Related papers: Euler Integration of Gaussian Random Fields and Pe…
The Dirichlet product of functions on a semi-Riemann domain and generalized Euler vector fields, which include the radial, $\bar \partial$-Euler, and the $\bar \partial$-Neumann vector fields, are introduced. The integral means and the…
Persistent homology is a method for computing the topological features present in a given data. Recently, there has been much interest in the integration of persistent homology as a computational step in neural networks or deep learning. In…
It is well known that the Euler characteristic of the cohomology of a complex algebraic variety coincides with the Euler characteristic of its cohomology with compact support. An old result of G. Laumon asserts that a relative version of…
We address the problem of estimating topological features from data in high dimensional Euclidean spaces under the manifold assumption. Our approach is based on the computation of persistent homology of the space of data points endowed with…
This article introduces an algorithm to compute the persistent homology of a filtered complex with various coefficient fields in a single matrix reduction. The algorithm is output-sensitive in the total number of distinct persistent…
We consider suitable weak solutions of 2-dimensional Euler equations on bounded domains, and show that the class of completely random measures is infinitesimally invariant for the dynamics. Space regularity of samples of these random fields…
It is well known that quantum mechanics admits a geometric formulation on the complex projective space as a Kahler manifold. In this paper we consider the notion of mutual information among continuous random variables in relation to the…
We show that recent results on randomized dimension reduction schemes that exploit structural properties of data can be applied in the context of persistent homology. In the spirit of compressed sensing, the dimension reduction is…
Let $\{(X(t), Y(s)): t\in T, s\in S\}$ be an $\mathbb{R}^2$-valued, centered, unit-variance smooth Gaussian vector field, where $T$ and $S$ are compact rectangles in $\mathbb{R}^N$. It is shown that, as $u\to \infty$, the joint excursion…
Tools of Topological Data Analysis provide stable summaries encapsulating the shape of the considered data. Persistent homology, the most standard and well studied data summary, suffers a number of limitations; its computations are hard to…
The notion of the orbifold Euler characteristic came from physics at the end of 80's. There were defined higher order versions of the orbifold Euler characteristic and generalized ("motivic") versions of them. In a previous paper the…
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…
Persistent homology is a common technique in topological data analysis providing geometrical and topological information about the sample space. All this information, known as topological features, is summarized in persistence diagrams, and…
We define a family {$\gamma(P)$} of generalized Euler constants indexed by finite sets of primes $P$ and study their distribution. These arise from partial sums of reciprocals of integers not divisible by any prime in $P$. An apparent…
The purpose of this paper is to derive the anisotropic averaged Euler equations and to study their geometric and analytic properties. These new equations involve the evolution of a mean velocity field and an advected symmetric tensor that…
Computational topology has recently known an important development toward data analysis, giving birth to the field of topological data analysis. Topological persistence, or persistent homology, appears as a fundamental tool in this field.…
Assume that a finite set of points is randomly sampled from a subspace of a metric space. Recent advances in computational topology have provided several approaches to recovering the geometric and topological properties of the underlying…
In this paper we consider probabilistic analogues of some classical integral geometric formulae: Weyl--Steiner tube formulae and the Chern--Federer kinematic fundamental formula. The probabilistic building blocks are smooth, real-valued…
This paper deals with the evaluation of some definite Euler-type integrals in terms of the Wright hypergeometric function. We obtain a theorem on the Wright hypergeometric function and then use this theorem to evaluate some definite…
Generating functions for the number of commuting m-tuples in the symmetric groups are obtained. We define a natural sequence of ``orbifold Euler characteristics'' for a finite group G acting on a manifold X. Our definition generalizes the…