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Think about music for a moment. When a band is in a recording studio putting together a song, a sound engineer records each singer and instrument individually on separate sound tracks. In fact, vocals and instruments often are recorded at different times with different musicians in attendance. No one really knows what the song sounds like until all the pieces are blended together on a mixing board and played back as a single musical piece. All these separate tracks give the producer a tremendous amount of creative control. If she doesn t like the way a particular instrument sounds, she can modify it or even toss it out and recreate it without affecting all the other pieces. If she had to work with a single track of all vocals and instruments recorded at the same time, massaging the sound would be very dif cult. This is very similar to the concept of layers in Photoshop. Photoshop has used layers since version 3.0. They allow you to isolate various elements of an image so one element can be managed separately from other elements. Just like the sound engineer in a recording studio, you control all the separate elements of an image by stacking image content in layers. If you don t like the way all these pieces t together, you can modify some of them or throw them away without compromising the whole image. This ability creates a huge amount of exibility when editing an image.

The drawing used in the following exercise on extending objects in 3D, ab24-l.dwg, is in the Drawings folder on the DVD.

(18.9)

forecasting, and the integration of suppliers and customers more closely in the supply chain through e-business. Each of these business processes involves intensive information exchange and an aligned IT strategy is essential. At least at a global level, the implication of IT in business strategy may not always be so evident. In these cases, the driver behind global IT strategy should still be business requirements, but more in the sense of the generic need to improve value for money and cut risk. This can lead to the following three, more introverted, categories of IT strategy.

for the phasor of the electric field (see below). 7.3.4 Time-harmonic Maxwell's equations Assume that all time-varying quantities of the electromagnetic field are harmonic with a frequency w > 0,

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Corollary A . l (Inverse operator & inverse matrix) Let V and M be real or complex . ' linear spaces of dimension n and let f E C ( V ,W ) , represented by matrix Aff, be a bijection. Then the matrix Aff is nonsingular and f E C ( W ,V ) is represented by the inverse matrix M F

QR methods. It is generally accepted that the QR method for solving and updating least-squares solutions is among the most reliable procedures for finite word-length implementations. The origin of the method can be traced back to the works of Householder (1953, pp. 72-73), Golub (1965), and Businger and Golub (1965) on least-squares problems - see Probs. VII.12 and VII.32 for a description of the QR method as usually derived in works on computational linear algebra. In the context of adaptive filtering, simulation studies in Yang and Bohme (1992) suggest that the QR method of Sec. 35.2 is reliable numerically for X < 1 and can diverge for X = 1. This is because the mechanism for the propagation of a single numerical error is exponentially stable for X < 1. Further experimental evidence for the numerical stability of the QR method can be found in Ward, Hargrave, and McWhirter (1986). However, as mentioned in the concluding remarks of Part VII (Least-Squares Methods), such conclusions on the numerical stability of RLS-type algorithms from single-error propagation models should be interpreted with care. The extended QR method of Sec. 35.3 was developed by Yang and Bohme (1992), and it also appears in Sayed (1992) and Sayed and Kailath (1994b). Unfortunately, the algorithm can face numerical difficulties in finite word-length implementations since its pre- and post-arrays involve both and its inverse.

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