Zeros of the Möbius function of permutations

We show that if a permutation \pi contains two intervals of length 2, where one interval is an ascent and the other a descent, then the Möbius function \mu[1,\pi] of the interval [1,\pi] is zero. As a consequence, we prove that the proportion of permutations of length $\textit{n}$ with principal Möbius function equal to zero is asymptotically bounded below by (1\ -\ \sfrac{1}{e)^2} \geq 0.3995. This is the first result determining the value of \mu\left[1,\pi\right] for an asymptotically positive proportion of permutations \pi. We further establish other general conditions on a permutation \pi that ensure \mu\left[1,\pi\right]\ =\ 0, including the occurrence in \pi of any interval of the form \alpha\oplus\ 1\ \oplus\ \beta

Postconcussive Symptoms After Early Childhood Concussion

IMPORTANCE: Research on postconcussive symptoms (PCS) following early childhood concussion has been hindered by a lack of measures suitable for this age group, resulting in a limited understanding of their evolution in young children. OBJECTIVE: To document PCS in the first 3 months after early childhood concussion using a developmentally appropriate measure. DESIGN, SETTING, AND PARTICIPANTS: This cohort study used data collected at 3 Canadian and 1 US urban pediatric emergency departments (EDs) and 8 Canadian daycares from December 2018 to December 2022 as part of the Kids\u27 Outcomes and Long-Term Abilities (KOALA) project, a prospective, multicenter, longitudinal cohort study. Participants included children aged 6 to 72 months with early childhood concussion or orthopedic injury (OI) or uninjured children from the community to serve as controls. Data were analyzed from March 2023 to January 2024. EXPOSURE: Concussion sustained between ages 6 and 72 months. MAIN OUTCOMES AND MEASURES: Primary outcomes were cognitive, physical, behavioral and total PCS assessed prior to injury (retrospectively), acutely (within 48 hours), and at 10 days, 1 month, and 3 months after injury or recruitment through caregiver observations using the Report of Early Childhood Traumatic Injury Observations & Symptoms inventory. Group comparisons were analyzed using ordinal regression models. RESULTS: The study included 303 children (mean [SD] age, 35.8 [20.2] months; 152 [50.2%] male). Of these, 174 children had a concussion (mean [SD] age,  33.3 [19.9] months), 60 children had an OI (mean [SD] age, 38.4 [19.8] months) and 69 children were uninjured controls (mean [SD] age, 39.7 [20.8] months). No meaningful differences were found between the concussion and comparison groups in retrospective preinjury PCS. Significant group differences were found for total PCS at the initial ED visit (concussion vs OI: odds ratio [OR], 4.33 [95% CI, 2.44-7.69]; concussion vs control: OR, 7.28 [95% CI, 3.80-13.93]), 10 days (concussion vs OI: OR, 4.44 [95% CI, 2.17-9.06]; concussion vs control: OR, 5.94 [95% CI, 3.22-10.94]), 1 month (concussion vs OI: OR, 2.70 [95% CI, 1.56-4.68]; concussion vs control: OR, 4.32 [95% CI, 2.36-7.92]), and 3 months (concussion vs OI: OR, 2.61 [95% CI, 1.30-5.25]; concussion vs control: OR, 2.40 [95% CI, 1.36-4.24]). Significant group differences were also found for domain-level scores (cognitive, physical, behavioral) at various time points. CONCLUSIONS AND RELEVANCE: In this early childhood cohort study, concussion was associated with more PCS than OIs or typical development up to 3 months after injury. Given the limited verbal and cognitive abilities typical of early childhood, using developmentally appropriate manifestations and behaviors is a valuable way of tracking PCS and could aid in concussion diagnosis in young children

Epidemiologic and clinical updates on impulse control disorders: a critical review

The article reviews the current knowledge about the impulse control disorders (ICDs) with specific emphasis on epidemiological and pharmacological advances. In addition to the traditional ICDs present in the DSM-IV—pathological gambling, trichotillomania, kleptomania, pyromania and intermittent explosive disorder—a brief description of the new proposed ICDs—compulsive–impulsive (C–I) Internet usage disorder, C–I sexual behaviors, C–I skin picking and C–I shopping—is provided. Specifically, the article summarizes the phenomenology, epidemiology and comorbidity of the ICDs. Particular attention is paid to the relationship between ICDs and obsessive–compulsive disorder (OCD). Finally, current pharmacological options for treating ICDs are presented and discussed

Accounting: A General Commentary on an Empirical Science

Many researchers have questioned the view of accounting as a science. Some maintain that it is a service activity rather than a science, yet others entertain the view that it is an art or merely a technology. While it is true that accounting provides a service and is a technology (a methodology for recording and reporting), that fact does not prevent accounting from being a science. Based upon the structure and knowledge base of the discipline, this paper presents the case for accounting as an empirical science

Alexander, Callisthenes and the Sources of the Nile

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