By Hartman, E.F.; Sandia National Laboratories.; United States. Dept. of Energy.; United States. Dept. of Energy. Office of Scientific and Technical Information
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76) mD1 where ω D (θ , ') is the aggregated angular coordinate of a point on the removed sphere; we denote the radius-vector of this point by ω. 73). 77) exp i k ω E 0 dEm 2π “ Á Á Á E 0 RE m d RE m . 79c) where ξ D sin θ cos ', η D sin θ sin '. The optimization problems formulated for the system considered, are different depending on the purpose for which the system is used. 2 can be formulated. 2 can be formulated. Below we formulate only three of these problems. Some other formulations will be considered in the next chapters where the mathematical properties of the problems will be investigated and algorithms for their solution will be described.
3 Phase optimization problems in waveguides and open resonators centrated on the mirror; part of it bypasses the mirror area. In order that such a wave can exist without damping, it must be fed by a source. As before, we call these waves eigenwaves or modes. The loss in the modes depends on the shape of the mirrors (more exactly, on the phase correction provided by them). The mode with the smallest loss is called the main mode; as a rule, its ﬁeld has the simplest structure in cross section. The optimization problems formulated for the systems considered consist in ﬁnding the shape of the mirrors (their phase corrections) such that the main mode has certain optimal properties.
20), we have AA f D AA [F exp(i arg f )] . 20) if the kernel of A is empty, that is, if there are no functions u 2 H1 such that Au D 0. 9). 20) both for analytical investigation and numerical solution. Remember, that the Lagrange–Euler equation is obtained from the necessary condition of a minimum of the functional σ(u), so that its solutions are not only the minima of the functional, but also all its stationary points. 6) is investigated. In general, the minimum of σ(u) may be unattainable, that is, there may be no function u on which the minimum value of σ is reached.
Aging and Radiation Effects in Stockpile Electronics by Hartman, E.F.; Sandia National Laboratories.; United States. Dept. of Energy.; United States. Dept. of Energy. Office of Scientific and Technical Information