Morse Function Research Articles

Elucidating the isotope effect in the Morse approximation of the ground state electronic term of hydrogen molecules nH2, n = 1 ÷ 7. Showcasing the Herzberg anomaly and anharmonicity

The Morse function is the simplest anharmonic approximation of the potential of a diatomic molecule, for which the vibrational Schrödinger equation could be solved almost precisely. Despite its crudeness, the Morse function has been widely used in solving various problems in molecular spectroscopy. In recent years, special attention was paid to the existence of two Morse approximations for the electronic terms U(r) of diatomic molecules, M1(r) and M2(r), which differ by the selection of primary fitting parameters and satisfactorily reproduce different parts of U(r). Some fine features of simple terms U(r), whose form does not differ much from the Morse function, could be highlighted by considering difference functions U(r) − M(r). The set of vibrational levels G(υ), predicted by Morse approximation, can be conveniently characterized by the difference functions Δ1G(υ) and Δ2G(υ), which are, respectively, the energies of vibrational quanta and the rates of their change. The function Δ2G(υ) can be considered as a generalization of the anharmonicity constant, determined experimentally by the position of levels υ = 0,1,2. Finally, analysing the anharmonicity, it is useful to compare the results obtained for various isotopologues, for which U(r) is the same, but the density of vibrational levels is different. The maximal range of isotope effects is expected for hydrogen, which has seven known isotopes, including extremely short-lived ones. In this work, the influence of the reduced mass on the results of the approximation of the electronic ground state potential U(r) by the Morse function is studied for seven isotopologues of hydrogen molecule nH2 (n = 1–7). For each isotopologue, alternative approximations M1(r) and M2(r) are considered. The systematic deviation of M1(r) from U(r) increases with r, reaching 1700–2770 cm−1 (for 7H2–1H2) in the region of the asymptote. The systematic deviation of M2(r) from U(r) decreases at larger distances: it is dome-shaped with a maximum of ca. 1500 cm−1 at r ∼ 2 Å, after which it falls to zero. The M2(r) curves for different isotopologues diverge by less than 20 cm−1. The U(r) − M(r) functions are used to analyse the so-called “Herzberg anomaly” in the region where M1(r) and M2(r) intersect U(r). For example, at around r = 1.08 Å the U(r) − M2(r) function shows anomalous negative values ranging from −125 cm−1 (for 1H2) to −142 cm−1 (for 7H2).

Some Aspects of Differential Topology of Subcartesian Spaces

In this paper, we investigate the differential topological properties of a large class of singular spaces: subcarteisan space. First, a minor further result on the partition of unity for differential spaces is derived. Second, the tubular neighborhood theorem for subcartesian spaces with constant structural dimensions is established. Third, the concept of Morse functions on smooth manifolds is generalized to differential spaces. For subcartesian space with constant structural dimension, a class of examples of Morse functions is provided. With the assumption that the subcartesian space can be embedded as a bounded subset of an Euclidean space, it is proved that any smooth bounded function on this space can be approximated by Morse functions. The infinitesimal stability of Morse functions on subcartesian spaces is studied. Classical results on Morse functions on smooth manifolds can be treated directly as corollaries of our results here.

The discrete flow category: structure and computation

In this article, we use concepts and methods from the theory of simplicial sets to study discrete Morse theory. We focus on the discrete flow category introduced by Vidit Nanda, and investigate its properties in the case where it is defined from a discrete Morse function on a regular CW complex. We design an algorithm to efficiently compute the Hom posets of the discrete flow category in this case. Furthermore, we show that in the special case where the discrete Morse function is defined on a simplicial complex, then each Hom poset has the structure of a face poset of a regular CW complex. Finally, we prove that the spectral sequence associated to the double nerve of the discrete flow category collapses on page 2.

Approximation of the electronic terms of diatomic molecules by the Morse function. The role of anharmonicity. II. Simple terms

This article continues the series of works by the authors on the approximation of the electronic terms of a diatomic molecules and their vibrational structure by the Morse formula, which is the simplest anharmonic approximation of the real term U ( r ) . Depending on the choice of parameters, the approximation has two alternative solutions M 1 ( r ) and M 2 ( r ) , with different patterns of deviations from the real term and its vibrational structure; some typical deviations for M 1 ( r ) and M 2 ( r ) are described. The difference δ ( r ) = U ( r ) − M ( r ) quantitatively shows the changes in the shape of the terms during approximation and can serve as a reference point when choosing its type for solving a specific problem. We introduced an empirical anharmonicity function − 2 ω e x ( v ) , which characterizes the positions of vibrational levels in the potential well; it demonstrates the distortion of the vibrational structure of the term U ( r ) during the approximation. Based on the data from literature, the functions δ ( r ) and − 2 ω e x ( v ) were constructed for more than 20 molecules and a preliminary classification of electronic terms according to their characteristic features is described. Here we present a group of simple terms with minimal deviations from the Morse shape.

