Study Ricci vector fields on 2D space with diagonal metrics.
problem Understanding Ricci vector fields on 2D space with specific metrics.
method Examined Ricci vector fields on R2 with a diagonal metric. result Characterized Ricci vector fields on R2 with a diagonal metric. Study of symmetries in a 2D space with specific metric properties.
problem Understanding symmetries in a 2D space with diagonal metrics.
method Analyzing Killing vector fields under specific restrictions on Lamé coefficients.
result Concretely described symmetries of the metric under given conditions.
Study of harmonic maps on 2D simplicial complexes, proving existence and regularity.
problem Existence and regularity of harmonic maps between 2D simplicial complexes.
method Extending previous work, study metrics conformal to flat or ideal hyperbolic, proving existence, uniqueness, and regularity of harmonic maps.
result Existence, uniqueness, and regularity results for harmonic maps between 2D simplicial complexes.
Study 2D spaces with curvature, finding a graph structure.
problem Understanding the geometry of 2D spaces with curvature constraints.
method Analyzing spaces as unions of disks, identifying singular points.
result Obtained a graph structure of topological singular points.
Computes a new metric quantity Y(M) for Riemannian 2d-manifolds.
problem No simple metric quantity exists for Riemannian manifolds.
method Defines Y(M) and Y_disc(M) involving sectional curvatures and computes them for specific manifolds.
result Y(M) and Y_disc(M) differ from the Euler characteristic and can be positive or negative.
Explicitly describes pluriclosed metrics on compact Lie groups.
problem Characterizing pluriclosed metrics on compact Lie groups.
method Explicit description using root systems and invariant structures.
result Explicit formulas for pluriclosed metrics in terms of root systems.
Study 2D spaces with curvature, focusing on structure and approximations.
problem Characterize and understand 2D metric spaces with curvature constraints.
method Lipschitz homotopy approximations, curvature measures, convergence analysis.
result Established Gauss-Bonnet Theorem and characterized spaces.
New probabilistic constructions for Kähler-Einstein metrics.
problem Finding Kähler-Einstein metrics on complex algebraic varieties.
method Microcanonical measures and maximum entropy principles.
result Novel characterizations and evolution equations.
Study on a specific type of Riemannian manifolds constructed from 2D space-forms.
problem Characterizing and understanding new types of Riemannian manifolds.
method Constructed as a product of a real line and a 2-dimensional Riemannian space-form, with metrics derived from cone and hyperbolic extensions.
result Characterized and studied in terms of their curvature properties.
Method finds domain of Laplace-Beltrami operator on 2D almost-Riemannian manifolds.
problem Determining the domain of the Laplace-Beltrami operator on 2D almost-Riemannian manifolds with tangency points.
method Using tools from Lie groupoids, natural domains of perturbations are found.
result Method allows treatment of geometries with tangency points.
In 2D, Finsler metrics with 3+ projective fields are projectively equivalent to Randers.
problem Characterizing Finsler metrics with multiple projective vector fields.
method Analyzing the Lie algebra of projective vector fields and showing equivalence to Randers metrics.
result A complete list of 2D Finsler metrics with at least 3 projective fields up to equivalence.
Formula for sectional curvature on 2D Lorentzian manifolds derived.
problem Calculating sectional curvature on 2D Lorentzian manifolds.
method Obtained a formula for sectional curvature.
result Formula for sectional curvature on 2D Lorentzian manifolds.
In 2D, Finsler metrics are Douglas and generalized Berwald if they are Berwald or Randers.
problem Characterizing Finsler metrics in 2D that are both Douglas and generalized Berwald.
method Proof that in dimension two, a Finsler metric is both Douglas and generalized Berwald if and only if it is Berwald or a Randers metric α+β with specific properties. result Finsler metrics in 2D are Douglas and generalized Berwald if they are Berwald or Randers metrics with specific properties.
Improved 2D cardiac MRI with less data using deep learning.
problem Reducing artefacts in undersampled 2D radial cine MRI.
method Modified U-net trained on spatio-temporal slices.
result Outperforms existing methods in image quality and training efficiency.
Study of 2D metrics with one projective symmetry leading to superintegrable systems.
problem Classifying 2D metrics with one projective symmetry and their integrable properties.
method Analyzing projective connections, partial differential equations, and geodesic flows.
result Superintegrable systems are parametrized by the 2-sphere, except for 6 exceptional points.
