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.

A generalization of the Hoffman-Lovász upper bound on the independence number of a regular graph

A family of quadratic programming problems whose optimal values are upper bounds on the independence number of a graph is introduced. Among this family, the quadratic programming problem which gives the best upper bound is identified. Also the proof that the upper bound introduced by Hoffman and Lovász for regular graphs is a particular case of this family is given. In addition, some new results characterizing the class of graphs for which the independence number attains the optimal value of the above best upper bound are given. Finally a polynomial-time algorithm for approximating the size of the maximum independent set of an arbitrary graph is described and the computational experiments carried out on 36 DIMACS clique benchmark instances are reported.

Besov spaces and the Laplacian on fractal H-sets

Nesta tese são estudados espaços de Besov de suavidade generalizada em espaços euclidianos, numa classe de fractais designados conjuntos-h e em estruturas abstractas designadas por espaços-h. Foram obtidas caracterizações e propriedades para estes espaços de funções. Em particular, no caso de espaços de Besov em espaços euclidianos, foram obtidas caracterizações por diferenças e por decomposições em átomos não suaves, foi provada uma propriedade de homogeneidade e foram estudados multiplicadores pontuais. Para espaços de Besov em conjuntos-h foi obtida uma caracterização por decomposições em átomos não suaves e foi construído um operador extensão. Com o recurso a cartas, os resultados obtidos para estes espaços de funções em fractais foram aplicados para definir e trabalhar com espaços de Besov de suavidade generalizada em estruturas abstractas. Nesta tese foi também estudado o laplaciano fractal, considerado a actuar em espaços de Besov de suavidade generalizada em domínios que contêm um conjunto-h fractal. Foram obtidos resultados no contexto de teoria espectral para este operador e foi estudado, à custa deste operador, um problema de Dirichlet fractal no contexto de conjuntos-h.

Sustainable viticulture in Bairrada appellation: vineyard and harvest year effects on grapes oenological potential

The sustainable viticulture of a region passes, among other aspects, for maximizing the varieties potential minimizing subsequent interventions during winemaking, which should contribute to the production of quality wines maintaining their typicity and rationalizing costs. The detailed knowledge of each Appellation specificities, namely vineyard parcel (soil type and topographical peculiarities) and harvest climatic conditions is crucial for sustainability in this sector. Thus, in line with this current trend, the starting point for the development of this PhD thesis was to evaluate the oenological potential of different varieties cultivated throughout Bairrada Appellation (Portugal). During maturation several changes in grape varieties occur, namely berries become sweeter, less acidic, and they develop flavour, aroma and colour properties. The development of these characteristics is essential to define grapes oenological potential, i.e. to estimate the possibility of their usage to produce specific wines. A three years sampling plan was designed to evaluate the effect of harvest year and parcel characteristics on V. vinifera cv. Arinto, Bical, Sauvignon Blanc, Baga, Castelão, Touriga Nacional, and Sousão grapes composition. For each variety, 3 parcels with different characteristics were selected. Several physicochemical parameters were evaluated, during maturation: berry weight, pH, titratable acidity, sugar and phenolic contents, antiradical activity, and volatile composition (free fraction). Special attention was devoted to grapes at technologic maturity, since, besides these parameters, glycosidically-linked fraction was also considered. By using the results obtained at technologic maturity, a comprehensive approach was applied to identify the significance of harvest and parcel characteristics effects on each variety composition. Considering all the parameters under study, it may be highlighted some significant differences. According to the obtained results determined during maturation, it was possible to conclude that Arinto, Bical and Sauvignon Blanc grapes from parcels with clay-sandy and clay-calcareous soils have higher phenolic content and antiradical activity. Otherwise, Sauvignon Blanc presented similar volatile composition for grapes cultivated in the 3 parcels, while Arinto and Bical exhibited higher volatile content in grapes from claysandy and clay-calcareous soils. For Baga, Castelão and Touriga Nacional red varieties, grapes with higher phenolic content, antiradical activity, and volatile content were obtained from clayey and clay-calcareous soils. Furthermore, for Touriga Nacional, parcels altitude seems also to modulate grapes composition. Beyond parcel effect, harvest year conditions also influence grapes composition: 2011 harvest was related with lower phenolic and volatile contents, as well as lower antiradical activity.For grapes collected at technologic maturity, analysis of variance-simultaneous component analysis (ASCA) was applied combining all the parameters under study, in order to assess the influence of harvest and parcel characteristics on each variety oenological potential. The results obtained using this comprehensive approach is closely related with those observed during maturation and revealed that harvest was the main factor that influenced grapes composition (53% to 68% of the total data set variance) followed by parcel characteristics, explaining ca. 15-19% of the total data set variance. The oenological potential of each variety may be different from one parcel to another, i.e., clay-sandy and clay-calcareous related-environments seem to favour Arinto and Bical white grapes composition, but for the red varieties, grapes composition was favoured by clayey and clay-calcareous soils. Besides, also higher altitude seems to favour Touriga Nacional grapes composition. Sauvignon Blanc seems to be a variety well adapted to the different parcel characteristics. In order to go forward in the valuation of these varieties, the aroma properties of 6 monovarietal wines were studied based on an aroma network-approach, linking molecular data related to volatile composition and aroma data about the key odor active molecules. This approach allowed to identify different wine aroma properties and to infer about the consumer’s sensory perception. It was found that aroma properties differ from one wine variety to another: while Arinto and Sauvignon Blanc wine exhibited higher tree fruity, sweety and flowery aromas, related essentially with ester compounds and C13 norisoprenoids, the opposite was obtained for Bical wine, corroborating the aroma sensory perceptions of the trained panel. Sauvignon Blanc also exhibited higher toasted aromas (related with thiols, mainly with 2-methyl-3-furanthiol). Touriga Nacional red wine exhibited higher tree, tropical, and berry fruits notes (sensory described as sweet fruits), toasted and flowery aromas, while these are similar for the other red wines under study. Besides Portuguese Bairrada wines, this aroma network approach is a tool that can be used to explain the aroma properties of wines worldwide. The grape and wine data generated under the present PhD thesis, in the context of Bairrada Appellation, shows the unique character of each variety, and may be used by growers and wine producers as a support for decisionmaking based on objective criteria, increasing the sustainability in this sector. For instance, it is possible to take advantage of the natural resources and produce products with different characteristics obtained from the same variety, minimizing costs during the winemaking process.