Study shows various weak solutions to complex flows match, proving viscosity equals pluripotential.
problem Comparing weak solutions to complex Monge-Ampère flows.
method Examined various notions of weak subsolutions and showed they coincide.
result Viscosity solution equals pluripotential solution.
Stability proven for complex equations on Kähler manifolds.
problem Stability of solutions to complex Monge-Ampère equations.
method Elliptic and parabolic complex Monge-Ampère equations on compact Kähler manifolds.
result Stability result applies to Kähler-Ricci flow.
Develops theory for Kähler-Ricci flow on singular varieties.
problem Analyzing Kähler-Ricci flow on varieties with log terminal singularities.
method Parabolic pluripotential theory and complex Monge-Ampère equations.
result Establishes a parabolic theory analogous to Bedford-Taylor's.
Study solves complex equation on specific types of manifolds.
problem Solving complex Monge-Ampère equation on Kähler manifolds.
method Flow-based arguments to establish existence of smooth solutions.
result Existence of smooth solutions under decreasing right-hand side.
Continuity of complex Monge-Ampère potentials on Kähler manifolds.
problem Continuity of solutions to complex Monge-Ampère equations on compact Kähler manifolds.
method Extending DiNezza-Lu's approach to big cohomology classes, proving continuity on Zariski open sets.
result Singular Kähler-Einstein metrics have continuous potentials on the ample locus outside of the non-klt part.
Develops parabolic pluripotential theory for complex flows.
problem Complex Monge-Ampère equations in degenerate settings.
method Study of semi-concave envelopes and unique solutions.
result Shows semi-concave envelopes as unique solutions.
Study complex Monge-Ampère flows on Kähler manifolds using Perron method.
problem Complex Monge-Ampère flows in big cohomology classes.
method Perron method for pluripotential subsolutions.
result Upper envelope of subsolutions is a unique pluripotential solution with regularity.
The paper constructs invariant Calabi-Yau structures on complexified symmetric spaces.
problem Constructing invariant Calabi-Yau structures on complexified symmetric spaces.
method Solutions of a Monge-Ampère type equation.
result Existence of solutions to the Monge-Ampère type equation.
The paper studies mean curvature flow of Lagrangian graphs in pseudo-Euclidean space.
problem Mean curvature flow of Lagrangian graphs in pseudo-Euclidean space.
method Analyzes the parabolic equation and Monge-Ampère type equation, proving smooth solutions and convergence to self-expanding solutions.
result Smooth solutions u(x,t) for specific nonlinear equations and convergence to self-expanding solutions. Kähler-Einstein metrics found on compactifications of groups.
problem Existence of Kähler-Einstein metrics on group compactifications.
method Continuity method, real Monge-Ampère equation, invariance under maximal compact subgroup.
result Necessary and sufficient condition for existence of Kähler-Einstein metrics.
In this paper, we investigate two hyperbolic flows obtained by adding forcing terms in direction of the position vector to the hyperbolic mean curvature flows in \cite{klw,hdl}. For the first hyperbolic flow, as in \cite{klw}, by using support function, we reduce it to a hyperbolic Monge-Ampeˋre equation …
N. V. Efimov \cite{Ef1} proved that there is no complete, smooth surface in R3 with uniformly negative curvature. We extend this to isometric immersions in a 3-manifold with pinched curvature: if M3 has sectional curvature between two constants K2 and K3, then there exists K1<min(K2,0) such that $M…
New methods solve complex equations, proving some have solutions.
problem Solving complex equations in infinite dimensions.
method Infinite-dimensional prequantum line bundles and moment maps.
result Proves some perturbed equations have solutions for small parameters.
Solves complex Monge-Ampère equations on Kähler manifolds.
problem Behavior of singularities in solutions to degenerate equations.
method Analyzes singularities of solutions to degenerate complex Monge-Ampère equations.
result Resolves unresolved problem from Yau's work.
Method solves high-dimensional nonlinear PDEs using neural networks.
problem Solving high-dimensional fully nonlinear PDEs.
method Backward induction with multi-layer neural networks to estimate solution and its gradient, with Hessian approximated by automatic differentiation.
result Method extends previous work on semi-linear PDEs to fully nonlinear cases, demonstrating accuracy on various examples.
Develops Kähler geometry on new varieties for canonical metrics.
problem No specific problem stated; focuses on new varieties.
method Introduces new varieties, develops Kähler geometry, associates convex functions with metrics.
result Provides expression for Mabuchi functional and combinatorial sufficient condition of properness.
