Reduction of the number

Large stepsizes result in a strong reduction of the number of force field evaluations per unit time (see left hand side of Fig. 4). This represents the major advantage of the adaptive schemes in comparison to structure conserving methods. On the right hand side of Fig. 4 we see the number of FFTs (i.e., matrix-vector multiplication) per unit time. As expected, we observe that the Chebyshev iteration requires about double as much FFTs than the Krylov techniques. This is due to the fact that only about half of the eigenstates of the Hamiltonian are essentially occupied during the process. This effect occurs even more drastically in cases with less states occupied. 

Also the arene-arene interactions, as encountered in Chapter 3, are partly due to hydrophobic effects, which can be ranked among enforced hydrophobic interactions. Simultaneous coordination of an aromatic oc amino acid ligand and the dienophile to the central copper(II) ion offers the possibility of a reduction of the number of water molecules involved in hydrophobic hydration, leading to a strengthening of the arene-arene interaction. Hence, hydrophobic effects can have a beneficial influence on the enantioselectivity of organic reactions. This effect is anticipated to extend well beyond the Diels-Alder reaction. 

Utilize periodic boundary conditions, which permit reduction of the number of nonbonded interactions at greater distances by involving only the "nearest neighbor" atoms from copies of the system which are in different but adjacent cells. 

In addition, detailed analysis of the chemical and physical properties of different oils used in the plant can, in some cases, allow consolidation or reduction of the number and types of lubricates required to maintain plant equipment. Elimination of unnecessary duplication can reduce required inventory levels and therefore maintenance costs. 

The current models for the mode of action of T-cell-specific monoclonal antibodies are based on the observation that the administration of CD3 antibodies leads to a significant reduction of the number of CD3-positive T cells. On the one hand, CD3+ T cells were... 

Hence use of the normalized distribution function allows a reduction of the number of independent parameters in the kernel (253). 

The rate of polymerization, as well as the mobility of macromolecules at the contact zone, is affected by the presence of the filler near the matrix. This results in a reduction of the number of possible conformations of molecules in the boundary layer developed, causing the formation of a denser material in the boundary layer than in the bulk. 

Equations (3.11) and (3.12) show that the friction factor of a rectangular micro-channel is determined by two dimensionless groups (1) the Reynolds number that is defined by channel depth, and (2) the channel aspect ratio. It is essential that the introduction of a hydraulic diameter as the characteristic length scale does not allow for the reduction of the number of dimensionless groups to one. We obtain... 

Fine chemicals are often manufactured in multistep conventional syntheses, which results in a high consumption of raw materials and, consequently, large amounts of by-products and wastes. On average, the consumption of raw materials in the bulk chemicals business is about 1 kg/kg of product. This figure in fine chemistry is much greater, and can reach up to 100 kg/kg for pharmaceuticals (Sheldon, 1994 Section 2.1). The high raw materials-to-product ratio in fine chemistry justifies extensive search for selective catalysts. Use of effective catalysts would result in a decrease of reactant consumption and waste production, and the simultaneous reduction of the number of steps in the synthesis. 

As mentioned earlier, a major cause of high costs in fine chemicals manufacturing is the complexity of the processes. Hence, the key to more economical processes is reduction of the number of unit operations by judicious process integration. This pertains to the successful integration of, for example, chemical and biocatalytic steps, or of reaction steps with (catalyst) separations. A recurring problem in the batch-wise production of fine chemicals is the (perceived) necessity for solvent switches from one reaction step to another or from the reaction to the product separation. Process simplification, e.g. by integration of reaction and separation steps into a single unit operation, will provide obvious economic and environmental benefits. Examples include catalytic distillation, and the use of (catalytic) membranes to facilitate separation of products from catalysts. 

