DESIGN:This was a prospective observational study.BACKGROUND AND AIMS:The characteristics of cannabis-involved motor vehicle collisions are poorly understood. This study of injured drivers identifies demographic and collision characteristics associated with high tetrahydrocannabinol (THC) concentrations.SETTING:The study was conducted in 15 Canadian trauma centres between January 2018 and December 2021.CASES:The cases (n = 6956) comprised injured drivers who required blood testing as part of routine trauma care.MEASUREMENTS:We quantified whole blood THC and blood alcohol concentration (BAC) and recorded driver sex, age and postal code, time of crash, crash type and injury severity. We defined three driver groups: high THC (THC ≥ 5 ng/ml and BAC = 0), high alcohol (BAC ≥ 0.08% and THC = 0) and THC/BAC-negative (THC = 0 = BAC). We used logistic regression techniques to identify factors associated with group membership.FINDINGS:Most injured drivers (70.2%) were THC/BAC-negative; 1274 (18.3%) had THC > 0, including 186 (2.7%) in the high THC group; 1161 (16.7%) had BAC > 0, including 606 (8.7%) in the high BAC group. Males and drivers aged less than 45 years had higher adjusted odds of being in the high THC group (versus the THC/BAC-negative group). Importantly, 4.6% of drivers aged less than 19 years had THC ≥ 5 ng/ml, and drivers aged less than 19 years had higher unadjusted odds of being in the high THC group than drivers aged 45-54 years. Males, drivers aged 19-44 years, rural drivers, seriously injured drivers and drivers injured in single-vehicle, night-time or weekend collisions had higher adjusted odds ratios (aORs) for being in the high alcohol group (versus THC/BAC-negative). Drivers aged less than 35 or more than 65 years and drivers involved in multi-vehicle, daytime or weekday collisions had higher adjusted odds for being in the high THC group (versus the high BAC group).CONCLUSIONS:In Canada, risk factors for cannabis-related motor vehicle collisions appear to differ from those for alcohol-related motor vehicle collisions. The collision factors associated with alcohol (single-vehicle, night-time, weekend, rural, serious injury) are not associated with cannabis-related collisions. Demographic factors (young drivers, male drivers) are associated with both alcohol and cannabis-related collisions, but are more strongly associated with cannabis-related collisions.

ABSTRACTImportanceRuling out pulmonary embolism (PE) among patients presenting to the Emergency Department (ED) with suspected or confirmed SARS-COV-2 is challenging due to symptom overlap, known increased pro-thrombotic risk, and unclear D-dimer test interpretation.ObjectiveOur primary objective was to assess the diagnostic accuracy of standard and age-adjusted D-dimer test thresholds for predicting 30-day pulmonary embolism (PE) diagnosis in patients with suspected SARS-COV-2 infection.Design, Setting, and ParticipantsThis was a retrospective observational study using data from 50 sites enrolling patients into the Canadian COVID-19 ED Rapid Response Network (CCEDRRN) registry between March 1, 2020 to July 2, 2021. Adults (≥18 years) with SARS-COV-2 testing performed at index ED visit were included if they had any of the following presenting complaints: chest pain, shortness of breath, hypoxia, syncope/presyncope, or hemoptysis. We excluded patients with duplicate records or no valid provincial healthcare number.Main Outcomes and MeasuresOur primary end point was 30-day PE diagnosis based on a positive computed tomography pulmonary angiogram (CTPA) or hospital discharge diagnosis code of PE. The outcome measure was the diagnostic accuracy of an age adjusted D-dimer strategy as compared to absolute D-dimer thresholds (500 – 5000 ng/mL).Results52,038 patients met inclusion criteria. Age-adjusted D-dimer had a sensitivity (SN) of 96% (95% CI 93-98%) and a specificity (SP) of 48% (95% CI 48-49%) which was comparable to the most sensitive absolute threshold of 500 ng/mL (SN 98%, 95% CI 96-99%; SP 41%, 95% CI 40-42%). Other absolute D-dimer thresholds did not perform well enough for clinical reliability (SN <90%). Both age-adjusted and absolute D-dimer performed better in SARS-COV-2 negative patients as compared to SARS-COV-2 positive patients for predicting 30-day PE diagnosis (c-statistic 0.88 vs 0.80).Conclusions and RelevanceIn this large Canadian cohort of ED patients with suspected SARS-COV-2 infection, an age-adjusted D-dimer strategy had similar sensitivity and superior specificity to the most sensitive D-dimer threshold of 500 ng/mL for predicting 30-day PE diagnosis irrespective of SARS-COV-2 infection status. Adopting an age-adjusted D-dimer strategy in patients with suspected SARS-COV-2 may help avoid unnecessary CTPA testing without compromising safety.Trial RegistrationClinicaltrials.gov, NCT04702945KEY POINTSQuestionWhat is the diagnostic accuracy of age-adjusted and absolute D-dimer thresholds for investigating PE in ED patients with suspected SARS-COV-2?FindingsAn age-adjusted D-dimer strategy had comparable sensitivity and higher specificity for 30-day PE diagnosis compared to the most sensitive absolute threshold of 500 ng/mL irrespective of patient’s SARS-COV-2 status.MeaningConsider using an age-adjusted D-dimer threshold for PE risk stratification in ED patients with suspected SARS-COV-2.