High-precision inverse potentials for neutron-proton scattering using piece-wise smooth Morse functions

Abstract In this work, our goal is to construct inverse potentials for various $\ell$-channels of neutron-proton scattering using piece-wise smooth Morse function as a reference. The phase equations for single-channel states and the coupled ones of multi-channel scattering have been solved numerically using the Runge-kutta 5$^{th}$ order method. We employ a piece-wise smooth reference potential comprising three Morse functions as initial input. Leveraging a machine learning-based Genetic Algorithm (GA), we optimize the model parameters to minimize the mean-squared error between simulated and anticipated phase shifts. Remarkably, our approach yields inverse potentials for both single and multi-channel scattering, achieving a convergence to a mean-squared error of $\leq 10^{-3}$. The resulting scattering lengths `$a_0$' and effective ranges `$r$' for $^3S_1$ and $^1S_0$ states, expressed as [$a_0$, $r$], are found to be [5.445(5.424), 1.770(1.760)] $fm$ and [-23.741(-23.749), 2.63(2.81)] $fm$, respectively, in excellent agreement with experimental values. Furthermore, the calculated total scattering cross-sections exhibit a remarkable accord with experimental counterparts, having a percentage error of less than 1$\%$. This computational approach is easily extendible to obtaining interaction potentials for charged particles scattering.

On the shape description of general solids using Morse theory

The automatic shape description of solids is a problem of interest in manufacturing engineering, amongst other related areas. This description can be either geometrical or topological in nature and can be applied to either surfaces or solids (embedded manifolds). Topological descriptions are specially interesting for the problem of shape comparison and retrieval, where one wants to know if a given shape resembles some other known shape. Some popular topological descriptions use Morse theory to study the topology of manifolds and encode their shape characteristics. A Morse function f is defined on the manifold and the manifold’s shape is indirectly studied by studying the behavior of the critical points of f. This family of methods is well defined for surfaces but does not consider the case of solids. In this paper we address the topological description of solids using Morse theory. Our methodology considers three cases: solids without internal boundaries, solids with internal boundaries and thin-walled solids. We present an algorithm to identify topological changes on these solids using the principle of shape decomposition by Morse handles. The presented algorithm deals with Morse functions that produce parallel planar level sets. Future endeavors should consider other candidate functions.

Beltrami fields with Morse proportionality factor

In this work we study Beltrami fields with non-constant proportionality factor on R3. More precisely, we analyze the existence of vector fields X satisfying the equations curl(X)=fX and div(X)=0 for a given f∈C∞(R3) in a neighborhood of a point p∈R3. Since the regular case has been treated previously, we focus on the case where p is a non-degenerate critical point of f. We prove that for a generic Morse function f, the only solution is the trivial one X≡0 (here generic refers to explicit arithmetic properties of the eigenvalues of the Hessian of f at p). Our results stem from the introduction of algebraic obstructions, which are discussed in detail throughout the paper.

Discrete Morse theory segmentation on high-resolution 3D lithic artifacts

Abstract Motivated by the question of understanding the roots of tool making by anatomically modern humans and coexisting Neanderthals in the Paleolithic, a number of shape classification methods have been tested on photographs and drawings of stone tools. Since drawings contain interpretation and photographs fool both human and computational methods by color and shadows on the surface, we propose an approach using 3D datasets as best means for analyzing shape, and rely on first open access repositories on lithic tools. The goal is to not only analyze shape on an artifact level, but allow a more detailed analysis of stone tools on a scar and ridge level. A Morse-Smale complex (MS complex) extracted from the triangular mesh of a 3D model is a reduced skeleton consisting of linked lines on the mesh. Discrete Morse theory makes it possible to obtain such a MS complex from a scalar function. Thus, we begin with Multi-Scale Integral Invariant filtering on the meshes of lithic artifacts, which provides curvature measures for ridges, which are convex, and scars, which are concave. The resulting values on the vertices can be used as our discrete Morse function and the skeleton we get is build up from lines that will coincide with the ridges and, implicitly, contains the scars as enclosed regions of those lines on the mesh. As this requires a few parameters, we provide a graphical user interface (GUI) to allow altering the predefined parameters to quickly find a good result. In addition, a stone tool may have areas that do not belong to the scar/ridge class. These can be masked and we use conforming MS complexes to ensure that the skeleton keeps these areas whole. Finally, results are shown on real and open access datasets. The source code and manually annotated ground truth for the evaluation are provided as Open Access with a Creative Commons license.

