The Gale–Nikaido–Debreu lemma plays an important role in establishing the existence of competitive equilibrium. In this paper, we use Sperner’s and basic elements topology to prove lemma.

This article tries to answer the question: How far can the algebro-geometric theory of rational descendant Gromov-Witten invariants be carried over to the tropical world? Given the fact that our moduli spaces are non-compact, the answer is surprisingly positive: We discuss universal families and the string, divisor and dilaton equations, we prove a splitting lemma describing the intersection wi...

Let nand m be integers with n = m 2 + m + 1. Then the projective plane of order m has n points and" lines with each line containing m + 1 ""n 1/2 points. In this paper, we consider the analogous problem for the Euclidean plane and show that there cannot be a comparably large collection of lines each of which contains approximately n 1/ 2 points from a given set of n points. More precisely, we s...