Sphere Research Articles

Construction of Smooth Source–Sink Arcs in the Space of Diffeomorphisms of a Two-Dimensional Sphere

Construction of Smooth Source–Sink Arcs in the Space of Diffeomorphisms of a Two-Dimensional Sphere

On uniform edge-n-colorings of tilings.

An edge-n-coloring of a uniform tiling {\cal T} is uniform if for any two vertices of {\cal T} there is a symmetry of {\cal T} that preserves the colors of the edges and maps one vertex onto the other. This paper gives a method based on group theory and color symmetry theory to arrive at uniform edge-n-colorings of uniform tilings. The method is applied to give a complete enumeration of uniform edge-n-colorings of the uniform tilings of the Euclidean plane, for which the results point to a total of 114 colorings, n = 1, 2, 3, 4, 5. Examples of uniform edge-n-colorings of tilings in the hyperbolic plane and two-dimensional sphere are also presented.

Deriving Schwarzschild metric from vacuum zero-point energy

In this paper, we study the correspondence between the three-dimensional and one-dimensional physical quantity in the spherically symmetric gravitational fields and we obtain a conjecture that the total energy enclosed by the two-dimensional sphere is equal to twice the change in vacuum zero-point energy of a one-dimensional circle that is the boundary of a hemispherical sphere, we can naturally obtain Schwarzschild metric without relying on Einstein's gravitational field equation under this conjecture, the result leads us to conclude that gravity originates from the change of zero-point energy in the one-dimensional vacuum.

Infrared structures of scattering on self-dual radiative backgrounds

The scattering of gluons and gravitons in trivial backgrounds is endowed with many surprising infrared features which have interesting conformal interpretations on the two-dimensional celestial sphere. However, the fate of these structures in more general asymptotically flat backgrounds is far from clear. In this paper, we consider holomorphic infrared structures in the presence of non-perturbative, self-dual background gauge and gravitational fields which are determined by freely specified radiative data. We make use of explicit formulae for tree-level gluon and graviton scattering in these self-dual radiative backgrounds, as well as chiral twistor sigma model descriptions of the classical dynamics. Remarkably, we find that the leading holomorphic part of tree-level collinear splitting functions — or celestial OPEs — and infinite-dimensional chiral soft algebras are undeformed by the background. We also compute all-order holomorphic celestial OPEs in the MHV sectors of gauge theory and gravity.

Asymptotic Antipodal Solutions as the Limit of Elliptic Relative Equilibria for the Two- and n-Body Problems in the Two-Dimensional Conformal Sphere

We consider the two- and n-body problems on the two-dimensional conformal sphere MR2, with a radius R>0. We employ an alternative potential free of singularities at antipodal points. We study the limit of relative equilibria under the SO(2) symmetry; we examine the specific conditions under which a pair of positive-mass particles, situated at antipodal points, can maintain a state of relative equilibrium as they traverse along a geodesic. It is identified that, under an appropriate radius–mass relationship, these particles experience an unrestricted and free movement in alignment with the geodesic of the canonical Killing vector field in MR2. An even number of bodies with pairwise conjugated positions, arranged in a regular n-gon, all with the same mass m, move freely on a geodesic with suitable velocities, where this geodesic motion behaves like a relative equilibrium. Also, a center of mass formula is included. A relation is found for the relative equilibrium in the two-body problem in the sphere similar to the Snell law.

Asymptotic distribution of nodal intersections for ARW against a surface

We investigate Gaussian Laplacian eigenfunctions (Arithmetic Random Waves) on the three-dimensional standard flat torus, in particular the asymptotic distribution of the nodal intersection length against a fixed regular reference surface. Expectation and variance have been addressed by Maffucci [Ann. Henri Poincaré 20(11), 3651–3691 (2019)] who found that the expected length is proportional to the square root of the eigenvalue times the area of the surface, while the asymptotic variance only depends on the geometry of the surface, the projected lattice points being equidistributed on the two-dimensional unit sphere in the high-energy limit. He also noticed that there are “special” surfaces, so-called static, for which the variance is of smaller order; however he did not prescribe the precise asymptotic law in this case. In this paper, we study second order fluctuations of the nodal intersection length. Our first main result is a Central Limit Theorem for “generic” surfaces, while for static ones, a sphere or a hemisphere e.g., our main results are a non-Central Limit Theorem and a precise asymptotic law for the variance of the nodal intersection length, conditioned on the existence of so-called well-separated sequences of Laplacian eigenvalues. It turns out that, in this regime, the nodal area investigated by Cammarota [Trans. Am. Math. Soc. 372(5), 3539–3564 (2019)] is asymptotically fully correlated with the length of the nodal intersections against any sphere. The main ingredients for our proofs are the Kac-Rice formula for moments, the chaotic decomposition for square integrable functionals of Gaussian fields, and some arithmetic estimates that may be of independent interest.

