Flow Polytopes of Partitions

Recent progress on flow polytopes indicates many interesting families with product formulas for their volume. These product formulas are all proved using analytic techniques. Our work breaks from this pattern. We define a family of closely related flow polytopes $F_{(\lambda, {\bf a})}$ for each partition shape $\lambda$ and netflow vector ${\bf a}\in Z^n_{> 0}$. In each such family, we prove that there is a polytope (the limiting one in a sense) which is a product of scaled simplices, explaining their product volumes. We also show that the combinatorial type of all polytopes in a fixed family $F_{(\lambda, {\bf a})}$ is the same. When $\lambda$ is a staircase shape and ${\bf a}$ is the all ones vector the latter results specializes to a theorem of the first author with Morales and Rhoades, which shows that the combinatorial type of the Tesler polytope is a product of simplices.

Product of Simplices and sets of positive upper density in ℝd

We establish that any subset of ℝd of positive upper Banach density necessarily contains an isometric copy of all sufficiently large dilates of any fixed two-dimensional rectangle provided d ⩾ 4.We further present an extension of this result to configurations that are the product of two non-degenerate simplices; specifically we show that if Δk1 and Δk2 are two fixed non-degenerate simplices of k1 + 1 and k2 + 1 points respectively, then any subset of ℝd of positive upper Banach density with d ⩾ k1 + k2 + 6 will necessarily contain an isometric copy of all sufficiently large dilates of Δk1 × Δk2.A new direct proof of the fact that any subset of ℝd of positive upper Banach density necessarily contains an isometric copy of all sufficiently large dilates of any fixed non-degenerate simplex of k + 1 points provided d ⩾ k + 1, a result originally due to Bourgain, is also presented.

Matching Ensembles (Extended Abstract)

International audience We introduce an axiom system for a collection of matchings that describes the triangulation of product of simplices. Nous introduisons un système d’axiomes pour une collection de couplages qui décrit la triangulation de produit de simplexes.

Small covers over a product of simplices

In this paper, we determine the number of equivariant homeomorphism classes of small covers over a product of m simplices for m ? 3 or for the dimension of each simplex being greater than 1 and m > 3. Moreover, we calculate the number of equivariant homeomorphism classes of all orientable small covers over a product of at most three simplices.

EQUILIBRIUM IN A DISCRETE EXCHANGE ECONOMY WITH REGIONAL SUB-ECONOMIES

In the first part of this note, we present a generalisation of a lemma due to Gale (1984) to a product of simplices. Then, we use it to derive a permutation-based extension of Browder Fixed Point Theorem proved by Bapat (1989). In the last part, we propose a model of an economy with regional sub-economies in the line of Gale's work (1984) and prove existence equilibrium results under different set of hypothesis.