Sharp comparison and maximum principles via horizontal normal mapping in the Heisenberg group

In this paper we solve a problem raised by Gutiérrez and Montanari about comparison principles for H−convex functions on subdomains of Heisenberg groups. Our approach is based on the notion of the sub-Riemannian horizontal normal mapping and uses degree theory for set-valued maps. The statement of the comparison principle combined with a Harnack inequality is applied to prove the Aleksandrov-type maximum principle, describing the correct boundary behavior of continuous H−convex functions vanishing at the boundary of horizontally bounded subdomains of Heisenberg groups. This result answers a question by Garofalo and Tournier. The sharpness of our results are illustrated by examples.

Recommendations for sex/gender neuroimaging research: Key principles and implications for research design, analysis and interpretation

Neuroimaging (NI) technologies are having increasing impact in the study of complex cognitive and social processes. In this emerging field of social cognitive neuroscience, a central goal should be to increase the understanding of the interaction between the neurobiology of the individual and the environment in which humans develop and function. The study of sex/gender is often a focus for NI research, and may be motivated by a desire to better understand general developmental principles, mental health problems that show female-male disparities, and gendered differences in society. In order to ensure the maximum possible contribution of NI research to these goals, we draw attention to four key principles—overlap, mosaicism, contingency and entanglement—that have emerged from sex/gender research and that should inform NI research design, analysis and interpretation. We discuss the implications of these principles in the form of constructive guidelines and suggestions for researchers, editors, reviewers and science communicators.