Paper proves extension of unit normal vector field from a hypersurface.
problem Need to extend unit normal vector field from a hypersurface.
method Provides an elementary proof of existence and uniqueness of such an extension.
result Elementary proof of existence and uniqueness of unit gradient field extension.
We provide some examples of harmonic unit vector fields as normalized gradients of isoparametric functions from a K-contact geometry setting.
Study k-almost Ricci solitons on contact metric manifolds.
problem Characterize k-almost Ricci solitons on contact metric manifolds.
method Prove isometric properties and extend results for k-almost gradient Ricci solitons and k-almost Ricci solitons.
result Compact K-contact metric manifolds that are k-almost gradient Ricci solitons are isometric to a unit sphere.
Improved mean-field theory for two-layer neural networks with stronger bounds and generalizations.
problem Learning dynamics of two-layer neural networks using stochastic gradient descent.
method Mean-field approximation and gradient flow in Wasserstein space.
result Stronger approximation guarantees for learning two-layer neural networks, independent of dimensionality.
The paper explores conditions for real holomorphic gradient fields on Kähler and conformally Kähler manifolds.
problem Conditions for real holomorphic gradient fields on Kähler and conformally Kähler manifolds.
method Investigation of real-valued weight functions with real holomorphic gradient fields on Kähler and conformally Kähler manifolds.
result Identification and determination of weight functions with real holomorphic gradient fields on specific metrics.
Study on neural networks' performance under different normalizations as N grows.
problem Characterizing neural networks' performance under various normalizations.
method Developed an asymptotic expansion to analyze statistical output of shallow neural networks.
result No bias-variance trade-off exists to leading order in N, and variance decreases as normalization approaches mean field.
New sampling method uses gradient-free IPS with RKHS velocity field.
problem Efficient sampling from unnormalized target densities.
method Gradient-free interacting particle systems (IPS) with RKHS velocity field.
result IPS produce high-quality samples from various target distributions.
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.
Characterizes magnetic unit vector fields on Lie groups.
problem Classifying magnetic unit vector fields on Lie groups.
method Characterization through critical points of Landau Hall and Dirichlet energy functionals.
result Classification of all magnetic left invariant unit vector fields on 3-dimensional Lie groups.
The paper simplifies the Fisher information matrix for random deep networks, speeding up learning.
problem Learning deep neural networks efficiently with large parameter spaces.
method Statistical neurodynamical method to reveal Fisher information properties, proving unit-wise block diagonal structure and explicit inverse.
result Explicit natural gradient formula without matrix inversion, speeding up learning.
The study proves biharmonic unit sections on 2-tori are always harmonic and exists in each homotopy class.
problem Characterizing biharmonic unit vector fields and sections on 2-tori.
method Analyzing variational problems for unit vector fields under conformal metrics, proving properties through homotopy classes.
result Biharmonic unit sections on 2-tori are always harmonic and exist in each homotopy class.
AOPU stabilizes NN training by approximating natural gradient, improving stability and convergence.
problem Stability and interpretability in online NN training for industrial soft sensors.
method AOPU truncates gradient backpropagation, optimizing trackable parameters, and approximating natural gradient.
result AOPU achieves stable convergence and superior performance on chemical process datasets.
We present a new equation with respect to a unit vector field on Riemannian manifold Mn such that its solution defines a totally geodesic submanifold in the unit tangent bundle with Sasaki metric and apply it to some classes of unit vector fields. We introduce a class of covariantly normal unit vector fields and pro…
Minimal vector fields on oscillator groups studied, with specific conditions for minimality.
problem Characterizing minimal left-invariant unit vector fields on oscillator groups.
method Analyzing structure constants and harmonic maps into the unit tangent bundle.
result Minimal vector fields defined by specific conditions on oscillator groups.
Proves a central limit theorem for neural networks with hidden layers.
problem Understanding the statistical behavior of neural networks with large numbers of hidden units and training iterations.
method Rigorous mathematical proof using weak convergence methods and stochastic analysis.
result Neural network fluctuations around mean-field limit follow a Gaussian distribution and satisfy a stochastic partial differential equation.
