Intereting Posts

About a paper by Gold & Tucker (characterizing twin primes)
Expression for Levi-Civita Connection
Property of $\{0^n, 1^n, \ldots\}$
Definite Integral and Constant of Integration
Non-revealing maximum
Proof: $\sum_{i=1}^pE_i \doteq \bigoplus_{i=1}^p E_i \leftrightarrow \forall i\in \{1,…,p\}(E_i \cap \sum_{t \in \{1,…,p\}-\{i\}}E_t=\{0\})$
Continuity conditions for multivariate functions.
How do you find the Lie algebra of a Lie group (in practice)?
Show $\frac{(2n)!}{n!\cdot 2^n}$ is an integer for $n$ greater than or equal to $0$
There exists no injective function from the power set of A to A
Example of set which contains itself
Proof of irrationality of square roots without the fundamental theorem of arithmetic
Refining Rudin's proof of $\lim \left (1+\frac 1 n\right)^n =\lim \sum_{k=1}^n \frac {1}{k!}$.
simplify $\sqrt{11+\sqrt{57}}$
Principle of Transfinite Induction

Let $S$ be a countable dense subset of $\mathbb R$. Must there exist a homeomorphism $f: \mathbb R \rightarrow \mathbb R$ such that $f(S) = \mathbb Q$? More weakly, must $S$ be homeomorphic to $\mathbb Q$?

- Prove $\csc(x)=\sum_{k=-\infty}^{\infty}\frac{(-1)^k}{x+k\pi}$
- Jensen's inequality for integrals
- Periodic functions and limit at infinity
- If $f$ continuous and $\lim_{x\to-\infty }f(x)=\lim_{x\to\infty }f(x)=+\infty $ then $f$ takes its minimum.
- Optimal assumptions for a theorem of differentiation under the integral sign
- If $Y$ is connected, why is $A\cup Y$ connected in this case?
- Is there a base in which $1 + 2 + 3 + 4 + \dots = - \frac{1}{12}$ makes sense?
- Complex projective line hausdorff as quotient space
- Does there exist a real Hilbert space with countably infinite dimension as a vector space over $\mathbb{R}$?
- Lipschitz continuity of atomless measures

Two countable totally ordered, densely ordered sets without endpoints are isomorphic—this is a theorem of Cantor (*Gesammelte Ahbandlungen*, chp. 9, page 303 ff. Springer, 1932) Thus two countable dense subsets of $\mathbb R$ are homeomorphic, since their topology is induced by their orders.

Now, if $A$, $B\subset\mathbb R$ are countable dense subsets, fix an order isomrphism $f:A\to B$ and extend it by continuity. What you get is an homeomorphism $\mathbb R\to\mathbb R$.

- Technical question about Strichartz estimate's proof.
- Problem with Abel summation
- Homology of disjoint union is direct sum of homologies
- How to prove the uniqueness of a continuous extension of a densely defined function?
- Are there infinitely many $A\in \mathbb{C}^{2 \times 2}$ satisfying $A^3 = A$?
- How to prove that $p-1$ is squarefree infinitely often?
- If $G$ is a groupe such that $|G|=p^m k$, does $G$ has a subgroup of order $p^n$ with $n<m$.
- the representation of a free group
- $ \int_1^2\int_1^2 \int_1^2 \int_1^2 \frac{x_1+x_2+x_3-x_4}{x_1+x_2+x_3+x_4}dx_1dx_2dx_3dx_4 $
- pigeonhole principle and division
- Prove that a connected graph not having $P_4$ or $C_3$ as an induced subgraph is complete bipartite
- Line integral over ellipse in first quadrant
- Proving $\mathbb{N}^k$ is countable
- Pointwise converging subsequence on countable set
- Abelianization of general linear group?