We propose two draw systems for the group round of sport tournaments where there are some geographical and/or seeding restrictions. One of the systems, related to the equal‐sum partition problem, is “perfect,” since it yields perfectly balanced groups. The other system, which uses the classical scheme of extracting teams from pots, is heuristic and gives results where the groups have very similar scores. We apply our results to Fédération Internationale de Football Association (FIFA) Soccer World Cups and show that our proposals are much better than the FIFA system and also outperform other recently developed systems.

INTRODUCTION: To evaluate the prescription patterns, effectiveness, and safety of adding probiotics to Helicobacter pylori eradication therapy, in Europe. METHODS: International, prospective, noninterventional registry of the clinical practice of the European gastroenterologists. Data were collected and quality reviewed until March 2021 at AEG-REDCap. The effectiveness was evaluated by modified intention-to-treat analysis, differentiating by geographic areas. Adverse events (AEs) were categorized as mild, moderate, and severe. RESULTS: Overall, 36,699 treatments were recorded, where 8,233 (22%) were prescribed with probiotics. Probiotics use was associated with higher effectiveness in the overall analysis (odds ratio [OR] 1.631, 95% confidence interval [CI] 1.456–1.828), as well as in triple (OR 1.702, 95% CI 1.403–2.065), quadruple (OR 1.383, 95% CI 0.996–1.920), bismuth quadruple (OR 1.248, 95% CI 1.003–1.554), and sequential therapies (OR 3.690, 95% CI 2.686–5.069). Lactobacillus genus was associated with a higher therapy effectiveness in Eastern Europe when triple (OR 2.625, 95% CI 1.911–3.606) and bismuth quadruple (OR 1.587, 95% CI 1.117–2.254) first-line therapies were prescribed. In Central Europe, the use of probiotics was associated with a decrease in both the overall incidence of AEs (OR 0.656, 95% CI 0.516–0.888) and severe AEs (OR 0.312, 95% CI 0.217–0.449). Bifidobacterium genus was associated with lower overall (OR 0.725, 95% CI 0.592–0.888) and severe (OR 0.254, 95% CI 0.185–0.347) AEs, and Saccharomyces was associated with reduced overall (OR 0.54, 95% CI 0.32–0.91) and severe (OR 0.257, 95% CI 0.123–0.536) AEs under quadruple-bismuth regimen. DISCUSSION: In Europe, the use of probiotics was associated with higher effectiveness and safety of H. pylori eradication therapy. Lactobacillus improved treatment effectiveness, whereas Bifidobacterium and Saccharomyces were associated with a better safety profile.

We consider the maximum likelihood and Bayesian estimation of parameters and prediction of future records of the Weibull distribution from $\delta$ -record data, which consists of records and near-records. We discuss existence, consistency and numerical computation of estimators and predictors. The performance of the proposed methodology is assessed by Montecarlo simulations and the analysis of monthly rainfall series. Our conclusion is that inferences for the Weibull model, based on $\delta$ -record data, clearly improve inferences based solely on records. This methodology can be recommended, more so as near-records can be collected along with records, keeping essentially the same experimental design.

We propose procedures for draws in sports tournaments that will achieve balanced groups of size 3. We start by developing an efficient algorithm that finds all possible balanced groupings; that is, allocations of teams to groups such that the strength of all groups lies between two bounds. The algorithm uses comparison results to discard large sets of unfeasible allocations and backtracking. We then define pot-and-ball draws that randomly pick one of the balanced groupings. Our draw procedures ensure that all groups are of (virtually) identical competitive levels, thereby enhancing the overall fairness and interest of the tournament.