The Hairy Ball Problem is PPAD-Complete.

When Brouwer continued his investigations into Hilbert 5, he discovered that his main topology source, Schoenflies Bericht, was far from correct. He set himself to straighten out the defective parts; the best known fall out of this research was his work on indecomposable continua, with the spectacular example of three domains with one common boundary. The chapter also contains the story of Brouwer’s research on fixed points on the sphere and his translation theorem (on fixed point free continuous maps of the plane onto itself). He simultaneously produced a number of papers on vector field on surfaces. The best known result was the hairy ball theorem: a continuous vector field on a sphere must be zero or infinite at at least one point.

Rotating and rolling rigid bodies exemplify a fascinating theorem of topology, jokingly called the “hairy ball” theorem, which demands that any continuous tangent vector field on the sphere has at least one point where the field is zero. We demonstrate via a gedanken experiment how drilling through a rotating ball, thereby converting it into a torus, leads to the elimination of zero-velocity points on the ball surface. Using the same reasoning, zero-velocity points can be removed from the surface of a drilled spinning top. We discuss the location of zero-velocity points on the surfaces of rigid bodies rolling with no slip and with slip. Observations made from different reference frames identify various zero-velocity points. Illustrative experiments visualizing zero-velocity points are presented.

The Klein bottle and hairy ball theorem are important concepts in advanced mathematics and they are both examples of the Poincaré–Hopf theorem. Complex theories such as these usually remain unrealized in the minds of mathematicians. In this collaborative work between a mathematician and a designer, we introduce the concept of the hairy Klein bottle, which is a fusion that explores a continuous non-vanishing vector field on a non-orientable surface. We strive to explain and shape these esoteric concepts accurately and to and describe the modelling process of creating a tangible, touchable sculpture designed to intrigue and invite simple understanding.

We study the solvability and removable singularities of the equation divF=µ , with measure data µ , in the class of continuous or Lp vector fields F , where 1=p=8 . In particular, we show that, for a signed measure µ , the equation divF=µ has a solution F?L8(Rn) if and only if |µ(U)|=CHn-1(?U) for any open set U with smooth boundary. For non-negative measures µ , we obtain explicit characterizations of the solvability of divF=µ in terms of potential energies of µ for p?8 , and in terms of densities of µ for continuous vector fields. These existence results allow us to characterize the removable singularities of the corresponding equation divF=µ with signed measures µ .

In this paper we present an n-dimensional canonical piecewise-linear electrical circuit. It contains 2n two-terminal elements: n linear dynamic elements (capacitors and inductors), n - 1 linear resistors and one nonlinear (piecewise-linear) resistor. This circuit can realize any prescribed eigenvalue pattern, except for a set of measure zero, associated with (i) any n-dimensional two-region continuous piecewise-linear vector fields and (ii) any n-dimensional three-region symmetric (with respect to the origin) piecewise-linear continuous vector fields. We also proved a theorem that specifies the conditions for a vector field, realized with our canonical circuit, to have two or three equilibrium points.

Surface instabilities and patterning at liquid/vapor interfaces: Exemplifications of the “hairy ball theorem”

Several recent works have combined discrete and continuous motion planning methods for robot navigation and control. The basic idea of some of these works is to plan a path, by determining a sequence of neighboring discrete regions of the configuration space, and to assign a vector field that drives the robots through these regions. This paper addresses the problem of efficiently computing vector fields over a sequence of consecutive triangles. Differently from previous numerical approaches, which were not able to compute fully continuous fields in triangulated spaces, this paper presents an algorithm that is able to compute guaranteed continuous vector fields over a sequence of adjacent triangles.

