In this paper, we introduce two generalizations of midpoint subdivision and analyze the smoothness of the resulting subdivision surfaces at regular and extraordinary points. The smoothing operators used in midpoint and mid-edge subdivision connect the midpoints of adjacent faces or of adjacent edges, respectively. An arbitrary combination of these two operators and the refinement operator that ...

The leapfrog method is popular because of its good stability when solving partial differential equations with oscillatory solutions. “It has the disadvantage that the solution at odd time steps tends to drift farther and farther from the solution for even time steps, so it is common to stop the integration every twenty time steps or so and reinitialize with the first order forward Euler method ...

The equivalent, second equivalent and (simply) modi®ed equations for the implicit midpoint rule are shown to be asymptotically equivalent in the sense that an asymptotic analysis of these equations with the time step size as small parameter yields exactly the same results; for linear problems with constant coecients, they are also equivalent to the original ®nite di€erence scheme. Straightforw...

The electrokinetic remediation method can function as a primary or secondary technology and be applied in conjunction with other physical biological methods, such soil washing, phytoremediation, bioremediation. Environmental impacts arising from the process determined using life cycle assessment analysis (LCA) tools. This study compared conventional strategy two hybrid strategies: electrokineti...

Molecular dynamic simulation is a powerful method that monitors all variations in the atomic level in explicit solvent. By this method we can calculate many chemical and biochemical properties of large scale biological systems. In this work all-atom molecular dynamics simulation of polyalanine (PA) was investigated in the presence of 0.224, 0.448, 0.673, 0.897 and 1.122 M of guanidinium chlorid...

These conditions do not provide a measure of driving force as needed for a Boltzmann fit to obtain conductance parameters of midpoint and slope factor. Therefore, we have analyzed the original dataset using the asymptote for the relative change in current amplitude to estimate driving force and plot G-V relations. G/GMAX was plotted as a function of voltage, and these curves were fit with a Bol...

As shown in Fig. 1, LCI results with similar impact pathways (e.g. all elementary flows influencing stratospheric ozone concentrations) are grouped into impact categories at midpoint level, also called midpoint categories. A midpoint indicator characterizes the elementary flows and other environmental interventions that contribute to the same impact. The term 'midpoint' expresses the fact that ...