The Dirichlet Laplacian in curved tubes of arbitrary cross-section rotating with respect to the Tang frame along infinite curves in Euclidean spaces of arbitrary dimension is investigated. If the reference curve is not straight and its curvatures vanish at infinity, we prove that the essential spectrum as a set coincid…
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48 results for “rotational spectrum”
Constructs coordinates to diagonalize Toda flow on matrices with simple spectrum.
problem Diagonalizing the Toda flow on matrices with simple spectrum.
method Lie theoretic methods applied to complex semisimple Lie algebras and their real forms.
result Decouples the Toda vector field into simpler components.
We consider the Dirichlet Laplacian in a two-dimensional strip composed of segments translated along a straight line with respect to a rotation angle with velocity diverging at infinity. We show that this model exhibits a "raise of dimension" at infinity leading to an essential spectrum determined by an asymptotic thre…
Study the spectrum of Poincaré operator in triaxial ellipsoids.
problem Spectrum of the Poincaré operator in triaxial ellipsoids.
method Microlocal analysis of partial differential equations and polynomial vector fields.
result Polynomial eigenvectors and large-degree asymptotics of the operator.
We explore how the spectrum of a 3-manifold's geometry can distinguish between non-isometric manifolds.
problem Determining if two non-isometric locally homogeneous 3-manifolds have the same spectrum.
method Analyzing the spectrum of locally homogeneous metrics on elliptic 3-manifolds and using geometric structures.
result If two locally homogeneous, non-isometric 3-manifolds have the same spectrum, they must have the same fundamental group and be locally isometric.
Let be a compact cmc rotational hypersurface of the -dimensional Euclidean unit sphere. Denote by the square of the norm of the second fundamental form and the stability or Jacobi operator. In this paper we compute the spectra of the…
Skein theory classifies UFCs with specific fusion rules.
problem Classifying unitary fusion categories with specific fusion rules.
method Graphical calculus and rotation operator action on a canonical basis.
result Explicit formulae for when and is ribbon.
A method for identifying joint and individual subspaces from multi-view data.
problem Unclear conditions for reliably identifying joint and individual subspaces from noisy, high-dimensional measurements.
method Rigorously quantifies conditions based on signal rank, principal angles, and noise levels. Characterizes spectrum perturbations of product of projection matrices.
result Estimates joint and individual subspaces more accurately than existing approaches in simulations and real-world applications.
Study on neural networks with non-normal interactions reveals unique spectral properties.
problem Understanding episodic memory encoding in the brain.
method Developed a neural network model with non-Hermitian couplings and applied random matrix theory.
result Spectral density of the model is non-uniform and can transition to chaos, providing computational benefits.
On the Consistency of Maximum Likelihood Estimation of Probabilistic Principal Component Analysisstat.ML
New approach resolves ambiguity in PPCA model's maximum likelihood estimation.
problem Ambiguity in maximum likelihood estimation of PPCA model due to rotational symmetry.
method Using quotient topological spaces, the approach resolves ambiguity and shows consistency of the maximum likelihood solution.
result Maximum likelihood solution is consistent in an appropriate quotient Euclidean space.
Analysis of DPPs and k-DPPs via spectral decomposition reveals identifiable parameters and non-identifiability gaps.
problem Identifying parameters of DPPs and k-DPPs through spectral decomposition.
method Spectral decomposition of the covariance matrix, analysis of invariances, and counting arguments.
result Identifiability of parameters changes fundamentally for k-DPPs, with specific invariances and non-identifiability gaps.
OMD monitors stock market dynamics through matrix trajectories and reveals crisis patterns.
problem Understanding and predicting stock market crises and sector rotations.
method Applying OMD to S\&P 500 returns over three crises, analyzing distance matrices and their spectra.
result Market dynamics show coherent changes during crises, with distinct sector leadership.
