Intereting Posts

A simple curve of positive area
torus by identifying two equivalent points (mod $\mathbb{Z^2}$)
Using the identity theorem: can there be an analytic function $f$ with $f\left(\frac{1}{n^2}\right) = \frac{1}{n}$
Calculate using residues $\int_0^\infty\int_0^\infty{\cos\frac{\pi}2\Big(nx^2-\frac{y^2}n\Big)\cos\pi xy\over\cosh\pi x\cosh\pi y}dxdy,n\in\mathbb{N}$
Euler function – all values of n give $\phi(n)=10$
Existence of a continuous function which does not achieve a maximum.
A Local Homeomorphism Between Compact Connected Hausdorff Topological Spaces
Finding a primitive root of a prime number
On the mean value of a multiplicative function: Prove that $\sum\limits_{n\leq x} \frac{n}{\phi(n)} =O(x) $
The proof of $e^x \leq x + e^{x^2}$
Why is this not a triangulation of the torus?
Is the complement of a countable set in $\mathbb{R}$ dense? Application to convergence of probability distribution functions.
Which of the following polynomials are subspaces of $\mathbb{P}_n$ for an appropriate value of n?
Probability of winning a game in tennis?
Prove the inequality $\frac 1a + \frac 1b +\frac 1c \ge \frac{a^3+b^3+c^3}{3} +\frac 74$

Consider a closed unit ball $\mathbb{B}^n = \{ x\in\mathbb{R^n} : \|x\|\le 1\} $

How do I show that $\mathbb{B}^n $ is a smooth manifold with its boundary ($\partial \mathbb{B^n}$) diffeomorphic to $S^{n-1}$ ?

- Applications of Principal Bundle Construction: Vague Question
- Recovering connection from parallel transport
- $SL(n)$ is a differentiable manifold
- Is this a valid way to show $\chi(SL_n(\mathbb{R}))=0$?
- Volume forms and volume of a smooth manifold
- Redundance of the Smoothness of the Inversion Map in the Definition of a Lie Group.

- Notion of a distribution as acting on tangent spaces
- Free and proper action
- Reference request: infinite-dimensional manifolds
- Are closed orbits of Lie group action embedded?
- Redundance of the Smoothness of the Inversion Map in the Definition of a Lie Group.
- Properly discontinuous action: equivalent definitions
- parallelizable manifolds
- What are examples of parallelizable complex projective varieties?
- Does every open manifold admit a function without critical point?
- Intuition for Smooth Manifolds

There is a lemma in Milnor’s “Topology from the differentiable viewpoint” (it’s the lemma 3 on the chapter 2) that says:

Let $M$ be a manifold without boundary and $g:M\to \mathbb R$ a smooth map with $0$ as regular value. The set $N=\{x\in M: g(x)\geq 0\}$ is a submanifold of $M$ with boundary and $\partial N = g^{-1}(0)$. Now, take $M=\mathbb R^m$ and $g(x)=1-|x|^2$ and you will obtain

$$N=\{x\in \mathbb R^m: 1-|x|^2\geq 0\} = \mathbb B^m$$

and $$\partial N = \mathbb S^{m-1}.$$

- Degree sequence of connected graphs
- This infinitely nested root gives me two answers $ \sqrt{4+\sqrt{8+\sqrt{32+\sqrt{512+\sqrt{\frac{512^2}{2}+\sqrt{…}}}}}} $
- Proving Binomial Identity without calculus
- dual of $H^1_0$: $H^{-1}$ or $H_0^1$?
- Distance between closed and compact sets.
- Summation using residues
- What is the most unfair set of three nontransitive dice?
- An application of the Inverse function theorem
- Does uniform continuity imply the interchangeability of the order of the limits?
- Modified Hermite interpolation
- Riemann rearrangement theorem
- Calculating $a^n\pmod m$ in the general case
- The method of proving the equality of integrals by showing they agree within $\epsilon$, for an arbitrary $\epsilon>0$
- C.H. Edwards “Advanced Calculus of Several Variables”, Problem 3.5 of page 194
- Intuition behind chain rule