Study biharmonic vector fields and unit vector fields on Riemannian manifolds.
problem Determine the equivalence of biharmonicity and harmonicity for vector fields and unit vector fields on Riemannian manifolds.
method Analyze biharmonic vector fields and unit vector fields on (M,g) with pseudo-Riemannian g-natural metrics on TM and T1M. result Contrary to Sasaki metric, biharmonicity and harmonicity are not equivalent for large classes of g-natural metrics on TM. New activation functions mimic neuronal biology to improve deep learning performance.
problem Vanishing gradients and suboptimal learning in deep learning models.
method Introducing bionodal root unit (BRU) activation functions based on neuronal cell properties.
result BRU activation functions lead to faster training and better generalization in deep learning models.
Lower bounds on unit vector fields' volume using Poincaré indices.
problem Finding volume bounds for unit vector fields on spheres with punctures.
method Using Poincaré indices to establish lower bounds and identify achieving fields.
result Identified vector fields achieving the determined volume bounds.
We give a complete list of those left invariant unit vector fields on three-dimensional Lie groups with the left-invariant metric that generate a totally geodesic submanifold in the unit tangent bundle of a group with the Sasaki metric. As a result, each class of three-dimensional Lie groups admits the totally geodesic…
The study classifies contact metric manifolds based on Ricci-Yamabe solitons.
problem Classifying contact metric manifolds based on Ricci-Yamabe solitons.
method Analyzing specific types of solitons in contact metric manifolds.
result Contact metric manifolds are classified based on the properties of Ricci-Yamabe solitons.
Sharp lower bound found for area of vector fields on spherical annuli.
problem Finding the minimum area of unit vector fields on spherical annuli.
method Established a sharp lower bound through mathematical analysis.
result Sharp lower bound for the area of unit vector fields on spherical annuli.
Characterizes loxodromic unit vector fields on punctured spheres.
problem Finding vector fields with a lower bound on volume functional.
method Characterization based on Poincaré indexes.
result Only loxodromic unit vector fields achieve the lower bound.
We give a full geometrical description of local totally geodesic unit vector field on Riemannian 2-manifold, considering the field as a local imbedding of the manifold into its unit tangent bundle with the Sasaki metric.
Study finds a minimum volume for vector fields on a punctured sphere.
problem Finding the minimum volume of unit vector fields on a punctured sphere.
method Analyzes the volume of vector fields tangent to an antipodally punctured unit 2-sphere.
result Provides a lower bound for the volume of unit vector fields.
We present an explicit formula for the mean curvature of a unit vector field on a Riemannian manifold, using a special but natural frame. As applications, we treat some known and new examples of minimal unit vector fields. We also give an example of a vector field of constant mean curvature on the Lobachevsky (n+1) s…
Study finds the volume of unit vector fields on a punctured sphere and shows their images match minimally immersed Klein bottles.
problem Finding the volume of unit vector fields on a punctured sphere.
method Analyzes the volume of unit vector fields on an antipodally punctured unit 2-sphere and shows their images coincide with minimally immersed Klein bottles.
result The images of minimizing vector fields on the punctured sphere match those of minimally immersed Klein bottles.
Minimal surfaces in S3(2) linked to vector fields on punctured sphere.
problem Connecting minimal surfaces in S3(2) to vector fields on a punctured sphere.
method Established a correspondence between minimal surfaces and area-minimizing vector fields.
result Stability relation for Lawson cylinders in S3(2).
We study the geometrical properties of a unit vector field on a Riemannian 2-manifold, considering the field as a local imbedding of the manifold into its tangent sphere bundle with the Sasaki metric. For the case of constant curvature K, we give a description of the totally geodesic unit vector fields for K=0 and K=1 …
Analyzes feature learning in neural networks using a self-consistent dynamical field theory.
problem Feature learning in infinite-width neural networks.
method Constructs deterministic dynamical order parameters as inner-product kernels for hidden unit activations and gradients.
result Reveals the hidden layer activation distribution, neural tangent kernel evolution, and output predictions.
Aguilar introduced isotropic almost complex structures Jδ,σ on the tangent bundle of a Riemannian manifold (M,g). In this paper, some results will be obtained on the integrability of these structures. These structures with the Liouville 1-form define a class of Riemannian metrics gδ,σ on TM which are …
We consider a unit normal vector field of (local) hyperfoliation on a given Riemannian manifold as a submanifold in the unit tangent bundle with Sasaki metric. We give an explicit expression of the second fundamental form for this submanifold and a rather simple condition its totally geodesic property in the case of a …
Investigates how SGD behaves in high-dimensional neural networks, distinguishing between global convergence and local minima.
problem Understanding the behavior of SGD in high-dimensional shallow neural networks.
method Extends statistical physics analysis to study SGD dynamics, focusing on mean-field/hydrodynamic regime and learning rate.
result Identifies the critical number of hidden units and learning rate for SGD to avoid local minima.