Table 37.3 shows the complete table of eight indicator variables for 10 triply substituted tetracyclines [31 ] that have been tested for bacteriostatic activity (1/Z), which is defined here as the ratio of the number of colonies grown with a substituted and with the unsubstituted tetracycline. In this application we have three substitution positions, labelled U, V and W. The number of substituents at the three sites equals 2,3 and 3, respectively. Arbitrarily, we chose the compound with substituents H, NOj and NO2 at the sites U, V and W as the reference compound. This leads to a reduction of the number of indicator variables from eight to five, as shown in Table 37.4. The solution of the Free-Wilson model can be obtained directly by means of multiple regression ...

Sets of spectroscopic data (IR, MS, NMR, UV-Vis) or other data are often subjected to one of the multivariate methods discussed in this book. One of the issues in this type of calculations is the reduction of the number variables by selecting a set of variables to be included in the data analysis. The opinion is gaining support that a selection of variables prior to the data analysis improves the results. For instance, variables which are little or not correlated to the property to be modeled are disregarded. Another approach is to compress all variables in a few features, e.g. by a principal components analysis (see Section 31.1). This is called... 

Reduction of the number of UI episodes (daytime and nighttime) and frequency of nocturia... 

The total curvature energy of a spherical vesicle is given by 4tt(2/cc + k). As all experimental data on phospholipids indicate that kc is not small, one is inclined to conclude that the vesicles are thermodynamically unstable the reduction of the number of vesicles, e.g. by vesicle fusion or by Ostwald ripening, will reduce the overall curvature energy. However, such lines of thought overlook the possibility that k is sufficiently negative to allow the overall curvature free energy of vesicles to remain small. 

ENDOR with circularly polarized rf fields 4.6 CP-ENDOR circularly polarized rf field 2.6 Reduction of the number of observed ENDOR lines... 

Figure 4-47 below, demonstrates the relationships between the dimensions of the matrices involved. The process can be summarised as a reduction of the number of wavelengths from the original nl to nc, the number of... 

The benefits of C2-symmetry are due to the reduction of the number of possible catalyst-substrate isomers that may form, and the number of enantiospecific pathways (Helmchen, Pfaltz [9]). As they reasoned, there is no fundamental reason why a C2-symmetric ligand is superior qualitate qua to a Ci-symmetric ligand. Two different co-ordinating ligands (P-X as shown above, or P-N, discussed here) could lead to more control, as the attack may... 

Prior knowledge allows to include fixed relations between some of the four parameters (amplitude, phase, frequency position, peak width) describing a symmetrical well-shaped resonance. Signal ratios, chemical shift difierences, linewidth relations and zero-order phase relations can be included. The reduction of the number of unknown parameters leads to a reduced calculation time, better convergence behaviour and improved results. However, the assumptions made to include the prior knowledge must be validated for each experiment. Differences between the parameter values set by the prior knowledge and the actual parameters could lead to systematic errors. 

The standard Colpodium campylum toxicity test developed by Dive and colleagues [98,99] measures the inhibition of growth of this ciliate, cultured monoxenically on E. coli. The reduction of the number of generations is measured in increasing concentrations of the toxicant, and the effects are expressed as 24 hour IC50 values. This bioassay is relatively easy to learn, to carry out, and to interpret. 

The authors, William Russell and Rex Burch (a zoologist and a microbiologist, respectively) recommended the reduction of the number of animals used in experiments to the minimum number required to obtain statistically relevant data, the refinement of procedures to minimize pain and distress in experimental animals and provide for their well-being based on their behavioral needs, and the replacement of whole, living animals with in vitro models like tissue and cell cultures when possible. 

The total vibrational energy is a sum of energies of 3N-6 distinct harmonic oscillators. Indeed, 3N-6 is the final number of coordinates in Equation 8. Namely, the number of 3N+3 coordinates of the initial equation has been reduced by three through the elimination of internal rotations. Furthermore, the equation of nuclear motion (mainly its potential) has to be invariant under rotations and translations of a molecule as a whole (which is equivalent to the momentum and angular momentum preservation laws). The latter requirement leads to a further reduction of the number of coordinates by six (five in the case of linear molecules for which there are only two possible rotations). 

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