Solves Christoffel-Minkowski problem for axially symmetric bodies.
problem Necessary and sufficient conditions for mixed area measures of axially symmetric convex bodies.
method Introduced a new method to transform mixed area measures and mixed volumes of axially symmetric bodies, refining Firey's classification and improving estimates.
result Complete solution to the mixed Christoffel-Minkowski problem for axially symmetric bodies without regularity assumptions.
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.
Solves Christoffel problem for disk area measures on spheres.
problem Conditions for a measure to be a disk area measure of convex bodies.
method Integral representation and differential equation reformulation.
result Reconstructs support function from disk area measure.
Solves a long-standing convex geometry problem about mixed volumes.
problem Characterizing the support of mixed area measures.
method Geometric approach to convex bodies in R^n and R^3.
result Resolved one direction of Schneider's conjecture for arbitrary convex bodies.
In this paper, we introduce several mixed Lp geominimal surface areas for multiple convex bodies for all p=−n. Our definitions are motivated from an equivalent formula for the mixed p-affine surface area. Some properties, such as the affine invariance, for these mixed Lp geominimal surface areas are prove…
Solves Christoffel-Minkowski problem and Hessian equations with radial symmetry.
problem Christoffel-Minkowski problem and Hessian equations under rotational symmetries.
method Constructing explicit convex solutions to mixed Monge-Ampère equations on \(\mathbb{R}^n\) under radial symmetry.
result Explicit representation formula for the support function of the resulting convex body.
Directional and pairwise measurements are often used to model inter-relationships in a social network setting. The Mixed-Membership Stochastic Blockmodel (MMSB) was a seminal work in this area, and many of its capabilities were extended since then. In this paper, we propose the \emph{Dynamic Infinite Mixed-Membership s…
Let S be a nonexceptional oriented surface of finite type. We construct an uncountable family of probability measures on the space of area on holomorphic quadratic differentials over the moduli space for S containing the usual Lebesgue measure. These measures are invariant under the Teichmueller geodesic flow, and they…
New Lp-Steiner quermassintegrals defined from Steiner formula.
problem Defining new Lp-Steiner quermassintegrals. method Analogy to classical Steiner formula, investigating properties in convex bodies.
result Rotation and reflection invariant valuations in convex bodies.
New PDEs of mixed type emerge in fluid mechanics and geometry.
problem Analysis of nonlinear PDEs of mixed type.
method Through historical problems and recent trends.
result Many PDEs are of mixed type, requiring new analysis.
Characterizes measures preserving compound mixed renewal process properties.
problem Preserving compound mixed renewal process properties under different probability measures.
method Characterization of progressively equivalent probability measures.
result Any compound mixed renewal process can be converted into a compound mixed Poisson process through a change of measures.
Unified Minkowski problem discussed for (p,q)-mixed quermassintegrals.
problem Unified Minkowski problem for (p,q)-mixed quermassintegrals.
method Introducing (p,q)-mixed quermassintegrals and (p,q)-dual mixed curvature measure to study the Minkowski problem.
result Derivation of important properties and geometric inequalities for (p,q)-mixed quermassintegrals.
Paper develops a novel approach for classifying high-dimensional mixed data.
problem Handling datasets with both categorical and continuous variables of high dimensions.
method Location model with Gaussian conditional distributions, kernel smoothing for bandwidth choice, penalized likelihood estimation.
result Competitive performance of the proposed classifier demonstrated through simulations and real data.
The Orlicz-Brunn-Minkowski theory receives considerable attention recently, and many results in the Lp-Brunn-Minkowski theory have been extended to their Orlicz counterparts. The aim of this paper is to develop Orlicz Lφ affine and geominimal surface areas for single convex body as well as for multiple convex bod…
The hermitian analog of Aleksandrov's area measures of convex bodies is investigated. A characterization of those area measures which arise as the first variation of unitarily invariant valuations is established. General smooth area measures are shown to form a module over smooth valuations and the module of unitarily …
Characterizes continuity of monotone functionals in mixed topology.
problem Continuity of monotone functionals in mixed topology.
method Characterization through lower semicontinuity and dual representations.
result Continuity in mixed topology is equivalent to dual representation in terms of countably additive measures.
