Alexandrov spaces have a special stratification that maps to spheres.
problem Characterizing the structure of Alexandrov spaces.
method Extremal stratification and space of directions analysis.
result Alexandrov spaces are homeomorphic to spheres in their space of directions.
The paper proves properties of strata of differentials, showing they are affine and extremal.
problem Properties of strata of differentials, particularly their geometry and tautological rings.
method Analyzing the tautological rings and using Teichmüller dynamics to prove properties.
result Strata of differentials are affine and their stratification is extremal.
Cryptocurrency markets show higher spreads during extreme fear and greed phases.
problem Understanding and predicting liquidity withdrawal in cryptocurrency markets.
method Analysis of Crypto Fear & Greed Index and Bitcoin daily data.
result Extreme fear and greed regimes exhibit significantly higher spreads than neutral periods.
Copula-based fusion improves breast cancer risk stratification.
problem Combining clinical and genomic risk scores using simple rules fails to capture their joint relationship.
method Used copulas to model the joint relationship between clinical and genomic risk scores.
result Copula-based fusion improves risk stratification, identifying subgroups with the worst prognosis.
We prove the existence of Verdier stratifications for sets definable in any o-minimal structure on (R, +, .). It is also shown that the Verdier condition (w) implies the Whitney condition (b) in o-minimal structures on (R, +, .). As a consequence the Whitney Stratification Theorem holds. The existence of (wf)-stratific…
The paper studies HKKN stratifications for non-compact spaces and proves convexity properties.
problem Proving convexity properties of moment maps for non-compact subsets.
method Algebraic and analytical study of HKKN stratifications for a vector space and compact Kähler manifold, then applying to non-compact subsets.
result Convexity properties of moment maps for invariant subsets are proven.
Stratifies representation varieties of twisted Hopf links.
problem Stratifying representation varieties of twisted Hopf links.
method Using stratification of AGLr(C)-representation varieties of the fundamental group of the complement of a twisted Hopf link. result Explicit description and computation of motives for ranks 1 and 2.
The paper defines a stratification for Lie groupoids in a tame topology context.
problem Presenting a tame topology counterpart to canonical stratification of Lie groupoids.
method Using Shiota's isotopy lemma and approximation theorem, the paper defines a canonical Whitney stratification of definable Lie groupoids into invariant strata.
result A canonical Whitney stratification of the Lie groupoid into definable strata invariant under the groupoid action.
Paper confirms MCS spaces are equivalent to CS sets.
problem Understanding the equivalence of MCS and CS sets.
method Analyzing the MCS stratification and intrinsic stratification.
result MCS spaces are equivalent to CS sets with respect to their stratification.
Investigates properties of moment maps and stratifications on Lie groups.
problem Understanding moment maps and stratifications on real reductive Lie groups.
method Functorial, algebraic approach to moment map and Kirwan-Ness stratification.
result Properties and properties of moment maps and stratifications established.
Optimizes biharmonic map regularity using stratification methods.
problem Improving the known almost optimal regularity of biharmonic maps.
method Quantitative stratification method.
result Optimal regularity results for minimizing biharmonic maps.
New stratification reveals intrinsic singularity types of orbit spaces.
problem Understanding the intrinsic structure of orbit spaces under Lie group actions.
method Introduced the isostabilizer decomposition and established a map to Klein strata.
result A new canonical stratification on the manifold clarifies the relationship with classical structures.
The aim of this paper is to compare stratifications of moduli spaces given by group actions in the case of similarity of matrices introduced by Arnold and the author's stratification by projective orbifolds, and its relation to deformations o elements in the moduli space.
Combines k-means and hill climbing for stratification and allocation.
problem Optimizing stratification and sample allocation for complex surveys.
method Combining k-means type algorithms with hill climbing.
result Multi-stage combination algorithms generally perform well compared to recent methods.
Social media enhances or diminishes scientific status, depending on usage.
problem Impact of social media on scientific stratification and mobility.
method Logistic Attribution Analysis combining statistical and machine learning methods.
result Social media promotes stratification and mobility, but beyond a threshold, it negatively impacts status.
We give a geometric proof of existence of Whitney stratifications of definable sets in o-minimal structures.
Study clarifies variance of stratification estimators for causal effects.
problem Estimating average causal effects with discrete covariates.
method Combines insights from potential outcomes, causal diagrams, and structural models.
result Derives expressions for the variance of stratification estimators.
