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are constant on equivalenceclasses is rigorously reproduced on the reduced manifold with the reduced geometric structure and Hamiltonian The possibility of fulfilling these conditions is usually related to symmetries which leave a geometric structure and a Hamiltonian invariant For a more concrete construction, we start with a Poisson manifold P on which a Lie group G acts by canonical transformations; that is, the action leaves the Poisson bracket on P invariant We then decompose P into equivalence classes, where two elements are equivalent if they are mapped into each other by the action of some group element We assume that we can select a representative of each equivalence class in such a way that P r e d = P/G is a manifold, on which we can define a reduced Poisson structure.

SNR (dB)

where, for the last term, we used the fact that Tr(AE) = XTF. For compactness of notation, we shall drop the diag{ .} notation from the subscripts and keep the vectors only, so that the above will be rewritten more compactly as

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4. Select the field next to source and click the Browse icon. When the Content Path dialog box appears, click the folder icon. In the Browse for FLV file dialog box, navigate to the 8 folder and select sheStories.flv; click Open. Click OK to confirm your choice and close the Content Path dialog. Again, you need to match the document size to the size of the video. Unfortunately, Flash doesn t recognize the skin as part of the contents, so you need to adjust the height. 5. Choose Modify u Document. In the Document dialog box, choose Match: Contents. In the Height field, change the value to 276 and click OK. The playback component is now properly configured it s time to add the captioning. 6. From the Component panel, drag the FLVPlaybackCaptioning component into the document area, but off the main stage. 228

For defining a Poisson bracket on F r e d , consider any two Ared ,B r e d E F ( P , e d ) Construct A, B E F(P) from A r e d , B r e d by mapping p E P to the value of Ared, B r e d on the equivalence class of p The function C = {A, B } is constant on equivalence because the bracket is invariant under group actions, and A, B are constant on equivalence classes We thus obtain a well-defined Poisson bracket {Ared, Bred}red on P r e d , which inherits all the defining properties from the Poisson bracket on P This construction of a Poisson structure is known as the Poisson Reduction Theorem [Marsden & Ratiu, p 3491 If a Hamiltonian H on P is invariant under 6.

+ T,(k)*'R(IC)-'T,(k)]P(k, l ) ~ , ( ~ ) * T ~ ( k ) +lI]-' -

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it must be constant on equivalence classes so that it defines a function H r e d on P r e d The Hamiltonian system given by H on P then reduces to that given by H r e d on P r e d The invariance of the Hamiltonian is indispensable for establishing the smaller space P r e d as an appropriate setting for reversible dynamics If we start from a symplectic manifold P on which a Lie group G acts canonically, that is, the action leaves the symplectic form on P invariant, we could try the above construction for Poisson manifolds because there exists a Poisson structure on any symplectic manifold However, a degeneracy problem arises so that one needs to select a symplectic leaf.

and the information estimate from the other component decoder, referred to as the a priori information of the decoded bits The decoder operates iteratively In the first iteration, the first component decoder provides a soft output and the so-called extrinsic output based on the soft channnel outputs alone The terminology extrinsic implies that this information is not based on the received information directly related to the bit concerned, it is rather based on information, which is indirectly related to the bit due to the code-constraints introduced by the encoder This extrinsic output generated by the first decoder - which constitutes the first decoder s opinion as to the bit concerned- is used by the second component decoder as a priori information, and this information together with the channel outputs is used by the second component decoder, in order to generate its soft output and extrinsic information.

Let us next collect the knowledge from previous resource allocation models in the form of a generic model to perform the resource

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