Base Size Sets and Determining Sets

Bridging the work of Cameron, Harary, and others, we examine the base size set B(G) and determining set D(G) of several families of groups. The base size set is the set of base sizes of all faithful actions of the group G on finite sets. The determining set is the subset of B(G) obtained by restricting the actions of G to automorphism groups of finite graphs. We show that for finite abelian groups, B(G)=D(G)={1,2,...,k} where k is the number of elementary divisors of G. We then characterize B(G) and D(G) for dihedral groups of the form D_{p^k} and D_{2p^k}. Finally, we prove B(G) is not equal to D(G) for dihedral groups of the form D_{pq} where p and q are distinct odd primes.Comment: 10 pages, 1 figur

Prolonged Use of the Etonogestrel Implant and Levonorgestrel Intrauterine Device - Two Years Beyond FDA-Approved Duration

The subdermal contraceptive implant, and the 52mg levonorgestrel intrauterine device (IUD) are currently FDA-approved for three and five years of use respectively. Limited available data has suggested both of these methods are effective beyond that time. Demonstration of prolonged effectiveness will improve the cost-effectiveness of the device, and potentially patient continuation and satisfaction. Objective To evaluate the effectiveness of the contraceptive implant and the 52-mg hormonal intrauterine device (IUD) in women using the method for two years beyond the current FDA-approved duration. Study Design We initiated this ongoing prospective cohort study in January 2012. We are enrolling women using the contraceptive implant or 52-mg levonorgestrel IUD for a minimum of 3 and 5 years, respectively (started IUD in 2007 or later or implant in 2009 or later). Demographic and reproductive health histories, as well as objective body mass index (BMI) were collected. Implant users were offered periodic venipuncture for analysis of serum etonogestrel levels. The primary outcome, unintended pregnancy rates, was calculated per 100 woman-years. We analyzed baseline demographic characteristics using chi-square test and Fisher Exact test, and compared serum ENG levels stratified by body mass index using the Kruskal-Wallis test. Results Implant users (n=291) have contributed 444.0 women-years of follow-up. There have been no documented pregnancies in implant users during the two years of post-expiration follow-up. Calculated failure rates in the fourth and fifth years for the implant are calculated as 0 (one-sided %97.5 confidence interval (CI) 0–1.48) per 100 woman years at four years and 0 (one-sided %97.5 CI 0–2.65) per 100 women years at five years. Among 496 levonorgestrel IUD users, 696.9 women-years of follow-up have been completed. Two pregnancies have been reported. The failure rate in the sixth year of use of the levonorgestrel IUD is calculated as 0.25(%95 CI 0.04–1.42) per 100 women year; failure rate during the seventh year is 0.43 (%95 CI 0.08–2.39) per 100 women years. Among implant users with serum etonogestrel results, the median etonogestrel level at the time of method expiration was 207.7 pg/mL (range 63.8–802.6 pg/mL), 166.1 pg/mL (range 67.9 25.0 – 470.5 pg/mL) at the end of the fourth year, and 153.0 pg/mL (range 72.1–538.8 pg/mL) at the end of the fifth year. Median ENG levels were compared by BMI at each time point and a statistical difference was noted at the end of four years of use with overweight women having the highest serum ENG (195.9 pg/ml: range 25.0–450.5) when compared to normal (178.9 pg/ml: range 87.0–463.7) and obese (137.9 pg/ml: range 66.0–470.5) women (p=0.04). Conclusion This study indicates that the contraceptive implant and 52-mg hormonal IUD continue to be highly effective for at least two additional years of use. Serum etonogestrel evaluation demonstrates median levels remain above the ovulation threshold of 90pg/ml for women of in all BMI classes

Homomesies on permutations -- an analysis of maps and statistics in the FindStat database

In this paper, we perform a systematic study of permutation statistics and bijective maps on permutations in which we identify and prove 122 instances of the homomesy phenomenon. Homomesy occurs when the average value of a statistic is the same on each orbit of a given map. The maps we investigate include the Lehmer code rotation, the reverse, the complement, the Foata bijection, and the Kreweras complement. The statistics studied relate to familiar notions such as inversions, descents, and permutation patterns, and also more obscure constructs. Beside the many new homomesy results, we discuss our research method, in which we used SageMath to search the FindStat combinatorial statistics database to identify potential homomesies