To determine if lockdown measures imposed during the first wave of the COVID19 pandemic affected trauma patterns, volumes, and outcomes in a western Canadian level 1 trauma center, we performed a retrospective cohort study assessing level 1 and 2 trauma patients presenting to our center during the initial COVID19 “lockdown” period (March 15–June 14, 2020) compared to a similar cohort of patients presenting during a “control” period 1 year prior (March 15–June 14, 2019). Overall, we saw a 7.8% reduction in trauma volumes during the lockdown period, and this was associated with a shorter average ED length of stay (6.2 ± 4.7 h vs. 9.7 ± 11.8 h, p = 0.003), reduced time to computed tomography (88.5 ± 68.2 min vs. 105.1 ± 65.5 min, p < 0.001), a reduction in intensive care unit admissions (11.0 ± 4.9% vs. 20.0 ± 15.5%, p = 0.001), and higher injury severity score (6.5 ± 7.6 vs. 6.2 ± 9.5, p = 0.04). Our findings suggest that lockdown measures imposed during the first wave of the COVID19 pandemic had a significant impact on trauma patients.

AbstractObjectivesTo develop and validate a clinical risk score that can accurately quantify an emergency department patient’s probability of SARS-CoV-2 infection without the need for laboratory testingDesignCohort study of participants in the Canadian COVID-19 Emergency Department Rapid Response Network (CCEDRRN) registry. Regression models were fitted to predict a positive SARS-CoV-2 test result using clinical and demographic predictors, as well as an indicator of local SARS-CoV-2 incidence.Setting32 emergency departments in eight Canadian provincesParticipants27,665 consecutively-enrolled patients who were tested for SARS-CoV-2 in participating emergency departments between March 1-October 30,2020Main outcome measuresPositive SARS-CoV-2 nucleic acid test result within 14 days of an index emergency department encounter for suspected COVID-19 diseaseResultsWe derived a 10-item CCEDRRN COVID-19 Infection Score using data from 21,743 patients. This score included variables from history and physical examination, and an indicator of local disease incidence. The score had a c-statistic of 0.838 with excellent calibration. We externally validated the rule in 5,295 patients. The score maintained excellent discrimination and calibration, and had superior performance compared to another previously published risk score. Score cutoffs were identified that can rule-in or rule-out SARS-CoV-2 infection without the need for nucleic acid testing with 97.4 % sensitivity (95% CI 96.4–98..3) and 95.9% specificity (95% CI 95.5-96.0).ConclusionsThe CCEDRRN COVID-19 Infection Score uses clinical characteristics and publicly available indicators of disease incidence to quantify a patient’s probability of SARS-CoV-2 infection. The score can identify patients at sufficiently high risk of SARS-CoV-2 infection to warrant isolation and empiric therapy prior to test confirmation, while also identifying patients at sufficiently low risk of infection that they may not need testing.Trial registrationCCEDRRN is registered at clinicaltrials.gov (NCT04702945).FundingThe network is funded by the Canadian Institutes of Health Research (447679), BC Academic Health Science Network Society, BioTalent Canada, Genome BC (COV024; VAC007), Ontario Ministry of Colleges and Universities (C-655-2129), the Saskatchewan Health Research Foundation (5357) and the Fondation CHU de Québec (Octroi #4007). These organizations are not-for-profit, and had no role in study conduct, analysis, or manuscript preparation.

