Study on existence and structure of P-area surfaces in Heisenberg group.
problem Existence and structure of P-area minimizing surfaces in the Heisenberg group.
method Characterization of existence and structure using an underlying vector field N, proving existence even without satisfying boundary conditions, and applying Barrier condition.
result Existence of P-area minimizing surfaces under certain conditions, providing new understanding of the Heisenberg group.
We prove the existence of a continuous B V BV B V minimizer with C 0 C^{0} C 0 boundary value for the p p p -area (pseudohermitian or horizontal area) in a parabolically convex bounded domain. We extend the domain of the area functional from B V BV B V functions to vector-valued measures. Our main purpose is to study the first and second v…
Researchers solve a specific case of the L p L^p L p Christoffel-Minkowski problem for 1 < p < k + 1 1 < p < k+1 1 < p < k + 1 .
problem Solving the L p L^p L p Christoffel-Minkowski problem for 1 < p < k + 1 1 < p < k+1 1 < p < k + 1 . method Establishing the existence of convex bodies with prescribed k k k -th even p p p -area measure under certain conditions. result Existence of convex bodies with prescribed k k k -th even p p p -area measure on S n \mathbb S^n S n under appropriate assumptions. Formula for Heisenberg group surface areas derived.
problem Deriving a formula for surface areas in Heisenberg groups.
method Analogy of Cauchy's surface area formula in Heisenberg groups.
result Formula for p-area of compact hypersurfaces in Heisenberg groups.
The paper extends Pappus-Guldin theorems to 3D-Heisenberg group surfaces.
problem Extending classical theorems to a new geometric setting.
method Deriving formulas for p-areas and volumes in the Heisenberg group.
result Pappus-Guldin theorems hold for surfaces in the Heisenberg group.
We study the uniqueness of generalized p p p -minimal surfaces in the Heisenberg group. The generalized p p p -area of a graph defined by u u u reads ∫ ∣ ∇ u + F ⃗ ∣ + H u \int |\nabla u+\vec{F}| + Hu ∫ ∣∇ u + F ∣ + H u . If u u u and v v v are two minimizers for the generalized p p p -area satisfying the same Dirichlet boundary condition, then we can only get $N_{\vec{F}}…
In \cite{CHMY04}, we studied p p p -mean curvature and the associated p p p -minimal surfaces in the Heisenberg group from the viewpoint of PDE and differential geometry. In this paper, we look into the problem through the variational formulation. We study a generalized p p p -area and associated ( p p p -) minimizers in general di…
We show the fundamental theorems of curves and surfaces in the 3-dimensional Heisenberg group and find a complete set of invariants for curves and surfaces respectively. The proofs are based on Cartan's method of moving frames and Lie group theory. As an application of the main theorems, a Crofton-type formula is prove…
In this paper, we study the structure of the singular set for a C 1 C^{1} C 1 smooth surface in the 3 3 3 -dimensional Heisenberg group H 1 \boldsymbol{H}_{1} H 1 . We discover a Codazzi-like equation for the p p p -area element along the characteristic curves on the surface. Information obtained from this ordinary differential equation …
Study invariant measures on measured laminations for subgroups of mapping class group.
problem Classify invariant Radon measures on space of measured laminations for subgroups of mapping class group.
method Geometric approach, focusing on recurrent measured laminations, explicitly constructing ergodic measures.
result Show uniquely ergodic for divergence-type subgroups, generalize results for full mapping class group.
New set-valued star-shaped risk measures introduced for better risk assessment.
problem Improving risk assessment in financial contexts.
method Developed new set-valued star-shaped risk measures and proved their representation theorems.
result Set-valued star-shaped risk measures can be represented as unions of set-valued convex risk measures.
The Bergman measure converges to the Zhang measure on a hybrid space.
problem Proving convergence of Bergman measures to Zhang measure.
method Analyzing convergence on a hybrid space and metrized curve complex.
result Bergman measure converges to Zhang measure on a hybrid space.
Bayesian approach to robust risk measures under model uncertainty.
problem Representing robust risk measures as a single probability measure.
method Introducing two types of risk measures and analyzing their relation to robust risk measures.
result Robust risk measures can be represented by a mixture probability measure, a Bayesian approach.
The paper classifies 1-dimensional uniform measures in various dimensions.
problem Classifying uniformly distributed measures of dimension 1 in general codimension.
method Analyzing measures with connected 1-dimensional support and providing a partial classification for general measures.
result Uniform measures with connected 1-dimensional support are homogeneous measures.
