Denominator values

The performance of NLP solvers is strongly influenced by the point from which the solution process is started. Points such as the origin (0, 0,...) should be avoided because there may be a number of zero derivatives at that point (as well as problems with infinite values). In general, any point where a substantial number of zero derivatives are possible is undesirable, as is any point where tiny denominator values are possible. Finally, for models of physical processes, the user should avoid starting points that do not represent realistic operating conditions. Such points may cause the solver to move toward points that are stationary points but unacceptable configurations of the physical system. 

The genera] idea is that every numerator value is multiplied and every denominator value is divided. For example ... 

The denominator in equation (2.5) is the estimate of the mean number of particles in a sample and is directly comparable with the denominator value A of equation (2.2). In order to estimate the limiting variance by equation (2.5) the size analysis of the components is required along with a knowledge of particle shape and specific gravity. The use of equation (2.5) is illustrated in section 2.4. 

In this case, the denominator value should be taken from the rate law appropriate to this mechanism (Eq. 3.36). 

To calculate allele frequencies for the class, therefore. 200 will be used as the denominator value. To calculate the allele frequency, we must look at all those students who have a + in their genotype. There are 20 students who are +/+" they are homozygous for the insertion. Since these 20 students have two copies of the Alu insert on their chromosomes, they contribute 40 alleles to the overall frequency ... 

The denominator is included in equation (A 1.1.115) because it is impossible to ensure tliat the trial fiinction is nomialized for arbitrarily chosen coefficients c. In order to minimize the value of e for the trial fiinction... 

If a Pfaff differential expression DF = Xdx + Tdy+Zdz has the property that every arbitrary neighbourhood of a point P(x, y, z) contains points that are inaccessible along a path corresponding to a solution of the equation DF = 0, then an integrating denominator exists. Physically this means that there are two mutually exclusive possibilities either a) a hierarchy of non-intersecting surfaces (x,y, z) = C, each with a different value of the constant C, represents the solutions DF = 0, in which case a point on one surface is inaccessible... 

The denominator in Eq. (13) can be interpreted as an average value over the momentum distribution from the initial wavepacket, that is,... 

Usually, the denominator, if present in a similarity measure, is just a normalizet it is the numerator that is indicative of whether similarity or dissimilarity is being estimated, or both. The characteristics chosen for the description of the objects being compared are interchangeably called descriptors, properties, features, attributes, qualities, observations, measurements, calculations, etc. In the formiilations above, the terms matches and mismatches" refer to qualitative characteristics, e.g., binary ones (those which take one of two values 1 (present) or 0 (absent)), while the terms overlap and difference" refer to quantitative characteristics, e.g., those whose values can be arranged in order of magnitude along a one-dimensional axis. 

For each combination of atoms i.j, k, and I, c is defined by Eq. (29), where X , y,. and Zj are the coordinates of atom j in Cartesian space defined in such a way that atom i is at position (0, 0, 0), atomj lies on the positive side of the x-axis, and atom k lies on the xy-plaiic and has a positive y-coordinate. On the right-hand side of Eq. (29), the numerator represents the volume of a rectangular prism with edges % , y ., and Zi, while the denominator is proportional to the surface of the same solid. If X . y ., or 2 has a very small absolute value, the set of four atoms is deviating only slightly from an achiral situation. This is reflected in c, which would then take a small absolute value the value of c is conformation-dependent because it is a function of the 3D atomic coordinates. 

An alternative to autoscaling is range scaling, where the denominator in Equation (12.20] equals the range (the difference between the maximum and minimum values). Range scaling gives a set of new values between —0.5 and +0.5. 

Change Program QWIEN so that the second term on the right of the line below statement 10 is x instead of x/5. Solve for this new equation. Change the line below statement 10 so that the second term on the left is xj2. Repeat with j/3, xj, etc. Tabulate the values of a and the values of the denominator. Is a a sensitive function of the denominator in the second term of Program WIEN ... 

Just as e takes its maximum value when x is at a minimum, the right side of proportion (3-3) is a maximum when its exponent is a minimum. To minimize a fraction with a constant denominator, one minimizes the numerator... 

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... 

There is an obvious similarity between equation 5.15 and the standard deviation introduced in Chapter 4, except that the sum of squares term for Sr is determined relative toy instead of y, and the denominator is - 2 instead of - 1 - 2 indicates that the linear regression analysis has only - 2 degrees of freedom since two parameters, the slope and the intercept, are used to calculate the values ofy . 

Again, the factor of 2 in the denominator is the result of using two values to determine T . 

This expression describes the variation of the pressure-temperature coordinates of a first-order transition in terms of the changes in S and V which occur there. The Clapeyron equation cannot be applied to a second-order transition (subscript 2), because ASj and AVj are zero and their ratio is undefined for the second-order case. However, we may apply L Hopital s rule to both the numerator and denominator of the right-hand side of Eq. (4.47) to establish the limiting value of dp/dTj. In this procedure we may differentiate either with respect to p. 

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