Example 1: Verify through dimensional analysis the following equation: d=vt. In words, we can write this as distance=(velocity)(time). In the international System, velocity is measured in units of meters per second (m/s), distance in meters (m) and time in seconds (s). Following this...
There are two main uses of dimensional analysis: It is used to get the units of a variable in an equation or formula It helps check the correctness of an equation or formula The following example will illustrate this: Example 4: Take this dimensional analysis formula and verify its correctness...
aThe following example shows how this analysis can work. In a three-dimensional plot such as that of Figure 2, one can use the k-nearest neighbor algorithm to ask for every uncharacterized protein in the three-dimensional space which of its knearest neighbors is known to be chromosomal. 以下...
Parametric statistical problems involving both large amounts of data and models with many parameters raise issues that are explicitly or implicitly differe
The previous example was helpful for presenting the dimensional analysis validation, but the equation itself only contained scalar operation. We will show here how vector, spinor, and matrix operations can be combined in a compact syntax to perform advanced calculations. In this example we reproduce...
Reverting now to equation (2.85) for a more general analysis of turbulent flow, it is noted that -uiuj‾ is often modelled mathematically. For example, by means of Boussinesq’s eddy viscosity concept, one may write (2.99)-uiuj‾=νt∂Ui∂xj+∂Uj∂xi-23Kδij, where K is ...
Numerical example In this section, various nonlinear mathematical problems are used for verifying the intelligent sampling-based MLS-HDMR. For all examples, the purpose is to minimize the nonlinear mathematical functions. The efficiency of modeling is indicated by the number of expensive sample points...
The system size used so far for exploration is not large enough to establish the true periodicity of some of the observed phases due to edge or finite size effects. For example, the square bilayer obtained from optimization appears with either parallel (\(\epsilon=0.04,\,\chi=0.6\)) or cr...
However, it was found that CE-AIS-GM practically does not work well for high dimensional problems with dimensions higher than 100. In fact, there are reasons to expect that Gaussian based IS density models would not work properly for high dimensional reliability analysis. This perspective will ...
The above example describes the six parameter pressure drop problem being replaced by a three parameter problem. Equation 7 states that the control of two parameters is enough to fix the third. While this discussion provides enough detail to allow the reader to perform some dimensional analysis it...