Using an integrable discrete Dirac operator, we construct a discrete version of the Weierstrass representation of time-like surfaces parametrized along isotropic directions in , and . The corresponding discrete surfaces have isotropic edges. We show that any discrete surface satisfying a gen…
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48 results for “isotropic direction”
Study of invariant surfaces in isotropic and pseudo-isotropic geometries.
problem Prescribed curvature for invariant surfaces in singular metrics.
method Analysis of one-parameter subgroups of isotropic rigid motions, computation of fundamental forms and curvatures.
result Generalization of revolution and helicoidal surfaces to singular metrics.
Sharp isoperimetric inequalities for the sine transform of even isotropic measures are established. The corresponding reverse inequalities are obtained in an asymptotically optimal form. These new inequalities have direct applications to strong volume estimates for convex bodies from data about their sections or projec…
Study geodesics on surfaces with changing metric.
problem Geodesic behavior on surfaces with a changing signature metric.
method Analyze geodesics at points where the isotropic direction is tangent to the discriminant curve.
result Properties of geodesics at points of isotropic tangency.
The paper develops methods to reduce deployment risk under dynamic covariate shifts.
problem Reduction of deployment risk under dynamic covariate shifts.
method Time-domain Poincare inequality and Jacobian-velocity theorem to identify and control directional tangent energy.
result Drift-aligned tangent regularization (DTR) reduces risk volatility and directional gain in low-rank drift regimes.
Four-dimensional complete gradient shrinking Ricci solitons with half positive isotropic curvaturemath.DG
Study on 4D Ricci solitons with specific curvature properties.
problem Characterizing 4D complete gradient shrinking Ricci solitons with half positive isotropic curvature.
method Curvature estimates, strong maximum principle, classification arguments.
result New classification results for gradient shrinking Kähler-Ricci solitons and 4D complete gradient shrinking Ricci solitons with half nonnegative isotropic curvature.
In the Friedmann Model of the universe, cosmologists assume that spacelike slices of the universe are Riemannian manifolds of constant sectional curvature. This assumption is justified via Schur's Theorem by stating that the spacelike universe is locally isotropic. Here we define a Riemannian manifold as almost locally…
A new exploration method for RL using parameter space noise.
problem Improving exploration in deep reinforcement learning.
method Switching isotropic and directional exploration in parameter space with parameter space noise.
result The proposed method achieves competitive results and better performance in sparse reward environments.
We classify six-dimensional Lie groups which admit a left-invariant half-flat SU(3)-structure and which split in a direct product of three-dimensional factors. Moreover, a complete list of those direct products is obtained which admit a left-invariant half-flat SU(3)-structure such that the three-dimensional factors ar…
New method uses stochastic gradients to escape saddle points in machine learning models.
problem Escaping saddle points in non-convex machine learning models.
method Proposes a new approach using isotropic noise to replace traditional saddle point escape methods.
result Derives the first convergence rate for plain SGD to a second-order stationary point, independent of problem dimension.
AniDS improves molecular force field modeling by learning anisotropic noise.
problem Molecular force field modeling suffers from oversimplified assumptions about atomic motions.
method AniDS introduces anisotropic noise generation for better modeling of directional and structural variability.
result AniDS outperforms existing methods on benchmarks, achieving significant improvements in force prediction accuracy.
Classifies surfaces in isotropic geometry with constant curvature.
problem Classifying surfaces with constant curvature in isotropic geometry.
method Classifies surfaces under the condition that at least one translating curve lies in a plane.
result Classification of surfaces in isotropic geometry with constant curvature.
The paper examines Randers metrics with isotropic scalar curvature properties.
problem Characterizing Randers metrics with specific scalar curvature properties.
method Analyzes properties of Randers metrics with isotropic scalar curvature.
result Proves that Randers metrics with weakly isotropic scalar curvature have isotropic -curvature and are either Minkowskian or Riemannian.
A particular Finsler-metric proposed in [1,2] and describing a geometry with a preferred null direction is characterized here as belonging to a subclass contained in a larger class of Finsler-metrics with one or more preferred directions (null, space- or timelike). The metrics are classified according to their group of…
A central theme in Riemannian geometry is understanding the relationships between the curvature and the topology of a Riemannian manifold. Positive isotropic curvature (PIC) is a natural and much studied curvature condition which includes manifolds with pointwisequarter-pinched sectional curvatures and manifolds with p…
Proposes Gaussian process priors on graph sets with geometric structure.
problem Defining Gaussian process priors on sets of graphs with geometric structure.
method Defines priors respecting graph geometric structure, analogous to Euclidean isotropic processes.
result Efficient computational technique for evaluating priors' kernels, making them usable in toolboxes.