Scenario of stable transition from diffeomorphism of torus isotopic to identity one to skew product of rough transformations of circle

In this paper, we consider gradient-like diffeomorphisms of a two-dimensional torus isotopic to the identical one. The isotopicity of diffeomorphisms $f_0$, $f_1$ on an $n$-manifold $M^n$ means the existence of some arc $\{f_t:M^n\to M^n,t\in[0,1]\}$ connecting them in the space of diffeomorphisms. If isotopic diffeomorphisms are structurally stable (qualitatively not changing their properties with small perturbations), then it is natural to expect the existence of a stable arc (qualitatively not changing its properties under small perturbations) connecting them. In this case, one says that the isotopic diffeomorphisms $f_0$, $f_1$ are stably isotopic or belong to the same class of stable isotopic connectivity. The simplest structurally stable diffeomorphisms on surfaces are gradient-like transformations having a finite hyperbolic non-wandering set, stable and unstable manifolds of various saddle points of which do not intersect. However, even on a two-dimensional sphere, where all orientation-preserving diffeomorphisms are isotopic, gradient-like diffeomorphisms are generally not stably isotopic. The countable number of pairwise different classes of stable isotopic connectivity is constructed on the base of a rough transformation of the circle $\phi_{\frac{k}{m}}$ with exactly two periodic orbits of the period $m$ and the rotation number $\frac{k}{m}$, which can be continued to a diffeomorphism $F_{\frac k m}:\mathbb S^2\to\mathbb S^2$ with two fixed sources at the North and South poles. On the torus $\mathbb T^2$, the model representative in the considered class is the skew products of rough transformations of a circle. We show that any isotopic gradient-like diffeomorphism of a torus is connected by a stable arc with some model transformation.

Fractional and scaled Brownian motion on the sphere: The effects of long-time correlations on navigation strategies.

We analyze fractional Brownian motion and scaled Brownian motion on the two-dimensional sphere S^{2}. We find that the intrinsic long-time correlations that characterize fractional Brownian motion collude with the specific dynamics (navigation strategies) carried out on the surface giving rise to rich transport properties. We focus our study on two classes of navigation strategies: one induced by a specific set of coordinates chosen for S^{2} (we have chosen the spherical ones in the present analysis), for which we find that contrary to what occurs in the absence of such long-time correlations, nonequilibrium stationary distributions are attained. These results resemble those reported in confined flat spaces in one and two dimensions [Guggenberger et al. New J. Phys. 21, 022002 (2019)1367-263010.1088/1367-2630/ab075f; Vojta et al. Phys. Rev. E 102, 032108 (2020)2470-004510.1103/PhysRevE.102.032108]; however, in the case analyzed here, there are no boundaries that affect the motion on the sphere. In contrast, when the navigation strategy chosen corresponds to a frame of reference moving with the particle (a Frenet-Serret reference system), then the equilibrium distribution on the sphere is recovered in the long-time limit. For both navigation strategies, the relaxation times toward the stationary distribution depend on the particular value of the Hurst parameter. We also show that on S^{2}, scaled Brownian motion, distinguished by a time-dependent diffusion coefficient with a power-scaling, is independent of the navigation strategy finding a good agreement between the analytical calculations obtained from the solution of a time-dependent diffusion equationon S^{2}, and the numerical results obtained from our numerical method to generate ensemble of trajectories.

EXISTENCE OF MAXIMUM OF TIME AVERAGED HARVESTING IN THE KPP-MODEL ON SPHERE WITH PERMANENT AND IMPULSE COLLECTION

On a two-dimensional sphere, a distributed renewable resource is considered, the dynamics of which is described by a model of the Kolmogorov–Petrovsky–Piskunov–Fisher type, and the exploitation of this resource, carried out by constant or periodic impulse harvesting. It is shown that after choosing an admissible exploitation strategy, the dynamics of the resource tend to the limiting dynamics corresponding to this strategy, and that there is an admissible harvesting strategy that maximizes the time averaged harvesting of the resource.

Celestial open strings at one-loop

We study celestial amplitudes in string theory at one-loop. Celestial amplitudes describe scattering processes of boost eigenstates and relate to amplitudes in the more standard basis of momentum eigenstates through a Mellin transform. They are thus sensitive to both the ultraviolet and the infrared, which raises the question of how to appropriately take the field theory limit of string amplitudes in the celestial basis. We address this problem in the context of four-dimensional genus-one scattering processes of gluons in open string theory which reach the two-dimensional celestial sphere at null infinity. We show that the Mellin transform commutes with the adequate limit in the worldsheet moduli space and reproduces the celestial one-loop field theory amplitude expressed in the worldline formalism. The dependence on α′ continues to be a simple overall factor in one-loop celestial amplitudes albeit with a power that is shifted with respect to tree-level, thus making manifest that the dimensionless parameter g102/α′3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {g}_{10}^2/{\\alpha}^{\\prime 3} $$\\end{document} organizes the loop expansion in the celestial basis. The precise way in which the amplitudes scale with this parameter depends on the number of non-compact dimensions in such a way that in 4 dimensions the scaling with α′ does agree with that at tree-level.

