A monic polynomial $f(x)\in {\mathbb Z}[x]$ of degree N is called monogenic if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,\theta ,\theta ^2,\ldots ,\theta ^{N-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(\theta )$, where $f(\theta )=0$. We use the classification of the Galois groups of quartic polynomials, due to Kappe and Warren [‘An elementary test for the Galois group of a quartic polynomial’, Amer. Math. Monthly 96(2) (1989), 133–137], to investigate the existence of infinite collections of monogenic quartic polynomials having a prescribed Galois group, such that each member of the collection generates a distinct quartic field. With the exception of the cyclic case, we provide such an infinite single-parameter collection for each possible Galois group. We believe these examples are new and we provide evidence to support this belief by showing that they are distinct from other infinite collections in the literature. Finally, we devote a separate section to the cyclic case.

Aviation passenger screening has been used worldwide to mitigate the translocation risk of SARS-CoV-2. We present a model that evaluates factors in screening strategies used in air travel and assess their relative sensitivity and importance in identifying infectious passengers. We use adapted Monte Carlo simulations to produce hypothetical disease timelines for the Omicron variant of SARS-CoV-2 for travelling passengers. Screening strategy factors assessed include having one or two RT-PCR and/or antigen tests prior to departure and/or post-arrival, and quarantine length and compliance upon arrival. One or more post-arrival tests and high quarantine compliance were the most important factors in reducing pathogen translocation. Screening that combines quarantine and post-arrival testing can shorten the length of quarantine for travelers, and variability and mean testing sensitivity in post-arrival RT-PCR and antigen tests decrease and increase with the greater time between the first and second post-arrival test, respectively. This study provides insight into the role various screening strategy factors have in preventing the translocation of infectious diseases and a flexible framework adaptable to other existing or emerging diseases. Such findings may help in public health policy and decision-making in present and future evidence-based practices for passenger screening and pandemic preparedness.

Let a and b be positive integers and let $\{U_n\}_{n\ge 0}$ be the Lucas sequence of the first kind defined by

$$ \begin{align*}U_0=0,\quad U_1=1\quad \mbox{and} \quad U_n=aU_{n-1}+bU_{n-2} \quad \mbox{for }n\ge 2.\end{align*} $$

We define an $(a,b)$-Wall–Sun–Sun prime to be a prime p such that $\gcd (p,b)=1$ and $\pi (p^2)=\pi (p),$ where $\pi (p):=\pi _{(a,b)}(p)$ is the length of the period of $\{U_n\}_{n\ge 0}$ modulo p. When $(a,b)=(1,1)$, such primes are known in the literature simply as Wall–Sun–Sun primes. In this note, we provide necessary and sufficient conditions such that a prime p dividing $a^2+4b$ is an $(a,b)$-Wall–Sun–Sun prime.

The Variables and Slow Transients Survey (VAST) on the Australian Square Kilometre Array Pathfinder (ASKAP) is designed to detect highly variable and transient radio sources on timescales from 5 s to $\sim\!5$ yr. In this paper, we present the survey description, observation strategy and initial results from the VAST Phase I Pilot Survey. This pilot survey consists of $\sim\!162$ h of observations conducted at a central frequency of 888 MHz between 2019 August and 2020 August, with a typical rms sensitivity of $0.24\ \mathrm{mJy\ beam}^{-1}$ and angular resolution of $12-20$ arcseconds. There are 113 fields, each of which was observed for 12 min integration time, with between 5 and 13 repeats, with cadences between 1 day and 8 months. The total area of the pilot survey footprint is 5 131 square degrees, covering six distinct regions of the sky. An initial search of two of these regions, totalling 1 646 square degrees, revealed 28 highly variable and/or transient sources. Seven of these are known pulsars, including the millisecond pulsar J2039–5617. Another seven are stars, four of which have no previously reported radio detection (SCR J0533–4257, LEHPM 2-783, UCAC3 89–412162 and 2MASS J22414436–6119311). Of the remaining 14 sources, two are active galactic nuclei, six are associated with galaxies and the other six have no multi-wavelength counterparts and are yet to be identified.

Data from the Seasonal Ice Zone Observing Network (SIZONet) acquired near Barrow, Alaska, during the 2009/10 ice season allow novel comparisons between measurements of ice thickness and velocity. An airborne electromagnetic survey that passed over a moored Ice Profiling Sonar (IPS) provided coincident independent measurements of total ice and snow thickness and ice draft at a scale of 10 km. Once differences in sampling footprint size are accounted for, we reconcile the respective probability distributions and estimate the thickness of level sea ice at 1.48 ± 0.1 m, with a snow depth of 0.12 ± 0.07 m. We also complete what we believe is the first independent validation of radar-derived ice velocities by comparing measurements from a coastal radar with those from an under-ice acoustic Doppler current profiler (ADCP). After applying a median filter to reduce high-frequency scatter in the radar-derived data, we find good agreement with the ADCP bottom-tracked ice velocities. With increasing regulatory and operational needs for sea-ice data, including the number and thickness of pressure ridges, coordinated observing networks such as SIZONet can provide the means of reducing uncertainties inherent in individual datasets.

