Rigidity shown for spherical product Ricci solitons.
problem Characterizing Ricci solitons on spherical products.
method Ricci flow analysis on and .
result Isolated rigidity of as a shrinking Ricci soliton.
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48 results for “spherical product”
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.
Study spherical T-duality and Massey products in iterated sphere bundles.
problem Understanding spherical T-duality and Massey products in iterated sphere bundles.
method Analyzing Gysin sequences and Massey products to find T-dual iterated sphere bundles.
result For certain iterated sphere bundles, spherical T-duality can be represented by Massey products.
Spherical T-duality for iterated sphere bundles
problem T-duality for iterated sphere bundles
method Repackaging cohomological data into Massey products
result Found T-dual iterated sphere bundles associated to Massey products
Study warped product metrics on hyperbolic and complex hyperbolic manifolds.
problem Understanding curvature of warped product metrics.
method Prove curvature formulas for warped product metrics on hyperbolic and complex hyperbolic manifolds.
result Curvature formulas expressed in spherical coordinates about totally geodesic submanifolds.
Study of Milnor invariants and ropelength of spherical links.
problem Understanding the relationship between the thickness of spherical links and their Milnor invariants.
method Generalized Massey products and Milnor invariants to spherical links, finding optimal asymptotic bounds.
result Optimal asymptotic bounds on Milnor invariants in terms of thickness, revealing a polynomial vs exponential regime.
We provide very general symmetrization theorems in arbitrary dimension and codimension, in products, warped products, and certain fiber bundles such as lens spaces, including Steiner, Schwarz, and spherical symmetrization and admitting density.
The study compares and finds Yamabe constants on warped products.
problem Comparing Yamabe constants on warped products.
method Using fiberwise spherical symmetrization.
result Existence of radially-symmetric Yamabe minimizers on product manifolds.
Study investigates minimal surfaces in spherical caps, extending previous findings.
problem Characterizing minimal surfaces with free boundaries and capillary conditions in spherical caps.
method Extending previous half-space intersection properties to warped products and capillary minimal surfaces in high codimension.
result Established a dual operation relating free boundary and capillary minimal surfaces.
Hyperbolic Knotoidsmath.GT
Study on hyperbolic knotoids, proving their volumes add and providing tables.
problem Defining and studying hyperbolic knotoids.
method Definitions and proofs for hyperbolicity of spherical and planar knotoids, including volume calculations.
result Volumes of hyperbolic spherical knotoids add and rational knotoids have least volume.
Entanglement and productsmath.DG
Quantum states are not entangled if submanifold is a product.
problem Understanding entanglement in quantum states associated with product submanifolds.
method Analyzing quantum states on submanifolds of product Kähler manifolds in the semiclassical limit.
result States are not entangled when submanifold is a product.
Formula derived for spherical growth series of specific groups.
problem Calculating the spherical growth series of specific groups.
method Developed a formula and derived a rational function expression for the spherical growth series.
result Explicit rational function expressions for the spherical growth series of specific groups.
We study special functions on euclidean spaces from the viewpoint of riemannian symmetric spaces. Here the euclidean space where is the semidirect product of the translation group with a closed subgroup of the orthogonal group O(n). We give exact parameterizations of the space of $(G,K…
New flow for capillary surfaces converges to spherical caps.
problem Optimizing capillary surfaces in space forms.
method Constrained mean curvature flow.
result Flow converges to spherical caps globally.
Corrected whitening restores orthogonality in high-dimensional spherical Gaussian mixtures.
problem In high-dimensional data, standard whitening fails to preserve orthogonality of mixture means.
method Derived exact limits for whitened means dot products using random matrix theory, constructed a corrected whitening matrix.
result Corrected whitening allows for improved estimation of spherical Gaussian mixtures in the large-dimensional regime.
New method uses spherical convolutional Wasserstein distance to validate climate models.
problem Ensuring the accuracy of global climate models.
method Spherical convolutional Wasserstein distance to measure model differences.
result Phase 6 models show modest improvements in realistic climatologies.