Solution to the Thomson Problem for Clifford Tori with an Application to Wigner Crystals.

In its original version, the Thomson problem consists of the search for the minimum-energy configuration of a set of point-like electrons that are confined to the surface of a two-dimensional sphere () that repel each other according to Coulomb's law, in which the distance is the Euclidean distance in the embedding space of the sphere, i.e., . In this work, we consider the analogous problem where the electrons are confined to an n-dimensional flat Clifford torus with n = 1, 2, 3. Since the torus can be embedded in the complex manifold , we define the distance in the Coulomb law as the Euclidean distance in , in analogy to what is done for the Thomson problem on the sphere. The Thomson problem on a Clifford torus is of interest because supercells with the topology of a Clifford torus can be used to describe periodic systems such as Wigner crystals. In this work, we numerically solve the Thomson problem on a square Clifford torus. To illustrate the usefulness of our approach, we apply it to Wigner crystals. We demonstrate that the equilibrium configurations we obtain for large numbers of electrons are consistent with the predicted structures of Wigner crystals. Finally, in the one-dimensional case, we analytically obtain the energy spectrum and the phonon dispersion law.

Almost representations of algebras and quantization

abstract: We introduce the notion of almost representations of Lie algebras and quantum tori, and establish an Ulam-stability type phenomenon: every irreducible almost representation is close to a genuine irreducible representation. As an application, we prove that geometric quantizations of the two-dimensional sphere and the two-dimensional torus are conjugate in the semi-classical limit up to a small error.

Integrability and dynamics of a simplified class B laser system.

A simplified class B laser system is a family of differential polynomial systems of degree two depending on the parameters a and b. Its rich dynamics has already been observed in 1980s, see Arecchi et al. [Opt. Commun. 51, 308-314 (1984)] and Politi et al. [Phys. Rev. A 33, 4055 (1986)], and still nowadays, it attracts the interest of the researchers. In this paper, we characterize its dynamics near infinity for all values of the parameters. When a=0, the partial integrability was already proved by Oppo and Politi [Z. Phys. B Con. Mat. 59, 111-115 (1985)]. Here, we prove that for a=0, it is completely integrable with two independent first integrals given by Liouvillian functions, and we present a complete study of its dynamics. When a≠0, we study its dynamics in the Poincaré ball B3, i.e., the interior of this ball is identified with R3 and its boundary the two-dimensional sphere S2 is identified with the infinity of R3.

Unified dark energy from Chiral-Quintom model with a mixed potential in Friedmann–Lemaître–Robertson–Walker cosmology

For a spatially Friedmann–Lemaître–Robertson–Walker cosmology, we propose a multi-scalar field gravitational model. Specifically, we consider a two-scalar field cosmological model in which the kinetic components of the scalar fields establish a two-dimensional sphere of Lorentzian signature. For our Chiral-Quintom model we choose a mixed potential term Vleft( phi ,psi right) =V_{0}e^{lambda phi }+U_{0}e^{kappa phi }psi ^{frac{1}{sigma } } and we investigate the asymptotic limits of the cosmological parameters. This model for U_{0}=0, provides a generalization of the hyperbolic inflation where the equation of the state parameter can cross the phantom divide line. When U_{0}ne 0 we observe that this cosmological model exhibits asymptotic solutions that encompass accelerated universes, big rip singularities, and dust-like solutions. Hence, this multi-scalar field model it can be regarded as as a dark energy unify model which describes a variety of asymptotic cosmological scenarios.

Ocular condition prognosis in Keratoconus patients after corneal ring implantation using artificial neural networks