Spectra of Coronae

We introduce a new invariant, the coronal of a graph, and use it to compute the spectrum of the corona $G\circ H$ of two graphs $G$ and $H$. In particular, we show that this spectrum is completely determined by the spectra of $G$ and $H$ and the coronal of $H$. Previous work has computed the spectrum of a corona only in the case that $H$ is regular. We then explicitly compute the coronals for several families of graphs, including regular graphs, complete $n$-partite graphs, and paths. Finally, we use the corona construction to generate many infinite families of pairs of cospectral graphs.Comment: 9 page

Hyperoxemia and excess oxygen use in early acute respiratory distress syndrome : Insights from the LUNG SAFE study

Publisher Copyright: © 2020 The Author(s). Copyright: Copyright 2020 Elsevier B.V., All rights reserved.Background: Concerns exist regarding the prevalence and impact of unnecessary oxygen use in patients with acute respiratory distress syndrome (ARDS). We examined this issue in patients with ARDS enrolled in the Large observational study to UNderstand the Global impact of Severe Acute respiratory FailurE (LUNG SAFE) study. Methods: In this secondary analysis of the LUNG SAFE study, we wished to determine the prevalence and the outcomes associated with hyperoxemia on day 1, sustained hyperoxemia, and excessive oxygen use in patients with early ARDS. Patients who fulfilled criteria of ARDS on day 1 and day 2 of acute hypoxemic respiratory failure were categorized based on the presence of hyperoxemia (PaO2 > 100 mmHg) on day 1, sustained (i.e., present on day 1 and day 2) hyperoxemia, or excessive oxygen use (FIO2 ≥ 0.60 during hyperoxemia). Results: Of 2005 patients that met the inclusion criteria, 131 (6.5%) were hypoxemic (PaO2 < 55 mmHg), 607 (30%) had hyperoxemia on day 1, and 250 (12%) had sustained hyperoxemia. Excess FIO2 use occurred in 400 (66%) out of 607 patients with hyperoxemia. Excess FIO2 use decreased from day 1 to day 2 of ARDS, with most hyperoxemic patients on day 2 receiving relatively low FIO2. Multivariate analyses found no independent relationship between day 1 hyperoxemia, sustained hyperoxemia, or excess FIO2 use and adverse clinical outcomes. Mortality was 42% in patients with excess FIO2 use, compared to 39% in a propensity-matched sample of normoxemic (PaO2 55-100 mmHg) patients (P = 0.47). Conclusions: Hyperoxemia and excess oxygen use are both prevalent in early ARDS but are most often non-sustained. No relationship was found between hyperoxemia or excessive oxygen use and patient outcome in this cohort. Trial registration: LUNG-SAFE is registered with ClinicalTrials.gov, NCT02010073publishersversionPeer reviewe

Effect of Hydrocortisone on Mortality and Organ Support in Patients With Severe COVID-19: The REMAP-CAP COVID-19 Corticosteroid Domain Randomized Clinical Trial.