A criterion in terms of differential invariants for a metric on a surface to be Liouville is established. Moreover, in this paper we completely solve in invariant terms the local mobility problem of a 2D metric, considered by Darboux: How many quadratic in momenta integrals does the geodesic flow of a given metric poss…
We reformulate the compatibility condition between a generalized metric and a small (non-maximal rank) Dirac structure in an exact Courant algebroid found in the context of the gauging of strings and formulated by means of two connections in purely Dirac-geometric terms. The resulting notion, a transverse generalized m…
We construct a counterexample to a conjectured inequality L<2D, relating the diameter D and the least length L of a nontrivial closed geodesic, for a Riemannian metric on the 2-sphere. The construction relies on Guillemin's theorem concerning the existence of Zoll surfaces integrating an arbitrary infinitesimal odd def…
Kernel-UCBVI algorithm balances exploration and exploitation in metric state-action spaces.
problem Exploration-exploitation dilemma in finite-horizon reinforcement learning with metric state-action spaces.
method Kernel-UCBVI, leveraging smoothness and kernel estimators of rewards and transitions.
result First regret bound for kernel-based RL using smoothing kernels, O(H3K2d/(2d+1)). This paper proposes grid cells encode position via a conformal isometric embedding of 2D physical space.
problem Hexagonal grid firing patterns in grid cells.
method Learning a distance-preserving position embedding in neural space using a recurrent neural network.
result The conformal isometric embedding of 2D physical space into neural space explains hexagonal grid firing patterns.
We define systems of pre-extremals for the energy functional of regular rheonomic Lagrange manifolds and show how they induce well-defined Hamilton orthogonal nets. Such nets have applications in the modelling of e.g. wildfire spread under time- and space-dependent conditions. The time function inherited from such a Ha…
The paper classifies Landsberg metrics on a 2D Lie group and proves a conjecture.
problem Classifying Landsberg metrics on a 2D Lie group.
method Analyzing left invariant conic Finsler metrics on a 2D non-Abelian Lie group.
result Any left invariant conic Landsberg metric on G must be Berwald. 2D CNNs approximate Korobov functions with near-optimal rates.
problem Approximating Korobov functions using 2D CNNs.
method Constructive approach for 2D CNNs with ReLU activations and fully connected layers.
result 2D CNNs achieve near-optimal approximation rates for Korobov functions.
We extend the notion of what it means for a complete Ricci flow to have a given initial metric, and consider the resulting well-posedness issues that arise in the 2D case. On one hand we construct examples of nonuniqueness by showing that surfaces with cusps can evolve either by keeping the cusps or by contracting them…
This research uses PointNets to detect 2D objects from radar data.
problem Detecting 2D objects from sparse radar data for automated driving.
method Adapting PointNets for radar data, performing 2D object classification and bounding box regression.
result Demonstrates the potential of PointNets for 2D object detection in radar data.
The Strominger conjecture is confirmed for compact Hermitian manifolds in 2D and special higher dimensions.
problem Determining conditions for a Hermitian metric to be Kähler based on the Strominger connection's curvature.
method Analyzing the Strominger connection's holomorphic sectional curvature in compact Hermitian manifolds.
result The Strominger conjecture is confirmed in 2D and special higher dimensions.
Proposes a model to generate 3D-aware images from 2D images.
problem Generating 3D-aware images from 2D images.
method Likelihood-based top-down model using Neural Radiance Fields and energy-based latent variables.
result Model can infer 3D object structures from 2D images and generate novel views.
Derive bihamiltonian structure for rational reduction of 2D-Toda hierarchy
problem Derive bihamiltonian structure for rational reduction of 2D-Toda hierarchy
method Direct computations
result Derive local bihamiltonian structure
We analyze the Ricci flow of a noncompact metric that describes a two-dimensional black hole. We consider entanglement entropy of a 2d black hole which is due to the quantum correlations between two subsystems: one is inside and the other is outside the black hole horizon. It is demonstrated that the entanglement entro…
Enhances 2D face recognition with 3D features using active illumination.
problem Improving robustness of 2D face recognition to spoofing attacks and low-light conditions.
method Projecting a high spatial frequency pattern onto the face to recover 3D information and a 2D image simultaneously.
result Significantly boosts face recognition performance and dramatically improves robustness to spoofing attacks.
