2D Sphere Research Articles

Hydrodynamics of superfluid quantum space: particle of spin-1/2 in a magnetic field

The modified Navier-Stokes equation describing the velocity field in the superfluid quantum space is loaded by the external Lorentz force introducing electromagnetic fields. In order to open the path for getting the \Schrodinger-Pauli equation describing the behavior of a particle with spin-1/2 in the magnetic field we need to extend the continuity equation to take into account conservation of spin flows on the 3D sphere. This extension includes conservation of the density distribution function in 6D space, that is a multiplication of the 3D Euclidean space by the 3D sphere of unit radius. The special unitary group SU(2) underlies the rotations of the spin on this sphere. This group is isomorphic to the group of quaternions containing the real 4x4 matrices of norm 1. Transition to the quaternion group opens up the way to the possibility of describing the spin-1/2 behavior in a magnetic field as a motion of a spin flag on the 2D sphere. Maxwell's electromagnetic field theory manifests itself in the quaternion group basis by the natural manner.

Gevrey regularity for Navier–Stokes equations under Lions boundary conditions

The Navier–Stokes system is considered in a compact Riemannian manifold. Gevrey class regularity is proven under Lions boundary conditions: in 2D for the Rectangle, Cylinder, and Hemisphere, and in 3D for the Rectangle. The cases of the 2D Sphere and 2D and 3D Torus are also revisited.

Convex billiards on convex spheres

In this paper we study the dynamical billiards on a convex 2D sphere. We investigate some generic properties of the convex billiards on a general convex sphere. We prove that C∞ generically, every periodic point is either hyperbolic or elliptic with irrational rotation number. Moreover, every hyperbolic periodic point admits some transverse homoclinic intersections. A new ingredient in our approach is Herman's result on Diophantine invariant curves that we use to prove the nonlinear stability of elliptic periodic points for a dense subset of convex billiards.

Ergodicity for a stochastic geodesic equation in the tangent bundle of the 2D sphere

We study ergodic properties of stochastic geometric wave equations on a particular model with the target being the 2D sphere while considering only solutions which are independent of the space variable. This simplification leads to a degenerate stochastic equation in the tangent bundle of the 2D sphere. Studying this equation, we prove existence and non-uniqueness of invariant probability measures for the original problem and obtain also results on attractivity towards an invariant measure. We also present a structure-preserving numerical scheme to approximate solutions and provide computational experiments to motivate and illustrate the theoretical results.

Effects of animation face on skin color perception

Face perception has a unique processing pathway in the visual system. Many previous studies showed that human face skin-color was precisely reproduced, compared with other objects without facial features. Although it was shown that cartoon face yielded responses similar to those of photo images in human FFA (the fusiform face area) (Tong et al., 2000), color perception of animation faces has seldom been investigated. In this study, we aim to clarify whether and what effect animation faces have on skin-color judgment. In experiment 1, skin-color judgment was compared with stimuli between animation facial and circular images. Test stimuli consisted of 2D face, 3D face, circle and sphere with three colors of A, B and C, which (redness, blueness, luminance (cd/m2)) in the Macleod-Boynton chromaticity diagram were (0.73, 0.65, 42), (0.72, 0.75, 40) and (0.72, 0.69, 59), respectively. Ten Japanese and Korean subjects performed memory color-matching between a test stimulus and a circular stimulus of which chromaticity and luminance the subject adjusted to achieve a matching point. The standard deviation (SD) of matching color points in animation faces was bigger than in circles. In experiment 2, the effects of face-looking stimuli on skin-color judgment were examined. Test stimuli were deformed animation faces, made in five steps (from face-like to non-face like) with three colors as in experiment 1. No difference was found in SD among five steps of deformed faces. In addition, SD of the ideal skin-color (C) matching point was smaller than that of other colors (A, B) only in face stimuli of experiment 1 and 2. These results indicate that animation faces make skin-color judgment difficult compared with images without facial features. Memory skin-color could hinder precise animation skin-color judgment. Moreover, ideal skin-color could make color judgment easily on animation face images. Meeting abstract presented at VSS 2015

A solvable model for fermion masses on a warped 6D world with the extra 2D sphere

In a warped 6D world with an extra two-dimensional sphere, we propose an exactly solvable model for fermion masses with zero mode. The warp factor is given by ϕ(θ, φ) = sin θ cos φ, which is a solution to the 6D Einstein equation with the bulk cosmological constant Λ and the energy–momentum tensor of the bulk matter fields. Our model provides another possibility of obtaining fermion zero mode, rather than traditional model based on Dirac's monopole.

