Sphere S2 Research Articles

Swarms on the 3-sphere with adaptive synapses: Hebbian and anti-Hebbian learning rule

We introduce and analyze several models of swarm dynamics on the sphere S3 with adaptive (state-dependent) interactions between agents. The equations describing the interaction dynamics are variations of the classical Hebbian principle from Neuroscience. We study asymptotic behavior in models with various realizations of Hebbian and anti-Hebbian learning rules. The swarm with the Hebbian rule and strictly nonnegative (attractive) interactions evolves towards consensus. If the Hebbian rule allows both attractive and repulsive interactions the swarm converges to bipolar configuration. The most interesting is the model with anti-Hebbian learning rule with both attractive and repulsive interactions. This model displays a rich variety of dynamical regimes and stationary formations, depending on the number of agents and system parameters. We prove that the model with such anti-Hebbian rule evolves towards a stable stationary configuration if the system parameter is above a certain bifurcation threshold. Finally, some simulation results are presented demonstrating how these theoretical results can be applied to coordination of rotating bodies in 3D space; this is done by mapping the trajectories from S3 to special orthogonal group SO(3).

Gauged six-dimensional supergravity from warped IIB reductions

We find a family of complete non-linear Kaluza-Klein reduction ansätze from type IIB supergravity to Romans’ 6D F (4) gauged supergravity in the bosonic sector. The reduction is over a sphere S2 and a Riemann surface Σ, and depends on a pair of arbitrary locally holomorphic functions {mathcal{A}}_{pm } on Σ. This family of reductions is inspired by the recent construction of 1/2 BPS supersymmetric warped AdS6 solutions of IIB supergravity that depend on these same functions {mathcal{A}}_{pm } .

Three Vortices in Spaces of Constant Curvature: Reduction, Poisson Geometry, and Stability

This paper is concerned with the problem of three vortices on a sphere S2 and the Lobachevsky plane L2. After reduction, the problem reduces in both cases to investigating a Hamiltonian system with a degenerate quadratic Poisson bracket, which makes it possible to study it using the methods of Poisson geometry. This paper presents a topological classification of types of symplectic leaves depending on the values of Casimir functions and system parameters.

Stochastic Landau–Lifshitz–Gilbert equation with anisotropy energy driven by pure jump noise

In this work we study a stochastic three-dimensional Landau–Lifshitz–Gilbert equation with non-zero anisotropy energy, which is drive by pure jump noise. We show existence of weak martingale solutions taking values in a two-dimensional sphere S2. The construction of the solution is based on the classical Faedo–Galerkin approximation, the compactness method and the Jakubowski version of the Skorokhod Theorem for nonmetric spaces.

Characterization of f-extremal disks

We show uniqueness for overdetermined elliptic problems defined on topological disks Ω with C2 boundary, i.e., positive solutions u to Δu+f(u)=0 in Ω⊂(M2,g) so that u=0 and ∂u∂η→=cte along ∂Ω, η→ the unit outward normal along ∂Ω under the assumption of the existence of a candidate family. To do so, we adapt the Gálvez–Mira generalized Hopf-type Theorem [19] to the realm of overdetermined elliptic problem.When (M2,g) is the standard sphere S2 and f is a C1 function so that f(x)>0 and f(x)≥xf′(x) for any x∈R+⁎, we construct such candidate family considering rotationally symmetric solutions. This proves the Berestycki–Caffarelli–Nirenberg conjecture in S2 for this choice of f. More precisely, this shows that if u is a positive solution to Δu+f(u)=0 on a topological disk Ω⊂S2 with C2 boundary so that u=0 and ∂u∂η→=cte along ∂Ω, then Ω must be a geodesic disk and u is rotationally symmetric. In particular, this gives a positive answer to the Schiffer conjecture D (cf. [33,35]) for the first Dirichlet eigenvalue and classifies simply-connected harmonic domains (cf. [28], also called Serrin Problem) in S2.

Stationary vacuum black holes in 5 dimensions

We study the problem of asymptotically flat bi-axially symmetric stationary solutions of the vacuum Einstein equations in 5-dimensional spacetime. In this setting, the cross section of any connected component of the event horizon is a prime 3-manifold of positive Yamabe type, namely the 3-sphere S3, the ring , or the lens space L(p, q). The Einstein vacuum equations reduce to an axially symmetric harmonic map with prescribed singularities from into the symmetric space . In this paper, we solve the problem for all possible topologies, and in particular the first candidates for smooth vacuum non-degenerate black lenses are produced. In addition, a generalization of this result is given in which the spacetime is allowed to have orbifold singularities. We also formulate conditions for the absence of conical singularities which guarantee a physically relevant solution.

