arXiv research

Unique optimal map found in curved spaces.

We find necessary and sufficient conditions for a Lipschitz map $f:\mathbb{R}E\to X$, into a metric space to have the image with the $k$-dimensional Hausdorff measure equal zero, $H^k(f(E))=0$. An interesting feature of our approach is that despite the fact that we are dealing with arbitrary metric spaces, we employ a …

We investigate the approximate j-dimensionality of the singularity sets of minimal surfaces prescribed by Simon. This leads to the clasification of 8 variations of approximately j-dimensional surfacs in terms of dimension and locally finite Hausdorff measure. We show that the singularity sets must either be well behave…

Geometric argument proves projection theorems in hyperbolic space.

We construct new examples of normal (metric) currents using inverse systems of cube complexes. For any $N\ge 2$ we provide examples of $N$-dimensional normal currents whose associated vector fields are simple, and whose supports are purely $2$-unrectifiable and have Nagata dimension $N$. We show that in $l^\infty$ norm…

Study contact 3-manifolds using sub-Riemannian geometry, proving new properties of their Lipschitz homotopy groups.

New insights into integrability and rectifiability in sub-Riemannian geometry.

The paper generalizes a theorem about rectifiability of sets.

Develops geometric integration for rough differential forms.

The paper explores rectifiability in sub-Riemannian geometry, finding a smooth hypersurface with unique properties.

The coarea formula is proven for Heisenberg group maps, addressing open questions.

Study invariant measures on measured laminations for subgroups of mapping class group.

New set-valued star-shaped risk measures introduced for better risk assessment.

The Bergman measure converges to the Zhang measure on a hybrid space.

Bayesian approach to robust risk measures under model uncertainty.

The paper classifies 1-dimensional uniform measures in various dimensions.

The paper studies dynamic star-shaped risk measures and their representation.

Introduces Star-Shaped deviation measures for risk analysis.

Transformers can interpolate between arbitrary measures.

Classifies invariant measures on specific character varieties.

Researchers compute the ratio between two normalizations of Thurston measure on measured laminations.

Paper characterizes star-shaped risk measures and their properties.

Paper compares fairness measures and feature importance measures using SHAP.

New Bayesian method for spectral deconvolution with Poisson noise.

Paper introduces quasi-logconvex risk measures and their properties.

Submodularity is studied for convex risk measures, including Expected Shortfall.

Paper compares graph and set partition measures for graph clustering.

New geometric measure simplifies complex analysis.

The paper explores non-convex risk measures and their characterizations.

The paper calculates extreme measures in continuous time conic finance.

Paper characterizes monotonic mean-deviation risk measures.

Introduces factor risk measures to assess risk relative to multiple factors.

Dual representations for robust risk measures and uncertainty sets.

A scalable approach to learning from probability measures using quantization.

Study on measurable pseudo-Anosov maps on surfaces.

The Cannon-Thurston map's pushed measures on the circle are singular with respect to sphere measures.

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 weighted surface area measures for convex bodies with applications.

Standardized fairness measures for continuous risk scores using Wasserstein distance.

Study SRB measures for Anosov actions on manifolds.

We study generalizations of Reifenberg's Theorem for measures in $\mathbb R^n$ under assumptions on the Jones' $β$-numbers, which appropriately measure how close the support is to being contained in a subspace. Our main results, which holds for general measures without density assumptions, give effective measure bounds…

Study proposes worst+gap measure for better DG evaluation.

The study evaluates AI model performance measures for medical use.

We introduce a weak notion of barycenter of a probability measure $μ$ on a metric measure space $(X, d, {\bf m})$, with the metric $d$ and reference measure ${\bf m}$. Under the assumption that optimal transport plans are given by mappings, we prove that our barycenter $B(μ)$ is well defined; it is a probability measur…

Study on convergence of Narasimhan-Simha measures on degenerating families of Riemann surfaces.

Stationary measures on hyperbolic surfaces with cusps are singular and stable under quasi-symmetries.

Study measures rigidity for random walks and flows via generalized u-Gibbs states.

The paper establishes a connection between different risk measures and their risk contributions.