A new integrable anisotropic oscillator on the two-dimensional sphere and the hyperbolic plane

A new integrable generalization to the two-dimensional (2D) sphere, , and to the hyperbolic space, , of the 2D Euclidean anisotropic oscillator Hamiltonian with Rosochatius (centrifugal) terms is presented, and its curved integral of motion is shown to be quadratic in the momenta. To construct such a new integrable Hamiltonian, , we will use a group theoretical approach in which the curvature, , of the underlying space will be treated as an additional (contraction) parameter, and we will make extensive use of projective coordinates and their associated phase spaces. When the oscillator parameters Ω1 and Ω2 are such that , the system turns out to be the well-known superintegrable oscillator on and . Nevertheless, numerical integration of the trajectories of suggests that for other values of the parameters Ω1 and Ω2, the system is not superintegrable. In this way, we support the conjecture that for each commensurate, and thus superintegrable, Euclidean oscillator there exists a two-parametric family of curved integrable oscillators that turns out to be superintegrable only when the parameters are tuned to the commensurability condition.

Magnetic fields in 2D and 3D sphere

In this note we study the Landau–Hall problem in the 2D and 3D unit sphere, that is, the motion of a charged particle in the presence of a static magnetic field. The magnetic flow is completely determined for any Riemannian surface of constant Gauss curvature, in particular, the unit 2D sphere. For the 3D case we consider Killing magnetic fields on the unit sphere, and we show that the magnetic flowlines are helices with the given Killing vector field as its axis.

The anisotropic oscillator on the 2D sphere and the hyperbolic plane

An integrable generalization on the 2D sphere S2 and the hyperbolic plane H2 of the Euclidean anisotropic oscillator Hamiltonian with ‘centrifugal’ terms given byis presented. The resulting generalized Hamiltonian depends explicitly on the constant Gaussian curvature κ of the underlying space, in such a way that all the results here presented hold simultaneously for S2 (κ > 0), H2 (κ < 0) and E2 (κ = 0). Moreover, is explicitly shown to be integrable for any values of the parameters δ, Ω, λ1 and λ2. Therefore, can also be interpreted as an anisotropic generalization of the curved Higgs oscillator, that is recovered as the isotropic limit Ω → 0 of . Furthermore, numerical integration of some of the trajectories for are worked out and the dynamical features arising from the introduction of a curved background are highlighted.The superintegrability issue for is discussed by focusing on the value Ω = 3δ, which is one of the cases for which the Euclidean Hamiltonian is known to be superintegrable (the 1 : 2 oscillator). We show numerically that for Ω = 3δ the curved Hamiltonian presents nonperiodic bounded trajectories, which seems to indicate that provides a non-superintegrable generalization of even for values of Ω that lead to commensurate frequencies in the Euclidean case. We compare this result with a previously known superintegrable curved analogue of the 1 : 2 Euclidean oscillator, which is described in detail, showing that the Ω = 3δ specialization of does not coincide with . Hence we conjecture that would be an integrable (but not superintegrable) curved generalization of the anisotropic oscillator that exists for any value of Ω and has constants of the motion that are quadratic in the momenta. Thus each commensurate Euclidean oscillator could admit another specific superintegrable curved Hamiltonian which would be different from and endowed with higher order integrals. Finally, the geometrical interpretation of the curved ‘centrifugal’ terms appearing in is also discussed in detail.

