Tom Howard | Search arXiv

arXiv2020

A simple demonstration of shear-flow instability

We describe a simple classroom demonstration of a fluid-dynamic instability. The demonstration requires only a bucket of water, a piece of string and some used tealeaves or coffee grounds. We argue that the mechanism for the instability, at least in its later stages, is two-dimensional barotropic (shear-flow) instability and we present evidence in support of this. We show results of an equivalent basic two-dimensional numerical non-linear model, which simulates behavior comparable to that observed in the bucket demonstration. Modified simulations show that the instability does not depend on the curvature of the domain, but rather on the velocity profile.

physics.ed-ph↗

arXiv2010

Representations of monomial algebras have poly-exponential complexities

We use directed graphs called "syzygy quivers" to study the asymptotic growth rates of the dimensions of the syzygies of representations of finite dimensional algebras. For any finitely generated representation of a monomial algebra, we show that this growth rate is poly-exponential, i.e. the product of a polynomial and an exponential function, and give a procedure for computing the corresponding degree and base from a syzygy quiver. We characterize the growth rates arising in this context: The bases of the occurring exponential functions are the real, nonnegative algebraic integers $b$ whose irreducible polynomial over $\mathbb{Q}$ has no root with with modulus larger than $b$. Moreover, we show that these growth rates are invariant under stable derived equivalences.

math.RT↗