Related papers: Morse functions statistics

This paper was motivated by work of Arnold where he explains how to count "snakes", i.e. Morse functions on the real axis with prescribed behavior at infinity. This leads immediately to a count of excellent Morse functions on the circle,…

We survey recent results and current challenges concerning the growth rate inequality for sphere endomorphisms, and present a number of open problems and conjectures arising in this context.

We give a Morse-theoretic characterization of simple closed geodesics on Riemannian $2$-spheres. On any Riemannian $2$-sphere endowed with a generic metric, we show there exists a simple closed geodesic with Morse index $1$, $2$ and $3$. In…

The partition function and the one- and two-body distribution functions are evaluated for two hard spheres with different sizes constrained into a spherical pore. The equivalent problem for hard disks is addressed too. We establish a…

We present an alternative application of discrete Morse theory for two-particle graph configuration spaces. In contrast to previous constructions, which are based on discrete Morse vector fields, our approach is through Morse functions,…

We construct random Morse functions on surfaces by random walk and compute related distributions. We study the space of Morse functions through these random variables. We consider subspaces characterized by the surfaces with boundary…

We consider different notions of equivalence for Morse functions on the sphere in the context of persistent homology, and introduce new invariants to study these equivalence classes. These new invariants are as simple, but more discerning…

In the 1950s Morse defined the analogue of Morse functions for topological manifolds. In many instances, when mathematicians are using techniques on topological manifolds that appear to be Morse-theoretic in nature, there is a topological…

We show by explicit example that local intersection multiplicities in holomorphic dynamical systems can grow arbitrarily fast, answering a question of V. I. Arnold. On the other hand, we provide results showing that such behavior is…

Sharpened forms of the concentration of measure phenomenon for classes of functions on the sphere are developed in terms of Hessians of these functions.

We investigate the permissible growth rates of functions that are distributionally chaotic with respect to differentiation operators. We improve on the known growth estimates for $D$-distributionally chaotic entire functions, where growth…

Given a smooth bounded domain ${\O}\subseteq \R^2$, we consider the equation $\D v = 2 v_x \wedge v_y$ in $\O$, where $v: {\O}\to \R^3$. We prescribe Dirichlet boundary datum, and consider the case in which this datum converges to zero. An…

We compute lower bounds for the Morse index and nullity of constant mean curvature tori of revolution in the three-dimensional unit sphere. In particular, all such tori have index at least five, with index growing at least linearly with…

We investigate a result on convergence of double sequences of numbers and how it extends to measurable functions.

In this note, we investigate estimates of the Morse index for F-harmonic maps into spheres, our results extend partially those obtained in ([14]) and ([15]) for harmonic and p-harmonic maps.

We consider the problem of two interacting particles on a sphere. The potential of the interaction depends on the distance between the particles. The case of Newtonian-type potentials is studied in most detail. We reduce this system to a…

Using an estimate on the number of critical points for a Morse-even function on the sphere $\mathbb S^m$, $m\ge1$, we prove a multiplicity result for orthogonal geodesic chords in Riemannian manifolds with boundary that are diffeomorphic to…

To investigate the topological structure of Morse functions on the projective plane we use the Reeb graphs. We describe it properties and prove that it is a complete topological invariant of simple Morse function on $\mathbb{R} P^2$. We…

In recent work the author investigates perfect matchings of a bipartite graph obtained from a knot diagram and demonstrates that these correspond to discrete Morse functions on a 2-complex for the 2-sphere. This relationship is expounded…

We study projectively self-dual polygons and curves in the projective plane. Our results provide a partial answer to problem No 1994-17 in the book of Arnold's problems.