Morse Function Research Articles

Uryson width of three dimensional mean convex domain with non-negative Ricci curvature

We prove that for every three dimensional manifold with nonnegative Ricci curvature and strictly mean convex boundary (non-empty), there exists a Morse function so that each connected component of its level sets has a uniform diameter bound, which depends only on the lower bound of mean curvature. This gives an upper bound of Uryson 1-width for those three manifolds with boundary.

Morse functions and contact convex surfaces

Let f be a Morse function on a closed surface Σ such that zero is a regular value and such that f admits neither positive minima nor negative maxima. In this note, we show that Σ×R admits an R-invariant contact form α=fdt+β whose characteristic foliation along the zero section is (negative) weakly gradient-like with respect to f. The proof is self-contained and gives explicit constructions of any R-invariant contact structure in Σ×R, up to isotopy. It provides evidence for these functions to be eigenfunctions of the Laplacian for some metric on the surface. As an application, we give an alternative geometric proof of the homotopy classification of R-invariant contact structures in terms of their dividing set.

Accuracy of the new modified Morse potential energy function for ground and excited states of diatomic molecules

This paper presents a detailed study on the performance of the new modified Morse potential in representing the vibrational motion of several diatomic molecules. The matrix Numerov method has been used to solve the time-independent Schrödinger equation for the new modified Morse potential, Morse potential, and Varshni potential. The vibrational energy eigenvalues of the 20 diatomic molecules in various electronic states were calculated. The vibrational energy eigenvalues and anharmonicity constants calculated using the new modified Morse potential were closer to the experimental values for all 20 molecules than the eigenvalues calculated from the Morse and Varshni potential functions.

A Variational Loop Shrinking Analogy for Handle and Tunnel Detection and Reeb Graph Construction on Surfaces

AbstractThe humble loop shrinking property played a central role in the inception of modern topology but it has been eclipsed by more abstract algebraic formalisms. This is particularly true in the context of detecting relevant non‐contractible loops on surfaces where elaborate homological and/or graph theoretical constructs are favored in algorithmic solutions. In this work, we devise a variational analogy to the loop shrinking property and show that it yields a simple, intuitive, yet powerful solution allowing a streamlined treatment of the problem of handle and tunnel loop detection. Our formalization tracks the evolution of a diffusion front randomly initiated on a single location on the surface. Capitalizing on a diffuse interface representation combined with a set of rules for concurrent front interactions, we develop a dynamic data structure for tracking the evolution on the surface encoded as a sparse matrix which serves for performing both diffusion numerics and loop detection and acts as the workhorse of our fully parallel implementation. The substantiated results suggest our approach outperforms state of the art and robustly copes with highly detailed geometric models. As a byproduct, our approach can be used to construct Reeb graphs by diffusion thus avoiding commonly encountered issues when using Morse functions.

Witten Complex of Transitive Digraph and Its Convergence

Digraphs are generalization of graphs in which each edge is given one or two directions. For each digraph, there exists a transitive digraph containing it. Moreover, all the formal linear combinations of allowed elementary paths form a basis of the path complex for a transitive digraph. Hence, the study of discrete Morse theory on transitive digraphs is very important for the further study of discrete Morse theory on general digraphs. As we know, the definition of discrete Morse function on a digraph is different from that on a simplical complex or a cell complex: each discrete Morse function on a digraph is a discrete flat Witten-Morse function. In this paper, we deform the usual boundary operator, replacing it with a boundary operator with parameters and consider the induced Laplace operators. In addition, we consider the eigenvectors of the eigenvalues of the Laplace operator that approach to zero when the parameters approach infinity, define the generation space of these eigenvectors, and further give the Witten complex of digraphs. Finally, we prove that for a transitive digraph, Witten complex approaches to its Morse complex. However, for general digraphs, the structure of Morse complex is not as simple as that of transitive digraphs and the critical path is not directly related to the eigenvector with zero eigenvalue of Laplace operator. This is explained in the last part of the paper.

Study on Intelligent Analysis and Processing Technology of Computer Big Data Based on Clustering Algorithm

Aim: Clustering belongs to unsupervised learning, which divides the data objects into the data set into multiple clusters or classes, so that the objects in the same cluster have high similarity Background: The clustering of spatial data objects can be solved by optimization based on the clustering objective function. Objective: Study on intelligent analysis and processing technology of computer big data based on clustering algorithm. Methods: First, a new dynamic self-organizing feature mapping model is proposed, and the training algorithm of the model is given. Then, the spectral clustering technology and related concepts are introduced. The spectral clustering algorithm is studied and analyzed, and a spectral clustering algorithm that automatically determines the number of clusters is proposed. Furthermore, an algorithm for constructing a discrete Morse function to find the optimal solution is proposed, proving that the constructed function is the optimal discrete Morse function. At the same time, two optimization models based on the discrete Morse theory are constructed. Finally, the optimization model based on discrete Morse theory is applied to cluster analysis, and a density clustering algorithm based on the discrete Morse optimization model is proposed. Results: This study is focused on designing and implementing a partitional-based clustering algorithm based on big data, that is suitable for clustering huge datasets to meet low computational requirements. The experiments are conducted in terms of time and space complexity and it is observed that the measure of clustering quality and the run time is capable of running in very less time without negotiating the quality of clustering. The results show that the experiments are carried out on the artificial data set and the UCI data set. Conclusion: The efficiency and superiority of the new model, are verified by comparing it with the clustering results of the DBSCAN algorithm.

