Modular knots, automorphic forms, and the Rademacher symbols for triangle groups.

É. Ghys proved that the linking numbers of modular knots and the "missing" trefoil in coincide with the values of a highly ubiquitous function called the Rademacher symbol for . In this article, we replace by the triangle group for any coprime pair (, ) of integers with .

Respiratory sarcopenia: A position paper by four professional organizations.

We defined respiratory sarcopenia as a coexistence of respiratory muscle weakness and decreased respiratory muscle mass. Although respiratory muscle function is indispensable for life support, its evaluation has not been included in the regular assessment of respiratory function or adequately evaluated in clinical practice. Considering this situation, we prepared a position paper outlining basic knowledge, diagnostic and assessment methods, mechanisms, involvement in respiratory diseases, intervention and treatment methods, and future perspectives on respiratory sarcopenia, and summarized the current consensus on respiratory sarcopenia.

Effect of tiotropium/olodaterol on sedentary and active time in patients with COPD: post hoc analysis of the VESUTO study.

Background: Patients with COPD are less physically active. This post hoc analysis of a randomized, double-blind, active-controlled, crossover trial assessed the efficacy of once-daily tiotropium/olodaterol combination therapy versus tiotropium monotherapy in Japanese patients with COPD.

Patients And Methods: Patients were provided with a three-axis accelerometer to measure sedentary and active behavior defined as 1.

Toward the harmonization of work with treatment and prevention for patients with chronic respiratory failure.

Home oxygen therapy (HOT), also known as long-term oxygen therapy, is prescribed to patients with chronic respiratory failure (CRF) due to advanced respiratory diseases, as it has been shown to improve the prognosis of patients with chronic obstructive pulmonary disease (COPD). However, the therapeutic impact of HOT does not fully reflect the "socialization" of patients, which is one of the final goals of "comprehensive pulmonary rehabilitation", proposed by the Japanese Respiratory Society. Since working is one form of socialization, we evaluated a 55-yr-old worker prescribed with HOT for the management of advanced COPD to elucidate and assess the social barriers experienced by HOT users.

Efficacy of tiotropium/olodaterol on lung volume, exercise capacity, and physical activity.

Purpose: This study evaluated the efficacy of tiotropium/olodaterol vs tiotropium on lung function, exercise capacity, and physical activity in patients with COPD.

Patients And Methods: A total of 184 patients aged ≥40 years with COPD (Global Initiative for Chronic Obstructive Lung Disease stage II-IV) received tiotropium/olodaterol for 6 weeks, then tiotropium for 6 weeks, or vice versa. The primary endpoint was inspiratory capacity (IC) at peak post-dose.