Dimensional Sphere Research Articles

Sasakian geometry on sphere bundles II: Constant scalar curvature

In a previous paper [18] the authors employed the fiber join construction of Yamazaki [38] together with the admissible construction of Apostolov, Calderbank, Gauduchon, and Tønnesen-Friedman [2] to construct new extremal Sasaki metrics on odd dimensional sphere bundles over smooth projective algebraic varieties. In the present paper we continue this study by applying a recent existence theorem [14] that shows that under certain conditions one can always obtain a constant scalar curvature Sasaki metric in the Sasaki cone. Moreover, we explicitly describe this construction for certain sphere bundles of dimension 5 and 7.

Circle Actions on Oriented Manifolds With 3 Fixed Points

Abstract Let the circle group act on a compact oriented manifold $M$ with a non-empty discrete fixed point set. Then the dimension of $M$ is even. If $M$ has one fixed point, $M$ is the point. In any even dimension, such a manifold $M$ with two fixed points exists, a rotation of an even dimensional sphere. Suppose that $M$ has three fixed points. Then the dimension of $M$ is a multiple of 4. Under the assumption that each isotropy submanifold is orientable, we show that if $\dim M=8$, then the weights at the fixed points agree with those of an action on the quaternionic projective space $\mathbb{H}\mathbb{P}^{2}$, and show that there is no such 12-dimensional manifold $M$.

Perfectly spherical Bloch hyper-spheres from quantum matrix geometry

Exploiting analogies between the precessing quantum spin system and the charge-monopole system, we construct Bloch hyper-spheres with exact spherical symmetries in arbitrary dimensions. Such Bloch hyper-spheres are realized as a collection of the orbits of a precessing quantum spin. The geometry of Bloch hyper-spheres is exactly equal to the quantum Nambu geometry of higher dimensional fuzzy spheres. The stabilizer group symmetry of the Bloch hyper-sphere necessarily introduces degenerate spin-coherent states, giving rise to the Wilczek-Zee geometric phase of non-Abelian monopoles associated with the hyper-sphere holonomy. The degenerate spin-coherent states induce matrix-valued quantum geometric tensors. While the minimal spin Bloch hyper-spheres exhibit similar properties in even and odd dimensions, their large spin counterparts differ qualitatively depending on the parity of the dimensions. Exact correspondences between spin-coherent states and monopole harmonics in higher dimensions are established. We also investigate density matrices described by Bloch hyper-balls and elucidate their corresponding statistical and geometric properties, such as von Neumann entropies and Bures quantum metrics.

The Equality of the Values of the Area and Perimeter of a Number of Two– Dimensional Figures, Volume and Area - Three-Dimensional

Possible variants of the equality of the values of the area and perimeter of a number of two–dimensional figures (square, circle, rectangular, obtuse and equilateral triangles), volume and area - three-dimensional (Platonic bodies, cone, cylinder, pyramid and sphere) are considered.

Dynamics-augmented cluster-based network model

In this study we propose a novel data-driven reduced-order model for complex dynamics, including nonlinear, multi-attractor, multi-frequency and multiscale behaviours. The starting point is a fully automatable cluster-based network model (CNM) (Li et al., J. Fluid Mech., vol. 906, 2021, A21) that kinematically coarse grains the state with clusters and dynamically predicts the transitions in a network model. In the proposed dynamics-augmented CNM (dCNM) the prediction error is reduced with trajectory-based clustering using the same number of centroids. The dCNM is first exemplified for the Lorenz system and then demonstrated for the three-dimensional sphere wake featuring periodic, quasi-periodic and chaotic flow regimes. For both plants, the dCNM significantly outperforms the CNM in resolving the multi-frequency and multiscale dynamics. This increased prediction accuracy is obtained by stratification of the state space aligned with the direction of the trajectories. Thus, the dCNM has numerous potential applications to a large spectrum of shear flows, even for complex dynamics.

