Abstract relates Lipschitz-Killing measures to polar volumes of definable sets.
problem Relating geometric measures of definable sets to their polar images.
method Relates Lipschitz-Killing measures to volumes of generic polar images for smooth submanifolds, extending to infinitesimal versions.
result Establishes a relation between polar invariants and densities of generic polar images.
Novel fusion network combines polarization and radiomics features for liver cancer classification.
problem Challenges in histopathological diagnosis of HCC and ICC.
method Two-tier fusion approach: feature-level and classification-level.
result Significantly enhances classification accuracy, even at reduced resolutions.
Study slopes of direct images in complex manifolds, proving a Mehta-Ramanathan type theorem.
problem Distribution of Harder-Narasimhan slopes in direct image sheaves.
method Analyzing asymptotic distributions of slopes under base changes of families of complex projective manifolds.
result Asymptotic distribution of slopes can be recovered from base changes over generic curves.
The paper studies polar normalizations of skew ruled surfaces in 3D space.
problem Understanding the properties and invariants of polar normalized skew ruled surfaces.
method Determination of invariants and analysis of Tchebychev and support vector fields.
result Special polar normalizations lead to degenerate curves.
The paper establishes lower bounds on Yang-Mills functionals for fibrations.
problem Analyzing the stability and nefness of direct image sheaves in fibrations.
method Generalizing mean curvature and Harder-Narasimhan filtrations to arbitrary polarized fibrations.
result Optimal lower bounds on fibered Yang-Mills functionals in terms of direct image sheaves.
Holomorphic families yield metrics with explicit curvature formulas.
problem Positivity of direct images with Poincaré type singularities.
method Analyzes holomorphic families and line bundles with Poincaré type singular metrics.
result Explicit formula for curvature of direct image metrics.
The paper analyzes how multiple classifiers' disagreement and polarization affect overall accuracy.
problem Improving accuracy through ensembling multiple classifiers.
method The paper derives an upper bound for polarization, proposes a neural polarization law, and presents a tight upper bound for the error of majority vote classifiers.
result Disagreement and polarization among classifiers are linearly correlated with the target, and polarization is nearly constant for a dataset.
Polarimetric images enhance object detection in adverse weather conditions.
problem Object detection in road scenes is challenging in adverse weather conditions.
method Combining polarimetric imaging and deep learning.
result Polarimetry improves object detection by 20% to 50% compared to conventional RGB images.
We study a compact invariant convex set E in a polar representation of a compact Lie group. Polar rapresentations are given by the adjoint action of K on p, where K is a maximal compact subgroup of a real semisimple Lie group G with Lie algebra g=k⊕p. If …
Neural network implementation of Brenier's polar factorization for vector fields.
problem Implementing Brenier's polar factorization theorem for vector fields using neural networks.
method Parameterizing the convex function u as an input convex neural network and estimating the measure-preserving map M. result Practical neural implementation of Brenier's polar factorization theorem.
Geometrically, Kostant's Convexity Theorem is extended to submetries with a fat section.
problem Extending Kostant's Convexity Theorem to a broader class of representations.
method Introducing a new concept of 'fat section' and proving the theorem for submetries with this property.
result Kostant's Convexity Theorem is partially extended to submetries with a fat section.
The paper studies curvature of geometric objects related to sheaves and line bundles.
problem Curvature of higher direct images of sheaves twisted by line bundles.
method Explicit curvature formulas derived for certain sheaves and line bundles.
result Proof of a result by Viehweg-Zuo in the context of a maximal variation family.
Deep learning model detects and flags artefacts in polarimetric images.
problem Artifacts in polarimetric images contaminate areas of interest.
method Convolutional Neural Network (CNN) for automatic artefact detection.
result Model achieves 98% true positive and 97% true negative rates.
The paper studies obstructions to solutions of the Wess-Zumino-Witten equation and its generalizations.
problem Existence of solutions to the Wess-Zumino-Witten equation and its generalizations.
method Identification of algebraic obstructions and construction of approximate solutions using Monge-Ampère type equations.
result Approximate solutions to the generalized Wess-Zumino-Witten equation are shown to be the closest to true solutions when the latter do not exist.
Enhanced X-ray polarimetry with deep learning for better exposure times.
problem Improving sensitivity of X-ray telescopic observations with imaging polarimeters.
method A weighted maximum likelihood combination of predictions from a deep ensemble of ResNet convolutional neural networks trained on Monte Carlo event simulations.
result Improves effective exposure times by ~45% for power-law source spectra.
