Related papers: Metric Limits in Categories with a Flow

We prove a compactness result for gradient flow lines in a general set-up which comprises both the situation of Morse gradient flow lines as well as Floer cylinders converging to a critical submanifold respectively. For the compactness…

Flow in porous media is difficult to address using standard analytical or numerical methods due to its complexity. However, since synthetic representations of porous media are easy to produce and data from physical experiments are becoming…

Flow matching has emerged as a powerful framework for generative modeling through continuous normalizing flows. We investigate a potential topological constraint: when the prior distribution and target distribution have mismatched topology…

In this work, we provide two novel approaches to show that incompressible fluid flow in a finite domain contains at most a finite number vortices. We use a recently developed geometric theory of incompressible viscous flows along with an…

We study the new geometric flow that was introduced in [11] that evolves a pair of map and (domain) metric in such a way that it changes appropriate initial data into branched minimal immersions. In the present paper we focus on the…

Flow Matching has emerged as a powerful framework for learning continuous transformations between distributions, enabling high-fidelity generative modeling. This work introduces Symmetrical Flow Matching (SymmFlow), a new formulation that…

Business process models abstract complex business processes by representing them as graphical models. Their layout, solely determined by the modeler, affects their understandability. To support the construction of understandable models it…

We consider point clouds obtained as random samples of a measure on a Euclidean domain. A graph representing the point cloud is obtained by assigning weights to edges based on the distance between the points they connect. Our goal is to…

We define a family of functionals generalizing the Yang-Mills functional. We study the corresponding gradient flows and prove long-time existence and convergence results for subcritical dimensions as well as a bubbling criterion for the…

This paper presents a distance function between sets based on an average of distances between their elements. The distance function is a metric if the sets are non-empty finite subsets of a metric space. It can be applied to produce various…

Evaluating the channel capacity is one of many key problems in information theory. In this work we derive rather-mild sufficient conditions under which the capacity is finite and achievable. These conditions are derived for generic,…

It is known that for a variety of choices of metrics, including the standard bottleneck distance, the space of persistence diagrams admits geodesics. Typically these existence results produce geodesics that have the form of a convex…

We study quasi-modular pseudometric spaces as asymmetric refinements of modular metric structures. To each such space we associate canonical forward and backward quasi-uniformities and the corresponding directional topologies. We introduce…

We discuss a purely variational approach to the total variation flow on metric measure spaces with a doubling measure and a Poincar\'e inequality. We apply the concept of parabolic De Giorgi classes together with upper gradients, Newtonian…

Flow matching has emerged as a simulation-free alternative to diffusion-based generative modeling, producing samples by solving an ODE whose time-dependent velocity field is learned along an interpolation between a simple source…

Competitive on-line prediction (also known as universal prediction of individual sequences) is a strand of learning theory avoiding making any stochastic assumptions about the way the observations are generated. The predictor's goal is to…

The aim of this paper is twofold. One is to give a definition of the Euler characteristic of infinite acyclic categories with filtrations and the other is to prove the invariance of the Euler characteristic under the subdivision of finite…

We study basic properties of flow equivalence on one-dimensional compact metric spaces with a particular emphasis on isotopy in the group of (self-) flow equivalences on such a space. In particular, we show that an orbit-preserving such map…

Bounds of total curvature and entropy are two common conditions placed on mean curvature flows. We show that these two hypotheses are equivalent for the class of ancient complete embedded smooth planar curve shortening flows, which are…

This paper investigates spaces equipped with a family of metric-like functions satisfying certain axioms. These functions provide a unified framework for defining topology, uniformity, and diffeology. The framework is based on a family of…