Question 2: As Mariano points out, we can lift $\Phi \in \operatorname{Diff}^+{(\mathbb S^1)}$ uniquely to a diffeomorphism $\phi : \mathbb{R} \to \mathbb{R}$ such that $\phi(0) \in [0,1)$ and $\phi(x+1) = \phi(x) + 1$ for all $x \in \mathbb{R}$. Lift $\Psi$ similarly to $\psi$. For $t \in [0,1]...

I am trying to read arXiv:1510.08739 on "Fourier Uniformity". Here is the abstract: Let $\mathbb{F}$ be a fixed finite field, and let $A \subset \mathbb{F}^n$. It is a well-known fact that there is a subspace $V \leq \mathbb{F}^n$, $\boxed{\color{#97C757}{\mathbf{\mbox{codim} V \ll_{\delta} ...

I always put it this way, which is probably not 100% accurate but gives a meaningful picture: All math you see in highschool and the first two undergrad years is more than 300 years old, with few exceptions (elementary linear algebra and elementary group theory are more like 150 years old, say)....

The home page of the Polish Virtual Library has a good search interface and the older issues of the classical Polish journals such as Fund. Math., Studia Math., etc. as well as the monograph series are available for free more or less in their entirety. Here's a link to the old repository which is...

I am exactly copying the answer given by Mr.Charles Matthews given at MO. So I request neither to up-vote this answer or give me bounty. ( As the credit goes to Charles Matthews ) Answer By Charles Matthews : The Hasse principle works for quadratic forms. As soon as you consider cubic forms, it d...