We construct a linear interval system Ax = b with a 4 × 4 interval matrix whose all proper interval coefficients (there are also some noninterval ones) are of the form [−ε, ε]. It is proved that for each ε > 0, the interval hull [x, x] and interval hull of the midpoint preconditioned system [x, x] satisfy x1 = 0.6 and x1 = 1.2, hence midpoint preconditioning produces a 100% overestimation of x1...

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 ...

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 constant K has the same meaning as above. 1CAS = Computer Algebra System. 2The interval [0, 2π] is for convenience only. Everything we say can easily be extended to an arbitrary interval [a, b]. 3One notices that the error bound for the midpoint rule is one half that of the trapezoidal rule; compare (2) with (3). For a pretty geometrical explanation of why one can expect the midpoint rule t...

Squeeze Film Dampers (SFD) are commonly used for passive vibration control of rotor-bearing systems. The Magnetorheological (MR) and Electrorheological (ER) fluids in SFDs give a varying damping characteristic to the bearing that can provide active control schemes for the rotor-bearing system. A common way to model an MR bearing is implementing the Bingham plastic model. Adding this model to th...