A chiral aperiodic monotile

The recently discovered "hat" aperiodic monotile mixes unreflected and reflected tiles in every tiling it admits, leaving open the question of whether a single shape can tile aperiodically using translations and rotations alone. We show that a close relative of the hat -- the equilateral member of the continuum to which it belongs -- is a weakly chiral aperiodic monotile: it admits only non-periodic tilings if we forbid reflections by fiat. Furthermore, by modifying this polygon's edges we obtain a family of shapes called Spectres that are strictly chiral aperiodic monotiles: they admit only chiral non-periodic tilings based on a hierarchical substitution system.Comment: 23 pages, 12 figure

A chiral aperiodic monotile

The recently discovered "hat" aperiodic monotile mixes unreflected and reflected tiles in every tiling it admits, leaving open the question of whether a single shape can tile aperiodically using translations and rotations alone. We show that a close relative of the hat--the equilateral member of the continuum to which it belongs--is a weakly chiral aperiodic monotile: it admits only non-periodic tilings if we forbid reflections by fiat. Furthermore, by modifying this polygon's edges we obtain a family of shapes called Spectres that are strictly chiral aperiodic monotiles: they admit only homochiral non-periodic tilings based on a hierarchical substitution system.Mathematics Subject Classifications: 05B45, 52C20, 05B50Keywords: Tilings, aperiodic order, polyform

An aperiodic monotile

A longstanding open problem asks for an aperiodic monotile, also known as an "einstein": a shape that admits tilings of the plane, but never periodic tilings. We answer this problem for topological disk tiles by exhibiting a continuum of combinatorially equivalent aperiodic polygons. We first show that a representative example, the "hat" polykite, can form clusters called "metatiles", for which substitution rules can be defined. Because the metatiles admit tilings of the plane, so too does the hat. We then prove that generic members of our continuum of polygons are aperiodic, through a new kind of geometric incommensurability argument. Separately, we give a combinatorial, computer-assisted proof that the hat must form hierarchical -- and hence aperiodic -- tilings.Comment: 89 pages, 57 figures; Minor corrections, renamed "fylfot" to "triskelion", added the name "turtle", added references, new H7/H8 rules (Fig 2.11), talk about reflection

Experiences of women who have lost young children to AIDS in KwaZulu-Natal, South Africa: a qualitative study

Background AIDS continues to be the leading cause of death in South Africa. Little is known about the experiences of mothers who have lost a young child to AIDS. The purpose of this qualitative study was to explore the attitudes and experiences of women who had lost a young child to HIV/AIDS in KwaZulu-Natal Province, South Africa. Methods In-depth interviews were conducted with 10 women who had lost a child to AIDS. The average age of the deceased children was six years. Interviews were also conducted with 12 key informants to obtain their perspectives on working with women who had lost a child to AIDS. A thematic analysis of the transcripts was performed. Results In addition to the pain of losing a child, the women in this study had to endure multiple stresses within a harsh and sometimes hostile environment. Confronted with pervasive stigma and extreme poverty, they had few people they could rely on during their child\u27s sickness and death. They were forced to keep their emotions to themselves since they were not likely to obtain much support from family members or people in the community. Throughout the period of caring for a sick child and watching the child die, they were essentially alone. The demands of caring for their child and subsequent grief, together with daily subsistence worries, took its toll. Key informants struggled to address the needs of these women due to several factors, including scarce resources, lack of training around bereavement issues, reluctance by people in the community to seek help with emotional issues, and poverty. Conclusions The present study offers one of the first perspectives on the experiences of mothers who have lost a young child to AIDS. Interventions that are tailored to the local context and address bereavement issues, as well as other issues that affect the daily lives of these mothers, are urgently needed. Further studies are needed to identify factors that promote resilience among these women

Price's Law on Nonstationary Spacetimes

In this article we study the pointwise decay properties of solutions to the wave equation on a class of nonstationary asymptotically flat backgrounds in three space dimensions. Under the assumption that uniform energy bounds and a weak form of local energy decay hold forward in time we establish a $t^{-3}$ local uniform decay rate (Price's law \cite{MR0376103}) for linear waves. As a corollary, we also prove Price's law for certain small perturbations of the Kerr metric. This result was previously established by the second author in \cite{Tat} on stationary backgrounds. The present work was motivated by the problem of nonlinear stability of the Kerr/Schwarzschild solutions for the vacuum Einstein equations, which seems to require a more robust approach to proving linear decay estimates.Comment: 32 pages, no figures, typos correcte

Usefulness of Lipid Apheresis in the Treatment of Familial Hypercholesterolemia

Lipid apheresis is used to treat patients with severe hyperlipidemia by reducing low-density lipoprotein cholesterol (LDL-C). This study examines the effect of apheresis on the lipid panel and cardiac event rates before and after apheresis. An electronic health record screen of ambulatory patients identified 11 active patients undergoing lipid apheresis with 10/11 carrying a diagnosis of FH. Baseline demographics, pre- and postapheresis lipid levels, highest recorded LDL-C, cardiac events, current medications, and first apheresis treatment were recorded. Patients completed a questionnaire and self-reported risk factors and interest in alternative treatment. There were significant reductions in mean total cholesterol (−58.4%), LDL-C (−71.9%), triglycerides (−51%), high-density lipoprotein (HDL) cholesterol (−9.3%), and non-HDL (−68.2%) values. Thirty-four cardiac events were documented in 8 patients before apheresis, compared with 9 events in 5 patients after apheresis. Our survey showed a high prevalence of statin intolerance (64%), with the majority (90%) of participants indicating an interest in alternative treatment options. Our results have shown that lipid apheresis primary effect is a marked reduction in LDL-C cholesterol levels and may reduce the recurrence of cardiac events. Apheresis should be compared to the newer alternative treatment modalities in a randomized fashion due to patient interest in alternative options

The inter-personal work of dental conscious sedation: a qualitative analysis

Aims Whilst there is a considerable body of literature examining the pharmacology of conscious sedation, the social tasks required to successfully provide conscious sedation have not been reported. This paper discusses data regarding the interpersonal work integral to effective conscious sedation provision, from a larger qualitative study exploring how patients and clinicians engage with secondary care conscious sedation provided within the UK. Method Semi-structured interviews were conducted with 13 conscious sedation providers and nine patients within UK-based secondary care sedation settings. Digital audio-recordings were transcribed verbatim and subsequently analysed using a constant comparative method within NVivo Data Analysis Software. Results Four main themes of interpersonal work were reported by participants: displaying care, containing emotions, demonstrating competence and maximizing the effect. Conclusion This study shows that performing conscious sedation requires more than technical delivery, and involves the projection of attributes in a literal “performance.” The importance of managing outward emotional appearance reflects previous dental research. The need to manage outward appearance, and the emotional impact this has, is of relevance to all clinicians