Sphere S2 Research Articles

Conjugate points along spherical harmonics

Utilizing structure constants, we present a version of the Misiolek criterion for identifying conjugate points. We propose an approach that enables us to locate these points along solutions of the quasi-geostrophic equations on the sphere S2. We demonstrate that for any spherical harmonics Ylm with 1≤|m|≤l, except for Y1±1 and Y2±1, conjugate points can be determined along the solution generated by the velocity field elm=∇⊥Ylm. Subsequently, we investigate the impact of the Coriolis force on the occurrence of conjugate points. Moreover, for any zonal flow generated by the velocity field ∇⊥Yl10, we demonstrate that proper rotation rate can lead to the appearance of conjugate points along the corresponding solution, where l1=2k+1.∈N Additionally, we prove the existence of conjugate points along (complex) Rossby-Haurwitz waves and explore the effect of the Coriolis force on their stability.

Deep probabilistic direction prediction in 3D with applications to directional dark matter detectors

We present the first method to probabilistically predict 3D direction in a deep neural network model. The probabilistic predictions are modeled as a heteroscedastic von Mises-Fisher distribution on the sphere S2 , giving a simple way to quantify aleatoric uncertainty. This approach generalizes the cosine distance loss which is a special case of our loss function when the uncertainty is assumed to be uniform across samples. We develop approximations required to make the likelihood function and gradient calculations stable. The method is applied to the task of predicting the 3D directions of electrons, the most complex signal in a class of experimental particle physics detectors designed to demonstrate the particle nature of dark matter and study solar neutrinos. Using simulated Monte Carlo data, the initial direction of recoiling electrons is inferred from their tortuous trajectories, as captured by the 3D detectors. For 40 keV electrons in a 70% He 30% CO2 gas mixture at STP, the new approach achieves a mean cosine distance of 0.104 (26∘) compared to 0.556 (64∘) achieved by a non-machine learning algorithm. We show that the model is well-calibrated and accuracy can be increased further by removing samples with high predicted uncertainty. This advancement in probabilistic 3D directional learning could increase the sensitivity of directional dark matter detectors.

Surface areas of equifacetal polytopes inscribed in the unit sphere 𝕊2

Surface areas of equifacetal polytopes inscribed in the unit sphere 𝕊2

Differentiable and accelerated spherical harmonic and Wigner transforms

Many areas of science and engineering encounter data defined on spherical manifolds. Modelling and analysis of spherical data often necessitates spherical harmonic transforms, at high degrees, and increasingly requires efficient computation of gradients for machine learning or other differentiable programming tasks. We develop novel algorithmic structures for accelerated and differentiable computation of generalised Fourier transforms on the sphere S2 and rotation group SO(3), i.e. spherical harmonic and Wigner transforms, respectively. We present a recursive algorithm for the calculation of Wigner d-functions that is both stable to high harmonic degrees and extremely parallelisable. By tightly coupling this with separable spherical transforms, we obtain algorithms that exhibit an extremely parallelisable structure that is well-suited for the high throughput computing of modern hardware accelerators (e.g. GPUs). We also develop a hybrid automatic and manual differentiation approach so that gradients can be computed efficiently. Our algorithms are implemented within the JAX differentiable programming framework in the S2FFT▪ software code. Numerous samplings of the sphere are supported, including equiangular and HEALPix sampling. Computational errors are at the order of machine precision for spherical samplings that admit a sampling theorem. When benchmarked against alternative C codes we observe up to a 400-fold acceleration. Furthermore, when distributing over multiple GPUs we achieve very close to optimal linear scaling with increasing number of GPUs due to the highly parallelised and balanced nature of our algorithms. Provided access to sufficiently many GPUs our transforms thus exhibit an unprecedented effective linear time complexity.

