Non-minimality of corners in subriemannian geometry

We give a short solution to one of the main open problems in subriemannian geometry. Namely, we prove that length minimizers do not have corner-type singularities. With this result we solve Problem II of Agrachev's list, and provide the first general result toward the 30-year-old open problem of regularity of subriemannian geodesics.Comment: 11 pages, final versio

Extremal polynomials in stratified groups

We introduce a family of extremal polynomials associated with the prolongation of a stratified nilpotent Lie algebra. These polynomials tre related to a new algebraic characterization of abnormal sub-Riemannian extremals in stratified nilpotent Lie groups. They satisfy a set of remarkable structure relations that are used to integrate the adjoint equations, in both normal and abnormal case

Besicovitch Covering Property on graded groups and applications to measure differentiation

We give a complete answer to which homogeneous groups admit homogeneous distances for which the Besicovitch Covering Property (BCP) holds. In particular, we prove that a stratified group admits homogeneous distances for which BCP holds if and only if the group has step 1 or 2. These results are obtained as consequences of a more general study of homogeneous quasi-distances on graded groups. Namely, we prove that a positively graded group admits continuous homogeneous quasi-distances satisfying BCP if and only if any two different layers of the associated positive grading of its Lie algebra commute. The validity of BCP has several consequences. Its connections with the theory of differentiation of measures is one of the main motivations of the present paper. As a consequence of our results, we get for instance that a stratified group can be equipped with some homogeneous distance so that the differentiation theorem holds for each locally finite Borel measure if and only if the group has step 1 or 2. The techniques developed in this paper allow also us to prove that sub-Riemannian distances on stratified groups of step 2 or higher never satisfy BCP. Using blow-up techniques this is shown to imply that on a sub-Riemannian manifold the differentiation theorem does not hold for some locally finite Borel measure

Sets with constant normal in Carnot groups: properties and examples

We analyze subsets of Carnot groups that have intrinsic constant normal, as they appear in the blowup study of sets that have finite subRiemannian perimeter. The purpose of this paper is threefold. First, we prove some mild regularity and structural results in arbitrary Carnot groups. Namely, we show that for every constant-normal set in a Carnot group its subRiemannian-Lebesgue representative is regularly open, contractible, and its topological boundary coincides with the reduced boundary and with the measure-theoretic boundary. We infer these properties from a metric cone property. Such a cone will be a semisubgroup with nonempty interior that is canonically associated with the normal direction. We characterize the constant-normal sets exactly as those that are arbitrary unions of translations of such semisubgroups. Second, making use of such a characterization, we provide some pathological examples in the specific case of the free-Carnot group of step 3 and rank 2. Namely, we construct a constant normal set that, with respect to any Riemannian metric, is not of locally finite perimeter; we also construct an example with non-unique intrinsic blowup at some point, showing that it has different upper and lower subRiemannian density at the origin. Third, we show that in Carnot groups of step 4 or less, every constant-normal set is intrinsically rectifiable, in the sense of Franchi, Serapioni, and Serra Cassano

Sub-Finsler Structures from the Time-Optimal Control Viewpoint for some Nilpotent Distributions

In this paper, we study the sub-Finsler geometry as a time-optimal control problem. In particular, we consider non-smooth and non-strictly convex sub-Finsler structures associated with the Heisenberg, Grushin, and Martinet distributions. Motivated by problems in geometric group theory, we characterize extremal curves, discuss their optimality, and calculate the metric spheres, proving their Euclidean rectifiability. © 2016, Springer Science+Business Media New York

Mass spectrometry of B. subtilis CopZ: Cu(I)-binding and interactions with bacillithiol

CopZ from Bacillus subtilis is a well-studied member of the highly conserved family of Atx1-like copper chaperones. It was previously shown via solution and crystallographic studies to undergo Cu(I)-mediated dimerisation, where the CopZ dimer can bind between one and four Cu(I) ions. However, these studies could not provide information about the changing distribution of species at increasing Cu(I) levels. To address this, electrospray ionisation mass spectrometry using soft ionisation was applied to CopZ under native conditions. Data revealed folded, monomeric CopZ in apo- and Cu(I)-bound forms, along with Cu(I)-bound dimeric forms of CopZ at higher Cu(I) loading. Cu4(CopZ)2 was the major dimeric species at loadings >1 Cu(I)/CopZ, indicating the cooperative formation of the tetranuclear Cu(I)-bound species. As the principal low molecular weight thiol in B. subtilis, bacillithiol (BSH) may play a role in copper homeostasis. Mass spectrometry showed that increasing BSH led to a reduction in Cu(I)-bound dimeric forms, and the formation of S-bacillithiolated apo-CopZ and BSH adducts of Cu(I)-bound forms of CopZ, where BSH likely acts as a Cu(I) ligand. These data, along with the high affinity of BSH for Cu(I), determined here to be β2(BSH) = ∼4 × 1017 M−2, are consistent with a role for BSH alongside CopZ in buffering cellular Cu(I) levels. Here, mass spectrometry provides a high resolution overview of CopZ–Cu(I) speciation that cannot be obtained from less discriminating solution-phase methods, thus illustrating the potential for the wider application of this technique to studies of metal–protein interactions

