New central set results prove special Kneser-Poulsen conjectures.
problem Volume increase in ball rearrangements.
method Central set analysis applied to Riemannian manifolds.
result New special cases of Kneser-Poulsen conjecture proved.
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48 results for “ball rearrangement”
The paper shows how to rearrange arcs to form closed curves.
problem Creating closed curves from planar arcs.
method Splitting a curve into arcs and rearranging them to form a closed curve.
result Closed curves can be formed by rearranging arcs under weak assumptions.
Sharp stability of Alexandrov's theorem for domains in the small-excess regime
problem Stability of Alexandrov's theorem for domains in the small-excess regime
method Combines a version of Fuglede's spectral-gap argument, a star-shaped rearrangement for sets of finite perimeter, quantitative estimates for the part of the boundary contained in the tentacles, and a polyhedral approximation argument for the non-graphical region
result Sharp stability estimate in a genuinely non-parametric regime
Simplified and extended a method for rearranging infinite configurations of cubes.
problem Constructing homotopies for isotopically rearranging cubes.
method Simplified and extended Eda and Kawamura's procedure.
result Simplified and extended the construction of homotopies.
This note proves a Gaussian version of a Pólya-Szegö conjecture using rearrangement techniques.
problem Finding the domain with the minimum Gaussian principal frequency when the Gaussian torsional rigidity is fixed.
method Adapted Kohler-Jobin rearrangement technique to the Gauss space, considering a modified torsional rigidity and rearranging layers to half-spaces.
result The Gaussian principal frequency is minimized for the half-space when the Gaussian torsional rigidity is fixed.
Paper explores closedness properties of convex sets in rearrangement invariant spaces.
problem Closedness properties of law-invariant convex sets in rearrangement invariant spaces.
method Analyzes equivalence of different closedness types in rearrangement invariant spaces.
result Order closedness, -closedness and -closedness of a law-invariant convex set are equivalent.
Numerical challenges inherent in algorithms for computing worst Value-at-Risk in homogeneous portfolios are identified and solutions as well as words of warning concerning their implementation are provided. Furthermore, both conceptual and computational improvements to the Rearrangement Algorithm for approximating wors…
Graphs represent gene segment organization, revealing complex interrelationships in a scrambled genome.
problem Understanding gene segment organization and interrelationships in a scrambled genome.
method Directed graphs representing gene segments and their relationships, with graph properties mapped to higher-dimensional space for analysis.
result Emerging star-like structures indicate complex interrelationships, including segments from multiple genes interleaving or overlapping.
Paper develops polynomial approximations for complex probability densities.
problem Approximating high-dimensional concentrated probability densities.
method Tensor-product spectral polynomials and KR rearrangements.
result Efficient approximation of complex densities using composite maps.
The paper proves a Moser-Trudinger inequality on metric spaces with curvature-dimension conditions.
problem Proving a Moser-Trudinger inequality on metric measure spaces.
method Rearrangement of functions on CD(k,n)-spaces satisfying a Polya-Szegö type inequality.
result Characterization of manifolds with lower bounded Ricci curvature admitting a Moser-Trudinger inequality.
Basket Options with Volatility Skew: Calibrating a Local Volatility Model by Sample Rearrangementq-fin.CP
Method calibrates basket options using rearranged samples from constituent processes.
problem Calibrate basket options with non-linear dependency structure.
method Propose a method to extract dependency structure from market data through systematic sampling rearrangement, then calibrate a local volatility model.
result Efficiently calibrates basket options with near-perfect accuracy.
The paper extends geometric inequalities from Euclidean space to Riemannian manifolds.
problem Proving geometric inequalities on smooth oriented Riemannian manifolds.
method Introducing symmetric decreasing rearrangement inequalities and testing their applicability to Riemannian manifolds.
result Smooth co-area formula and re-formulated geometric inequalities on Riemannian manifolds.
Hypergeometric function and Modular Curvature I. --Hypergeometric functions in Heat Coefficientsmath.DG
New proof of rearrangement lemma for noncommutative tori using hypergeometric functions.
problem Proving rearrangement lemma in noncommutative tori.
method Using Lauricella functions of type D and Gauss hypergeometric functions.
result Full reduction of spectral functions to Gauss hypergeometric functions.
The strong Fatou property is crucial for risk measures' dual representations.
problem Ensuring nice dual representations of risk measures.
method Exploring Fatou-type properties and inf-convolutions of law-invariant or surplus-invariant risk measures.
result Every quasiconvex law-invariant functional on a rearrangement invariant space with the strong Fatou property is σ(X, L∞)-lower semicontinuous.
Gaussian process regression cuts energy evaluations for atomic rearrangement paths.
problem Reducing computational effort for minimum energy paths in complex systems.
method Gaussian process regression to approximate energy surfaces and converge to minimum energy paths.
result Significant reduction in energy evaluations (less than a fifth for a test problem).
