Hairy Ball Research Articles

Isotropic antenna based on Rydberg atoms.

Governed by the hairy ball theorem, classical antennas with isotropic responses to linearly polarized radio waves are unrealizable. Also, their calibrations face a causal dilemma. Therefore, radio wave measurements based on classical antennas are challenging to achieve high accuracy. This work shows that the antenna based on Rydberg atoms can theoretically achieve an ideal isotropic response to linearly polarized radio waves; that is, it has zero isotropic deviation. Although this conclusion is straightforward, it is not theoretically clear when complex atomic energy levels are taken into account. Experimental results of isotropic deviation within 5 dB and 0.3 dB possible with optimization in microwave and terahertz wave measurements support the theory and is at least 15 dB improvement than the classical omnidirectional antenna. Combined with the SI traceable and ultrawideband property, the ideal isotropic response will make radio wave measurement based on atomic antenna much more accurate and reliable than the traditional method. This isotropic atomic antenna is an excellent example of what a tailored quantum sensor can realize, but a classical sensor cannot. It has crucial applications in fields such as radio wave electrometry.

Detecting Rewiring Events in Protein-Protein Interaction Networks Based on Transcriptomic Data.

Proteins rarely carry out their cellular functions in isolation. Instead, eukaryotic proteins engage in about six interactions with other proteins on average. The aggregated protein interactome of an organism forms a “hairy ball”-type protein-protein interaction (PPI) network. Yet, in a typical human cell, only about half of all proteins are expressed at a particular time. Hence, it has become common practice to prune the full PPI network to the subset of expressed proteins. If RNAseq data is available, one can further resolve the specific protein isoforms present in a cell or tissue. Here, we review various approaches, software tools and webservices that enable users to construct context-specific or tissue-specific PPI networks and how these are rewired between two cellular conditions. We illustrate their different functionalities on the example of the interactions involving the human TNR6 protein. In an outlook, we describe how PPI networks may be integrated with epigenetic data or with data on the activity of splicing factors.

The Hairy Ball problem is PPAD-complete

The Hairy Ball Theorem states that every continuous tangent vector field on an even-dimensional sphere must have a zero. We prove that the associated computational problem of (a) computing an approximate zero is PPAD-complete, and (b) computing an exact zero is FIXP-hard. We also consider the Hairy Ball Theorem on toroidal instead of spherical domains and show that the approximate problem remains PPAD-complete. On a conceptual level, our PPAD-membership results are particularly interesting, because they heavily rely on the investigation of multiple-source variants of End-of-Line, the canonical PPAD-complete problem. Our results on these new End-of-Line variants are of independent interest and provide new tools for showing membership in PPAD. In particular, we use them to provide the first full proof of PPAD-completeness for the Imbalance problem defined by Beame et al. in 1998.

Topology of eeg wave fronts

Whenever one attempts to comb a hairy ball flat, there will always be at least one tuft of hair at one point on the ball. This seemingly worthless sentence is an informal description of the hairy ball theorem, an invaluable mathematical weapon that has been proven useful to describe a variety of physical/biological processes/phenomena in terms of topology, rather than classical cause/effect relationships. In this paper we will focus on the electrical brain field-electroencephalogram (EEG). As a starting point we consider the recently-raised observation that, when electromagnetic oscillations propagate with a spherical wave front, there must be at least one point of the tangential components of the vector fields where the electromagnetic field vanishes. We show how this description holds also for the electric waves produced by the brain and detectable by EEG. Once located these zero-points in EEG traces, we confirm that they are able to modify the electric wave fronts detectable in the brain. This sheds new light on the functional features of a nonlinear, metastable nervous system at the edge of chaos, based on the neuroscientific model of Operational Architectonics of brain-mind functioning. As an example of practical application of this theorem, we provide testable previsions, suggesting the proper location of transcranial magnetic stimulation's coils to improve the clinical outcomes of drug-resistant epilepsy.

Using design to develop an in-depth understanding of mathematical surfaces: The hairy Klein bottle

ABSTRACT 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.