This paper provides a stratification of semi-algebraic sets in the plane with finitely many geodesic segments.
problem How to stratify semi-algebraic sets in the plane with finitely many geodesic segments.
method Develops a semi-algebraic stratification of a real semi-algebraic set in the plane with open cells having the finiteness property.
result Provides insights for high-dimensional stratifications of semi-algebraic sets in connection with geodesics.
The Bialynicki-Birula decomposition of the space of lambda-connections restricts to the Morse stratification on the moduli space of Higgs bundles and to the partial oper stratification on the de Rham moduli space of holomorphic connections. For both the Morse and partial oper stratifications, every stratum is a holomor…
We study the topology of the inertia space of a smooth G-manifold M where G is a compact Lie group. We construct an explicit Whitney stratification of the inertia space, demonstrating that the inertia space is a triangulable differentiable stratified space. In addition, we demonstrate a de Rham theorem for differ…
Machine learning models for medical image analysis often suffer from poor performance on important subsets of a population that are not identified during training or testing. For example, overall performance of a cancer detection model may be high, but the model still consistently misses a rare but aggressive cancer su…
The paper studies harmonic map flows and proves rectifiability of singular sets.
problem Understanding the structure of singular sets in harmonic map flows.
method Investigates the stratification theory for suitable solutions using tangent measures.
result Each time slice of the singular set is rectifiable.
The paper describes a stratification of a compactified Hurwitz space using combinatorial trees.
problem Describing the boundary stratification of a compactified Hurwitz space.
method Using decorated trees to describe the boundary strata and their containment relations.
result The boundary strata of the compactified Hurwitz space are in bijection with decorated trees, and containment is given by edge contraction.
The complement of a complex hyperplane arrangement is known to be homotopic to a minimal CW complex. There are several approaches to the minimality. In this paper, we restrict our attention to real two dimensional cases, and introduce the "dual" objects so called minimal stratifications. The strata are explicitly descr…
Let G be a Lie group, and let (M,ω) be a symplectic manifold. If G admits a Hamiltonian action on (M,ω) with momentum map μ, then M, the zero-level set of μ, the orbit space, and the corresponding symplectic quotient all have induced stratifications. We push this setting into the language of differential …
The paper offers simple, near-optimal algorithms for multi-group learning.
problem Learning predictors within subgroups of a population, addressing fairness and hidden stratification.
method Studies the structure of solutions and provides simple, near-optimal algorithms.
result Simple and near-optimal algorithms for multi-group learning.
The monster tower is a tower of spaces over a specified base; each space in the tower is a parameter space for curvilinear data up to a specified order. We describe and analyze a natural stratification of these spaces.
Decomposes smooth manifolds into algebraic submanifolds.
problem Understanding the structure of smooth manifolds induced by continuous selections.
method Generic continuous selection of smooth functions provides stratification of the manifold.
result Stratification leads to local topological structure with nondegenerate critical points.
We introduce a natural stratification of the space of projective classes of measured laminations on a complete hyperbolic surface of finite area. We prove a rigidity result, namely, the group of self-homeomorphisms of the space of projective measured laminations that preserve such a stratification is in general identif…
Study chaotic dynamics in social stratification models leading to thermalization and turbulence.
problem Understanding social stratification dynamics through chaotic nonlinear systems.
method Modeling social network links with oscillators and energies, studying Hamiltonian evolution and nonlinear interactions.
result Chaotic dynamics leads to dynamical thermalization and Kolmogorov-Zakharov turbulence, with implications for wealth inequality.
Study almost rigidity of super Ricci flow with non-negative Muller quantity.
problem Almost rigidity properties of super Ricci flow with non-negative Muller quantity.
method Almost splitting and quantitative stratification theorems established by Bamler for Ricci flow.
result Obtained almost constancy for a certain integral quantity concerning scalar curvature at an almost self-similar point.
Study identifies and estimates treatment effect heterogeneity within principal stratification subpopulations.
problem Causal inference with intermediate outcomes and treatment effect heterogeneity.
method Proposes a novel doubly cross-fit doubly robust machine learner to efficiently learn conditional principal causal effects under principal ignorability.
result Demonstrates informative patterns of treatment effect heterogeneity within the always-survivor subpopulation in an acute lung injury trial.
