Rectifies flat singular points for area-minimizing currents.
problem Understanding singularities of area-minimizing currents.
method Analyzes countably (m−2)-rectifiable singular points with flat tangent cones. result The set of singular density-Q points is countably (m−2)-rectifiable and has finite upper Minkowski content. Study bounds singular set of minimal hypersurfaces with index control.
problem Estimating singular set size of minimal hypersurfaces.
method Finite index and null singular set conditions on integral varifolds.
result Local measure bounds on singular set and upper Minkowski content.
This article proves that the zero locus of a Z/2 harmonic spinor on a 4 dimensional manifold is 2-rectifiable and has locally finite Minkowski content.
The study bounds Hausdorff measure of flat singular points in area-minimizing currents.
problem Bounding Hausdorff measure of flat singular points in area-minimizing currents.
method Proving locally finite (m−2)-dimensional Hausdorff measure and Minkowski content bounds. result The set of flat singular points has locally finite (m−2)-dimensional Hausdorff measure. In this article we extend to generic p-energy minimizing maps between Riemannian manifolds a regularity result which is known to hold in the case p=2. We first show that the set of singular points of such a map can be quantitatively stratified: we classify singular points based on the number of almost-symmetries of…
Upper bound on singular set dimension for area-minimizing currents.
problem Bounding the dimension of singular points in area-minimizing currents.
method Using upper Minkowski dimension and properties of blow-up scales.
result Upper Minkowski bound of m−2 for the interior singular set. Perturbs area-minimizing hypersurfaces to reduce singular set's dimension.
problem Reduces the dimension of the singular set of area-minimizing hypersurfaces.
method Perturbs a smooth hypersurface to minimize the Minkowski dimension of the singular set.
result The singular set of the perturbed minimizing current has Minkowski dimension less than n-9.
Paper solves capillary Orlicz-Minkowski problem with new inequalities.
problem Finding capillary convex bodies with prescribed Orlicz surface area measures.
method Continuity method and inequalities to solve the capillary even Orlicz-Minkowski problem.
result Volume-normalized smooth solutions and inequalities established.
We obtain upper bounds on the heat content and on the torsional rigidity of a complete Riemannian manifold M, assuming a generalized Hardy inequality for the Dirichlet Laplacian on M.
Derives exact formula for Minkowski sum of ellipsoids in N-space.
problem Finding volume bounds for Minkowski sum of ellipsoids.
method Closed-form parametric equation derivation and volume bounds calculation.
result Upper and lower volume bounds for Minkowski sum of ellipsoids.
Study an anisotropic capillary flow to solve capillary Orlicz-Minkowski problem.
problem Solve capillary Orlicz-Minkowski problem without evenness assumption.
method Analyze an anisotropic capillary Gauss curvature flow to prove convergence and establish existence.
result Establish existence result for capillary Orlicz-Minkowski problem without evenness assumption.
Sharp upper bounds derived for Alexandrov-Fenchel deficit using weighted Minkowski integral formulas.
problem Deriving upper bounds for the Alexandrov-Fenchel deficit.
method Using weighted Minkowski integral formulas and an integral formula for the deficit in Jensen's inequality.
result Quantitative estimates under weaker convexity assumptions, including a distance term.
We prove that the results regarding the Isoperimetric inequality and Cheeger constant formulated in terms of the Minkowski content, obtained by the authors in previous papers in the framework of essentially non-branching metric measure spaces verifying the local curvature dimension condition, also hold in the stronger …
Sharp inequality for eigenvalues of convex bodies, proving ellipsoid uniqueness.
problem Eigenvalue bounds for convex bodies and ellipsoid uniqueness.
method Established a sharp upper-bound for eigenvalues of a specific operator.
result Equality holds only for ellipsoids, complementing conjectural lower-bound.
Derives hyperbolic laws of cosines and sines with fermionic corrections.
problem Deriving hyperbolic laws of cosines and sines with new mathematical corrections.
method Using Minkowski supergeometry, the laws of cosines and sines are derived in the super hyperbolic plane.
result Identical formulae to classical cases with fermionic corrections for cosines and sines.
