Related papers: Complexity and Algorithms for Euler Characteristic…
In this article, we study Euler characteristic techniques in topological data analysis. Pointwise computing the Euler characteristic of a family of simplicial complexes built from data gives rise to the so-called Euler characteristic…
Given a finite simplicial complex L and a collection of pairs of spaces indexed by its vertex set, one can define their polyhedral product. We record a simple formula for its Euler characteristic. In special cases the formula simplifies…
Everyone knows that the Euler characteristic of a combinatorial manifold is given by the alternating sum of its numbers of simplices. It is shown that there are other linear combinations of the numbers of simplices which are combinatorial…
We consider different generalizations of the Euler formula and discuss the properties of the associated trigonometric functions. The problem is analyzed from different points of view and it is shown that it can be formulated in a natural…
Let $\mathrm{R}$ be a real closed field and $\mathrm{D} \subset \mathrm{R}$ an ordered domain. We consider the algorithmic problem of computing the generalized Euler-Poincar\'e characteristic of real algebraic as well as semi-algebraic…
It is well known that the Euler characteristic of an odd dimensional compact manifold is zero. An Euler complex is a combinatorial analogue of a compact manifold. We present here an elementary proof of the corresponding result for Euler…
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…
We prove that that the number p of positive eigenvalues of the connection Laplacian L of a finite abstract simplicial complex G matches the number b of even dimensional simplices in G and that the number n of negative eigenvalues matches…
We use some basic properties of binomial and Stirling numbers to prove that the Euler characteristic is, essentially, the unique numerical topological invariant for compact polyhedra which can be expressed as a linear combination of the…
We study the use of the Euler characteristic for multiparameter topological data analysis. Euler characteristic is a classical, well-understood topological invariant that has appeared in numerous applications, including in the context of…
We present an algorithm which takes as input a closed semi-algebraic set, $S \subset \R^k$, defined by \[ P_1 \leq 0, ..., P_\ell \leq 0, P_i \in \R[X_1,...,X_k], \deg(P_i) \leq 2, \] and computes the Euler-Poincar\'e characteristic of $S$.…
If a real value invariant of compact combinatorial manifolds (with or without boundary) depends only on the number of simplices in each dimension on the manifold, then the invariant is completely determined by Euler characteristics of the…
The aim of this article is to define some new families of the special numbers. These numbers provide some further motivation for computation of combinatorial sums involving binomial coefficients and the Euler kind numbers of negative order.…
A simple and efficient algorithm to numerically compute the genus of surfaces of three-dimensional objects using the Euler characteristic formula is presented. The algorithm applies to objects obtained by thresholding a scalar field in a…
Viewing optimization methods as numerical integrators for ordinary differential equations (ODEs) provides a thought-provoking modern framework for studying accelerated first-order optimizers. In this literature, acceleration is often…
A closed form formula (generating function) for the Euler characteristic of the configuration space of $\scriptstyle n$ particles in a simplicial complex is given.
We prove characterizations of Appell polynomials by means of symmetric property. For these polynomials, we establish a simple linear expression in terms of Bernoulli and Euler polynomials. As applications, we give interesting examples. In…
The purpose of this note is to complete the study, begun in the first author's PhD thesis, of the topology of the poset of generalized noncrossing partitions associated to real reflection groups. In particular, we calculate the Euler…
This study proposes a novel approach to extract topological properties, specifically the Euler characteristic, from input images using neural networks without relying on large pre-existing datasets but with a single geometric image.…
Euler diagrams are a tool for the graphical representation of set relations. Due to their simple way of visualizing elements in the sets by geometric containment, they are easily readable by an inexperienced reader. Euler diagrams where the…