A k-permutation family on n vertices is a set-system consisting of the intervals of k permutations of the integers 1 to n. The discrepancy of a set-system is the minimum over all red–blue vertex colourings of the maximum difference between the number of red and blue vertices in any set in the system. In 2011, Newman and Nikolov disproved a conjecture of Beck that the discrepancy of any 3-permutation family is at most a constant independent of n. Here we give a simpler proof that Newman and Nikolov’s sequence of 3-permutation families has discrepancy $\Omega (\log \,n)$. We also exhibit a sequence of 6-permutation families with root-mean-squared discrepancy $\Omega (\sqrt {\log \,n} )$; that is, in any red–blue vertex colouring, the square root of the expected squared difference between the number of red and blue vertices in an interval of the system is $\Omega (\sqrt {\log \,n} )$.

Understanding individual differences in attitudes to autism is crucial for improving attitudes and reducing stigma towards autistic people, yet there is limited and inconsistent research on this topic. This is compounded by a lack of appropriate measures and multivariate analyses. Addressing these issues, using up-to-date measures and multiple linear regression, we examined the relative contributions of participant age, sex, autism knowledge, level of contact with autistic people, and autistic traits to attitudes towards autistic people. We found that greater autism knowledge and higher levels of contact, but no other variables, were uniquely predictive of attitudes towards autistic people. We conclude that, in addition to public awareness campaigns to raise knowledge of autism, it may be important to increase contact between autistic and non-autistic people to improve public attitudes towards autistic people.

The outbreak of the novel coronavirus severe acute respiratory syndrome-coronavirus-2 has raised major health policy questions and dilemmas. Whilst respiratory droplets are believed to be the dominant transmission mechanisms, indirect transmission may also occur through shared contact of contaminated common objects that is not directly curtailed by a lockdown. The conditions under which contaminated common objects may lead to significant spread of coronavirus disease 2019 during lockdown and its easing is examined using the susceptible-exposed-infectious-removed model with a fomite term added. Modelling the weekly death rate in the UK, a maximum-likelihood analysis finds a statistically significant fomite contribution, with 0.009 ± 0.001 (95% CI) infection-inducing fomites introduced into the environment per day per infectious person. Post-lockdown, comparison with the prediction of a corresponding counterfactual model with no fomite transmission suggests fomites, through enhancing the overall transmission rate, may have contributed to as much as 25% of the deaths following lockdown. It is suggested that adding a fomite term to more complex simulations may assist in the understanding of the spread of the illness and in making policy decisions to control it.

Although handwashing is an effective way to prevent infections, there is scarce evidence on predictors of handwashing during a pandemic. This paper aims to identify behavioural and demographic predictors of handwashing. The study surveyed 674 adults in Malaysia in May 2020 regarding whether the time spent on social media predicted handwashing contingent on gender and number of children. More time spent on social media was positively associated with handwashing for males with three or more children. However, for males without children, social media use was negatively associated with handwashing. The association was not significant for males with one or two children. For females, more time spent on social media was significantly linked to more handwashing only for females with one child. Gender, a traditional predictor of handwashing, was a useful predictor only for those who spent more than three hours per day on social media and had at most one child. Number of children was a novel negative predictor for males who did not use social media and who averaged one hour per day on social media, a positive predictor for males who spent lots of time on social media, but not a predictor for females. In sum, social media use predicts handwashing, and is thus a helpful variable for use in targeted health communication during a pandemic – particularly through social media. Further, more conventional predictors like gender and number of children exhibit contingency effects with social media use.

Organo-modified clay nanoparticles were mixed at 1 and 5 wt% concentrations with a molten blend of 75 wt% of polylactide (PLA) and 25 wt% poly[(butylene adipate)-co-terephthalate] (PBAT). Three mixing strategies were used to control the localization of nanoclay. Small amplitude oscillatory shear (SAOS) and stress growth tests were conducted to clarify the nanoclay interactions with the blend components and its effect on the molecular relaxation behavior. SAOS and weighted relaxation spectra properties were determined before and after pre-shearing at a rate of 0.01 s−1. Molecular relaxation and its characteristics were influenced by PLA degradation, PBAT droplet coalescence, and nanoclay localization.

The number of people who receive a stable income for life from a closed pooled annuity fund is studied. Income stability is defined as keeping the income within a specified tolerance of the initial income in a fixed proportion of future scenarios. The focus is on quantifying the effect of the number of members, which drives the level of idiosyncratic longevity risk in the fund, on the income stability. To do this, investment returns are held constant, and systematic longevity risk is omitted. An analytical expression that closely approximates the number of fund members who receive a stable income is derived and is seen to be independent of the mortality model. An application of the result is to calculate the length of time for which the pooled annuity fund can provide the desired level of income stability.

