arXiv research

Rotation systems can't always be drawn in surfaces.

RotEqNet preserves rotation symmetry in fluid systems using high-order tensors.

Study Vassiliev invariants for virtual knots, expanding quantum theory.

Rotates MFVI for better Gaussian approximations.

Convolutional networks are successful due to their equivariance/invariance under translations. However, rotatable data such as images, volumes, shapes, or point clouds require processing with equivariance/invariance under rotations in cases where the rotational orientation of the coordinate system does not affect the m…

Classifies geodesic flows on projective plane with potential field.

Study uses geometric algebra to analyze credit cycles, revealing dangerous feedback loops.

Investigates the rotating Kepler problem for energy values ≤ -3/2.

We study variational systems for space curves, for which the Lagrangian or action principle has a Euclidean symmetry, using the Rotation Minimising frame, also known as the Normal, Parallel or Bishop frame. Such systems have previously been studied using the Frenet-Serret frame. The Rotation Minimising frame has many a…

New MCMC method improves sampling efficiency across diverse structural models.

Minimal surfaces found in 4D space.

The scalar curvature equation for rotation invariant Kähler metrics on $\mathbb{C}^n \backslash \{0\}$ is reduced to a system of ODEs of order 2. By solving the ODEs, we obtain complete lists of rotation invariant zero or positive csck on $\mathbb{C}^n \backslash \{0\}$ in lower dimensions. We also prove that there doe…

Minimalistic model captures head direction system properties.

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…

New tree structure for pseudo-Anosovs from interval maps.

Study invariant $λ$-translators in Lorentz-Minkowski space.

The introduction of automated flight control and management systems have made possible aircraft designs that sacrifice arodynamic stability in order to incorporate stealth technology intro their shape, operate more efficiently, and are highly maneuverable. Therefore, modern flight management systems are reliant on mult…

Quantum mechanics models for financial Black-Scholes model.

Study of motion control systems on Lie groups with specific geometric constraints.

The homotopy theory of topological defects in ordered media fails to completely characterize systems with broken translational symmetry. We argue that the problem can be understood in terms of the lack of rotational Goldstone modes in such systems and provide an alternate approach that correctly accounts for the intera…

The existence of the theory of `twisted cotangent bundles' (symplectic groupoids) allows to study classical mechanical systems which are generalized in the sense that their configurations form a Poisson manifold. It is natural to study from this point of view first such systems which arise in the context of some basic …

The paper studies dynamical systems with evolving geometric structure using numerical methods.

We borrow a classical construction from the study of rational billiards in dynamical systems known as the "unfolding construction" and show that it can be used to study the automorphism group of a Platonic surface. More precisely, the monodromy group, or deck group in this case, associated to the cover of a regular pol…

Generalizes CNNs for Lie group equivariance across various data types.

An example of mechanical system whose configuration space is direct product of a curved space and the local group of rotations, is presented. The system is considered as a model of spinning particle moving in the space. The Hamiltonian formalism for this system and possible method for its quantization are discussed. It…

Study on rotating surfaces in 4D space with matrices.

Sequences of Levy transformations for the Darboux system of conjugates nets in multidimensions are studied. We show that after a suitable number of Levy transformations, with at least a Levy transformation in each direction, we get closed formulae in terms of multi-Wroński determinants. These formulae are for the tange…

Predict missing movie ratings or graph embeddings with low rank matrices.

In this work, we are interested in the differential geometry of curves in the simply isotropic and pseudo-isotropic 3-spaces, which are examples of Cayley-Klein geometries whose absolute figure is given by a plane at infinity and a degenerate quadric. Motivated by the success of rotation minimizing (RM) frames in Eucli…

A new transform links rotating calorons to solutions of a differential equation.

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…

In the present article we study a special class of surfaces in the four-dimensional Euclidean space, which are one-parameter systems of meridians of the standard rotational hypersurface. They are called meridian surfaces. We classified semi-parallel meridian surface in 4-dimensional Euclidean space $E^4$.

Equivariance is a nice property to have as it produces much more parameter efficient neural architectures and preserves the structure of the input through the feature mapping. Even though some combinations of transformations might never appear (e.g. an upright face with a horizontal nose), current equivariant architect…

Study cohomological equation for robotic screw motions on SE(3).

We prove that 2-dimensional simplicial complexes whose first homology group is trivial have topological embeddings in 3-space if and only if there are embeddings of their link graphs in the plane that are compatible at the edges and they are simply connected.

Method proves connection stability of vector fields on noncompact manifolds.

The paper defines and analyzes homotopic rotation sets for surfaces of higher genus.

Geometric GNNs model 3D atomic systems with rotations and translations.

Study of timelike surfaces in Minkowski space with specific geometric properties.

The study characterizes loxodromes on specific rotational surfaces in 3D space.

This paper addresses the problem of blind demixing of instantaneous mixtures in a multiple-input multiple-output communication system. The main objective is to present efficient blind source separation (BSS) algorithms dedicated to moderate or high-order QAM constellations. Four new iterative batch BSS algorithms are p…

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…

We construct a special class of Lorentz surfaces in the pseudo-Euclidean 4-space with neutral metric which are one-parameter systems of meridians of rotational hypersurfaces with lightlike axis and call them meridian surfaces. We give the complete classification of the meridian surfaces with constant Gauss curvature an…

We develop a framework for analyzing extreme values in correlated financial data.

Proposes a robust 3D classification method for sparse point clouds.

This paper proposes a set of rules to revise various neural networks for 3D point cloud processing to rotation-equivariant quaternion neural networks (REQNNs). We find that when a neural network uses quaternion features under certain conditions, the network feature naturally has the rotation-equivariance property. Rota…

The rotation prediction (Rotation) is a simple pretext-task for self-supervised learning (SSL), where models learn useful representations for target vision tasks by solving pretext-tasks. Although Rotation captures information of object shapes, it hardly captures information of textures. To tackle this problem, we intr…

The paper applies Clairaut's theorem to rotational surfaces in pseudo-Euclidean 4-space.