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Back-Propagation Algorithm An algorithm known as back propagation may be applied to nd weighting factors in a multilayer arti cial neural network. This is mathematical back propagation which has nothing to do with the back propagation of electrical solitons. Our focus will be on two layers. If one can design weighting factors for two layers, one should be able to design for any number of layers. Back propagation is an example of mathematical optimization that applies to a variety of design problems. It is going to be derived from basic considerations in the hope that the algorithm will become understood. It is an iterative algorithm. To simplify the notation, the index of iteration is not going to be shown, although it will be necessary when writing a computer program to run the algorithm. We begin with step n=1. Error is de ned to be the difference between the dk desired and the output yk of neuron k; the error equation is e k dk yk The equation for yk is similar to those above for the perceptron except that the inputs are de ned here to be yi and not xi, because having the same symbol with a different subscript is a convenient notation for multilayers. Note that the subscript i does double duty. It identi es input signals and it also serves as an index to count the inputs. yk F vk with vk

E2.2 Given the circui ts in Fig. E2.2, find Ca) R and Vs in the circuit in Fig. E2.2a, and C find b) I and R in the circuit in Fig. E2.2b.

No matter how many segments it contains, the entire multiline is one object. Many editing commands simply do not work with multilines. Table 16.1 shows the editing commands and whether they work with multilines. TABLE 16.1

At this point, you likely have two questions: How do you use the Displace filter, and why in the world would you possibly want to The hows of the Displace filter are covered in the following section.

(6.43)

L2;2Ca)

1. Set up the conditions that you want to apply for the macro. The macro does not record all drawing settings. You can include settings in the macro by using the SETVAR command. For more information, see 12. 2. Choose Manage tab Action Recorder panel Record. 3. Carry out the actions you want to include in the macro. A red circle follows the cursor to remind you that you are recording. 4. Choose Manage tab Action Recorder panel Stop. 5. The Action Macro dialog box opens. There you do the following:

Command:LlNEAR Specify first extension line origin .... >:END of (piok 1) Specify second extension line origin:CEN of (pick 2) Specify dimension line location or [ .... Rotated]: (pick 3) Command:.J Specify first extension line 4 origin or ... >:CEN of (piok 4) Specify second extension line origin:CEN of (pick 5) Specify dimension line location or [Mtext/Text/Angle/Horizontal/Verticall Rotated]: (pick 6)

Since we are dealing with a single server, E[N,] can at most be equal to 1 and must at least equal 0. In fact, E[N8] not only indicates the average number of jobs in the server but also the average time the server is busy. Therefore, E[N,] is also called the utilisation or traJJic intensity and denoted as p = XE[S] = XE[S] (where the last equality holds when there are no losses). Returning to our overall queueing system, the average number of jobs in the queueing system must be equal to the sum of the average numbers of jobs in the components of the queueing system, i.e., E[N] = E[N,] + E[N,]. Applying the earlier derived versions of Little s law, we obtain E[N] = E[N,] + E[N,] = XE[W] + XE[S] = X(E[W] + E[S]) = XE[R], (24

where p denotes the eigenvalue spread of R,, p = Xmax/Amin. optimal slowest modes; one negative and one positive.