Simplified approach to pseudo-Anosov flows on 3-manifolds.
problem Complexity in understanding pseudo-Anosov flows on 3-manifolds.
method Streamlined framework called Anosov-like group actions.
result Unified and simplified presentation of pseudo-Anosov flows.
In this paper, we study the backward Ricci flow on locally homogeneous 3-manifolds. We describe the long time behavior and show that, typically and after a proper re-scaling, there is convergence to a sub-Riemannian geometry. A similar behavior was observed by the authors in the case of the cross curvature flow.
We investigate how to obtain various flows of Kähler metrics on a fixed manifold as variations of Kähler reductions of a metric satisfying a given static equation on a higher dimensional manifold. We identify static equations that induce the geodesic equation for the Mabuchi's metric, the Calabi flow, the pseudo-Calabi…
It is natural to ask: what kinds of matrices satisfy the Restricted Eigenvalue (RE) condition? In this paper, we associate the RE condition (Bickel-Ritov-Tsybakov 09) with the complexity of a subset of the sphere in Rp, where p is the dimensionality of the data, and show that a class of random matrices with indep…
Given a complex analytic function f on a Whitney stratified complex analytic variety of complex dimension n, whose real part Re(f) is Morse, we prove the existence of a stratified gradient-like vector field for Re(f) such that the unstable set of a critical point p on a stratum S of complex dimension s has real dimensi…
DeepWeightFlow generates diverse neural network weights efficiently.
problem Generating complete neural network weights efficiently and accurately.
method Flow Matching in weight space with Git Re-Basin and TransFusion.
result DeepWeightFlow generates high-accuracy neural networks without fine-tuning.
Study uses neural networks to predict wall quantities in turbulent flows.
problem Predicting wall quantities in turbulent open channel flows.
method Training convolutional neural networks (FCN) and a proposed R-Net architecture to predict wall-shear-stress and wall pressure.
result R-Net architecture performs better and predicts wall quantities with around 10% error.
Study eigenvalues of Laplace operator on specific 3D manifolds under Ricci flow.
problem Analyze eigenvalues of Laplace operator with potential under backward Ricci flow.
method Use backward Ricci flow on locally homogeneous 3-manifolds, derive bounds and convergence results.
result Eigenvalue λ+(t) approaches zero as flow converges to sub-Riemannian geometry. Flow doesn't get wider near singularities if they're convex.
problem Preventing the fattening of surfaces during flow.
method Analyzing mean curvature flow with mean convex singularities.
result The level set flow of a mean convex initial surface doesn't get wider near singularities.
Let X be a closed manifold with zero Euler characteristic, and let f: X --> S^1 be a circle-valued Morse function. We define an invariant I which counts closed orbits of the gradient of f, together with flow lines between the critical points. We show that our invariant equals a form of topological Reidemeister torsion …
In [Centro-affine invariants for smooth convex bodies, Int. Math. Res. Notices. doi: 10.1093/imrn/rnr110, 2011] Stancu introduced a family of centro-affine normal flows, p-flow, for 1≤p<∞. Here we investigate the asymptotic behavior of the planar p-flow for p=∞ in the class of smooth, origin-symme…
New framework analyzes deep neural networks using feature probabilities.
problem Degenerate situation in over-parameterized DNNs.
method Mean-field framework representing DNNs by feature probabilities and functions.
result Global convergence proof for over-parameterized Res-Net training.
Convolutional networks predict turbulence from wall quantities.
problem Predicting turbulence fields from wall-shear-stress components and wall pressure.
method Two CNN models: FCN and FCN-POD, trained on DNS data.
result FCN and FCN-POD models outperform EPOD in predicting turbulence fields.
The study finds conditions for a third rank Killing tensor field on a 2D Riemannian torus.
problem Conditions for the existence of a third rank Killing tensor field on a 2D Riemannian torus.
method Analyzes the metric of the torus and uses Fourier coefficients to derive conditions for the function λ.
result Equations relating Fourier coefficients of the function λ determine the existence of a third rank Killing tensor field.
Neural network predicts turbulence near-wall regions efficiently.
problem Reducing computational cost in turbulent flow simulations.
method Fully-convolutional neural network trained on DNS data.
result FCN predicts velocity fluctuations at y+=50 with less than 20% error. The paper studies practical estimation and interpretation of Rényi transfer entropy.
problem Challenges in accurately estimating and interpreting Rényi transfer entropy.
method Systematic study of k-nearest neighbor estimator for Rényi entropy and transfer entropy.
result Effective estimates of effective Rényi transfer entropy can accurately capture directional information flow.
