It follows from [1] and [7] that any closed n-codimensional subspace (n ≥ 1 integer) of a real Banach space X is the kernel of a projection X → X, of norm less than f(n) + ε (ε > 0 arbitrary), where f(n) = 2 + (n − 1) √ n + 2 n + 1 . We have f(n) < √ n for n > 1, and f(n) = √ n − 1 √ n + O (

We find all polynomials f, g, h over a field K such that g and h are linear and f(g(x)) = h(f(x)). We also solve the same problem for rational functions f, g, h, in case the field K is algebraically closed.

Let R = (R,<,+, ·, . . . ) be a non-valuational weakly o-minimal real closed field, I a definable convex open subset of R and f : I → R a definable function. We prove that {x ∈ I : f ′(x) exists in R} is definable and f ′ is definable if f is differentiable.

Thompson’s group F is the group of all increasing dyadic PL homeomorphisms of the closed unit interval. We compute Σm(F ) and Σm(F ;Z), the homotopical and homological Bieri-Neumann-Strebel-Renz invariants of F , and show that Σm(F ) = Σm(F ;Z). As an application, we show that, for every m, F has subgroups of type Fm−1 which are not of type FPm (thus certainly not of type Fm).

Let F be an algebraically closed field of characteristic 0 and f(x) a polynomial of degree strictly greater than one in F [x]. We show that the number of degree k polynomials with coefficients in F that commute with f (under composition) is either zero or equal to the number of degree one polynomials with coefficients in F that commute with f . As a corollary, we obtain a theorem of E. A. Bertr...

In this paper, we prove that on every Finsler n-sphere (S, F ) for n ≥ 6 with reversibility λ and flag curvature K satisfying ( λ λ+1 )2 < K ≤ 1, either there exist infinitely many prime closed geodesics or there exist [ 2 ] − 2 closed geodesics possessing irrational average indices. If in addition the metric is bumpy, then there exist n − 3 closed geodesics possessing irrational average indice...

Let M be a closed oriented 3-manifold which can be triangulated with N simplices. We prove that any map from M to a genus 2 surface has Hopf invariant at most C . Let X be a closed oriented hyperbolic 3-manifold with injectivity radius less than ǫ at one point. If there is a degree non-zero map from M to X, then we prove that ǫ is at least C . In the 1970’s, Thurston invented simplex straighten...

We prove the existence of smooth closed hypersurfaces of prescribed mean curvature homeomorphic to S for small n, n ≤ 6, provided there are barriers. 0. Introduction In a complete (n+1)-dimensional manifold N we want to find closed hypersurfaces M of prescribed mean curvature. To be more precise, let Ω be a connected open subset of N , f ∈ C(Ω̄), then we look for a closed hypersurface M ⊂ Ω such...

Let k be an algebraically closed field, let R be an associative kalgebra, and let F = {Mα : α ∈ I} be a family of orthogonal points in Mod(R) such that EndR(Mα) ∼= k for all α ∈ I. Then Mod(F), the minimal full subcategory of Mod(R) which contains F and is closed under extensions, is a full exact Abelian sub-category of Mod(R) and a length category in the sense of Gabriel [8]. In this paper, we...