Partial integration

The lower bound confidence limit is the probability that a parameter, x, is less than some value x . The upper bound confidence limit is the probability that a parameter, x, is greater than some value x . Figure 2.5-1 shows that confidence may be obtained from the discrete curve by simply adding the probabilities below or above x, for the lower or upper bound confidence respectively. If the curve is continuous it must be integrated above or below x. These results are normalized, by dividing the partial integral or partial sum by the full integral of the curve or complete sum. 

Confidence limits are partial integrations over a probability density function. There are two special cases failure with time and failure with demand. 

Confidence is calculated as the partial integral over the chi-squared distribution, i.e., the partial integral over equation 2.5-31 which is equation 2.5-32. where is the cumulative... 

To obtain the last equality one uses a partial integration so that the derivative by e is now taken on d y/de instead of on S. Consequently, the variational derivative gives the free energy variation as... 

There are already a few companies that have wholly or partially integrated Quality Management into their ESH functions the following examples illustrate the savings achieved by several of these organizations. 

J. M. Appeil et ai., Partial Integral Operators and Integro-Differential Equations (2000)... 

The relationships presented thus far for partial, integral and relative partial molar free energies are applicable in a similar manner to entropy, enthalpy and also volume. 

The integral can be related to the error function by partial integration, which gives... 

We will be interested not only in the total absorption cross section [Eqs. (4.1) and (4.2)], which gives us a measure of the total probability that the molecule will absorb light and dissociate, but also in the probability that different product quantum states will be formed. This probability is given by a partial cross section cj/( ). From Eq. (4.1) we see that this partial integral cross section may be written as... 

The density matrices are by definition partial integrals of the corresponding trial wave functions FcQrs ( ti Xi, X2,..., xjv) obtained for the given composition C and nuclear configuration Q so that they have the specified total spin S and spatial symmetry T ... 

For convergent integrals Jqp we thus have to analyze the integral JqQ. Partial integration with respect to 0 yields... 

Turning next to the magnetic moment (A. 11) and its integral, partial integration with respect to p yields... 

Combining expressions (B.26) with Eqs. (B.15) and (B.18)-(B.20), partial integration with respect to 0 finally results in... 

Exercise. Derive (7.18) without using the eigenfunction expansion. [Hint Apply successive partial integrations to (III.3.4) and use (IV.3.7).]... 

Rather than solving the master equation and subsequently taking the Laplace transform, it is often easier to make use of the equation for pB>IB(A) itself. Multiply (VI.1.2) with e-A, integrate over t from 0 to oo, and use partial integration in the left-hand member,... 

To find the escape rate we compute the net probability flow across the top and find, using partial integration,... 

For the most part, only the steady state condition will be of concern here, in which the case the partial integral of Eq. (8.1) becomes... 

If both ends of the one-dimensional system are still unaffected by the diffusion process, partial integration of Eqn. (4.7) yields... 

To obtain equations for them the equations (3) are multiplied by yp and integrated from — to < , giving, after a reduction by partial integration,... 

The (VS)2 term was transformed by partial integration, assuming S(r) to vanish at the boundaries of the integration volume. Equation (A 7.21) has the form of a functional Gaussian integral yielding... 

When the overlap between kinetic energy AT10 will be considerable. This may intuitively be seen from the nodal plane in the antibonding orbital yu, when one remembers that the kinetic energy, which can be written in different forms by a partial integration,... 

---