The Binomial Theorem

In addition to composition factors, a sampling theoiy is available in sampling for size distribution. Quantity of sample needed to reach a specified error in determining size fraction retained on a designated screen is estimated by application of the binomial theorem (Gayle). 

Stresses are usually related to strains through an effective modulus. If the components of stress are nondimensionalized by a suitable scalar modulus c, then they are also of order c. Using (A.94), (A.lOl), and the binomial theorem in (A.39), the relation between the normalized spatial stress s = s/c and the normalized referential stress S = S/c becomes... 

Since there is no inverse of equation 10.136 in its present form, it is necessary to expand using the binomial theorem. Noting that, since 2y/(p/D)L is positive, e-2V<

[Pg.614]

This is the BINOMIAL THEOREM. Using a Taylor Expansion, we can find the total probability for items taken r at a time as ... 

It is clear from the definition (2) that the coefficients P (/i) are polynomials in ft. The first one or two can readily he calculated directly from the definiton. By the binomial theorem we have... 

Expanding the right-hand side using the binomial theorem gives l nCST =... 

Only the root with the negative sign is of any practical significance, the other solution yielding an exceedingly high temperature. Using the binomial theorem to expand the square-root term shows that... 

If the immediate neighbor atoms (coordination number) m is known such as in a crystalline solid, the question can be answered readily by invoking the binomial theorem. Since in the liquid state the coordination number m varies dynamically throughout the system, the answer to this question requires more thought. We know that for a given P (the atomic composition of B), there exists an average number of B s surrounding A and that the number n (P) can be written as ... 

Equation (10.31) applies to a fluid where compressibility may be disregarded. Using the preceding subscripts in Eq. (10.18) and rearranging so as to solve this equation for ps - p0, and expanding by the binomial theorem, the final result is ... 

Two level full factorial designs (also sometimes called saturated factorial designs), as presented in this section, take into account all linear terms, and all possible k way interactions. The number of different types of terms can be predicted by the binomial theorem [given by k /(k — m) m for mth-order interactions and k factors, e.g. there are six two factor (=m) interactions for four factors (= )]. Hence for a four factor, two level design, there will be 16 experiments, the response being described by an equation with a total of 16 terms, of which... 

The relative intensities of the 2nl + 1 lines can be determined from the number of ways each spin state may be formed. For the case of / = V2, the intensities of the n + 1 lines correspond to the coefficients of the binomial theorem, as indicated in Table 6.2. 

An alternative pulse sequence that provides the same multiplicities as INEPT but with intensity ratios that follow the binomial theorem is DEPT (distortionless enhancement by polarization transfer).The pulse sequence, depicted in Fig. 12.3a, can be used, like refocused INEPT, for sensitivity enhancement but is usually employed as an editing technique. The three evolution periods T are chosen to approximate 1/2/, but the length of the pulse labeled 0 can be varied. As we show in the following, CH has maximum intensity at 0 = 90° CH2 has zero... 

Applying the binomial theorem and neglecting terms in s of powers higher than the first, we have... 

The normalization condition is satisfied by the binomial theorem since PW = (p + We discuss properties of this distribution in Section 7.3.3. 

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