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where scalar values of An for which non-trivial solutions exist are referred to as the eigenvalues, and the corresponding solutions x # 0 are the n eigenvectors (en). For example, A and en for (i ) are computed as follows. Solving the following , y equation gives the eigenvalues of the matrix A .

ActiveX is an interface that exposes objects to the user, a programmer, or an application. AutoLISP supports ActiveX, giving you more information and flexibility in working with your drawings. You can also use ActiveX to work with objects in other Windows applications that support ActiveX. ActiveX is a programming interface that is used within a programming language that supports it. For example, you can also use ActiveX with Visual Basic for Applications (see the next chapter) and C++.

You can change the view size the size at which an image appears on screen so you can either see more of an image or concentrate on individual pixels. Each change in view size is expressed as a zoom ratio, which is the ratio between screen pixels and image pixels. Photoshop displays the zoom ratio as a percentage value in the title bar as well as in the magnification box. The 100-percent zoom ratio shows one image pixel for each screen pixel (and is therefore equivalent to the old 1:1 zoom ratio in Photoshop 3 and earlier). A 200 percent zoom ratio doubles the size of the image pixels on screen, and so on.

When you check the Google Mail account, you should see the message there. (Be aware that it could take some time for the message to show up.) If you can both send and receive mail using Postfix (or just send if your ISP restricts outgoing mail), Postfix is configured, and you can now move on to setting up your POP/IMAP server.

This is a startlingly small number; compare for example the density of water which is 103 kg m -3. If there is any more matter than this apparently tiny amount, it is enough to tip the balance beyond a flat Universe to a closed one with k > O. So only a very tiny density of matter is needed in order to provide enough gravitational attraction to halt and reverse the expansion of the Universe. However, let us write that another way, since kilograms and metres are rather small and inconvenient units for dealing with something as big as the Universe. Let's try measuring masses in solar masses and distances in megaparsecs. It becomes

Since the least squares solution t^ for t is the absolute minimum of the least squares criterion, it is a stationary point. Therefore, a necessary but not a sufficient condition for t^ to be the least squares solution is that at t = t^ the gradient vector of J ( t ) vanishes. The gradient vector of the weighted least squares criterion (5.173) is equal to (5.179) where ag(t)/atTis the N x K Jacobian matrix of g ( t ) with respect to t . This expression may also be written

5.1 INTRODUCTION In 4, the concepts estimator and estimate have been introduced. In this chapter, the properties and use of two specific estimators will be discussed. These are the maximum likelihood estimator and the least squares estimator. They have been chosen because, in the author s opinion, these are the most important estimators for practice. The maximum likelihood estimator is defined in Section 5.2. Properties described in Section 5.3 show that this estimator is not only practically feasible but also optimal in a number of respects. In Section 5.4, maximum likelihood estimation from normally distributed observations is discussed and its relation to least squares estimation is explained. Maximum likelihood estimation from Poisson distributed observations and from multinomially distributed observations are the subjects of Section 5.5 and Section 5.6, respectively. Maximum likelihood estimation from exponential family distributed observations is discussed in Section 5.7. Maximum likelihood estimation is based on the assumptions that the distribution of the observations is known and that the expectation model is correct. A test if the latter assumption has to be rejected is the likelihood ratio test discussed in Section 5.8. Most of the remainder of the chapter is devoted to least squares estimation. After a general introduction to least squares estimation in Section 5.9, theoretical results on nonlinear least squares estimation are presented in Section 5.10. This is least squares estimation of parameters of expectation models that are nonlinear in one or more of these parameters. Sections 5.11-5.19 are devoted to linear least squares estimation. This is least squares estimation of parameters of expectation models that are linear in all of these parameters.

18 Check and Test, Check and Test Again!

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the system. Additional Hazop studies should be performed as necessary to examine the effects of proposed and implemented changes.

where M is the matrix for one turn or repetition period, must remain finite for arbitrarily large n. Let V l ,V, be the two eigenvectors of M, corresponding to the eigenvalues A,, A,. Any initial condition can be expressed in terms , of v, v,: