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Campos localmente resolúveis, espaços de Hardy e extensão de funções CR
(Universidade Federal de São Carlos, 2012-05-23) Liboni Filho, Paulo Antonio; Hounie, Jorge Guillermo; https://lattes.cnpq.br/7302904386484949; https://lattes.cnpq.br/2231598823550786
Suppose that M is a smooth submanifold of CN and that L = ∪z∈CnLz is the Cauchy- Riemann structure associated to the N-dimensional complex space. For each p ∈ M we can consider the vector space Ap = CTpM ∩ Lp. If the reunion of those fibers originates a locally integrable structure, then we are going to say that M is a CR submanifold with CR structure A = ∪p∈MAp. It immediately follows that if h is a holomorphic application defined in a certain neighborhood U ⊃ M, then A(h|M) = 0. If we consider a distribution u ∈ D′(M) such that Au = 0, then one can asks: is there a holomorphic application h defined in certain open set U such that h|M = u? The question, as it is, can be paraphrased as: is there any analytic extension of the CR distribution u? The answer is negative and there are several examples one can create. Consider a quadric application q : Cm × Cm −→ Cd and the manifold given by M = {(w, t) ∈ Cm × Cd,ℑt = q(w,w)}. Boggess has proved in [Bog01] that all Lp CR distributions in M (with p ≥ 1) admit a holomorphic extension to the interior of the convex hull of M. In this work, we are going to address the same question, but we are going to deal with CR distributions that are in hp with p > 0. Since hp(M) = Lp(M) if p > 1, then our result can be understood as an extension of the original Boggess theorem. The main ingredient of your work is a version of the Baouendi-Treves Approximation Theorem.
Decomposição de macrófitas aquáticas em dois reservatórios com diferentes estados tróficos
(Universidade Federal de São Carlos, 2012-05-23) Rocha, Rinaldo José da Silva; Bianchini Júnior, Irineu; https://lattes.cnpq.br/0659731944736389; https://lattes.cnpq.br/3916467877386034
The macrophytes are essential for the development and maintenance of food webs, to carbon dynamics and nutrient cycling in tropical aquatic systems. After senescence and plant death, decomposition of organic detritus depends heavily on their elemental compositions. Thus, the present study aimed to evaluate the effect of seasonality and trophic state of reservoirs and Vigário Lajes on the mass losses of detritus of Salvinia auriculata, Pistia stratiotes, Eichhornia crassipes, Sagittaria montevidensis and Brachiaria subquadripara, the influence of elemental composition of debris on the coefficients decay of carbon, nitrogen and phosphorus, and the stoichiometric ratios between these elements during the decomposition process. Over 120 days, the richest detritus in nutrients and less refractory (S. montevidensis and P. stratiotes) showed a biphasic pattern of decay and the greater decay of carbon (70-90%), phosphorus (90-96%) and nitrogen (68-94%). The poorest and refractories detritus (S. auriculata and E. crassipes) had lower rates of decay, with monophasic and biphasic patterns and lower mass loss in carbon-based (20-34%), phosphorus-based (26%), and nitrogen-based (10- 37%). As for the difference between the environments, the total mass loss of carbon was up to 70% higher during the rainy season (Sept/10 to Jan/11) and 27% higher in the dry season (may/11 to Sep/11) for detritus S. auriculata incubated in the reservoir Vigário; for the detritus of B. subquadripara total mass loss was also greater in this reservoir, reaching 58% in the rainy season and 36% during the dry season. The minor differences between the environments occurred in the mass loss of carbon detritus of S. montevidensis, (ca. 1.3% and 0.3% in rainfall in the dry season) and P. stratiotes (ca. 5.3% in the rainy season and 5.5% in the dry season). Yet these detritus showed the greatest losses of phosphorus (90-95%) and nitrogen (68-94%) in both experiments. The minor differences between the environments occurred in the mass loss of carbon detritus of S. montevidensis, (ca. 1.3% and 0.3% in the rainy season in the dry season) and P. stratiotes (ca. 5.3% and 5.5% in the rainy season in the dry season). Yet these detritus showed the greatest losses of phosphorus (90-95%) and nitrogen (68-94%) in both experiments. The ratio between carbon and nitrogen was higher in detritus of S. auriculata and E. crassipes in oligotrophic reservoir, and lower in detritus of S. montevidensis and P. stratiotes, whose mass loss coefficients (k) were higher.