Counting fixed points and rooted closed walks of the singular map x↦xxnx \mapsto x^{x^n} modulo powers of a prime

The "self-power" map $x \mapsto x^x$ modulo $m$ and its generalized form $x \mapsto x^{x^n}$ modulo $m$ are of considerable interest for both theoretical reasons and for potential applications to cryptography. In this paper, we use $p$-adic methods, primarily $p$-adic interpolation, Hensel's lemma, and lifting singular points modulo $p$, to count fixed points and rooted closed walks of equations related to these maps when $m$ is a prime power. In particular, we introduce a new technique for lifting singular solutions of several congruences in several unknowns using the left kernel of the Jacobian matrix.Comment: 18 pages. Version 2 shortens proofs, reduces redundancy, and introduces new technique for counting rooted closed walks. Version 3 updates title to agree with journal publicatio

Counting Fixed Points, Two-Cycles, and Collisions of the Discrete Exponential Function using p-adic Methods

Brizolis asked for which primes p greater than 3 does there exist a pair (g, h) such that h is a fixed point of the discrete exponential map with base g, or equivalently h is a fixed point of the discrete logarithm with base g. Zhang (1995) and Cobeli and Zaharescu (1999) answered with a "yes" for sufficiently large primes and gave estimates for the number of such pairs when g and h are primitive roots modulo p. In 2000, Campbell showed that the answer to Brizolis was "yes" for all primes. The first author has extended this question to questions about counting fixed points, two-cycles, and collisions of the discrete exponential map. In this paper, we use p-adic methods, primarily Hensel's lemma and p-adic interpolation, to count fixed points, two cycles, collisions, and solutions to related equations modulo powers of a prime p.Comment: 14 pages, no figure

Role of the AP-5 adaptor protein complex in late endosome-to-Golgi retrieval

The AP-5 adaptor protein complex is presumed to function in membrane traffic, but so far nothing is known about its pathway or its cargo. We have used CRISPR-Cas9 to knock out the AP-5 ζ subunit gene, AP5Z1, in HeLa cells, and then analysed the phenotype by subcellular fractionation profiling and quantitative mass spectrometry. The retromer complex had an altered steady-state distribution in the knockout cells, and several Golgi proteins, including GOLIM4 and GOLM1, were depleted from vesicle-enriched fractions. Immunolocalisation showed that loss of AP-5 led to impaired retrieval of the cation-independent mannose 6-phosphate receptor (CIMPR), GOLIM4, and GOLM1 from endosomes back to the Golgi region. Knocking down the retromer complex exacerbated this phenotype. Both the CIMPR and sortilin interacted with the AP-5–associated protein SPG15 in pull-down assays, and we propose that sortilin may act as a link between Golgi proteins and the AP-5/SPG11/SPG15 complex. Together, our findings suggest that AP-5 functions in a novel sorting step out of late endosomes, acting as a backup pathway for retromer. This provides a mechanistic explanation for why mutations in AP-5/SPG11/SPG15 cause cells to accumulate aberrant endolysosomes, and highlights the role of endosome/lysosome dysfunction in the pathology of hereditary spastic paraplegia and other neurodegenerative disorders.The Wellcome Trust https://wellcome.ac.uk/ (grant number 086598). Received by MSR. The funder had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. German Research Foundation http://www.dfg.de/ (grant number DFG/Gottfried Wilhelm Leibniz Prize MA 1764/2-1). Contributed to GHHB's research. The funder had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. The Wellcome Trust https://wellcome.ac.uk/ (grant number 100140). Strategic Award to the CIMR. The funder had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript

Social inequalities in wheezing in children: findings from the UK Millennium Cohort Study

