Study on surfaces in pseudo-Euclidean space with neutral metric.
problem Characterizing surfaces in pseudo-Euclidean 4-space with neutral metrics.
method Defined and studied Lorentz general rotational surfaces with specific properties.
result Complete classification of various types of general rotational surfaces.
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.
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…
Minimal sets of moves for rotational Reidemeister diagrams are identified.
problem Understanding the minimal sets of moves for rotational Reidemeister diagrams.
method Detailed description and proof of minimal generating sets for rotational Reidemeister moves.
result Minimal generating sets for oriented, framed links contain 5 moves.
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.
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.
Study Wick-rotations of Lie groups using GIT results.
problem Understanding Wick-rotations of pseudo-Riemannian Lie groups.
method Using results from real GIT, analyze invariant metrics and Lie algebras.
result Existence and conjugacy of Cartan involutions for pseudo-Riemannian Lie groups.
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.
Smoothness of Minkowski sum depends on rotations of convex bodies.
problem Can the Minkowski sum of two convex bodies be made smoother by rotating one of them?
method Constructing and analyzing infinitely differentiable strictly convex plane bodies and their rotations.
result Generic rotations of convex bodies can make the Minkowski sum less smooth, but under certain conditions, the sum can become as smooth as the summands.
Study on rotational hypersurfaces with constant Gauss-Kronecker curvature.
problem Exploring hypersurfaces with constant Gauss-Kronecker curvature.
method Solving ODE for generating curves and analyzing geometric properties.
result Discovery of non-compact rotational hypersurfaces with negative Gauss-Kronecker curvature and finite volume.
In this paper we study general rotational surfaces in the 4- dimensional Euclidean space E4 and give a characterization of flat general rotation surface with pointwise 1-type Gauss map. Also, we show that a non-planar flat general rotation surface with pointwise 1-type Gauss map is a Lie group if and only if it is a Cl…
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.
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.
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.
In this paper, we study general rotational surfaces in the 4- dimensional pseudo-Euclidean space E4-2 and obtain a characterization of flat general rotation surfaces with pointwise 1-type Gauss map in E4-2 and give an example of such surfaces.
The study explores special surfaces in a normed space.
problem Constant Gaussian and mean curvature surfaces in normed spaces.
method Analyzes rotational surfaces with specific curvature properties.
result Generalizes catenoid, pseudo-sphere, and Delaunay surfaces.
Layer rotation predicts model generalization, improving test accuracy by up to 30%.
problem Predicting model generalization in deep networks.
method Monitoring the cosine distance between layer weights and their initial values during training.
result Training procedures that maximize layer rotation consistently lead to better generalization performance.
Study rotational surfaces with prescribed Gauss curvature in 3D space.
problem Classify and analyze rotational surfaces with prescribed Gauss curvature.
method Phase plane analysis and mild assumptions on the prescribed function.
result Existence of singular radial solutions intersecting orthogonally the axis of rotation.
This paper shows how rotating, cropping, and translating images improves a reinforcement learning agent's ability to generalize.
problem Reinforcement learning agents struggle to generalize to slight variations of their training environments.
method The authors investigate the impact of rotation, translation, and cropping on the input representation of reinforcement learning agents.
result Cropped, translated, and rotated observations lead to better generalization in reinforcement learning agents.
The study characterizes helices in Euclidean and hyperbolic spaces.
problem Characterizing helices in Euclidean and hyperbolic spaces.
method Analyzing Killing vector fields associated with rotations in both spaces.
result Helices in hyperbolic space are geodesics on suitable surfaces.
Abstract: Proves generic torus diffeomorphisms act parabolically and non-properly on fine curve graph and have generalized rotation sets.
problem Generic torus diffeomorphisms on fine curve graph.
method Proves generic torus diffeomorphisms act parabolically and non-properly on fine curve graph.
result Generic torus diffeomorphisms have generalized rotation sets of any point-symmetric compact convex homothety type.
The paper classifies and constructs rotational surfaces with constant astigmatism in space forms.
problem Classifying surfaces with constant astigmatism in space forms.
method Classification and construction of surfaces using variational problems and binormal evolution.
result Locally constructed all rotational surfaces of constant astigmatism.
Harmonic Networks make images equivariant to translation and rotation.
problem Making CNNs equivariant to rotation.
method Replacing CNN filters with circular harmonics to achieve patch-wise translation and 360-degree rotation equivariance.
result Deep feature maps encode complicated rotational invariants.
The paper characterizes special curves and generalizes rectifying-type curves in n-dimensional space.
problem Characterizing and generalizing special curves in higher dimensions.
method Characterization through Rotation minimizing frame (RMF) and generalization of rectifying-type curves.
result Rectifying-type curves are generalized in n-dimensional space.
