Applications of frames of subspaces in Richardson and Chebyshev methods for solving operator equations

This paper is concerned with the construction of two iterative methods by frames of subspaces for solving an operator equation on Hilbert spaces. We present two algorithms based on Richardson and Chebyshev methods, and investigate their convergence and optimality

Does cocoa/dark chocolate supplementation have favorable effect on body weight, body mass index and waist circumference? A systematic review, meta-analysis and dose-response of randomized clinical trials

<p><i>Background:</i> Cocoa and dark chocolate (DC) have been reported to be effective for health promotion; however the exact effect of cocoa/DC on anthropometric measures have not been yet defined. <i>Methods:</i> A comprehensive search to identify randomized clinical trials investigating the impact of cocoa/DC on body weight, body mass index (BMI) and waist circumference (WC) was performed up to December 2017. A meta-analysis of eligible studies was performed using random effects model to estimate pooled effect size. Fractional polynominal modeling was used to explore dose-response relationships. <i>Results:</i> A total of 35 RCTs investigated the effects of cocoa/DC on weight, BMI and WC were included. Meta-analysis did not suggest any significant effect of cocoa/DC supplementation on body weight (−0.108 kg, 95% CI −0.262, 0.046 P = 0.168), BMI (−0.014 kg/m² 95% CI −0.105, 0.077, P: 0.759,) and WC (0.025 cm 95% CI −0.083, 0.129, P = 0.640). Subgroup analysis revealed that that weight and BMI were reduced with cocoa/DC supplementation ≥ 30 g chocolate per day in trials between 4-8 weeks. Cocoa/DC consumption resulted in WC reduction in non-linear fashion (r = 0.042, P-nonlinearity = 0.008).<i>Conclusion:</i> Cocoa/DC supplementation does not reduce anthropometric measures significantly. However subgroup analysis regarding dose (≥ 30 g/day) and duration (between 4 to 8 weeks) revealed significant reduction of body weight and BMI.</p