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...

A‎s we know, developing mathematical models and numerical procedures that would appropriately treat and solve systems of linear equations where some of the system's parameters are proposed as fuzzy numbers is very important in fuzzy set theory. For this reason, many researchers have used various numerical methods to solve fuzzy linear systems. In this paper, we define the concepts of midpoint a...

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 ...

It is a striking fact that differential calculus exists not only in analysis (based on the real numbers R), but also in algebraic geometry, where no limit processes are available. In algebraic geometry, one rather uses the idea of nilpotent elements in the “affine line” R; they act as infinitesimals. (Recall that an element x in a ring R is called nilpotent if xk = 0 for suitable non-negative i...

The crystal structure of the bovine Rieske iron-sulfur protein indicates a sulfur atom (S-1) of the iron-sulfur cluster and the sulfur atom (Sgamma) of a cysteine residue that coordinates one of the iron atoms form hydrogen bonds with the hydroxyl groups of Ser-163 and Tyr-165, respectively. We have altered the equivalent Ser-183 and Tyr-185 in the Saccharomyces cerevisiae Rieske iron-sulfur pr...