Persistent pairs and connectedness in discrete Morse functions on simplicial complex I

In this paper, we study some useful properties of persistent pairs in a discrete Morse function on a simplicial complex K. In case of dim⁡K=1 (i.e., a graph), by using the properties, we characterize strongly connectedness of critical simplices between two distinct discrete Morse functions, and relate the number of such pairs to the Euler characteristic of K.

Complex vs. convex Morse functions and geodesic open books

Suppose that [Formula: see text] is a closed and oriented surface equipped with a Riemannian metric. In the literature, there are three seemingly distinct constructions of open books on the unit (co)tangent bundle of [Formula: see text], having complex, contact and dynamical flavors, respectively. Each one of these constructions is based on either an admissible divide or an ordered Morse function on [Formula: see text]. We show that the resulting open books are pairwise isotopic provided that the ordered Morse function is adapted to the admissible divide on [Formula: see text]. Moreover, we observe that if [Formula: see text] has positive genus, then none of these open books are planar and furthermore, we determine the only cases when they have genus one pages.

MORSE AND CODIMENSION 1 FUNCTIONS ON THE PROJECTIVE PLANE

The topological properties of functions on two-dimensional spaces are described using an invariant called the Reeb graph. In the case of Morse functions on closed oriented two-dimensional manifolds, this serves as a complete topological invariant for homotopy equivalence. For non-oriented two-dimensional manifolds and manifolds with boundaries, additional information is required to construct a complete topological invariant. If a linear order is imposed on the set of vertices, it becomes a complete topological invariant with respect to topological equivalence. The research explores the topological classification of Morse functions and functions of corank 1 on the projective plane. In these cases, the Reeb graph is an oriented rooted tree. An analytical expression is derived for the number of oriented Reeb graphs with a root. For functions of corank 1 and a fixed number of critical points, all possible Reeb graphs are constructed.

Fast Computation of Neck-Like Features.

Locating neck-like features, or locally narrow parts, of a surface is crucial in various applications such as segmentation, shape analysis, path planning, and robotics. Topological methods are often utilized to find the set of shortest loops around handles and tunnels. However, there are abundant neck-like features on genus-0 shapes without any handles. While 3D geometry-aware topological approaches exist to find neck loops, their construction can be cumbersome and may even lead to geometrically wide loops. Thus we propose a "topology-aware geometric approach" to compute the tightest loops around neck features on surfaces, including genus-0 surfaces. Our algorithm starts with a volumetric representation of an input surface and then calculates the distance function of mesh points to the boundary surface as a Morse function. All neck features induce critical points of this Morse function where the Hessian matrix has precisely one positive eigenvalue, i.e., type-2 saddles. As we focus on geometric neck features, we bypass a topological construction such as the Morse-Smale complex or a lower-star filtration. Instead, we directly create a cutting plane through each neck feature. Each resulting loop can then be tightened to form a closed geodesic representation of the neck feature. Moreover, we offer criteria to measure the significance of a neck feature through the evolution of critical points when smoothing the distance function. Furthermore, we speed up the detection process through mesh simplification without compromising the quality of the output loops.

A topology constrained phase field model

This paper presents a topology constrained phase field model based on a variable mobility Cahn-Hilliard equation. Mostly in the level set framework, this issue has been addressed using concepts from digital topology. However, point based specifications can not be effective in phase fields where the interface is represented by a diffuse region. In phase field models, topology constraints are included in the energy as a penalty term. However, there are no studies which provide topological constraints in the Cahn-Hilliard model to our knowledge. In this work, we use the mobility parameter to enforce topology constraints; this allows local, explicit control of topological events, unlike the optimization based solutions. We define topologically critical regions in phase fields based on a topological analysis of phase field evolution. This analysis shows that diffuse layers formed at topological encounters behave like a Morse function and the critical points of these transitional layers give topologically critical points in phase fields. Using this property, we first detect critical points using an efficient image analysis method. Then we detect the transitional region (critical region) around each critical point using a local morphological segmentation algorithm. We define a mobility function which both has a spatial and concentration dependence. This function is designed to reverse the effect of diffuse layers formed at topological encounters and prevent the further development of topological events. Being extracted from phase field values, the mobility map naturally aligns with phase field evolution. Together with the use of an unconditionally stable semi-implicit Fourier spectral method for the variable mobility Cahn-Hilliard equation, an efficient model is provided. Local, explicit control mechanism allows the design of free energy functional for modeling different materials; this extends the use of Cahn-Hilliard equation to applications which require topology control.

