Cup Stacking in Graphs

Here we introduce a new game on graphs, called cup stacking, following a line of what can be considered as $0$-, $1$-, or $2$-person games such as chip firing, percolation, graph burning, zero forcing, cops and robbers, graph pebbling, and graph pegging, among others. It can be more general, but the most basic scenario begins with a single cup on each vertex of a graph. For a vertex with $k$ cups on it we can move all its cups to a vertex at distance $k$ from it, provided the second vertex already has at least one cup on it. The object is to stack all cups onto some pre-described target vertex. We say that a graph is stackable if this can be accomplished for all possible target vertices. In this paper we study cup stacking on many families of graphs, developing a characterization of stackability in graphs and using it to prove the stackability of complete graphs, paths, cycles, grids, the Petersen graph, many Kneser graphs, some trees, cubes of dimension up to 20, "somewhat balanced" complete $t$-partite graphs, and Hamiltonian diameter two graphs. Additionally we use the Gallai-Edmonds Structure Theorem, the Edmonds Blossom Algorithm, and the Hungarian algorithm to devise a polynomial algorithm to decide if a diameter two graph is stackable. Our proof that cubes up to dimension 20 are stackable uses Kleitman's Symmetric Chain Decomposition and the new result of Merino, M\"utze, and Namrata that all generalized Johnson graphs (excluding the Petersen graph) are Hamiltonian. We conjecture that all cubes and higher-dimensional grids are stackable, and leave the reader with several open problems, questions, and generalizations

Promotion, prevention and protection: interventions at the population- and community-levels for mental, neurological and substance use disorders in low- and middle-income countries

Background In addition to services within the health system, interventions at the population and community levels are also important for the promotion of mental health, primary prevention of mental, neurological and substance use (MNS) disorders, identification and case detection of MNS disorders; and to a lesser degree treatment, care and rehabilitation. This study aims to identify “best practice” and “good practice” interventions that can feasibly be delivered at these population- and community-levels in low- and middle-income countries (LMICs), to aid the identification of resource efficiencies and allocation in LMICs. Methods A narrative review was conducted given the wide range of relevant interventions. Expert consensus was used to identify “best practice” at the population-level on the basis of existing quasi-experimental natural experiments and cost effectiveness, with small scale emerging and promising evidence comprising “good practice”. At the community-level, using expert consensus, the ACE (Assessing Cost-Effectiveness in Prevention Project) grading system was used to differentiate “best practice” interventions with sufficient evidence from “good practice” interventions with limited but promising evidence. ResultsAt the population-level, laws and regulations to control alcohol demand and restrict access to lethal means of suicide were considered “best practice”. Child protection laws, improved control of neurocysticercosis and mass awareness campaigns were identified as “good practice”. At the community level, socio-emotional learning programmes in schools and parenting programmes during infancy were identified as “best practice”. The following were all identified as “good practice”: Integrating mental health promotion strategies into workplace occupational health and safety policies; mental health information and awareness programmes as well as detection of MNS disorders in schools; early child enrichment/preschool educational programs and parenting programs for children aged 2–14 years; gender equity and/or economic empowerment programs for vulnerable groups; training of gatekeepers to identify people with MNS disorders in the community; and training non-specialist community members at a neighbourhood level to assist with community-based support and rehabilitation of people with mental disorders. Conclusion Interventions provided at the population- and community-levels have an important role to play in promoting mental health, preventing the onset, and protecting those with MNS disorders. The importance of inter-sectoral enga

Polarized blazar X-rays imply particle acceleration in shocks

Most of the light from blazars, active galactic nuclei with jets of magnetized plasma that point nearly along the line of sight, is produced by high-energy particles, up to around 1 TeV. Although the jets are known to be ultimately powered by a supermassive black hole, how the particles are accelerated to such high energies has been an unanswered question. The process must be related to the magnetic field, which can be probed by observations of the polarization of light from the jets. Measurements of the radio to optical polarization—the only range available until now—probe extended regions of the jet containing particles that left the acceleration site days to years earlier1,2,3, and hence do not directly explore the acceleration mechanism, as could X-ray measurements. Here we report the detection of X-ray polarization from the blazar Markarian 501 (Mrk 501). We measure an X-ray linear polarization degree ΠX of around 10%, which is a factor of around 2 higher than the value at optical wavelengths, with a polarization angle parallel to the radio jet. This points to a shock front as the source of particle acceleration and also implies that the plasma becomes increasingly turbulent with distance from the shock

