Related papers: Euler Characteristic Tools For Topological Data An…
We provide a natural interpretation of the secondary Euler characteristic and introduce higher Euler characteristics. For a compact oriented manifold of odd dimension, the secondary Euler characteristic recovers the Kervaire…
Techniques from computational topology, in particular persistent homology, are becoming increasingly relevant for data analysis. Their stable metrics permit the use of many distance-based data analysis methods, such as multidimensional…
The Euler-Poincar\'e characteristic of a finite-dimensional Lie algebra vanishes. If we want to extend this result to Lie superalgebras, we should deal with infinite sums. We observe that a suitable method of summation, which goes back to…
Determination of the nature of the dynamical state of a system as a function of its parameters is an important problem in the study of dynamical systems. This problem becomes harder in experimental systems where the obtained data is…
We extend the notion of the distance to a measure from Euclidean space to probability measures on general metric spaces as a way to do topological data analysis in a way that is robust to noise and outliers. We then give an efficient way to…
Topological features based on persistent homology capture high-order structural information so as to augment graph neural network methods. However, computing extended persistent homology summaries remains slow for large and dense graphs and…
Individual-based models are complex and they have usually an elevated number of input parameters which must be tuned for reproducing the observed population data or the experimental results as accurately as possible. Thus, one of the…
A methodology is introduced which uses three simple objective function features to predict effective control parameters for differential evolution. This is achieved using cluster analysis techniques to classify objective functions using…
We investigate the qualitative dynamics of smooth solutions to the radially symmetric isentropic compressible Euler equations, focusing specifically on the evolution of rarefactive and compressive wave characters across three distinct…
We develop a general method for computing the homological Euler characteristic of finite index subgroups G of GL_m(O_K) where O_K is the ring of integers in a number field K. With this method we find, that for large, explicitly computed…
The Euler characteristic $\chi =|V|-|E|$ and the total length $\mathcal{L}$ are the most important topological and geometrical characteristics of a metric graph. Here, $|V|$ and $|E|$ denote the number of vertices and edges of a graph. The…
We develop a framework for detecting regime transitions in dynamical systems using the Mixup Euler Characteristic Profile (Mixup ECP) -- the Euler characteristic of the geometric intersection of ball unions around adjacent delay-embedded…
This paper examines the properties of the homentropic Euler equations when the characteristics of the equations have been spatially averaged. The new equations are referred to as the characteristically averaged homentropic Euler (CAHE)…
Each complex network (or class of networks) presents specific topological features which characterize its connectivity and highly influence the dynamics of processes executed on the network. The analysis, discrimination, and synthesis of…
Revealing the structural features of a complex system from the observed collective dynamics is a fundamental problem in network science. In order to compute the various topological descriptors commonly used to characterize the structure of…
The current paper focuses on fundamental groups and Euler characteristics of various digital models of the 2-dimensional sphere. For all models that we consider, we show that the fundamental groups are trivial, and compute the Euler…
Topological data analysis, as a tool for extracting topological features and characterizing geometric shapes, has experienced significant development across diverse fields. Its key mathematical techniques include persistent homology and the…
This paper introduces progressive algorithms for the topological analysis of scalar data. Our approach is based on a hierarchical representation of the input data and the fast identification of topologically invariant vertices, which are…
In this paper we propose a solution to the problem of parameter estimation of nonlinearly parameterized regressions--continuous or discrete time--and apply it for system identification and adaptive control. We restrict our attention to…
Topological Data Analysis (TDA) involves techniques of analyzing the underlying structure and connectivity of data. However, traditional methods like persistent homology can be computationally demanding, motivating the development of neural…