Eating behavior associated with food intake in European adolescents participating in the HELENA study

Adolescence is recognized as a time of rapid physiological and behavioral change. In this transition, eating behavior is still being formed and remains an integral part of a person’s lifestyle throughout his or her life. This study aims to assess eating behavior and associations with food intake in European adolescents. We included 2194 adolescents (45.9% boys), aged 12.5 to 17.5 years, from the cross-sectional HELENA study, with two completed 24 h recalls and complete questionnaire data on their eating behavior (Eating Behavior and Weight Problems Inventory for Children- EWI-C). Three subscales of the EWI were evaluated; they measured Strength and motivation to eat (EWI 1), Importance and impact of eating (EWI 2), and Eating as a means of coping with emotional stress (EWI 3). Since these subscales were specially focused on eating behavior, participants were classified as either Low (=P75) or High (>P75) on each of the subscales. Our results showed a higher consumption of different types of food, in the EWI 1 scales, linked to the hunger and interest in eating, and we observed a relationship with the consumption of energy-dense products. This result was repeated in EWI 3, the subscale linked to emotional eating, where we also found higher consumption of energy-dense products. This study suggests that special features of eating behavior are associated with food intake in adolescents

Relationship between self-reported dietary intake and physical activity levels among adolescents: The HELENA study

Background Evidence suggests possible synergetic effects of multiple lifestyle behaviors on health risks like obesity and other health outcomes. Therefore it is important to investigate associations between dietary and physical activity behavior, the two most important lifestyle behaviors influencing our energy balance and body composition. The objective of the present study is to describe the relationship between energy, nutrient and food intake and the physical activity level among a large group of European adolescents. Methods The study comprised a total of 2176 adolescents (46.2% male) from ten European cities participating in the HELENA (Healthy Lifestyle in Europe by Nutrition in Adolescence) study. Dietary intake and physical activity were assessed using validated 24-h dietary recalls and self-reported questionnaires respectively. Analyses of covariance (ANCOVA) were used to compare the energy and nutrient intake and the food consumption between groups of adolescents with different physical activity levels (1st to 3rd tertile). Results In both sexes no differences were found in energy intake between the levels of physical activity. The most active males showed a higher intake of polysaccharides, protein, water and vitamin C and a lower intake of saccharides compared to less active males. Females with the highest physical activity level consumed more polysaccharides compared to their least active peers. Male and female adolescents with the highest physical activity levels, consumed more fruit and milk products and less cheese compared to the least active adolescents. The most active males showed higher intakes of vegetables and meat, fish, eggs, meat substitutes and vegetarian products compared to the least active ones. The least active males reported the highest consumption of grain products and potatoes. Within the female group, significantly lower intakes of bread and cereal products and spreads were found for those reporting to spend most time in moderate to vigorous physical activity. The consumption of foods from the remaining food groups, did not differ between the physical activity levels in both sexes. Conclusion It can be concluded that dietary habits diverge between adolescents with different self-reported physical activity levels. For some food groups a difference in intake could be found, which were reflected in differences in some nutrient intakes. It can also be concluded that physically active adolescents are not always inclined to eat healthier diets than their less active peers.The HELENA study took place with the financial support of the European Community Sixth RTD Framework Programme (Contract FOOD-CT: 2005-007034). This work was also partially supported by the European Union, in the framework of the Public Health Programme (ALPHA project, Ref: 2006120), the Swedish Council for Working Life and Social Research (FAS), the Spanish Ministry of Education (EX-2007-1124, and EX-2008-0641), and the Spanish Ministry of Health, Maternal, Child Health and Development Network (number RD08/0072) (JPRL, LAM)