Wheezing in childhood is socially patterned, but it is unclear what factors explain the social differences.Regression analysis of the UK Millennium Cohort Study, based on 11 141 singleton children who participated at ages 9 months and 3, 5 and 7 years. Relative risk ratios (RRR) for early and persistent/relapsing wheeze were estimated using multinomial regression, according to measures of socioeconomic circumstances. Maternal, antenatal and early-life characteristics were assessed as potential mediators.Children of mothers with no educational qualifications were more likely to have both wheeze types, compared to children of mothers with degree-level qualifications (RRR 1.53, 95% CI 1.26-1.86 for early wheeze; 1.32 95% CI 1.04-1.67 for persistent/relapsing wheeze). Controlling for maternal age, smoking during pregnancy and breastfeeding removed the elevated risk of wheezing. Male sex, maternal age, body mass index, atopy, smoking during pregnancy, preterm birth, breastfeeding, exposure to other children and furry pets were independently associated with wheezing, but the pattern of association varied between wheezing types.In this representative UK cohort, adjustment for maternal smoking during pregnancy and breastfeeding removed the socioeconomic inequalities in common wheezing phenotypes. Policies to reduce the social gradient in these risk factors may reduce inequalities in wheezing and asthma

Identifying the participant characteristics that predict recruitment and retention of participants to RCTs involving children : a systematic review

Background: Randomised controlled trials (RCTs) are recommended as the gold standard in evaluating healthcare interventions. The conduct of RCTs is often impacted by difficulties surrounding recruitment and retention of participants in both adult and child populations. Factors influencing recruitment and retention of children to RCTs can be more complex than in adults. There is little synthesised evidence of what influences participation in research involving parents and children. Aim: To identify predictors of recruitment and retention in RCTs involving children. Methods/ Design: A systematic review of RCTs was conducted to synthesise the available evidence. An electronic search strategy was applied to four databases and restricted to English language publications. Quantitative studies reporting participant predictors of recruitment and retention in RCTs involving children aged 0-12 were identified. Data was extracted and synthesised narratively. Quality assessment of articles was conducted using a structured tool developed from two existing quality evaluation checklists. Results: 28 studies were included in the review. Of the 154 participant factors reported, 66 were found to be significant predictors of recruitment and retention in at least one study. These were classified as parent, child, family and neighbourhood characteristics. Parent characteristics (e.g. ethnicity, age, education, socio economic status) were the most commonly reported predictors of participation for both recruitment and retention. Being young, less educated, of an ethnic minority and low socio economic status (SES) appear to be barriers to participation in RCTs although there was little agreement between studies. When analysed according to setting and severity of the child’s illness there appeared to be little variation between groups. The quality of the studies varied. Articles adhered well to reporting guidelines around provision of a scientific rationale for the study and background information as well as displaying good internal consistency of results. However, few studies discussed the external validity of the results or provided recommendations for future research. Conclusion: Parent characteristics may predict participation of children and their families to RCTs, however, there was a lack of consensus. Whilst socio-demographics variables may be useful in identifying which groups are least likely to participate they do not provide insight into the processes and barriers to participation for children and families. Further studies that explore variables that can be influenced are warranted. Reporting of studies in this field need greater clarity as well as agreed definitions of what is meant by retention

Fit-for-work or fit-for-unemployment? Does the reassessment of disability benefit claimants using a tougher work capability assessment help people into work?

Background Many governments have introduced tougher eligibility assessments for out-of-work disability benefits, to reduce rising benefit caseloads. The UK government initiated a programme in 2010 to reassess all existing disability benefit claimants using a new functional checklist. We investigated whether this policy led to more people out-of-work with long-standing health problems entering employment. Method We use longitudinal data from the Labour Force Survey linked to data indicating the proportion of the population experiencing a reassessment in each of 149 upper tier local authorities in England between 2010 and 2013. Regression models were used to investigate whether the proportion of the population undergoing reassessment in each area was independently associated with the chances that people out-of-work with a long-standing health problem entered employment and transitions between inactivity and unemployment. We analysed whether any effects differed between people whose main health problem was mental rather than physical. Results There was no significant association between the reassessment process and the chances that people out-of-work with a long-standing illness entered employment. The process was significantly associated with an increase in the chances that people with mental illnesses moved from inactivity into unemployment (HR=1.22, 95% CI 1.03 to 1.45). Conclusions The reassessment policy appears to have shifted people with mental health problems from inactivity into unemployment, but there was no evidence that it had increased their chances of employment. There is an urgent need for services that can support the increasing number of people with mental health problems on unemployment benefits