We provide an asymptotically tight, computationally efficient approximation of the joint spectral radius of a set of matrices using sum of squares (SOS) programming. The approach is based on a search for an SOS polynomial that proves simultaneous contractibility of a finite set of matrices. We provide a bound on the quality of the approximation that unifies several earlier results and is indepe...

We provide an asymptotically tight, computationally efficient approximation of the joint spectral radius of a set of matrices using sum of squares (SOS) programming. The approach is based on a search for a SOS polynomial that proves simultaneous contractibility of a finite set of matrices. We provide a bound on the quality of the approximation that unifies several earlier results and is indepen...

In this paper we prove that the problem of deciding contractibility an arbitrary closed curve on boundary a 3-manifold is in NP. We emphasize manifold and are both inputs to problem. Moreover, our algorithm also works if given as compressed word. Previously, such was known for simple (non-compressed) curves, and, very limited cases, curves with self-intersections. Furthermore, fixed-parameter t...

In this note, we will introduce a notion of modularity of exact categories due to Masana Harada [Har05]. The naming is coming from the classical modular lattices theory [Bir48]. We will also state and prove so-called “homotopy Grayson-Staffeldt-Jordan-Hölder theorem” which is implicitly appeared in [Gra87] and [Sta89]. The theorem says contractibility of a simplicial set associated to a certain...

We extend and strengthen both Athey’s (2001) and McAdams’ (2003) results on the existence of monotone pure strategy equilibria in Bayesian games. We allow action spaces to be compact locally-complete metrizable semilattices, type spaces to be partially ordered complete separable metric spaces, and can handle both a weaker form of quasisupermodularity than is employed by McAdams and a weaker sin...

In [4], given a binary 26-adjacency voxel-based digital volume V , the homological information (that related to n-dimensional holes: connected components, ”tunnels” and cavities) is extracted from a linear map (called homology gradient vector field) acting on a polyhedral cell complex P (V ) homologically equivalent to V . We develop here an alternative way for constructing P (V ) based on homo...