Discovery of X-ray polarization angle rotation in active galaxy Mrk 421

The magnetic field conditions in astrophysical relativistic jets can be probed by multiwavelength polarimetry, which has been recently extended to X-rays. For example, one can track how the magnetic field changes in the flow of the radiating particles by observing rotations of the electric vector position angle $\Psi$. Here we report the discovery of a $\Psi_{\mathrm x}$ rotation in the X-ray band in the blazar Mrk 421 at an average flux state. Across the 5 days of Imaging X-ray Polarimetry Explorer (IXPE) observations of 4-6 and 7-9 June 2022, $\Psi_{\mathrm x}$ rotated in total by $\geq360^\circ$. Over the two respective date ranges, we find constant, within uncertainties, rotation rates ($80 \pm 9$ and $91 \pm 8 ^\circ/\rm day$) and polarization degrees ($\Pi_{\mathrm x}=10\%\pm1\%$). Simulations of a random walk of the polarization vector indicate that it is unlikely that such rotation(s) are produced by a stochastic process. The X-ray emitting site does not completely overlap the radio/infrared/optical emission sites, as no similar rotation of $\Psi$ was observed in quasi-simultaneous data at longer wavelengths. We propose that the observed rotation was caused by a helical magnetic structure in the jet, illuminated in the X-rays by a localized shock propagating along this helix. The optically emitting region likely lies in a sheath surrounding an inner spine where the X-ray radiation is released

X-ray Polarization Observations of BL Lacertae

Blazars are a class of jet-dominated active galactic nuclei with a typical double-humped spectral energy distribution. It is of common consensus the Synchrotron emission to be responsible for the low frequency peak, while the origin of the high frequency hump is still debated. The analysis of X-rays and their polarization can provide a valuable tool to understand the physical mechanisms responsible for the origin of high-energy emission of blazars. We report the first observations of BL Lacertae performed with the Imaging X-ray Polarimetry Explorer ({IXPE}), from which an upper limit to the polarization degree $\Pi_X<$12.6\% was found in the 2-8 keV band. We contemporaneously measured the polarization in radio, infrared, and optical wavelengths. Our multiwavelength polarization analysis disfavors a significant contribution of proton synchrotron radiation to the X-ray emission at these epochs. Instead, it supports a leptonic origin for the X-ray emission in BL Lac.Comment: 17 pages, 5 figures, accepted for publication in ApJ

BHPR research: qualitative1. Complex reasoning determines patients' perception of outcome following foot surgery in rheumatoid arhtritis

Background: Foot surgery is common in patients with RA but research into surgical outcomes is limited and conceptually flawed as current outcome measures lack face validity: to date no one has asked patients what is important to them. This study aimed to determine which factors are important to patients when evaluating the success of foot surgery in RA Methods: Semi structured interviews of RA patients who had undergone foot surgery were conducted and transcribed verbatim. Thematic analysis of interviews was conducted to explore issues that were important to patients. Results: 11 RA patients (9 ♂, mean age 59, dis dur = 22yrs, mean of 3 yrs post op) with mixed experiences of foot surgery were interviewed. Patients interpreted outcome in respect to a multitude of factors, frequently positive change in one aspect contrasted with negative opinions about another. Overall, four major themes emerged. Function: Functional ability & participation in valued activities were very important to patients. Walking ability was a key concern but patients interpreted levels of activity in light of other aspects of their disease, reflecting on change in functional ability more than overall level. Positive feelings of improved mobility were often moderated by negative self perception ("I mean, I still walk like a waddling duck”). Appearance: Appearance was important to almost all patients but perhaps the most complex theme of all. Physical appearance, foot shape, and footwear were closely interlinked, yet patients saw these as distinct separate concepts. Patients need to legitimize these feelings was clear and they frequently entered into a defensive repertoire ("it's not cosmetic surgery; it's something that's more important than that, you know?”). Clinician opinion: Surgeons' post operative evaluation of the procedure was very influential. The impact of this appraisal continued to affect patients' lasting impression irrespective of how the outcome compared to their initial goals ("when he'd done it ... he said that hasn't worked as good as he'd wanted to ... but the pain has gone”). Pain: Whilst pain was important to almost all patients, it appeared to be less important than the other themes. Pain was predominately raised when it influenced other themes, such as function; many still felt the need to legitimize their foot pain in order for health professionals to take it seriously ("in the end I went to my GP because it had happened a few times and I went to an orthopaedic surgeon who was quite dismissive of it, it was like what are you complaining about”). Conclusions: Patients interpret the outcome of foot surgery using a multitude of interrelated factors, particularly functional ability, appearance and surgeons' appraisal of the procedure. While pain was often noted, this appeared less important than other factors in the overall outcome of the surgery. Future research into foot surgery should incorporate the complexity of how patients determine their outcome Disclosure statement: All authors have declared no conflicts of interes