Dimensional continuation

For the strongly contracting phase volumes associated with chemical reactions, the three-dimensional continuous-... 

In an axisymmetric flow regime all of the field variables remain constant in the circumferential direction around an axis of symmetry. Therefore the governing flow equations in axisymmetric systems can be analytically integrated with respect to this direction to reduce the model to a two-dimensional form. In order to illustrate this procedure we consider the three-dimensional continuity equation for an incompressible fluid written in a cylindrical (r, 9, 2) coordinate system as... 

Diffraction of x-rays from many atoms with many electrons results in a two-dimensional continuous pattern of peaks and valleys. Diffraction from a hquid or an amorphous (noncrystalline) soHd results in a continuous pattern with few features, mainly broad peaks and valleys. 

The time evolution of the discrete-valued CA rule, F —> F, is thus converted into a two-dimensional continuous-valued discrete-time map, 3 xt,yt) —> (a y+i, /y+i). This continuous form clearly facilitates comparisons between the long-time behaviors of CA and their two-dimensional discrete mapping counter-... 

We recall from our earlier discussion of chaos in one-dimensional continuous systems (see section 4.1) that period-doubling is not the only mechanism by which chaos can be generated. Another frequently occurring route to chaos is intermittency. But while intermittency in low dimensional dynamical systems appears to be constrained to purely temporal behavior [pomeau80], CMLs exhibit a spatio-temporal intermittency in which laminar eddies are intermixed with turbulent regions in a complex pattern in space-time. 

Figure 9.12 shows the modified objective function for a one-dimensional continuous object parameter with first- and second-stage infeasibilities. 

Silica sol is a stable dispersion of fine particles, while gel has a three-dimensional continuous structure. Si02 content in sol range between 10-50%, while that in dry sdica gels is between 96.5 to 99.6%. Density of dry gels is 2.22 g/cm3 and sols 2.20 to 2.30 g/cm. Weight loss in sols at 105°C is between... 

One-Dimensional Continuous Lake Models (Advanced Topic)... 

Next in complexity are the more general n-dimensional continuous spaces. Such a space can be cut up into a finite set of pieces, each of which can be mapped onto... 

Thus, in order to remove illumination effects we first need to transform the input image to frequency space using a Fourier transform. The one-dimensional continuous Fourier transform is defined as (Bronstein et al. 2001)... 

Optimization of resolution as a function of solvent velocity, development distance, and temperature Possibility of using high-viscosity eluents and poorly wettable stationary phases Possibility of both quantitative evaluation and preparative applications Efficient separation of multicomponent samples Different development modes unidirectional, bi-dimensional, continuous on-line, and off-line Long migration distances on fine-particle layers with short development times No air interactions... 

It is permissible to assume that all steady flows in HCP are incompressible. By using the one-dimensional continuity equation for incompressible flow, the following holds ... 

The time-dependent decline of reaction rate (and potential) of a porous electrocatalyst could be modeled in analogy to gas phase deactivation (267-268. Figure 19 shows a partly poisoned pore of a gas diffusion electrode. If Wp is the surface concentration of the poison in the poisoned part of the pore (moles per unit area of the catalyst) and Cp(x) is the local poison concentration in the pore, the one-dimensional continuity equation for the poison yields... 

At 5 dynes/cm (Figure 3C), the micrographs show large areas of a two-dimensional continuous phase with discontinuous so-called uncovered areas. Micro-islands and individual molecules beyond the limit of resolution of the microscope may be present in the so-called uncovered area. At 15 dynes/cm (3D), further compression increases the ratio of the covered to the uncovered area, but the small holes persist. 

Equation (2.2-7), known as the one-dimensional continuity equation, expresses the fact that Ihe density can only change at a particular location if there is a change in the mass flux with position. Converesly. if the density does not change with time, the mass flux is Independent of position. 

Wassmuth et al. (1990) modelled this type of surface instabilities, denoted as Linde instability. The resulting hair needle-like flow pattern in the surface is shown in Fig. 3C.6. The condition of two dimensional continuity of flow is fulfilled in the surface. 

The stream function defined this way automatically satisfies the two-dimensional continuity equation (2-21). The original set of two partial differential equations may be reduced to a single ordinary differential equation by transforming the independent variables x and y to T) and the dependent variables from V(x, y) to /(r)), where... 

Method materialization One-dimensional Multi-dimensional (two-dimensional) Continuous... 

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