Space proof complexity for random 3-CNFs

We investigate the space complexity of refuting 3-CNFs in Resolution and algebraic systems. We prove that every Polynomial Calculus with Resolution refutation of a random 3-CNF φ in n variables requires, with high probability, distinct monomials to be kept simultaneously in memory. The same construction also proves that every Resolution refutation of φ requires, with high probability, clauses each of width to be kept at the same time in memory. This gives a lower bound for the total space needed in Resolution to refute φ. These results are best possible (up to a constant factor) and answer questions about space complexity of 3-CNFs

Sampling grid colourings with fewer colours

We provide an optimally mixing Markov chain for 6-colourings of the square grid. Furthermore, this implies that the uniform distribution on the set of such colourings has strong spatial mixing. 4 and 5 are now the only remaining values of k for which it is not known whether there exists a rapidly mixing Markov chain for k-colourings of the square grid

Rapid Mixing for Lattice Colorings with Fewer Colors

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.Comment: Appeared in Proc. LATIN 2004, to appear in JSTA

The Avon Longitudinal Study of Parents and Children - A resource for COVID-19 research:Antibody testing results, April – June 2021

The Avon Longitudinal Study of Parents and Children (ALSPAC) is a prospective population-based cohort which recruited pregnant women in 1990-1992 and has followed these women, their partners (Generation 0; G0) and their offspring (Generation 1; G1) ever since. The study reacted rapidly and repeatedly to the coronavirus disease 2019 (COVID-19) pandemic, deploying multiple online questionnaires and a previous home-based antibody test in October 2020. A second antibody test, in collaboration with ten other longitudinal population studies, was completed by 4,622 ALSPAC participants between April and June 2021. Of 4,241 participants with a valid spike protein antibody test result (8.2% were void), indicating antibody response to either COVID-19 vaccination or natural infection, 3,172 were positive (74.8%). Generational differences were substantial, with 2,463/2,555 G0 participants classified positive (96.4%) compared to 709/1,686 G1 participants (42.1%). Of 4,199 participants with a valid nucleocapsid antibody test result (9.2% were void), suggesting potential and recent natural infection, 493 were positive (11.7%); 248/2,526 G0 participants (9.8%) and 245/1,673 G1 participants (14.6%) tested positive, respectively. We also compare results for this round of testing to that undertaken in October 2020. Future work will combine these test results with additional sources of data to identify participants’ COVID-19 infection and vaccination status. These ALSPAC COVID-19 serology data are being complemented with linkage to health records and Public Health England pillar testing results as they become available, in addition to four previous questionnaire waves and a prior antibody test. Data have been released as an update to the previous COVID-19 datasets. These comprise: 1) a standard dataset containing all participant responses to all four previous questionnaires with key sociodemographic factors; and 2) individual participant-specific release files enabling bespoke research across all areas supported by the study. This data note describes the second ALSPAC antibody test and the data obtained from it