In this paper, we study Riemannian functionals defined by -norms of Ricci curvature, scalar curvature, Weyl curvature, and Riemannian curvature. We try to understand stability of their critical points that are products of Einstein metrics. In particular, we prove that the product of a spherical space form and a co…
Sharp bounds for eigenfunctions on product spaces restricted to submanifolds.
problem Establishing bounds for eigenfunctions on product spaces.
method Combining asymptotics of Jacobi polynomials and positivity of Fourier coefficients of spherical functions.
result Sharp bounds for eigenfunctions on products of rank-one symmetric spaces.
The paper constructs examples of coupled Dirac-Yang-Mills pairs on Riemannian manifolds.
problem Constructing coupled Dirac-Yang-Mills pairs on Riemannian manifolds.
method Constructing spherically symmetric Dirac-Yang-Mills pairs on Riemannian 3-manifolds with SU(2) structure group.
result The construction yields coupled solutions, including on S^1(r_1) x S^2(r_2) for certain radii.
The paper shows how to use hyperplanes and hyperballs interchangeably using inversive geometry.
problem Tackles the interchangeability of hyperplanes and hyperballs in discriminative boundaries.
method Applies inversive geometry to transform Euclidean data into spherical data and back, providing explicit formulae.
result Shows a duality between hyperspherical caps and hyperballs, providing explicit formulae to map between them.
The paper verifies deep neural networks' ability to approximate functions on spheres.
problem Theoretical verification of deep neural networks' performance on spherical functions.
method Spherical analysis using reproducing kernels and convolutional factorizations.
result Rates of uniform approximation for functions in Sobolev spaces and additive ridge forms.
We prove generalized lower Ricci curvature bounds for warped products over complete Finsler manifolds. On the one hand our result covers a theorem of Bacher and Sturm concerning euclidean and spherical cones. On the other hand it can be seen in analogy to a result of Bishop and Alexander in the setting of Alexandrov sp…
This paper classifies Kaehler submanifolds in hyperbolic space with low codimension.
problem Local classification of Kaehler submanifolds in hyperbolic space with low codimension.
method Intrinsic assumptions and extrinsic product of two-dimensional umbilical spheres in S^3n-1.
result Generalization of results for spherical submanifolds to hyperbolic ambient space.
We study the spherical cap packing problem with a probabilistic approach. Such probabilistic considerations result in an asymptotic sharp universal uniform bound on the maximal inner product between any set of unit vectors and a stochastically independent uniformly distributed unit vector. When the set of unit vectors …
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.
A new overlapping space solves the configuration search problem for graph embeddings.
problem Configuring product spaces for graph embeddings is resource-intensive and impractical.
method Introducing overlapping spaces that share subsets of coordinates between different types of spaces (Euclidean, hyperbolic, spherical).
result Overlapping spaces achieve nearly optimal results without configuration tuning, reducing training time.
New Fourier features improve high-precision approximation in large-scale problems.
problem Designing scalable, high-precision Fourier features for large-scale kernel methods.
method Introducing a new family of quadrature rules that accurately approximate the Gaussian measure in higher dimensions.
result Improved approximation bounds with new Fourier features.
We will show that if a proper complete CAT(0) space X has a visual boundary homeomorphic to the join of two Cantor sets, and X admits a geometric group action by a group containing a subgroup isomorphic to Z^2, then its Tits boundary is the spherical join of two uncountable discrete sets. If X is geodesically complete,…
We show that canonical Carnot-Caratheodory spherical and horospherical metrics, which are defined on the boundary at infinity of every rank one symmetric space of non-compact type, are visual, i.e., they are bilipschitz equivalent with universal bilipschitz constants to the inverse exponent of Gromov products based in …
Triple linking numbers were defined for 3-component oriented surface-links in 4-space using signed triple points on projections in 3-space. In this paper we give an algebraic formulation using intersections of homology classes (or cup products on cohomology groups). We prove that spherical links have trivial triple lin…
We prove the existence of Ricci flow starting from a class of metrics with unbounded curvature, which are doubly-warped products over an interval with a spherical factor pinched off at an end. These provide a forward evolution from some known and conjectured finite-time local singularities of Ricci flow, generalizing p…
Given a closed orientable Euclidean cone 3-manifold C with cone angles less than or equal to pi, and which is not almost product, we describe the space of constant curvature cone structures on C with cone angles less than pi. We establish a regeneration result for such Euclidean cone manifolds into spherical or hyperbo…
Non-positively curved spaces admitting a cocompact isometric action of an amenable group are investigated. A classification is established under the assumption that there is no global fixed point at infinity under the full isometry group. The visual boundary is then a spherical building. When the ambient space is geode…
We prove new kinematic formulas for tensor valuations and simplify previously known Crofton formulas by using the recently developed algebraic theory of translation invariant valuations. The heart of the paper is the computation of the Alesker-Fourier transform on the large class of spherical valuations, which is achie…
Warped-product black hole spacetimes are -inextendible.