We prove a generalization of Hsiung-Minkowski formulas for closed submanifolds in semi-Riemannian manifolds with constant curvature. As a corollary, we obtain volume and area upper bounds for k-convex hypersurfaces in terms of a weighted total k-th mean curvature of the hypersurface. We also obtain some Alexandrov-type…
By means of a family of counter-examples, it is shown that the Reilly upper bound for the first eigenvalue of the Laplace operator for a compact submanifold in Euclidean space does not work for n-dimensional compact spacelike submanifolds of Lorentz-Minkowski spacetime Lm, m≥n+2. We develop a new su…
A method for disentangling text representations without supervision.
problem Challenges in learning disentangled representations of natural language.
method Information-theoretic guidance to induce independent style and content embeddings.
result High quality disentangled representations in terms of content and style preservation.
Study complete space-like stationary surfaces with graphical Gauss image, estimating exceptional values and classifying degenerate surfaces.
problem Estimating exceptional values and classifying degenerate surfaces in Minkowski spacetime.
method Generalizing Fujimoto's theorem, estimating upper bounds, introducing conjugate similarity, and establishing structure theorems.
result Sharp contrast to Bernstein type results for minimal surfaces, estimating upper bounds of exceptional values.
Harmonic maps to Euclidean buildings have rectifiable singular strata.
problem Understanding the structure of singular points for harmonic maps.
method Defining singular strata and proving rectifiability using the rectifiable Reifenberg program.
result Rectifiability of singular strata for harmonic maps into F-connected complexes. We quantify content availability and user discovery opportunities in recommender systems.
problem Determining the maximum probability of recommending content to users.
method Stochastic reachability to compute upper bounds on recommendation likelihood.
result Reachability metrics can detect biases and diagnose user discovery limitations.
Study Poincaré inequality in metric spaces via separating sets.
problem Geometric characterization of Poincaré inequality in metric spaces.
method Properties of separating sets and various notions of energy.
result Equivalence of conditions for 1-Poincaré inequality.
Study quantifies properties of PMC hypersurfaces with area bounds.
problem Understanding the topology and singular set of PMC hypersurfaces.
method Established quantitative topological and singularity properties for PMC hypersurfaces.
result Quantitative bounds on Betti numbers and Minkowski content of singular sets.
Extends Penrose's method to null shells with pressure and energy flux.
problem Constructing null thin shells with arbitrary gravitational/matter content.
method Derive locally Lipschitz metric and coordinate transformation.
result Example of null shell with non-trivial energy density, flux, and pressure in Minkowski space.
LOLA uses LLMs to optimize content delivery, outperforming traditional methods.
problem Identifying the most engaging headlines for user engagement.
method LOLA integrates LLMs with adaptive experimentation to optimize content delivery.
result LOLA outperforms traditional methods in optimizing user engagement.
The goal of homomorphic encryption is to encrypt data such that another party can operate on it without being explicitly exposed to the content of the original data. We introduce an idea for a privacy-preserving transformation on natural language data, inspired by homomorphic encryption. Our primary tool is {\em obfusc…
New model improves website ranking by considering user choices as a whole.
problem Optimizing content ordering for user clicks in website design.
method Introduced multinomial logit (MNL) choice model to LTR framework, proposing UCB algorithms.
result Proved theoretical bounds on regret for UCB algorithms in both known and unknown position parameter settings.
New estimate reduces overfitting risk in machine learning models.
problem Error rate on test data may not reflect true population error due to adaptive data analysis practices.
method Introduces Rip van Winkle's Razor, a simple estimate of overfit to test data based on information content.
result Shows non-vacuous estimate of deviation in many modern settings.
The problem of content search through comparisons has recently received considerable attention. In short, a user searching for a target object navigates through a database in the following manner: the user is asked to select the object most similar to her target from a small list of objects. A new object list is then p…
We study a Riemannian manifold equipped with a density which satisfies the Bakry--Émery Curvature-Dimension condition (combining a lower bound on its generalized Ricci curvature and an upper bound on its generalized dimension). We first obtain a Poincaré-type inequality on its boundary assuming that the latter is local…
The purpose of this paper is to study the validity of Stokes' Theorem for singular submanifolds and differential forms with singularities in Euclidean space. The results are presented in the context of Lebesgue Integration, but their proofs involve techniques from gauge integration in the spirit of R.~Henstock, J.~Kurz…
The paper proves a theorem about earthquake extensions of vector fields on circles.
problem Proving a theorem about earthquake extensions of vector fields on circles.
method Using the geometry of the dual of Minkowski three-space and Half-pipe three-geometry.
result A generalization of Kerckhoff's and Gardiner's infinitesimal earthquake theorems to a broader setting.
