Interactions in biology and social systems are not restricted to pairwise but can take arbitrary sizes. Extensive studies have revealed that the arbitrary-sized interactions significantly affect spreading dynamics on networked systems. Competing dynamics, i.e., several epidemics spread simultaneously compete with each other, been widely observed real world, yet way competing still lacks systema...

For a finite real reflection group W with Coxeter element γ we give a case-free proof that the closed interval, [I, γ], forms a lattice in the partial order on W induced by reflection length. Key to this is the construction of an isomorphic lattice of spherical simplicial complexes. We also prove that the greatest element in this latter lattice embeds in the type W simplicial generalised associ...

We show that for all n ≥ 3 and all primes p there are infinitely many simplicial toric varieties of codimension n in the 2n-dimensional affine space whose minimum number of defining equations is equal to n in characteristic p, and lies between 2n − 2 and 2n in all other characteristics. In particular, these are new examples of varieties which are settheoretic complete intersections only in one ...

Towards a Spectral Theory for Simplicial Complexes

The relation between algebraic shifting and join which was conjectured by Nevo [8] is studied. Let σ ∗ τ denote a join of two simplicial complexes σ and τ . Let ∆(σ) denote the exterior algebraic shifting of a simplicial complex σ. In the present paper, we will prove ∆(σ ∗ τ) ≤L ∆(∆(σ) ∗∆(τ)).

In this article, we present a set of musical transformations based on chord spaces representations derived from the Tonnetz. These chord spaces are formalized as simplicial complexes. A piece is represented in such a space by a trajectory. Spatial transformations are applied on these trajectories and induce a transformation of the original piece. These concepts are implemented in two applicatio...

The Upper Bound Conjecture is verified for a class of odd-dimensional simplicial complexes that in particular includes all Eulerian simplicial complexes with isolated singularities. The proof relies on a new invariant of simplicial complexes—a short simplicial h-vector.

A flag complex can be defined as a simplicial complex whose simplices correspond to complete subgraphs of its 1-skeleton taken as a graph. In this article, by introducing the notion of s-dismantlability, we shall define the s-homotopy type of a graph and show in particular that two finite graphs have the same s-homotopy type if, and only if, the two flag complexes determined by these graphs hav...

We introduce finite spaces and explain their relations to sets with reflexive and transitive relations and to simplicial complexes. We introduce Quillen’s conjecture: If Ap(G) is weakly contractible then G has a nontrivial normal p-subgroup. We use Quillen’s method to show this is true for solvable groups.