Roots of number

Another scale of measurement of precision is standard error of mean (.M) which is the ratio of the standard deviation to the square root of number of measurements ( ). 

Powers and Roots When numbers expressed in exponential notation are raised to a power, the exponents are multiplied by the power. When the roots of numbers expressed in exponential notation are taken, the exponents are divided by the root. 

This expression is especi ly useful for extracting the roots of numbers. Thus, to determine (2.5 x 10 ), we write... 

The S/N of any light intensity measurement varies as tire square root of tire intensity (number of photons) produced by tire source during tire time of tire measurement. The intensities typical of xenon arc lamps are sufficient for measurements of reasonable S/N on time scales longer tlian about a microsecond. However, a cw lamp will... 

It remains to investigate the zeros of Cg t) arising from having divided out by. The position and number of these zeros depend only weakly on G, but depends markedly on the fomi that the time-dependent Hamiltonian H(x, () has. It can be shown that (again due to the smallness of ci,C2,...) these zeros are near the real axis. If the Hamiltonian can be represented by a small number of sinusoidal terms, then the number of fundamental roots will be small. However, in the t plane these will recur with a period characteristic of the periodicity of the Hamiltonian. These are relatively long periods compared to the recurrence period of the roots of the previous kind, which is characteristically shorter by a factor of... 

This equation is a quadratic and has two roots. For quantum mechanical reasons, we are interested only in the lower root. By inspection, x = 0 leads to a large number on the left of Eq. (1-10). Letting x = leads to a smaller number on the left of Eq. (1-10), but it is still greater than zero. Evidently, increasing a approaches a solution of Eq. (1-10), that is, a value of a for which both sides are equal. By systematically increasing a beyond 1, we will approach one of the roots of the secular matrix. Negative values of x cause the left side of Eq. (1-10) to increase without limit hence the root we are approaching must be the lower root. 

If two square matrices of the same size can be multiplied, then a square matrix can be multiplied into itself to obtain A, A, or A". A is the square root of A and the nth root of A". A number has only two square roots, but a matrix has infinitely many square roots. This will be demonstrated in the problems at the end of this chapter. 

The solubility parameter is not calculated directly. It is calculated as the square root of the cohesive energy density. There are a number of group additivity techniques for computing cohesive energy. None of these techniques is best for all polymers. 

The logarithm of a root of a number is equal to the logarithm of the number divided by the index of the root thus... 

The geometric mean of a set of N numbers is the A th root of the product of the numbers The root mean square (RMS) or quadratic mean of a set of numbers is defined by ... 

This result shows that the square root of the amount by which the ratio M /M exceeds unity equals the standard deviation of the distribution relative to the number average molecular weight. Thus if a distribution is characterized by M = 10,000 and a = 3000, then M /M = 1.09. Alternatively, if M / n then the standard deviation is 71% of the value of M. This shows that reporting the mean and standard deviation of a distribution or the values of and Mw/Mn gives equivalent information about the distribution. We shall see in a moment that the second alternative is more easily accomplished for samples of polymers. First, however, consider the following example in which we apply some of the equations of this section to some numerical data. 

The concentration [MB] constantly experiences tiny fluctuations, the duration of which can determine linewidths. It is also possible to adopt a traditional kinetic viewpoint and measure the time course of such spontaneous fluctuations directly by monitoring the time-varying concentration in an extremely small sample (6). Spontaneous fluctuations obey exactly the same kinetics of return to equiUbrium that describe relaxation of a macroscopic perturbation. Normally, fluctuations are so small they are ignored. The relative ampHtude of a fluctuation is inversely proportional to the square root of the number of AB entities being observed. Consequently, fluctuations are important when concentrations are small or, more usehiUy, when volumes are tiny. 

Flow cytometer cell counts are much more precise and more accurate than hemocytometer counts. Hemocytometer cell counts are subject both to distributional (13) and sampling (14—16) errors. The distribution of cells across the surface of a hemocytometer is sensitive to the technique used to charge the hemocytometer, and nonuniform cell distribution causes counting errors. In contrast, flow cytometer counts are free of distributional errors. Statistically, count precision improves as the square root of the number of cells counted increases. Flow cytometer counts usually involve 100 times as many cells per sample as hemocytometer counts. Therefore, flow cytometry sampling imprecision is one-tenth that of hemocytometry. 

Upper Bound for the Real Roots Any number that exceeds all the roots is called an upper bound to the real roots. If the coefficients of a polynomial equation are all of hke sign, there is no positive root. Such equations are excluded here since zero is the upper oound to the real roots. If the coefficient of the highest power of P x) = 0 is negative, replace the equation by —P x) = 0. 

Descartes Rule of Signs The number of positive real roots of a polynomial equation with real coefficients either is equal to the number V of its variations in sign or is less than i by a positive even integer. The number of negative roots of P(x) = 0 either is equal to the number of variations of sign of P - ) or is less than that number by a positive even integer. 

The existing data indicate that fcja is proportional to the square root of the solute-diffusion coefficient, and since the interfacial area a does not depend on Dl, it follows that /cl is proportional to Dl. An analysis of the design variables involved indicates that /cl should be proportional to Nsc when the Reynolds number is held constant. 

For the purposes of understanding this concept and formula, there s nothing mathematically significant about the square root of the flow, or the NPSHr to 1 power. These mathematical manipulations simply give us Nss values that are e. y understood and recognizable. For example, the health inspector might j. . a restaurant s cleanliness on a scale from 1 to 100. We might ask you to rate this Lc on a scale from 1 to 10. Those are easy numbers to deal with. How would yc- this book on a scale from 2,369 to 26,426,851 This doesn t make sense. Likewi , the mathematical manipulations in the Nss formula serve simply to convert weird v . into a scale from 1,000 to 20,000 that cover most impellers and pumps. Values at... 

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