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Both u-separation and d-separation satisfy the graphoid axioms. For u-separation this is evident from the illustration shown in Figure 4.2 [Pearl 1988] (symmetry is trivial and thus neglected). Therefore the graphoid axioms are sound w.r.t. inferences about u-separation in graphs. However, they are not complete, because they are much weaker than u-separation. To be more precise, the weak union axiom allows us only to extend the separating set Z by specific sets of nodes, namely those for which it is known that they are separated by Z from one of the sets X and Y (which are separated by Z), whereas u-separation is monotonic, that is, any superset of a separating set is also a separating set. That is, u-separation cannot be destroyed by enlarging a separating set by any set of nodes, while the graphoid axioms do not exclude this. A set of axioms for u-separation that has been shown to be sound and complete [Pearl and Paz 1987] are the graphoid axioms with the weak union axiom replaced by the following two axioms: strong union: transitivity: (X Y | Z) (X Y | Z W) (X Y | Z) A V (X Y Z) : ({A} Y | Z) (X {A} | Z)



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A digit x 3 in the BEC code corresponding to edges of a hexagon such that one of the edges belong only to this hexagon can be replaced by a5b, where a + b + 1 = x and a 1 and b 1 2 A sequence xy in the BEC code with x 2 and y 2 can be replaced by (x 1)4(y 1) 3 A sequence x1y with x 2 and y 2 in the BEC code can be replaced by (x 1)3(y 1) In each of the above cases, we must make sure that the addition of the hexagon does not produce holes This can be accomplished by checking for the presence of a hexagon in up to three adjacent positions, as shown in Figure 28b; if any of these hexagons is present, this addition is not valid.





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. This formula is again very intuitive: where = 1 / Ai Ao Ao = ao in an undirected tree, any attribute Ao u-separates all attributes in a subgraph reached through one of its neighbors from all attributes in a subgraph reached through any other of its neighbors. Consequently, if Ao is instantiated, all paths through Ao are blocked and thus no information should be passed from one neighbor to any other. Note that in an implementation we can neglect , because it is the same for all values ai dom(Ai ). Hence it can be implicitly incorporated into the normalization constant introduced above. The above formulae are all written in terms of single values of attributes. However, we obviously need to determine the probability of all values of the goal attribute and we have to evaluate, even in the above formulae, the messages for all values of the attributes that are message arguments. Therefore it is convenient to write the equations in vector form, with a vector for each attribute that has as many elements as the attribute has values. The factor potentials can then be represented as matrices. Rewriting the formulae is straightforward, though, and therefore we do not restate them here.

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Procedure GenerateKids that generates, from an HS P with j hexagons, its children in the search with j + 1 hexagons is outlined next 1 Addition of hexagons: Any attempt to add a hexagon in the steps below is preceded by a test that guarantees that no holes are created..

CSS is one of the most important technologies used with XUL. CSS is used to attach behavioral bindings to XML elements with XBL. Style sheets are also used in conjunction with the skinning of the Mozilla browser. Before XUL, the only control over the browser interface in a standards-based Web browser was in the document display area. XUL and CSS enable you to skin, or change, the whole look and feel of the browser creating a Netscape theme. The menus, windows, and everything viewable to a user are easily modified in a Netscape theme. The potential for this is tremendous. In the future, imagine that a visitor to your music-oriented Web site sees a browser that transforms

s3 were 48, 1.25 and 0.16 ps, respectively (see Table 1.4). Note that they are considerably different from 51.5, 7.09 and 1.12 ps, respectively, reported by Barthel et al. (Table 2.2 in Ref. [6a]).

(February 25, 2002), A-3.

The context diagram in Exhibit 6.7 shows the parties involved in the field of interest (inside the framework), and the external parties involved (outside the framework).

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