Photocatalytic and fouling resistant MXene/3D-S-COF for efficient oil-water emulsion separation

Three-dimensional sphere covalent organic framework (3D-S-COF) membranes have received increasing attention in the field of oily wastewater and industrial organic matter due to their rich pore structure as well as having many exposed functional group sites. However, the synthesis of conventional imine COF through the Schiff base method, which suffers from the easy hydrolysis under organic solvent conditions. Herein, a novel super-wettable MXene/3D-S-COF composite membrane for oil–water emulsion separation was prepared by depositing MXene nanosheets on oxidized amide COF from the imine COF. The hydroxyl groups of the COF membrane not only improved the water permeability but also formed hydrogen bonds with the MXene layer, which enhanced the interaction and formed the high stability composite membrane. The porous 3D-S-COF and the zigzag MXene layer work synergistically to enable emulsion fluxes up to 240 L·m−2h−1 bar−1 and the separation efficiency is more than 98 %. In addition, the residual mass of the composite membrane remained above 95 % after 10 days of immersion in various solvents, and it also has photocatalytic properties and good resistance to fouling. This study provides a brand new idea for the use of MXene/3D-S-COF based composite membrane for oil–water emulsion separation.

Möbius homogeneous hypersurfaces in [formula omitted]

Let M(Sn+1) denote the Möbius transformation group of the (n+1)-dimensional sphere Sn+1. A hypersurface x:Mn→Sn+1 is called a Möbius homogeneous hypersurface if there exists a subgroup G of M(Sn+1) such that the orbit G⋅p=x(Mn),p∈x(Mn). In this paper, the Möbius homogeneous hypersurfaces are classified completely up to a Möbius transformation of Sn+1.

The characterization of (𝑛 − 1)-spheres with 𝑛 + 4 vertices having maximal Buchstaber number

Abstract We present a computationally efficient algorithm that is suitable for graphic processing unit implementation. This algorithm enables the identification of all weak pseudo-manifolds that meet specific facet conditions, drawn from a given input set. We employ this approach to enumerate toric colorable seeds. Consequently, we achieve a comprehensive characterization of ( n − 1 ) (n-1) -dimensional PL spheres with n + 4 n+4 vertices that possess a maximal Buchstaber number. A primary focus of this research is the fundamental categorization of non-singular complete toric varieties of Picard number 4. This classification serves as a valuable tool for addressing questions related to toric manifolds of Picard number 4. Notably, we have determined which of these manifolds satisfy equality within an inequality regarding the number of minimal components in their rational curve space. This addresses a question posed by Chen, Fu, and Hwang in 2014 for this specific case.

On conformally flat minimal Legendrian submanifolds in the unit sphere

This paper is concerned with the study on an open problem of classifying conformally flat minimal Legendrian submanifolds in the $(2n+1)$ -dimensional unit sphere $\mathbb {S}^{2n+1}$ admitting a Sasakian structure $(\varphi,\,\xi,\,\eta,\,g)$ for $n\ge 3$ , motivated by the classification of minimal Legendrian submanifolds with constant sectional curvature. First of all, we completely classify such Legendrian submanifolds by assuming that the tensor $K:=-\varphi h$ is semi-parallel, which is introduced as a natural extension of $C$ -parallel second fundamental form $h$ . Secondly, such submanifolds have also been determined under the condition that the Ricci tensor is semi-parallel, generalizing the Einstein condition. Finally, as direct consequences, new characterizations of the Calabi torus are presented.

Spin textures of coherent photons with SU(4) symmetry

We show coherent photons with spin and orbital angular momentum have symmetry of special unitary group of degree four, SU(4), both theoretically and experimentally. We show the degree of polarisation could be a measure to characterise the quantum coherency of a spin state, which was significantly affected by the partial trace out of orbital degree of freedom. By changing the amplitudes of the classically entangled state, we observed a crossover from quantum Heisenberg spin to classical Ising spin. We also demonstrated projections of coherent states, which recover the coherency of the spin state, while the SU(4) state is projected to be a simple product state of SU(2)×SU(2). Based on the developed platform, we have explored spin texture of coherent photons, and found a unique spin profile reminiscent of the Bengal cat upon wrapping to a three-dimensional sphere. We have controlled the phase between singlet and triplet states and found the spin texture rotated upon increasing the phase.

On the separation cut-off phenomenon for Brownian motions on high dimensional spheres

On the separation cut-off phenomenon for Brownian motions on high dimensional spheres

On stability of subelliptic harmonic maps with potential

In this paper, we investigate the stability problem of subelliptic harmonic maps with potential. First, we derive the first and second variation formulas for subelliptic harmonic maps with potential. As a result, it is proved that a subelliptic harmonic map with potential is stable if the target manifold has nonpositive curvature and the Hessian of the potential is nonpositive definite. We also give Leung type results which involve the instability of subelliptic harmonic maps with potential when the target manifold is a sphere of dimension ≥3.

New multiplicity results in prescribing Q-curvature on standard spheres

Abstract In this paper, we study the problem of prescribing Q-Curvature on higher dimensional standard spheres. The problem consists in finding the right assumptions on a function K so that it is the Q-Curvature of a metric conformal to the standard one on the sphere. Using some pinching condition, we track the change in topology that occurs when crossing a critical level (or a virtually critical level if it is a critical point at infinity) and then compute a certain Euler-Poincaré index which allows us to prove the existence of many solutions. The locations of the levels sets of these solutions are determined in a very precise manner. These type of multiplicity results are new and are proved without any assumption of symmetry or periodicity on the function K.

Dynamics of the atom-polyatomic molecule conversion with quantum decoherence

Based on the mean-field approximation and the two-mode Bose–Hubbard model, we investigate the dynamical properties of atom-polyatomic molecule conversion systems under dissipation and dephasing. For such a many-body system problem, we partition phase space based on the states of fixed points, study the movement of fixed points and the transition to attractors in an open system. Additionally, for all the atoms that constitute a molecule, depending on whether the species of these atoms are identical or not, we introduce the innovative concepts of unbiased and biased systems. We discover that in biased conversion systems under particle losses, the evolution trajectories are distributed on a three-dimensional Bloch sphere with an internal structure. There exists a significant difference from the two-dimensional Hilbert space of an unbiased system. Finally, we discuss the sweeping rate of the time-varying magnetic field applied externally to the conversion system, and analyze a feasible scheme for the quantum coherent synthesis of triatomic molecules.

Characterization of a special type of Ricci–Bourguignon soliton on sequential warped product manifold

In this paper, we aim to characterize the sequential warped product [Formula: see text]-almost gradient conformal Ricci–Bourguignon soliton. We derive applications of some vector fields like conformal vector field, torse-forming vector field, torqued vector field on [Formula: see text]-almost conformal Ricci–Bourguignon soliton. The inheritance properties of the Einstein-like sequential warped product [Formula: see text]-almost gradient conformal Ricci–Bourguignon solitons of class types [Formula: see text] are investigated in this paper. Later we show that if [Formula: see text] is a sequential warped product of a complete connected [Formula: see text]-dimensional Riemannian manifold [Formula: see text] and one-dimensional Riemannian manifolds [Formula: see text] and [Formula: see text] admitting [Formula: see text]-almost conformal Ricci–Bourguignon soliton, then [Formula: see text] becomes a [Formula: see text]-dimensional sphere of radius [Formula: see text]. We have discovered that for a [Formula: see text]-almost gradient conformal Ricci–Bourguignon soliton sequential warped product, the warping functions are constants under certain conditions.

Nonlinear (3, 4, 1) multiplet of N=4 , d=1 supersymmetry as a semidynamical spin multiplet

We consider a new type of N=4, d=1 semidynamical multiplet based on the nonlinear version of the mirror multiplet (3,4,1), with the triplet of bosonic physical fields parametrizing a three-dimensional sphere S3 of the radius R. The limit R→∞ amounts to the contraction S3→R3 and leads to the linear mirror multiplet (3,4,1). Spin degrees of freedom described by a Wess-Zumino action specify a two-dimensional surface embedded in the sphere S3. A pair of the examples considered correspond to the round and squashed “fuzzy” 2-spheres. We couple the squashed 2-sphere model to the dynamical mirror multiplet (2,4,2). A notable feature of this coupling is the dependence of the squashing parameter on the bosonic fields z, z¯ of the chiral multiplet. Published by the American Physical Society 2024

A discrete basis for celestial holography

Celestial holography provides a reformulation of scattering amplitudes in four dimensional asymptotically flat spacetimes in terms of conformal correlators of operators on the two dimensional celestial sphere in a basis of boost eigenstates. A basis of massless particle states has been previously identified in terms of conformal primary wavefunctions labeled by a boost weight ∆ = 1+iλ with λ ∈ ℝ. Here we show that a discrete orthogonal and complete basis exists for ∆ ∈ ℤ. This new basis consists of a tower of discrete memory and Goldstone observables, which are conjugate to each other and allow to reconstruct gravitational signals belonging to the Schwartz space. We show how generalized dressed states involving the whole tower of Goldstone operators can be constructed and evaluate the higher spin Goldstone 2-point functions. Finally, we recast the tower of higher spin charges providing a representation of the w1+∞ loop algebra (in the same helicity sector) in terms of the new discrete basis.

Revisiting anapoles in a single high-index dielectric structure

High-index dielectric structures support electric and magnetic Mie resonance. Through careful manipulation of geometric parameters, destructive interference can be induced between electric multipole moments and toroidal multipole moments. This leads to the formation of anapoles, which are characterized by quenched scattering in the far field and giant enhancement in the near field. Here, we revisit the formation mechanism of anapole states in a single dielectric structure with a high refractive index from an eigenmode perspective. We find that scattering efficiency is mainly determined by the intrinsic phase governed by the leaky mode of the structure and the extrinsic phase induced by the frequency deviation from resonance. It is also demonstrated that the anapole modes in a two-dimensional cylinder and a three-dimensional sphere can only occur in the following two situations: (1) when only one mode is involved, the combined phase of intrinsic and extrinsic phase should be equal to 2π at a certain frequency (anapole frequency), which is very close to the resonance frequency. Generally, these types of anapoles are low-order anapoles since low-order resonant modes (i.e., magnetic (electric) dipole and quadrupole) are well separated. (2) If two or more leaky modes are involved, the combined phase for each mode must be 2π at the same frequency located between the two resonances. This corresponds to the high-order anapoles. It is also found that more anapole states will emerge with increasing refractive index. Our results may provide new perspectives for designing high-order anapoles with more freedom.

Divisibility of spheres with measurable pieces

For an r -tuple (\gamma_1,\ldots,\gamma_r) of special orthogonal d\times d matrices, we say that the Euclidean (d-1) -dimensional sphere \mathbb S^{d-1} is (\gamma_1,\ldots,\gamma_r) -divisible if there is a subset A\subseteq \mathbb S^{d-1} such that its translations by the rotations \gamma_1,\ldots,\gamma_r partition the sphere. Motivated by some old open questions of Mycielski and Wagon, we investigate the version of this notion where the set A has to be measurable with respect to the spherical measure. Our main result shows that measurable divisibility is impossible for a "generic" (in various meanings) r -tuple of rotations. This is in stark contrast to the recent result of Conley, Marks and Unger which implies that, for every "generic" r -tuple, divisibility is possible with parts that have the property of Baire.

On Newton polytopes of Lagrangian augmentations

AbstractThis note explores the use of Newton polytopes in the study of Lagrangian fillings of Legendrian submanifolds. In particular, we show that Newton polytopes associated to augmented values of Reeb chords can distinguish infinitely many distinct Lagrangian fillings, both for Legendrian links and higher dimensional Legendrian spheres. The computations we perform work in finite characteristic, which significantly simplifies arguments and also allows us to show that there exist Legendrian links with infinitely many nonorientable exact Lagrangian fillings.