Related papers: An Isometry Theorem for Generalized Persistence Mo…
In spaces of nonpositive curvature the existence of isometrically embedded flat (hyper)planes is often granted by apparently weaker conditions on large scales. We show that some such results remain valid for metric spaces with non-unique…
Multiparameter persistence modules come up naturally in topological data analysis and topological robotics. Given a metric graph $(X,\delta)$, the second configuration space of $(X,\delta)$ with proximity parameters (for example, the…
We prove the equivalence of the two important facts about finite metric spaces and universal Urysohn metric spaces $\Bbb U$, namely theorem A and theorem B below: Theorem A (Approximation): The group of isometry $ISO(\Bbb U)$ contains…
We study in this paper the well-posedness and stability of a linear system of a thermoelastic Cosserat medium with infinite memory, where the Cosserat medium is a continuum in which each point has the degrees of freedom of a rigid body.
While persistent homology has taken strides towards becoming a wide-spread tool for data analysis, multidimensional persistence has proven more difficult to apply. One reason is the serious drawback of no longer having a concise and…
For any finite poset we define a generating polynomial counting upsets, downsets, and their intersection. We investigate the behaviour of this polynomial with respect to poset operations, show that it distinguishes series-parallel posets,…
Consider a standard representation $\pi_{st}$ of a quasi-split reductive p-adic group G. The generalized injectivity conjecture, posed by Casselman and Shahidi, asserts that any generic irreducible subquotient $\pi$ of $\pi_{st}$ is…
The generalized Kostka polynomials are the Poincare polynomials of isotypic components of certain graded GL(n)-modules. The former satisfy a monotonicity property arising from natural surjections of the corresponding modules. This…
A standard way of approximating or discretizing a metric space is by taking its Rips complexes. These approximations for all parameters are often bound together into a filtration, to which we apply the fundamental group or the first…
In the present paper, we study a set that can be treated as a generalised set of subsums for a geometric series. This object was discovered independently in various mathematical aspects. For instance, it is closely related to various…
A significant part of modern topological data analysis is concerned with the design and study of algebraic invariants of poset representations -- often referred to as multi-parameter persistence modules. One such invariant is the minimal…
It has been shown that $1$-parameter persistence modules have a very simple classification, namely there is a discrete invariant called a barcode that completely characterizes $1$-parameter persistence modules up to isomorphism. In…
In the early 80's, Alain Quilliot presented an approach of ordered sets and graphs in terms of metric spaces, where instead of positive real numbers, the values of the distance are elements of an ordered monoid equipped with an involution.…
We establish a transversality theorem for multiple-point crossings under generic linear perturbations with explicit Hausdorff measure estimates for the exceptional parameter set, and hence explicit upper bounds on its Hausdorff dimension.…
We introduce topological conditions on a broad class of functionals that ensure that the persistent homology modules of their associated sublevel set filtration admit persistence diagrams, which, in particular, implies that they satisfy…
In this paper, a decomposition theorem for (covariant) unitary group representations on Kaplansky-Hilbert modules over Stone algebras is established, which generalizes the well-known Hilbert space case (where it coincides with the…
In topological data analysis (TDA), persistence diagrams have been a succesful tool. To compare them, Wasserstein and Bottleneck distances are commonly used. We address the shortcomings of these metrics and show a way to investigate them in…
We show that a pointwise finite-dimensional persistence module indexed over a small category decomposes into a direct sum of indecomposables with local endomorphism rings. As an application of this result we give new, short proofs of…
Motivated by persistent homology and topological data analysis, we consider formal sums on a metric space with a distinguished subset. These formal sums, which we call persistence diagrams, have a canonical 1-parameter family of metrics…
This paper establishes connections between three of the most prominent metrics used in the analysis of persistence diagrams in topological data analysis: the bottleneck distance, Patel's erosion distance, and Bubenik's landscape distance.…