We prove an analogue of the classical Steiner formula for the Lp affine surface area of a Minkowski outer parallel body for any real parameters p. We show that the classical Steiner formula and the Steiner formula of Lutwak's dual Brunn Minkowski theory are special cases of this new Steiner formula. This new Stein…
We study probability measures induced by set functions with constraints. Such measures arise in a variety of real-world settings, where prior knowledge, resource limitations, or other pragmatic considerations impose constraints. We consider the task of rapidly sampling from such constrained measures, and develop fast M…
This paper studies convergence behavior of latent mixing measures that arise in finite and infinite mixture models, using transportation distances (i.e., Wasserstein metrics). The relationship between Wasserstein distances on the space of mixing measures and f-divergence functionals such as Hellinger and Kullback-Leibl…
Bayesian econometrics improves nowcasting during pandemics.
problem Improving nowcasting during extreme economic events like pandemics.
method Bayesian econometric methods using non-parametric mixed frequency VARs with additive regression trees.
result Significant improvements in nowcasting performance compared to linear models.
The paper proves a unique conformal measure for Anosov groups and shows local mixing.
problem Proving the uniqueness of conformal measures for Anosov groups.
method Analogue of Sullivan's theorem for Anosov subgroups of semisimple groups.
result Uniqueness of conformal measures and local mixing for Anosov groups.
New proof for weak mixing in polygonal billiards.
problem Proving weak mixing in polygonal billiards.
method Using Baire category and eigenvalue analysis.
result Billiard flow is weakly mixing for non-rational polygons.
A flag area measure on an n-dimensional euclidean vector space is a continuous translation-invariant valuation with values in the space of signed measures on the flag manifold consisting of a unit vector v and a (p+1)-dimensional linear subspace containing v with 0≤p≤n−1. Using local parallel sets, …
A new method combines machine learning with mixed-effects models for better repeated measurement analysis.
problem Inference of linear coefficients in partially linear mixed-effects models with complex interactions and high-dimensional variables.
method Double machine learning approach to estimate nonparametrically nonlinear variables, then use standard linear mixed-effects techniques to estimate the linear coefficient.
result The estimated fixed effects coefficient converges at the parametric rate and is semiparametrically efficient.
Rapid mixing of Langevin dynamics on Riemannian manifolds
problem Mixing time of Langevin dynamics on Riemannian manifolds
method Relation between Langevin processes in domain and image
result Achievable polynomial mixing times
A new method for community detection in networks is presented.
problem Community detection in network analysis.
method Mixed regularized spectral clustering (Mixed-RSC) based on the regularized Laplacian matrix.
result The method is asymptotically consistent under mild conditions.
New tools for estimating and inferring Wasserstein distance in topic models.
problem Estimating and inferring the Wasserstein distance between mixing measures in topic models.
method New canonical interpretation and tools for inference on Wasserstein distance in topic models.
result First minimax lower bounds and fully data-driven inferential tools for the Wasserstein distance in topic models.
Researchers prove formulas for flag area measures, extending previous work.
problem Proving additive kinematic formulas for flag area measures.
method Introducing an algebraic framework to compute these formulas explicitly.
result Existence and explicit computation of additive kinematic formulas for flag area measures.
The existence of kinematic formulas for area measures with respect to any connected, closed subgroup of the orthogonal group acting transitively on the unit sphere is established. In particular, the kinematic operator for area measures is shown to have the structure of a co-product. In the case of the unitary group the…
We show that the discrete principal nets in quadrics of constant curvature that have constant mixed area mean curvature can be characterized by the existence of a Königs dual in a concentric quadric.
This paper develops basic setting for the dual Orlicz-Brunn-Minkowski theory for star bodies. An Orlicz φ-radial addition of two or more star bodies is proposed and related dual Orlicz-Brunn-Minkowski inequality is established. Based on a linear Orlicz φ-radial addition of two star bodies, we derive a f…
Unique entropy measure found for convex projective manifolds.
problem Entropy measure for convex projective manifolds.
method Developed Patterson--Sullivan densities and mixing theory.
result Unique mixing measure of maximal entropy exists.
