RotEqNet preserves rotation symmetry in fluid systems using high-order tensors.
problem Lack of rotational symmetry in machine learning models for fluid systems.
method Introduces RotEqNet, a network that guarantees rotation-equivariance for high-order tensors.
result RotEqNet reduces errors and maintains rotation-equivariance in fluid systems.
In this paper is considered the differential equation Ric(g)=T, where Ric(g) is the Ricci tensor of the metric g and T is a rotational symmetric tensor on R^n. A new, geometric, proof of the existence of smooth solutions of this equation, based on qualitative theory of implicitdifferential equations, is presented here.…
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 show that a metric of arbitrary dimension and signature which allows for a standard Wick-rotation to a Riemannian metric necessarily has a purely electric Riemann and Weyl tensor.
The paper develops GPR models for hyperelastic materials, improving accuracy and rotational invariance.
problem Modeling stress tensors of hyperelastic materials with fewer training examples and higher accuracy.
method Developed three approaches: direct stress tensor modeling, embedding rotational invariance, and recovering strain-energy density.
result Improved GPR models achieve higher accuracy and rotational invariance with fewer training examples.
Paper derives formulas for higher-order curvature derivatives of framed space curves.
problem Deriving exact formulas for higher-order derivatives of curvature of framed space curves.
method Parametrizing rotation tensor using Gibbs vector, deriving closed-form formulas for derivatives, and formulating a linearized updating algorithm.
result Closed-form formulas and a linearized updating algorithm for curvature and its derivatives of framed space curves.
Classical dynamical equations describing a certain version of the nonHamiltonian interaction of two rotators (Euler tops with completely degenerate inertia tensors) are considered. The simplest case is integrated. It is shown that the dynamics is almost periodic with periods depending on the initial data.
The paper classifies CMC free boundary hypersurfaces in rotational domains.
problem Existence and uniqueness of free boundary constant mean curvature hypersurfaces in rotational domains.
method Classification and construction of CMC free boundary hypersurfaces under specific conditions.
result Classification of CMC free boundary hypersurfaces as topological disks or annuli.
We investigate the relationship between the algebra of tensor categories and the topology of framed 3-manifolds. On the one hand, tensor categories with certain algebraic properties determine topological invariants. We prove that fusion categories of nonzero global dimension are 3-dualizable, and therefore provide 3-di…
We study a class of Riemannian manifolds with respect to the covariant derivative of their curvature tensors. We introduce geometrically the class of directed Riemannian manifolds of pointwise constant relative sectional curvature and give a tensor characterization for such manifolds. We prove that all rotational hyper…
The paper is devoted to vector fields on the spaces R^2 and R^3, their flow and invariants. Attention is plaid on the tensor representations of the group GL(2,R) and on fundamental vector fields. The rotation group on R^3 is generalized to rotation groups with arbitrary quadrics as orbits.
Optimizing over the set of orthogonal matrices is a central component in problems like sparse-PCA or tensor decomposition. Unfortunately, such optimization is hard since simple operations on orthogonal matrices easily break orthogonality, and correcting orthogonality usually costs a large amount of computation. Here we…
RENNs protect input privacy by rotating d-ary features.
problem Protecting input privacy from intermediate-layer features.
method Rotation-equivariant neural networks using d-ary vectors/tensors.
result RENNs effectively hide input information without degrading output accuracy.
In this paper we introduce an algorithm to determine the equivalence of five dimensional spacetimes, which generalizes the Karlhede algorithm for four dimensional general relativity. As an alternative to the Petrov type classification, we employ the alignment classification to algebraically classify the Weyl tensor. To…
The higher-dimensional Kerr-NUT-de Sitter spacetime describes the general rotating asymptotically de Sitter black hole with NUT parameters. It is known that such a spacetime possesses a rank-2 closed conformal Killing-Yano (CKY) tensor as a ``hidden'' symmetry which provides the separation of variables for the geodesic…
Classifies weakly Einstein hypersurfaces in spaces of constant curvature.
problem Identifying weakly Einstein hypersurfaces in spaces of constant curvature.
method Complete classification through tensor analysis and geometric properties.
result Hypersurfaces are either products of spaces of constant curvature or rotation hypersurfaces.
Tensor measures chirality for curves, even those with rough edges.
problem Quantifying chirality for complex, possibly irregular curves.
method Developed a tensorial chirality measure for rigid filaments and curves.
result A curve's chirality can be determined by its twist about perpendicular axes.
