The problem of determining the set of possible eigenvalues of 3 Hermitian matrices that sum up to zero is known as the Horn problem. The answer is a polyhedral cone, which, following Knutson and Tao, can be described as the projection of a simpler cone in the space of triangular tableaux (or hives) to the boundary nodes of the tableau.
In this paper, we introduce a combinatorial problem defined in terms of certain weighted planar graphs giving rise to exactly the same polyhedral cone. In our framework, the values at the inner nodes of the triangular tableaux receive a natural interpretation. Other problems of linear algebra fit into the same scheme, among them the Gelfand--Zeitlin problem. Our approach is motivated by the works of Fomin and Zelevinsky on total positivity and by the ideas of tropicalization.
Circular Planar Resistor Networks with Nonlinear and Signed Conductors
本文研究含非线性电阻的离散电网络的边值问题。Curtis，Ingerman，Morrow，Colin de Verdiere，Gitler和Vertigan的推广工作，我们刻画了圆形可平面图的逆边值问题有一个解决方案，在这个广义的非线性设置。答案与线性设置相同。我们的证明方法从不要求电阻器以连续或单调的方式工作;这使我们能够在许多情况下恢复有符号电导。我们将这一问题的恢复图，不是圆形平面。我们还使用我们的结果，使一个轻浮的纽结理论的声明，并稍微推广了一个事实证明了林和Pylyavskyy关于因式分解计划在他们的电气线性群。
