Narrow Results

We provide an elementary polyhedral approach to study and deduce results about the arithmeticity and commensurability of an infinite family of hyperbolic link complements $M_n$ for $n\ge 3$. The manifold $M_n$ is the complement of ${𝕊}^3$ by the ($2n$)-link chain ${{𝒟}}_{2n}$ and has $2n$ cusps. We show that $M_n$ is closely related to a hyperbolic Coxeter orbifold that is commensurable to an orbifold with a single cusp. Vinberg’s arithmeticity criterion and certain cusp density and volume computations allow us to reproduce some of the main results in [20] and [18] about $M_n$ in a comparatively elementary and direct way.

As a by-product, we give a rigorous proof of Thurston’s volume formula for $M_n$ and deduce that, for $n\ge 6$, the volume of $M_n$ is strictly bigger than the volume of the ($2n-1$)-cyclic cover over one component of the Whitehead link.

We give algorithms for the asymptotic expansions of the almost sure and moment Lyapunov exponents associated with the two-dimensional stochastic differential equation obtained as a small perturbation of the deterministic rotation with rate ω. The matrices in the perturbation terms are all assumed to be periodic functions of time with period l. The form of the algorithms depends on whether or not the periods 2π/ω and l of the unperturbed system and the perturbation coefficients are commensurable (i.e. the ratio of the periods is rational). In the commensurable case certain resonances may cause jumps in the Lyapunov exponents. We give an example of a stochastically perturbed Hill's oscillator which is almost surely stable when 2ω is not an integer, but is almost surely unstable at resonant frequencies ω = m/2. This work extends recent results of Imkeller and Milstein.

This paper performs a tendency judgment on the time symmetry of earthquake disasters in the Liaoning region through ternary, quarternary and pentanary commensurability calculations as well as analysis of butterfly structural drawings and commensurability structural systems based on the data of Ms≥5.7 earthquakes in Liaoning and the adjacent regions since 1900. The results demonstrate that the occurrence signal of Ms≥5.7 earthquake in Liaoning and adjacent regions in 2016 and 2018 are stronger, between which the possibility of occurrence of an earthquake in 2016 is greater than that in 2018. The research on spatial symmetry and epicenter migration characteristics find: longitudinal and latitudinal directions of the spatial migration have a certain synchronism and symmetry; the symmetry axis of latitudinal migration is at around 40.5 degrees north latitude while the symmetry axis of longitudinal migration is at around 122 degrees east longitude. Therefore, it is judged that the next earthquake is more likely to occur in the Yingkou-Haicheng region. The study may provide some reference and basis for earthquake tendencies in Liaoning region during the next 1∼3 years.