We propose Cormorant, a rotationally covariant neural network architecture for learning the behavior and properties of complex many-body physical systems. We apply these networks to molecular systems with two goals: learning atomic potential energy surfaces for use in Molecular Dynamics simulations, and learning ground…
New RL method designs 3D molecules with improved symmetry.
problem Lack of 3D information in molecular design.
method Symmetry-aware actor-critic architecture using spherical harmonics.
result Improves generalization and molecule quality.
New method for cross-validation in high-dimensional data with dependent or heavy-tailed covariates.
problem Inconsistent cross-validation in high-dimensional settings with dependent or heavy-tailed covariates.
method ROTI-GCV framework for cross-validation under proportional asymptotics regime.
result Demonstrated accuracy of ROTI-GCV in synthetic and semi-synthetic settings.
Rotationally symmetric solutions persist after mean curvature flow starts from a double cone.
problem Understanding the symmetry of solutions to mean curvature flow.
method Analyzing solutions coming out of a double cone.
result Rotationally symmetric solutions persist.
Rotationally invariant Ricci flows are constructed and shown to converge to spacetimes.
problem Constructing and understanding Ricci flows through surgery on rotationally invariant manifolds.
method Rotationally invariant Ricci flow through surgery, convergence to spacetimes, blowup rate analysis.
result Rotationally invariant Ricci flows converge to spacetimes with controlled curvature blowup.
The paper classifies hypersurfaces in Heisenberg groups with rotational symmetry.
problem Classifying hypersurfaces in Heisenberg groups with rotational symmetry.
method Fundamental theorems and earlier results in [3] and [4] were used to classify umbilic hypersurfaces and generate curves for hypersurfaces with constant p-mean curvature. result Complete classification of umbilic hypersurfaces and generating curves in Heisenberg groups Hn. New Ricci flow solutions found with rotational symmetry and cone-like singularities.
problem Finding Ricci flow solutions with specific symmetry and singularity properties.
method Rotationally symmetric Ricci flow with scaling-invariant curvature bounds, using approximation method.
result Complete Ricci flow solution with cone-like singularity at the origin.
Sharp inequalities found for orbifold metrics.
problem Bounding systolic ratios on rotationally symmetric orbifolds.
method Analyzing spindle orbifolds and Besse metrics.
result Upper bounds on systolic ratios are attained at Besse metrics.
We investigate existence and stability of rotationally symmetric critical immersions of variational problems of higher order which were considered by Nitsche.
New theorems on compactness and finiteness for specific types of self-shrinkers.
problem Characterizing rotationally symmetric self-shrinkers with constraints.
method Compactness and finiteness theorems for self-shrinkers with specific symmetries and constraints.
result Existence of entropy minimizing self-shrinkers diffeomorphic to S1imesSn−1 for each n≥2. Estimates for p-capacities on symmetric manifolds.
problem Estimating relative p-capacities on symmetric manifolds. method Rotationally symmetric manifolds and novel volumetric estimates.
result Sharp weak (p,q)-embeddings and precise lower bounds of principal p-frequencies. In this paper, we show that the nonexistence of rotationally symmetric harmonic diffeomorphism between the unit disk without the origin and a punctured disc with hyperbolic metric on the target.
We investigate the properties of the Cheeger sets of rotationally invariant, bounded domains Ω⊂Rn. For a rotationally invariant Cheeger set C, the free boundary ∂C∩Ω consists of pieces of Delaunay surfaces, which are rotationally invariant surfaces of constant mean curvature. We show…
Paper proves March's criterion for transience on symmetric manifolds.
problem Determining transience on rotationally symmetric manifolds.
method Analyzes bounded non-constant harmonic functions and Dirichlet problem at infinity.
result March's criterion is necessary and sufficient for transience.
Noise-cleaning fMRI brain activity matrices for better precision estimation.
problem Denoise precision matrices of fMRI time series to estimate true matrices.
method Comparison of various noise-cleaning algorithms on synthetic and real fMRI data.
result Optimal Rotationally Invariant Estimator outperforms others in fMRI data.
New AMP algorithms for rotationally invariant models with reduced complexity.
problem Signal estimation in generalized linear models with arbitrary spectral design matrices.
method Rotationally invariant approximate message passing (AMP) algorithms.
result Performance close to Vector AMP with significantly lower complexity.