Study fine Pólya-Szegő inequalities in metric spaces with applications.
problem Fine Pólya-Szegő rearrangement inequalities in metric spaces.
method Theory of Sobolev and BV functions, synthetic Ricci bounds, isoperimetric inequality.
result New geometric and functional inequalities under Ricci lower bounds.
Pixle attacks images by rearranging pixels, bypassing neural networks.
problem Vulnerability of neural networks to black-box adversarial attacks.
method A novel attack that rearranges a small number of pixels in images.
result Successfully attacks a high percentage of samples on various datasets and models.
Paper compares solutions of Poisson equations on Riemannian manifolds with Robin boundary.
problem Comparing solutions of Poisson equations on Riemannian manifolds with Robin boundary.
method Using Schwarz rearrangement and isoperimetric inequalities.
result Extends results on Poisson equations with .
Study Hamiltonian diffeomorphisms on symplectic manifolds and properties of invariant convex functions.
problem Properties of invariant convex functions under Hamiltonian diffeomorphisms.
method Analysis of the adjoint action and properties of invariant convex functions.
result Continuous convex functions invariant under Hamiltonian diffeomorphisms are also invariant under strict rearrangements.
Frank and Lieb proved sharp Sobolev inequalities without rearrangements.
problem Proving sharp Sobolev inequalities for function spaces.
method Using conformal covariance and commutator identities from the Fefferman-Graham ambient metric.
result Direct proof of sharp Sobolev inequalities and new nonlinear inequality.
Machine learning models simulate molecular spectra and reactions in solvents.
problem Accurate simulation of molecular spectra and reactions in solvent environments.
method Introduced FieldSchNet, a deep neural network for modeling molecular interactions with external fields.
result Demonstrated significant lowering of Claisen rearrangement reaction activation barrier using FieldSchNet.
Extends subspace detour method to Gromov-Wasserstein problem.
problem Matching shapes using Gromov-Wasserstein distance.
method Project measures onto a subspace, then compute optimal transport plan.
result Connections with Knothe-Rosenblatt rearrangement.
Examines how algorithms affect user autonomy and information choice.
problem Impact of algorithmic recommendations on user autonomy and free choice.
method Double dichotomy analysis of user intentions and actions, prior and posterior information rearrangement.
result Algorithms can expand or limit user cognitive and social horizons.
An affine rearrangement inequality is established which strengthens and implies the recently obtained affine Pólya--Szegö symmetrization principle for functions on . Several applications of this new inequality are derived. In particular, a sharp affine logarithmic Sobolev inequality is established which i…
Optimal DP mechanisms for vector queries are found to be staircase distributions.
problem Designing optimal additive mechanisms for vector-valued queries under differential privacy.
method Reduction to radially symmetric distributions and convex rearrangement theory.
result Staircase mechanisms are optimal for any norm and cost function.
We equip many non compact non simply connected surfaces with smooth Riemannian metrics whose isoperimetric profile is smooth, a highly non generic property. The computation of the profile is based on a calibration argument, a rearrangement argument, the Bol-Fiala curvature dependent inequality, together with new result…
Optimized financial exposure networks reduce systemic risk by 3.5x without increasing capital requirements.
problem Reducing systemic risk in financial markets without raising capital requirements.
method Optimizing network topology to minimize systemic risk.
result Systemic risk reduced by a factor of approximately 3.5.
OmicsMapNet converts omics data into 2D images for deep learning analysis.
problem Analyzing high-dimensional omics data for phenotype classification.
method Reorganize omics data into 2D images, apply deep learning to classify, identify key features.
result Deep learning models accurately classify TCGA glioma samples based on molecular features.
Sharp Sobolev inequality on circle proven with Lorentz invariance.
problem Proving a sharp Sobolev inequality on the circle.
method Rearrangement inequality, variational method, Lorentz transformation.
result Sharp inequality and global minimizer on circle.
Link concordance equals homotopy for high-dimensional spheres.
problem Understanding when immersions of high-dimensional spheres are homotopically trivial.
method Developed stratified Morse theory for generic immersions, using gradient-like vector fields and Cerf theory.
result Every link of high-dimensional spheres is homotopically trivial, resolving a long-standing conjecture.
This note proves sharp affine Gagliardo-Nirenberg inequalities which are stronger than all known sharp Euclidean Gagliardo-Nirenberg inequalities and imply the affine Sobolev inequalities. The logarithmic version of affine Sobolev inequalities is verified. Moreover, An alternative proof of the affine Mo…
Study of spectral gaps in non-smooth spaces with bounded Ricci curvature.
problem Analyzing spectral gaps in non-smooth metric measure spaces.
method Establishing a Polya-Szego type inequality and applying it to show spectral gaps for the p-Laplace operator.
result Sharp spectral gap results for the p-Laplace operator on various non-smooth spaces.
