Smarandache Sequences: Explorations and Discoveries with a Computer Algebra System

We study Smarandache sequences of numbers, and related problems, via a Computer Algebra System. Solutions are discovered, and some conjectures presented.Comment: To appear in the 2003 December issue of the Smarandache Notions Journa

Smarandache Sequences: Explorations and Discoveries with a Computer Algebra System

We study Smarandache sequences of numbers, and related problems, via a Computer Algebra System. Solutions are discovered, and some conjectures presented

Symbolic Computation in the Calculus of Variations: Determination of Symmetries and Conservation Laws (in Portuguese)

English version of abstract: The dynamic optimization problems treated by the calculus of variations are usually solved with the help of the 2nd order Euler-Lagrange differential equations. These equations are, generally speaking, nonlinear, and very hard to solve. One way to address the problem is to obtain conservation laws of lower order than those of the corresponding Euler-Lagrange equations. While in Physics and Economics the question of existence of conservation laws is treated in a rather natural way, because the application itself suggest the conservation laws (e.g., conservation of energy, income/health law), from a strictly mathematical point of view, given a problem of the calculus of variations, it is not obvious how one might derive a conservation law or, for that matter, if it even has a conservation law. The question we address is thus to develop computational facilities, based on a systematic method, which permits to identify functionals that have conservation laws. The central result we use is the celebrated Noether's theorem. This theorem links conservation laws with the invariance properties of the problem (with symmetries), and provides an algorithm for finding conservation laws. Thus the problem is reduced to the one of finding the variational symmetries. We show how a Computer Algebra System can help to find the symmetries and the conservation laws in the calculus of variations. Several illustrative examples are given.Comment: 20 pages, in Portuguese. Partially presented at XXVII CNMAC (Brazilian Congress of Applied Mathematics and Computation), FAMAT/PUCRS, Porto Alegre, RS, Brasil, 13-16 September 2004. Supported by the program PRODEP III/5.3/2003 and the R&D unit CEOC of the University of Aveir

Release of volatile compounds from polymeric microcapsules mediated by photocatalytic nanoparticles

In this study we propose a suitable method for the solar-activated controlled release of volatile compounds from polymeric microcapsules bonded with photocatalytic nanoparticles. These reservoirs can find applications, for example, in the controlled release of insecticides, repellents, or fragrances, amongst other substances. The surfaces of the microcapsules have been functionalized with TiO2 nanoparticles.Upon ultraviolet irradiation, redox mechanisms are initiated on the semiconductor surface resulting in the dissociation of the polymer chains of the capsule wall and, finally, volatilization of the encapsulated compounds. The quantification of the output release has been performed by gas chromatography analysis coupled with mass spectroscopy.Strategic Project PEST-C/FIS/UI607/2011 and PTDC/CTMNAN/119979/2010 Projec