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A survey on the Szlenk index and some of its applications.

Gilles Lancien (2006)

RACSAM

We describe how the Szlenk index has been used in various areas of the geometry of Banach spaces. We cover the following domains of application of this notion: non existence of universal spaces, linear classification of C(K) spaces, descriptive set theory, renorming problems and non linear classification of Banach spaces.

Fegen und Dünnheit mit Anwendungen auf die Laplace-und Wärmeleitungsgleichung

Wolfhard Hansen (1971)

Annales de l'institut Fourier

Several properties of balayage of measures in harmonic spaces are studied. In particular, characterisations of thinness of subsets are given. For the heat equation the following result is obtained: suppose that E = R m + 1 is given the presheaf of solutions of ∑ i = 1 m ∂ u ∂ x i = ∂ u ∂ x m + 1 and B is a subset of R m × [ - ∞ , 0 ] satisfying { ( α x , α 2 t ) : ( x , t ) ∈ B , x ∈ R m , t ∈ R } ⊂ B for α > 0 arbitrarily small. Then B is thin at 0 if and only if B is polar. Similar result for the Laplace equation. At last the reduced of measures is defined and several approximation theorems on reducing and balayage...

Fredholm determinants

Henry McKean (2011)

Open Mathematics

The article provides with a down to earth exposition of the Fredholm theory with applications to Brownian motion and KdV equation.

Functions locally dependent on finitely many coordinates.

Petr Hájek, Václav Zizler (2006)

RACSAM

The notion of functions dependent locally on finitely many coordinates plays an important role in the theory of smoothness and renormings on Banach spaces, especially when higher smoothness (C∞) is involved. In this note we survey most of the main results in this area, and indicate many old as well as new open problems.

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