The common disorder, Keratoconus (KC), is distinguished by cumulative corneal slimming and steepening. The corneal ring implantation has become a successful surgical procedure to correct the KC patient’s vision. The determination of suitable patients for the surgery alternative is among the paramount concerns of ophthalmologists. To reduce the burden on them and enhance the treatment, this research aims to previse the ocular condition of KC patients after the corneal ring implantation. It focuses on predicting post-surgical corneal topographic indices and visual characteristics. This study applied an efficacious artificial neural network approach to foretell the aforementioned ocular features of KC subjects 6 and 12 months after implanting KeraRing and MyoRing based on the accumulated data. The datasets are composed of sufficient numbers of corneal topographic maps and visual characteristics recorded from KC patients before and after implanting the rings. The visual characteristics under study are uncorrected visual acuity (UCVA), sphere (SPH), astigmatism (Ast), astigmatism orientation (Axe), and best corrected visual acuity (BCVA). In addition, the statistical data of multiple KC subjects were registered, including three effective indices of corneal topography (i.e., Ast, K-reading, and pachymetry) pre- and post-ring embedding. The outcomes represent the contribution of practical training of the introduced models to the estimation of ocular features of KC subjects following the implantation. The corneal topographic indices and visual characteristics were estimated with mean errors of 7.29% and 8.60%, respectively. Further, the errors of 6.82% and 7.65% were respectively realized for the visual characteristics and corneal topographic indices while assessing the predictions by the leave-one-out cross-validation (LOOCV) procedure. The results confirm the great potential of neural networks to guide ophthalmologists in choosing appropriate surgical candidates and their specific intracorneal rings by predicting post-implantation ocular features.

Kaehler submanifolds of the real hyperbolic space

AbstractThe local classification of Kaehler submanifolds $M^{2n}$ of the hyperbolic space $\mathbb{H}^{2n+p}$ with low codimension $2\leq p\leq n-1$ under only intrinsic assumptions remains a wide open problem. The situation is quite different for submanifolds in the round sphere $\mathbb{S}^{2n+p}$, $2\leq p\leq n-1$, since Florit et al. [7] have shown that the codimension has to be $p=n-1$ and then that any submanifold is just part of an extrinsic product of two-dimensional umbilical spheres in $\mathbb{S}^{3n-1}\subset\mathbb{R}^{3n}$. The main result of this paper is a version for Kaehler manifolds isometrically immersed into the hyperbolic ambient space of the result in [7] for spherical submanifolds. Besides, we generalize several results obtained by Dajczer and Vlachos [5].

Well-posed problems for the Laplace-Beltrami operator on a punctured two-dimensional sphere

An arbitrary point is removed from a three-dimensional Euclidean space on a two-dimensional sphere. The new well-posed solvable boundary value problems for the corresponding Laplace-Beltrami operator on the resulting punctured sphere are presented. To formulate the well-posed problems some properties of Green's function of the Laplace-Beltrami operator on a two-dimensional sphere are previously studied in detail.

Spreading Solution Additives Governs the Quality of Polystyrene Particle-Based Two-Dimensional Opals.

Two-dimensional polystyrene sphere opals are important materials for nanotechnology applications and fundamental nanoscience research. They are a facile and inexpensive nanofabrication tool, but the quality factor of these opals has drastic differences between reports. Additives like ethanol, ions, and organic molecules in the aqueous particle spreading solution are known to affect the quality factor and growth efficiency of the produced opals. However, a systematic study on the effect and optimization of some of the most effective additives has not been reported until now. Here, we investigate the influence of additives on the growth efficiency and quality factor of such monolayers formed at the air-water interface without the use of a Langmuir-Blodgett trough. The additives induced large variations in the monolayer quality factor and growth efficiency, and we found that the ideal additive content of the spreading agents is 30 wt % < cethanol < 70 wt %, 0 < cH2SO4 < 30.5 mM, and 0 < csty < 255.0 mM. This study provides a guideline for the rational composition and additive content of the spreading solution to obtain high-quality two-dimensional opals for further applications in nanofabrication and photonics and will enable researchers and application engineers to produce standardized nanofabrication materials.

Collision trajectories and regularisation of two-body problem on S2

In this paper, we investigate collision orbits of two identical bodies placed on the surface of a two-dimensional sphere and interacting via an attracting potential of the form V(q)=−cot⁡(q), where q is the angle formed by the position vectors of the two bodies. We describe the ω-limit set of the variables in the symplectically reduced system corresponding to initial data that lead to collisions. Lastly, we regularise the system and investigate its behaviour on near collision orbits. This involves the study of completely degenerate equilibria and the use of high-dimensional non-homogeneous blow-ups.

The spherical ensemble and quasi-Monte-Carlo designs

The spherical ensemble is a well-known ensemble of N repulsive points on the two-dimensional sphere, which can realized in various ways (as a random matrix ensemble, a determinantal point process, a Coulomb gas, a Quantum Hall state...). Here we show that the spherical ensemble enjoys remarkable convergence properties from the point of view of numerical integration. More precisely, it is shown that the numerical integration rule corresponding to N nodes on the two-dimensional sphere sampled in the spherical ensemble is, with overwhelming probability, nearly a quasi-Monte-Carlo design in the sense of Brauchart-Saff-Sloan-Womersley for any smoothness parameter s≤2.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$s\\le 2.$$\\end{document} The key ingredient is a new explicit sub-Gaussian concentration of measure inequality for the spherical ensemble.