Related papers: Algorithms and topological invariants for dynamic …

In this article, a new method for segmentation and restoration of images on two-dimensional surfaces is given. Active contour models for image segmentation are extended to images on surfaces. The evolving curves on the surfaces are…

This paper analyzes the robustness of recent 3D shape descriptors to SO(3) rotations, something that is fundamental to shape modeling. Specifically, we formulate the task of rotated 3D object instance detection. To do so, we consider a…

This chapter discusses the interplay between structure and dynamics in complex networks. Given a particular network with an endowed dynamics, our goal is to find partitions aligned with the dynamical process acting on top of the network. We…

We review recent developments in differential topology with special concern for their possible significance to physical theories, especially general relativity. In particular we are concerned here with the discovery of the existence of…

Invariant manifolds facilitate the understanding of nonlinear stochastic dynamics. When an invariant manifold is represented approximately by a graph for example, the whole stochastic dynamical system may be reduced or restricted to this…

There are many fundamental algorithmic problems on triangulated 3-manifolds whose complexities are unknown. Here we study the problem of finding a taut angle structure on a 3-manifold triangulation, whose existence has implications for both…

We study random 2-dimensional complexes in the Linial - Meshulam model and find torsion in their fundamental groups at various regimes. We find a simple algorithmically testable criterion for a subcomplex of a random 2-complex to be…

Classes of branched surfaces extend the classes of surfaces or 2-dimensional manifolds satisfying suitable properties and defined in various manners. Reeb spaces of smooth maps of suitable classes into surfaces whose codimensions are…

We investigate the topological properties of invariant sets associated with the dynamics of scattering systems with three or more degrees of freedom. We show that the asymptotic separation of one degree of freedom from the rest in the…

Random invariant manifolds are geometric objects useful for understanding complex dynamics under stochastic influences. Under a nonuniform hyperbolicity or a nonuniform exponential dichotomy condition, the existence of random pseudo-stable…

The space-like hypersurface of the Universe at the present cosmological time is a three-dimensional manifold. A non-trivial global topology of this space-like hypersurface would imply that the apparently observable universe (the sphere of…

In this chapter, we utilize dynamical systems to analyze several aspects of machine learning algorithms. As an expository contribution we demonstrate how to re-formulate a wide variety of challenges from deep neural networks, (stochastic)…

We investigate a generic dynamical theory to characterize topological quantum phases by quantum quenches, and study the emergent topology of quantum dynamics when the quenches start from a deep or shallow trivial phase to topological…

In this paper we explore the mathematical structure of hierarchical organization in smooth dynamical systems. We start by making precise what we mean by a level in a hierarchy, and how the higher le vels need to respect the dynamics on the…

Integral invariants obtained from Principal Component Analysis on a small kernel domain of a submanifold encode important geometric information classically defined in differential-geometric terms. We generalize to hypersurfaces in any…

In the realm of computer-aided design (CAD) software, the intersection of B-spline surfaces stands as a fundamental operation. Despite the extensive history of surface intersection algorithms, the challenge of handling complex intersection…

In this article we study abstract and embedded invariants of reduced curve germs via topological techniques. One of the most important numerical analytic invariants of an abstract curve is its delta invariant. Our primary goal is to develop…

Active matter encompasses different nonequilibrium systems in which individual constituents convert energy into non-conservative forces or motion at the microscale. This review provides an elementary introduction to the role of topology in…

The global behaviour of nonlinear systems is extremely important in control and systems theory since the usual local theories will only give information about a system in some neighbourhood of an operating point. Away from that point, the…

We introduce an information-theoretic framework for smooth structures on topological manifolds, replacing coordinate charts with small-scale entropy data of local probability probes. A concise set of axioms identifies admissible coordinate…