Spectral analysis of an open $q$-difference Toda chain with two-sided boundary interactions on the finite integer lattice

A quantum n -particle model consisting of an open q -difference Toda chain with two-sided boundary interactions is placed on a finite integer lattice. The spectrum and eigenbasis are computed by establishing the equivalence with a previously studied q -boson model from which the quantum integrability is inherited. Specifically, the q -boson-Toda correspondence in question yields Bethe Ansatz eigenfunctions in terms of hyperoctahedral Hall–Littlewood polynomials and provides the pertinent solutions of the Bethe Ansatz equations via the global minima of corresponding Yang–Yang-type Morse functions.

Prescribing Morse Scalar Curvatures: Incompatibility of Non Existence

Given a closed manifold of positive Yamabe invariant and for instance positive Morse functions upon it, the conformally prescribed scalar curvature problem raises the question of whether or not such functions can by conformally changing the metric be realized as the scalar curvature of this manifold. As we shall quantify, depending on the shape and structure of such functions, every lack of a solution for some candidate function leads to existence of energetically uniformly bounded solutions for entire classes of related candidate functions.

Topological Inference of the Conley Index

AbstractThe Conley index of an isolated invariant set is a fundamental object in the study of dynamical systems. Here we consider smooth functions on closed submanifolds of Euclidean space and describe a framework for inferring the Conley index of any compact, connected isolated critical set of such a function with high confidence from a sufficiently large finite point sample. The main construction of this paper is a specific index pair which is local to the critical set in question. We establish that these index pairs have positive reach and hence admit a sampling theory for robust homology inference. This allows us to estimate the Conley index, and as a direct consequence, we are also able to estimate the Morse index of any critical point of a Morse function using finitely many local evaluations.

Uniqueness, multiplicity and nondegeneracy of positive solutions to the Lane-Emden problem

In this paper, we study the nearly critical Lane-Emden equations(⁎){−Δu=up−εinΩ,u>0inΩ,u=0on∂Ω, where Ω⊂RN with N≥3, p=N+2N−2 and ε>0 is small. Our main result is that when Ω is a smooth bounded convex domain and the Robin function on Ω is a Morse function, then for small ε the equation (⁎) has a unique solution, which is also nondegenerate. As for non-convex domain, we also obtain exact number of solutions to (⁎) under some conditions.In general, the solutions of (⁎) may blow-up at multiple points a1,⋯,ak of Ω as ε→0. In particular, when Ω is convex, there must be a unique blow-up point (i.e., k=1). In this paper, by using the local Pohozaev identities and blow-up techniques, even having multiple blow-up points (non-convex domain), we can prove that such blow-up solution is unique and nondegenerate. Combining these conclusions, we finally obtain the uniqueness, multiplicity and nondegeneracy of solutions to (⁎).

Bifiltrations and persistence paths for 2–Morse functions

Bifiltrations and persistence paths for 2–Morse functions

Discrete Morse Functions and Watersheds

Discrete Morse Functions and Watersheds

Inverse potentials for all ℓ channels of neutron-proton scattering using reference potential approach

Reference potential approach (RPA) has been successful in obtaining inverse potentials for weakly bound diatomic molecules using the Morse function. In this work, our goal is to construct inverse potentials for all the available ℓ-channels of np-scattering using RPA. Riccati-type phase equations for various ℓ-channels are solved using 5th order Runge-Kutta method to obtain the scattering phase shifts (SPS) in tandem with an optimisation procedure to minimize mean squared error (MSE). Interaction potentials for a total of 18 states have been constructed using only a three parameter Morse interaction model. The obtained MSE is <1% for 1 S 0, 3 P 1 and 3 D 1 channels and <2% for 1 P 1 channel and <0.1% for rest of the 14 channels. Total scattering cross-sections at various lab energies obtained using RPA are found to match well with experimental ones. A complete study of the np-scattering using RPA has been undertaken using Morse function as zeroth reference for the first time.