Related papers: Retrieving biparameter persistence modules from mo…

We associate to a bimonoidal functor, i.e. a bifunctor which is monoidal in each variable, a nonabelian version of a biextension. We show that such a biextension satisfies additional triviality conditions which make it a bilinear analog of…

Nanocomposite materials made of block copolymer and nanoparticles display properties which can be different from the purely polymeric matrix. The resulting material is a crossover of the original properties of the block copolymer and the…

We define pullback and separated presentations of modules over pullback rings, and, if the ring is a pullback of epimorphisms over a semisimple ring, an algorithm reducing such a presentation of a module to an $R$-diagram. The latter is the…

We study the relation between the persistent homology and the spectral sequence of a filtered chain complex over a field. Our method is based on a decomposition of the persistent homology. We demonstrate that, under fairly general…

The homotopy theory of higher categorical structures has become a relevant part of the machinery of algebraic topology and algebraic K-theory, and this paper contains contributions to the study of the relationship between B\'enabou's…

From the work of Bauer and Lesnick, it is known that there is no functor from the category of pointwise finite-dimensional persistence modules to the category of barcodes and overlap matchings. In this work, we introduce sub-barcodes and…

We define biprops as a generalization of coloured props and of symmetric weak multicategories. These are bicategories whose objects form a free monoid. They are equipped with some structure resembling a symmetric strict tensor product. We…

We prove that, in characteristic 0, any Hasse-Schmidt module structure can be recovered from its underlying integrable connection, and consequently Hasse--Schmidt modules and modules endowed with an integrable connection coincide.

We present a method for synthesizing recursive functions that provably satisfy a given specification in the form of a polymorphic refinement type. We observe that such specifications are particularly suitable for program synthesis for two…

When working with (multi-parameter) persistence modules, one usually makes some type of tameness assumption in order to obtain better control over their algebraic behavior. One such notion is Ezra Millers notion of finite encodability,…

Based on the novel notion of `weakly counital fusion morphism', regular weak multiplier bimonoids in braided monoidal categories are introduced. They generalize weak multiplier bialgebras over fields and multiplier bimonoids in braided…

Modularity is one of the most prominent properties of real-world complex networks. Here, we address the issue of module identification in two important classes of networks: bipartite networks and directed unipartite networks. Nodes in…

We present in this paper the preliminary design of a module system based on a notion of components such as they are found in COM. This module system is inspired from that of Standard ML, and features first-class instances of components,…

We study primary submodules and primary decompositions from a differential and computational point of view. Our main theoretical contribution is a general structure theory and a representation theorem for primary submodules of an arbitrary…

Multigraded Betti numbers are one of the simplest invariants of multiparameter persistence modules. This invariant is useful in theory -- it completely determines the Hilbert function of the module and the isomorphism type of the free…

We consider the property of unique parallel decomposition modulo branching and weak bisimilarity. First, we show that infinite behaviours may fail to have parallel decompositions at all. Then, we prove that totally normed behaviours always…

In topological data analysis (TDA), one often studies the shape of data by constructing a filtered topological space, whose structure is then examined using persistent homology. However, a single filtered space often does not adequately…

For safety and robustness of AI systems, we introduce topological parallax as a theoretical and computational tool that compares a trained model to a reference dataset to determine whether they have similar multiscale geometric structure.…

This paper suggests a [email protected] of composable specification of concurrent programs that permits: (1) verification of program code for a given specification, and (2) composition of the specifications of the components to yield…

A weak entwining structure in a 2-category K consists of a monad t and a comonad c, together with a 2-cell relating both structures in a way that generalizes a mixed distributive law.A weak entwining structure can be characterized as a…