The Generation of Random Equilateral Polygons

Freely jointed random equilateral polygons serve as a common model for polymer rings, reflecting their statistical properties under theta conditions. To generate equilateral polygons, researchers employ many procedures that have been proved, or at least are believed, to be random with respect to the natural measure on the space of polygonal knots. As a result, the random selection of equilateral polygons, as well as the statistical robustness of this selection, is of particular interest. In this research, we study the key features of four popular methods: the Polygonal Folding, the Crankshaft Rotation, the Hedgehog, and the Triangle Methods. In particular, we compare the implementation and efficacy of these procedures, especially in regards to the population distribution of polygons in the space of polygonal knots, the distribution of edge vectors, the local curvature, and the local torsion. In addition, we give a rigorous proof that the Crankshaft Rotation Method is ergodic.     

Crystallographic studies and structural systematics of the C2c alkali metal metavanadates

Four compounds in the system (Li, Na)VO₃ were synthesized and their structures refined in the space group $\text{C2}\text{c}$. Site population analysis showed that their compositions are (Na_0.5Li_0.5)VO₃, (Na_0.62Li_0.38)VO₃, (Na_0.71Li_0.29)VO₃ and (Na_0.84Li_0.16)VO₃. All have the structure of LiVO₃ and α-NaVO₃ which are related to the $\text{C2}\text{c}$ silicate pyroxenes. Structural data of eight compounds in the system (K, Na, Li)VO₃ were compiled, and correlations were established by multiple regression analyses between the effective ionic radii r_M1 and r_M2 of the alkali metal ions and various structural parameters. The size of the M2 site and the relative displacement of the (VO₃)_∞ chains are found to depend primarily on r_M2 only. The size of the M1 site and the amount of chain rotation are affected by both r_M1 and r_M2. Changes in lattice parameters are related to the chain movements. The anomalous chain configuration of LiVO₃ is also discussed.     

Solving the capacitated location-routing problem by a GRASP complemented by a learning process and a path relinking

As shown in recent researches, the costs in distribution systems may be excessive if routes are ignored when locating depots. The location routing problem (LRP) overcomes this drawback by simultaneously tackling location and routing decisions. This paper presents a new metaheuristic to solve the LRP with capacitated routes and depots. A first phase executes a GRASP, based on an extended and randomized version of Clarke and Wright algorithm. This phase is implemented with a learning process on the choice of depots. In a second phase, new solutions are generated by a post-optimization using a path relinking. The method is evaluated on sets of randomly generated instances, and compared to other heuristics and a lower bound. Solutions are obtained in a reasonable amount of time for such a strategic problem. Furthermore, the algorithm is competitive with a metaheuristic published for the case of uncapacitated depots.     

Photoluminescence from Er-doped silicon rich oxide thin films

 Photoluminescence (PL) properties of Er-doped silicon rich oxide thin films deposited on Si substrate by co-evaporation of silicon monoxide and Er under different atmospheres are investigated. The samples exhibit luminescence peak at 1.54 μm which could be assigned to the recombination in intra-4f Er³⁺ transition. PL shows that this transition is highest when ammonia atmosphere is used during deposition followed by an annealing temperature at 850 °C in 95% N₂+5% H₂ gas (forming gas). In fact, we believe that the presence of the N atoms around Er ions increases the intensity of the 1.54 μm luminescence.     

Order conditions for canonical Runge-Kutta-Nyström methods

We are concerned with Runge-Kutta-Nyström methods for the integration of second order systems of the special formd ² y/dt ²=f(y). If the functionf is the gradient of a scalar field, then the system is Hamiltonian and it may be advantageous to integrate it by a so-called canonical Runge-Kutta-Nyström formula. We show that the equations that must be imposed on the coefficients of the method to ensure canonicity are simplifying assumptions that lower the number of independent order conditions. We count the number of order conditions, both for general and for canonical Runge-Kutta-Nyström formulae.This research has been supported by Junta de Castilla y León under project 1031-89 and by Dirección General de Investigación Científica y Técnica under project PB89-0351.     

Fractals related to Pascal's triangle

A precise definition of a fractalF _p ^{r 1} derived from Pascal's triangle modulop ^r (p prime) is given. The number of nonzero terms in the firstp ^s lines of Pascal's triangle modulop ^r is computed. From this result the Hausdorff dimension and Hausdorff measure ofF _p ^{r 1} are deduced. The nonself-similarty ofF _p ^{r 1},r2, is also discussed.     

Simultaneous titration of ternary alkaline-earth mixtures employing a potentiometric electronic tongue

A fast novel complexometric titration procedure is proposed, in which equivalence concentrations were determined using a potentiometric electronic tongue. The titration consists in a reduced number of fixed additions of titrant and the recording of the potentials of an array of ion-selective electrodes with cross-selectivity response. The generated data (number of sensors × number of additions) were used as input data to an artificial neural network response model previously trained with standards. The application of this procedure is demonstrated with automated EDTA titrations of Ca²⁺, Mg²⁺ and Sr²⁺ mixtures, up to 3.3 mM total ion concentration. In the studied case, good results predicting the three concentrations were obtained using five 1 mL fixed volume additions plus the readings of eight sensors. The unique additional step required was the pH buffering of the sample to pH 9.5. Good comparisons were observed between obtained and expected concentrations for the three cations with the external validation samples (n = 17, with RMSE values of 0.18, 0.28 and 0.32 mM for Ca²⁺, Mg²⁺ and Sr²⁺, respectively). The proposed procedure was applied to mineral waters and compared with reference methods.     

On the change of step size in multistep codes

In this paper the effect of changing step size on the local discretization error of BDF and Adams type methods is considered. According to Shampine and Bogacki the usual assumption for variable step size multistep methods of orderp, that the local discretization error changes by ^p+1 as the step size changes by a factor of , is incorrect and may lead to unreliability in step size selection algorithms. Here, by using the true expression of the local discretization error for variable step size BDF-, Adams- and FLC methods, new algorithms for step size control are proposed. It is shown that the new algorithms are more accurate and reliable than those employed in usual codes. to confirm the advantages of the new algorithms some numerical experiments based on a modified version of EPISODE are presented.