If Gamma is a nonuniform, irreducible lattice in a semisimple Lie group whose real rank is greater than 1, we show Gamma contains a subgroup that is isomorphic to a nonuniform, irreducible lattice in either SL(3,R), SL(3,C), or a direct product SL(2,R)^m x SL(2,C)^n$, with m + n > 1. (In geometric terms, this can be in…
Paper establishes a relation between Berwald scalar curvature and S-curvature.
problem Understanding the relationship between Finsler metrics' curvature properties.
method Proved conditions for isotropic Berwald scalar curvature and weakly isotropic S-curvature.
result Finsler metrics with isotropic Berwald scalar curvature have weakly isotropic S-curvature.
This paper counts special Lagrangian fibrations on K3 surfaces with a quadratic growth rate.
problem Counting special Lagrangian fibrations on K3 surfaces.
method Counting primitive isotropic vectors in indefinite lattices, deduced from equidistribution results in homogeneous dynamics.
result The number of special Lagrangian fibrations grows like \( V^{20} \) with a power-saving term.
The paper classifies surfaces in isotropic space satisfying Weingarten conditions.
problem Classifying surfaces in isotropic space with specific curvature conditions.
method Analyzing rotational surfaces in isotropic 3-space with Weingarten conditions.
result Classification of surfaces of linear Weingarten type in isotropic space.
The paper classifies pseudo-isotropic and lightlike pseudo-isotropic Lagrangian surfaces.
problem Characterizing Lagrangian surfaces in complex pseudo spaces.
method Analyzing properties of Lagrangian surfaces in complex pseudo spaces.
result Classification of Lagrangian surfaces based on pseudo-isotropy.
Isotropic tori in revisitedmath.SG
New isotropic tori found in complex space, not Hamiltonian isotopic.
problem Finding non-Hamiltonian isotopic isotropic tori in complex space.
method Analyzing isotropic tori in for .
result At least two exact isotropic -tori in are not Hamiltonian isotopic.
In this paper, we find a condition on -metrics under which the notions of isotropic S-curvature, weakly isotropic S-curvature and isotropic mean Berwald curvature are equivalent.
Improved loss functions adapt to weight-space anisotropy, outperforming isotropic counterparts.
problem Adapting to the anisotropic nature of deep weight spaces for better performance.
method Refined local entropic loss functions restricted to a subset of weights, exploiting anisotropy.
result Partial local entropies outperform isotropic counterparts on image classification tasks.
Study physical work done by isotropic vector forces along isotropic curves.
problem Investigate physical work done by isotropic vector forces.
method Analyze forces represented by isotropic vectors acting along isotropic curves on a manifold with specific metric structures.
result Calculate the work done by isotropic vector forces along isotropic curves.
Study isotropic Riemannian maps and helices along them.
problem Understanding Riemannian maps and their associated helices.
method Presented isotropic Riemannian maps and characterized helices along them.
result Characterization of helices along Riemannian maps.
Forward gradients improve neural network training without backpropagation issues.
problem Training neural networks without backpropagation's locking and memorization problems.
method Using directional derivatives in forward differentiation mode, with biased guesses based on feedback from small auxiliary networks.
result Using gradients from a local loss as a candidate direction improves Forward Gradient methods.
The paper explores algebraic structures on diffeological vector spaces.
problem Defining and understanding algebraic structures on diffeological vector spaces.
method Formal checks, verification of properties, and analysis of decompositions.
result The maximal isotropic subspace is invariant and unique, while characteristic subspaces are not.
We study two types of isotropic planes: weakly isotropic and strongly isotropic planes. We prove that a Riemannian manifold of indefinite metric is conformally flat if and only if its curvature tensor vanishes on all the strongly isotropic planes. We specialize the plane axiom for Riemannian manifolds of indefinite met…
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.
Rotation minimizing frames and spherical curves in simply isotropic and pseudo-isotropic 3-spacesmath.DG
The paper develops RM frames for isotropic and pseudo-isotropic spaces and characterizes spherical curves.
problem Differential geometry of curves in isotropic and pseudo-isotropic 3-spaces.
method Developing rotation minimizing frames and applying them to spherical curves in isotropic and pseudo-isotropic spaces.
result Characterization of spherical curves via linear equations involving curvatures and RM frames.
Study of surfaces in isotropic and pseudo-isotropic spaces using Gauss map and shape operator.
problem Differential geometry of surfaces in degenerate metric spaces.
method Introducing isotropic Gauss map and shape operator, computing curvature tensors and geodesics.
result New curvature tensor not vanishing identically, related to relative Gaussian curvature.
