Factorial notation

The manufacture of solventicss powder tn German factories Notation Manufacture Incorporating Storage Rolling... 

In general, for a sequence of N distinguishable objects, the number of different permutations IV can be expressed in factorial notation... 

The gamma function is a generalization of the factorial introduced in Section 1.4. There, toe notation n = X- 2-3-4-was employed, with n a positive integer (or zero). The gamma function in this case is chosen so that r(n) = (n -1) . However, a general definition due to Euler states that... 

A prime on Z denotes the configurational integral without the factorial coefficients. This notation is introduced to simplify expressions which appear later. 

THE MANUFACTURE OF SOLVENTLESS POWDER IN GERMAN FACTORIES [42J Notation... 

A "matrix design" is a suitable way of obtaining all the treatment combinations implicated in a 2 factorial design, but it is not a handy system to notate... 

Table 1. Basic Quantities in Analyses of CW Laser Scattering for Probability Density Function. In Eq. 1 within the table, F(J) is the photon count distribution obtained over a large number of consecutive short periods. For example, F(3) expresses the fraction of periods during which three photons are detected. The PDF, P(x), characterizes the statistical behavior of a fluctuating concentration. Eq. 1 describes the relationship between Fj and P(x) provided that the effects of dead time and detector imperfections such as multiple pulsing can be neglected. In order to simplify notation, the concentration is expressed in terms of the equivalent average number of counts per period, x. The normalized factorial moments and zero moments of the PDF can be shown to be equal by substitution of Eq.l into Eq.2. The relationship between central and zero moments is established by expansion of (x-a)m in Eq.(4). The trial PDF [Eq.(5)] is composed of a sum of k discrete concentration components of amplitude Ak at density xk. [The functions 5 (x-xk) are delta functions.]...

The first designs for mixture experiments were described by Scheffe [3] in the form of a grid or lattice of points uniformly distributed on the simplex. They are called q, i j simplex-lattice designs. The notation q, v implies a simplex lattice for q components used to construct a mixture polynomial of degree v. The term mixture polynomial is introduced to distinguish it from the polynomials applicable for mutually independent or process variables, which are described later in our discussion of factorial designs (section 8.4). In this way, we distinguish mixture polynomials from classical polynomials. 

One of the most useful symmetrical designs is based on the 4 factorial design (table 2.17), described (using the previous notation) as 4V/4 5 factors at 4 levels are screened in 4 = 16 experiments. We again emphasize that the numbers 0, 1,2, and 3 identify qualitative levels of each variable and have no quantitative significance whatsoever. 

This notation 2 for the half factorial design for 4 variables at 2 levels indicates the fractional nature of this design. 

The columns X, Xi and X2 can be seen to be identical for the factorial experiments. Following the same reasoning as in chapter 3, we conclude that the estimator bo for the constant term, obtained from the factorial points, is biased by any quadratic effects that exist. On the other hand the estimate b o obtained only from the centre point experiments is unbiased. Whatever the polynomial model, the values at the centre of the domain are direct measurements of Pq. So the difference between the estimates, bo - b o, (which we can write as U+22 using the same notation as in chaptw 3) is a measure of the curvature P, + P22- The standard deviation o o is s/v 8 (as it is the mean of 8 data of the factorial design) and that of b o is s/ /2. (being the mean of 2 centre points). We define a function t as ... 

This notation indicates that the design has four factors, each at two levels, but we perform only eight runs. The presence of the -1 superscript means the full factorial was divided by 2. If it had been divided into four parts, the exponent would be 4—2. 

Further reduction of the number of observations can be achieved by employing the fractional factorial design (FFD). The notation 2 is used to denote a 2 fraction of a 2 fractional design. See Table 8 for the one-half fraction of the 2 FD of the weight-watch experiment. 

The notation n, called n factorial, is mathematical shorthand for the result of multiplying n by all the numbers between it and zero. 

It will be assumed that a 2 -factorial experiment has been designed with rij full replicates. Furthermore, it will be assumed that all the factors have been coded so that —1 and 1 represent the upper and lower levels in the experiment. The same notation as presented in Chap. 4 will be used. Thus, instead of calculating inverses and transposes, the following simplifications work for a 2 -factorial experiment ... 

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