On Liouville-type theorems and the uniqueness of the positive Cauchy problem for a class of hypoelliptic operators

This note contains a representation formula for positive solutions of linear degenerate second-order equations of the form $$\partial_t u (x,t) = \sum_{j=1}^m X_j^2 u(x,t) + X_0 u(x,t) \qquad (x,t) \in \mathbb{R}^N \times\, ]- \infty ,T[,$$ proved by a functional analytic approach based on Choquet theory. As a consequence, we obtain Liouville-type theorems and uniqueness results for the positive Cauchy problem.Comment: The results of the present version recover most of the ones in the previous version, but, on top of it, this new version contains some further new and interesting result

Harnack Inequality for Hypoelliptic Second Order Partial Differential Operators

We consider non-negative solutions (Formula presented.) of second order hypoelliptic equations(Formula presented.) where \u3a9 is a bounded open subset of (Formula presented.) and x denotes the point of \u3a9. For any fixed x0 08 \u3a9, we prove a Harnack inequality of this type(Formula presented.) where K is any compact subset of the interior of the (Formula presented.)-propagation set ofx0 and the constant CK does not depend on u

Pesquisas apontam o aumento da eficiência do uso de rochas fosfatadas pela associação com microrganismos solubilizadores.

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Conservation Status of Marine Biodiversity in Oceania: An Analysis of Marine Species on the IUCN Red List of Threatened Species

Given the economic and cultural dependence on the marine environment in Oceania and a rapidly expanding human population, many marine species populations are in decline and may be vulnerable to extinction from a number of local and regional threats. IUCN Red List assessments, a widely used system for quantifying threats to species and assessing species extinction risk, have been completed for 1190 marine species in Oceania to date, including all known species of corals, mangroves, seagrasses, sea snakes, marine mammals, sea birds, sea turtles, sharks, and rays present in Oceania, plus all species in five important perciformfish groups. Many of the species in these groups are threatened by themodification or destruction of coastal habitats, overfishing fromdirect or indirect exploitation, pollution, and other ecological or environmental changes associated with climate change. Spatial analyses of threatened species highlight priority areas for both site- and species-specific conservation action. Although increased knowledge and use of newly available IUCN Red List assessments for marine species can greatly improve conservation priorities for marine species in Oceania,many important fish groups are still in urgent need of assessment

Market outlook of mineral-organic fertilizers in Brazil.

The aim of this work is to identify producers of organic-mineral fertilizers in Brazil and analyze the characteristics of the manufactured products