Fabrication of Heterogeneous Double-Ring-Like Structure Arrays by Combination of Colloidal Lithography and Controllable Dewetting

We report a novel technique for fabricating the heterogeneous double-ring-like structural array by colloidal lithography and two-step dewetting process. First, the 2D non-closed-packed (ncp) silica sphere arrays were obtained by lift-up lithography. Then, the ncp sphere array transferred onto the Rhodamine B (RB)@poly(vinyl alcohol) (PVA) film was used for the mask during reactive ion etching (RIE) process. Sequentially, the substrate with RB@PVA ring-like structure arrays under the silica sphere was dip-coated from poly(N-vinylcarbazole) (PVK) chloroform solution with certain concentration. Due to the presence of ordered 2D sphere arrays, the two-step dewetting behavior happened on top of the sphere and the silicon wafer between adjacent spheres, respectively. After removing the silica sphere arrays by hydrofluoric acid, the RB@PVA/PVK heterogeneous double-ring-like structure array was exhibited on the substrate. We characterized this particular structure by SEM, AFM, and fluorescence spectrum, which prove that both the inner RB@PVA ring and outer PVK ring are independent without any reaction. Accordingly, this method could be extended to other materials owing to its universality. These unique structural arrays have potential application in optoelectronic devices, surface photocatalysis, and surface enhanced Raman scattering (SERS).

Standard model vacua for two-dimensional compactifications

We examine the structure of lower-dimensional standard model vacua for two-dimensional compactifications (on a 2D torus and on a 2D sphere). In the case of the torus we find a new standard model vacuum for a large range of neutrino masses consistent with experiment. Quantum effects play a crucial role in the existence of this vacuum. For the compactification on a sphere the classical terms dominate the effective potential for large radii and a stable vacuum is achieved only by introducing a large magnetic flux. We argue that there are no two-dimensional standard model vacua for compactifications on a surface of genus greater than one.

The primordial non-Gaussianity of local type (flocalNL) in the WMAP 5-year data: the length distribution of CMB skeleton

We present skeleton studies of non-Gaussianity in the CMB temperature anisotropy observed in the WMAP5 data. The local skeleton is traced on the 2D sphere by cubic spline interpolation which leads to more accurate estimation of the intersection positions between the skeleton and the secondary pixels than conventional linear interpolation. We demonstrate that the skeleton-based estimator of non-Gaussianity of the local type (f_NL) - the departure of the length distribution from the corresponding Gaussian expectation - yields an unbiased and sufficiently converged f_NL-likelihood. We analyse the skeleton statistics in the WMAP5 combined V- and W-band data outside the Galactic base-mask determined from the KQ75 sky-coverage. The results are consistent with Gaussian simulations of the the best-fitting cosmological model, but deviate from the previous results determined using the WMAP1 data. We show that it is unlikely that the improved skeleton tracing method, the omission of Q-band data, the modification of the foreground-template fitting method or the absence of 6 extended regions in the new mask contribute to such a deviation. However, the application of the Kp0 base-mask in data processing does improve the consistency with the WMAP1 results. The f_NL-likelihoods of the data are estimated at 9 different smoothing levels. It is unexpected that the best-fit values show positive correlation with the smoothing scales. Further investigation argues against a point-source or goodness-of-fit explanation but finds that about 30% of either Gaussian or f_NL samples having better goodness-of-fit than the WMAP5 show a similar correlation. We present the estimate f_NL=47.3+/-34.9 (1sigma error) determined from the first four smoothing angles and f_NL=76.8+/-43.1 for the combination of all nine. The former result may be overestimated at the 0.21sigma-level because of point sources.

DIMENSIONAL REDUCTION ON A SPHERE

The question of the dimensional reduction of two-dimensional (2d) quantum models on a sphere to one-dimensional (1d) models on a circle is addressed. A possible application is to look at a relation between the 2d anyon model and the 1d Calogero–Sutherland model, which would allow for a better understanding of the connection between 2d anyon exchange statistics and Haldane exclusion statistics. The latter is realized microscopically in the 2d LLL anyon model and in the 1d Calogero model. In a harmonic well of strength ω or on a circle of radius R — both parameters ω and R have to be viewed as long distance regulators — the Calogero spectrum is discrete. It is well known that by confining the anyon model in a 2d harmonic well and projecting it on a particular basis of the harmonic well eigenstates, one obtains the Calogero–Moser model. It is then natural to consider the anyon model on a sphere of radius R and look for a possible dimensional reduction to the Calogero–Sutherland model on a circle of radius R. First, the free one-body case is considered, where a mapping from the 2d sphere to the 1d chiral circle is established by projection on a special class of spherical harmonics. Second, the N-body interacting anyon model is considered: it happens that the standard anyon model on the sphere is not adequate for dimensional reduction. One is thus led to define a new spherical anyon-like model deduced from the Aharonov–Bohm problem on the sphere where each flux line pierces the sphere at one point and exits it at its antipode.

