We classify hyperbolic monopoles with continuous symmetries and construct new examples.
problem Classifying and constructing hyperbolic monopoles with continuous symmetries.
method Developed a Structure Theorem and used representation theory to simplify the problem.
result Found constraints on structure groups and constructed novel spherically symmetric Sp(n) hyperbolic monopoles. Researchers found multiple spherical Ricci metrics on tori with rotational symmetry.
problem Constructing and analyzing spherical Ricci metrics with rotational symmetry.
method Explicitly constructed a two-parameter family of metrics with rotational symmetry and showed their existence on tori.
result Infinitely many non-isometric spherical Ricci metrics can be realized on the same torus.
New symmetries found in Riemann-Cartan geometries.
problem Investigating symmetries in geometries with curvature and torsion.
method Mathematical tools to determine symmetries and subclasses of geometries.
result Determined all static and stationary spherically symmetric Riemann-Cartan geometries and subclasses with specific symmetries.
Study of continuous symmetries in Nahm data and BPS monopoles.
problem Solutions to Nahm's equations with continuous symmetries.
method Classification of Ansätze and construction of Nahm data.
result Construction of new BPS monopoles with spherical symmetry.
Published in 1999, Christodoulou proved that the naked singularities of a self-gravitating scalar field are not stable in spherical symmetry and therefore the cosmic censorship conjecture is true in this context. The original proof is by contradiction and sharp estimates are obtained strictly depending on spherical sym…
Unique steady and expanding solitons with spherical links identified.
problem Characterizing steady and expanding Ricci solitons with specific asymptotic symmetries.
method Symmetry principle applied to asymptotically cylindrical and conical GRSs, proving uniqueness for Bryant solitons.
result Bryant steady and expanding solitons are the unique asymptotically cylindrical and conical GRSs with spherical links under certain conditions.
Study biharmonic functions on vector bundles with spherical symmetry.
problem Investigate biharmonic functions on vector bundles with spherically symmetric metrics.
method Analyze vertical lifts and radial functions of functions on vector bundle manifolds.
result Construct an infinite two-parameter family of proper biharmonic functions.
Proves local existence and extension principle for Einstein Yang--Mills system with spherical symmetry.
problem Local existence and stability of the spherically symmetric Einstein Yang--Mills system.
method Employed an L2-based method to prove local existence and establish an extension principle. result Established local existence and extension principle for the SSEYM with H1 data. Characterizes conical angles for metrics with dihedral symmetry.
problem Understanding metrics with specific symmetry properties.
method Using recent results on local invariants of quadratic differentials.
result Complete characterization of conical angles for dihedral spherical metrics.
New method uses scalar-based models to approximate spherical tensors efficiently.
problem Efficiently approximating spherical tensors with equivariant functions.
method Expressing equivariant functions as the product of a scalar function and a small tensor basis.
result Approximations are fast, simple to implement, and accurate in practical settings.
We give a complete classification of intertwining operators (symmetry breaking operators) between spherical principal series representations of G=O(n+1,1) and G'=O(n,1). We construct three meromorphic families of the symmetry breaking operators, and find their distribution kernels and their residues at all poles explic…
Novikov equation symmetries, solutions, and pseudo-spherical surfaces studied.
problem Analyzing geometrically integrable Novikov equation properties.
method Lie symmetries, group-invariant solutions, conservation laws, unique continuation, pseudo-spherical surfaces.
result Classification of invariant solutions and existence of analytic metrics for pseudo-spherical surfaces.
We provide a direct proof for the positivity of Chen-Nester-Tung quasi-local energy with analytic reference in spherical symmetry. A hoop-type theorem for this energy is also established. Finally, the relation between Chen-Nester-Tung and Brown-York quasi-local energies will be discussed.
Abstract properties of hypersurface data analyzed in spherical symmetry.
problem Analyzing hypersurface data in spherical symmetry.
method Study of hypersurface data properties, gauge group, and curvature tensor.
result General solution of Einstein field equations in vacuum and Lorentzian ambient signature.
Paper proves trapped surface formation for Einstein-Maxwell-charged scalar field system.
problem Formation of trapped surfaces in Einstein-Maxwell-charged scalar field system.
method Generalized Christodoulou's approach for spherical symmetry and improved for Minkowskian data.
result Improved bound on trapped surface formation for Minkowskian data.
The paper examines properties of spherical Finsler metrics, proving their semi-C-reducibility and conditions for vanishing mean stretch curvature.
problem Analyzing curvature properties of spherical Finsler metrics.
method Proving semi-C-reducibility and finding conditions for vanishing mean stretch curvature.
result Conditions for a general spherically symmetric Finsler metric to have vanishing mean stretch curvature.