Mutual embeddability in groups, trees, and spheres

Two subsets in a group are called twins if each is contained in a left translate of the other, though the two sets themselves are not translates of each other. We show that in the free group F{α,β}, there exist maximal families of twins of any finite cardinality. This result is used to show that in the context of embeddings of trees, there exist maximal families of twin trees of any finite cardinality. These are counterexamples to the “tree alternative” conjecture, which supplement the first counterexamples published by Kalow, Laflamme, Tateno, and Woodrow. We also investigate twin sets in the sphere S2, where the embeddings considered are isometries of S2. We show that there exist maximal families of twin sets in S2 of any finite cardinality.

Betti Functionals as Probes for Cosmic Topology

The question of the global topology of the Universe (cosmic topology) is still open. In the ΛCDM concordance model, it is assumed that the space of the Universe possesses the trivial topology of R3, and thus that the Universe has an infinite volume. As an alternative, in this paper, we study one of the simplest non-trivial topologies given by a cubic 3-torus describing a universe with a finite volume. To probe cosmic topology, we analyze certain structure properties in the cosmic microwave background (CMB) using Betti functionals and the Euler characteristic evaluated on excursions sets, which possess a simple geometrical interpretation. Since the CMB temperature fluctuations δT are observed on the sphere S2 surrounding the observer, there are only three Betti functionals βk(ν), k=0,1,2. Here, ν=δT/σ0 denotes the temperature threshold normalized by the standard deviation σ0 of δT. The analytic approximations of the Gaussian expectations for the Betti functionals and an exact formula for the Euler characteristic are given. It is shown that the amplitudes of β0(ν) and β1(ν) decrease with an increasing volume V=L3 of the cubic 3-torus universe. Since the computation of the βk’s from observational sky maps is hindered due to the presence of masks, we suggest a method that yields lower and upper bounds for them and apply it to four Planck 2018 sky maps. It is found that the βk’s of the Planck maps lie between those of the torus universes with side-lengths L=2.0 and L=3.0 in units of the Hubble length and above the infinite ΛCDM case. These results give a further hint that the Universe has a non-trivial topology.

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.

Moment methods for the radiative transfer equation based on [formula omitted]-divergences

The method of moments is widely used for the reduction of kinetic equations into fluid models. It consists in extracting the moments of the kinetic equation with respect to a velocity variable, but the resulting system is a priori underdetermined and requires a closure relation. In this paper, we adapt the φ-divergence based closure, recently developed for rarefied gases i.e. with a velocity variable describing Rd, to the radiative transfer equation where velocity describes the unit sphere S2. This closure is analyzed and a numerical method to compute it is provided. Eventually, it provides the main desirable properties to the resulting system of moments: Similarly to the entropy minimizing closure (MN), it dissipates an entropy and captures exactly the equilibrium distribution. However, contrarily to MN, it remains computationally tractable even at high order and it relies on an exact quadrature formula which preserves exactly symmetry properties, i.e. it does not trigger ray effects. The purely anisotropic regimes (beams) are not captured exactly but they can be approached as close as desired and the closures remains again tractable in this limit.

Branched coverings of the sphere having a completely invariant continuum with infinitely many Wada Lakes

We construct a family of smooth branched coverings of degree 2 of the sphere S2 having a completely invariant indecomposable continuum K and infinitely many Wada Lakes.

Numerical accuracy of principal geodesic analysis on the sphere in director‐based dynamics of hybrid mechanical systems

AbstractWe aim at nonlinear model order reduction (MOR) in hybrid mechanical systems by means of Principal Geodesic Analysis (PGA) on the Riemannian manifolds S2 (the sphere) and SO(3) (the rotation group). MOR requires highly accurate and efficient implementations of the logarithm maps and the resulting lifts across multiple branches. However, in our cases these maps have singularities due to periodicity. In this work we focus on the logarithm and lift maps for the sphere S2. We conduct detailed numerical experiments on mechanical systems to achieve maximal accuracy in spite of the singularities.

