Duality between Ahlfors-Liouville and Khas'minskii properties for nonlinear equations

In recent years, the study of the interplay between (fully) non-linear potential theory and geometry received important new impulse. The purpose of this work is to move a step further in this direction by investigating appropriate versions of parabolicity and maximum principles at infinity for large classes of non-linear (sub)equations $F$ on manifolds. The main goal is to show a unifying duality between such properties and the existence of suitable $F$-subharmonic exhaustions, called Khas'minskii potentials, which is new even for most of the "standard" operators arising from geometry, and improves on partial results in the literature. Applications include new characterizations of the classical maximum principles at infinity (Ekeland, Omori-Yau and their weak versions by Pigola-Rigoli-Setti) and of conservation properties for stochastic processes (martingale completeness). Applications to the theory of submanifolds and Riemannian submersions are also discussed.Comment: 67 pages. Final versio

Dirichlet parabolicity and L1L^1-Liouville property under localized geometric conditions

We shed a new light on the $L^1$-Liouville property for positive, superharmonic functions by providing many evidences that its validity relies on geometric conditions localized on large enough portions of the space. We also present examples in any dimension showing that the $L^1$-Liouville property is strictly weaker than the stochastic completeness of the manifold. The main tool in our investigations is represented by the potential theory of a manifold with boundary subject to Dirichlet boundary conditions. The paper incorporates, under a unifying viewpoint, some old and new aspects of the theory, with a special emphasis on global maximum principles and on the role of the Dirichlet Green's kernel

Representational organization of novel task sets during proactive encoding

Recent multivariate analyses of brain data have boosted our understanding of the organizational principles that shape neural coding. However, most of this progress has focused on perceptual visual regions (Connolly et al., 2012), whereas far less is known about the organization of more abstract, action-oriented representations. In this study, we focused on humans{\textquoteright} remarkable ability to turn novel instructions into actions. While previous research shows that instruction encoding is tightly linked to proactive activations in fronto-parietal brain regions, little is known about the structure that orchestrates such anticipatory representation. We collected fMRI data while participants (both males and females) followed novel complex verbal rules that varied across control-related variables (integrating within/across stimuli dimensions, response complexity, target category) and reward expectations. Using Representational Similarity Analysis (Kriegeskorte et al., 2008) we explored where in the brain these variables explained the organization of novel task encoding, and whether motivation modulated these representational spaces. Instruction representations in the lateral prefrontal cortex were structured by the three control-related variables, while intraparietal sulcus encoded response complexity and the fusiform gyrus and precuneus organized its activity according to the relevant stimulus category. Reward exerted a general effect, increasing the representational similarity among different instructions, which was robustly correlated with behavioral improvements. Overall, our results highlight the flexibility of proactive task encoding, governed by distinct representational organizations in specific brain regions. They also stress the variability of motivation-control interactions, which appear to be highly dependent on task attributes such as complexity or novelty.SIGNIFICANCE STATEMENTIn comparison with other primates, humans display a remarkable success in novel task contexts thanks to our ability to transform instructions into effective actions. This skill is associated with proactive task-set reconfigurations in fronto-parietal cortices. It remains yet unknown, however, how the brain encodes in anticipation the flexible, rich repertoire of novel tasks that we can achieve. Here we explored cognitive control and motivation-related variables that might orchestrate the representational space for novel instructions. Our results showed that different dimensions become relevant for task prospective encoding depending on the brain region, and that the lateral prefrontal cortex simultaneously organized task representations following different control-related variables. Motivation exerted a general modulation upon this process, diminishing rather than increasing distances among instruction representations

Maximum principle for semi-elliptic trace operators and geometric applications

Based on ideas of L. Al\'ias, D. Impera and M. Rigoli developed in "Hypersurfaces of constant higher order mean curvature in warped products", we develope a fairly general weak/Omori-Yau maximum principle for trace operators. We apply this version of maximum principle to generalize several higher order mean curvature estimates and to give an extension of Alias-Impera-Rigoli Slice Theore

Objetivos do ensino de ciências no discurso de professores : um olhar sociocultural

O debate sobre as contribuições da Educação em Ciências no sentido de melhor servir aos interesses da coletividade nos levou ao objetivo de investigar o discurso de professores de ciências sobre as finalidades da educação e sua relação com a realidade social. Analisamos os enunciados de 27 professores (de Física, Química, Biologia e Matemática) e a influência da linguagem acadêmica no discurso comprometido com o social. Apenas dez professores indicaram objetivos relacionados à realidade social, tendo os demais enfatizado a realidade natural ou o indivíduo. A influência de contextos acadêmicos de formação continuada parece influenciar a preocupação social dos professores. De toda a forma, os resultados apontam para a necessidade de envidar esforços na formação dos professores das ciências naturais no sentido de favorecer a abordagem de problemas sociais

Phase portraits of Bernoulli quadratic polynomial differential systems

In this paper we study a new class of quadratic polynomial differential systems. We classify all global phase portraits in the Poincaré disk of Bernoulli quadratic polynomial differential systems in R2