New comparison theorems for rotationally symmetric self-shrinkers help in proving the uniqueness of the Angenent torus.
problem Uniqueness of the Angenent torus in rotationally symmetric self-shrinkers
method Analyzing profile curves and vertical points of rotationally symmetric self-shrinkers
result Proving the existence and monotonicity of horizontal-point trajectories
New kernel interprets 3D anisotropic data with rotations and improved predictions.
problem Capturing rotated anisotropy in 3D spatial fields.
method Introduces a Lie-algebraic kernel with three principal length-scales and an explicit rotation.
result Posterior recovers rotated anisotropy and improves prediction over axis-aligned kernels.
Researchers set entropy limits for specific types of self-shrinkers.
problem Understanding entropy limits for self-shrinkers with symmetries.
method Derived explicit entropy bounds for two specific classes of self-shrinkers using isoparametric foliations and symmetry analysis.
result Entropy bounds generalized to new classes of self-shrinkers, extending previous findings.
We study a second order differential equation corresponding to rotationally symmetric F-harmonic maps between certain noncompact manifolds. We show unique continuation and Liouville's type theorems for positive solutions. Asymptotic properties and the existence of bounded positive solutions are investigated.
We study a second order ordinary differential equation corresponding to rotationally symmetric p-harmonic maps. We show unique continuation and Liouville's type theorems for positive solutions. We discuss the existence of bounded positive entire solutions. Asymptotic properties of the positive solutions are investiga…
Motivated by the rich theory of harmonic maps from a 2-sphere, we study biharmonic maps from a 2-sphere in this paper. We first derive biharmonic equation for rotationally symmetric maps between rotationally symmetric 2-manifolds. We then apply the equation to obtain a classification of biharmonic maps in a family of r…
We consider a class of overdetermined problems in rotationally symmetric spaces, which reduce to the classical Serrin's overdetermined problem in the case of the Euclidean space. We prove some general integral identities for rotationally symmetric spaces which imply a rigidity result in the case of the round sphere.
The paper classifies 3D complete gradient Yamabe solitons.
problem Classifying nontrivial 3D complete gradient Yamabe solitons.
method Analyzing properties of Yamabe solitons to show rotationally symmetry.
result Nontrivial non-flat 3D complete steady gradient Yamabe solitons are rotationally symmetric.
In this paper we study the gradient Ricci shrinking soliton equation on rotationally symmetric manifolds of dimension three and higher and prove that the only complete examples of such metrics on Sn, Rn and R×Sn−1 are, respectively, the round, flat, and standard cylindrical metrics.
This work investigates how gradient-based learning performs with structured data, revealing issues and improvements.
problem Gradient-based learning under structured data, particularly with a spiked covariance structure.
method Investigates the effect of a spiked covariance structure on gradient-based feature learning and proposes weight normalization.
result Gradient-based dynamics may fail to recover the true direction in anisotropic settings, but weight normalization can improve performance.
In this note, using Calabi's method, we construct rotationally symmetric Kahler-Ricci solitons on the total space of direct sum of fixed hermitian line bundle and its projective compactification, where the curvature of hermitian line bundle is Kahler-Einstein. These examples generalize the construction of Koiso, Cao an…
The study proves geometric inequalities for static convex domains in static rotationally symmetric spaces.
problem Proving geometric inequalities for static convex domains in static rotationally symmetric spaces.
method Locally constrained curvature flow in a static rotationally symmetric space Nn+1, proving graphical solutions and static convexity preservation. result Proves weighted geometric inequalities for static convex domains close to a slice of Nn+1. Rotationally equivariant convolutions improve molecular property prediction.
problem Predicting molecular properties using graph neural networks.
method Ablation study with rotationally equivariant and invariant convolutions on QM9 data set.
result Rotationally equivariant layers decrease test error by an average of 23%.
Extends Milnor's criterion to biharmonic functions.
problem Deciding surface type for biharmonic functions.
method Generalizes Milnor's criterion to biharmonic functions.
result Characterizes whether a surface is hyperbolic or parabolic for biharmonic functions.
3-manifold curvature comparison with rotationally symmetric bodies.
problem Comparing scalar curvature of 3-manifolds with rotationally symmetric boundaries.
method Inspired by Gromov, comparing mean curvatures and induced metrics.
result Flatness of 3-manifolds under certain curvature conditions.
The paper finds rotationally symmetric critical metrics for Laplace eigenvalues on tori.
problem Maximizing the first normalized Laplace-Beltrami eigenvalue on tori.
method Constructing equivariant harmonic maps to spheres and analyzing their properties.
result Rotationally symmetric critical metrics for the first eigenvalue are found and characterized.