The fixed point set of the inverse involution on a Lie group

The inverse involution on a Lie group $G$ is the periodic $2$ transformation $\gamma $ that sends each element $g\in G$ to its inverse $g^{-1}$. The variety of the fixed point set $\Fix(\gamma )$ is of importance for the relevances with Morse function on the Lie group $G$, and the $G$-representations of the cyclic group $\mathbb{Z}_{2}$. In this paper we develop an approach to calculate the diffeomorphism types of the fixed point sets $\Fix(\gamma)$ for the simple Lie groups.

Flattening knotted surfaces

A knotted surface in $$S^{4}$$ may be described by means of a hyperbolic diagram that captures the 0-section of a special Morse function, called a hyperbolic decomposition. We show that every hyperbolic decomposition of a knotted surface $${\mathcal {K}}$$ defines a projection of $${\mathcal {K}}$$ onto a 2-sphere $$\Sigma $$ , whose set of critical values is the hyperbolic diagram of $${\mathcal {K}}$$ . We apply such projections, called flattenings, to define three invariants of knotted surfaces: the layering, the trunk, and the partition number. The basic properties of flattenings and their derived invariants are obtained. Our construction is used to study flattenings of satellite 2-knots.

Integration of vector fields on cell complexes and Morse theory

In this paper, we investigate vector fields on polyhedral complexes and their associated trajectories. We study vector fields which are analogues of the gradient vector field of a function in the smooth case. Our goal is to define a nice theory of trajectories of such vector fields, so that the set of them captures the topology of the polyhedral complex, as in classical Morse theory. Since we do not assume the polyhedral complex to be a manifold, the definition of vector fields on it is very different from the smooth case. Nevertheless, we will show that there exist nice classes of functions and metrics which give gradient vector fields with desired properties. Our construction relies on Forman's discrete Morse theory. In particular, the class of functions we use is an improvement of Forman's discrete Morse functions. A notable feature of our theory is that our gradient vector fields are defined purely from functions and metrics as in the smooth case, contrary to the case of discrete Morse theory where we need the data of dimension of cells. This allows us to implement several useful constructions which were not available in the discrete case.

Computing Persistent Homology by Spanning Trees and Critical Simplices.

Topological data analysis can extract effective information from higher-dimensional data. Its mathematical basis is persistent homology. The persistent homology can calculate topological features at different spatiotemporal scales of the dataset, that is, establishing the integrated taxonomic relation among points, lines, and simplices. Here, the simplicial network composed of all-order simplices in a simplicial complex is essential. Because the sequence of nested simplicial subnetworks can be regarded as a discrete Morse function from the simplicial network to real values, a method based on the concept of critical simplices can be developed by searching all-order spanning trees. Employing this new method, not only the Morse function values with the theoretical minimum number of critical simplices can be obtained, but also the Betti numbers and composition of all-order cavities in the simplicial network can be calculated quickly. Finally, this method is used to analyze some examples and compared with other methods, showing its effectiveness and feasibility.

Analytic approach to S1–equivariant Morse inequalities

It is well known that the cohomology groups of a closed manifold $M$ can be reconstructed using the gradient dynamical of a Morse-Smale function $f\colon M\to \R$. A direct result of this construction are Morse inequalities that provide lower bounds for the number of critical points of $f$ in term of Betti numbers of $M$. These inequalities can be deduced through a purely analytic method by studying the asymptotic behaviour of the deformed Laplacian operator. This method was introduced by E. Witten and has inspired a numbers of great achievements in Geometry and Topology in few past decades. In this paper, adopting the Witten approach, we provide an analytic proof for; the so called; equivariant Morse inequalities when the underlying manifold is acted on by the Lie group $G=S^1$ and the Morse function $f$ is invariant with respect to this action.

On merge trees and discrete Morse functions on paths and trees

In this work we answer an open question asked by Johnson–Scoville. We show that each merge tree is represented by a discrete Morse function on a path. Furthermore, we present explicit constructions for two different but related kinds of discrete Morse functions on paths that induce any given merge tree. A refinement of the used methods allows us to define notions of equivalence of discrete Morse functions on trees which give rise to a bijection between equivalence classes of discrete Morse functions and isomorphism classes of certain labeled merge trees. We also compare our results to similar ones from the literature, in particular to work by Curry.

Simplicity of Augmentations of Codimension 1 Germs and by Morse Functions

We study the simplicity of map-germs obtained by the operation of augmentation and describe how to obtain their versal unfoldings. When the augmentation comes from an {mathscr {A}}_e-codimension 1 germ or the augmenting function is a Morse function, we give a complete characterisation for simplicity. These characterisations yield all the simple augmentations in all explicitly obtained classifications of {mathscr {A}}-simple monogerms except for one (F_4 in Mond’s list from mathbb C^2 to mathbb C^3). Moreover, using our results we produce a list of corank 1 simple augmentations from mathbb C^4 to mathbb C^4.