Importance: Evidence regarding corticosteroid use for severe coronavirus disease 2019 (COVID-19) is limited. Objective: To determine whether hydrocortisone improves outcome for patients with severe COVID-19. Design, Setting, and Participants: An ongoing adaptive platform trial testing multiple interventions within multiple therapeutic domains, for example, antiviral agents, corticosteroids, or immunoglobulin. Between March 9 and June 17, 2020, 614 adult patients with suspected or confirmed COVID-19 were enrolled and randomized within at least 1 domain following admission to an intensive care unit (ICU) for respiratory or cardiovascular organ support at 121 sites in 8 countries. Of these, 403 were randomized to open-label interventions within the corticosteroid domain. The domain was halted after results from another trial were released. Follow-up ended August 12, 2020. Interventions: The corticosteroid domain randomized participants to a fixed 7-day course of intravenous hydrocortisone (50 mg or 100 mg every 6 hours) (n = 143), a shock-dependent course (50 mg every 6 hours when shock was clinically evident) (n = 152), or no hydrocortisone (n = 108). Main Outcomes and Measures: The primary end point was organ support-free days (days alive and free of ICU-based respiratory or cardiovascular support) within 21 days, where patients who died were assigned -1 day. The primary analysis was a bayesian cumulative logistic model that included all patients enrolled with severe COVID-19, adjusting for age, sex, site, region, time, assignment to interventions within other domains, and domain and intervention eligibility. Superiority was defined as the posterior probability of an odds ratio greater than 1 (threshold for trial conclusion of superiority >99%). Results: After excluding 19 participants who withdrew consent, there were 384 patients (mean age, 60 years; 29% female) randomized to the fixed-dose (n = 137), shock-dependent (n = 146), and no (n = 101) hydrocortisone groups; 379 (99%) completed the study and were included in the analysis. The mean age for the 3 groups ranged between 59.5 and 60.4 years; most patients were male (range, 70.6%-71.5%); mean body mass index ranged between 29.7 and 30.9; and patients receiving mechanical ventilation ranged between 50.0% and 63.5%. For the fixed-dose, shock-dependent, and no hydrocortisone groups, respectively, the median organ support-free days were 0 (IQR, -1 to 15), 0 (IQR, -1 to 13), and 0 (-1 to 11) days (composed of 30%, 26%, and 33% mortality rates and 11.5, 9.5, and 6 median organ support-free days among survivors). The median adjusted odds ratio and bayesian probability of superiority were 1.43 (95% credible interval, 0.91-2.27) and 93% for fixed-dose hydrocortisone, respectively, and were 1.22 (95% credible interval, 0.76-1.94) and 80% for shock-dependent hydrocortisone compared with no hydrocortisone. Serious adverse events were reported in 4 (3%), 5 (3%), and 1 (1%) patients in the fixed-dose, shock-dependent, and no hydrocortisone groups, respectively. Conclusions and Relevance: Among patients with severe COVID-19, treatment with a 7-day fixed-dose course of hydrocortisone or shock-dependent dosing of hydrocortisone, compared with no hydrocortisone, resulted in 93% and 80% probabilities of superiority with regard to the odds of improvement in organ support-free days within 21 days. However, the trial was stopped early and no treatment strategy met prespecified criteria for statistical superiority, precluding definitive conclusions. Trial Registration: ClinicalTrials.gov Identifier: NCT02735707

Effect of angiotensin-converting enzyme inhibitor and angiotensin receptor blocker initiation on organ support-free days in patients hospitalized with COVID-19

IMPORTANCE Overactivation of the renin-angiotensin system (RAS) may contribute to poor clinical outcomes in patients with COVID-19. Objective To determine whether angiotensin-converting enzyme (ACE) inhibitor or angiotensin receptor blocker (ARB) initiation improves outcomes in patients hospitalized for COVID-19. DESIGN, SETTING, AND PARTICIPANTS In an ongoing, adaptive platform randomized clinical trial, 721 critically ill and 58 non–critically ill hospitalized adults were randomized to receive an RAS inhibitor or control between March 16, 2021, and February 25, 2022, at 69 sites in 7 countries (final follow-up on June 1, 2022). INTERVENTIONS Patients were randomized to receive open-label initiation of an ACE inhibitor (n = 257), ARB (n = 248), ARB in combination with DMX-200 (a chemokine receptor-2 inhibitor; n = 10), or no RAS inhibitor (control; n = 264) for up to 10 days. MAIN OUTCOMES AND MEASURES The primary outcome was organ support–free days, a composite of hospital survival and days alive without cardiovascular or respiratory organ support through 21 days. The primary analysis was a bayesian cumulative logistic model. Odds ratios (ORs) greater than 1 represent improved outcomes. RESULTS On February 25, 2022, enrollment was discontinued due to safety concerns. Among 679 critically ill patients with available primary outcome data, the median age was 56 years and 239 participants (35.2%) were women. Median (IQR) organ support–free days among critically ill patients was 10 (–1 to 16) in the ACE inhibitor group (n = 231), 8 (–1 to 17) in the ARB group (n = 217), and 12 (0 to 17) in the control group (n = 231) (median adjusted odds ratios of 0.77 [95% bayesian credible interval, 0.58-1.06] for improvement for ACE inhibitor and 0.76 [95% credible interval, 0.56-1.05] for ARB compared with control). The posterior probabilities that ACE inhibitors and ARBs worsened organ support–free days compared with control were 94.9% and 95.4%, respectively. Hospital survival occurred in 166 of 231 critically ill participants (71.9%) in the ACE inhibitor group, 152 of 217 (70.0%) in the ARB group, and 182 of 231 (78.8%) in the control group (posterior probabilities that ACE inhibitor and ARB worsened hospital survival compared with control were 95.3% and 98.1%, respectively). CONCLUSIONS AND RELEVANCE In this trial, among critically ill adults with COVID-19, initiation of an ACE inhibitor or ARB did not improve, and likely worsened, clinical outcomes. TRIAL REGISTRATION ClinicalTrials.gov Identifier: NCT0273570