Paper defines continuous rotation representations for neural networks.
problem Discontinuous representations of rotations in neural networks.
method Definition of continuous representations, relating to topological concepts.
result Continuous representations for 3D rotations in 5D and 6D are more suitable for neural networks.
We discuss which Kleinian groups are commensurable with Kleinian groups generated by rotations, with particular emphasis on Kleinian groups that arise from Dehn surgery on a knot.
Study on unique minimal hypersurfaces in rotational domains.
problem Existence of compact free-boundary minimal hypersurfaces in rotational domains.
method Integral identity for compact free-boundary minimal hypersurfaces, applied to rotational domains.
result Existence of minimal hypersurfaces in rotational domains without topological restrictions.
We consider an infinite 3-dimensional elastic continuum whose material points experience no displacements, only rotations. This framework is a special case of the Cosserat theory of elasticity. Rotations of material points are described mathematically by attaching to each geometric point an orthonormal basis which give…
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.
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.
Paper proves Legendrian knots with same GL-rack have similar invariants.
problem Classifying Legendrian knots based on their invariants.
method Examined fundamental GL-racks and their relationship to Thurston-Bennequin and rotation numbers.
result Two Legendrian knots with isomorphic fundamental GL-racks have similar invariants.
Finite translation surfaces can be classified by the order of their singularities. When generalizing to infinite translation surfaces, however, the notion of order of a singularity is no longer well-defined and has to be replaced by new concepts. This article discusses the nature of two such concepts, recently introduc…
New kernel interprets 3D anisotropic data with rotations and improved predictions.
problem Capturing rotated anisotropy in 3D spatial fields.
method Introduces a Lie-algebraic kernel with three principal length-scales and an explicit rotation.
result Posterior recovers rotated anisotropy and improves prediction over axis-aligned kernels.
New approach to rotational Weingarten surfaces using geometric momentum.
problem Classifying and characterizing rotational Weingarten surfaces.
method Introducing geometric linear momentum of a plane curve to reduce Weingarten conditions to differential equations.
result Classification of non-degenerate quadric surfaces and elasticoids.
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.
We solve Bayesian PCA's rotational symmetry issue by rotation-invariant parameterization.
problem Bayesian PCA's rotational symmetry complicates inference and interpretation.
method Rotation-invariant Householder parameterization using random matrix theory.
result Efficient rotation-invariant probabilistic PCA implementation.
Optimal transport aligns rotated linear regression models across domains.
problem Aligning rotated linear regression models across domains with differing statistical properties.
method Combines K-means clustering, OT, and SVD to estimate rotation angle and adapt regression model.
result Optimal transport map recovers underlying rotation in R2. 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.
RotDCF decomposes CNN filters for rotation-equivariant deep networks.
problem Handling global deformations in images for vision tasks.
method Decomposes convolutional filters over joint steerable bases for rotation-equivariance.
result Significantly reduces model size and computational complexity while preserving performance.
Study curve shortening flows on specific surfaces, proving properties and existence.
problem Analyzing curve shortening flows on rotational surfaces with negative Gauss curvatures.
method Assume negative Gauss curvatures and conditions on Gauss curvature and curve curvature. Prove curve remains a graph and establish flow properties.
result Prove the curve remains a graph over parallels and establish long-time existence of the flow.
This paper studies rotational virtual knot theory and its relationship with quantum link invariants. Every quantum link invariant for classical knots and links extends to an invariant of rotational virtual knots and links. The paper sets up the background virtual knot theory, defines rotational virtual knot theory, stu…
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.
3DGCN predicts molecular properties and biochemical activities using 3D molecular graph.
problem Predicting molecular properties and biochemical activities from 3D molecular graphs.
method Unified graph convolution with learning operations to handle spatial information, distinguishing 3D rotations.
result Significantly higher performance on various molecular tasks compared to other deep-learning models.
The paper classifies surfaces with constant mean curvature and rotational symmetry.
problem Surfaces with specific geometric properties in Euclidean space.
method Analyzes surfaces parametrized on an annular domain with rotational symmetry.
result Surfaces with constant mean curvature are of rotational symmetry.
We apply the invariant theory of surfaces in the four-dimensional Euclidean space to the class of general rotational surfaces with meridians lying in two-dimensional planes. We find all minimal super-conformal surfaces of this class.
We prove that any piece of a rotational hypersurface with prescribed mean curvature function in a Euclidean space can be uniquely extended infinitely, which generalizes the results by Euler and Delaunay for surfaces of revolution with constant mean curvautre. Next, we prove the same kind of theorem for generalized rota…