Leading Digit Laws on Linear Lie Groups

We determine the leading digit laws for the matrix components of a linear Lie group $G$. These laws generalize the observations that the normalized Haar measure of the Lie group $\mathbb{R}^+$ is $dx/x$ and that the scale invariance of $dx/x$ implies the distribution of the digits follow Benford's law, which is the probability of observing a significand base $B$ of at most $s$ is $\log_B(s)$; thus the first digit is $d$ with probability $\log_B(1 + 1/d)$). Viewing this scale invariance as left invariance of Haar measure, we determine the power laws in significands from one matrix component of various such $G$. We also determine the leading digit distribution of a fixed number of components of a unit sphere, and find periodic behavior when the dimension of the sphere tends to infinity in a certain progression.Comment: Version 1.0, 17 pages, 1 figur

Chronic Migraine Epidemiology and Outcomes – International (CaMEO-I) Study: Methods and multi-country baseline findings for diagnosis rates and care

BACKGROUND: The Chronic Migraine Epidemiology and Outcomes-International study provides insight into people with migraine in multiple countries. METHODS: This cross-sectional, observational, web-based cohort study was conducted in Canada, France, Germany, Japan, United Kingdom, and United States. An initial Screening Module survey solicited general healthcare information from a representative sample and identified participants with migraine based on modified International Classification of Headache Disorders-3 criteria; those with migraine completed a detailed survey based on validated migraine-specific assessments. RESULTS: Among 90,613 people who correctly completed the screening surveys, 76,121 respondents did not meet the criteria for migraine, while 14,492 did. Among respondents with migraine, mean age ranged from 40 to 42 years. The median number of monthly headache days ranged from 2.33 to 3.33 across countries, while the proportion of respondents with moderate-to-severe disability (measured by Migraine Disability Assessment) ranged from 30% (Japan) to 52% (Germany). The proportion of respondents with ≥15 monthly headache days ranged from 5.4% (France) to 9.5% (Japan). Fewer than half of respondents with migraine in each country reported having received a migraine diagnosis. CONCLUSION: These results demonstrated high rates of migraine-related disability and underdiagnosis of migraine across six countries. This study will characterize country-level burden, treatment patterns, and geographical differences in care

Defining the Differences Between Episodic Migraine and Chronic Migraine

Chronic migraine (CM) and episodic migraine (EM) are part of the spectrum of migraine disorders, but they are distinct clinical entities. Population-based studies have shown that those with CM demonstrate higher individual and societal burden because they are significantly more disabled than those with EM and have greater impaired quality of life both inside and outside the home. Proper diagnosis of both conditions requires clearly defined clinical criteria. Diagnosis enables the initiation of appropriate treatments and risk-factor modification, which ultimately improve functional status and quality of life for persons with migraine. Recognizing that both disorders are on the spectrum of migraine, this review serves as a guide to define the disease state of CM as distinct from EM in terms of clinical, epidemiological, sociodemographic, and comorbidity profiles

桂林寺下附近地図（堀割再検分のため寺社奉行鳥居小八郎）

OnabotulinumtoxinA has been shown to reduce headache-days among patients with chronic migraine (CM). The objective of this analysis was to determine whether onabotulinumtoxinA has an impact on headache-day severity in patients with CM among those patients who were deemed non-responders based on reduction in the frequency of headache days alone. Data from the Phase 3 REsearch Evaluating Migraine Prophylaxis Therapy (PREEMPT) clinical trial program (a 24-week, 2-treatment cycle, double-blind, randomized, placebo-controlled, parallel-group phase, followed by a 32-week, 3-treatment cycle, open-label phase) were pooled for analysis. Patients kept a daily diary to record headache severity on a 4-point scale (from none to severe), and a 6-domain Headache Impact Test (HIT-6) was used to determine the clinical impact of headaches. Analysis was undertaken to assess whether the subset of patients that were headache-day frequency non-responders at week 24 (patients with <50% reduction in headache-day frequency) experienced a reduction in headache severity whilst receiving onabotulinumtoxinA. For headache-day frequency non-responders, significant reductions in the number of severe headache days, average daily headache severity, pooled percentage of severe headache days and headache severity score were observed at week 24 for patients who had received onabotulinumtoxinA compared with those who had received placebo. The between-group differences were reduced and non-significant at week 56. Similarly, headache-day frequency non-responders receiving onabotulinumtoxinA were found to have an improvement in the clinical impact of headaches using results from the HIT-6. These results suggest that even those patients with CM who are deemed non-responders based on analysis of headache frequency alone experience clinically meaningful relief from headache intensity following treatment with onabotulinumtoxinA. The online version of this article (doi:10.1186/s10194-017-0784-4) contains supplementary material, which is available to authorized users

Optimized Acute Treatment of Migraine Is Associated With Greater Productivity in People With Migraine: Results From the Chronic Migraine Epidemiology and Outcomes (CaMEO) Study.

OBJECTIVE: This study aimed to ascertain whether level of optimization of acute treatment of migraine is related to work productivity across the spectrum of migraine. METHODS: Data were from the Chronic Migraine Epidemiology and Outcomes (CaMEO) Study, an internet-based longitudinal survey. Respondents with migraine who reported full-time employment and use of ≥1 acute prescription medication for migraine were included. We determined relationships among lost productive time (LPT; measured with the Migraine Disability Assessment Scale), acute treatment optimization (Migraine Treatment Optimization Questionnaire- ), and monthly headache days (MHDs). RESULTS: There was a direct relationship between LPT and MHD category. Greater acute treatment optimization was associated with lower total LPT, less absenteeism, and less presenteeism within each MHD category. CONCLUSIONS: Optimizing acute treatment for migraine may reduce LPT in people with migraine and reduce indirect costs