Spatial independence

Let us consider the genera class of systems described by a system of n nonlinear parabolic or hyperbolic partial differential equations. For simplicity vve assume that we have only one spatial independent variable, z. 

Distributed if time and spatial independent variables are required described by partial differential equations. 

As argued earlier in the text, 72 (t) serves as an appropriate spatially independent measure of the impact of the dynamical fluctuations on the kinetics of the reaction. The absence of secular terms in 73(1) [cf. Eq. (36)] enables us to treat it as a purely perturbative correction to Jo(t). In such a case, we can discern the impact of the dynamical fluctuations by analyzing... 

Dusek (1986a) characterizes network-formation models into the following categories spatially independent and spatially dependent models. Of the spatially independent models, there are statistical models (in which network structure is developed from various interacting monomer units) and kinetic models (in which each concentration of species is modelled by a kinetic differential equation). 

Overall it is considered that the spatially independent network models, whilst simpler, should strictly be used for lightly crosslinked, homogeneous networks, whereas the spahally dependent models, although computationally intensive and limited by pre-defined lattice structure, provide a better understanding of network heterogeneities in highly crosslinked systems. 

It is important to state that the derivation of (2.17) requires the medium to be locally homogeneous i.e., it is assumed that all constitutive parameters are spatially independent. The small-valued (At)3/24 term in (2.15) and (2.16) permits further simplifications. Since all the derivatives encountered on the right-hand side of (2.17) are multiplied by it, they can be amply discretized by the common second-order finite differences. Therefore, one avoids ambiguous mathematical complexities and at the same time manages to construct a significantly improved FDTD approach, presenting lower phase and propagation errors. 

Variations in OC values in soil samples collected along the N-S and W-E transects at the Florida site are shown in Figure 4. Note that for both the transects, OC in soils from the 0-15 cm depth were more variable than in samples collected from the 15-30 cm depth. Variograms ( 14 ) calculated using these data indicated that for the 0-15 cm depth, OC values in soil samples collected within a separation distance (i.e., lag) of 15 m would be spatially correlated. On the other hand, OC values for the 15-30 cm depth are spatially independent. OC data for both depth increments could be fitted to a normal frequency distribution the normality was confirmed by the Kolmogorov- Smirnov D-statistic ( 23 ). The coefficient of variation (CV) in OC data for both depths was less than 20%. 

Light-induced processes are described quite differently in molecular photochemistry and solid-state photophysics. In photochemistry one is used to an atomistic picture in which the arrangement of the atoms in the structure of a single molecule determines the electronic levels and thus the photochemical behavior. In contrast, the electronic levels of a solid are determined by the infinite periodicity of the atomic sequence in the crystal lattice. This leads to a basic concept according to which the solid can be treated as a dielectric continuum. Atomistic irregularities in the crystalline structure, such as lattice defects or impurities, are treated as perturbations of the spatially independent states in the energy bands. 

With further simplifications, such as spatial independence of the rate and the transport coefficients, in one-dimensional Cartesian coordinates (positional coordinate x), they read as Equations (2.2-4) and (2.2-5) ... 

Gaussian plume models These models use a time and spatially independent horizontal wind field, time-dependent point source, and no chemical reactions or loss mechanisms. A bell-shaped downwind distribution is assumed. The answer is a function of source strength, average wind velocity, and two diffusion parameters Easy to use Sanctioned by EPA for developing implementation plans for ambient air-quality standards Much experience with use Assumes wind field constant and uniform Limits use to 1 h and 10 km Not useful for reactive pollutants 114... 

The hydrodynamic properties, i.e., gas holdup, in-terfacial area, heat and mass transfer coefficients and dispersion coefficients are assumed to be spatially independent. 

In the remainder of this paper, I will review our understanding of the spatially independent, slow neutron spectrum. In II the general features of the problem will be discussed, independent of the detailed behavior of the actual cross section. In III the energy-transfer cross sections for various moderators will be considered in terms of explicit models, and neutron spectra calculated from these models will be presented. Comparison with... 

The spatial average of which over the volume V containing N particles gives the spatial independent stress components ... 

Note that the above expression for 0 also follows immediately from Eq. (7.242) for the case of a homogeneous, nonmultiplying medium with no direct external sources at lethargy u. It is of further interest to note that this relation for reduces to the usual coupling equation when q is spatially independent [cf. Eq. (4.93)]. For the present case... 

Additionally, because the spectra collected are spatially independent, a localized particle as small as a single pixel can be identified among hundreds or thousands of... 

Hypervisor The hypervisor provides interference freeness (bounded time and spatial independence) among partitions, can fail in an arbitrary failure mode when it is affected by a fault and it is a compliant item as defined... 

The hypervisor is considered to be a software compliant item (see Section 5.4) that guarantees temporal and spatial independence among partitions (IEC-61508-2 Annex F), ensuring interference freeness among safety and non safety partitions, and interference freeness among safety partitions. Partitioning and multicore allocation is used to support a reasonable balance between performance (e.g., supervision functional group is divided in multiple partition(s)) and temporal interference due to shared resources. 

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