We classify the polar actions on the complex hyperbolic plane up to orbit equivalence. Apart from the trivial and transitive polar actions, there are five polar actions of cohomogeneity one and four polar actions of cohomogeneity two.
Paper develops machine learning to translate SAR to optical images for easier interpretation.
problem Difficulty in human interpretation of SAR images due to non-adapted human vision to microwave scattering.
method Develops a novel reciprocal GAN scheme to train machine intelligence on co-registered SAR and optical images.
result The proposed translation network works well under various SAR and optical image resolutions and polarizations.
Uniform K-stability holds for close polarizations if original is canonical or anti-canonical.
problem Conditions for uniform K-stability of projective varieties.
method Analysis of stability conditions for close polarizations.
result Uniform K-stability is preserved for close polarizations if original is canonical or anti-canonical.
Study polar actions on Damek-Ricci spaces, proving existence and finding examples.
problem Characterize polar actions on Damek-Ricci spaces.
method Prove criteria for isometric actions to be polar, find examples, and classify actions.
result Non-trivial polar actions exist on all Damek-Ricci spaces.
The paper studies how Kähler polarizations degenerate to mixed polarizations on toric varieties.
problem Degeneration of Kähler polarizations to mixed polarizations on toric varieties.
method Constructing polarizations by Hamiltonian actions, finding one-parameter families of Kähler polarizations, and analyzing convergence of spaces of holomorphic sections.
result Kähler polarizations degenerate to mixed polarizations as k increases, with specific convergence results for one-parameter families. The point of the paper is to show some limitations of geometrical optics in the analysis of subwavelength focusing. We analyze the resolution of the image of a line source radiating in the Maxwell fisheye and the Veselago-Pendry slab lens. The former optical medium is deduced from the stereographic projection of a virt…
Formula for curvature of higher direct images in complex manifolds.
problem Computing the curvature of higher direct images in complex manifolds.
method Explicit formula for curvature tensor of Rn−pf∗ΩX/Sp(L). result Generalization of Schumacher's result on Nakano positivity.
Polarized and G-polarized CR manifolds are smooth manifolds endowed with a double structure: a real foliation $\Cal F$ (given by the action of a Lie group G in the G-polarized case) and a transverse CR distribution (E,J). Polarized means that (E,J) is roughly speaking invariant by $\Cal F$. Both structures ar…
A polarity of a projective plane is a map, often assumed to be involutive, mapping a generic point to a generic line and reciprocally. The most classical polarity is the polarity with respect to a conic, but other exist: the harmonic polarity with respect to a triangle, the polarities with respect to high-degree algebr…
The generic fiber of a Lagrangian fibration on an irreducible holomorphic symplectic manifold is an abelian variety. Associate a polarization type to such Lagrangian fibrations coming from polarizations on a generic fiber. We prove that this polarization type is constant in families of Lagrangian fibrations. Further, w…
Paper uses ResUNet-CMB to reconstruct cosmic polarization rotation from CMB data.
problem Reconstructing anisotropic cosmic polarization rotation from CMB data.
method Extended ResUNet-CMB to handle gravitational lensing and patchy reionization.
result ResUNet-CMB outperforms standard quadratic estimator in reconstructing all three effects.
New method polarizes anisotropic Heisenberg groups.
problem Polarizing anisotropic Heisenberg groups.
method Implementing a technique to polarize anisotropic Heisenberg groups.
result New class of polarizable Carnot groups expanded.
The study introduces polarization of generalized Nijenhuis torsions and their relevance in operator fields.
problem Characterization of Haantjes C∞(M)-modules of operator fields. method Introducing polarization of generalized Nijenhuis torsions and proving algebraic identities.
result Polarizations of generalized Nijenhuis torsions are relevant in the characterization of Haantjes C∞(M)-modules of operator fields. The study classifies Riemannian homogeneous spaces with polar isotropy actions.
problem Characterizing Riemannian homogeneous spaces with polar isotropy actions.
method Analyzing simply connected Riemannian homogeneous spaces of compact semisimple Lie groups and various non-compact spaces.
result Classification and non-polar isotropy actions for specific spaces.
Classifies polar foliations on symmetric spaces.
problem Classifying polar foliations on symmetric spaces.
method Orbit equivalence and classification up to codimension two.
result Foliations are either hyperpolar or extensions of rank one foliations.
Paper examines trading polarity to predict market crashes.
problem Insufficient investigation of trading imbalance at high frequency.
method Investigates trading polarity from Shenzhen Stock Exchange data.
result Trading polarity correlates with market crashes and returns.