Paper introduces a new cosmological volume function and its properties.
problem Introducing a new cosmological volume function.
method Introduces and analyzes the cosmological volume function τ_V, showing it's continuously differentiable.
result τ_V leads to a canonical splitting of the metric tensor and a canonical Wick-rotated Riemannian metric.
In this paper we provide a new method for establishing the rotational symmetry of the solutions to a couple of very classical overdetermined problems arising in potential theory, in both the exterior and the interior punctured domain. Thanks to a conformal reformulation of the problems, we obtain Riemannian manifolds w…
Proves existence and uniqueness of rotating fluid bodies in GR to second order.
problem Understanding rotating fluid bodies in GR, especially beyond Newtonian limits.
method Second order perturbation theory, derived from first principles, with rigidly rotating finite perfect fluid ball assumptions.
result Equatorially symmetric spacetime determined by central pressure and uniform angular velocity.
Carter tensor analysis aids wave equation on Kerr-Newman spacetime.
problem Analyzing perturbations of Kerr-Newman spacetime using wave equation.
method Physical-space analysis adapted to Kerr-Newman spacetime, leveraging Carter operator commutation.
result Carter operator commutes with wave equation on Kerr-Newman spacetime, enabling wave equation analysis.
In this paper we use the relationship between conformal metrics on the sphere and horospherically convex hypersurfaces in the hyperbolic space for giving sufficient conditions on a conformal metric to be radial under some constrain on the eigenvalues of its Schouten tensor. Also, we study conformal metrics on the spher…
TSGO optimizes gradients in tensor networks to avoid vanishing/exploding issues.
problem Gradient vanishing and exploding problems in deep learning models.
method TSGO rotates parameters towards gradient direction in tangent space of normalized state.
result TSGO naturally determines learning rate based on angle between parameters and gradient.
Using a theorem of Lück-Reich-Rognes-Varisco, we show that the Whitehead group of Thompson's group T is infinitely generated, even when tensored with the rationals. To this end we describe the structure of the centralizers and normalizers of the finite cyclic subgroups of T, via a direct geometric approach based on rot…
Study of SO(3)-irreducible geometry in complex 5D and ternary Pauli exclusion principle.
problem Exploring SO(3)-irreducible geometry in complex 5D.
method Defined a ternary skew-symmetric tensor, split the 10D space into irreducible SO(3) subspaces, found invariants and defined geometric structures.
result Defined a SO(3)-irreducible geometric structure on a 5D complex Hermitian manifold.
New method uses scalars to approximate physics functions.
problem Designing neural networks that respect physical symmetries.
method Parameterizing polynomial functions equivariant to various symmetries using scalars.
result Universal approximation of polynomial functions under various symmetries using scalars.
A machine learning model captures non-Newtonian fluid dynamics from molecular details.
problem Creating accurate non-Newtonian fluid models from molecular data.
method Developed a machine learning framework that maps micro-scale polymer configurations to macro-scale fluid dynamics, preserving molecular fidelity.
result The deep non-Newtonian model (DeePN2) accurately predicts fluid behavior without empirical closures. Algorithm learns polynomial transformations of Gaussian distributions.
problem Learning high-dimensional polynomial transformations of Gaussian distributions.
method Polynomial-time algorithms for smoothed settings, tensor ring decomposition.
result First end-to-end guarantees for learning pushforwards under neural networks.
Som-Raychaudhuri spacetime is a stationary cylindrical symmetric solution of Einstein field equation corresponding to a charged dust distribution in rigid rotation. The main object of the present paper is to investigate the curvature restricted geometric structures admitting by the Som-Raychaudhuri spacetime and it is …
Study on rotating surfaces in 4D space with matrices.
problem Understanding rotational surfaces in pseudo-Euclidean 4-space.
method Defined hyperbolic and elliptic rotational surfaces using curves and matrices in 4D semi-Euclidean space.
result Generated rotated surfaces using specific rotation matrices.
Characterizes null Lagrangians in Cosserat elasticity.
problem Understanding null Lagrangians in micropolar elasticity.
method Applying Olver and Sivaloganathan's theorem to characterize null Lagrangians.
result Complete characterization of null Lagrangians for three-dimensional bodies and shells.
