It is known that Szpiro’s conjecture, or equivalently the ABC-conjecture, implies Lang’s conjecture giving a uniform lower bound for the canonical height of nontorsion points on elliptic curves. In this note we show that a significantly weaker version of Szpiro’s conjecture, which we call “prime-depleted,” suffices to prove Lang’s conjecture.

In this paper we show that under some fairly general conditions the Overfull Conjecture about the chromatic index of a graph G implies the Conformability Conjecture about the total chromatic number of G. We also show that if G has even order and high maximum degree, then G is conformable unless the de0ciency is very small. c © 2001 Elsevier Science B.V. All rights reserved.

The conjecture is true in dimension 2 since the only surfaces which have positive Euler characteristic are S2 and RP2 and they are the only two which are not aspherical. In the special case where M2k is a nonpositively curved Riemannian manifold this conjecture is usually attributed to Hopf by topologists and either to Chern or to both Chern and Hopf by differential geometers. When I first hear...

In this note we generalize and unify two results on connectivity of graphs: one by Balinsky and Barnette, one by Athanasiadis. This is done through a simple proof using commutative algebra tools. In particular we use bounds for the Castelnuovo–Mumford regularity of their Stanley–Reisner rings. As a result, if ∆ is a simplicial d-pseudomanifold and s is the largest integer such that ∆ has a miss...

If N = qkn2 is an odd perfect number, where q is the Euler prime, then we show that σ(n) ≤ qk is necessary and sufficient for Sorli’s conjecture that k = νq(N) = 1 to hold. It follows that, if k = 1 then the Euler prime q is the largest prime factor of N and that q > 10500. We also prove that qk < 23n 2.

We give a review of the mathematical and physical properties of the celebrated family of Calogero-like models and related spin chains.

We show that it is consistent that the Borel Conjecture and the dual Borel Conjecture hold simultaneously.