TDmat–mathematics diagnosis evaluation test for engineering sciences students

Since 1989, the Mathematics Education Project (PmatE–Projecto Matemática Ensino) has developed several strategies to improve the success of students in Mathematics. The most important of these are mathematical games for all grades above primary school. The online evaluation of Mathematics subjects is one of PmatE's goals. The implementation of an on-line diagnosis test–TDmat (Teste Diagnóstico de Matemática), mandatory for all students enrolling for the first time in Science and Engineering courses at the University of Aveiro, was the approach chosen to evaluate the mathematical competences of these students. TDmat is a computer test with automatic evaluation in which the analysis of results is done automatically. This paper describes how it is conceived as well as the main results and conclusions drawn from evaluations of the school years 2003/2004 and 2004/2005.     

Initial-value problems for linear partial difference equations

The method presented in [4] for the solution of linear difference equations in a single variable is extended to some equations in two variables. Every linear combination of a given functionf and of its partial differences can be obtained by the discrete convolution product off by a suitable functionl (which depends on the considered linear combination), and we want to solve in a convolutional form difference equations in the whole plane. However, the convolution of two functions may not be possible if their supports contain half straight lines with opposite directions. To avoid this, we take four sets of functions corresponding to the quadrants such thatl belong to every set, every set endowed with the convolution and with the usual addition is a ring, and there is an inverse ofl in each of the four rings. This is attained by taking, for each ring, a set of functions whose supports belong to suitable cones. After choosing such rings, a very natural initial-value first-order Cauchy Problem (in partial differences) is reduced to a convolutional form. This is done either by a direct method or by introducing the forward difference functions _i f(i=1,2) in a general way depending on the shape of the support off so that Laplace-like formulas with initial and final values) hold. Applications to difference equations in the whole plane and to partial differential problems are made.     

Stepwise disproportionation in polyelectrolyte complexes

Structural properties and the topology of polyelectrolyte complexes (PECs) formed in solution have been investigated under different conditions by Monte Carlo simulations using a coarse-grained model. The extension of individual polyions has been characterized by their radius of gyration, whereas the composition of the complexes has been investigated by their net charge and their internal topological structure by a novel analysis describing how the shorter polycations link to monomers of the longer polyanion. Conditions have been found at which the polyanion and a given number of polycations form distinguishable complexes differing in (i) the polyanion conformation and (ii) the fraction of polycations being in extended and collapsed states. Thus, at equilibrium, these PECs display a stepwise variation of the degree of intrachain disproportionation within the polyanion (also referred to as intrachain segregation), concomitant with the interchain disproportionation of the polycations, which is in agreement with previous theoretical predictions. The coexistence of the different polyelectrolyte complex structures appears, generally, at mixing ratios close to but different from charge equivalence and, as a consequence, broad polyelectrolyte size distributions are commonly obtained.     

Interplay of electrostatic and hydrophobic effects with binding of cationic gemini surfactants and a conjugated polyanion: experimental and molecular modeling studies

Understanding factors responsible for the fluorescence behavior of conjugated polyelectrolytes and modulation of their behavior are important for their application as functional materials. The interaction between the anionic poly{1,4-phenylene-[9,9-bis(4-phenoxy-butylsulfonate)]fluorene-2,7-diyl}copolymer (PBS-PFP) and cationic gemini surfactants alpha,omega-(CmH2m+1N+(CH3)2)2(CH2)s(Br-)2 (m-s-m; m=12, s=2, 3, 5, 6, 10, and 12) has been studied experimentally in aqueous solution. These surfactants are chosen to see whether molecular recognition and self-assembly occurs between the oppositely charged conjugated polyelectrolyte and gemini surfactant when the spacer length on the surfactant is similar to the intercharge separation on the polymer. Without surfactants, PBS-PFP exists as aggregates. These are broken up upon addition of gemini surfactants. However, as anticipated, the behavior strongly depends upon spacer length (s). Fluorescence measurements show three surfactant concentration regimes: At low concentrations (<2x10(-6) M) quenching occurs and is most marked with the small spacer 12-2-12; at intermediate concentrations (approximately 2x10(-6)-10(-3) M), fluorescence intensity is constant, with a 12-carbon spacer 12-12-12 showing the strongest fluorescence; above the critical micelle concentration (CMC; approximately 10(-3) M) increases in emission intensity are seen in all cases and are largest with the intermediate spacers 12-5-12 and 12-6-12, where the spacer length most closely matches the distance between monomer units on the polymer. With longer spacer length surfactants, surface tension measurements for concentrations below the CMC reveal the presence of polymer-surfactant aggregates at the air-water interface, possibly reflecting increased hydrophobicity. Above the CMC, small-angle neutron scattering experiments for the 12-6-12 system show the presence of spherical aggregates, both for the pure surfactant and for polyelectrolyte/gemini mixtures. Molecular dynamics simulations help rationalize these observations and show that there is a very fine balance between electrostatic and hydrophobic interactions. With the shortest spacer 12-2-12, Coulombic interactions are dominant, while for the longest spacer 12-12-12 the driving force involves hydrophobic interactions. Qualitatively, with the intermediate 12-5-12 and 12-6-12 systems, the optimum balance is observed between Coulombic and hydrophobic interactions, explaining their strong fluorescence enhancement.