Asymptotic behavior of averaged and firmly nonexpansive mappings in geodesic spaces

We further study averaged and firmly nonexpansive mappings in the setting of geodesic spaces with a main focus on the asymptotic behavior of their Picard iterates. We use methods of proof mining to obtain an explicit quantitative version of a generalization to geodesic spaces of result on the asymptotic behavior of Picard iterates for firmly nonexpansive mappings proved by Reich and Shafrir. From this result we obtain effective uniform bounds on the asymptotic regularity for firmly nonexpansive mappings. Besides this, we derive effective rates of asymptotic regularity for sequences generated by two algorithms used in the study of the convex feasibility problem in a nonlinear setting

A Vanishing Theorem and Asymptotic Regularity of Powers of Ideal Sheaves

Let $\mathscr{I}$ be an ideal sheaf on $P^n$. In the first part of this paper, we bound the asymptotic regularity of powers of $\mathscr{I}$ as ps-3\leq \reg \mathscr{I}^p\leq ps+e, where $e$ is a constant and $s$ is the $s$-invariant of $\mathscr{I}$. We also give the same upper bound for the asymptotic regularity of symbolic powers of $\mathscr{I}$ under some conditions. In the second part, by using multiplier ideal sheaves, we give a vanishing theorem of powers of $\mathscr{I}$ when it defines a local complete intersection subvariety with log canonical singularities.Comment: 13 pages; Corrected typos, added references, improve one of the main theorem

Effective results on compositions of nonexpansive mappings

This paper provides uniform bounds on the asymptotic regularity for iterations associated to a finite family of nonexpansive mappings. We obtain our quantitative results in the setting of $(r,\delta)$-convex spaces, a class of geodesic spaces which generalizes metric spaces with a convex geodesic bicombing

Bifurcation values and monodromy of mixed polynomials

We study the bifurcation values of real polynomial maps f: \bR^{2n} \to \bR^2 which reflect the lack of asymptotic regularity at infinity. We formulate real counterparts of some structure results which have been previously proved in case of complex polynomials by Kushnirenko, N\'emethi and Zaharia and other authors, emphasizing the typical real phenomena that occur

Rates of asymptotic regularity for Halpern iterations of nonexpansive mappings

In this paper we obtain new effective results on the Halpern iterations of nonexpansive mappings using methods from mathematical logic or, more specifically, proof-theoretic techniques. We give effective rates of asymptotic regularity for the Halpern iterations of nonexpansive self-mappings of nonempty convex sets in normed spaces. The paper presents another case study in the project of {\em proof mining}, which is concerned with the extraction of effective uniform bounds from (prima-facie) ineffective proofs.Comment: in C.S. Calude, G. Stefanescu, and M. Zimand (eds.), Combinatorics and Related Areas. A Collection of Papers in Honour of the 65th Birthday of Ioan Tomesc