Towards Robust Direction Invariance in Character Animation

AbstractIn character animation, direction invariance is a desirable property. That is, a pose facing north and the same pose facing south are considered the same; a character that can walk to the north is expected to be able to walk to the south in a similar style. To achieve such direction invariance, the current practice is to remove the facing direction's rotation around the vertical axis before further processing. Such a scheme, however, is not robust for rotational behaviors in the sagittal plane. In search of a smooth scheme to achieve direction invariance, we prove that in general a singularity free scheme does not exist. We further connect the problem with the hairy ball theorem, which is better‐known to the graphics community. Due to the nonexistence of a singularity free scheme, a general solution does not exist and we propose a remedy by using a properly‐chosen motion direction that can avoid singularities for specific motions at hand. We perform comparative studies using two deep‐learning based methods, one builds kinematic motion representations and the other learns physics‐based controls. The results show that with our robust direction invariant features, both methods can achieve better results in terms of learning speed and/or final quality. We hope this paper can not only boost performance for character animation methods, but also help related communities currently not fully aware of the direction invariance problem to achieve more robust results.

Motion of the liquid on the surface of Leidenfrost droplets and the hairy ball theorem

Leidenfrost droplets are non-stick droplets, rapidly self-propelling above the hot surface. Leidenfrost effect is a phenomenon in which a liquid, in close contact with a mass significantly hotter than the liquid's boiling point, produces an insulating vapor layer which keeps that liquid from boiling rapidly. The field of the liquid velocities arising on the surface of Leidenfrost droplets is considered. The pattern of surface velocities demonstrates the zero points predicted by the Poincaré–Brouwer theorem for surfaces possessing the Euler characteristic χ = 2. Evaporation of droplets does not change their topology. Zero-velocity points appear when the Leidenfrost droplets are at rest and when they are self-propelled. Topological transformations of bouncing Leidenfrost-like non-stick droplets (called liquid marbles) converted under high values of Weber numbers into tori (χ = 0) results in disappearing of zero-velocity points in their surfaces. Zero-velocity points should also necessarily appear at the surfaces of rolling droplets, placed on superhydrophobic surfaces or liquid marbles.

Making the world from topological order.

Topology used to be a term confined to a branch of pure mathematics, where it referred to an invariant property of shape. The classic example was the way objects containing a single hole, like a torus and a coffee cup with handle, can be smoothly moulded into one another without tearing. But topological considerations have long played a role in the physics of matter, where for example they might dictate particular arrangements of component parts that can’t be erased from the system. The classic example here is the fact that a ‘hairy ball’ can’t be combed flat without having at least two pointy tufts. Such ‘defects’ in organization can be considered ‘topologically protected’, since they are robust against any recombing of the hair. They are universal features that don’t depend on the material specifics of the system: topological defects in liquid crystals are analogous to defects in spacetime called cosmic strings. In the past several decades in particular, properties of matter arising from topological considerations have become a major theme, reflected for example in the award of the 1985 and 1998 Nobel Prizes in Physics for discoveries involving the quantum Hall effect. Here the ‘Hall conductance’, quantifying the passage of electrical current in a 2D conductor in the presence of a transverse magnetic field, takes precise integral or fractional multiples of a particular quantized value related to the electron charge. This behaviour persists regardless of how we modify the material, for example by adding impurities. Topological phases and transitions were also a feature of the work that won the 2016 Nobel Prize. It has become recognized that the topological properties of the quantum-mechanical electronic structures of certain materials can give them unusual and perhaps useful properties. Some researchers think, for example, that ‘topological matter’ might supply quantum bits for quantum computation that resist the randomizing effects of noise. Xiao-Gang Wen of the Massachusetts Institute of Technology has been developing ideas about ‘topological order’ in fundamental physics for several decades. His notions of how topology in the underlying structure of spacetime might give rise to fundamental particles and forces make a connection to the topological phases recognized in condensed matter, revealing a new unifying principle in physics. National Science Review spoke to him about his work.

Another Short Proof of the Hairy Ball Theorem

We show how the assumption of the existence of a continuous unit tangent vector field on the sphere leads to an explicit formula for a homotopy between curves of winding number 1 and − 1 about the origin, thus proving the hairy ball theorem by contradiction.

Rotating and rolling rigid bodies and the “hairy ball” theorem

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.