problem Future -inextendibility of warped-product black hole spacetimes
method Adapting Sbierski's proof for a broad class of warped-product black hole spacetimes
result Future -inextendibility established for spacetimes with a static exterior region
In this work, it is shown that a simply-connected, rationally-elliptic torus orbifold is equivariantly rationally homotopy equivalent to the quotient of a product of spheres by an almost-free, linear torus action, where this torus has rank equal to the number of odd-dimensional spherical factors in the product. As an a…
Crocheting Mathematicsmath.HO
Crochet creates precise 2D shapes from 1D material.
problem Creating precise 2D shapes from 1D material.
method Using crochet to generate constant flat, spherical, or hyperbolic shapes.
result Crochet is the most flexible and precise method for building dynamical systems with high curvature precision.
We show that complete conformally flat manifolds of dimension n>2 with nonnegative Ricci curvature enjoy nice rigidity properties: they are either flat, or locally isometric to a product of a sphere and a line, or are globally conformally equivalent to flat space or to a spherical spaceform. This extends previous works…
Researchers prove inner product recovery is impossible in latent space models.
problem Recovering inner products in latent space models with random geometric graphs.
method Rate-distortion theory applied to Gaussian or spherical latent locations.
result Impossible to recover inner products if dimensionality exceeds , matching positive results' conditions.
Many classical results in relativity theory concerning spherically symmetric space-times have easy generalizations to warped product space-times, with a two-dimensional Lorentzian base and arbitrary dimensional Riemannian fibers. We first give a systematic presentation of the main geometric constructions, with emphasis…
Linear classifiers in product space forms improve scRNA-seq data classification.
problem Linear classification in products of Euclidean, spherical, and hyperbolic spaces.
method Novel formulations of linear classifiers on Riemannian manifolds, proving expressive power, and formalizing perceptron and SVM classifiers.
result Linear classifiers in product space forms have the same expressive power as in Euclidean space of the same dimension.
In this paper we show that the cohomology of a connected CW complex is periodic if and only if it is the base space of an orientable spherical fibration with total space that is homotopically finite dimensional. As applications we characterize those discrete groups that act freely and properly on a cartesian product of…
Given a group and a subset , an element is called quasi-positive if it is equal to a product of conjugates of elements in the semigroup generated by . This notion is important in the context of braid groups, where it has been shown that the closure of quasi-positive braids coincides with t…
Spherical Knot Mosaicsmath.GT
Paper introduces spherical knot mosaics for knot and link invariants.
problem Representing knots on a sphere with tiles.
method Tiling a 2-sphere with 11 knot mosaic tiles to define new invariants.
result New knot invariants derived from spherical mosaic tiling.
The paper classifies hypersurfaces with special curvature properties in various spaces.
problem Characterizing hypersurfaces with specific curvature properties.
method Introducing Ribaucour partial tubes and using them to derive applications.
result Classification of hypersurfaces of Enneper type in .
Study geodesics on spherical polyhedra, estimating their number.
problem Counting simple closed geodesics on spherical polyhedra.
method Examined regular spherical octahedra, cubes, and tetrahedra.
result Estimated the number of simple closed geodesics on spherical polyhedra.
LR and L+R Systemsmath-ph
We consider coupled nonholonomic LR systems on the product of Lie groups. As examples, we study -dimensional variants of the spherical support system and the rubber Chaplygin sphere. For a special choice of the inertia operator, it is proved that the rubber Chaplygin sphere, after reduction and a time reparametrizat…
We consider an application involving a financial quadratic portfolio of options, when the joint underlying log-returns changes with multivariate elliptic distribution. This motivates the needs for methods for the approximation of multiple integrals over hyperboloids. A transformation is used to reduce the hyperboloid i…