Surface or shape reconstruction from 3D digitizations performed in planar samplings are frequent in product design, reverse engineering, rapid prototyping, medical and artistic applications, etc. The planar slicing of the object offers an opportunity to recover part of the neighborhood information essential to reconstruct the topological 2-manifold embedded in R3 that approximates the object su...

Let c be a positive integer. In this paper, we use the method of Tzanakis to transform the quartic Thue inequality |x − 4xy − (2c− 2)xy + (4c+ 4)xy − (2c− 1)y| ≤ max { c 4 , 4 } into systems of Pellian equations. Then we find all primitive solutions of this inequality using continued fractions.

In this paper, we study merge trees induced by a discrete Morse function on tree. Given function, provide method to constructing an tree and define new notion of equivalence functions based the We then relate matching number certain invariant Finally, count that can be star graph characterize

For Morse-Smale pairs on a smooth, closed manifold the MorseSmale-Witten chain complex can be defined. The associated Morse homology is isomorphic to the singular homology of the manifold and yields the classical Morse relations for Morse functions. A similar approach can be used to define homological invariants for isolated invariant sets of flows on a smooth manifold, which gives an analogue ...

Exact cellular decompositions represent a robot’s free space by dividing it into regions with simple structure such that the sum of the regions fills the free space. These decompositions have been widely used for path planning between two points, but can be used for mapping and coverage of free spaces. In this paper, we define exact cellular decompositions where critical points of Morse functio...

Exact cellular decompositions represent a robot’s free space by dividing it into regions with simple structure such that the sum of the regions fills the free space. These decompositions have been widely used for path planning between two points, but can be used for mapping and coverage of free spaces. In this paper, we define exact cellular decompositions where critical points of Morse functio...

We propose a new method for representing the topology and geometry of 2D compact surfaces embedded in a three-dimensional space. Topology is modeled through a skeletal graph, where a distance function is employed as a Morse function. Invariance to rigid transformations follows directly from this choice of Morse function. Geometry is captured by modeling the evolution of level curves of the dist...

This paper, as a continuing work of [1], focus on establishing the fact that if we equip a reciprocal multi-agent (RMA) system with a triangulated Laman graph (TLG), then the associated potential function is generically an equivariant Morse function, i.e., there are only finitely many critical orbits each of which is nondegenerate. Though this assumption on the potential function of being an eq...