We show that if a graph admits packing and covering both consisting of ? many spanning trees, where is some infinite cardinal, then the also decomposition into trees. For finite analogous question remains open, however, slightly weaker statement proved.

This short note gives an alternative proof of the M-convex intersection theorem, which is one of the central results in discrete convex analysis. This note is intended to provide a direct simpler proof accessible to nonexperts. 1 M-Convex Intersection Theorem The M-convex intersection theorem [3, Theorem 8.17] reads as follows, where V is a nonempty finite set, and Z and R are the sets of integ...

When solving the inverse scattering problem for a discrete Sturm–Liouville operator with a rapidly decreasing potential, one gets reflection coefficients s± and invertible operators I +Hs± , where Hs± is the Hankel operator related to the symbol s±. The Marchenko–Faddeev theorem [8] (in the continuous case, for the discrete case see [4, 6]), guarantees the uniqueness of the solution of the inve...

Mg,n ψ1 1 · · ·ψ dn n . Witten-Kontsevich theorem [9, 4] provides a recursive way to compute all these intersection numbers. However explicit and effective recursion formulae for computing intersection indices are still very rare and very welcome. Our n-point function formula [5] computes intersection indices recursively by decreasing the number of marked points. So it is natural to ask whether...