Spatial variability

HyperChem allows the visualization of two-dimensional contour plots for a certain number of variables, fh esc contour plots show the values of a spatial variable (a property f(x,y,z) in normal th rce-dimensional Cartesian space ) on a plane that is parallel to the screen. To obtain these contour plots the user needs to specify ... 

The number of terms of a complete polynomial of any given degree will hence correspond to the number of nodes in a triangular element belonging to this family. An analogous tetrahedral family of finite elements that corresponds to complete polynomials in terms of three spatial variables can also be constructed for three-dimensional analysis. 

Figure 3 shows the substantial spatial variability in NFlj emissions, particularly in source regions. It is clear that maps at a lower resolution, which are often made (e.g. 20 km grids for the 150 km grids for Europewill artificially smooth... 

In contrast, because of the spatially variable (inhomogeneous) nature of material in a composite stiffener, the bending stiffness cannot be separated into a material factor times a geometric term as in Equation (7.6). Instead, the composite stiffener bending stiffness is... 

Here the integration J dr is over the coordinates of both electrons. Such integrals are therefore eight-dimensional (three spatial variables and one spin variable per electron). Integration over the spin variables is straightforward, but the spatial variables are far from easy a particular source of trouble arises from the electron repulsion term. 

In our discussion of the electron density in Chapter 5, I mentioned the density functions pi(xi) and p2(xi,X2). I have used the composite space-spin variable X to include both the spatial variables r and the spin variable s. These density functions have a probabilistic interpretation pi(xi)dridii gives the chance of finding an electron in the element dri d i of space and spin, whilst P2(X], X2) dt] d i dt2 di2 gives the chance of finding simultaneously electron 1 in dri dii and electron 2 in dr2di2- The two-electron density function gives information as to how the motion of any pair of electrons is correlated. For independent particles, these probabilities are independent and so we would expect... 

It is a first-order differential equation in time, but second-order in the spatial variables. Space and time do not enter on an equal footing, as required by the special theory of relativity. 

The above models consider only one spatial variable which is the bonding distance. It is clear that, for a molecule anything more complex than diatomic, many parameters are needed to define even approximately the potential energy surface. The enormous advances in computational chemistry during the last few years have allowed quantum mechanical calculations on fairly large size molecules. The first attempt to apply quantum mechanics on deformed polymer chains was made... 

The inherent temporal variability of climate thus obscures the search for long-term climatic signals. In addition, the search is plagued by the great spatial variability of climatic change. 

How well do GCMs simulate the spatial variability of climatic change Today s GCMs utilize data grids that partition the atmosphere into cells, each covering an area about the size of Colorado. A mean state of the atmosphere (temperature, humidity, cloud cover, for example) is computed for each cell. Consequently, any ou ut statistics (the prediction) has a lower spatial resolution (more genei ized, less detailed) than the real atmosphere is likely to manifest. 

Hamrud, M. (1983). Residence time and spatial variability for gases in the atmosphere. Tellus 35B, 295-303. 

Alternatively, we could organize the list by variability in which we would see that N2, O2, and the noble gas concentrations are extremely stable, with increasing variability for substances of low concentration and for chemically reactive substances. Both the temporal and spatial variability are influenced by the same factors source strength and its variability, sink mechanisms... 

Recently, the ocean-basin distribution of marine biomass and productivity has been estimated by satellite remote sensing. Ocean color at different wavelengths is determined and used to estimate near-surface phytoplankton chlorophyll concentration. Production is then estimated from chlorophyll using either in situ calibration relationships or from empirical functional algorithms (e.g., Platt and Sathyendranth, 1988 Field et al., 1998). Such studies reveal a tremendous amount of temporal and spatial variability in ocean biological production. 

Legates, D. R. and Willmott, C. J. (1990a). Mean seasonal and spatial variability in global surface air temperature, Theor. Appl. Climatol. 41,11-21. 

The above derivation assumes straight streamlines and a monotonic velocity profile that depends on only one spatial variable, r. These assumptions substantially ease the derivation but are not necessary. Anal5dical expressions for the residence time distributions have been derived for noncircular ducts,... 

Let us stress that the integro-interpolational method is a rather flexible and general tool in designing difference schemes relating to stationary and nonstationary problems with one or several spatial variables. 

In this chapter difference schemes for the simplest time-dependent equations are studied, namely, for the heat conduction equation with one or more spatial variables, the one-dimensional transfer equation and the equation of vibrations of a string. Two-layer and three-layer schemes are designed for the first, second and third boundary-value problems. Stability is investigated by different methods such as the method of separation of variables and the method of energy inequalities as well as by means of the maximum principle. Asymptotic stability of difference schemes is discovered for the heat conduction equation in ascertaining the viability of difference approximations. Finally, stability theory is being used, increasingly, to help us understand a variety of phenomena, so it seems worthwhile to discuss it in full details. 

SCHEMES FOR THE HEAT CONDUCTION EQUATION WITH SEVERAL SPATIAL VARIABLES... 

The explicit difference scheme. The schemes considered in Section 1 may be generalized to the case of the heat conduction equation with several spatial variables. 

Heat conduction equation with several spatial variables... 

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