Study shows directional convergence for neural networks under spherical symmetry.
problem Learning linear predictors with neural networks under spherically symmetric data.
method Analysis of gradient flow and gradient descent for two-layer and deep linear networks.
result Directional convergence guarantees with exact convergence rate for specific network architectures.
It is known that every ribbon category with unimodality allows symmetrized 6j-symbols with full tetrahedral symmetries while a spherical category does not in general. We give an explicit counterexample for this, namely the category E. We define the mirror conjugate symmetry of 6j-symbols instead and sho…
The paper explores geometric properties of interception curves on planes and spheres.
problem Geometric properties of interception curves defined by differential equations.
method Parametric representation and spherical curve defined by Gudermannian function.
result Symmetry/asymmetry between spherical and planar cases, connections to lemniscate constants.
New method uses spherical harmonics to simplify learning single-index models.
problem Learning single-index models with unknown one-dimensional projections.
method Proposes using spherical harmonics instead of Hermite polynomials to capture rotational symmetry.
result Characterizes the complexity of learning single-index models under arbitrary spherically symmetric input distributions.
Presented spherical symmetric teleparallel geometry frames and field equations.
problem Teleparallel geometry with spherical symmetry.
method Developed proper and diagonal co-frames, spin connections, and field equations.
result Advantage of diagonal co-frame over proper in f(T) teleparallel gravity.
In this paper, we give the general form of spherically symmetric Finsler metrics in Rn and surprisedly find that many well-known Finsler metrics belong to this class. Then we explicitly express projective metrics of this type. The necessary and sufficient conditions that projective Finsler metrics with spherical sym…
It is shown that the initial singularities in spatially compact spacetimes with spherical, plane or hyperbolic symmetry admitting a compact constant mean curvature hypersurface are crushing singularities when the matter content of spacetime is described by the Vlasov equation (collisionless matter) or the wave equation…
New RL method designs 3D molecules with improved symmetry.
problem Lack of 3D information in molecular design.
method Symmetry-aware actor-critic architecture using spherical harmonics.
result Improves generalization and molecule quality.
We prove some related results concerning blow-up solutions for the Jang equation. First: it has been shown that, given an outermost marginally outer trapped surface (MOTS) Σ, there exists a solution to Jang's equation which blows up at Σ. Here we show that in addition, large classes of spherically symmetric initial dat…
The paper proves the existence and uniqueness of Calabi-Yau metrics on affine spherical varieties.
problem Existence and uniqueness of Calabi-Yau metrics on affine spherical varieties.
method Explicit K-stability condition, degeneration, and asymptotic cone analysis.
result Uniqueness of K-invariant Calabi-Yau metrics on affine spherical manifolds. Small deformations of marginally (outer) trapped surfaces are considered by using their stability operator. In the case of spherical symmetry, one can use these deformations on any marginally trapped round sphere to prove several interesting results. The concept of 'core' of a black hole is introduced: it is a minimal …
Characterizes spherical Finsler metrics satisfying a specific condition.
problem Spherically symmetric Finsler metrics satisfying the σT-condition. method Complete characterization and subclass investigation within Landsberg category.
result Identifies precise conditions for metrics satisfying the T-condition. The Regge symmetry is a set of remarkable relations between two tetrahedra whose edge lengths are related in a simple fashion. It was first discovered as a consequence of an asymptotic formula in mathematical physics. Here we give a simple geometric proof of Regge symmetries in Euclidean, spherical, and hyperbolic geom…
Study spherically symmetric Finsler metrics with specific curvature properties.
problem Characterize Finsler metrics with scalar and constant flag curvature.
method Analyze spherically symmetric metrics on symmetric spaces with given curvature properties.
result Provide families of Finsler metrics with scalar and constant flag curvature.
Proves stability of spacetime Penrose inequality for spherical symmetric initial data.
problem Stability of the Penrose inequality for spherical symmetric spacetimes.
method Formulated and proved stability statement using spherical symmetry and asymptotically flat initial data.
result Initial data must arise from an isometric embedding into a static spacetime close to Schwarzschild spacetime.
New rigidity found for 3D warped product domains.
problem Finding rigidity conditions for warped product domains.
method Developed scalar curvature rigidity for a general class of domains.
result Identified domains satisfying a boundary condition analogous to logarithmic concavity.