The well-known Cauchy integral decomposition in the plane is a representation of a continuous function on a curve as a sum of two functions analytic inside and outside the curve. These analytic components inherit smoothness of the original function almost independently of the smoothness of the curve itself. Let G be a domain in homeomorphic to a ball with C 1-smooth boundary S. Let F be a continuous vector field on S. When it is possible to represent it as a sumF = F + + F −, where vector fields F ± are harmonic in G and respectively, and smooth of prescribed order s up to the boundary? The following two conditions on F turn out to be necessary and sufficient: 1. The circulation of F along any closed path on S vanishes. 2. For any ball B with radius r centered on S there exists a polynomial harmonic vector field H of degree m = [s] such that on B∩S. This is a multidimensional analog of the Cauchy integral decomposition. The result does not presuppose any extra smoothness of S.

In the present paper we are concerned with the existence of Γ-periodic solutions for the differential equation x(t) = f(t, .x( r)), t e R, where / is a continuous time dependent Γ-periodic tangent vector field defined on an ^-dimensional differentiable manifold M possibly with boundary. We prove that if the Euler characteristic of the average vector field w(p) = (l/T)fίff(t,p)dt is defined and nonzero and if all the possible orbits of the parametrized equation x(t) = λf(ί,x(t)), /eR and λ £ (0,1], lie in a compact set and do not hit the boundary of M, then the given equation admits a Γ-periodic solution. 0. Introduction. Let M be an w-dimensional differentiable manifold, possibly with boundary, and let /: RxM->Γ(M) be a time dependent Γ-periodic tangent vector field on M. In this paper we give a topological result concerning the existence of Γ-periodic solutions for the differential equation

In this work a theorical framework to apply the Poincare compactification technique to locally Lipschitz continuous vector fields is developed. It is proved that these vectors fields are compactifiable in the n-dimensional sphere, though the compactified vector field can be identically null in the equator. Moreover, for a fixed projection to the hemisphere, all the compactifications of a vector field, which are not identically null on the equator are equivalent. Also, the conditions determining the invariance of the equator for the compactified vector field are obtained. Up to the knowledge of the authors, this is the first time that the Poincare compactification of locally Lipschitz continuous vector fields is studied. These results are illustrated applying them to some families of vector fields, like polynomial vector fields, vector fields defined as a sum of homogeneous functions and vector fields defined by piecewise linear functions.

Stabilizability of Bimodal Piecewise Linear Systems With Continuous Vector Field

Many algorithms for scientific visualization and image analysis are rooted in the world of continuous scalar, vector, and tensor fields, but are programmed in low-level languages and libraries that obscure their mathematical foundations. Diderot is a parallel domain-specific language that is designed to bridge this semantic gap by providing the programmer with a high-level, mathematical programming notation that allows direct expression of mathematical concepts in code. Furthermore, Diderot provides parallel performance that takes advantage of modern multicore processors and GPUs. The high-level notation allows a concise and natural expression of the algorithms and the parallelism allows efficient execution on real-world datasets.

Geometric, coordinate-free approaches are widely used to control quadrotors on the special Euclidean group (SE(3)). These approaches rely on the construction of an element of the special orthogonal group (SO(3)) from a desired thrust vector direction which lies on a sphere (\(S^2\)) and a desired yaw angle which lies on a circle (\(S^1\)). The Hairy Ball Theorem can be applied to show that any construction of this type has to be discontinuous or degenerate somewhere. We propose a new geometric control algorithm based on the Hopf fibration, which allows us to place the point of discontinuity as far away from the hover configuration as possible and further than existing approaches. We then use multiple maps from \(S^2\times S^1\) to SO(3) to be able to control the quadrotor through any position and orientation. The proposed Hopf Fibration Control Algorithm (HFCA) is compared to existing geometric control algorithms in experiments and simulation. The HFCA employs multiple charts to allow the quadrotor to execute arbitrary dynamically feasible trajectories on SE(3), including those through configurations in which the vehicle is inverted.

(1978). Analytic Proofs of the “Hairy Ball Theorem” and the Brouwer Fixed Point Theorem. The American Mathematical Monthly: Vol. 85, No. 7, pp. 521-524.

Nontangential a.e. vanishing of the oblique derivative of a harmonic function with respect to a Hölder continuous vector field on a Lipschitz boundary is shown to imply that the harmonic function is constant.