OMD monitors stock market dynamics through matrix trajectories, revealing crisis patterns and sector rotations.
problem Understanding and predicting stock market dynamics during crises.
method Applying OMD to S&P 500 returns over three crises, analyzing distance matrices and their spectra.
result Market dynamics show coherent changes during crises, with sector-specific patterns and volatility clustering.
The paper studies eigenvalues and stability of hypersurfaces in spheres.
problem Finding eigenvalues and stability of hypersurfaces in spheres.
method Derives equations for mean curvature and uses numerical methods to compute eigenvalues.
result Numerical computation of eigenvalues and stability indices for specific hypersurfaces.
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.
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…
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.
Basic Classes of Timelike General Rotational Surfaces in the Four-dimensional Minkowski Spacemath.DG
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…
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.
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.
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.
We consider -dimensional discrete motions such that any two neighbouring positions correspond in a pure rotation ("rotating motions"). In the Study quadric model of Euclidean displacements these motions correspond to quadrilateral nets with edges contained in the Study quadric ("rotation nets"). The main focus of ou…
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.
New method studies moving points on curves using rotating frames.
problem Understanding the motion of points on curves.
method Constructing rotating frames for curves and analyzing the motion of points within these frames.
result A new binary mathematical formation mechanism for curves based on linear and rotational motion.
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-plane drill rotations are impossible for smooth shells.
problem In-plane drill rotations on smooth shells are impossible.
method Analyzing the differential geometry of surfaces and isometries.
result Any isometry that coincides with the given surface at a portion of the boundary is the identity.
Positive factorization found for a specific map on surfaces.
problem Balanced superelliptic rotation on surfaces.
method Positive factorization approach.
result Positive factorization for balanced superelliptic rotation.
Study of rotation angles in a rotating disc model.
problem Understanding geometric phase in rotating systems.
method Analyzes a simple kinematic model of rotating discs.
result Explicit form of geometric phase found using Baumkuchen lemma.
The study disproves rotating ancient flows in 4D space.
problem The existence of rotating ancient flows in .
method Analysis of ancient noncollapsed flows in .
result Nonexistence of rotating ancient flows among ancient noncollapsed flows in .
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.
In this work, we study a class of rotational surfaces in the pseudo-Euclidean space whose profile curves lie in two-dimensional planes. We solve the differential equation that characterizes the rotational surfaces with zero mean curvature to determine the profile curves of such rotational surfaces. The…
Rotational elasticitymath-ph
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…
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.
Helicoidal surfaces rotate and translate under mean curvature flow.
problem Existence of helicoidal surfaces under mean curvature flow.
method One-parameter families of helicoidal surfaces rotating and translating.
result Existence of helicoidal surfaces under mean curvature flow.
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.
Extends Euler class result to symplectic group.
problem Relationship between bounded Euler class and symplectic rotation number.
method Extends Ghys's result to symplectic group.
result Establishes relationship between bounded Euler class and symplectic rotation number.
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.
We study the problem of learning representations of entities and relations in knowledge graphs for predicting missing links. The success of such a task heavily relies on the ability of modeling and inferring the patterns of (or between) the relations. In this paper, we present a new approach for knowledge graph embeddi…
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…
New quadratic forms expand and rotate linear endomorphisms in geometric theory.
problem Understanding the expansion and rotation properties of linear endomorphisms.
method Constructing new quadratic forms based on two-plane rotations.
result Established relations among eigenvalues, eigendirections, and matrix invariants.
In the four-dimensional pseudo-Euclidean space with neutral metric there are three types of rotational surfaces with two-dimensional axis - rotational surfaces of elliptic, hyperbolic or parabolic type. A surface whose mean curvature vector field is lightlike is said to be quasi-minimal. In this paper we classify all q…
In neural networks, it is often desirable to work with various representations of the same space. For example, 3D rotations can be represented with quaternions or Euler angles. In this paper, we advance a definition of a continuous representation, which can be helpful for training deep neural networks. We relate this t…