A general method for obtaining minimal interatomic distance in molecule conformation problems is introduced. The method can be applied to a wide family of potential energy functions having reasonable properties. Using this method new lower bounds for the minimal inter-particle distance for the optimal Lennard-Jones and Morse potential functions are derived which are independent from the number ...

We study topology change in (2+1)D gravity coupling with non-Abelian SO(2, 1) Higgs field from the point of view of Morse theory. It is shown that the Higgs potential can be identified as a Morse function. The critical points of the latter (i.e. loci of change of the spacetime topology) coincide with zeros of the Higgs field. In these critical points twodimensional metric becomes degenerate, bu...

Scalar-valued functions are ubiquitous in scientific research. Analysis and visualization of scalar functions defined on low-dimensional and simple domains is a well-understood problem, but complications arise when the domain is high-dimensional or topologically complex. Topological analysis and Morse theory provide tools that are effective in distilling useful information from such difficult s...

Critical points of a scalar function (minima, saddle points and maxima) are important features to characterize large scalar datasets, like topographic data. But the acquisition of such datasets introduces noise in the values. Many critical points are caused by the noise, so there is a need to delete these extra critical points. The Morse-Smale complex is a mathematical object which is studied i...

Morse and Morse-Smale complexes have been recognized as a suitable tool for modeling the topology of a manifold M through a decomposition of M induced by a scalar field f defined over M. We consider here the problem of representing, constructing and simplifying Morse and Morse-Smale complexes in 3D. We first describe and compare two data structures for encoding 3D Morse and Morse-Smale complexe...

Novikov [25] generalized this to multiple-valued functions, i.e. closed 1-forms. In the simplest version of Novikov’s construction, we consider a circle-valued Morse function / :XPS1. There is an analogue of the Morse complex which counts gradient flow lines of /. In this circle-valued case, to obtain a finite count we need to classify flow lines using some information about their homotopy clas...

Let (M, f,G) be a manifold, a function and a Riemann metric on the manifold. Topologists would use these data in order to analyze the manifold by means of Morse theory, that is by studying the dynamical system ẋ = ±∇f . Many recent applications of physics to topology are based on another point of view suggested in E. Witten’s paper Supersymmetry and Morse theory J. Diff. Geom. (1982). Given the...

In this paper we give a new simple proof of a result of Luigi De Pascale, which states that the Morse-Sard Theorem holds under the hypothesis of Sobolev regularity. Moreover, as our proof is independent of the MorseSard Theorem with Ck regularity, our result implies the classical Morse-Sard Theorem. The Morse-Sard Theorem is concerned with the size of the image of the critical values of a diffe...

For an integral parameter t ∈ Z we investigate the family of Thue equations F (x, y) = x + (t− 1)xy − (2t + 4t + 4)xy + (t + t + 2t + 4t − 3)xy + (t + t + 5t+ 3)xy + y = ±1 , originating from Emma Lehmer’s family of quintic fields, and show that for |t| ≥ 3.28 ·1015 the only solutions are the trivial ones with x = 0 or y = 0. Our arguments contain some new ideas in comparison with the standard ...

We prove that a sequence satisfying a certain symmetry property is 2-regular in the sense of Allouche and Shallit, i.e., the Z-module generated by its 2-kernel is finitely generated. We apply this theorem to develop a general approach for studying the `-abelian complexity of 2-automatic sequences. In particular, we prove that the period-doubling word and the Thue–Morse word have 2-abelian compl...