Introduction: CAEP recently developed the acute atrial fibrillation (AF) and flutter (AFL) [AAFF] Best Practices Checklist to promote optimal care and guidance on cardioversion and rapid discharge of patients with AAFF. We sought to assess the impact of implementing the Checklist into large Canadian EDs. Methods: We conducted a pragmatic stepped-wedge cluster randomized trial in 11 large Canadian ED sites in five provinces, over 14 months. All hospitals started in the control period (usual care), and then crossed over to the intervention period in random sequence, one hospital per month. We enrolled consecutive, stable patients presenting with AAFF, where symptoms required ED management. Our intervention was informed by qualitative stakeholder interviews to identify perceived barriers and enablers for rapid discharge of AAFF patients. The many interventions included local champions, presentation of the Checklist to physicians in group sessions, an online training module, a smartphone app, and targeted audit and feedback. The primary outcome was length of stay in ED in minutes from time of arrival to time of disposition, and this was analyzed at the individual patient-level using linear mixed effects regression accounting for the stepped-wedge design. We estimated a sample size of 800 patients. Results: We enrolled 844 patients with none lost to follow-up. Those in the control (N = 316) and intervention periods (N = 528) were similar for all characteristics including mean age (61.2 vs 64.2 yrs), duration of AAFF (8.1 vs 7.7 hrs), AF (88.6% vs 82.9%), AFL (11.4% vs 17.1%), and mean initial heart rate (119.6 vs 119.9 bpm). Median lengths of stay for the control and intervention periods respectively were 413.0 vs. 354.0 minutes (P < 0.001). Comparing control to intervention, there was an increase in: use of antiarrhythmic drugs (37.4% vs 47.4%; P < 0.01), electrical cardioversion (45.1% vs 56.8%; P < 0.01), and discharge in sinus rhythm (75.3% vs. 86.7%; P < 0.001). There was a decrease in ED consultations to cardiology and medicine (49.7% vs 41.1%; P < 0.01), but a small but insignificant increase in anticoagulant prescriptions (39.6% vs 46.5%; P = 0.21). Conclusion: This multicenter implementation of the CAEP Best Practices Checklist led to a significant decrease in ED length of stay along with more ED cardioversions, fewer ED consultations, and more discharges in sinus rhythm. Widespread and rigorous adoption of the CAEP Checklist should lead to improved care of AAFF patients in all Canadian EDs.

Introduction: Many drugs, including cannabis and alcohol, cause impairment and contribute to motor vehicle collisions (MVCs). Policy makers require knowledge of the prevalence of drug use in crash-involved drivers, and types of drugs used in order to develop effective prevention programs. This issue is particularly relevant with the recent legalization of cannabis. We aim to study the prevalence of alcohol, cannabis, sedating medications, and other drugs in injured drivers from 4 Canadian Provinces. Methods: This prospective cohort study obtained excess clinical blood samples from consecutive injured drivers who attended a participating Canadian trauma centre following a MVC. Blood samples were analyzed using a broad spectrum toxicology screen capable of detecting cannabinoids, cocaine, amphetamines (including their major analogues), and opioids as well as psychotropic pharmaceuticals (including antihistamines, benzodiazepines, other hypnotics, and sedating antidepressants). Alcohol and cannabinoids were quantified. Health records were reviewed to extract demographic, medical, and MVC information using a standardized data collection tool. Results: This study has been collecting data in 4 trauma centres in British Columbia (BC) since 2011 and was launched in 2 trauma centres in Alberta (AB), 1 in Saskatchewan (SK), and 2 in Ontario (ON) in 2018. In preliminary results from BC (n = 2412), 8% of injured drivers tested positive for THC and 13% for alcohol. Preliminary results from other provinces (n = 301) suggest a regional variation in prevalence of drivers testing positive for THC (10% - 27%), alcohol (17% - 29%), and other drugs. By May 2018, an estimated 4500 cases from BC, 600 from AB, 150 from SK, and 650 from ON will have been analyzed. We will report the prevalence of positive tests for alcohol, THC, other recreational drugs, and sedating medications, pre and post cannabis legalization. The number of cases with alcohol and/or THC levels above Canadian per se limits will also be reported. Results will be reported according to province, driver sex, age, single vs. multi vehicle crashes, and requirement for hospital admission. Conclusion: This will be among the largest international datasets on drug use by injured drivers. Our findings will provide patterns of drug and alcohol impairment in 4 Canadian provinces pre and post cannabis legalization. The significance of these findings and implication for impaired driving policy and prevention programs in Canada will be discussed.

