Related papers: Certified Computation of planar Morse-Smale Comple…
Let $\mathcal C$ be a real plane algebraic curve defined by the resultant of two polynomials (resp. by the discriminant of a polynomial). Geometrically such a curve is the projection of the intersection of the surfaces $P(x,y,z)=Q(x,y,z)=0$…
In this paper, we consider the problem of planar graph-based simultaneous localization and mapping (SLAM) that involves both poses of the autonomous agent and positions of observed landmarks. We present CPL-SLAM, an efficient and…
Intending to introduce a method for the topological analysis of fields, we present a pipeline that takes as an input a weighted and based chain complex, produces a factored chain complex, and encodes it as a barcode of tagged intervals…
In this work, we design a nearly linear time discrete Morse theory based algorithm for computing homology groups of 2-manifolds, thereby establishing the fact that computing homology groups of 2-manifolds is remarkably easy. Unlike previous…
Contour integrals of rational functions over ${\cal M}_{0,n}$, the moduli space of $n$-punctured spheres, have recently appeared at the core of the tree-level S-matrix of massless particles in arbitrary dimensions. The contour is determined…
We compute the minimum number of critical points of a small codimension smooth map between two manifolds. We give as well some partial results for the case of higher codimension when the manifolds are spheres.
We construct random Morse functions on surfaces by random walk and compute related distributions. We study the space of Morse functions through these random variables. We consider subspaces characterized by the surfaces with boundary…
We study the task of smoothing a circuit, i.e., ensuring that all children of a plus-gate mention the same variables. Circuits serve as the building blocks of state-of-the-art inference algorithms on discrete probabilistic graphical models…
We investigate properties of the set of discrete Morse functions on a simplicial complex as defined by Forman. It is not difficult to see that the pairings of discrete Morse functions of a finite simplicial complex again form a simplicial…
We consider the problem of approximating a two-dimensional shape contour (or curve segment) using discrete assembly systems, which allow to build geometric structures based on limited sets of node and edge types subject to edge length and…
We present an algorithm for computing the main topological characteristics of three-dimensional bodies. The algorithm is based on a discretization of Morse theory and uses discrete analogs of smooth functions with only nondegenerate (Morse)…
In our previous work, we have constructed explicit smooth real algebraic functions which may have both compact and non-compact preimages on smooth real algebraic manifolds. This paper presents its variant. Our result is new in obtaining…
We rely on the framework of Morse sequences to enable the direct computation of gradient vector fields on simplicial complexes. A Morse sequence is a filtration from a subcomplex $L$ to a complex $K$ via elementary expansions and fillings,…
Inspired by the works of Forman on discrete Morse theory, which is a combinatorial adaptation to cell complexes of classical Morse theory on manifolds, we introduce a discrete analogue of the stratified Morse theory of Goresky and…
The efficient evaluation of high-dimensional integrals is of importance in both theoretical and practical fields of science, such as data science, statistical physics, and machine learning. However, exact computation methods suffer from the…
The interaction between discrete and continuous mathematics lies at the heart of many fundamental problems in applied mathematics and computational sciences. In this paper we discuss the problem of discretizing vector-valued functions…
1) We introduce random discrete Morse theory as a computational scheme to measure the complicatedness of a triangulation. The idea is to try to quantify the frequence of discrete Morse matchings with a certain number of critical cells. Our…
In this paper, we establish strong holomorphic Morse inequalities on non-compact manifolds under the condition of optimal fundamental estimates. We show that optimal fundamental estimates are satisfied and then strong holomorphic Morse…
In this paper, we study semilinear elliptic equations in domains where there is a natural class of solutions, which depend only on one variable, and whose simple geometry reflects the geometry of the domain. We prove that under quite…
An explicit isomorphism between Morse homology and singular homology is constructed via the technique of pseudo-cycles. Given a Morse cycle as a formal sum of critical points of a Morse function, the unstable manifolds for the negative…