Pivoting

Partial pivoting not only eliminates the problem of zero on the diagonal line, it also reduces the round-off error since the pivot element (i.e., the diagonal element) is the divisor in the elimination process. To demonstrate the pivoting procedure, we use an example of three linear equations. 

Putting this set of equations into the augmented matrix form, we have 

We note that the coefficient n is zero therefore, there is a need to carry out the pivoting procedure. The largest element of the first column is 4. Therefore, upon interchanging the first and the second rows, we will get 

multiplying the third row by 4 and subtracting the first row to get the new third row will yield 

Although the pivot element in the second row is 1 ( 0), it is not the largest element in that column (second column). Hence, we carry out pivoting again, and this process is done with rows underneath the pivot element, not with rows above it. This is because the rows above the pivot element have already gone through the elimination process. Using them will destroy the elimination completed so far. 

The use of a uniform scale in partial pivoting can also significantly reduce round off eiTors (Gerald and Wheatley, 1984). 

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Following the similar structure - similar property principle", high-ranked structures in a similarity search are likely to have similar physicochemical and biological properties to those of the target structure. Accordingly, similarity searches play a pivotal role in database searches related to drug design. Some frequently used distance and similarity measures are illustrated in Section 8.2.1. 

Interactions between hydrogen-bond donor and acceptor groups in different molecules play a pivotal role in many chemical and biological problems. Hydrogen bonds can be studied with quantum chemical calculations and empirical methods. 

The development of a new drug is both a time-consuming and a cost-intensive process. It takes 12 to 15 years and costs up to 800 million to bring a new drug to the market. As measured by the market capitalization, the pharmaceutical companies play a pivotal role in the global economy. In February 2003 Pfizer was ranked at position five worldwide, with a market capitalization of 163 billion. Ranking third as far as the market capitalization in Europe is concerned was GlaxoSmithKline, with a current value of 101 billion. Novartis was number five in Europe with 82 billion. 

The tetroxide has been used to detect fingerprints and to stain fatty tissue for miscroscope slides. The metal is almost entirely used to produce very hard alloys, with other metals of the platinum group, for fountain pen tips, instrument pivots, phonograph needles, and electrical contacts. 

In the final chapter ofi this thesis, the work described in the preceding chapters is evaluated. Furthermore, two pivotal themes ofi this work, Lewis acid - Lewis base interactions in water and hydrophobic fiects, are reviewed. Finally, the prospects ofi Lewis-acid catalysis in aqueous solution are discussed. 

Up to this point all our attention has been directed toward aldoses carbohydrates hav ing an aldehyde function in their open chain form Aldoses are more common than ketoses and their role m biological processes has been more thoroughly studied Nev ertheless a large number of ketoses are known and several of them are pivotal inter mediates m carbohydrate biosynthesis and metabolism Examples of some ketoses include d nbulose l xylulose and d fructose... 

As a result of these considerations, the primary difference between a spectrophotometer and a light-scattering photometer is the fact that the photodetector is mounted on an arm which pivots at the sample so that intensity measurements can be made at various angles. 

Fig. 10. Cross section of an EMFC weighing cell. Item 1, pan 2, hanger 3, parallel guide 4, chassis 5, flexure 6, lever 7, flexible link 8, pivot point 9, position indicator 10, position sensor 11, permanent magnet 12, permanent magnet air gap 13, compensation cod. See text.

Buoyant Effect of Air. Weighing operations performed m vacuo are not affected by buoyancy forces. An object in air, however, is subject to a buoyancy force that is equal and opposite to the gravitational force on the mass of air the object displaces (10). If the equal arm balance of Figure 1 is in balance with a test weight of mass, in one pan, and material of mass, m, in the other, m = m if they have the same density. If the densities are different, then the buoyancy forces acting on each pan affect the result. Taking moments about the center pivot point gives... 

A variation on this type of filter is the double tipping pan filter, which is a semicontinuous type consisting of two rectangular pans fitted with a filter cloth and pivoted about a horizontal axis. Slurry is first fed onto one pan, which is turned over for cake discharge at the end of the cycle. The second pan is used for filtration while the first is being discharged. 

Another resonant frequency instmment is the TA Instmments dynamic mechanical analy2er (DMA). A bar-like specimen is clamped between two pivoted arms and sinusoidally oscillated at its resonant frequency with an ampHtude selected by the operator. An amount of energy equal to that dissipated by the specimen is added on each cycle to maintain a constant ampHtude. The flexural modulus, E is calculated from the resonant frequency, and the makeup energy represents a damping function, which can be related to the loss modulus, E". A newer version of this instmment, the TA Instmments 983 DMA, can also make measurements at fixed frequencies as weU as creep and stress—relaxation measurements. 

The traction thickener includes a traction mechanism where the movement of the rake is suppHed by a single long arm pivoted around the center column and driven by a troUey that moves on a peripheral rail around the basin. Such units have diameters of 60—130 m. 

Studies confirm that inhibin plays a role in regulating FSH secretion. However, the importance of this role in the human has not yet been determined. If inhibin-regulated FSH secretion is pivotal in foUicular recmitment and growth, then it may be possible to block ovulation by means of inhibin antagonists. 

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