Computational space

Zuse Introduced concept of computing spaces, or digital models of mechanics... 

It will be convenient to define a time and solution dependent transformation which proportions grid points on the derivative, will require that the grid will be tmiform in this so called computational space. In order to normalize, allow for optimization, and remove singularities we can write for the transformed coordinate C(x,t) (11) ... 

Here the constant C takes care of the relative importance of the second derivative influence. Instead of solving a front problem in the coordinates (x,t) (physical space) we perform the calculations in the computational space (C t). For one dimensional problems this adaptive grid transformation proved to be very successful. We can perform a transformation in a similar spirit for a two dimensional domain (x,y,t) -> A general sketch of this transformation... 

After formulating the problem in the physical (x,y)-space, the governing equations are transformed to the uniform computational space (C>n)> using relationships such as... 

For the example in Figure 2.14 it would be possible to perform the coordinate transformation analytically by introducing cylindrical coordinates. However, in general, geometries are too complex to be described by a simple analytical transformation. There are a variety of methods related to numerical curvilinear coordinate transformations relying on ideas of tensor calculus and differential geometry [94]. The fimdamental idea is to establish a numerical relationship between the physical space coordinates and the computational space curvilinear coordinates The local basis vectors of the curvilinear system are then given as... 

Figure 2.14 Example of a grid structure in physical space (left) and in computational space (right).

Computational space curvilinear coordinates Zeta potential... 

Table 3.4 summarizes similar data, but also includes the sodium dodecylbenzene sulfonate type and the resultant effect on the specific surface area and computed spacing factor of the bubbles [14],... 

Surface active agent Concentration (% by wt of cement) Air content (% by volume) Specific surface area of bubble (mm2 mm-2) Computed spacing factor (mm)... 

Table 3.5 The effect of water-cement ratio of cement pastes on the air content, specific surface area and computed spacing factor...

Progress in the application of sensor arrays to gas analysis will be made through increasingly independent data channels using novel combinations of sensors and operating modes. Computational approaches will be modified to suit specific types of sensor arrays and to make economical use of computational space for portable instrument applications. The primary challenges of the near future will be to solve the "needle-in-the-haystack" problem and to proceed to complex mixture analysis using a plurality of sensor responses. 

Table 3.4 The effect of various air-entraining agents at different concentrations on surface area and computed spacing factor of air bubbles in cement paste the specific... 

Leopold, P. E., Motal, M., Onuchic, J. N. Protein folding funnels kinetic pathways through compact computational space Proc. Natl. Acad. Set. USA 1992 89, 8721-8725. 

The development environment for expert systems includes both hardware and software. Initially, most of the expert systems were developed on microcomputers. However, as shown in Figure 2, there is a steady increase in the number of minicomputer or workstation based systems. This is due to a variety of factors, the most common of which is that the PC systems run out of computer "space" before they can solve a complex problem due to the size of the code and other operating requirements. The IBM compatible PC-AT is still the most common development platform because it is a very widely distributed system and provides the broadest user base. However, the limit of 640K of random access memory is causing programmers to undertake ingenious solutions to fit their code into this space. 

In the case of an inhomogeneous dielectric, serving as a boundary or an intermediate layer, onesided difference operators have to be reformulated in order to circumvent possible instabilities. The key concept for these amendments, which lies on the efficient analysis of [20,23], presumes an explicit (2, 4) FDTD approach in the homogeneous areas of the computational space and... 

As an example, consider the original PML absorber in a 2-D computational space. Regarding the propagation of the TE case, the fourth-order electric field expression for Ex in the interior of the layer, becomes... 

FIGURE 8.1 (a) Transverse cut of the computational space with the PEC scatterer terminated by a spherical PML. (b) Normalized electric field Er component of the z-polarized electric dipole... 

FIGURE 25.6 Coordinate transformation for uneven terrain (a) two-dimensional terrain in x — z space (b) same as (a) but with contours of constant superimposed (c) same as (a) but with contours of constant z superimposed (d) two-dimensional terrain inx — C, computational space (the terrain is indicated by the shaded region). 

Lack of Robustness to High Dimensionality of Biological Data Highdimensional data require clustering algorithms to take more time and computational space to process. 

Locating droplets in a generalized coordinate structured code is straightforward since the physical coordinates can be transformed into a uniform computational space. This is not the case for unstructured grids. The approach used in this work... 

This equation refers to the strong orthogonality operator Eq. (75), where the projection space is spanned by the full computational space of MOs. In the case of An3 the analogous space involves the full occupied space (active + inactive MOs), therefore the summation index q in terms on the r.h.s. of Eq. (96) is replaced with J, where J belongs to the full occupied space. The An3 formula can be written as... 

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