Homotopic Distance and Generalized Motion Planning

We prove that the homotopic distance between two maps defined on a manifold is bounded above by the sum of their subspace distances on the critical submanifolds of any Morse–Bott function. This generalizes the Lusternik–Schnirelmann theorem (for Morse functions) and a similar result by Farber for the topological complexity. Analogously, we prove that, for analytic manifolds, the homotopic distance is bounded by the sum of the subspace distances on any submanifold and its cut locus. As an application, we show how navigation functions can be used to solve a generalized motion planning problem.

Fiber of persistent homology on morse functions

Let f be a Morse function on a smooth compact manifold M with boundary. The path component mathrm {PH}^{-1}_f(D) containing f of the space of Morse functions giving rise to the same Persistent Homology D=mathrm {PH}(f) is shown to be the same as the orbit of f under pre-composition phi mapsto fcirc phi by diffeomorphisms of M which are isotopic to the identity. Consequently we derive topological properties of the fiber mathrm {PH}^{-1}_f(D): In particular we compute its homotopy type for many compact surfaces M. In the 1-dimensional settings where M is the unit interval or the circle we extend the analysis to continuous functions and show that the fibers are made of contractible and circular components respectively.

Influence of Temperature and Pressure on the Lattice Constant of SrTiO3 Perovskite by the Statistical Moment Method with Improved Interatomic Potential

Temperature and pressure dependence of lattice constants of a cubic Strontium Titanate (SrTiO3) has been investigated using the statistical moment method. The lattice constants at various temperatures is derived in closed analytic form by including the anharmonic effects of the lattice vibrations explicitly. The potential with the partial charge model and Morse function is used. The numerical lattice constants at high temperatures by the statistical moment method are in good and reasonable agreement with the other theories and the experimental data. Variations of the lattice parameter of SrTiO3 with the temperature are obtained at 1 atm, 8.2 GPa, and 15.2 GPa. Increasing of the lattice constants with increasing temperature is due to enhancing atomic anharmonic fluctuations in SrTiO3 lattice crystal at higher temperature. A decrease of the lattice constants with increasing pressure can be demonstrated by the reduction of atomic vibrations in SrTiO3 crystal lattice at higher pressures.

Are Daojing and Dejing stylistically independent of each other: A stylometric analysis with activity and descriptivity

Abstract It has been qualitatively evidenced that Daojing and Dejing, the two sections of DaoDejing, are independent of each other in content, style, and authorship. However, this claim remains controversial because there has been scant quantitative study of them that probes into their syntactic structure to quantitatively evidence the independence due to methodological challenges to quantitative approaches. To address this problem, the present study employed two indexes of quantitative linguistics, activity and descriptivity, to capture the syntactic features of the inner structure of Daojing and Dejing to validate the independence as of the above. Results of this study reveal that (1) the chi-square test shows that both sections are significantly active texts; (2) the u-test and nonparametric two-related-sample test show that the two texts do not differ from each other significantly either holistically or by the 300-character segments in terms of activity and descriptivity; and (3) the activity sequences (Q-sequence) in the two sections can be both better fitted by two different functions, viz. the beta-function and the Morse function. Moreover, with the development of the text, a synergic relationship is found in the distribution of verbs and adjectives, indicative of a dynamic and complex self-regulating process of the text development. To conclude, contrary to the claim of prior studies that they were written by different authors, our stylometric study has quantitatively demonstrated that Daojing and Dejing are not independent of each other stylistically, especially in terms of activity and descriptivity.

Circuit Complexity from Supersymmetric Quantum Field Theory with Morse Function

Computation of circuit complexity has gained much attention in the theoretical physics community in recent times, to gain insights into the chaotic features and random fluctuations of fields in the quantum regime. Recent studies of circuit complexity take inspiration from Nielsen’s geometric approach, which is based on the idea of optimal quantum control in which a cost function is introduced for the various possible path to determine the optimum circuit. In this paper, we study the relationship between the circuit complexity and Morse theory within the framework of algebraic topology, which will then help us study circuit complexity in supersymmetric quantum field theory describing both simple and inverted harmonic oscillators up to higher orders of quantum corrections. We will restrict ourselves to N=1 supersymmetry with one fermionic generator Qα. The expression of circuit complexity in quantum regime would then be given by the Hessian of the Morse function in supersymmetric quantum field theory. We also provide technical proof of the well known universal connecting relation between quantum chaos and circuit complexity of the supersymmetric quantum field theories, using the general description of Morse theory.

Discrete Morse Theory on Join of Digraphs

For given two digraphs, we can construct a larger digraph through join. The two digraphs that make up the join are called the factors of the join. In this paper, we give a necessary and sufficient condition that the function on the join determined by the discrete Morse functions on factors is a discrete Morse function. Moreover, we further prove the discrete Morse theory on join when the factors satisfy certain conditions.

Merge trees in discrete Morse theory

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