New methods improve multi-agent reinforcement learning by addressing rotational dynamics.
problem Reproducibility crisis in multi-agent reinforcement learning.
method Reframing MARL approaches using Variational Inequalities (VIs) and proposing gradient-based VI methods.
result Significant performance improvements across benchmarks, including better convergence to equilibrium strategies in zero-sum games.
Study homeomorphisms on fine curve graph of surfaces, revealing new types of dynamics.
problem Understanding dynamics of homeomorphisms on fine curve graphs of surfaces.
method Analyzing the action of homeomorphisms on the fine curve graph and relating to classical curve graphs.
result Homeomorphisms induce parabolic isometries, and all positive reals are realized as asymptotic translation lengths.
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.
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…
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 Δg found using Baumkuchen lemma. 3D capsule network for point clouds handles rotations and translations.
problem Processing 3D point clouds with equivariance and invariance.
method Quaternion equivariant capsule network with dynamic routing on quaternions.
result Parsimonious capsule network disentangles geometry and pose.
Investigates the rotating Kepler problem for energy values ≤ -3/2.
problem Understanding periodic orbits and symplectic structures in rotating celestial mechanics.
method Ligon-Schaaf and Levi-Civita symplectic regularizations, special concave toric domain construction.
result Identification of a special concave toric domain (SCTD) for the RKP phase space.
Study on Langevin dynamics on planar motion group, highlighting geometric mechanism.
problem Understanding Langevin dynamics on the Lie group SE(2).
method Intrinsic formulation on SE(2), using invariant vector fields and natural projection.
result Effective macroscopic diffusion on R^2 emerges through averaging.
Study uses geometric algebra to analyze credit cycles, revealing dangerous feedback loops.
problem Understanding and predicting dangerous feedback loops in credit cycles.
method Represent economic states as multi-vectors in Clifford algebra, focusing on bivector elements for rotational coupling.
result Geometric relationship between unemployment and credit contraction shifts from simple correlation to dangerous rotational dynamics during crises.
The rotation angle of a rolling disc is shown to be a geometric phase related to the Hopf fibration.
problem Understanding the geometric nature of rotation angles in kinematic models.
method Using the Hopf fibration and Gauss map, the geometric phase is decomposed into dynamical and geometric components.
result The geometric phase of rotation is described as the holonomy of the Hopf fibration.
The paper develops a new algorithm for RBMs using dynamical mean-field theory.
problem Learning in Restricted Boltzmann Machines (RBMs) with complex dependencies.
method Dynamical mean-field theory applied to RBMs with rectangular coupling matrices drawn from a bi-rotation invariant ensemble.
result The algorithm converges globally under a stability criterion, with rates matching numerical simulations.
Study dynamic portfolio choice under rotating drivers, revealing a new geometric structure.
problem Investment under changing drivers with mutual independence.
method Analyzes geometric structure of portfolio choice, focusing on drivers and their rotation.
result Optimal policy separates into static and hedging components, reflecting the dynamic nature of drivers.
New tree structure for pseudo-Anosovs from interval maps.
problem Understanding pseudo-Anosovs from interval maps.
method Tree structure on pseudo-Anosovs using rational numbers.
result Deepened dictionary between invariants.
A new algorithm computes elastic shape distances between curves efficiently.
problem Computing elastic shape distances between curves in high dimensions.
method Dynamic Programming for optimal diffeomorphisms and Kabsch-Umeyama algorithm for optimal rotation matrices.
result Efficient computation of elastic shape distances with improved efficiency for closed curves.
We revisit the non-rotating massive BTZ black hole within a pseudo-Riemannian symmetric space context. Using classical symmetric space techniques we find that every such space intrinsically carries a regular Poisson structure whose symplectic leaves are para-hermitian symmetric surfaces. We also obtain a global express…
A new method for 3D surface registration using dynamic programming.
problem Elastic shape registration of 3D surfaces.
method Optimization over a subset of reparametrizations using dynamic programming.
result Proposes an algorithm that produces a solution closer to optimal than gradient-based methods.