A generalisation of the four-dimensional Kerr-de Sitter metrics to include a NUT charge is well known, and is included within a class of metrics obtained by Plebanski. In this paper, we study a related class of Kerr-Taub-NUT-de Sitter metrics in arbitrary dimensions D \ge 6, which contain three non-trivial continuous p…
A new transform links rotating calorons to solutions of a differential equation.
problem Existence and characterization of rotating calorons.
method Formulated a Nahm transform to relate rotating calorons to solutions of a delayed-differential equation.
result Existence of an eight-parameter family of rotating calorons with nontrivial holonomy.
We define general rotational surfaces of elliptic and hyperbolic type in the pseudo-Euclidean 4-space with neutral metric which are analogous to the general rotational surfaces of C. Moore in the Euclidean 4-space. We study Lorentz general rotational surfaces with plane meridian curves and give the complete classificat…
Study finds whitepaper narratives do not predict market factor structure.
problem Predicting market behavior from cryptocurrency whitepaper claims.
method Zero-shot NLP classification combined with CP tensor decomposition of market data.
result Weak alignment between whitepaper claims and market statistics and latent factors.
The paper defines and analyzes homotopic rotation sets for surfaces of higher genus.
problem Defining and analyzing homotopic rotation sets for surfaces of higher genus.
method Developed a definition and proved several results using the theory of Le Calvez and Tal.
result Found that the homotopic rotation set can imply the existence of infinitely many periodic orbits under certain conditions.
Study of timelike surfaces in Minkowski space with specific geometric properties.
problem Characterizing geometric properties of timelike surfaces in Minkowski space.
method Analytical study of two types of timelike general rotational surfaces.
result Explicit descriptions of minimal and surfaces with specific curvature properties.
The study characterizes loxodromes on specific rotational surfaces in 3D space.
problem Characterizing loxodromes on rotational surfaces with special geometric properties.
method Parametrizations and curvature/torsion calculations for loxodromes on various rotational surfaces.
result The loxodrome on a flat rotational surface is a general helix.
Overview of methods for rotating 2D and 3D data.
problem Processing data with equivariance/invariance under rotations.
method An overview of methods for 2D and 3D rotations.
result Identification of commonalities and links between methods.
Enhanced rotation prediction improves SSL models by capturing both shape and texture information.
problem Rotation prediction misses texture information, limiting model performance.
method Introduces image enhanced rotation prediction (IE-Rot) that combines rotation and image enhancement tasks.
result IE-Rot models outperform Rotation on various benchmarks.
General rotational surfaces as a source of examples of surfaces in the four-dimensional Euclidean space have been introduced by C. Moore. In this paper we consider the analogue of these surfaces in the Minkowski 4-space. On the base of our invariant theory of spacelike surfaces we study general rotational surfaces with…
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.
The paper introduces REQNNs for robust 3D point cloud processing.
problem 3D point cloud processing robustness to rotations.
method Revised neural networks using quaternion features for rotation-equivariance.
result REQNNs exhibit higher rotation robustness compared to original networks.
Rotation systems can't always be drawn in surfaces.
problem Rotation systems and simple drawings in surfaces.
method Extended the plane result to all fixed surfaces.
result Existence of rotation systems not arising from simple drawings in any fixed surface.
Paper learns to rotate filters for group convolutions.
problem Difficult to rotate 3x3 filters on pixel grids.
method Learn filter basis and rotation-invariant coefficients; switch basis for rotation.
result Produces feature maps insensitive to input rotations.
Cormorant learns molecular properties via rotationally covariant neural networks.
problem Learning molecular potential energy surfaces and properties.
method Rotationally covariant neural network architecture with tensor products and Clebsch-Gordan decomposition.
result Significantly outperforms competing algorithms in learning molecular Potential Energy Surfaces.
RotatE embeds knowledge graphs using rotations in complex space for better link prediction.
problem Predicting missing links in knowledge graphs.
method RotatE models relations as rotations in complex vector space, using self-adversarial negative sampling.
result RotatE models and infers various relation patterns, outperforming existing models.
The paper applies Clairaut's theorem to rotational surfaces in pseudo-Euclidean 4-space.
problem Exploring geodesic curves on rotational surfaces in pseudo-Euclidean 4-space.
method Expressing Clairaut's theorem and deriving equations for geodesic curves.
result Characterization of geodesic curves on hyperbolic and elliptic surfaces of rotation.