The common fixed point of single-valued generalized φf-weakly contractive mappings

Fixed point and coincidence results are presented for single-valued generalized ${\phi }_f$-weakly contractive mappings on complete metric spaces $(X,d)$, where $\phi :[0,\infty )\to [0,\infty )$ is a lower semicontinuous function with $\phi \left(0\right)=0$ and $\phi \left(t\right)>0$ for all $t>0$ and $f:E\to X$ is a function such that $E?X$ is nonempty and closed. Our results extend previous results given by Rhoades (2001) [1] and by Zhang and Song (2009) [2].     

On functors preserving skeletal maps and skeletally generated compacta

A map $f:X\to Y$ between topological spaces is skeletal if the preimage $f^{?1}(A)$ of each nowhere dense subset $A?Y$ is nowhere dense in X. We prove that a normal functor $F:\mathbf{Comp}\to \mathbf{Comp}$ is skeletal (which means that F preserves skeletal epimorphisms) if and only if for any open surjective map $f:X\to Y$ between metrizable zero-dimensional compacta with two-element non-degeneracy set $N^f=\{x\in X:|f^{?1}(f(x))|>1\}$ the map $Ff:FX\to FY$ is skeletal. This characterization implies that each open normal functor is skeletal. The converse is not true even for normal functors of finite degree. The other main result of the paper says that each normal functor $F:\mathbf{Comp}\to \mathbf{Comp}$ preserves the class of skeletally generated compacta. This contrasts with the known ??epin?s result saying that a normal functor is open if and only if it preserves the class of openly generated compacta.     

Generalized differentiation and bi-Lipschitz nonembedding in L1

We consider Lipschitz mappings, $f:X\to V$, where X is a doubling metric measure space which satisfies a Poincaré inequality, and V is a Banach space. We show that earlier differentiability and bi-Lipschitz nonembedding results for maps, $f:X\to {\mathbb{R}}^N$, remain valid when ${\mathbb{R}}^N$ is replaced by any separable dual space. We exhibit spaces which bi-Lipschitz embed in $L^1$, but not in any separable dual V. For certain domains, including the Heisenberg group with its Carnot–Caratheodory metric, we establish a new notion of differentiability for maps into $L^1$. This implies that the Heisenberg group does not bi-Lipschitz embed in $L^1$, thereby proving a conjecture of J. Lee and A. Naor. When combined with their work, this has implications for theoretical computer science. To cite this article: J. Cheeger, B. Kleiner, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

The group of invariants of an inner function with finite spectrum

This paper determines the group of continuous invariants corresponding to an inner function Θ with finitely many singularities on the unit circle $\mathbb{T}$; that is, the continuous mappings $g:\mathbb{T}\to \mathbb{T}$ such that $\Theta °g=\Theta$ on $\mathbb{T}$. These mappings form a group under composition.     

Multiple summing maps: Coordinatewise summability,inclusion theorems and p-Sidon sets

We discuss the multiple summability of a multilinear map $T:X_1\times ?\times X_m\to Y$ when we have information on the summability of the maps it induces on each coordinate. Our methods have applications to inclusion theorems for multiple summing multilinear mappings and to the product of p-Sidon sets.     

Explicit formulas for C1,1 and Cconv1,ω extensions of 1-jets in Hilbert and superreflexive spaces

Given X a Hilbert space, ω a modulus of continuity, E an arbitrary subset of X, and functions $f:E\to \mathbb{R}$, $G:E\to X$, we provide necessary and sufficient conditions for the jet $(f,G)$ to admit an extension $(F,\mathrm{?}F)$ with $F:X\to \mathbb{R}$ convex and of class $C^{1,\omega }(X)$, by means of a simple explicit formula. As a consequence of this result, if ω is linear, we show that a variant of this formula provides explicit $C^{1,1}$ extensions of general (not necessarily convex) 1-jets satisfying the usual Whitney extension condition, with best possible Lipschitz constants of the gradients of the extensions. Finally, if X is a superreflexive Banach space, we establish similar results for the classes $C_{\mathrm{conv}}^{1,\alpha }(X)$.     

Parabolic subgroups and automorphism groups of Schubert varieties

Let G be a simple algebraic group of adjoint type over the field $\mathbb{C}$ of complex numbers, B be a Borel subgroup of G containing a maximal torus T of G. Let w be an element of the Weyl group W and $X(w)$ be the Schubert variety in $G/B$ corresponding to w. In this article we show that given any parabolic subgroup P of G containing B properly, there is an element $w\in W$ such that P is the connected component, containing the identity element of the group of all algebraic automorphisms of $X(w)$.