The paper characterizes risk measures with the Fatou property in function spaces.
problem Investigating the Fatou property of law-invariant risk measures in function spaces.
method Characterization of the Fatou property using the AOCEA property and dual representations.
result Risk measures with the Fatou property exist under the AOCEA property in most classical model spaces.
We prove that an embedded cobordism between manifolds with boundary can be split into a sequence of right product and left product cobordisms, if the codimension of the embedding is at least two. This is a topological counterpart of the algebraic splitting theorem for embedded cobordisms of the first author, A. Nemethi…
We consider the effect of recovery rates on a pool of credit assets. We allow the recovery rate to depend on the defaults in a general way. Using the theory of large deviations, we study the structure of losses in a pool consisting of a continuum of types. We derive the corresponding rate function and show that it has …
New framework for learning KR maps from data, ensuring stable generalization.
problem Learning monotone triangular transport maps efficiently and accurately.
method General framework using invertible transformations of smooth functions, ensuring no spurious local minima.
result Unique global minimizer corresponds to the KR map under certain conditions.
We study two systems of tangle equations that arise when modeling the action of the Integrase family of proteins on DNA. These two systems--direct and inverted repeats--correspond to two different possibilities for the initial DNA sequence. We present one new class of solutions to the tangle equations. In the case of i…
We build a new probability measure on closed space and plane polygons. The key construction is a map, given by Knutson and Hausmann using the Hopf map on quaternions, from the complex Stiefel manifold of 2-frames in n-space to the space of closed n-gons in 3-space of total length 2. Our probability measure on polygon s…
3-balls in 4-sphere become isotopic in 5-ball.
problem Whether 3-balls in 4-sphere become isotopic in 5-ball.
method Analyzing the embedding of 3-balls in 4-sphere and 5-ball.
result Affirmative answer to Gay, Hughes, Kim, and Miller's question.
Motivated by a geometric problem, we introduce a new non-convex graph partitioning objective where the optimality criterion is given by the sum of the Dirichlet eigenvalues of the partition components. A relaxed formulation is identified and a novel rearrangement algorithm is proposed, which we show is strictly decreas…
New graph Fourier transform distinguishes directions in multi-dimensional signals.
problem Existing graph Fourier transform fails to distinguish directions in multi-dimensional signals.
method Algebraic properties of Cartesian products rearrange 1-D spectra into multi-dimensional frequency domain.
result Solves multi-valuedness of spectra and enables directional frequency analysis.
Gaussian process regression speeds up nudged elastic band calculations for transitions.
problem Reducing computational effort for calculating minimum energy paths in thermalized systems.
method Approximate energy surface generation and refinement using Gaussian process regression.
result The number of energy and force evaluations can be reduced by an order of magnitude.
Several isoperimetric inequalities of Dirichlet and Neumann eigenvalues of the Witten-Laplacianmath.AP
Paper derives inequalities for eigenvalues of Witten-Laplacian under fixed volume constraint.
problem Eigenvalue inequalities of Witten-Laplacian on bounded domains.
method Rearrangement technique and trial functions under fixed weighted volume constraint.
result Several isoperimetric inequalities for eigenvalues of Witten-Laplacian.
Study on ball widths and minimal submanifolds in space forms.
problem Understanding widths of balls and minimal submanifolds.
method Analyzing the area of equatorial balls and related bounds for minimal submanifolds.
result Lower bounds for the area of free boundary minimal submanifolds.
This paper describes a method to construct standard 4-balls from homotopy 4-balls in .
problem The problem is whether every homotopy 4-ball in is standard.
method The approach is to use Stein surfaces and pseudoconvex domains to construct a diffeomorphic domain that is the union of three pseudoconvex domains, ensuring it is a standard 4-ball.
result The construction method ensures that the domain is a standard 4-ball, providing a compelling reimbedding construction for homotopy 4-balls in .
Diameters of ball intersections decrease as centers move apart.
problem Behavior of intersections of moving balls in Riemannian manifolds.
method Continuous decrease of intersection diameter as centers move apart.
result Diameter of intersections decreases continuously.
Some Properties of the Intersection of Free Boundary Minimal Hypersurfaces in Euclidean Ballsmath.DG
Two minimal hypersurfaces in a ball intersect in any half-ball.
problem Intersection properties of minimal hypersurfaces in a ball.
method Analyzing the intersection of two minimal hypersurfaces in a unit Euclidean ball.
result Intersection point in any half-ball, strong Frankel property.
A Dirichlet problem on ballsmath.DG
Many solutions found for a ball boundary problem.
problem Finding solutions for a Dirichlet problem on balls.
method Infinitely many solutions provided.
result Many solutions found for a Dirichlet problem on balls.