Distribution theory and fundamental solutions of differential operators

The objective of this dissertation is to study the theory of distributions and some of its applications. Certain concepts which we would include in the theory of distributions nowadays have been widely used in several fields of mathematics and physics. It was Dirac who first introduced the delta function as we know it, in an attempt to keep a convenient notation in his works in quantum mechanics. Their work contributed to open a new path in mathematics, as new objects, similar to functions but not of their same nature, were being used systematically. Distributions are believed to have been first formally introduced by the Soviet mathematician Sergei Sobolev and by Laurent Schwartz. The aim of this project is to show how distribution theory can be used to obtain what we call fundamental solutions of partial differential equations.

"Heroes" and "Villains" of World History across Cultures

Emergent properties of global political culture were examined using data from the World History Survey (WHS) involving 6,902 university students in 37 countries evaluating 40 figures from world history. Multidimensional scaling and factor analysis techniques found only limited forms of universality in evaluations across Western, Catholic/Orthodox, Muslim, and Asian country clusters. The highest consensus across cultures involved scientific innovators, with Einstein having the most positive evaluation overall. Peaceful humanitarians like Mother Theresa and Gandhi followed. There was much less cross-cultural consistency in the evaluation of negative figures, led by Hitler, Osama bin Laden, and Saddam Hussein. After more traditional empirical methods (e.g., factor analysis) failed to identify meaningful cross-cultural patterns, Latent Profile Analysis (LPA) was used to identify four global representational profiles: Secular and Religious Idealists were overwhelmingly prevalent in Christian countries, and Political Realists were common in Muslim and Asian countries. We discuss possible consequences and interpretations of these different representational profiles.

Dietary animal and plant protein intakes and their associations with obesity and cardio-metabolic indicators in European adolescents: the HELENA cross-sectional study

Background: Previous studies suggest that dietary protein might play a beneficial role in combating obesity and its related chronic diseases. Total, animal and plant protein intakes and their associations with anthropometry and serum biomarkers in European adolescents using one standardised methodology across European countries are not well documented. Objectives: To evaluate total, animal and plant protein intakes in European adolescents stratified by gender and age, and to investigate their associations with cardio-metabolic indicators (anthropometry and biomarkers). Methods: The current analysis included 1804 randomly selected adolescents participating in the HELENA study (conducted in 2006-2007) aged 12.5-17.5 y (47% males) who completed two non-consecutive computerised 24-h dietary recalls. Associations between animal and plant protein intakes, and anthropometry and serum biomarkers were examined with General linear Model multivariate analysis. Results: Average total protein intake exceeded the recommendations of World Health Organization and European Food Safety Authority. Mean total protein intake was 96 g/d (59% derived from animal protein). Total, animal and plant protein intakes (g/d) were significantly lower in females than in males and total and plant protein intakes were lower in younger participants (12.5-14.9 y). Protein intake was significantly lower in underweight subjects and higher in obese ones; the direction of the relationship was reversed after adjustments for body weight (g/(kg.d)). The inverse association of plant protein intakes was stronger with BMI z-score and body fat percentage (BF%) compared to animal protein intakes. Additionally, BMI and BF% were positively associated with energy percentage of animal protein. Conclusions: This sample of European adolescents appeared to have adequate total protein intake. Our findings suggest that plant protein intakes may play a role in preventing obesity among European adolescents. Further longitudinal studies are needed to investigate the potential beneficial effects observed in this study in the prevention of obesity and related chronic diseases.