable manifolds
- The cone is not immersed in $\mathbb{R}^3$
- “Immediate” Applications of Differential Geometry

I don’t get it very well what does it mean about ** A differentiable manifold of class $\mathcal{C}^{r}$ tangent to $E^{\pm}$ and representable as graph**.

Given the below imagen, how can it be interpreted with such definition?

- Showing that a level set is not a submanifold
- Sanity check about Wikipedia definition of differentiable manifold as a locally ringed space
- Properly discontinuous action: equivalent definitions
- Simple question on perturbation theory for a function with two small parameters
- Proof that two spaces that are homotopic have the same de Rham cohomology
- Definition of Hamiltonian system through integral invariant
- When does gradient flow not converge?
- Why can we always take the zero section of a vector bundle?
- Proving Nonhomogeneous ODE is Bounded
- Software to draw links or knots

Since stable and unstable subspaces complement each other (I mean, $\mathbb{R}^2 = E^{+} \oplus E^{-}$ and any vector $v = \pi_{E^{+}} v + \pi_{E^{-}} v$), the two dimensional case could be interpreted this way:

There exists a (1) $C^r$-function $\psi\, \colon E^{+} \rightarrow E^{-}$ such that (2) $\psi(0) = 0$, (3) $D\psi = 0$ and set (4) $x + \psi ( x)$ is a local unstable manifold.

- (1) tells you that set $x + \psi(x)$ would be a $C^r$ differentiable manifold; also, since this manifold has special analytical form ($x + \psi(x)$, $x \in E^{+}$, $\psi(x) \in E^{-}$), it’s called
**graph** - (2) tells that this manifold passes through the origin
- (3) means that manifold is tangent to $E^{+}$ at origin

(it’s very easy to see through straightforward differentiation) - (4) just means that all trajectories that start at $x + \psi(x)$ in some small neighbourhood of origin

tend to origin in backward time

Hope this illustration will help:

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- We define a sequence of rational numbers {$a_n$} by putting $a_1=3$ and $a_{n+1}=4-\frac{2}{a_n}$ for all natural numbers. Put $\alpha = 2 + \sqrt{2}$
- probability circle determined by chord determined by two random points is enclosed in bigger circle
- Finite index subgroup $G$ of $\mathbb{Z}_p$ is open.
- What's the difference between saying that there is no cardinal between $\aleph_0$ and $\aleph_1$ as opposed to saying that…
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- Axiom of Choice and Determinacy
- Example of composition of two normal field extensions which is not normal.
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- How not to prove the Riemann hypothesis
- Entire function. Prove that $f(\bar{z})=\overline{f(z)}, \forall z\in C$