In calculus if we want to find the area under a curve, there is often a simple technique of calculus for us to use. Consider the curve in Figure 3-1. This is the curve of y = 2x – x2.

The events recorded in this chapter took place almost seventy years ago. This chapter is therefore what Benjamin Disraeli, novelist, wit, and twice prime minister of England, has called an instance of one’s “anecdotage.” My first knowledge of the details of scientific computation came from a book discarded by the MIT Library and brought home by my elder brother, then an undergraduate at MIT. That book derived from the computation laboratory of the University of Edinburgh run by Sir Edmund Whittaker (1873–1956). It is interesting and amusing to read how the individual computer’s desks were outfitted. (Incidentally, in those years a computer was not an instrument, but a person, and a computation laboratory was a rarity on university campuses.) “Each desk. . . contains a locker in which computing paper can be kept without being folded. . . .Each desk is supplied with a copy of Barlow’s tables (which give the square, square root, cube and cube root, and the reciprocal of all the numbers up to 10,000) and with tables giving the values of trigonometric functions and logarithms. These may, of course, be supplemented by a slide rule or any of the various calculating machines now in use. . . .” In the pre-electronic days, then, scientific computations were carried out by a variety of means. There was the pencil and paper method employing the rules of arithmetic taught in elementary school. There were slide rules, both the ten and twenty inch and circular varieties. There were special purpose slide rules adapted to special technologies. There were electro-mechanical computing machines such as the Marchant or the Frieden. There were mathematical tables of logarithms, exponentials, and special functions such as the Bessel functions. (One of the very first jobs that the electric relay computers carried out circa 1944 was to compute tables of the Bessel and related functions.) There were sets of French curves used for interpolation, approximation or smoothing. (Smoothing was often called “fairing”.) Very large French curves were employed in both auto and ship design. There were nomograms galore. There were planimeters—simple ones that were used to get areas and more complicated ones that would also yield moments. Computational mechanisms and devices have a very long history.

There are many paradoxes associated with the concept of infinity. I would like to set forth and comment on one which, though well-known in principle, I have not seen stated in the following form.

Foreword by Philip J. Davis vii Preface ix Acknowledgments xiii Sources xv One: Lars V. Ahlfors 1 Two: Tom Apostol 17 Three: Harold M. Bacon 43 Four: Tom Banchoff 52 Five: Leon Bankoff 79 Six: Alice Beckenbach 96 Seven: Arthur Benjamin 107 Eight: Dame Mary L. Cartwright 129 Nine: Joe Gallian 146 Ten: Richard K. Guy 165 Eleven: Fern Hunt 193 Twelve: Dusa McDuff 215 Thirteen: Donald G. Saari 240 Fourteen: Atle Selberg 254 Fifteen: Jean Taylor 274 Sixteen: Philippe Tondeur 294 Biographical Notes 319 Glossary 321 Index 325

The notion of entropy was first introduced in the 1850's by Rudolf Clausius in the context of thermodynamics. Focusing on the meaning that an increase of entropy means an increase of disorder, dissipation and decay, subsequent generations of authors have imported the notion into practically every area of intellectual discussion.This paper will mention numerous definitions of entropy and consider the virtues and ambiguities of the concept of entropy. It will pass judgement on some old and new interpretations of certain current societal developments that have been made along entropic lines and that conflict with the current Western social ethos.

Group Theory and the Classification of Finite Simple Groups. Non-Cantorian Set Theory. Non-Euclidean Geometry. The Prime Number Theorem. Appendix A. Nonstandard Analysis. Fourier Analysis.

TED WILLIAMS (pseudonym) is the Chairman of the Mathematics Departement in a fine private school in New England. He was interviewed in April, 1978.