Related papers: Quillen-McCord theorem for persistence finite pose…

We introduce a persistence-like pseudo-distance on Tamarkin's category and prove that the distance between an object and its Hamiltonian deformation is at most the Hofer norm of the Hamiltonian function. Using the distance, we show a…

In this paper we investigate the continuity of $\mathbb{Q}_+$-homogeneous superadditive correspondences defined on the cones with finite basis in real topological vector spaces. It is also shown that every lower semicontinuous and…

Persistent homology is a way of determining the topological properties of a data set. It is well known that each persistence module admits the structure of a representation of a finite totally ordered set. In previous work, the authors…

The interleaving distance, although originally developed for persistent homology, has been generalized to measure the distance between functors modeled on many posets or even small categories. Existing theories require that such a poset…

We prove a persistence result for noncompact normally hyperbolic invariant manifolds in the setting of Riemannian manifolds of bounded geometry. Bounded geometry of the ambient manifold is a crucial assumption required to control the…

The minimal Quillen model is a free Lie model for rational spaces proposed by Quillen. Meanwhile, persistence modules are theoretical abstractions of persistent homology. In this paper, we integrate the ideas of rational homotopy theory and…

We extend the Quillen Theorem Bn for homotopy fibers of Dwyer, et al. to similar results for homotopy pullbacks and note that these results imply similar results for zigzags in the categories of relative categories and k-relative…

We consider profunctors $f : P \promap Q$ between posets and introduce their {\em graph} and {\em ascent}. The profunctors $\Pro(P,Q)$ form themselves a poset, and we consider a partition $\cI \sqcup \cF$ of this into a down-set $\cI$ and…

Let $\mathfrak{P}_r$ be a representation system of the non-isomorphic finite posets, and let ${\cal H}(P,Q)$ be the set of order homomorphisms from $P$ to $Q$. For finite posets $R$ and $S$, we write $R \sqsubseteq_G S$ iff, for every $P…

Interleaving distances provide a fundamental tool for comparing persistence modules and have been widely used in topological data analysis. Their definitions are typically based on translation structures (shift operations) on the indexing…

We prove a generalization of Gotzmann's persistence theorem in the case of modules with constant Hilbert polynomial. As a consequence, we show that the defining equations that give the embedding of a Quot scheme of points into a…

In this paper we prove FG-coupled fixed point theorems for different contractive mappings and generalized quasi- contractive mappings in partially ordered complete metric spaces. We prove the existence of FG-coupled fixed points of…

The persistent homology of a stationary point process on ${\bf R}^N$ is studied in this paper. As a generalization of continuum percolation theory, we study higher dimensional topological features of the point process such as loops,…

We examine the lattice of all order congruences of a finite poset from the viewpoint of combinatorial algebraic topology. We will prove that the order complex of the lattice of all nontrivial order congruences (or order-preserving…

In this article, we prove the existence of common fixed points for a pair of maps on a $q$-spherically complete $T_0$-ultra-quasi-metric space. The present article is a generalization, in the assymmetric setting of the paper of Rao et…

Let $P$ and $Q$ be bounded posets. In this note, a lemma is introduced that provides a set of sufficient conditions for the proper part of $P$ being homotopy equivalent to the suspension of the proper part of~$Q$. An application of this…

A $1$-Lipschitz map between compact metric spaces $f\colon X\to Y$ induces a homomorphism of persistence modules on degree-$d$ Vietoris--Rips persistent homology. We define the persistent cost of $f$ from this induced homomorphism by…

The theory of persistence, which arises from topological data analysis, has been intensively studied in the one-parameter case both theoretically and in its applications. However, its extension to the multi-parameter case raises numerous…

Let G be a complex reductive Lie group acting on a compact K\"ahler manifold X and assume that the action of a maximal compact subgroup K of G is Hamiltonian. For each extreme point of the convex hull of the momentum map image, there is an…

In this article we establish two fundamental results for the sublevel set persistent homology for stationary processes indexed by the positive integers. The first is a strong law of large numbers for the persistence diagram (treated as a…