The paper studies dynamic star-shaped risk measures and their representation.
problem Representing dynamic star-shaped risk measures and their properties.
method Representation theorems for dynamic monetary and star-shaped risk measures.
result Dynamic star-shaped risk measures can be represented as the lower envelope of a family of dynamic convex risk measures.
Introduces Star-Shaped deviation measures for risk analysis.
problem Risk measurement and analysis in finance.
method Characterizes Star-Shaped deviation measures through acceptance sets and convex deviation measures.
result Exposes the relationship between Star-Shaped risk measures and deviation measures.
Transformers can interpolate between arbitrary measures.
problem Understanding the expressive power of Transformers as measure-to-measure maps.
method Provided an explicit choice of parameters for a single Transformer to match N arbitrary input measures to N arbitrary target measures.
result A single Transformer can interpolate between arbitrary measures.
Classifies invariant measures on specific character varieties.
problem Classifying invariant probability measures on character varieties.
method Measure disintegration along transverse Lagrangian tori fibrations.
result Ergodic measures are either counting measures on finite orbits or Liouville measures.
Researchers compute the ratio between two normalizations of Thurston measure on measured laminations.
problem Computing the ratio between two normalizations of Thurston measure.
method Using the integral and symplectic structures on the space of measured laminations.
result Computed the ratio between two normalizations of Thurston measure.
Paper characterizes star-shaped risk measures and their properties.
problem Characterizing risk measures in the presence of liquidity risk and competitive delegation.
method Characterization of star-shaped risk measures, study of their properties.
result Star-shaped risk measures include all practically used risk measures.
Theorem generalizes Reifenberg's for measures with bounds on β-numbers.
problem Bounding measures away from k-rectifiable sets with β-numbers.
method Assumptions on Jones' β-numbers to measure closeness to subspaces.
result Effective measure bounds on μ away from a closed k-rectifiable set.
Paper compares fairness measures and feature importance measures using SHAP.
problem Comparing fairness measures and feature importance measures.
method Focus on SHAP, a game-theoretic measure of feature importance.
result Results for unfairness-prone datasets.
A new method calculates a barycenter for probability measures using Wasserstein distance.
problem Finding a central measure for a set of probability distributions.
method Regularizing the pushforward measure of a set of probability distributions into the Wasserstein space and then finding the barycenter.
result The method yields a uniquely defined barycenter measure supported on the barycentric points of the input measures.
New Bayesian method for spectral deconvolution with Poisson noise.
problem Estimating physical model parameters from noisy spectral data.
method Bayesian measurement framework applied to Poisson noise model.
result Clarifies relationship between measurement time and estimation limits.
Paper introduces quasi-logconvex risk measures and their properties.
problem Characterizing and understanding new risk measures.
method Characterization through dual representation and properties of acceptance sets.
result Established dual representation and taxonomy of quasi-logconvex risk measures.
Submodularity is studied for convex risk measures, including Expected Shortfall.
problem Characterizing submodularity in convex risk measures.
method Analyzing submodularity properties of law-invariant coherent risk measures, including Expected Shortfall and Value-at-Risk.
result AES is submodular only when it reduces to ES, and empirical analysis shows AES violations are less frequent than VaR and ES violations.
Paper compares graph and set partition measures for graph clustering.
problem Comparing graph clustering methods using different similarity measures.
method Introduces graph-aware partition similarity measures and compares them with set partition measures.
result Graph-aware measures provide complementary information to set partition measures.
New geometric measure simplifies complex analysis.
problem Complex geometric analysis challenges.
method Geometric integration and convergence methods.
result Smallest measure satisfying Area Formula.
The paper explores non-convex risk measures and their characterizations.
problem Characterizing non-convex risk measures without convexity or weak convexity.
method Characterizes monetary risk measures as lower envelopes of families of convex or coherent risk measures, considering law-invariance and SSD-consistency.
result Unified representation theorems for law-invariant risk measures, including VaR.
The paper calculates extreme measures in continuous time conic finance.
problem Determining valuation bounds for financial claims.
method Using dynamic spectral risk measures and estimating extreme measures from market data.
result Explicit formulas for extreme measures' Radon-Nykodim derivatives and estimation methods.
Paper characterizes monotonic mean-deviation risk measures.
problem Developing consistent risk measures from mean-deviation models.
method Applying a risk-weighting function to the deviation part of a mean-deviation model.
result Characterizes monotonic mean-deviation measures as consistent risk measures.
Introduces factor risk measures to assess risk relative to multiple factors.
problem Measuring risk relative to multiple factors.
method Introduces a double-argument mapping as a risk measure to assess risk relative to a vector of factors.
result Characterizes various types of factor risk measures including distortion, quantile, linear, and coherent measures.
