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Mathoverflow.net news digest

  • 0 days

    Can the plane be this union of smooth injective curves, exactly two meeting at e...

    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)?

  • 9 months

    Extremal regular bipartite directed graph

    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...

  • 5 years

    What are some of the earliest examples of analytic continuation?

    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?

  • 0 days

    Spectral orthogonality between $e_n$ and $\mathbf{1}_{[t,\infty)}(|S_k|)$ in a t...

    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...

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