Population analyses of functional connectivity have provided a rich understanding of how brain function differs across time, individual, and cognitive task. An important but challenging task in such population analyses is the identification of reliable features that describe the function of the brain, while accounting for individual heterogeneity. Our work is motivated by two particularly important challenges in this area: first, how can one analyze functional connectivity data over populations of individuals, and second, how can one use these analyses to infer group similarities and differences. Motivated by these challenges, we model population connectivity data as a multilayer network and develop the multi-node2vec algorithm, an efficient and scalable embedding method that automatically learns continuous node feature representations from multilayer networks. We use multi-node2vec to analyze resting state fMRI scans over a group of 74 healthy individuals and 60 patients with schizophrenia. We demonstrate how multilayer network embeddings can be used to visualize, cluster, and classify functional regions of the brain for these individuals. We furthermore compare the multilayer network embeddings of the two groups. We identify significant differences between the groups in the default mode network and salience network—findings that are supported by the triple network model theory of cognitive organization. Our findings reveal that multi-node2vec is a powerful and reliable method for analyzing multilayer networks. Data and publicly available code are available at https://github.com/jdwilson4/multi-node2vec.

During the first months of the severe acute respiratory syndrome-coronavirus-2 (SARS-CoV-2) epidemic in 2020, Spain implemented an initial lockdown period on 15 March followed by a strengthened lockdown period on 30 March when only essential workers continued to commute to work. However, little is known about the epidemic dynamics in different age groups during these periods.

We used the daily number of coronavirus 2019 cases (by date of symptom onset) reported to the National Epidemiological Surveillance Network among individuals aged 15–19 years through 65–69 years. For each age group g, we computed the proportion PrE(g) of individuals in age group g among all reported cases aged 15–69 years during the pre-lockdown period (1−10 March 2020) and the corresponding proportion PrL(g) during two lockdown periods (initial: 25 March−3 April; strengthened: 8–17 April 2020). For each lockdown period, we computed the proportion ratios PR(g) = PrL(g)/PrE(g). For each pair of age groups g1, g2, PR(g1)>PR(g2) implies a relative increase in the incidence of detected SARS-CoV-2 infection in the age group g1 compared with g2 for the lockdown period vs. the pre-lockdown period.

For the initial lockdown period, the highest PR values were in age groups 50–54 years (PR = 1.21; 95% CI: 1.12,1.30) and 55–59 years (PR = 1.19; 1.11,1.27). For the second lockdown period, the highest PR values were in age groups 15–19 years (PR = 1.26; 0.95,1.68) and 50–54 years (PR = 1.20; 1.09,1.31).

Our results suggest that different outbreak control measures led to different changes in the relative incidence by age group. During the initial lockdown period, when non-essential work was allowed, individuals aged 40–64 years, particularly those aged 50–59 years, had a higher relative incidence compared with the pre-lockdown period. Younger adults/older adolescents had an increased relative incidence during the later, strengthened lockdown. The role of different age groups during the epidemic should be considered when implementing future mitigation efforts.

Measurement errors are omnipresent in network data. Most studies observe an erroneous network instead of the desired error-free network. It is well known that such errors can have a severe impact on network metrics, especially on centrality measures: a central node in the observed network might be less central in the underlying, error-free network. The robustness is a common concept to measure these effects. Studies have shown that the robustness primarily depends on the centrality measure, the type of error (e.g., missing edges or missing nodes), and the network topology (e.g., tree-like, core-periphery). Previous findings regarding the influence of network size on the robustness are, however, inconclusive. We present empirical evidence and analytical arguments indicating that there exist arbitrary large robust and non-robust networks and that the average degree is well suited to explain the robustness. We demonstrate that networks with a higher average degree are often more robust. For the degree centrality and Erdős–Rényi (ER) graphs, we present explicit formulas for the computation of the robustness, mainly based on the joint distribution of node degrees and degree changes which allow us to analyze the robustness for ER graphs with a constant average degree or increasing average degree.