Dirac equation for a spin-$frac12$ charged particle on the 2D sphere S2 and the hyperbolic plane H2

Using the idea of shape invariance with respect to the main quantum number n, we represent Lie algebras u(2) and u(1, 1). The induced metric by the Casimir operator of Lie algebras u(2) and u(1, 1) leads us to obtain new solutions of the Dirac equation corresponding to a spin-½ charged particle on the 2D sphere S2 and the hyperbolic plane H2 in the presence of a magnetic monopole. It is shown that the related new spinors represent the supersymmetry algebra, and that they satisfy shape invariance equations with respect to n.

Scaling Lee-Yang Model on a Sphere I. Partition Function

Some general properties of perturbed (rational) CFT in the background metric of symmetric 2D sphere of radius $R$ are discussed, including conformal perturbation theory for the partition function and the large $R$ asymptotic. The truncated conformal space scheme is adopted to treat numerically perturbed rational CFT's in the spherical background. Numerical results obtained for the scaling Lee-Yang model lead to the conclusion that the partition function is an entire function of the coupling constant. Exploiting this analytic structure we are able to describe rather precisely the ``experimental'' truncated space data, including even the large $R$ behavior, starting only with the CFT information and few first terms of conformal perturbation theory.

Simple tools to study global dynamics in non-axisymmetric galactic potentials – I

In a rst part we discuss the well-known prob- lem of the motion of a star in a general non-axisymmetric 2D galactic potential by means of a very simple but al- most universal system: the pendulum model. It is shown that both loop and box families of orbits arise as a natural consequence of the dynamics of the pendulum. An approx- imate invariant of motion is derived. A critical value of the latter sharply separates the domains of loops and boxes and a very simple computation allows to get a clear pic- ture of the distribution of orbits on a given energy surface. Besides, a geometrical representation of the global phase space using the natural surface of section for the prob- lem, the 2D sphere, is presented. This provides a better visualization of the dynamics. In a second part we introduce a new indicator of the basic dynamics, the Mean Exponential Growth fac- tor of Nearby Orbits (MEGNO), that is suitable to inves- tigate the phase space structure associated to a general Hamiltonian. When applied to the 2D logarithmic poten- tial it is shown to be eective to obtain a picture of the global dynamics and, also, to derive good estimates of the largest Lyapunov characteristic number in realistic physi- cal times. Comparisons with other techniques reveal that the MEGNO provides more information about the dynam- ics in the phase space than other wide used tools. Finally, we discuss the structure of the phase space as- sociated to the 2D logarithmic potential for several values of the semiaxis ratio and energy. We focus our attention on the stability analysis of the principal periodic orbits and on the chaotic component. We obtain critical energy values for which connections between the main stochastic zones take place. In any case, the whole chaotic domain appears to be always conned to narrow laments, with a Lyapunov time about three characteristic periods.

Locally Supported Wavelets on Manifolds with Applications to the 2D Sphere

In this paper we present a construction principle for locally supported wavelets on manifolds once a multiresolution analysis is given. The wavelets provide a stable (or unconditional) basis for a scale of Sobolev spaces Hs, 0 ≤ s ≤ s̄. We examine a fast wavelet transform with almost optimal complexity. For the two-dimensional sphere we construct a multiresolution analysis generated by continuous splines that are bilinear with respect to some special spherical grid. In our approach the poles are not exceptional points concerning the approximation power or the stability of the wavelet basis. Finally we present some numerical applications to singularity detection and the analysis of observed atmospheric data.