Asymptotic velocity for four celestial bodies.

Asymptotic velocity is defined as the Cesàro limit of velocity. As such, its existence has been proved for bounded interaction potentials. This is known to be wrong in celestial mechanics with four or more bodies. Here, we show for a class of pair potentials including the homogeneous ones of degree -α for α∈(0, 2), that asymptotic velocities exist for up to four bodies, dimension three or larger, for any energy and almost all initial conditions on the energy surface.This article is part of the theme issue 'Finite dimensional integrable systems: new trends and methods'.

Laparoscopic anterior hemifundoplication and hiatoplasty for the treatment of upside-down stomach: mid- and long-term results after 40 patients.

BACKGROUND: Treatment of type 4 hiatal hernia using a minimally invasive approach is challenging and requires good familiarity with this technique. METHODS: From October 1992 to August 2010, 40 patients with a median age of 68 years underwent laparoscopic anterior hemifundoplication surgery for upside-down stomach and were included in our prospective study. The median symptoms duration was 5 years. The leading clinical symptoms were postprandial, epigastric, or retrosternal pain (80%), heartburn (78%), regurgitation (80%), dysphagia (53%), and anemia (48%). Preoperative evaluation included blood test, chest X-ray, upper endoscopy, and barium swallow. In some patients an esophageal 24-h pH study and esophageal manometry were performed. The median follow-up was 46 months using a standardized questionnaire, including Smiley score, modified Visick score, gastrointestinal quality-of-life index (GQLI), and specific reflux symptoms score. RESULTS: Surgery was finished laparoscopically in 39 patients (97%). One patient had to be converted to an open procedure because of severe adhesions. Mesh hiatoplasty had to be performed in one patient due to a large hiatal defect. Median operative time was 160 min (range=90-275) and median blood loss was 5 ml (range=0-300). Seven patients (18%) presented with acute symptoms. Intraoperative technical complications occurred in four patients (10%) and nontechnical complications in two cases (5%). Median postoperative hospital stay was 5 days (range=2-17). Postoperative complications occurred in two patients (5%): one pleural effusion and one surgical emphysema. There was no mortality or symptomatic recurrence. All scores showed significant improvement and patient satisfaction. CONCLUSION: Laparoscopic treatment of type 4 hiatal hernia is safe. With respect to the quality of life, anterior hemifundoplication is highly effective.