Coronavirus disease-2019 precipitated the rapid deployment of novel therapeutics, which led to operational and logistical challenges for healthcare organizations. Four health systems participated in a qualitative study to abstract lessons learned, challenges, and promising practices from implementing neutralizing monoclonal antibody (nMAb) treatment programs. Lessons are summarized under three themes that serve as critical building blocks for health systems to rapidly deploy novel therapeutics during a pandemic: (1) clinical workflows, (2) data infrastructure and platforms, and (3) governance and policy. Health systems must be sufficiently agile to quickly scale programs and resources in times of uncertainty. Real-time monitoring of programs, policies, and processes can help support better planning and improve program effectiveness. The lessons and promising practices shared in this study can be applied by health systems for distribution of novel therapeutics beyond nMAbs and toward future pandemics and public health emergencies.

In 2010, Turaev introduced knotoids as a variation on knots that replaces the embedding of a circle with the embedding of a closed interval with two endpoints which here we call poles. We define generalised knotoids to allow arbitrarily many poles, intervals and circles, each pole corresponding to any number of interval endpoints, including zero. This theory subsumes a variety of other related topological objects and introduces some particularly interesting new cases. We explore various analogs of knotoid invariants, including height, index polynomials, bracket polynomials and hyperbolicity. We further generalise to knotoidal graphs, which are a natural extension of spatial graphs that allow both poles and vertices.

In response to Timothy Darvill's article, ‘Mythical rings?’ (this issue), which argues for an alternative interpretation of Waun Mawn circle and its relationship with Stonehenge, Parker Pearson and colleagues report new evidence from the Welsh site and elaborate on aspects of their original argument. The discovery of a hearth at the centre of the circle, as well as further features around its circumference, reinforces the authors’ original interpretation. The authors explore the evidence for the construction sequence, which was abandoned before the completion of the monument. Contesting Darvill's argument that the Aubrey Holes at Stonehenge originally held posts, the authors reassert their interpretation of this circle of cut features as Bluestone settings.

A 20-month-old girl presented with severe dilated cardiomyopathy and decompensated congestive cardiac failure. Despite escalating inotropic and mechanical ventilation support, she required placement on extracorporeal membrane oxygenation and transfer to the transplant centre in Newcastle, England. She was placed on biventricular assist device and then Berlin Heart but failed to show any recovery of ventricular function. She underwent orthotopic heart transplantation at 2 years of age. She developed bacterial endocarditis with Enterococcus faecalis resulting in severe aortic valve regurgitation requiring aortic valve replacement with a 19 mm On-X valve (Airtivion) 11 days after her transplant. Given the size of the donor heart, it was possible to implant a 19-mm valve in this 12 kg child with minimal risk of patient prosthesis mismatch. She was anticoagulated with warfarin (On-X valve INR 2-3 for first 3 months; INR 1.5-2.0 thereafter). Although she suffered several other post-operative complications, including malabsorption, nasojejunal feeding, liver dysfunction, vertebral fractures, renal impairment and renal calcification, and need for repeat opening of her tracheostomy site following her initial decannulation, her aortic valve function has remained stable.

The discovery of a dismantled stone circle—close to Stonehenge's bluestone quarries in west Wales—raises the possibility that a 900-year-old legend about Stonehenge being built from an earlier stone circle contains a grain of truth. Radiocarbon and OSL dating of Waun Mawn indicate construction c. 3000 BC, shortly before the initial construction of Stonehenge. The identical diameters of Waun Mawn and the enclosing ditch of Stonehenge, and their orientations on the midsummer solstice sunrise, suggest that at least part of the Waun Mawn circle was brought from west Wales to Salisbury Plain. This interpretation complements recent isotope work that supports a hypothesis of migration of both people and animals from Wales to Stonehenge.

For an integer $n\geq 8$ divisible by $4$ , let $R_n={\mathbb Z}[\zeta _n,1/2]$ and let $\operatorname {\mathrm {U_{2}}}(R_n)$ be the group of $2\times 2$ unitary matrices with entries in $R_n$ . Set $\operatorname {\mathrm {U_2^\zeta }}(R_n)=\{\gamma \in \operatorname {\mathrm {U_{2}}}(R_n)\mid \det \gamma \in \langle \zeta _n\rangle \}$ . Let $\mathcal {G}_n\subseteq \operatorname {\mathrm {U_2^\zeta }}(R_n)$ be the Clifford-cyclotomic group generated by a Hadamard matrix $H=\frac {1}{2}[\begin {smallmatrix} 1+i & 1+i\\1+i &-1-i\end {smallmatrix}]$ and the gate $T_n=[\begin {smallmatrix}1 & 0\\0 & \zeta _n\end {smallmatrix}]$ . We prove that $\mathcal {G}_n=\operatorname {\mathrm {U_2^\zeta }}(R_n)$ if and only if $n=8, 12, 16, 24$ and that $[\operatorname {\mathrm {U_2^\zeta }}(R_n):\mathcal {G}_n]=\infty $ if $\operatorname {\mathrm {U_2^\zeta }}(R_n)\neq \mathcal {G}_n$ . We compute the Euler–Poincaré characteristic of the groups $\operatorname {\mathrm {SU_{2}}}(R_n)$ , $\operatorname {\mathrm {PSU_{2}}}(R_n)$ , $\operatorname {\mathrm {PU_{2}}}(R_n)$ , $\operatorname {\mathrm {PU_2^\zeta }}(R_n)$ , and $\operatorname {\mathrm {SO_{3}}}(R_n^+)$ .

Fitness is enhanced by determining when to behave prosocially. Elevation, an uplifting emotion elicited by witnessing exemplary prosociality, upregulates prosociality in the presence of prosocial others, as such contexts render prosociality profitable and/or antisociality costly. Prior research examines responses to a single highly prosocial individual. However, the profitability of enhancing prosociality hinges not only on potential interactions with a single actor, but also on the actions of others. Accordingly, information regarding how others respond to the prosocial exemplar may influence elevation elicitation and corresponding changes in prosocial motivation. If others reciprocate the exemplar's prosociality, or pay prosociality forward, this expands opportunities for the observer to profit by increasing prosociality, and thus could enhance elevation elicitation. Conversely, if others exploit the exemplar, this may diminish the profitability of prosociality, as the observer who acts prosocially may similarly be exploited and/or the resources with which the exemplar could reciprocate will be depleted. Conducting three online studies of Americans in which information regarding the responses of others to a prosocial exemplar was manipulated, we find that, against predictions, prosocial responses by the beneficiaries of prosociality generally do not enhance elevation among observers, whereas, consonant with predictions, antisocial responses markedly diminish elevation among observers.

Geologists and archaeologists have long known that the bluestones of Stonehenge came from the Preseli Hills of west Wales, 230km away, but only recently have some of their exact geological sources been identified. Two of these quarries—Carn Goedog and Craig Rhos-y-felin—have now been excavated to reveal evidence of megalith quarrying around 3000 BC—the same period as the first stage of the construction of Stonehenge. The authors present evidence for the extraction of the stone pillars and consider how they were transported, including the possibility that they were erected in a temporary monument close to the quarries, before completing their journey to Stonehenge.