Obstructions imposed by the Poincaré–Brouwer (“hairy ball”) theorem on the propagation of electromagnetic waves

The Poincaré–Brouwer (“hairy ball”) theorem is applied for the analysis of the propagation of electromagnetic waves, in the case when the wavefront forms a surface, topologically equivalent to a sphere (the surface possessing the Euler characteristic χ = 2). At least one point on the surface at which vectors of electric and magnetic fields equal zero will appear. The Poynting vector in this point is also zero. Restrictions imposed by the Poincaré–Brouwer theorem on the reflection and refraction of electromagnetic waves are discussed, in the case when the media are separated by the interface topologically equivalent to a sphere.

An Extremely Short Proof of the Hairy Ball Theorem

Using winding numbers, we give an extremely short proof that every continuous field of tangent vectors on S2 must vanish somewhere.

Geometric optics and the “hairy ball theorem”

Applications of the hairy ball theorem to the geometrical optics are discussed. When the ideal mirror, topologically equivalent to a sphere, is illuminated at every point, the “hairy ball theorem” prescribes the existence of at least one point at which the incident light will be normally reflected. For the more general case of the surface, topologically equivalent to a sphere, which is both reflecting and refracting the “hairy ball theorem” predicts the existence of at least one point, at which the incident light will be normally reflected and also normally refracted.

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

Application of the “hairy ball theorem” to the analysis of the surface instabilities inherent for liquid/vapor interfaces is reported. When a continuous tangential velocity field exists on the surface of the liquid sample which is homeomorphic to a ball, zero velocity points will be necessarily present at the surface. The theorem is exemplified with the analysis of the instability occurring under the rapid evaporation of polymer solutions. Zero velocity points, accumulating pores, enable direct visualization of the instability. The patterning may be essentially different on the surface of a torus.

Active swarms on a sphere.

We show that coupling to curvature nontrivially affects collective motion in active systems, leading to motion patterns not observed in flat space. Using numerical simulations, we study a model of self-propelled particles with polar alignment and soft repulsion confined to move on the surface of a sphere. We observe a variety of motion patterns with the main hallmarks being polar vortex and circulating band states arising due to the incompatibility between spherical topology and uniform motion-a consequence of the "hairy ball" theorem. We provide a detailed analysis of density, velocity, pressure, and stress profiles in the circulating band state. In addition, we present analytical results for a simplified model of collective motion on the sphere showing that frustration due to curvature leads to stable elastic distortions storing energy in the band.

Probing the anisotropic vortex lattice in the Fe-based superconductor KFe2As2using small-angle neutron scattering

Using small angle neutron scattering, the anisotropy of the magnetic vortex lattice (VL), in the heavily hole-doped pnictide superconductor ${\text{KFe}}_{2}{\text{As}}_{2}$, was studied. Well-ordered VL scattering patterns were measured with fields applied in directions between $\mathbf{B}\ensuremath{\parallel}\mathbf{c}$ and the basal plane, rotating either towards $[100]$ or $[110]$. Slightly distorted hexagonal patterns were observed when $\mathbf{B}\ensuremath{\parallel}\mathbf{c}$. However, the scattering pattern distorted more strongly as the field was rotated away from the c axis. At low field, the arrangement of vortices is affected by the anisotropy of penetration depth in the plane perpendicular to the field. By fitting the distortion with the anisotropic London model, we obtain an estimate of $\ensuremath{\sim}$$3.4$ for the anisotropy factor $\ensuremath{\gamma}$ between the in-plane and c-axis penetration depths at the lowest temperature studied. The results further reveal VL phase transitions as a function of field direction. We discuss these transitions using the ``hairy ball'' theorem.

