Sparse

We use Equation (2) primarily with five parameters, or with four parameters, excluding C. When data were sparse or of poor precision, a linear two-parameter fit (C = = 0) was... 

Volumetric estimates are required at all stages of the field life cycle. In many instances a first estimate of how big an accumulation could be is requested. If only a back of the envelope estimate is needed or if the data available is very sparse a quick look estimation can be made using field wide averages. 

When an oil or gas field has just been discovered, the quality of the information available about the well stream may be sparse, and the amount of detail put into the process design should reflect this. However, early models of the process along with broad cost estimates are needed to progress, and both design detail and cost ranges narrow as projects develop through the feasibility study and field development planning phases (see Section 12.0 for a description of project phases). 

The conceptually simplest approach to solve for the -matrix elements is to require the wavefimction to have the fonn of equation (B3.4.4). supplemented by a bound function which vanishes in the asymptote [32, 33, 34 and 35] This approach is analogous to the fiill configuration-mteraction (Cl) expansion in electronic structure calculations, except that now one is expanding the nuclear wavefimction. While successfiti for intennediate size problems, the resulting matrices are not very sparse because of the use of multiple coordinate systems, so that this type of method is prohibitively expensive for diatom-diatom reactions at high energies. 

An alternative to split operator methods is to use iterative approaches. In these metiiods, one notes that the wavefiinction is fomially "tt(0) = exp(-i/7oi " ), and the action of the exponential operator is obtained by repetitive application of //on a function (i.e. on the computer, by repetitive applications of the sparse matrix... 

For a very large number of variables, the question of storing the approximate Hessian or inverse Hessian F becomes important. Wavefunction optimization problems can have a very large number of variables, a million or more. Geometry optimization at the force field level can also have thousands of degrees of freedom. In these cases, the initial inverse Hessian is always taken to be diagonal or sparse, and it is best to store the... 

In tlie previous section we showed tliat because tlie stmcture space is very sparse tliere have to be many sequences tliat map onto tlie countable number of basins in tlie stmcture space. The kinetics here shows tliat not all tlie sequences, even for highly designable stmctures, are kinetically competent. Consequently, the biological requirements of stability and speed of folding severely restrict tlie number of evolved sequences for a given fold. This very important result is schematically shown in figure C2.5.4. 

This left 20 X 20 X 8 = 3200 classes, with some classes being very sparsely populated. For such classes, the error term is unacceptably large,... 

The LIN method (described below) was constructed on the premise of filtering out the high-frequency motion by NM analysis and using a large-timestep implicit method to resolve the remaining motion components. This technique turned out to work when properly implemented for up to moderate timesteps (e.g., 15 Is) [73] (each timestep interval is associated with a new linearization model). However, the CPU gain for biomolecules is modest even when substantial work is expanded on sparse matrix techniques, adaptive timestep selection, and fast minimization [73]. Still, LIN can be considered a true long-timestep method. 

S. Ranka, Analyzing images containing multiple sparse patterns with neural networks, in Proceedings of IJCAI-91 1991. 

They then compared measured and predicted fluxes for diffusion experiments in the mixture He-N. The tests covered a range of pressures and a variety of compositions at the pellet faces but, like the model itself, they were confined to binary mixtures and isobaric conditions. Feng and Stewart [49] compared their models with isobaric flux measurements in binary mixtures and with some non-isobaric measurements in mixtures of helium and nitrogen, using data from a variety of sources. Unfortunately the information on experimental conditions provided in their paper is very sparse, so it is difficult to assess how broadly based are the conclusions they reached about the relative merits oi their different models. 

Note that in equation system (2.64) the coefficients matrix is symmetric, sparse (i.e. a significant number of its members are zero) and banded. The symmetry of the coefficients matrix in the global finite element equations is not guaranteed for all applications (in particular, in most fluid flow problems this matrix will not be symmetric). However, the finite element method always yields sparse and banded sets of equations. This property should be utilized to minimize computing costs in complex problems. 

As the number of elements in the mesh increases the sparse banded nature of the global set of equations becomes increasingly more apparent. However, as Equation (6,4) shows, unlike the one-dimensional examples given in Chapter 2, the bandwidth in the coefficient matrix in multi-dimensional problems is not constant and the main band may include zeros in its interior terms. It is of course desirable to minimize the bandwidth and, as far as possible, prevent the appearance of zeros inside the band. The order of node numbering during... 

Acid Deposition. Acid deposition, the deposition of acids from the atmosphere to the surface of the earth, can be dry or wet. Dry deposition involves acid gases or their precursors or acid particles coming in contact with the earth s surface and thence being retained. The principal species associated with dry acid deposition are S02(g), acid sulfate particles, ie, H2SO4 and NH HSO, and HN02(g). Measurements of dry deposition are quite sparse, however, and usually only speciated as total and total NO3. In general, dry acid deposition is estimated to be a small fraction of the total... 

Any one of the five basic processes may be responsible for limiting the extraction rate. The rate of transfer of solvent from the bulk solution to the soHd surface and the rate into the soHd are usually rapid and are not rate-limiting steps, and the dissolution is usually so rapid that it has only a small effect on the overall rate. However, knowledge of dissolution rates is sparse and the mechanism may be different in each soHd (1). 

X-ray diffraction work (11,15) shows that there is an ionomer peak at 4°C which is absent in the acid precursor. This low, broad peak is not affected by annealing or ion type and persists up to 300°C. Since the 4°C peak corresponds to a spacing of about 2.5 nm, it is reasonable to propose a stmctural feature of this dimension in the ionomer. The concept of ionic clusters was initially suggested to explain the large effects on properties of relatively sparse ionic species (1). The exact size of the clusters has been the subject of much debate and has been discussed in a substantial body of Hterature (3,4,18—20). A theoretical treatment has shown that various models can give rise to supramoleculat stmctures containing ionic multiplets which ate about 10 nm in diameter (19). 

Fig. 1. Schematic of the cross section of a mammal s skin. The relative size and function of the parts depend on the species and breed of the animal. For goats, where the wool or hair is sparse because it is not needed for warmth, the skin is dense to provide protection for sheep protected primarily by heavy wool, the skin contains more oil (sebaceous) glands to lubricate the wool for catde, both the hair and the heavy hide stmcture protect the animal (3).

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