We consider a Morse function f and a Morse-Smale gradient-like vector field X on a compact connected oriented 3-manifold M such that f has only one critical point of index 3. Based on Laudenbach's ideas, we will show that the flow of X can be isotoped into one so that the trajectory spaces of the new flow pro…
New method improves compatibility of risk stratification models without sacrificing accuracy.
problem Compatibility issues arise when updating clinical machine learning models.
method Proposes rank-based compatibility measure and new loss function.
result Increased compatibility of models by 0.019 with no loss in discriminative performance.
A homology stratification is a filtered space with local homology groups constant on strata. Despite being used by Goresky and MacPherson [Intersection homology theory: II, Inventiones Mathematicae, 71 (1983) 77-129] in their proof of topological invariance of intersection homology, homology stratifications do not appe…
Survey on hyperplane arrangements and their topology.
problem Topology of hyperplane arrangements.
method Focus on the relationship between topology and real structure.
result Relationship between topology and real structure of hyperplane arrangements.
Deep learning model creates patient representations for scalable EHR-based stratification.
problem Challenges in summarizing and representing patient data from EHRs prevent scalable stratification analysis.
method Unsupervised framework based on deep learning (ConvAE) using word embeddings, CNNs, and autoencoders.
result ConvAE significantly outperformed baselines in clustering diverse patient cohorts, identifying clinically relevant subtypes.
The analysis of the USA 2001 income distribution shows that it can be described by at least two main components, which obey the generalized Tsallis statistics with different values of the q parameter. Theoretical calculations using the gas kinetics model with a distributed saving propensity factor and two ensembles rep…
We reduce variance in monetization metrics for ranking experiments.
problem Heavy-tailed monetization metrics lead to unreliable conclusions in A/B experiments.
method Post-stratification combined with CUPED.
result Significant reduction in variance and improved decision stability.
Clarifies the structure of quantum states using algebraic methods.
problem Unclear stratification of quantum states in physics literature.
method Analyzes the state space S(A) of a finite-dimensional C*-algebra A, focusing on unitary orbits and their properties.
result Identifies a natural Whitney stratification of the state space into matrices of fixed rank, providing a pseudo-manifold structure.
Within its traditional range of perversity parameters, intersection cohomology is a topological invariant of pseudomanifolds. This is no longer true once one allows superperversities, in which case intersection cohomology may depend on the choice of the stratification by which it is defined. Topological invariance also…
We consider the problem of adaptive stratified sampling for Monte Carlo integration of a noisy function, given a finite budget n of noisy evaluations to the function. We tackle in this paper the problem of adapting to the function at the same time the number of samples into each stratum and the partition itself. More p…
Improved optimal regularity for harmonic almost complex structures.
problem Establishing optimal regularity for harmonic almost complex structures.
method Quantitative stratification method and rectifiability of singular strata.
result Optimal regularity theory for energy minimizing harmonic almost complex structures.
The moduli space of Higgs bundles is stratified into complex symplectic submanifolds.
problem Constructing a complex Whitney stratification of the moduli space of Higgs bundles.
method Showed that the orbit type decomposition is a complex Whitney stratification with each stratum being a complex symplectic submanifold.
result The moduli space of Higgs bundles is a stratified complex symplectic space.
By attaching a Lie algebra of germs of analytic vector fields to every point of a (real or complex) analytic variety V we construct the Nagano foliation of the variety. We prove that the Nagano foliation of V is a stratification. The treatment of the subject is totally coordinate free but relies on the Oka-Cartan-Serre…
New method tests risk measures for various distortions.
problem Testing risk measures for different distortions.
method Stratification and randomization of risk levels.
result Method performs well in numerical case studies.
Study fractional Allen-Cahn equation and nonlocal minimal surfaces, improving energy and perimeter estimates.
problem Properties of solutions to fractional Allen-Cahn equation and stationary nonlocal minimal surfaces.
method Quantitative stratification principle applied to fractional Allen-Cahn equation, leading to optimal estimates.
result Sharp potential energy and perimeter estimates for fractional Allen-Cahn equation and nonlocal minimal surfaces.
In this paper we investigate how germs of real functions can change under deformation. In particular we look at deformations of germs of isolated singularities from R_n to R_k (n >= k) and the relation with there natural stratification in some tame categorie (algebraic, analytic, semi-algebraic, subanalytic, o-minimal …