The Pi Day Massacre

I was in bed, dreaming about a theorem, when the incessant ringing of my phone dragged me back to reality. ‘‘Hello?’’ I said groggily. ‘‘Yeah, Mangum, it’s Schmishschmitt. I got a case for you.’’ Schmishschmitt was my contact at LAPD. My name’s Mangum, Mangum P.I. That’s right. I’m a principal investigator on an NSF grant. I’m the guy they call when there’s a mathematical crime. And times being what they are, I get a lot of calls. ‘‘All right Schmishschmitt,’’ I said. ‘‘I was just dreaming up a new proof of the Hairy Ball Theorem. I’ll have to start over again, tonight.’’ ‘‘Sorry to disturb naptime, Mangum, but this is important. Come down to the Math Common Room on the second floor of Millikan Laboratory at Pomona College. I’ll meet you there.’’ A half hour later, I strolled into the wreckage. Students were sobbing and holding onto each other for comfort. A busload of grief counselors was working the room. Schmishschmitt was in the corner talking to a distraught woman with short grey hair, reading glasses hanging on a chain around her neck, and red cherry stains down the front of her white blouse. ‘‘Mangum, this is Freida Warminster, chair of the Pomona Math Department. She can explain what happened.’’ ‘‘Don’t bother,’’ I said, holding up a hand. ‘‘It’s obvious.’’ ‘‘What do you mean?’’ asked Shmishschmitt. ‘‘Well,’’ I said pointing to the broken pies littering the floor, the whipped cream sprayed around the room and the fresh blueberry stains dripping down the walls, ‘‘given that today is March 14, and the pies, one can deduce that the Pomona Math Department was putting on a Pi Day celebration. Timed to start at the traditional time of 3/14 at 1:59 and 26 and a half seconds, corresponding to the first nine digits of pi.’’ ‘‘You know about Pi Day?’’ said Schmishschmitt skeptically. ‘‘How could you? You’re at UCLA.’’ ‘‘Not surprised you might wonder, Schmishscmitt,’’ I said. ‘‘There are two types of math departments, the ones that celebrate Pi Day and the ones that don’t.’’ Schmishschmitt rolled his eyes, as he often did when I went into lecture mode, so I focused my attention on Warminster. ‘‘Which ones don’t, you ask. Sometimes it’s a department with a chair who’s allergic to shortening. Sometimes it’s a

Topological interface physics of defects and textures in spinor Bose-Einstein condensates

We provide a detailed description of our previously proposed scheme for topological interface engineering with constructed defects and textures perforating across coherent interfaces between different broken symmetries [Borgh and Ruostekoski, Phys. Rev. Lett. 109, 015302 (2012)]. We consider a spin-1 Bose-Einstein condensate, in which polar and ferromagnetic phases are prepared in spatially separated regions. We show that a stable coherent interface is established between the two phases, allowing defects of different topology to connect continuously across the boundary. We provide analytic constructions of interface-crossing defect solutions that could be experimentally phase imprinted using existing technology. By numerically minimizing the energy, we calculate the core structures of interface-crossing defect configurations. We demonstrate nontrivial core deformations to considerably more complex structures, such as the formation of an arch-shaped half-quantum line defect, an Alice arch, at the interface, with the topological charge of a point defect, whose emergence may be understood by the ``hairy ball'' theorem. Another example of an energetically stable object is the connection of a coreless vortex to a pair of half-quantum vortices. We show that rotation leads to spontaneous nucleation of defects in which a coreless vortex continuously transforms to a half-quantum vortex across the interface.

A Case of Rapunzel Syndrome

Rapunzel syndrome refers to a very rare condition in which swallowed hair forms a gastric trichobezoar that has a long tail extending into the small bowel. We describe a case of Rapunzel syndrome in an 8-year-old girl who presented with abdominal mass, epigastric pain and vomiting. Abdominal computed tomography scan showed a markedly dilated stomach filled with coarse heterogeneous materials. Upper gastrointestinal endoscopy revealed a huge hairy ball with a tail extending through the pylorus. We performed a surgical laparotomy and successfully removed a huge trichobezoar with a long tail extending into the middle portion of jejunum. Psychiatric consultation with review showed her past history of trichotillomania and trichophagia 4 years ago. But her parents denied further psychiatric therapy and she was lost to the follow-up. Rapunzel syndrome should be included in the differential diagnosis in children with chronic abdominal pain and trichophagia.

Forbidden substructures and combinatorial dichotomies: WQO and universality

We discuss two combinatorial problems concerning classes of finite or countable structures of combinatorial type. We consider classes determined by a finite set of finite constraints (forbidden substructures). Questions about such classes of structures are naturally viewed as algorithmic decision problems, taking the finite set of constraints as the input. While the two problems we consider have been studied in a number of natural contexts, it remains far from clear whether they are decidable in their general form. This broad question leads to a number of more concrete problems. We discuss twelve open problems of varying levels of concreteness, and we point to the “Hairy Ball Problem” as a particularly concrete problem, which we give first in direct model theoretic terms, and then decoded as an explicit graph theoretic problem.