By Achtziger W., Bletzinger K.-U., Svanberg K.

This publication comprises the lecture notes ready by means of the international teachers on the DCAMM complex tuition 'Advanced themes in Structural Optimization1 held on the Technical collage of Denmark, June 25 to Juli three, 1998. the fabric coated by way of the notes isn't easily accesible in present literature as unified displays directed in the direction of the strucural optimization neighborhood. the aim of this booklet is therefore to make the cloth to be had to a broader audience.We want to thank the authors, Wolfgang Achtziger, Kai-Uwe Bletzinger, andKrister Svanberg for taking the effort and time in getting ready their contributions and for permitting DCAMM to print their notes within the DCAMM particular record Series.The DCAMM complicated college 'Advanced issues in Structural Optimization' was once held less than the auspices of the DCAMM foreign Graduate study college in utilized Mechanics. The aid got from the Danish study Academy is gratefully stated.

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Star products which are invariant and covariant are used in the problem of reduction: this is a device in symplectic geometry which allows one to reduce the number of variables. An important issue in quantization is to know if and how quantization commutes with reduction. This problem has been studied by Fedosov for the action of a compact group on the particular star products constructed by him with trivial characteristic class ( Ãr, 0 ). Here, one indeed obtains some ‘‘quantization commutes with reduction’’ statements.

For example, the selection of pure potential two-dimensional Schro¨dinger operators originally was not so evident. To formulate the answer, it is convenient to introduce a new function b(), T() = b() " sgn(" À 1)=. For E0 < 0, the following constraints select real potential operators:     1 1 " b À ¼ bðÞ; b ¼ bðÞ ½50Š "  " In some situations, the problem of finding appropriate reductions is the most difficult part of the integration procedure. It is true not only for the " @-approach, but also for other techniques including the finite-gap method.

The wave function É(
, x, y) of the perturbed operator L = @y À @x2 þ u(x, y) is defined at the same spectral curve À, but it is not holomorphic any more. It has the following properties: 1. At the point 1, the wave function É(
, x, y) has an essential singularity: É(
, x, y) = É0 (
, x, y) (1 þ o(1)). 2. In the neighborhoods of the points k , É(
, x, y) can be written as a product of a continuous function by a simple pole at k . 3. The wave function É(
, x, y) satisfies the @" equation @Éð
; x; y; tÞ d" ¼ Tð
ÞÉð
; x; y; tÞ @ " ½53Š where the (0, 1)-form T(
) = t(
)d" is regular outside the divisor points k and in the neighborhood of k it possible to define local coordinate such that t(
) = sgn(=
)t1 (
)( À k )=( À k ), t1 (
) is regular.

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