The paper studies dynamical systems with evolving geometric structure using numerical methods.
problem Qualitative behavior of ODEs with varying geometric structure.
method Fourth-order Runge-Kutta scheme for numerical analysis.
result Qualitative transitions in system dynamics as rotation parameter varies.
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…
Develops computational methods for simulating rigid body dynamics on SO(3).
problem Simulating rotational dynamics of rigid bodies on SO(3).
method Discrete Mechanics, Variational Integrators, Newton-Raphson algorithm.
result Preserves symplectic structure of SO(3) manifold dynamics.
This paper proposes an improved design of the perceptron unit to mitigate the vanishing gradient problem. This nuisance appears when training deep multilayer perceptron networks with bounded activation functions. The new neuron design, named auto-rotating perceptron (ARP), has a mechanism to ensure that the node always…
Unified geometric description of Kepler flow across all energies.
problem Understanding the Kepler flow across different energy levels.
method Revisiting Ligon--Schaaf regularization and identifying geometric origins of anomalies.
result Unified geometric description of Kepler flow for all energies.
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.
We initiate a series of works where we study the interior of dynamical rotating vacuum black holes without symmetry. In the present paper, we take up the problem starting from appropriate Cauchy data for the Einstein vacuum equations defined on a hypersurface already within the black hole interior, representing the exp…
The development of a metric for structural data is a long-term problem in pattern recognition and machine learning. In this paper, we develop a general metric for comparing nonlinear dynamical systems that is defined with Perron-Frobenius operators in reproducing kernel Hilbert spaces. Our metric includes the existing …
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…
New MCMC method improves sampling efficiency across diverse structural models.
problem Low sampling efficiency in generic MCMC methods for specific problems.
method Adaptive Principal-Component (PC) Meta-learning Stochastic Gradient Hamiltonian Monte Carlo (APM-SGHMC) algorithm.
result Universal samplers achieve zero-shot generalization across structurally distinct models.
Generative model improves noise estimation in stochastic rotating shallow water models.
problem Improving noise estimation in stochastic partial differential equations for fluid dynamics.
method Replaced PCA with a generative model to avoid constraints on stochastic increments.
result Generative model produces better RMSE, CRPS score, and forecast rank histograms.
The paper develops algorithms and topological invariants for distinguishing dynamic systems.
problem Distinguishing the topological type of surfaces and functions in dynamic systems.
method Construction of algorithms and topological invariants using discrete topological structures.
result The development of discrete topological structures for topological equivalence of dynamic systems.
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.
In recent years, convolutional neural networks (CNN) have played an important role in the field of deep learning. Variants of CNN's have proven to be very successful in classification tasks across different domains. However, there are two big drawbacks to CNN's: their failure to take into account of important spatial h…
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.
We consider cocycles of isometries on spaces of nonpositive curvature H. We show that the supremum of the drift over all invariant ergodic probability measures equals the infimum of the displacements of continuous sections under the cocycle dynamics. In particular, if a cocycle has uniform sublinear drift, then there…
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.
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.
In this paper we relate the study of actions of discrete groups over connected manifolds to that of their orbit spaces seen as differentiable stacks. We show that the orbit stack of a discrete dynamical system on a simply connected manifold encodes the dynamics up to conjugation and inversion. We also prove a generaliz…
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 present a numerical model for the dynamics of thin viscous threads based on a discrete, Lagrangian formulation of the smooth equations. The model makes use of a condensed set of coordinates, called the centerline/spin representation: the kinematical constraints linking the centerline's tangent to the orientation of …
We prove, for a class of contact manifolds, that the universal cover of the group of contact diffeomorphisms carries a natural partial order. It leads to a new viewpoint on geometry and dynamics of contactomorphisms. It gives rise to invariants of contactomorphisms which generalize the classical notion of the rotation …
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.
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…
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.
We consider n-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.