Networks of well-known dynamical units but unknown interaction topology arise across various fields of biology, including genetics, ecology, and neuroscience. The collective dynamics of such networks is often sensitive to the presence (or absence) of individual interactions, but there is usually no direct way to probe for their existence. Here we present an explicit method for reconstructing interaction networks of leaky integrate-and-fire neurons from the spike patterns they exhibit in response to external driving. Given the dynamical parameters are known, the approach works well for networks in simple collective states but is also applicable to networks exhibiting complex spatio-temporal spike patterns. In particular, stationarity of spiking time series is not required.

We provide an optimally mixing Markov chain for 6-colorings of the square lattice on rectangular regions with free, fixed, or toroidal boundary conditions. This implies that the uniform distribution on the set of such colorings has strong spatial mixing, so that the 6-state Potts antiferromagnet has a finite correlation length and a unique Gibbs measure at zero temperature. Four and five are now the only remaining values of q for which it is not known whether there exists a rapidly mixing Markov chain for q-colorings of the square lattice.

A compartmental model is proposed to predict the coronavirus 2019 (Covid-19) spread. It considers: detected and undetected infected populations, social sequestration, release from sequestration, plus reinfection. This model, consisting of seven coupled equations, has eight coefficients which are evaluated by fitting data for eight US states that make up 43% of the US population. The evolution of Covid-19 is fairly similar among the states: variations in contact and undetected recovery rates remain below 5%; however, variations are larger in recovery rate, death rate, reinfection rate, sequestration adherence and release rate from sequestration. Projections based on the current situation indicate that Covid-19 will become endemic. If lockdowns had been kept in place, the number of deaths would most likely have been significantly lower in states that opened up. Additionally, we predict that decreasing contact rate by 10%, or increasing testing by approximately 15%, or doubling lockdown compliance (from the current ~15% to ~30%) will eradicate infections in Texas within a year. Extending our fits for all of the US states, we predict about 11 million total infections (including undetected), and 8 million cumulative confirmed cases by 1 November 2020.

We provide an optimally mixing Markov chain for 6-colourings of the square lattice on rectangular regions with free, fixed, or toroidal boundary conditions. This implies that the uniform distribution on the set of such colourings has strong spatial mixing, so the six-state Potts antiferromagnet has a finite correlation length and a unique Gibbs measure at zero temperature. Four and five are now the only remaining values of q for which it is not known whether there exists a rapidly mixing Markov chain for q-colourings of the square lattice.