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This system of equations can be solved by hand, but let us look at the matrix solution: Sb = y. S-1Sb = S-1y Multiplying each side by S-1. Ib = S-1y Since S-1S = I. b = S-1y. Since any matrix times the identity is itself. This solution works for any number of equations in the system, and therefore will be of value when the equations get too numerous to solve by hand. In the example above, 5 3 2 -4 S= , Cofactor(S) = -3 5 , det(S) = 10 - 12 = -2. 4 2 2 -3 -1 -1 1.5 -1 -1 1.5 19 2 = Adj(S) = ,S = ,S y= . -4 5 2 -2.5 2 -2.5 14 3 So b1 = 2 and b2 = 3 is the solution. 8.1.5 Characteristic Polynomial, Eigenvalues, and Eigenvectors

Prerequisite Data Transparency A system allows end-to-end support for any of the scripts, character sets, and encoding discussed in 6. Ideally, someone anywhere on the planet can input his name, have it stored, managed, and otherwise pass through the system without any corruption or loss of information. Localizable A system can manage all aspects of data processing such as displaying error messages in the local language and presenting prices or calculations in local currency. Ideally, developers can ag individual pieces of content or whole classes for use in speci c locales using XML tags. Rule Server A company can de ne content rules to specify adaptations and usage for each market where that content is used. The software automatically executes these rules whenever changes are made, thus enforcing consistency and correctness across all sites.

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For instance, by referring to a color class P r = {1, 2, 3, 4} of four colors: For a state whose atomic propositions are W t1 , W t2 , W t3 , W t4 , the partition is composed of a singleton, { 1 , 2 , 3 , 4 } (ie every color permutation is allowed, here !4) For a state whose atomic proposition is W t2 , the partition is composed of a singleton, { 2 }, and another subset composed of the remaining colors, here { 1 , 3 , 4 } (ie the color permutations must leave 2 unchanged) For a state whose atomic propositions are W t2 , W t3 , Co4 , the partition is composed of four subsets: { 1 }, { 2 }, { 3 }, { 4 }.

where a is the element spacing of the array, Vs represents the modal voltage of the aperture source, and es x is the modal vector of the source field, which is assumed to be a real function of x. For instance, if the source is an array of waveguide apertures, then es x will be equal to the modal electric field vector [1] of a waveguide. The above array of apertures will produce an incident wave to the scatterers at an angle with the z-axis, where is given by = k0 a sin (4.2)

In this case, only the identity permutation is allowed (ie each color only represents itself) Note that the building of local partitions is analogous to splitting static subclasses Therefore, the elements of a local partition of colors are named local static subclasses As already used in well-formed Petri nets, such a color partition offers a highly compact representation of an isotropic subgroup according to a state, hence can be used to ef ciently code the orbit of a state We will take advantage of this when building the symbolic synchronized product 14332.

The states of the symbolic synchronized product Let us recall that in the dynamic symmetry method, the representation of an equivalence class of nodes is described according to a triplet (subgroup of permutations, subset of markings, state of the automaton), such that the subset of markings is the image of any marking of the subset, under the actions of the subgroup The permutations subgroup is implicitly represented by a local partition that is not necessarily the same as that of the state automaton The subset of markings is represented by a symbolic marking whose static subclasses (here local) belong to the aforementioned local partition Let us recall that a symbolic marking built with respect to a partition in static subclasses of colors is in fact built by using a re ned anonymous partition whose elements are called dynamic subclasses.

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