MathOverflow

Let $I$ be an ideal of the ring of convergent power series $\mathbb{C}\{x,y\}$, and let $(f_1, \dots, f_k)$ be a system of generators of $I$. Consider the ideal$$\operatorname{Jac}(f_1, \dots, f_k) = \left( \frac{\partial f_i}{\partial x}\frac{\partial...

Theorem: For any odd integer $N > 1$ such that $N \equiv 1 \pmod 4$, the sequence of odd numbers generated by the Collatz odd-to-odd map will eventually contain either an odd number $M \equiv 3 \pmod 4$ or the number 1.
Proof:Let $N_0 = N$ be an odd integer with $N_0 > 1$ and $N_0 \equiv 1 \pmod 4$. Define the sequence of odd numbers $N_i$ using the odd-to-odd Collatz map $T(n)$: $N_{i+1} = T(N_i)$. Since $N_0$ is odd, all subsequent terms $N_i$ are also odd. Thus...

I am currently trying to understand Strichartz estimates for linear disperive equations on the circle:$$\begin{cases} i \frac{\partial u}{\partial t}= \Phi(\sqrt{-\partial^2_x})u\:,\: &\text{ in } \mathbb{T} \times \mathbb{T} \\u(\cdot,0)=f\:,&#...

The prototypical example of an abelian category is that of modules over a ring. Mitchell's embedding theorem tells us that we can essentially pretend that every abelian category is of this form.
The prototypical example of a dg-category is that of chain complexes over a field. Is it true that, up to some suitable precise statement, we can essentially pretend that every dg-category is of this form?...