Equalization of the populations

Saturation If the Rf field is applied continuously, or if the pulse repetition rate is too high, then a partial or complete equalization of the populations of the energy levels of an ensemble of nuclei can occur and a state of saturation is reached. 

Depolarization. The equalization of the populations of two or more spin states. 

The method used to excite the nuclei and achieve resonance must clearly be capable of covering all of the Larmor frequencies in the sample. This is achieved in the Fourier transform (FT) method by simultaneously exciting all the Larmor frequencies by application of a pulse (short burst) of rf signal (Bj) at or near all vqS, which results in the equalization of the populations of the nuclear spin energy levels. Equilibrium spin populations are reestablished in a free-induction-decay (FID) process following the rf pulse. The vector diagram in Fig. 20.5 can be used to visualize the effect of the rf pulse (61) on the nuclear spins and their subsequent FID to equilibrium. 

Again, when the saturation is sufficiently strong ( sat), there occurs equalization of the populations of the two levels. 

When a high-power decoupling field is applied to one of the two nuclei, it results in an equalization of the populations of levels 1 and 3, as well as of levels 2 and 4. What this means, in effect, is that half the excess d) of nuclei in levels 3 and 4 are promoted to levels 1 and 2. The populations of the levels at equilibrium and immediately after application of the decoupling field are shown in Table 3.3. 

There will be incidences when the foregoing assumptions for a two-tailed test will not be true. Perhaps some physical situation prevents p from ever being less than the hypothesized value it can only be equal or greater. No results would ever fall below the low end of the confidence interval only the upper end of the distribution is operative. Now random samples will exceed the upper bound only 2.5% of the time, not the 5% specified in two-tail testing. Thus, where the possible values are restricted, what was supposed to be a hypothesis test at the 95% confidence level is actually being performed at a 97.5% confidence level. Stated in another way, 95% of the population data lie within the interval below p + 1.65cr and 5% lie above. Of course, the opposite situation might also occur and only the lower end of the distribution is operative. 

Defining the sample s variance with a denominator of n, as in the case of the population s variance leads to a biased estimation of O. The denominators of the variance equations 4.8 and 4.12 are commonly called the degrees of freedom for the population and the sample, respectively. In the case of a population, the degrees of freedom is always equal to the total number of members, n, in the population. For the sample s variance, however, substituting X for p, removes a degree of freedom from the calculation. That is, if there are n members in the sample, the value of the member can always be deduced from the remaining - 1 members andX For example, if we have a sample with five members, and we know that four of the members are 1, 2, 3, and 4, and that the mean is 3, then the fifth member of the sample must be... 

Throughout this discussion we have used the numerical fraction of molecules in a class as the weighting factor for that portion of the population. This restriction is not necessary some other weighting factor could be used equally well. As a matter of fact, one important type of average encountered in polymer chemistry is the case where the mass fraction of the ith component is used as the weighting factor. Defining the mass of material in the ith class as mj, we write... 

For all practical purposes, source testing can be considered as simple random sampling (2). The source may be considered to be composed of such a large population of samples that the populahon N is infinite. From this population, n units are selected in such a manner that each unit of the population has an equal chance of being chosen. For the sample, determine the sample mean, y ... 

For solution of the population balanee equation, many forms exist for the partiele disruption terms Ba and Da respeetively (Randolph and Larson, 1988 Petanate and Glatz, 1983) but a partieularly simple form, whieh requires no integration of a fragment distribution, is the two-body equal-volume breakage funetion. It is assumed that eaeh partiele breaks into two smaller pieees, eaeh of half the original volume from whieh it follows that... 

The schema population can be estimated using a simple mean-field-like argument, Let S represent a schema in a size-A population V t) at time t, and Z V,t) instances of the schema at time t. Let f s) be the fitness of the string s, fs be the average fitness of instances of S at time t, and / = Y2ifi be the average fitness of the population. Then the expected number of instances of S at time t + l, Z V, t -f 1), is equal to... 

For k = J and for degenerate eigenvalues the elements of f are taken equal to zero. Let us next consider the variation of the population-bond-order matrix, which, in the orthogonal case of eqn 1, is just ... 

Bias Systemic error as opposed to a sampling error. For example, selection bias may occur when each member of the population does not have an equal chance of being selected for the sample... 

There are three assumptions about sampling which are common to most of the statistical analysis techniques that are used in toxicology. These are that the sample is collected without bias, that each member of a sample is collected independently of the others and that members of a sample are collected with replacements. Precluding bias, both intentional and unintentional, means that at the time of selection of a sample to measure, each portion of the population from which that selection is to be made has an equal chance of being selected. Ways of precluding bias are discussed in detail in the section on experimental design. 

In devising a model for an analytical operation, we identify a target population to which we want our conclusions to apply This will differ from the parent population from which the samples are actually taken The difference may be reduced by random selection of individual portions (increments) for analysis so that each part of the population has an equal chance of selection Genuinely random sampling is difficult because bias, unconscious or deliberate, is readily introduced Untrained individuals often have difficulty in accepting that an apparently unsystematic sampling pattern must be followed to be valid ... 

The simulation of the population structure and dynamics of autotrophs and phagotrophs is another important interaction that can be modeled to test for effects of pollutant stress. A standard approach is the use of a flnite-population-difference model. The model assumes that the population change of a species in a specific period is equal to the species population multiplied by an intrinsic coefficient of rate of change. The rate coefficients are difficult to define without extensive data. The task is further complicated because a consistent feature of... 

Now consider a small sample (n = 9, say) drawn from an infinite population. The responses in this sample can be used to calculate the sample mean, y and the sample variance, s. It is highly improbable that the sample mean will equal exactly the population mean (yj = p), or that the sample variance will equal exactly the population variance (s = cr ). It is true that the sample mean will be approximately equal to the population mean, and that the sample variance will be approximately equal to the population variance. It is also true (as would be expected) that as the number of responses in the sample increases, the closer the sample mean approximates the population mean, and the closer the sample variance approximates the population variance. The sample mean, yy, is said to be an estimate of the population mean, p, and the sample variance, is said to be an estimate of the population variance,... 

Any absorption signal observed in a spectroscopic experiment must originate from excess of the population in the lower energy state, the so called Boltzmann excess, which is equal to where TVp and A are the populations in the lower (P) and... 

The definition of what constitutes a rare disease varies in different regions of the world. The prevalence figure accepted in the European Union is no more than five individuals per 10 000 of the EU population. In the United States, it is defined as a disease that affects less than 200 000 of the population. This size of population is approximately equal to a prevalence of 7.5 individuals per 10 000 of the United States population. Similarly, the prevalence figures accepted in Japan and Australia are no more than 4.2 and 1.1 individuals respectively per 10 000 of their corresponding populations. 

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