### A degree theory for compact perturbations of proper ${C}^{1}$ Fredholm mappings of index 0.

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We present a geometric approach to diffeomorphism invariant full Colombeau algebras which allows a particularly clear view of the construction of the intrinsically defined algebra $\widehat{}\left(M\right)$ on the manifold M given in [gksv].

This paper deals with homeomorphisms F: X → Y, between Banach spaces X and Y, which are of the form $F\left(x\right):=F\u0303{x}^{(2n+1)}$ where $F\u0303:{X}^{2n+1}\to Y$ is a continuous (2n+1)-linear operator.

We prove a uniform version of the converse Taylor theorem in infinite-dimensional spaces with an explicit relation between the moduli of continuity for mappings on a general open domain. We show that if the domain is convex and bounded, then we can extend the estimate up to the boundary.

We show that a ${C}^{k}$-smooth mapping on an open subset of ${\mathbb{R}}^{n}$, $k\in \mathbb{N}\cup \{0,\infty \}$, can be approximated in a fine topology and together with its derivatives by a restriction of a holomorphic mapping with explicitly described domain. As a corollary we obtain a generalisation of the Carleman-Scheinberg theorem on approximation by entire functions.

We answer a question of Aharoni by showing that every separable metric space can be Lipschitz 2-embedded into c₀ and this result is sharp; this improves earlier estimates of Aharoni, Assouad and Pelant. We use our methods to examine the best constant for Lipschitz embeddings of the classical ${\ell}_{p}$-spaces into c₀ and give other applications. We prove that if a Banach space embeds almost isometrically into c₀, then it embeds linearly almost isometrically into c₀. We also study Lipschitz embeddings into...

Bourgain’s discretization theorem asserts that there exists a universal constant $C\in (0,\infty )$ with the following property. Let $X,Y$ be Banach spaces with $dimX=n$. Fix $D\in (1,\infty )$ and set $\delta ={e}^{-{n}^{Cn}}$. Assume that $\mathcal{N}$ is a $\delta $-net in the unit ball of $X$ and that $\mathcal{N}$ admits a bi-Lipschitz embedding into $Y$ with distortion at most $D$. Then the entire space $X$ admits a bi-Lipschitz embedding into $Y$ with distortion at most $CD$. This mostly expository article is devoted to a detailed presentation of a proof of Bourgain’s theorem.We also obtain an improvement...

In this paper we study the notions of finite turn of a curve and finite turn of tangents of a curve. We generalize the theory (previously developed by Alexandrov, Pogorelov, and Reshetnyak) of angular turn in Euclidean spaces to curves with values in arbitrary Banach spaces. In particular, we manage to prove the equality of angular turn and angular turn of tangents in Hilbert spaces. One of the implications was only proved in the finite dimensional context previously, and equivalence of finiteness...

Given a compact manifold ${N}^{n}\subset {\mathbb{R}}^{\nu}$ and real numbers $s\ge 1$ and $1\le p\<\infty $, we prove that the class ${C}^{\infty}({\overline{Q}}^{m};{N}^{n})$ of smooth maps on the cube with values into ${N}^{n}$ is strongly dense in the fractional Sobolev space ${W}^{s,p}({Q}^{m};{N}^{n})$ when ${N}^{n}$ is $\lfloor sp\rfloor $ simply connected. For $sp$ integer, we prove weak sequential density of ${C}^{\infty}({\overline{Q}}^{m};{N}^{n})$ when ${N}^{n}$ is $sp-1$ simply connected. The proofs are based on the existence of a retraction of ${\mathbb{R}}^{\nu}$ onto ${N}^{n}$ except for a small subset of ${N}^{n}$ and on a pointwise estimate of fractional derivatives of composition of maps in ${W}^{s,p}\cap {W}^{1,sp}$.

Let f be a smooth function defined on a finite union U of open convex sets in a locally convex Lindelöf space E. If, for every x ∈ U, the restriction of f to a suitable neighbourhood of x admits a smooth extension to the whole of E, then the restriction of f to a union of convex sets that is strictly smaller than U also admits a smooth extension to the whole of E.