Calculus of variations on time scales and applications to economics

We consider some problems of the calculus of variations on time scales. On the beginning our attention is paid on two inverse extremal problems on arbitrary time scales. Firstly, using the Euler-Lagrange equation and the strengthened Legendre condition, we derive a general form for a variation functional that attains a local minimum at a given point of the vector space. Furthermore, we prove a necessary condition for a dynamic integro-differential equation to be an Euler-Lagrange equation. New and interesting results for the discrete and quantum calculus are obtained as particular cases. Afterwards, we prove Euler-Lagrange type equations and transversality conditions for generalized infinite horizon problems. Next we investigate the composition of a certain scalar function with delta and nabla integrals of a vector valued field. Euler-Lagrange equations in integral form, transversality conditions, and necessary optimality conditions for isoperimetric problems, on an arbitrary time scale, are proved. In the end, two main issues of application of time scales in economic, with interesting results, are presented. In the former case we consider a firm that wants to program its production and investment policies to reach a given production rate and to maximize its future market competitiveness. The model which describes firm activities is studied in two different ways: using classical discretizations; and applying discrete versions of our result on time scales. In the end we compare the cost functional values obtained from those two approaches. The latter problem is more complex and relates to rate of inflation, p, and rate of unemployment, u, which inflict a social loss. Using known relations between p, u, and the expected rate of inflation π, we rewrite the social loss function as a function of π. We present this model in the time scale framework and find an optimal path π that minimizes the total social loss over a given time interval.

Biology, ecology, population dynamics and distribution of metals in Sepia officinalis on a typical estuarine coastal lagoon - Ria de Aveiro

The common cuttlefish, Sepia officinalis, is a necto-benthic cephalopod that can live in coastal ecosystems, with high influence of anthropogenic pressures and thus be vulnerable to exposure to various types of contaminants. The cuttlefish is a species of great importance to the local economy of Aveiro, considering the global data of catches of this species in the Ria de Aveiro. However, studies on this species in Ria de Aveiro are scarce, so the present study aims to fill this information gap about the cuttlefish in the Ria de Aveiro. The cuttlefish enters Ria de Aveiro in the spring and summer to reproduce, returning to deeper waters in the winter. In terms of abundance, the eastern and center regions of the lagoon, closer to the sea, showed the highest values of abundance, while the northern and southern regions of the main channel had the lowest abundance. This fact may be related to abiotic factors, as well as depth, salinity and temperature. In the most southern point of the Ria de Aveiro (Areão) no cuttlefish was caught. This site had the lowest values of salinity and depth. The cuttlefish has an allometric the females being heavier than males to mantle lengths greater than 82.4 mm. Males reach sexual maturity first than females. In Ria de Aveiro in a generation of parents was found. The cuttlefish, presents itself as opportunistic predators, consuming a wide variety of prey from different taxa. The diet was similar in different sampling locations observing significant differences for the seasons. S. officinalis was captured at 10 sites in the Ria de Aveiro with different anthropogenic sources of contamination. Thus, levels of metals analyzed were similar at all sampling sites, with the exception of a restricted area, Laranjo, which showed higher values. The cuttlefish has the ability to accumulate metals in your body. The levels of Fe, Zn, Cu, Cd, Pb and Hg found in the digestive gland and mantle reflect a differential accumulation of metals in the tissues. This accumulation is related to the type and function of tissue analyzed and the type of metal analysis (essential and non-essential). The metal concentrations in the digestive gland are higher than in the mantle, with the exception of mercury. This may be due to the high affinity of the mantle for the incorporation of methylmercury (MeHg), the most abundant form of mercury. The accumulation of metals can vary over a lifetime, depending on the metal. The concentrations of Zn, Cd and Hg increases throughout life, while Pb decreases and essential metals such as Fe and Cu remain constant. The data collected suggest that the cuttlefish (Sepia officinalis) can be used as a bioindicator of environmental contamination for some metals.

Functional analysis of String/CDC25 phosphatase in post-mitotic neurons

Cell cycle and differentiation are two highly coordinated processes during organ development. Recent studies have demonstrated that core cell cycle regulators also play cell cycle-independent functions in post-mitotic neurons, and are essential for the maintenance of neuronal homeostasis. CDC25 phosphatases are well-established CDK activators and their activity is mainly associated to proliferating tissues. The expression and activity of mammalian CDC25s has been reported in adult brains. However, their physiological relevance and the potential substrates in a non-proliferative context have never been addressed. string (stg) encodes the Drosophila CDC25 homolog. Previous studies from our group showed that stg is expressed in photoreceptors (PRs) and in lamina neurons, which are two differentiated cell types that compose the fly visual system. The aims of this work are to uncover the function of stg and to identify its potential neuronal substrates, using the Drosophila visual system as a model. To gain insight into the function of stg in a non-dividing context we used the GAL4/UAS system to promote downregulation of stg in PR-neurons, through the use of an RNAi transgene. The defects caused by stg loss-of-function were evaluated in the developing eye imaginal disc by immunofluorescence, and during adult stages by scanning electron microscopy. This genetic approach was combined with a specific proteomic method, two-dimensional difference gel electrophoresis (2D-DIGE), to identify the potential substrates in PR-cells. Our results showed that stg downregulation in PRs affects the well-patterned retina organization, inducing the loss of apical maintenance of PR-nuclei on the eye disc, and ommatidia disorganization. We also detected an abnormal accumulation of cytoskeletal proteins and a disruption of the axon structure. As a consequence, the projection of PR-axons into the lamina and medulla neuropils of the optic lobe was impaired. Upon stg downregulation, we also detected that PR-cells accumulate Cyclin B. Although the rough eye phenotype observed upon stg downregulation suggests neurodegeneration, we did not detect neuronal death during larval stages, suggesting that it likely occurs during pupal stages or during adulthood. By 2D-DIGE, we identified seven proteins which were differentially expressed upon stg downregulation, and are potential neuronal substrates of Stg. Altogether, our observations suggest that Stg phosphatase plays an essential role in the Drosophila visual system neurons, regulating several cell components and processes in order to ensure their homeostasis.