Small data scattering of semirelativistic Hartree equation

In this paper we study the small data scattering of Hartree type semirelativistic equation in space dimension 3. The Hartree type nonlinearity is [V∗|u|2]u and the potential V which generalizes the Yukawa has some growth condition. We show that the solution scatters to linear solution if an initial data given in Hs,1 is sufficiently small and s>14. Here, Hs,1 is Sobolev type space taking in angular regularity with norm defined by ‖φ‖Hs,1=‖φ‖Hs+‖∇Sφ‖Hs. To establish the results we employ the recently developed Strichartz estimate which is Lθ2-averaged on the unit sphere S2 and construct the resolution space based on Up-Vp space.

Some solutions for scalar models and conformal invariance

It is discussed solutions in scalar conformal field theories with finite action. One can observe such solutions in flat and curved spaces. As example is considered sphere S4. Another solution with non-zero action is given. Also it is discussed a similar solution in cosmology.

Left Lie reduction for curves in homogeneous spaces

Let H be a closed subgroup of a connected finite-dimensional Lie group G, where the canonical projection π : G → G/H is a Riemannian submersion with respect to a bi-invariant Riemannian metric on G. Given a C∞ curve x : [a, b] → G/H, let $\tilde {x}:[a,b]\rightarrow G$ be the horizontal lifting of x with $\tilde {x}(a)=e$ , where e denotes the identity of G. When (G, H) is a Riemannian symmetric pair, we prove that the left Lie reduction $V(t):=\tilde x(t)^{-1}\dot {\tilde x}(t)$ of $\dot {\tilde x}(t)$ for t ∈ [a, b] can be identified with the parallel pullbackP(t) of the velocity vector $\dot {x}(t)$ from x(t) to x(a) along x. Then left Lie reductions are used to investigate Riemannian cubics, Riemannian cubics in tension and elastica in homogeneous spaces G/H. Simplifications of reduced equations are found when (G, H) is a Riemannian symmetric pair. These equations are compared with equations known for curves in Lie groups, focusing on the special case of Riemannian cubics in the 3-dimensional unit sphere S3.

On topological actions of finite groups on S3

We consider orientation-preserving actions of a finite group G on the 3-sphere S3 (and also on Euclidean space R3). By the geometrization of finite group actions on 3-manifolds, if such an action is smooth then it is conjugate to an orthogonal action, and in particular G is isomorphic to a subgroup of the orthogonal group SO(4) (or of SO(3) in the case of R3). On the other hand, there are topological actions with wildly embedded fixed point sets; such actions are not conjugate to smooth actions but one would still expect that the corresponding groups G are isomorphic to subgroups of the orthogonal groups SO(4) (or of SO(3), resp.). In the present paper, we obtain some results in this direction; we prove that the only finite, nonabelian simple group with a topological action on S3, or on any homology 3-sphere, is the alternating or dodecahedral group A5 (the only finite, nonabelian simple subgroup of SO(4)), and that every finite group with a topological, orientation-preserving action on Euclidean space R3 is in fact isomorphic to a subgroup of SO(3).

Equilibrium States for Expanding Thurston Maps

In this paper, we use the thermodynamical formalism to show that there exists a unique equilibrium state $${\mu_\phi}$$ for each expanding Thurston map $${f : S^2\rightarrow S^2}$$ together with a real-valued Holder continuous potential $${\phi}$$ . Here the sphere S2 is equipped with a natural metric induced by f, called a visual metric. We also prove that identical equilibrium states correspond to potentials that are co-homologous up to a constant, and that the measure-preserving transformation f of the probability space $${(S^2,\mu_\phi)}$$ is exact, and in particular, mixing and ergodic. Moreover, we establish versions of equidistribution of preimages under iterates of f, and a version of equidistribution of a random backward orbit, with respect to the equilibrium state. As a consequence, we recover various results in the literature for a postcritically-finite rational map with no periodic critical points on the Riemann sphere equipped with the chordal metric.

Strong local nondeterminism of spherical fractional Brownian motion

Let B=Bx,x∈S2 be the fractional Brownian motion indexed by the unit sphere S2 with index 0<H≤12, introduced by Istas 2015. We establish optimal upper and lower bounds for its angular power spectrum {dℓ,ℓ=0,1,2,…}, and then exploit its high-frequency behavior to establish the property of its strong local nondeterminism of B.

