MathOverflow

Can $\mathbb R^2$ be the union of a family of infinitely differentiable, injective, unit-speed images of the real line such that at each point of the plane, precisely two of these curves merge (they coincide from that point on)?

Assign exactly one $0$ and one $1$ to the two ends of each edge of a regular simple undirected bipartite graph of $e$ edges, uniform degree $d$ and number of vertices $2v$. The assignment of the pair $\{0,1\}$ to an edge can be considered as orienting...

I'm wondering how Riemann knew that $\zeta(z)$ could be extended to a larger domain. In particular, who was the first person to explicitly extend the domain of a complex valued function and what was the function?

Let $(M, \tau)$ be a von Neumann algebra with faithful normal tracialstate, and let $(x_i)_{i=1}^n$ be a sequence of self-adjoint elementsin $M$ with $\tau(x_i) = 0$. Set $S_k = \sum_{i=1}^k x_i$.
Define projections recursively:$$e_0 = \mathbf{1}, \qquad e_k = e_{k-1}\,\mathbf{1}_{[0,t)}\!\left(e_{k-1}|S_k|e_{k-1}\right), \quad k = 1, \ldots, n.$$This gives a decreasing sequence of projections$e_0 \geq e_1 \geq \cdots \geq e_n$. What I want to...