We establish a uniform estimate for the injectivity radius of the past null cone of a point in a general Lorentzian manifold foliated by spacelike hypersurfaces and satisfying an upper curvature bound. Precisely, our main assumptions are, on one hand, upper bounds on the null curvature of the spacetime and the lapse fu…
Böröczky, Lutwak, Yang and Zhang recently proved the log-Brunn-Minkowski inequality which is stronger than the classical Brunn-Minkowski inequality for two origin-symmetric convex bodies in the plane. This paper establishes the log-Brunn-Minkowski, log-Minkowski, Lp-Minkowski and Lp-Brunn-Minkowski inequalities f…
The paper solves a generalized Christoffel-Minkowski problem using a curvature flow.
problem Solving the (p,q)-Christoffel-Minkowski problem.
method Investigating the problem via an expanding curvature flow.
result Existence and uniqueness of smooth solutions to the (p,q)-Christoffel-Minkowski problem.
Study anisotropic flows solving Orlicz-Minkowski problems, proving existence and new results.
problem Anisotropic non-homogeneous Gauss curvature flows and Orlicz-Minkowski problems.
method Long-time existence and behavior analysis, parabolic approximation method, curvature flow.
result Existence and new results for Orlicz-Minkowski problems, including Lp versions. Researchers prove uniqueness and continuity of solution to L_p dual Minkowski problem.
problem Proving uniqueness and continuity of solution to L_p dual Minkowski problem.
method Established new Minkowski-type inequalities related to optimization problem.
result Uniqueness and continuity of solution for general convex bodies when q<p. Developed a sub-Riemannian version of Minkowski problem in Heisenberg groups.
problem Minkowski type problem in Heisenberg groups.
method Variational method.
result Positive answer to sub-Riemannian Minkowski type problem.
Smooths metrics with nonnegative scalar curvature near singular sets.
problem Approximating metrics with nonnegative scalar curvature near singularities.
method Ricci-DeTurck flow to approximate metrics.
result Approximated metrics converge to the original metric in C∞ away from the singular set. The paper develops a method to identify LLM-generated text without training.
problem Identifying LLM-generated text among human-generated content.
method Modeling LLM text as a sequential process and designing zero-shot statistical tests.
result The method can distinguish between text from known LLMs and non-sanctioned models with low false positive rates.
Paper solves Minkowski problem for anisotropic p-torsional rigidity.
problem Solving the Minkowski problem for anisotropic p-torsional rigidity.
method Using the anisotropic p-Laplacian equation, presenting sufficient and necessary conditions for existence. result Presented sufficient and necessary conditions for the existence of a solution.
Study shows online learning algorithms incentivize low-quality content, proposing new algorithms to improve quality.
problem Online learning algorithms in content recommender systems incentivize producers to create low-quality content.
method Analyzed the game between producers and content quality, designed new learning algorithms to incentivize high effort and quality.
result New algorithms incentivize producers to invest high effort and achieve high user welfare, improving content quality.
Develops Brunn-Minkowski theory in hyperbolic space using hyperbolic p-sum.
problem No universally accepted sum for sets in hyperbolic space.
method Introduces hyperbolic p-sum and develops horospherical p-Brunn-Minkowski theory.
result Solves p-Minkowski and p-Christoffel-Minkowski problems for various p.
Paper proves anisotropic Minkowski inequality and related inequalities.
problem Proving anisotropic Minkowski inequality and related inequalities.
method Utilizes a nonlinear potential theoretic approach.
result Sharp anisotropic Minkowski inequality and related inequalities proved.
Proposes a model to optimize feedback for content creators on social media.
problem Nurturing content creation on social media platforms.
method Modeling feedback distribution to incentivize creators and optimize newsfeed experience.
result Optimized newsfeed for content creators without compromising user experience.
Characterizes Lorentzian manifolds embeddable in Minkowski spacetime.
problem Identifying Lorentzian manifolds embeddable in Minkowski spacetime.
method Characterization and proof of embeddability conditions.
result Lorentzian manifolds embeddable in Minkowski spacetime coincide with globally hyperbolic spacetimes.
Proposes a VAE variant for ordinal content factors.
problem Isolating ordinal-valued content factors in deep latent variable models.
method Introduces a partially ordered set (poset) structure and a conditional Gaussian spacing prior model.
result Significant improvements in content-style separation over previous non-ordinal approaches.
Sensory data are often comprised of independent content and transformation factors. For example, face images may have shapes as content and poses as transformation. To infer separately these factors from given data, various ``disentangling'' models have been proposed. However, many of these are supervised or semi-super…