We present an avatar of the Euler obstruction to foliated structures on certain non-metric surfaces. This adumbrates (at least for the simplest 2D-configurations) that the standard mechanism---to the effect that the devil of algebra sometimes barricades the existence of angelic geometric structures (obstruction theory …
For generic torus-invariant metrics, eigenspaces are 2D and nodal sets are connected hypersurfaces.
problem Understanding spectral multiplicity and nodal sets for generic torus-invariant metrics.
method Analyzing real Δg-eigenspaces and nodal sets for generic T-invariant metrics. result For generic T-invariant metrics, real Δg-eigenspaces are irreducible and have dimension at most 2, and nodal sets are connected hypersurfaces with specific properties. First order Hamiltonian operators of differential-geometric type were introduced by Dubrovin and Novikov in 1983, and thoroughly investigated by Mokhov. In 2D, they are generated by a pair of compatible flat metrics g and g~ which satisfy a set of additional constraints coming from the skew-symmetry condition…
A novel method compresses point cloud attributes by folding them onto a 2D grid.
problem Efficiently compressing point cloud attributes for storage and transmission.
method Interpreting point clouds as 2D manifolds, folding onto a grid, and mapping attributes to the grid using optimized methods.
result The proposed folding-based approach achieves performance comparable to state-of-the-art codecs.
Paper classifies brain signals using eigenvalues for 2D and 3D educational content questions.
problem Classifying brain signals for 2D and 3D educational content questions.
method Eigenvalues of covariance matrix used as features; KNN and SVM classifiers applied.
result No significant difference in learning, memory retention, and recall between 2D and 3D educational content.
New perspective on Ricci flow on spheres using Minkowski spacetime.
problem Classifying singularity models for null mean curvature flow in Minkowski spacetime.
method Equivalence of 2d-Ricci flow and null mean curvature flow on lightcones.
result Classification of singularity models for null mean curvature flow.
2D tissue model predicts neurotoxicity more accurately and robustly.
problem Fast and accurate prediction of developmental neurotoxicity.
method Machine learning on 2D bio-engineered tissue models.
result 2D model outperforms 3D model in accuracy and robustness.
New method reconstructs 3D shapes from 2D images using Kendall's shape space.
problem Reconstruct 3D shapes from 2D images, especially for rare specimens.
method Kendall's shape space approach with prior information.
result More robust and plausible shapes compared to previous methods.
Geometric interpretation of 2d-4d wall-crossing formulas.
problem Understanding wall-crossing phenomena in coupled 2d-4d systems.
method Deformation theory of holomorphic pairs and relation to scattering diagrams.
result Geometric interpretation of wall-crossing formulas.
The paper establishes T-duality for 2D σ-models with H-flux.
problem T-duality for 2D σ-models with H-flux.
method Localization and graded T-duality map (graded Hori morphism).
result Establishes the most general version of T-duality for Type II String Theory.
New theorem shows embedding restrictions for manifold skeletons.
problem Embedding restrictions for triangulated manifolds.
method Proves van Kampen-Flores theorem for manifolds with specific Stiefel-Whitney classes.
result Triangulated manifolds with non-trivial Stiefel-Whitney classes cannot embed into R2d. The paper finds a family of 2D surfaces where Laplacian and Dirichlet-to-Neumann commute.
problem Finding surfaces with commuting boundary Laplacian and Dirichlet-to-Neumann map.
method Investigated 2D submanifolds of R^2, classified surfaces of genus 0 or with k≥3 boundary components.
result One-parameter family of 2D submanifolds with commuting Laplacian and Dirichlet-to-Neumann map.
We propose the Legendrian web in a contact three manifold as a second order generalization of the planar web. An Abelian relation for a Legendrian web is analogously defined as an additive equation among the first integrals of its foliations. For a class of Legendrian d-webs defined by simple second order ODE's, w…
Instantiation-Net reconstructs 3D mesh from single 2D image for right ventricle.
problem Reconstructing 3D shape from limited 2D images for surgical navigation.
method Combines DCNN for feature extraction and GCN for mesh reconstruction.
result Demonstrates practical strength and potential clinical use.
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.
2d dualities linked to 4-simplex triangulation.
problem Exploring dualities between 2d and 3d theories via 4-manifold geometry.
method Using Pachner moves and supersymmetric half-indices.
result Identified IR dualities and abelian dualities.
Transformer-M learns molecular data in 2D or 3D formats.
problem Learning models for molecules are limited to specific data formats.
method Developed a Transformer-based model that can handle 2D and 3D molecular data.
result Transformer-M achieves strong performance on both 2D and 3D molecular tasks.
DISPR uses diffusion models to predict 3D cell shapes from 2D images.
problem Predicting 3D cell shapes from 2D microscopy images.
method Diffusion model trained to predict 3D shapes from 2D microscopy images as a prior.
result Adding DISPR predictions to minority cell classes improves classification accuracy.