Study local invariants of divergence-free webs in geometry.
problem Characterize triviality of divergence-free webs.
method Introduce two local invariants: differential and geometric.
result Triviality of either invariant characterizes trivial divergence-free web-germs.
Upper bound found for divergence-free Killing 2-tensors on manifolds.
problem Bounding the space of divergence-free symmetric Killing 2-tensors.
method Witten deformation and Morse function analysis.
result Explicit calculation of dimension for p=2. New proof finds three divergence-free vector fields for any 3D manifold.
problem Proving the existence of divergence-free vector fields on 3D manifolds.
method Using geometric properties of eigenspinors in three dimensions.
result Found three divergence-free vector fields that are orthogonal and have the same length at every point.
New Lie-group methods preserve geometric divergence-free features on manifolds.
problem Designing divergence-free Lie-group methods on manifolds.
method Introducing planar aromatic trees to span the free tracial post-Lie-Rinehart algebra.
result New Lie-group methods derived for high-order accuracy.
Researchers solve a 25-year-old conjecture about vector fields.
problem Proving a 25-year-old conjecture about divergence-free vector fields.
method Analysis of a Leibniz algebra underlying these vector fields.
result Construction of the universal central extension for divergence-free vector fields and diffeomorphisms.
We prove that a gradient shrinking Ricci soliton with fourth order divergence-free Riemannian tensor is rigid. For the 4-dimensional case, we show that any gradient shrinking Ricci soliton with fourth order divergence-free Riemannian tensor is either Einstein, or a finite quotient of the Gaussian shrinking soliton $\…
The study classifies flat solvmanifolds and finds G2-structures on them.
problem Classifying and finding G2-structures on flat solvmanifolds. method Classification of flat splittable solvmanifolds and search for compatible G2-structures. result Examples of compact flat manifolds with specific G2-structures. Unified framework for data-free sampling using Wasserstein gradient flows.
problem Efficient sampling from unnormalized distributions without data.
method Unified theoretical framework based on Wasserstein gradient flows.
result Unified form of velocity field for various f-divergences.
New model-free DR-RL algorithm with finite sample complexity.
problem Limited model-free DR-RL methods with convergence guarantees or sample complexities.
method Integrates Multi-level Monte Carlo (MLMC) technique with threshold mechanism.
result First model-free DR-RL approach with finite sample complexity for total variation and Chi-square divergence.
In this paper, we classify 3-dimensional complete gradient Yamabe solitons with divergence-free Cotton tensor. We also give some classifications of complete gradient Yamabe solitons with nonpositively curved Ricci curvature in the direction of the gradient of the potential function.
SRFE clarifies KL divergences without unifying learning frameworks.
problem Inductive biases of KL divergences and their limitations.
method Introducing SRFE, a log-moment-based functional of the likelihood ratio.
result SRFE recovers KL divergences as limits and reveals a mean-variance tradeoff.
New method proves dimension-free convergence for ULD in KL divergence.
problem Polynomial scaling of existing convergence guarantees in high dimensions.
method Refined KL local error framework, focusing on tr(H) instead of d.
result First dimension-free KL divergence bounds for discretized ULD.
We study the problem of finding strain-minimising stream surfaces in a divergence-free vector field. These surfaces are generated by motions of seed curves that propagate through the field in a strain minimising manner, i.e., they move without stretching or shrinking, preserving the length of their arbitrary arc. In ge…
In this review paper, we present several results on central extensions of the Lie algebra of symplectic (Hamiltonian) vector fields, and compare them to similar results for the Lie algebra of (exact) divergence free vector fields. In particular, we comment on universal central extensions and integrability to the group …
This paper deals with the problem of describing the vector spaces of divergence-free, natural tensors on a pseudo-Riemannian manifold that are second-order; i.e., that are defined using only second derivatives of the metric. The main result establishes isomorphisms between these spaces and certain spaces of tensors (at…
Researchers found G2-structures with zero torsion on specific Lie groups.
problem Finding G2-structures with divergence-free torsion.
method Isometric flow to evolve G2-structures with divergence-free torsion as critical points.
result Three families of G2-structures on solvable Lie groups have zero torsion.
New rigidity results for quasi-Einstein metrics with non-zero divergence-free vector fields.
problem Classifying quasi-Einstein metrics with specific vector field properties.
method Analyzing quasi-Einstein metrics on closed manifolds and near-horizon geometries of extreme black holes.
result These metrics always admit a one-parameter group of isometries generated by the divergence-free vector field.
