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Orienteering and Coverage
Published in Yasmina Bestaoui Sebbane, Multi-UAV Planning and Task Allocation, 2020
The Reeb graph can be used as input to the CPP to calculate a Eulerian circuit, which consists of a closed path traversing every cell at least once. The Reeb graph is a construction that originated in Morse theory to study a real-valued function defined on a topological space. The structure of a Morse function can be made explicit by plotting the evolution of the component of the level set. The Reeb graph is a fundamental data structure that encodes the topology of a shape. It is obtained by contracting to a point the connected components of the level-sets (also called contours) of a function defined on a mesh. Reeb graphs can determine whether a surface has been reconstructed correctly, can indicate problem areas and can be used to encode and animate a model. The Reeb graph has been used in various applications to study noisy data, which creates a desire to define a measure of similarity between these structures. A Eulerian circuit can be achieved by doubling selected edges of the Reeb graph, although no edge needs to be duplicated more than once. The Eulerian circuit is the solution of the linear programming problem: Minimizez=∑e∈Ece⋅xe
Topological Analysis of Local Structure in Atomic Systems
Published in Jeffrey P. Simmons, Lawrence F. Drummy, Charles A. Bouman, Marc De Graef, Statistical Methods for Materials Science, 2019
Emanuel A. Lazar, David J. Srolovitz
As topology focuses on studying the connectivity of shapes such as spheres and tori, it is not immediately clear that it would have much relevance to studying sets of discrete points, such as those encountered in studying atomic systems. In what meaningful way can points in space be considered connected? Over the last decade or so, however, powerful tools such as discrete Morse theory and persistent homology [288, 1134] have been developed to analyze data of diverse kinds [360]. Voronoi topology continues in this spirit. In what follows we show how considering the topology of a Voronoi cell can provide keen insight into the manner in which a set of points is arranged in space. In this sense, Voronoi topology forms a bridge between the discrete and continuous, and enables the application of ideas from topology to the study of atomic systems.
Orienteering and Coverage
Published in Yasmina Bestaoui Sebbane, Intelligent Autonomy of Uavs, 2018
The Reeb graph can be used as input to the Chinese postperson problem to calculate a Eulerian circuit, which consists of a closed path traversing every cell at least once. The Reeb graph is a construction which originated in Morse theory to study a real-valued function defined on a topological space. The structure of a Morse function can be made explicit by plotting the evolution of the component of the level set. The Reeb graph is a fundamental data structure that encodes the topology of a shape. It is obtained by contracting to a point the connected components of the level-sets (also called contours) of a function defined on a mesh. Reeb graphs can determine whether a surface has been reconstructed correctly, indicate problem areas, and can be used to encode and animate a model. The Reeb graph has been used in various applications to study noisy data which creates a desire to define a measure of similarity between these structures. A Eulerian circuit can be achieved by doubling selected edges of the Reeb graph, although no edge needs to be duplicated more than once. The Eulerian circuit is the solution of the linear programming problem: Minimizez=∑e∈Ece.xe $$ \begin{aligned} \text{ Minimize} z = \sum _{e \in E}{c_e.x_e} \end{aligned} $$
Index theory and multiple solutions for asymptotically linear wave equation
Published in Applicable Analysis, 2023
This kind of conditions were also widely used in the study of periodic solutions of Hamiltonian systems. By using Morse theory and critical point theory, many authors obtained the existence of nontrivial periodic solutions under a twist condition for the Hamiltonian function in terms of the difference of the Conley-Zehnder index (or Maslov index) at the origin and at infinity. The index theory exhibited as a powerful tool to measure the difference of the nonlinearity at the origin and at infinity. In [9], Ekeland established an index theory for convex linear Hamiltonian system. By the results of Amann, Zehnder and Long [10–13], an index theory for symplectic paths was introduced. In [14, 15], Long and Zhu defined spectral flows for paths of linear operators and defined the relative Morse index between two linear operators, and redefined Maslov index for symplectic paths. Liu in [16] introduced an index theory for symplectic paths using the algebratic methods. Dong [17] developed an index theory for abstract operator equations with compact resolvent. We refer to the books of Ekeland [18] and Long [19] for a more detailed account of the concepts.
Topological classification of global magnetic fields in the solar corona
Published in Dynamical Systems, 2018
A flow f t ∈ G has a self-indexing energy function which is a Morse function with Ωi points of index i, i = 0, 1, 2, 3. By Morse theory [17] the alternating sum |Ω0| − |Ω1| + |Ω2| − |Ω3| equals to the Euler characteristic of , which is 0 since is an odd-dimensional, closed oriented manifold. Hence, |Ω0| − |Ω1| = |Ω3| − |Ω2|. Since f t ∈ G we have |Ω0| = |Ω3|, therefore, |Ω1| = |Ω2|. Thus
Constant sign and nodal solutions for parametric anisotropic (p, 2) -equations
Published in Applicable Analysis, 2023
Nikolaos S. Papageorgiou, Dušan D. Repovš, Calogero Vetro
Let be a bounded domain with a -boundary . In this paper, we study the following anisotropic -equation In this problem, the exponent is Lipschitz continuous and . By , we denote the variable exponent (anisotropic) p-Laplacian, defined by The reaction of the problem is parametric, with being the parameter. The function is measurable in , continuous in . We assume that is -superlinear as () but without satisfying the usual in such cases Ambrosetti-Rabinowitz condition (the AR-condition for short). Our goal is to prove a multiplicity theorem for problem () providing sign information for all the solutions produced. Using variational tools from the critical point theory, together with suitable truncation and comparison techniques and also Morse Theory (critical groups), we show that for all small values of the parameter the problem has at least five nontrivial smooth solutions (four of constant sign and the fifth nodal (sign-changing)).