Fundamental Theory and R-linear Convergence of Stretch Energy Minimization for Spherical Equiareal Parameterization

Abstract Here, we extend the finite distortion problem from bounded domains in ℝ2 to closed genus-zero surfaces in ℝ3 by a stereographic projection. Then, we derive a theoretical foundation for spherical equiareal parameterization between a simply connected closed surface M and a unit sphere 𝕊2 by minimizing the total area distortion energy on ̅ℂ. After the minimizer of the total area distortion energy is determined, it is combined with an initial conformal map to determine the equiareal map between the extended planes. From the inverse stereographic projection, we derive the equiareal map between M and 𝕊2. The total area distortion energy is rewritten into the sum of Dirichlet energies associated with the southern and northern hemispheres and is decreased by alternatingly solving the corresponding Laplacian equations. Based on this foundational theory, we develop a modified stretch energy minimization function for the computation of spherical equiareal parameterization between M and 𝕊2. In addition, under relatively mild conditions, we verify that our proposed algorithm has asymptotic R-linear convergence or forms a quasi-periodic solution. Numerical experiments on various benchmarks validate that the assumptions for convergence always hold and indicate the efficiency, reliability, and robustness of the developed modified stretch energy minimization function.

A Fast and Reliable Solution to PnP, Using Polynomial Homogeneity and a Theorem of Hilbert

One of the most-extensively studied problems in three-dimensional Computer Vision is “Perspective-n-Point” (PnP), which concerns estimating the pose of a calibrated camera, given a set of 3D points in the world and their corresponding 2D projections in an image captured by the camera. One solution method that ranks as very accurate and robust proceeds by reducing PnP to the minimization of a fourth-degree polynomial over the three-dimensional sphere S3. Despite a great deal of effort, there is no known fast method to obtain this goal. A very common approach is solving a convex relaxation of the problem, using “Sum Of Squares” (SOS) techniques. We offer two contributions in this paper: a faster (by a factor of roughly 10) solution with respect to the state-of-the-art, which relies on the polynomial’s homogeneity; and a fast, guaranteed, easily parallelizable approximation, which makes use of a famous result of Hilbert.

Absolute minima of potentials of certain regular spherical configurations

We use methods of approximation theory to find the absolute minima on the sphere of the potential of spherical (2m−3)-designs with a non-trivial index 2m that are contained in a union of m parallel hyperplanes, m≥2, whose locations satisfy certain additional assumptions. The interaction between points is described by a function of the dot product, which has positive derivatives of orders 2m−2, 2m−1, and 2m. This includes the case of the classical Coulomb, Riesz, and logarithmic potentials as well as a completely monotone potential of the distance squared. We illustrate this result by showing that the absolute minimum of the potential of the set of vertices of the icosahedron on the unit sphere S2 in R3 is attained at the vertices of the dual dodecahedron and the one for the set of vertices of the dodecahedron is attained at the vertices of the dual icosahedron. The absolute minimum of the potential of the configuration of 240 minimal vectors of E8 root lattice normalized to lie on the unit sphere S7 in R8 is attained at a set of 2160 points on S7 which we describe.

On the Borromean arithmetic orbifolds

We revisit the fundamental groups Gmnp of the orbifolds Bmnp, where the underlying manifold is the 3-sphere S3 and the Borromean rings are the singular set with isotropies of order m, n and p. We correct an omission in [2] and show that Gmnp is arithmetic if and only if (m,n,p) is one of the 12 triples (3,3,3), (3,3,∞), (3,4,4), (3,4,∞), (3,6,6), (3,∞,∞), (4,4,4), (4,4,∞), (4,∞,∞), (6,6,6), (6,6,∞), (∞,∞,∞). The main purpose of the paper is to present each Gmnp, arithmetic, as a group of 4×4 matrices with entries in the ring of integers of a totally real number field K, and which are automorphs of a quaternary form F with entries in K of Sylvester type (+,+,+,−).

The Heat Equation on Submanifolds in Lie Groups and Random Motions on Spheres

We studied the random variable Vt=volS2(gtB∩B), where B is a disc on the sphere S2 centered at the north pole and (gt)t≥0 is the Brownian motion on the special orthogonal group SO(3) starting at the identity. We applied the results of the theory of compact Lie groups to evaluate the expectation of Vt for 0≤t≤τ, where τ is the first time when Vt vanishes. We obtained an integral formula using the heat equation on some Riemannian submanifold ΓB seen as the support of the function f(g)=volS2(gB∩B) immersed in SO(3). The integral formula depends on the mean curvature of ΓB and the diameter of B.

