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I will survey various results on applications and extensions of Hofer-Wysocki-Zehnder's polyfold theory: - fiber products of polyfold Fredholm sections - equivariant transversality - existence and obstructions - equivariant fundamental class -

14] V. S. Rabinovich, S. Roch and J. Roe: Fredholm indices of band-dominatedoperators, Integral Equations Operator Theory 49 (2004), 221–238. ... 15] V. S. Rabinovich, S. Roch and B. Silbermann: Fredholm Theory andFinite Section Method for

Since there is no small parameter. in the equation, its solution is based on the Fredholm theory [9]. ... Illner and M. Pulvirenti, “The Mathematical Theory of Dilute Gases”,. Springer-Verlag (1994).

Therefore, our existence proof mainly relies on a simple fixed point iterationand standard Fredholm theory. ... The standard Fredholm theory therefore gives the result.Proof of Corollary 3.4 Because of the Fredholm alternative in Proposition 3.3 weonly

infinite matrices, limit operators, collective compactness, Fredholm. operators, spectral theory. viii. ... overcome. The matrix [A] that corresponds to an operator A BDO(Y ) now hasoperator entries aij L(U ) which are infinite dimensional operators

There exists a close connection between the study ofToeplitz operators and the theory of WH factorisation. ... dim kerTG codim ImTG = 0, (17). i.e., the Fredholm index of TG is zero.

Remark 3.1 In the classical theory of optimal design, convex optimalitycriteria are almost always considered. ... H(u,v) =1. |u v|α(1 v2)(1α)/2.(4.9). It appears in the theory of Fredholm-Volterra integral equations of the firstkind with special kernel,

The following statements are routine, due to the direct integral ex-pansion (4.17) of the operator, perturbation theory [157], and somestandard properties of analytic Fredholm operator-functions (e.g., ... An easier statement, without estimates on growth