Search results for "Homotopy"

Fixed point and homotopy results for mixed multi-valued mappings in 0-complete partial metric spaces*

2015

We give sufficient conditions for the existence of common fixed points for a pair of mixed multi-valued mappings in the setting of 0-complete partial metric spaces. An example is given to demonstrate the usefulness of our results over the existing results in metric spaces. Finally, we prove a homotopy theorem via fixed point results.

Common fixed points of generalized contractions on partial metric spaces and an application

2011

Abstract In this paper, common fixed point theorems for four mappings satisfying a generalized nonlinear contraction type condition on partial metric spaces are proved. Presented theorems extend the very recent results of I. Altun, F. Sola and H. Simsek [Generalized contractions on partial metric spaces, Topology and its applications 157 (18) (2010) 2778–2785]. As application, some homotopy results for operators on a set endowed with a partial metric are given.

Homotopy limits for 2-categories

2008

AbstractWe study homotopy limits for 2-categories using the theory of Quillen model categories. In order to do so, we establish the existence of projective and injective model structures on diagram 2-categories. Using these results, we describe the homotopical behaviour not only of conical limits but also of weighted limits. Finally, pseudo-limits are related to homotopy limits.

Stochastic factorizations, sandwiched simplices and the topology of the space of explanations

2003

We study the space of stochastic factorizations of a stochastic matrix V, motivated by the statistical problem of hidden random variables. We show that this space is homeomorphic to the space of simplices sandwiched between two nested convex polyhedra, and use this geometrical model to gain some insight into its structure and topology. We prove theorems describing its homotopy type, and, in the case where the rank of V is 2, we give a complete description, including bounds on the number of connected components, and examples in which these bounds are attained. We attempt to make the notions of topology accessible and relevant to statisticians.

Fronto-parietal homotopy in resting-state functional connectivity predicts task-switching performance

2021

Homotopic functional connectivity reflects the degree of synchrony in spontaneous activity between homologous voxels in the two hemispheres. Previous studies have associated increased brain homotopy and decreased white matter integrity with performance decrements on different cognitive tasks across the life-span. Here, we correlated functional homotopy, both at the whole-brain level and specifically in fronto-parietal network nodes, with task-switching performance in young adults. Cue-to-target intervals (CTI: 300 vs. 1200 ms) were manipulated on a trial-by-trial basis to modulate cognitive demands and strategic control. We found that mixing costs, a measure of task-set maintenance and moni…

Inductive types in homotopy type theory

2012

Homotopy type theory is an interpretation of Martin-L\"of's constructive type theory into abstract homotopy theory. There results a link between constructive mathematics and algebraic topology, providing topological semantics for intensional systems of type theory as well as a computational approach to algebraic topology via type theory-based proof assistants such as Coq. The present work investigates inductive types in this setting. Modified rules for inductive types, including types of well-founded trees, or W-types, are presented, and the basic homotopical semantics of such types are determined. Proofs of all results have been formally verified by the Coq proof assistant, and the proof s…

On the proper homotopy invariance of the Tucker property

2006

A non-compact polyhedron P is Tucker if, for any compact subset K ⊂ P, the fundamental group π1(P − K) is finitely generated. The main result of this note is that a manifold which is proper homotopy equivalent to a Tucker polyhedron is Tucker. We use Poenaru’s theory of the equivalence relations forced by the singularities of a non-degenerate simplicial map.

Isotopy classes of diffeomorphisms of (k-1)-connected almost-parallelizable 2k-manifolds

1979

The “λ-medial axis”

2005

Medial axis is known to be unstable for nonsmooth objects. For an open set O, we define the weak feature size, wfs, minimum distance between Oc and the critical points of the function distance to Oc. We introduce the "lambda-medial axis" Mλ of O, a subset of the medial axis of O which captures the homotopy type of O when λ < wfs. We show that, at least for some "regular" values of λ, Mλ remains stable under Hausdorff distance perturbations of Oc.

Motivic Complexes and Relative Cycles

2019

This part is based on Suslin and Voevodsky’s theory of relative cycles that we develop in categorical terms, in the style of EGA. The climax of the theory is obtained in the study of a pullback operation for suitable relative cycles which is the incarnation of intersection theory in this language. Properties of this pullback operation, and on the conditions necessary to its definition, are made again inspired by intersection theory. We study the compatibility of this pullback operation with projective limits of schemes. In Section 9, the theory of relative cycles is exploited to introduce Voevodsky’s category of finite type schemes over an arbitrary base with morphisms finite correspondence…