Electrostatic systems with specific tensors are locally conformally flat.
problem Understanding the geometry of electrostatic systems with special tensors.
method Proving local conformal flatness for electrostatic manifolds with divergence-free Bach tensor.
result Three-dimensional electrostatic manifolds with divergence-free Bach tensor are locally conformally flat.
New theorem links symmetries to first integrals in plasma physics.
problem Understanding the relationship between symmetries and first integrals in divergence-free fields.
method Developed a Noether-type Theorem reformulation for three-dimensional divergence-free vector fields.
result Converse of the Noether-type Theorem holds on the toroidal region, proving the existence of flux coordinates.
Formula derived for sample complexity in binary hypothesis testing.
problem Determine the minimum number of samples to distinguish between two distributions.
method Developed a formula for sample complexity in both prior-free and Bayesian settings, using Jensen-Shannon and Hellinger divergences.
result Formula characterizes sample complexity for a wide range of error parameters, up to multiplicative constants.
We prove that, in a space-time of dimension n>3 with a velocity field that is shear-free, vorticity-free and acceleration-free, the covariant divergence of the Weyl tensor is zero if the contraction of the Weyl tensor with the velocity is zero. The other way, if the covariant divergence of the Weyl tensor is zero, then…
We generalise the average asymptotic linking number of a pair of divergence-free vector fields on homology three-spheres by considering the linking of a divergence-free vector field on a manifold of arbitrary dimension with a codimension two foliation endowed with an invariant transverse measure. We prove that the aver…
We classify complete gradient Ricci solitons satisfying a fourth-order vanishing condition on the Weyl tensor, improving previously known results. More precisely, we show that any n-dimensional (n≥4) gradient shrinking Ricci soliton with fourth order divergence-free Weyl tensor is either Einstein, or a finite q…
New method learns disentangled signals without prior or model constraints.
problem Learning disentangled signals from data without prior or model constraints.
method Minimizes conditional KL divergence using a sequential algorithm to learn de-mixing flow models.
result Method learns self-sufficient signals that can reconstruct missing values.
Proves helicity is the only regular Casimir for 3D hydrodynamics.
problem Identifying unique Casimir functions for 3D hydrodynamics.
method Normal forms, Poincaré-Birkhoff theorem, division lemma.
result Helicity is the only regular Casimir for volume-preserving diffeomorphisms.
We prove a CR Obata type result that if the first positive eigenvalue of the sub-Laplacian on a compact strictly pseudoconvex pseudohermitian manifold with a divergence free pseudohermitian torsion takes the smallest possible value then, up to a homothety of the pseudohermitian structure, the manifold is the standart S…
Conditions found for linearizing divergence-free fields on invariant tori.
problem Linearizing divergence-free vector fields on invariant tori.
method Assuming a solution to the cohomological equation and using Bers' results in pseudo-analytic function theory.
result The field B on S is either identically zero or nowhere vanishing, with a linearizable form. The paper proves rigidity of certain solitons with specific properties.
problem Classifying and understanding Bach-flat gradient Schouten solitons.
method Analyzing the properties of Schouten solitons and their Ricci tensors.
result Rigidity of certain Schouten solitons under specific conditions.
Model-free preference under ambiguity defined and applied.
problem Understanding and quantifying ambiguity aversion and prudence.
method Introduces a new model-free definition of ambiguity attitudes and applies it in various contexts.
result New definition of ambiguity prudence equivalent to specific mathematical functions.
Study finds rigidity of biconservative hypersurfaces in space forms without curvature assumptions.
problem Investigating biconservative hypersurfaces in space forms without scalar curvature assumptions.
method Introduced a novel divergence-free tensor to derive results without curvature assumptions.
result Rigidity results for biconservative hypersurfaces in space forms without scalar curvature assumptions.
We compute all 2-covariant tensors naturally constructed from a semiriemannian metric which are divergence-free and have weight greater than -2. As a consequence, it follows a characterization of the Einstein tensor as the only, up to a constant factor, 2-covariant tensor naturally constructed from a semiriemannian met…
Reformulates divergence map for Turaev cobracket in non-commutative geometry.
problem Algebraic description of Turaev cobracket on surfaces.
method Non-commutative geometry, flat connection, associative algebras, Lie operad.
result Algebraic description of Turaev cobracket on surfaces.