New method for summarizing Bayesian mixture models using sliced Wasserstein distances.
problem Estimating the mixing measure in nonparametric Bayesian mixture models.
method Decision-theoretic approach using sliced Wasserstein distances for Gaussian mixtures.
result Effective estimation of the mixing measure and mixture density.
New surface area measures defined for ball-convex bodies, leading to entropy and inequalities.
problem Defining and analyzing surface area measures for ball-convex bodies.
method Introducing Lp relative surface areas, proving invariance and inequalities, and using geometric interpretations. result Established inequalities and a new notion of entropy for ball-convex bodies.
New geometric measure simplifies complex analysis.
problem Complex geometric analysis challenges.
method Geometric integration and convergence methods.
result Smallest measure satisfying Area Formula.
Common perpendiculars equidistribute in negatively curved spaces.
problem Equidistribution of common perpendiculars in negatively curved spaces.
method Analyzing the Bowen-Margulis measure and geodesic flow properties.
result Lebesgue measures of common perpendiculars equidistribute to the Bowen-Margulis measure.
New illumination bodies defined for ball-convex shapes, proving convexity and establishing surface area measures.
problem Characterizing properties of ball-convex shapes.
method Introducing illumination bodies and weighted illumination bodies, proving convexity, and establishing surface area measures.
result Illumination bodies are convex and provide surface area measures for ball-convex shapes.
New method for mixed data FI controls type I error and achieves high power.
problem Statistical inadequacy of feature importance measures for mixed data.
method Combining CPI framework with sequential knockoffs for mixed data.
result Our method controls type I error and achieves high power for mixed data.
Paper estimates area covered by a line-sweep sensor in robotics.
problem Accurately estimating the area covered by a line-sweep sensor.
method Relies on coverage measure and topological degree in the plane.
result Guaranteed characterization of the explored area using interval analysis.
Paper extends FOFC algorithm to work with mixed data types.
problem Designing causal discovery algorithms for mixed data types.
method Proves tetrad constraint can be entailed for mixed data types and applies FOFC algorithm.
result FOFC algorithm can work on mixed data types.
In the first part of the paper we survey some nonlocal flows of convex plane curves ever studied so far and discuss properties of the flows related to enclosed area and length, especially the isoperimetric ratio and the isoperimetric difference. We also study a new nonlocal flow of convex plane curves and discuss its e…
Frame flows on certain symmetric spaces mix exponentially.
problem Exponential mixing of frame flows in convex cocompact locally symmetric spaces.
method Generalized local non-integrability and non-concentration properties to apply Dolgopyat's method.
result Exponential mixing of frame flows proved for convex cocompact locally symmetric spaces.
The paper proposes a mixed-frequency quantile regression model for VaR and ES forecasting.
problem Forecasting VaR and ES with mixed-frequency data.
method Mixed-frequency quantile regression model to estimate VaR and ES.
result The proposed model outperforms other models in VaR and ES backtesting tests.
Short introduction to discrete flat fronts in hyperbolic space with a Weierstrass representation proof.
problem Understanding discrete flat fronts in hyperbolic space.
method Proving a Weierstrass representation for discrete flat fronts.
result Any discrete flat front in the mixed area sense admits a Weierstrass representation.
Understanding urban growth is one with understanding how society evolves to satisfy the needs of its individuals in sharing a common space and adapting to the territory. We propose here a quantitative analysis of the historical development of a large urban area by investigating the spatial distribution and the age of c…
Forré introduces a new conditional independence notion for mixed variables.
problem Unified framework for random and non-stochastic variables.
method Unified framework of transitional conditional independence and causal calculus for iDMGs.
result Unified framework connects conditional independencies to graphical separation criteria.
New study on No-U-Turn Sampler for accelerated mixing in Hamiltonian Monte Carlo.
problem Achieving accelerated convergence in Hamiltonian Monte Carlo.
method Combining concentration of measure and coupling analysis for mixing.
result Rigorous mixing guarantees for the No-U-Turn Sampler in certain Gaussian distributions.
Introduces fractional k-dimensional measure bridging fractional length and area.
problem Defining fractional measures for dimensions between 0 and n-1.
method Introduces a parameterized fractional measure σ that converges to Hausdorff measure. result Fractional measure converges to Hausdorff measure with a known constant factor.