Data on the prevalence of extrapulmonary tuberculosis (EPTB) patients are limited in many African countries including Malawi. We conducted a retrospective review of all histology reports for cancer suspected patients at Mzuzu Central Hospital (MZCH) between 2013 and 2018 to determine the proportion of EPTB cases among cancer suspected patients and characterised them epidemiologically. All reports with inconclusive findings were excluded. In total, 2214 reports were included in the review, 47 of which reported EPTB, representing 2.1% (95% CI 1.6−2.8). The incidence of EPTB was significantly associated with sex, age and HIV status. Men were more than twice (OR 2.1; 95% CI 1.2–3.9) as likely to have EPTB as women while those with HIV were more than six times (OR 6.4; 95% CI 1.7–24.8) as likely to have EPTB compared to those who were HIV-negative. EPTB demonstrated an inverse relationship with age. The highest proportion of EPTB was found from neck lymph nodes (10.3% (5.4–17.2)). A reasonable number of EPTB cases are diagnosed late or missed in Malawi's hospitals. There is a need for concerted efforts to increase EPTB awareness and likely come up with a policy to consider EPTB as a differential diagnosis in cancer suspected patients.

The validity of network observations is sometimes of concern in empirical studies, since observed networks are prone to error and may not represent the population of interest. This lack of validity is not just a result of random measurement error, but often due to systematic bias that can lead to the misinterpretation of actors’ preferences of network selections. These issues in network observations could bias the estimation of common network models (such as those pertaining to influence and selection) and lead to erroneous statistical inferences. In this study, we proposed a simulation-based sensitivity analysis method that can evaluate the robustness of inferences made in social network analysis to six forms of selection mechanisms that can cause biases in network observations—random, homophily, anti-homophily, transitivity, reciprocity, and preferential attachment. We then applied this sensitivity analysis to test the robustness of inferences for social influence effects, and we derived two sets of analytical solutions that can account for biases in network observations due to random, homophily, and anti-homophily selection.

Aircraft performance models play a key role in airline operations, especially in planning a fuel-efficient flight. In practice, manufacturers provide guidelines which are slightly modified throughout the aircraft life cycle via the tuning of a single factor, enabling better fuel predictions. However, this has limitations, in particular, they do not reflect the evolution of each feature impacting the aircraft performance. Our goal here is to overcome this limitation. The key contribution of the present article is to foster the use of machine learning to leverage the massive amounts of data continuously recorded during flights performed by an aircraft and provide models reflecting its actual and individual performance. We illustrate our approach by focusing on the estimation of the drag and lift coefficients from recorded flight data. As these coefficients are not directly recorded, we resort to aerodynamics approximations. As a safety check, we provide bounds to assess the accuracy of both the aerodynamics approximation and the statistical performance of our approach. We provide numerical results on a collection of machine learning algorithms. We report excellent accuracy on real-life data and exhibit empirical evidence to support our modeling, in coherence with aerodynamics principles.

We study and classify proper q-colourings of the ℤd lattice, identifying three regimes where different combinatorial behaviour holds. (1) When $q\le d+1$, there exist frozen colourings, that is, proper q-colourings of ℤd which cannot be modified on any finite subset. (2) We prove a strong list-colouring property which implies that, when $q\ge d+2$, any proper q-colouring of the boundary of a box of side length $n \ge d+2$ can be extended to a proper q-colouring of the entire box. (3) When $q\geq 2d+1$, the latter holds for any $n \ge 1$. Consequently, we classify the space of proper q-colourings of the ℤd lattice by their mixing properties.

Let G be a graph on n vertices and with maximum degree Δ, and let k be an integer. The k-recolouring graph of G is the graph whose vertices are k-colourings of G and where two k-colourings are adjacent if they differ at exactly one vertex. It is well known that the k-recolouring graph is connected for $k\geq \Delta+2$. Feghali, Johnson and Paulusma (J. Graph Theory 83 (2016) 340–358) showed that the (Δ + 1)-recolouring graph is composed by a unique connected component of size at least 2 and (possibly many) isolated vertices.

In this paper, we study the proportion of isolated vertices (also called frozen colourings) in the (Δ + 1)-recolouring graph. Our first contribution is to show that if G is connected, the proportion of frozen colourings of G is exponentially smaller in n than the total number of colourings. This motivates the use of the Glauber dynamics to approximate the number of (Δ + 1)-colourings of a graph. In contrast to the conjectured mixing time of O(nlog n) for $k\geq \Delta+2$ colours, we show that the mixing time of the Glauber dynamics for (Δ + 1)-colourings restricted to non-frozen colourings can be Ω(n2). Finally, we prove some results about the existence of graphs with large girth and frozen colourings, and study frozen colourings in random regular graphs.