On a compact n-dimensional manifold M, it is well known that a critical metric of the total scalar curvature, restricted to the space of metrics with unit volume, is Einstein. It has been conjectured that a critical metric of the total scalar curvature, restricted to the space of metrics with constant scalar curvat…
The paper proves rigidity results for manifolds with special holonomy.
problem Proving rigidity results for compact Riemannian manifolds with special holonomy.
method Using divergence free Weyl tensors and curvature operators, the paper proves similar results for manifolds with special holonomy.
result The paper proves that manifolds with special holonomy are locally symmetric or conformally equivalent to a quotient of the sphere.
New differential geometry perspective on orthogonal RNNs.
problem Mitigating exploding and vanishing gradients in RNNs.
method Using tools from differential geometry, parameterizing vector fields via directional derivatives of scalar functions.
result Our approach achieves comparable or better results on benchmark tasks.
Generative models tackle incompressible fluid flows by enforcing divergence-free constraints.
problem Simulating incompressible fluid flows with generative models.
method Score-based diffusion models with divergence-free constraint.
result Models can reproduce Kolmogorov turbulence characteristics.
On a compact n-dimensional manifold, it is well known that a critical metric of the total scalar curvature, restricted to the space of metrics with unit volume is Einstein. It has been conjectured that a critical metric of the total scalar curvature, restricted to the space of metrics with constant scalar curvature o…
This work proposes robust reinforcement learning methods using both offline and online data.
problem Designing robust policies against parameter uncertainties in high-dimensional systems.
method Proposes RPQ for model-free learning with historical data and HyTQ for hybrid learning with both historical and online data.
result Unified analysis and theoretical guarantees for robust optimal policies in high-dimensional systems.
New GPs model edge functions on complex networks, capturing divergence and curl.
problem Modeling flow data on networks with independent learning of Hodge components.
method Developed Hodge-compositional edge GPs using Hodge decomposition.
result Hodge-compositional edge GPs can represent any edge function and capture flow relevance.
New RL method nearly optimally learns policies with generative models.
problem Finding optimal policies in reinforcement learning with generative models.
method Mirror descent value iteration with KL divergence and entropy regularization.
result The method is nearly minimax-optimal for small ε-optimal policies. On manifolds with an even Riemannian conformally compact Einstein metric, the resolvent of the Lichnerowicz Laplacian, acting on trace-free, divergence-free, symmetric 2-tensors is shown to have a meromorphic continuation to the complex plane, defining quantum resonances of this Laplacian. For higher rank symmetric ten…
New tensors reveal full curvature structure from Riemann tensor.
problem Limited information from Ricci contraction of Riemann tensor.
method Contracting double dual of Riemann tensor to reveal full curvature.
result New tensors provide canonical parents of Einstein tensor.
CRBMs improve financial regime detection with PCD and free energy analysis.
problem Detecting systemic risk regimes in financial time series.
method Extended RBM to CRBM with autoregressive conditioning and PCD. Decomposed free energy into magnitude and correlation components.
result CRBM's free energy metric distinguishes between magnitude shocks and market regimes.
A new method stabilizes deep reinforcement learning by using QGraphs to retain replay memory information.
problem Stabilizing model-free off-policy deep reinforcement learning with soft divergence.
method Representing past experiences as a QGraph, selecting a subgraph with favorable structure, and using lower bounds for temporal difference learning.
result QG-DDPG method is less prone to soft divergence and more robust to hyperparameters.
Study characterizes non-collapsed RCD(K, N) spaces using heat kernel metrics.
problem Characterize non-collapsed RCD(K, N) spaces via heat kernel metrics.
method Investigate the second principal term in heat kernel metrics and prove divergence free property.
result Proves non-collapsed property via divergence free property of heat kernel metrics.
We introduce a flow of G2-structures defining the same underlying Riemannian metric, whose stationary points are those structures with divergence-free torsion. We show short-time existence and uniqueness of the solution.
New method for privacy amplification without sampling for matrix factorization.
problem Privacy amplification for differentially private model training with matrix factorization.
method Sampling-free bounds based on Rényi divergence and conditional composition.
result Stronger privacy guarantees for small ε, applicable to various matrices.
We introduce the notion of biconservative hypersurfaces, that is hypersurfaces with conservative stress-energy tensor with respect to the bienergy. We give the (local) classification of biconservative surfaces in 3-dimensional space forms.