Related papers: Reliable Topology for Dynamic Data: Mathematical F…
Accurate simulations of ice sheet dynamics, mantle convection, lava flow, and other highly viscous free-surface flows involve solving the coupled Stokes/free-surface equations. In this paper, we theoretically analyze the stability and…
For deterministic monotone cellular automata on the $d$-dimensional integer lattice, Toom (1980) has given necessary and sufficient conditions for the all-one fixed point to be stable against small random perturbations. We are interested in…
In this work, we investigate the analysis of generators for dynamic graphs, which are defined as graphs whose topology changes over time. We introduce a novel concept, called ''sustainability,'' to qualify the long-term evolution of dynamic…
Rock discontinuities critically govern the mechanical behavior and stability of rock masses. Their internal distributions remain largely unobservable and are typically inferred from surface-exposed discontinuities using generative…
This work establishes a rigorous connection between stability properties of discrete-time algorithms (DTAs) and corresponding continuous-time dynamical systems derived through $ O(s^r) $-resolution ordinary differential equations (ODEs). We…
The surge of data available on the Internet has driven the adoption of a wide range of computational methods for analyzing and extracting insights from large-scale data. Among these, Machine Learning (ML) has become a central paradigm,…
This paper explores the theoretical limits of using discrete abstractions for nonlinear control synthesis. More specifically, we consider the problem of deciding continuous-time control with temporal logic specifications. We prove that…
We start with a simple introduction to topological data analysis where the most popular tool is called a persistent diagram. Briefly, a persistent diagram is a multiset of points in the plane describing the persistence of topological…
This paper develops a new approach to the estimation of the degree of boundedness or stability of multidimensional nonlinear systems with time-dependent nonperiodic coefficients-an essential task in various engineering and natural science…
Dynamical system properties give rise to effects in Statistical Mechanics. Topological index changes can be the basis for phase transitions. The Euler characteristic is a versatile topological invariant that can be evaluated for model…
We develop a framework to give upper bounds on the "practical" computational complexity of stability problems for a wide range of nonlinear continuous and hybrid systems. To do so, we describe stability properties of dynamical systems using…
Persistence diagrams play a fundamental role in Topological Data Analysis where they are used as topological descriptors of filtrations built on top of data. They consist in discrete multisets of points in the plane $\mathbb{R}^2$ that can…
Analyzing flight trajectory data sets poses challenges due to the intricate interconnections among various factors and the high dimensionality of the data. Topological Data Analysis (TDA) is a way of analyzing big data sets focusing on the…
This paper analyzes the stability of interconnected continuous-time (CT) and discrete-time (DT) systems coupled through sampling and zero-order hold mechanisms. The DT system updates its output at regular intervals $T>0$ by applying an…
Optimization under uncertainty and risk is indispensable in many practical situations. Our paper addresses stability of optimization problems using composite risk functionals which are subjected to measure perturbations. Our main focus is…
Topological data analysis involves the statistical characterization of the shape of data. Persistent homology is a primary tool of topological data analysis, which can be used to analyze topological features and perform statistical…
Persistent homology is a method for probing topological properties of point clouds and functions. The method involves tracking the birth and death of topological features (2000) as one varies a tuning parameter. Features with short…
Preserving stability is a central problem in data-driven model order reduction of dynamical systems. For linear systems whose dynamics depend on geometric or physical parameters, multivariate rational approximation algorithms such as the…
In this chapter, we discuss applications of topological data analysis (TDA) to spatial systems. We briefly review the recently proposed level-set construction of filtered simplicial complexes, and we then examine persistent homology in two…
Impulsive systems are a very flexible class of systems that can be used to represent switched and sampled-data systems. We propose to extend here the previously obtained results on deterministic impulsive systems to the stochastic setting.…