Dual representations for robust risk measures and uncertainty sets.
problem Characterizing continuity of robust risk measures and their uncertainty sets.
method Develop dual representations for robust risk measures and uncertainty sets based on distinct geometric assumptions.
result Two dual frameworks for consolidated uncertainty sets are complementary, not interchangeable.
A scalable approach to learning from probability measures using quantization.
problem Efficiently comparing and manipulating large sets of probability measures.
method Quantization of probability measures to a fixed support, followed by optimal transport computations.
result Consistency and convergence guarantees for quantized measures in various OT-based tasks.
Study on measurable pseudo-Anosov maps on surfaces.
problem Characterize dynamics of pseudo-Anosov maps on surfaces.
method Analyze measurable pseudo-Anosov homeomorphisms with specific properties.
result Prove transitivity, dense periodic points, sensitivity, and ergodicity.
The Cannon-Thurston map's pushed measures on the circle are singular with respect to sphere measures.
problem Understanding the behavior of geodesics and measures on fibered hyperbolic 3-manifolds.
method Properties of geodesics and measures on the circle and sphere are analyzed to prove singularity.
result Natural measures on the circle become singular with respect to measures on the sphere.
Risk measures for multivariate financial positions are studied in a utility-based framework. Under a certain incomplete preference relation, shortfall and divergence risk measures are defined as the optimal values of specific set minimization problems. The dual relationship between these two classes of multivariate ris…
New measure detects direct causal influences and captures strong dependencies.
problem Shortcomings of existing dependency measures in detecting direct causal influences and group selection.
method Inspired by Dobrushin's coefficients, the measure uses conditional distribution properties.
result Advantages over related measures in detecting dependencies and causal influences.
New weighted surface area measures for convex bodies with applications.
problem Generalizing surface area measures to weighted Borel measures.
method Formulating and analyzing weighted surface area measures, proving integral formula and Bézout-type inequality.
result New integral formula for mixed measure of three bodies, generalizing Bézout-type inequality.
Standardized fairness measures for continuous risk scores using Wasserstein distance.
problem Quantifying and interpreting group disparities in continuous risk scores.
method Proposes standardized fairness measures based on Wasserstein distance for continuous scores.
result Proposed measures outperform ROC-based fairness measures by being more explicit and quantifying significant biases.
Study SRB measures for Anosov actions on manifolds.
problem Characterize SRB measures for Anosov actions.
method Use Ruelle-Taylor resonances and properties of Sinai-Ruelle-Bowen measures.
result SRB measures have properties like smooth disintegrations, positive basins, and are unique under certain conditions.
Study proposes worst+gap measure for better DG evaluation.
problem Lack of comprehensive exploration of average measure in DG evaluation.
method Introduced worst+gap measure and compared it with average measure.
result Worst+gap measure provides a more accurate approximation of true DG performance.
The study evaluates AI model performance measures for medical use.
problem Selecting appropriate performance measures for AI models in medical practice.
method Assessed 32 performance measures across five domains for binary outcomes.
result 17 measures are both proper and reflect decision-analytic performance.
Study on convergence of Narasimhan-Simha measures on degenerating families of Riemann surfaces.
problem Analyzing the convergence of measures on degenerating families of Riemann surfaces.
method Hybrid space approach, using metrized curve complex and Hermitian pairing.
result Convergence of measures on hybrid space, extending to singular curves.
Stationary measures on hyperbolic surfaces with cusps are singular and stable under quasi-symmetries.
problem Understanding stationary measures on hyperbolic surfaces with cusps.
method Analyzing exponential decay of cusp excursions and proving quasi-symmetry stability.
result Stationary measures on hyperbolic surfaces with cusps are quasi-symmetrically stable and singular.
Study measures rigidity for random walks and flows via generalized u-Gibbs states.
problem Measure rigidity for stationary measures of random walks and flows.
method Factorization method applied to generalized u-Gibbs states.
result Established extra invariance of generalized u-Gibbs states.
The paper establishes a connection between different risk measures and their risk contributions.
problem Understanding the relationship between conditional coherent and deviation risk measures.
method Axiomatic framework and continuous-time risk contribution analysis.
result Risk contributions of time-consistent risk measures are also time-consistent.
The study shows finite measure-preserving isometry groups for certain metric measure spaces.
problem Understanding the structure of isometry groups in metric measure spaces.
method Analyzing synthetic negative Ricci curvature and Bakry-Émery Ricci curvature.
result The measure-preserving isometry group is finite for compact metric measure spaces with specific curvature conditions.