Classifies totally geodesic submanifolds and polar actions on Stiefel manifolds.
problem Classifying totally geodesic submanifolds and polar actions on Stiefel manifolds.
method Classification through polar actions and cohomogeneity-one actions.
result Classification of orbits of polar actions on Stiefel manifolds.
Classifies hexagonal circular 3-webs with cubic polar curves.
problem Classifying hexagonal circular 3-webs with algebraic polar curves of degree three.
method Analyzes hexagonal circular 3-webs on unit sphere with polar points on a twisted cubic.
result Completes the classification of hexagonal circular 3-webs with algebraic polar curves of degree three.
The main result of this paper is that a polar action on a compact irreducible homogeneous Kaehler manifold is coisotropic. This is then used to give new examples of polar actions and to classify coisotropic and polar actions on quadrics.
POLAR framework interprets word embeddings using polar opposites.
problem Lack of interpretability in pre-trained word embeddings.
method Adopt semantic differentials and polar opposites to transform embeddings.
result Interpretable word embeddings maintain performance comparable to original embeddings.
Study empty polar varieties' impact on singular function-germs.
problem Topology of singular function-germs with nonisolated singularities.
method Analysis of empty polar varieties.
result Topology implications of nonempty polar varieties.
For complex projective manifolds we introduce polar homology groups, which are holomorphic analogues of the homology groups in topology. The polar k-chains are subvarieties of complex dimension k with meromorphic forms on them, while the boundary operator is defined by taking the polar divisor and the Poincare residue …
Totally geodesic sections found in polar actions.
problem Understanding sections of polar actions on Riemannian manifolds.
method Elementary proof of a folklore result.
result Sections of polar actions are totally geodesic.
We prove a criterion for an isometric action of a Lie group on a Riemannian manifold to be polar. From this criterion, it follows that an action with a fixed point is polar if and only if the slice representation at the fixed point is polar and the section is the tangent space of an embedded totally geodesic submanifol…
Carnot groups can be polarized if they have specific coordinate systems.
problem Understanding when Carnot groups can be polarized.
method Proving Carnot groups with certain coordinate systems are polarizable.
result Carnot groups with suitable horizontal polar coordinates are polarizable.
Study Mabuchi rays on toric Kähler manifolds to understand quantization.
problem Understanding quantization in toric Kähler manifolds.
method Analyzing Mabuchi rays associated with test configurations and moment polytopes.
result Quantization in limit polarizations corresponds to restrictions of monomial sections.
Polar manifolds are Riemannian G-manifolds admitting a "section", i.e., a complete submanifold passing through every orbit and doing so orthogonally. We consider compact simply-connected polar manifolds and achieve an equivariantly diffeomorphic classification in dimensions 5 or less. As an application, we determine wh…
The paper bounds abnormal and Goh-abnormal sets for metabelian Lie groups with polarizations.
problem Bounding abnormal and Goh-abnormal sets for metabelian Lie groups.
method Analyzing rank 2 polarizations and sub-Riemannian structures on metabelian Lie groups.
result Metabelian Lie groups with polarizations satisfy the minimizing Sard property.
In this paper, we generalize the polar transforms of spacelike isothermic surfaces in Q14 to n-dimensional pseudo-Riemannian space forms Qrn. We show that there exist c−polar spacelike isothermic surfaces derived from a spacelike isothermic surface in Qrn, which are into Srn+1(c), Hr−1n+1(c)…
Classifies polar actions on 3D homogeneous spaces.
problem Classifying polar isometric actions on 3D homogeneous spaces.
method Orbit equivalence classification and study of cohomogeneity one actions.
result Classification of extrinsically homogeneous surfaces and orbit foliations.
Study shows horizontal diameter of unit spheres is π for specific foliations.
problem Understanding horizontal diameter in unit spheres with specific foliations.
method Analyzing singular Riemannian foliations on unit spheres.
result Horizontal diameter of unit spheres is π for polar and infinitesimally polar actions.
The paper introduces polarizations in symplectic and orthogonal settings.
problem Understanding polarizations in symplectic and orthogonal contexts.
method Exposition of symplectic and orthogonal polarizations, emphasizing symmetry.
result Grassmannians of polarizations in symplectic and orthogonal settings.
We classify infinitesimally polar actions on compact Riemannian symmetric spaces of rank one. We also prove that every polar action on one of those spaces has the same orbits as an asystatic action.