Here, we report reproducible and accurate measurement of crystallographic parameters using scanning transmission electron microscopy. This is made possible by removing drift and residual scan distortion. We demonstrate real-space lattice parameter measurements with <0.1% error for complex-layered chalcogenides Bi2Te3, Bi2Se3, and a Bi2Te2.7Se0.3 nanostructured alloy. Pairing the technique with atomic resolution spectroscopy, we connect local structure with chemistry and bonding. Combining these results with density functional theory, we show that the incorporation of Se into Bi2Te3 causes charge redistribution that anomalously increases the van der Waals gap between building blocks of the layered structure. The results show that atomic resolution imaging with electrons can accurately and robustly quantify crystallography at the nanoscale.

Purpose: There is limited research on the costs of social and emotional learning (SEL) interventions [Crowley, Jones, Greenberg, Feinberg & Spoth (2012). Resource Consumption of a Diffusion Model for Prevention Programs: The PROSPER Delivery System. Journal of Adolescent Health, 50 (3), 256–263]. This paper describes a comprehensive methodology for determining the costs of a successful universal, school-based SEL intervention that was implemented in nine public schools over 3 years.

Methods: Resource costs were identified using the Cost–Procedure–Process–Outcome Analysis Model [Yates (1996). Analyzing Costs, Procedures, Processes, and Outcomes in Human Services. Thousand Oaks, CA, US: Sage Publications, Inc.; Yates (1999). Measuring and Improving Cost, Cost-Effectiveness, and Cost-Benefit for Substance Abuse Treatment Programs. No. NIH 99-4518, 135] and the ingredients model [Levin (Ed.) (1983). Cost-Effectiveness A Primer (Vol. 4). Beverly Hills, CA: Sage; Levin & McEwan (2001). Cost-Effectiveness Analysis: Methods and Applications. (2nd ed.). Thousand Oaks, CA: Sage Publications]. This involved careful identification of resource use, finding the cost per unit for each resource by intervention activity, and ultimately calculating the total resource cost (resource use $\times$ cost per unit).

Results: Our analysis estimated the overall cost of this 3-year SEL and literacy intervention to be $1,831,296 for nine schools. This averages to $67,825 yearly per school and $130 yearly for each student. The analysis estimated the first year of the intervention to be the costliest ($683,106) and then decreasing in Year 2 ($581,764) and Year 3 ($566,426).

Conclusion: This research emphasizes the need to study the costs of SEL interventions. By providing a detailed and standardized methodology, this cost analysis can provide added support for implementing an effective social and emotional learning intervention in a school setting. Furthermore, it provides groundwork for more advanced cost analyses, such as a cost–effectiveness analysis or a benefit-cost analysis (BCA).

In this article we develop a test to determine whether a sextic polynomial that is irreducible over $\mathbb{Q}$ has Galois group isomorphic to the alternating group ${{A}_{4}}$. This test does not involve the computation of resolvents, and we use this test to construct several infinite families of such polynomials.

Illegal exploitation of resources is a cause of environmental degradation worldwide. The effectiveness of conservation initiatives such as marine protected areas relies on users' compliance with regulations. Although compliance can be motivated by social norms (e.g. peer pressure and legitimacy), some enforcement is commonly necessary. Enforcement is expensive, particularly in areas far from land, but costs can be reduced by optimizing enforcement. We present a case study of how enforcement could be optimized at Cocos Island National Park, Costa Rica, an offshore protected area and World Heritage Site. By analysing patrol records we determined the spatial and temporal distribution of illegal fishing and its relationship to patrol effort. Illegal fishing was concentrated on a seamount within the Park and peaked during the third year-quarter, probably as a result of oceanographic conditions. The lunar cycle in conjunction with the time of year significantly influenced the occurrence of incursions. The predictability of illegal fishing in space and time facilitates the optimization of patrol effort. Repeat offenders are common in the Park and we suggest that unenforced regulations and weak governance are partly to blame. We provide recommendations for efficient distribution of patrol effort in space and time, establishing adequate governance and policy, and designing marine protected areas to improve compliance. Our methods and recommendations are applicable to other protected areas and managed natural resources.

(See the commentary by Pfeiffer and Beldavs, on pages 984–986.)