Study on colored Jones polynomial of figure-eight knot for complex parameters.
problem Asymptotic behavior of colored Jones polynomial for figure-eight knot.
method Analyzing the asymptotic growth rate of the polynomial for complex parameters with small imaginary part.
result Growth rate of polynomial is related to the Chern-Simons invariant for large real part of the parameter and to the reciprocal of Alexander polynomial for small real part.
GLASS Flows improves flow and diffusion model performance by optimizing sampling efficiency.
problem Efficiency bottleneck in sampling Markov transitions for flow and diffusion models.
method Introduces GLASS Flows, a new sampling paradigm that simulates a 'flow matching model within a flow matching model' to sample Markov transitions efficiently.
result Eliminates the trade-off between stochastic evolution and efficiency in large-scale text-to-image models.
This paper tackles the dynamics of singularities in geometric flows.
problem Understanding the behavior of singularities in geometric flows over long time.
method By incorporating dynamical properties, the paper shows smoothing for long time for generic initial conditions.
result The singularities are shown to be the simplest possible in an important special case.
Study reduces complexity and uncertainty in human atrial cell models.
problem Uncertainty in parameter estimates from gating kinetics models.
method Approximate Bayesian computation to re-calibrate models, investigate two approaches: more complete datasets and less complex formulations.
result Less complex model with fewer parameters gives better fit and lower uncertainty.
In this note we prove the following result: Let X be a complete, connected 4-manifold with uniformly positive isotropic curvature, with bounded geometry and with no essential incompressible space form. Then X is diffeomorphic to S4, or RP4, or S3×S1, or $\mathbb{S…
The paper studies a special Grassmannian space and shows it's an orbit of a unitary group.
problem Investigating a specific Grassmannian space of infinite-dimensional subspaces.
method Analyzing the restricted p-Schatten class Grassmannian and showing it's an affine coadjoint orbit of a unitary group. result The restricted p-Schatten class Grassmannian is shown to be an affine coadjoint orbit of an infinite-dimensional restricted unitary group. In this short paper, we re-derive the Bochner formula for the Laplacian by considering local variations of volume. The derivation is rooted in the fact that the Laplacian of a function measures the volume variation along the flow of the gradient vector of the function. Possible extensions of this approach/technique are…
Improved exploration in SAC using Normalizing Flows policies.
problem Brittleness and inefficiency of DRL algorithms in continuous action spaces.
method Introducing Normalizing Flow policies within the SAC framework to learn more expressive policies.
result Increased stability and better exploration in sparse reward settings.
RDL-Net improves speech enhancement with fewer parameters and better performance.
problem Improving speech enhancement with fewer parameters and better performance.
method Proposes RDL-Net, a CNN combining residual and dense aggregations without over-allocating parameters.
result RDL-Net achieves higher speech enhancement performance with fewer parameters and lower computational requirements.
The geometric constructions are elaborated on (semi) Riemannian manifolds and vector bundles provided with nonintegrable distributions defining nonlinear connection structures induced canonically by metric tensors. Such spaces are called nonholonomic manifolds and described by two equivalent linear connections also ind…
The Willmore flow is well known problem from the differential geometry. It minimizes the Willmore functional defined as integral of the mean-curvature square over given manifold. For the graph formulation, we derive modification of the Willmore flow with anisotropic mean curvature. We define the weak solution and we pr…
The paper provides approximation guarantees for neural networks trained with gradient flow.
problem Approximating neural networks trained with gradient flow in continuous L2(Sd−1)-norm. method NTK argument for non-convex second but last layer, under-parametrized regime.
result Gradient flow convergence guarantees for neural networks under Sobolev smoothness assumptions.
New method for inference on covariates in NMF with random effects.
problem Formal inference for covariate effects in NMF with non-negativity constraints.
method NMF-RE model with random effects, ridge updates, df-based cap, asymptotic linearization, wild bootstrap.
result Valid inference on covariates with non-negativity constraint, avoiding degeneracy.
New framework transforms labeled datasets for various machine learning tasks.
problem Lack of principled methods to transform labeled datasets.
method Wasserstein gradient flows in probability space for optimization of data-generating distributions.
result Framework can impose constraints, adapt for transfer learning, or re-purpose models.
A framework combines diverse power grid data for a unified view.
problem Unified view of complex power grids with distributed resources.
method Belief Propagation for probabilistic data fusion.
result Efficient distributed inference algorithm for grid state quantification.
Normalizing Flows improve prediction interval efficiency in CP.
problem Inefficient prediction intervals in CP due to non-uniform error distribution.
method Train a Normalizing Flow to optimize the distance metric between errors and inputs.
result Optimized prediction intervals are more efficient and valid.