We show that for a real-analytic connected holomorphically nondegenerate 5-dimensional CR-hypersurface M and its symmetry algebra s one has either: (i) dims=15 and M is spherical (with Levi form of signature either (2,0) or (1,1) everywhere), or (ii) dims≤11 where $\di…
New exact spherically symmetric vacuum solutions found in Finsler gravity.
problem Finding exact vacuum solutions in Finsler gravity.
method Spherically symmetric, asymptotically flat Berwald spacetimes solved for Finsler gravity vacuum equation.
result Only one class of spherically symmetric Berwald spacetimes is compatible with asymptotic flatness and a well-defined causal structure.
In this paper, we initiate the study of the instability of naked singularities without symmetries. In a series of papers, Christodoulou proved that naked singularities are not stable in the context of the spherically symmetric Einstein equations coupled with a massless scalar field. We study in this paper the next simp…
Paper relaxes symmetry conditions for universal feature selection in noisy data.
problem Feature selection in noisy data with weak symmetry.
method Developed a universal feature selection framework using singular value decomposition of canonical dependence matrix.
result Selected features achieve asymptotically optimal error exponents up to a residual term.
Locally classifies 4D spherical symmetric Finsler spaces.
problem Classifying 4D spherical symmetric Finsler spaces.
method Local classification of pseudo-Finsler Berwald structures.
result Six classes of non-Riemannian SO(3)-symmetric pseudo-Finsler Berwald functions.
The present paper considers volume formulae, as well as trigonometric identities, that hold for a tetrahedron in 3-dimensional spherical space of constant sectional curvature +1. The tetrahedron possesses a certain symmetry: namely rotation through angle π in the middle points of a certain pair of its skew edges.
Spherical data is found in many applications. By modeling the discretized sphere as a graph, we can accommodate non-uniformly distributed, partial, and changing samplings. Moreover, graph convolutions are computationally more efficient than spherical convolutions. As equivariance is desired to exploit rotational symmet…
The study finds regular null hypersurfaces in a perturbed Schwarzschild black hole exterior.
problem Existence of regular null hypersurfaces in a perturbed Schwarzschild black hole.
method Proof of existence for null hypersurfaces in a perturbed Schwarzschild spacetime.
result Existence of many foliations by regular null hypersurfaces in the exterior region of a perturbed Schwarzschild black hole.
Derives a formula for fermion dimensions in spherically symmetric monopole backgrounds.
problem Calculating the dimension of the plane-wave normalizable kernel for massless fermions in spherically symmetric monopole backgrounds.
method Derives a formula for the dimension of the plane-wave normalizable kernel of the Dirac operator for fermions of any representation of SU(N) in the presence of any spherically symmetric monopole background.
result Derives a formula for the dimension of the plane-wave normalizable kernel of the Dirac operator.
Study proposes initial data sets for solving gravitational equations, proving energy estimates.
problem Solving the constraint equations in the evolutionary form.
method Proposes a family of initial data sets, proving Penrose-like energy estimates.
result Established existence of solutions for specific cases.
Study on naked singularities without symmetry, forming incomplete future null infinity and singular inner Cauchy horizon.
problem Formation of naked singularities in Einstein-scalar field system without symmetry assumptions.
method Employing four-type differences and scale-invariant weighted norms to control geometry.
result Global naked singularity structure with incomplete future null infinity and singular inner Cauchy horizon.
The study proves strong cosmic censorship violation for spherically symmetric dust clouds.
problem Violation of strong cosmic censorship for spherically symmetric dust clouds.
method Derived an ordinary differential equation for light rays and used it to prove strong cosmic censorship violation.
result Generic violation of strong cosmic censorship for spherically symmetric dust clouds.
Stability of extremal Reissner-Nordström black holes proven in spherical symmetry.
problem Stability of extremal Reissner-Nordström black holes in spherical symmetry.
method Proved nonlinear asymptotic stability through spherically symmetric characteristic data.
result Existence of a submanifold Mstab leading to stable solutions. The study constructs free boundary CMC annuli in spherical and hyperbolic balls.
problem Finding free boundary CMC annuli in spherical and hyperbolic balls.
method Constructing free boundary CMC annuli with constant mean curvature H in geodesic balls of S^3 and H^3.
result Embedded free boundary CMC annuli exist for certain mean curvatures in both spaces.
We use the notion of intrinsic flat distance to address the almost rigidity of the positive mass theorem for asymptotically hyperbolic manifolds. In particular, we prove that a sequence of spherically symmetric asymptotically hyperbolic manifolds satisfying the conditions of the positive mass theorem converges to hyper…
We treat the problem of estimation of orientation parameters whose values are invariant to transformations from a spherical symmetry group. Previous work has shown that any such group-invariant distribution must satisfy a restricted finite mixture representation, which allows the orientation parameter to be estimated u…