A conjeccture concerning strongly connected graphs

AbstractLet Cl(n;d be the class of all directed graphsG, without loops and without multiple arcs, such that each graphG hasn vertices andd arcs. A primal subgraph ofG is generated by deleting one vertex and all the arcs going out from this vertex or into it. We conjecture that ifG ε Cl(n;d) where (n(n − 1)/2) + 1 ⩽ d ⩽ n(n − 1), and ifG is strongly connected, then it has a strongly connected primal subgraph. This conjecture is verified forn = 3, 4, and 5 (Theorems 1, 3′ and 5). Two related results hold for all n (Theorems 2 and 4)

Totally positive differential systems Binyamin Schwarz.

Mathematics Technical Repor

Norm conditions for disconjugacy of complex differential systems

Mathematics Technical Repor

Carathéodory balls and norm balls in Hp,n=z∈Cn:∥z∥p<1H_{p,n} = {z ∈ ℂ^{n} :∥z∥ _{p} < 1}

It is shown that for n ≥ 2 and p > 2, where p is not an even integer, the only balls in the Carathéodory distance on $H_{p,n} = {z ∈ ℂ^{n}: ∥ z∥_{p} < 1 }$ which are balls with respect to the complex $l_{p}$ norm in $ℂ^{n}$ are those centered at the origin