On 3-submanifolds of S3 which admit complete spanning curve systems

Let M be a compact connected 3-submanifold of the 3-sphere S3 with one boundary component F such that there exists a collection of n pairwise disjoint connected orientable surfaces S = {S1, · · ·, S n } properly embedded in M, ∂S = {∂S1, · · ·, ∂S n } is a complete curve system on F. We call S a complete surface system for M, and ∂S a complete spanning curve system for M. In the present paper, the authors show that the equivalent classes of complete spanning curve systems for M are unique, that is, any complete spanning curve system for M is equivalent to ∂S. As an application of the result, it is shown that the image of the natural homomorphism from the mapping class group M(M) to M(F) is a subgroup of the handlebody subgroup H n .

Sasakian manifolds with purely transversal Bach tensor

We show that a (2n + 1)-dimensional Sasakian manifold (M, g) with a purely transversal Bach tensor has constant scalar curvature ≥2n(2n+1), equality holding if and only if (M, g) is Einstein. For dimension 3, M is locally isometric to the unit sphere S3. For dimension 5, if in addition (M, g) is complete, then it has positive Ricci curvature and is compact with finite fundamental group π1(M).

Exact self-dual skyrmions

We introduce a Skyrme type model with the target space being the sphere S3 and with an action possessing, as usual, quadratic and quartic terms in field derivatives. The novel character of the model is that the strength of the couplings of those two terms are allowed to depend upon the space–time coordinates. The model should therefore be interpreted as an effective theory, such that those couplings correspond in fact to low energy expectation values of fields belonging to a more fundamental theory at high energies. The theory possesses a self-dual sector that saturates the Bogomolny bound leading to an energy depending linearly on the topological charge. The self-duality equations are conformally invariant in three space dimensions leading to a toroidal ansatz and exact self-dual Skyrmion solutions. Those solutions are labelled by two integers and, despite their toroidal character, the energy density is spherically symmetric when those integers are equal and oblate or prolate otherwise.

Closed Willmore minimal hypersurfaces with constant scalar curvature in [formula omitted] are isoparametric

In this paper, we will prove that any closed minimal Willmore hypersurface M4 of S5 with constant scalar curvature must be isoparametric. To be precise, M4 is either an equatorial 4 sphere, a product of sphere S2(22)×S2(22) or a Cartan's minimal hypersurface. In particular, the value of the second fundamental form S can only be 0, 4, 12. This result strongly supports Chern's Conjecture.

Generalized Lorenz equations on a three-sphere

Edward Lorenz is best known for one specific three-dimensional differential equation, but he actually created a variety of related N-dimensional models. In this paper, we discuss a unifying principle for these models and put them into an overall mathematical framework. Because this family of models is so large, we are forced to choose. We sample the variety of dynamics seen in these models, by concentrating on a four-dimensional version of the Lorenz models for which there are three parameters and the norm of the solution vector is preserved. We can therefore restrict our focus to trajectories on the unit sphere S 3 in ℝ4. Furthermore, we create a type of Poincare return map. We choose the Poincare surface to be the set where one of the variables is 0, i.e., the Poincare surface is a two-sphere S 2 in ℝ3. Examining different choices of our three parameters, we illustrate the wide variety of dynamical behaviors, including chaotic attractors, period doubling cascades, Standard-Map-like structures, and quasiperiodic trajectories. Note that neither Standard-Map-like structure nor quasiperiodicity has previously been reported for Lorenz models.

The region index and the unknotting number of a knot

For any knot in the 3-sphere S3, there exists a diagram such that we have the unknot if we change all crossings on the boundary of some region of the diagram. The minimal number of the crossing changes over all such diagrams is called the region index of a knot. Clearly, the unknotting number is less than or equal to the region index. In this paper, we show that there exists a knot which has a gap between the unknotting number m and the region index for any positive integer m (m≥2) by using the Goeritz invariant. We also show that there exists a knot which has the unknotting number and the region index that are equal to n for any positive integer n (n≥2) by using the Rasmussen invariant.

Ruled three-dimensional CR submanifolds of the sphere S6(1)

We investigate proper, three-dimensional CR submanifolds of the nearly Kahler sphere S6(1) ruled by totally geodesic spheres S2(1), and classify them by using a sphere curve and a vector field along that curve.

SU(2) particle sigma model: the role of contact symmetries in global quantization

In this paper we achieve the quantization of a particle moving on the SU(2) group manifold, that is, the three-dimensional sphere S3, by using group-theoretical methods. For this purpose, a fundamental role is played by contact symmetries, i.e., symmetries that leave the Poincaré–Cartan form semi-invariant at the classical level, although not necessarily the Lagrangian. Special attention is paid to the role played by the basic quantum commutators, which depart from the canonical, Heisenberg–Weyl ones, as well as the relationship between the integration measure in the Hilbert space of the system and the non-trivial topology of the configuration space. Also, the quantization on momentum space is briefly outlined.