The stability of the 3-dimensional Hopf vector field, as a harmonic section of the unit tangent bundle, is viewed from a number of different angles. The spectrum of the vertical Jacobi operator is computed, and compared with that of the Jacobi operator of the identity map on the 3-sphere. The variational behaviour of t…
Study spherical images of modified vector fields on a unit sphere.
problem Understanding spherical images of modified vector fields.
method Analysis of spherical images using modified orthogonal vector fields and Darboux vector.
result Characterization of spherical indicatrices with modified orthogonal frame.
Automates feature selection and weighting in molecular systems.
problem Optimal feature selection and alignment in molecular systems.
method Differentiable Information Imbalance (DII) method for automated feature ranking and scaling.
result Automated feature selection and scaling that preserves information content and interpretability.
Study classifies harmonic vector fields on 3-manifolds.
problem Classifying harmonic unit vector fields on 3-manifolds.
method Investigates under mild curvature assumptions, classifying vector fields and manifolds.
result Classifies both vector fields and manifolds supporting them.
Calibrations help estimate volumes on odd spheres without gaps.
problem Estimating the minimum volume of tangent vector fields on odd spheres.
method Using a specific calibration and analyzing stable mass in the section class.
result No smooth unit field on Sn has a graph that is ω-calibrated everywhere. We prove that the Hopf vector field is a unique one among geodesic covariantly normal unit vector fields on spheres such that the submanifold generated by the field is totally geodesic in the unit tangent bundle with Sasaki metric. As application, we give a new proof of stability (instability) of the Hopf vector field …
The paper "Minimal unit vector fields" by O. Gil-Medrano and E. Llinares-Fuster \cite{GilLli1}. is a seminal paper in the field that has been cited by many authors. It contains, however, a minor technical mistake in Theorem 14 that is important to fix. In this short note, we will provide a correction to that result.
We obtain the explicit representation of Legendre surfaces in the unit 5-sphere with harmonic mean curvature vector field, under the condition that the mean curvature function is constant along a certain special direction.
New method estimates sigmoids' parameters using gradient estimation.
problem Learning parameters of models with hidden variables.
method Estimate gradients at random points, cluster, use cluster centers as parameter estimates.
result Proven that estimated gradients concentrate around true parameter vectors.
Paper solves curvature prescription on unit ball with sign-changing functions.
problem Prescribing mean curvature on the unit ball with sign-changing functions.
method Negative gradient flow method to realize f as mean curvature. result Proves that a sign-changing function f can be realized as the boundary mean curvature of a conformal metric. A new recurrent unit alleviates vanishing gradients for long-term dependencies.
problem Vanishing gradients in recurrent neural networks make long-term dependencies hard to model.
method Proposes a new NRU architecture that avoids saturating activation functions and gates.
result Demonstrates superior performance across various tasks with and without long-term dependencies.
The paper explores how the unit inclusion affects topological quantum field theories in non-semisimple categories.
problem Understanding the effects of unit inclusion in non-semisimple braided tensor categories on topological quantum field theories.
method Analyzes the dualizability of the unit inclusion morphism in Morita 4-category of braided tensor categories and applies the Cobordism Hypothesis.
result Shows that the unit inclusion in non-semisimple modular categories leads to non-compact relative 3D topological quantum field theories.
Outer billiards maps on foliated surfaces with specific vector fields.
problem Characterizing vector fields that induce outer billiards maps on foliated surfaces.
method Analyzing necessary and sufficient conditions for foliation of the exterior of a hypersurface.
result Explicit periodic and unbounded orbits in a specific outer billiard map.
We study 6-dimensional nearly Kahler manifolds admitting a Killing vector field of unit length. In the compact case it is shown that up to a finite cover there is only one geometry possible, that of the 3--symmetric space S3×S3.
Study on Ricci solitons on tangent and unit tangent bundles.
problem Characterizing Ricci solitons on tangent and unit tangent bundles.
method Analyzing pseudo-Riemannian g-natural metrics and their Ricci soliton properties. result Classification of conformal vector fields and existence of non-Einstein Ricci solitons.
New examples of harmonic unit vector fields on hyperbolic 3-space are constructed by exploiting the reduction of symmetry arising from the foliation by horospheres. This is compared and contrasted with the analogous construction in Euclidean 3-space, using a foliation by planes, which produces some new examples of harm…