New findings on magnetic geodesic flows and periodic motions.
problem Characterizing superintegrable systems in magnetic geodesic flows.
method Analyzing rotationally symmetric magnetic geodesic flows.
result All sufficiently slow motions in a central magnetic field are periodic under specific curvature and homogeneity conditions.
Paper analyzes holdout cross-validation for large non-Gaussian covariance estimation.
problem Estimating large covariance matrices for non-Gaussian data.
method Use of Weingarten calculus and Ledoit-Péché formula for theoretical error derivation.
result Optimal train-test split ratio is proportional to square root of matrix dimension.
We give a full classification of complete rotationally invariant surfaces with constant Gauss curvature in Berger spheres: they are either Clifford tori, which are flat, or spheres of Gauss curvature K≥K0 for a positive constant K0, which we determine explicitly and depends on the geometry of the ambient Ber…
Proves conjecture about sphere widths under rotational symmetry.
problem Width stability of rotationally symmetric metrics.
method Proof of conjecture and extensions to higher dimensions.
result Stability of min-max width under rotational symmetry.
The flow of a torus by inverse mean curvature keeps total curvature bounded until singularity.
problem Understanding the behavior of a torus under inverse mean curvature flow until singularity.
method Analyzing the evolution of a rotationally symmetric embedded torus in R3 by inverse mean curvature flow. result The total curvature remains bounded until the singular time Tmax. Numerical simulations show stability of Type-II singularities in noncompact hypersurfaces.
problem Stability of Type-II singularities in noncompact hypersurfaces with rotationally-symmetric perturbations.
method Adaptation of the overlap method to include angular dependence.
result MCF of noncompact hypersurfaces with angular dependence behaves similarly to rotationally-symmetric perturbations, developing Type-II or Type-I singularities.
We consider the mean curvature evolution of rotationally symmetric surfaces. Using numerical methods, we detect critical behavior at the threshold of singularity formation resembling the one of gravitational collapse. In particular, the mean curvature simulation of a one-parameter family of initial data reveals the exi…
In this paper we study the classification of ancient convex solutions to the mean curvature flow in Rn+1. An open problem related to the classification of type II singularities is whether a convex translating solution is k-rotationally symmetric for some integer 2≤k≤n, namely whether its level set is a …
New minimal tori found in curved spaces.
problem Existence of minimal tori in curved spaces.
method Generalized Angenent's shrinking tori to minimal n-dimensional tori. result Existence of rotationally symmetric embedded f-minimal tori.
In this note we prove that a (anti-)self dual quasi Yamabe soliton with positive sectional curvature is rotationally symmetric. This generalizes a recent result of G. Huang and H. Li in dimension four. Whence, (anti-) self dual gradient Yamabe solitons with positive sectional curvature is rotationally symmetric. We als…
New AMP algorithms improve multi-layer signal reconstruction.
problem Reconstructing signals and hidden variables from multi-layer networks with rotationally invariant weights.
method Developed multi-layer rotationally invariant generalized AMP (ML-RI-GAMP) algorithms and state evolution recursion.
result ML-RI-GAMP outperforms existing methods in terms of lower complexity and similar performance.
New algorithm for signal estimation in noisy matrix models.
problem Signal estimation in rectangular spiked matrix models with rotationally invariant noise.
method Orthogonal Approximate Message Passing (OAMP) algorithm for signal estimation.
result Optimal OAMP algorithm minimizes mean-squared error and achieves Bayes-optimal performance.
Upper bound on index of rotationally symmetric self-shrinking tori.
problem Stability of singularities in mean curvature flow.
method Entropy functional and eigenvalue analysis.
result Upper bound on the index of rotationally symmetric self-shrinking tori.
The study proves rotationally symmetric property of certain shrinking gradient Yamabe solitons.
problem Understanding the rotational symmetry of specific shrinking gradient Yamabe solitons.
method Analyzing nontrivial complete shrinking gradient Yamabe solitons with bounded scalar curvature.
result The assumption of bounded scalar curvature and strict inequality at some point is necessary and sufficient for rotational symmetry.
Upper bounds on Laplacian eigenvalues on manifolds with non-negative curvature.
problem Bounding Laplacian eigenvalues on manifolds with non-negative scalar curvature.
method Investigation of invariant spectrum on compact Riemannian manifolds with large isometry groups.
result Upper bounds for eigenvalues of the invariant spectrum assuming non-negative scalar curvature.
We consider a diffuse interface approximation for the lipid phases of rotationally symmetric two-phase bilayer membranes and rigorously derive its Γ-limit. In particular, we prove that limit vesicles are C1 across interfaces, which justifies a regularity assumption that is widely made in formal asymptotic and nume…