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Additional info for Fundamentals of Solid-State Electronics [SOLUTIONS]

Example text

The function F is assumed to be zero in the complement of the angular domain Ω to the period. 1, respectively. 2. 58) is the outgoing energy part outside the angle domain Ω (including the complex angles up to the period), with respect to the complex current distribution U on the array. 59) with respect to the complex current distribution U on the array. 60) at a given t > 0 which describes a compromise demand of simultaneously decreasing both the approximation of a given amplitude pattern in the angular area Ω and the outgoing energy outside it (two-criterion optimization), with respect to the complex current distribution U on the array.

As a rule, such problems are reduced to homogeneous equations; the modulus of their maximal or minimal eigenvalue describes the extreme value of the objective function. g. as arguments of some complex functions). Each section contains a general formulation of the variational problem and its reduction to the Lagrange–Euler equation which is a necessary condition for the functional to be extremal. The equations are nonlinear; their solutions are, as a rule, nonunique. The simplest iterative procedures are proposed for their solution and the relaxation properties of these procedures are established.

All the functionals use the mean-square metric in the optimization criteria. This fact suggests the functional spaces in which the operators act in the problems considered. This is a class of complex functional Hilbert spaces in which, besides the usual properties of the inner product, certain additional conditions hold. We call such spaces spaces of L 2 -type. 1) for any u 1 2 H , u 2 2 H , jv j Ä C < 1. 2) is assumed to be valid for this space. 2) becomes an equality if and only if arg (u 1 ) arg (u 2 ) D c o n s t almost everywhere in support of the functions from H.

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