The paper classifies surfaces with constant curvature in isotropic space.
problem Classifying surfaces with constant curvature in isotropic space.
method Classification based on constant Gaussian and mean curvature.
result Non-existence result for surfaces with H/K=const.
The classification of all possible holonomy algebras of Einstein and vacuum Einstein Lorentzian manifolds is obtained. It is shown that each such algebra appears as the holonomy algebra of an Einstein (resp., vacuum Einstein) Lorentzian manifold, the direct constructions are given. Also the holonomy algebras of totally…
Study isotropic curves on complex quadric with geometric relations.
problem Characterize isotropic curves on the complex quadric.
method Analyze geometric properties and relations of isotropic curves.
result Discovers relations between isotropic curves and surfaces in spaceforms.
In this paper, we construct a new class of Finsler manifolds called generalized isotropic Berwald manifolds which is an extension of the class of isotropic Berwald manifolds. We prove that every generalized isotropic Berwald manifold is a generalized Douglas-Weyl manifold. On a compact generalized isotropic Berwald man…
Constructs a moment map flow for isotropic maps on surfaces.
problem Understanding isotropic maps on surfaces and their properties.
method Develops a Kähler moment map geometry and a modified moment map flow.
result Polyhedral modified moment map flow induces a strong deformation retraction.
The study finds conditions for flat isotropic non-homogeneous tori to be Hamiltonian minimal.
problem Finding conditions for Hamiltonian minimality of isotropic non-homogeneous tori.
method Constructing a family of flat isotropic non-homogeneous tori and finding necessary and sufficient conditions.
result Necessary and sufficient conditions for Hamiltonian minimality of isotropic non-homogeneous tori in and .
Spinor representation in isotropic space via Laguerre geometry.
problem Representing conformal and constant mean curvature surfaces in isotropic space.
method Developing Laguerre geometry of isotropic space, defining spin transformations, and constructing Weierstrass and Kenmotsu representations.
result Explicit constructions of zero mean curvature and constant mean curvature surfaces.
Paper calculates curvatures of metrics from isotropic structures.
problem Calculating curvatures of metrics induced by isotropic structures.
method Calculating curvature tensors of induced Riemannian metrics.
result Curvatures of the metrics are determined.
The paper studies hanging chains and surfaces in degenerate geometries.
problem Investigating hanging chains and surfaces in simply isotropic plane and space.
method Characterizing catenaries and proving them as minimal surfaces in the simply isotropic space.
result The simply isotropic catenary is the generating curve of a minimal surface of revolution.
We create real-time geodesic rendering for non-isotropic geometries.
problem Challenging visualization of non-isotropic geometries.
method Novel methods for real-time native geodesic rendering.
result Methods can be applied to visualization, machine learning, and video games.
The paper approximates smooth isotropic surfaces with piecewise linear ones.
problem Approximating smooth isotropic surfaces with piecewise linear ones.
method Using analogies with infinite dimensional moment map geometry, the authors prove the approximation of smooth isotropic immersions by piecewise linear ones.
result Smooth isotropic immersions can be approximated by piecewise linear isotropic maps.
Developed a new concept of isometric surfaces in isotropic space.
problem Missing notion of isometric deformations in isotropic space.
method Using Gauss' Theorema Egregium as a condition, developed a new concept.
result Natural analogues of Euclidean space isometries found in isotropic space.
Study extends Ramanathan's result to isotropic surfaces in spheres of any dimension.
problem Minimal isometric immersions of isotropic surfaces in spheres.
method Analyzes compact and non-compact isotropic surfaces in spheres of arbitrary dimension.
result Extends Ramanathan's result to isotropic surfaces in spheres of any dimension.
The paper classifies isotropic Lagrangian submanifolds in a specific nearly Kähler manifold.
problem Characterizing isotropic Lagrangian submanifolds in nearly Kähler manifolds.
method Analyzing the geometric properties and using the nearly Kähler structure to classify submanifolds.
result Complete classification of -isotropic Lagrangian submanifolds in the homogeneous nearly Kähler .
The paper classifies surfaces in the isotropic 3-space with constant curvature and mean curvature.
problem Classifying surfaces in the isotropic 3-space with specific curvature properties.
method Analyzing helicoidal surfaces and special curves on them.
result Classification of surfaces with constant curvature and mean curvature in the isotropic 3-space.
Paper shows isotropic - and -curvatures are equivalent in warped Finsler metrics.
problem Equivalence of non-Riemannian curvatures in warped product Finsler metrics.
method Study of warped product Finsler metrics.
result Isotropic - and -curvatures are equivalent for these metrics.