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By differentiating conjugation at the group identity, we obtain the corresponding adjoint action on the Lie algebra G (according to (B.l l), the infinitesimal action of g on G is then given by g I+ - [ g , g ] for all g E G), and, by keeping the natural pairing invariant, we find the coadjoint action of G on G*[Marsden & Ratiu, p. 3 1 1 and p. 3751. After identifying the natural action of the group of space transformations G on all components of our configuration space P = G*x V*, which has previously been studied in detail (but on a more intuitive level) to construct Poisson brackets,16 one can verify that the momentum map is given by a simple projection,



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The nal bone age assessment (BAA) is determined by three fuzzy bone ages from the phalangeal subsystem (BA1), carpal subsystem (BA2), and distal radius subsystem (BA3). These three fuzzy output membership functions were aggregated into the nal fuzzy bone age as shown in Figure 26.23. A nal bone age is then obtained through defuzzi cation, the conversion of fuzzy output from the integration procedure to a precise bone age. The center of gravity method is used. The completed BAA program with a three-region analysis and the corresponding fuzzy logic classi ers is applied to the digital hand atlas cases to obtain a computeraided bone age assessment.





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Specify axis endpoint of ellipse or [Arc/Center]: Right-click and choose Arc. Specify axis endpoint of elliptical arc or [Center]: Use the Endpoint object snap to pick the free endpoint of the line on the right. Specify other endpoint of axis: Use the Endpoint object snap to pick the free endpoint of the line on the left. Specify distance to other axis or [Rotation]: 3<90 (If you don t have Dynamic Input on and set to the default of relative coordinates, then type @ before the distance.)

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wavelength and decay time arising from each nucleus. The one-dimensional NMR spectrum of small molecules is structurally informative since resonances resulting from the experiment are well-resolved and arise from individual protons. However, large biomolecules such as proteins and nucleic acids yield complex spectra with extensive overlap between individual peaks which makes structural interpretation of these samples from one-dimensional NMR spectra alone impossible. Two-dimensional NMR (2-D NMR) overcomes this dif culty by extending measurements of these effects into a second dimension and, as will be described later, three- and four-dimensional NMR are also possible. These measurements can be used to detect interactions between magnetized protons which are near together in space and are of two general types; through-space or through-bond. Through-space interactions are called nuclear overhauser effects (NOEs) while through-bond interactions result in correlation spectroscopy. The two most important types of 2-D NMR spectroscopy are therefore referred to as NOESY (nuclear overhauser effect spectroscopy) and COSY (correlation spectroscopy). Both of these effects are due to interactions between magnetic dipoles in a pair of nuclei which are spatially near to each other and both types of experiment are used in the determination of structure in large biomolecules by NMR.

FIGURE 22-7

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The REVOLVE command allows you to select more than one object to revolve, something that you can t do with the REVSURF command. The prompts of the two commands ask for the same information but in a slightly different way. Figure 23-27 shows two examples of the REVOLVE command, one that creates a surface and the other that creates a solid. The arc is revolved around the line in each case, but when the line and the arc create a closed figure, the result is a solid.

Parabolic partial differential equation, 526 Parallelogram equality, 15 Parametric ampli er, 473 Parseval s equality, 216 Partial differential equation (PDE), 525 PASCAL, 38 Perfectly matched layer (PML), 540,544,553 Permeability, 535 Permittivity, 535 Permutation matrix, 205

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