Unique points

When we plot the sample concentrations in this way, we begin to see that each sample with a unique combination of component concentrations occupies a unique point in this concentration space. (Since this is the concentration space of a training set, it sometimes called the calibration space.) If we want to construct a training set that spans this concentration space, we can see that we must do it in the multivariate sense by including samples that, taken as a set, will occupy all the relevant portions of the concentration space. 

When we regard each of our spectra as a unique point in the n-dimensional absorbance space, we can say that the error in our data is isotropic. By this, we mean that the net effect of the errors in a given spectrum is to displace that spectrum some random distance in some random direction in the n-dimensional data space. As a result, when we find the eigenvectors for our data, each eigenvector will span its equivalent share of the error. But recall, we said that we must take degrees-of-ffeedom into account in order to understand what is meant by equivalent share. 

For the origin of the unit cell a geometrically unique point is selected, with priority given to an inversion center. 

It has also been shown [254] that a commercial petroleum sulfonate surfactant which consists of a diverse admixture of monomers does not exhibit behavior typically associated with micelle formation (i.e., a sharp inflection of solvent properties as the concentration of surfactant reaches CMC). These surfactants exhibit gradual change in solvent behavior with added surfactant. This gradual solubility enhancement indicates that micelle formation is a gradual process instead of a single event (i. e., CMC does not exist as a unique point, rather it is a continuous function of molecular properties). This type of surfactant can represent humic material in water, and may indicate that DHS form molecular aggregates in solution, which comprise an important third phase in the aqueous environment. This phase can affect an increase in the apparent solubility of very hydrophobic chemicals. 

We first examine the final state of the reaction, i.e. the chemical equilibrium composition. This is not of great relevance to oscillatory behaviour but is an important first check that the model is chemically reasonable . Equilibrium arises when all three rates of change become zero simultaneously. Equations (2.1)—(2.3) have a unique point satisfying this condition, as required chemically, given by... 

Replication Begins at an Origin and Usually Proceeds Bidirectionally Following the confirmation of a semiconservative mechanism of replication, a host of questions arose. Are the parent DNA strands completely unwound before each is replicated Does replication begin at random places or at a unique point After initiation at any point in the DNA, does replication proceed in one direction or both ... 

The determination of whether the replication loops originate at a unique point in the DNA required landmarks along the DNA molecule. These were provided... 

Butler [33] extended the treatment for the case where the intermediate H" is adsorbed on the electrode surface. In these theories, the electron transfer is considered to take place at a unique point in time and space. 

When data obtained from similar measurements on different gases and at different pressures are plotted, it is found that the straight lines can be extended to the same point (Fig. 4.12). The extension of a graph beyond the data is called extrapolation, and we say that, in this case, the straight lines extrapolate to the same point. For Charles s law, this unique point corresponds to zero volume and —273.15°C. Because a volume cannot be negative, this temperature must be the lowest possible temperature. It is the value corresponding to 0 on the Kelvin scale. It follows that if we use temperature, T, on the Kelvin scale, then we can write Charles s law as... 

DNA replication proceeds by the synthesis of one new strand on each of the parental strands. This mode of replication is called semiconservative, and it appears to be universal. DNA synthesis initiates from a primer at a unique point on a prokaryotic template such as the E. coli chromosome. From the initiation point, DNA synthesis proceeds bidirectionally on the circular bacterial chromosome. The bidirectional mode of synthesis is not followed by all chromosomes. For some chromosomes, usually small in size, replication is unidirectional. 

In goose grass (E. indica), two alleles of a-tubulin 1 (each is the result of a single unique point mutation) have been described, which confer either an intermediate or high level of tolerance to a number of antimicrotubule herbicides, for example, dinitroanilines and phosphoroamidates (Anthony and Hussey, 1999 Yamamoto et al, 1998 Yamamoto and Bird, 1999 Zeng and Baird, 1997). A positive transformation system may be developed using these herbicides. 

The significance of die value/= 0 is that we are not allowed to freely choose either T or P. In other words the system can only exist at a unique temperature and a unique pressure. This unique point is called the triple point, which is discussed in Section 3.2.2. in more detail. 

For instance, if the solids flow rate is specified at Gs = 50 kg/(m2s), choking will take place at Ug = 3.21 m/s for system FCC/air as indicated in the figure. Throughout the entire regime spectrum, only at this unique point (l/pl, K ) can both dense-phase fluidization and dilute-phase transport coexist. At velocities higher than Upt, only dilute transport can exist, shown as Mode FD in Fig. 4 at velocities lower than l/pt, only dense-phase fluidization can take place, shown as Mode PFC in Fig. 4. The transition point at l/pt identifies the unique Mode PFC/FD on the curve of Fig. 5 for the coexistence of both modes, the relative proportion of which depends on other external conditions such as the imposed pressure APimp as reported by Weinstein et al. (1983). 

Soit V le sous-schAma de X image rdciproque par u d une section de Y au-dessus de S passant par u(x). Done V est quasi-fini sur S et contient x comme unique point au-dessus de s. Comme S est henseiien, V = V JlV", avec V fini sur S et V" au-dessus de S-s, de sorte que V1 est spectre d un anneau local. Montrons que V —> S est surjectif. Pour cela, on peut remplacer S par le sous-schema ferme reduit sous-jacent A une composante irrdductible de S, puis supposer S normal, done geometriquement unibranche. Comme f est universellement ouvert en z, il existe une composante irreductible de X contenant z et equidimen-sionnelle sur S (de dimension relative n) au point z. Comme u est quasi-fini et Y de dimension relative n, ndeessairement domine Y. Toujours parce que u est quasi-fini, va Stre aussi equidimensionnelle sur Y au point... 

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