Small-time bilinear control of Schrödinger equations with application to rotating linear molecules

In Duca and Nersesyan (2023), a small-time controllability property of nonlinear Schrödinger equations is proved on a d-dimensional torus Td. In this paper we study a similar property, in the linear setting, starting from a closed Riemannian manifold. We then focus on the 2-dimensional sphere S2, which models the bilinear control of a rotating linear top: as a corollary, we obtain the approximate controllability in arbitrarily small times among particular eigenfunctions of the Laplacian of S2.

Some evaluations of the Jones polynomial for certain families of weaving knots

In this paper, we derive formulae for the determinant of weaving knots W(3,n) and W(p,2). We calculate the dimension of the first homology group with coefficients in Z3 of the double cyclic cover of the 3-sphere S3 branched over W(3,n) and W(p,2) respectively. As a consequence, we obtain a lower bound of the unknotting number of W(3,n) for certain values of n.

Link as a complete invariant of Morse-Smale 3-diffeomorphisms

In this paper we consider gradient-like Morse-Smale diffeomorphisms defined on the three-dimensional sphere S3 . For such diffeomorphisms, a complete invariant of topological conjugacy was obtained in the works of C. Bonatti, V. Grines, V. Medvedev, E. Pecu. It is an equivalence class of a set of homotopically non-trivially embedded tori and Klein bottles embedded in some closed 3-manifold whose fundamental group admits an epimorphism to the group Z . Such an invariant is called the scheme of the gradient-like diffeomorphism f:S3→S3 . We single out a class G of diffeomorphisms whose complete invariant is a topologically simpler object, namely, the link of essential knots in the manifold S2×S1 . The diffeomorphisms under consideration are determined by the fact that their non-wandering set contains a unique source, and the closures of stable saddle point manifolds bound three-dimensional balls with pairwise disjoint interiors. We prove that, in addition to the closure of these balls, a diffeomorphism of the class G contains exactly one nonwandering point, which is a fixed sink. It is established that the total invariant of topological conjugacy of class G diffeomorphisms is the space of orbits of unstable saddle separatrices in the basin of this sink. It is shown that the space of orbits is a link of non-contractible knots in the manifold S2×S1 and that the equivalence of links is tantamount to the equivalence of schemes. We also provide a realization of diffeomorphisms of the considered class along an arbitrary link consisting of essential nodes in the manifold S2×S1 .

Long-time principal geodesic analysis in director-based dynamics of hybrid mechanical systems

In this article, we investigate an extended version of principal geodesic analysis for the unit sphere S2 and the special orthogonal group SO(3). In contrast to prior work, we address the construction of long-time smooth lifts of possibly non-localized data across branches of the respective logarithm maps. To this end, we pay special attention to certain critical numerical aspects such as singularities and their consequences on the numerical accuracy. Moreover, we apply principal geodesic analysis to investigate the behavior of several mechanical systems that are very rich in dynamics. The examples chosen are computationally modeled by employing a director-based formulation for rigid and flexible mechanical systems. Such a formulation allows to investigate our algorithms in a direct manner while avoiding the introduction of additional sources of error that are unrelated to principal geodesic analysis. Finally, we test our numerical machinery with the examples and, at the same time, we gain deeper insight into their dynamical behavior.

Some classifications of 2-dimensional self-shrinkers

It is well-known that self-shrinkers play an important role in the study of mean curvature flows. In this paper, we develop new techniques to study the rigidity of self-shrinkers. We prove that any self-shrinker X:M→R3 with nonzero constant Gauss curvature is the round sphere S2(2). Moreover, we prove that any flat self-shrinker X:M→R3 is a plane R2, a cylinder S1(1)×R, a generalized cylinder Γ×R, where Γ is an Abresch-Langer curve. At last, we show that any self-shrinker X:M→R3 with constant mean curvature is a plane R2, a cylinder S1(1)×R or the round sphere S2(2).