Related papers: Morse index of a cyclic polygon
Morse theoretical ideas are applied to the study of relative equilibria in the planar $n$-vortex problem. For the case of positive circulations, we prove that the Morse index of a critical point of the Hamiltonian restricted to a level…
We construct Morse-Smale-Witten complex for an effective orientable orbifold. For a global quotient orbifold, we also construct a Morse-Bott complex. We show that certain type of critical points of a Morse function has to be discarded to…
We will discuss a method for visual presentation of knotted surfaces in the four space, by examining a number and a position of its Morse's critical points. Using this method, we will investigate surface-knot with one critical point of…
We examine the conditions under which a signed graph contains an edge or a vertex that is contained in a unique negative circle or a unique positive circle. For an edge in a unique signed circle, the positive and negative case require the…
This paper presents a general method for obtaining radius of the corresponding circumference to a cyclical polygon $n$ sides given the lengths of said sides, using the notion of complex number. As of radius $r$, obtained, can then be…
This contains a new version of the so-called non-commutative Gauss algorithm for polycyclic groups. Its results allow to read off the order and the index of a subgroup in an (possibly infinite) polycyclic group.
The classical approach to the study of dynamical systems consists in representing the dynamics of the system in the form of a "source-sink", that means identifying an attractor-repeller pair, which are attractor-repellent sets for all other…
The basic object of the paper is the moduli space $M_{2,3}(L)$ of a closed polygonal linkage either in $\mathbb{R}^2$ or in $\mathbb{R}^3$. As was originally suggested by G. Khimshiashvili, the space $M_{2}(L)$ is equipped with the oriented…
A $configuration$ of a linkage $\Gamma$ is a possible positioning of $\Gamma$ in $\mathbb{R}^d$ and the collection of all such forms the configuration space $\mathcal{C}(\Gamma)$ of $\Gamma$. We here introduce the notion of the $symmetric…
Signed Minkowski decomposition is an expression of a polytope as a Minkowski sum and difference of smaller polytopes. Signed Minkowski decompositions of a polytope can be interpreted as factorizations of a max-plus (tropical) function. We…
We study a two dimensional analogue of the Roe-Higson index theorem for a partitioned manifold. We prove that Connes' pairing of some invertible element with Roe's cyclic one-cocycle coincides to the Fredholm index of a Toeplitz operator.…
Recently, Mullins calculated the Casson-Walker invariant of the 2-fold cyclic branched cover of an oriented link in S^3 in terms of its Jones polynomial and its signature, under the assumption that the 2-fold branched cover is a rational…
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.
The (negative) gradient vector fields of Morse functions on a compact manifold provide an important example in dynamical system. In this note we prove two important properties of this kind of vector field: Connectedness of critical points…
We provide a recursive construction of an acyclic matching (also known as a gradient vector field, an equivalent notion to a discrete Morse function) on the independence complex of a graph with a simplicial vertex using given acyclic…
Polymatroids can be considered as "fractional matroid" where the rank function is not required to be integer valued. Many, but not every notion in matroid terminology translates naturally to polymatroids. Defining cyclic flats of a…
We prove that the index of a CMC surface with capillary boundary is bounded from above linearly by its genus, number of boundary components, and branching order, and also by some Willmore-type energy involving the area, mean curvature,…
Index maps taking values in the $K$-theory of a mapping cone are defined and discussed. The resulting index theorem can be viewed in analogy with the Freed-Melrose index theorem. The framework of geometric $K$-homology is used in a…
We prove upper bounds for the Morse index and number of intersections of min-max geodesics achieving the $p$-widths of a closed surface. A key tool in our analysis is a proof that for a generic set of metrics, the tangent cone at any vertex…
In his study of periodic orbits of the 3 body problem, Hill obtained a formula relating the characteristic polynomial of the monodromy matrix of a periodic orbit and an infinite determinant of the Hessian of the action functional. A…