Paper controls shape stability in infinite Riemannian manifolds.
problem Characterizing optimal shapes in infinite-dimensional Riemannian manifolds.
method Uses Riemannian manifold framework and mean curvature analysis.
result Control on shape stability depends only on mean curvature.
The paper proves stability of Wulff shapes using anisotropic curvature functionals.
problem Stability of Wulff shapes under anisotropic curvature.
method Estimates distance to Wulff shape using Lp-norm of traceless F-Hessian of a foliating function. result Quantitative stability results for anisotropic inequalities and problems.
The purpose of this paper is to study the shapes and stabilities of bio-membranes within the framework of exterior differential forms. After a brief review of the current status in theoretical and experimental studies on the shapes of bio-membranes, a geometric scheme is proposed to discuss the shape equation of closed…
An important "stability" theorem in shape theory, due to D.A. Edwards and R. Geoghegan, characterizes those compacta having the same shape as a finite CW complex. In this note we present straightforward and self-contained proof of that theorem.
Stability of hypersurface immersions in Riemannian manifolds proved for Lp perturbations.
problem Stability of isometric immersions of hypersurfaces in Riemannian manifolds under Lp perturbations of their fundamental forms. method Young measure approach, relaxation of energy, regularity result for immersions.
result Sequence of immersions converges to an isometric immersion with the reference shape operator.
Flat minimal hypersurfaces found in wedge-shaped domains.
problem Finding minimal surfaces in wedge-shaped domains.
method Proving stability and flatness of C1,1-to-edge minimal hypersurfaces. result Stable minimal hypersurfaces are flat in wedge-shaped domains.
The paper solves a thermodynamics problem about crystal shape.
problem Understanding if minimizing free energy with convex potential and mass constraint generates a convex crystal.
method Utilized a stability theorem, convexity, and a new maximum principle approach to prove a three-dimensional convexity theorem.
result Completely settled the Almgren problem in R3 under generic conditions. Mathematical methods of population genetics and framework of exchangeability provide a Markov chain model for analysis and interpretation of stochastic behaviour of equity markets, explaining, in particular, market shape formation, statistical equilibrium and temporal stability of market weights.
After a brief introduction to several variational problems in the study of shapes of thin thickness structures, we deal with variational problems on 2-dimensional surface in 3-dimensional Euclidian space by using exterior differential forms. The morphological problems of lipid bilayers and stabilities of cell membranes…
Study proves inequality for hypersurfaces and shows almost extremals are close to Wulff shape.
problem Proving anisotropic extrinsic radius pinching inequality for hypersurfaces.
method Analyzes anisotropic mean curvatures and studies equality cases.
result Almost extremal hypersurfaces are close to Wulff shape.
We prove a qualitative and a quantitative stability of the following rigidity theorem: an anisotropic totally umbilical closed hypersurface is the Wulff shape. Consider n≥2, p∈(1,+∞) and Σ an n-dimensional, closed hypersurface in Rn+1, boundary of a convex, open set. We show that …
The paper proves inequalities for star-shaped and F-mean convex hypersurfaces in Rn+1.
problem Proving geometric inequalities for specific types of hypersurfaces.
method Using anisotropic p-momentum, perimeter, and volume, the paper derives inequalities for star-shaped and F-mean convex hypersurfaces. result The Wulff shape of F is the unique minimizer of the corresponding functionals among all star-shaped and F-mean convex sets. Found a stable 3D shape with specific properties.
problem Finding K-stable Fano threefolds.
method Analyzing specific Fano threefolds with given properties.
result Identified a K-stable Fano threefold with Picard rank 3 and anti-canonical degree 28.
We study the stability of closed, not necessarily smooth, equilibrium surfaces of an anisotropic surface energy for which the Wulff shape is not necessarily smooth. We show that if the Cahn Hoffman field can be extended continuously to the whole surface and if the surface is stable, then the surface is, up to rescaling…
Study on stability of network flow shrinkers with findings on instability of specific shapes.
problem Stability of regular shrinkers in network flow.
method Analysis of self-similarly shrinking solutions called regular shrinkers.
result All regular shrinkers with two or more enclosed regions can be perturbed away. Specific shapes like 4-ray star, 5-ray star, fish, and rocket are unstable among those with one enclosed region.
EoS selectively shapes learning, affecting some groups more than others.
problem EoS affects learning differently across the data distribution.
method Branching intervention to enter or exit EoS regime, controlled perturbation to isolate mechanisms.
result EoS redistributes learning, amplifying progress on some groups and suppressing others.
Given a positive function F on S n satisfying an appropriate con-vexity assumption, we consider hypersurfaces for which a linear combination of some higher order anisotropic curvatures is constant. We define the varia-tional problem for which these hypersurfaces are critical points and we prove that, up to translations…
By introducing a shape manifold as a solution set to solve inverse obstacle scattering problems we allow the reconstruction of general, not necessarily star-shaped curves. The bending energy is used as a stabilizing term in Tikhonov regularization to gain independence of the parametrization. Moreover, we discuss how se…
A framework for generating 3D shapes by sequentially assembling primitives.
problem Combinatorial complexity in generating 3D shapes.
method Bayesian optimization for efficient exploration and exploitation of feasible combinations.
result Successfully generates realistic combinatorial 3D shapes.
Two-sample tests improve on existing methods for microtubule data.
problem Testing differences between two groups of filament data.
method Optimal lifts and manifold stability theorem applied to microtubule data.
result New tests outperform existing methods on simulated and real data.
Paper studies stability of curved surfaces in a half-space.
problem Stability of anisotropic capillary hypersurfaces in a half-space.
method Analyzes weak stability and proves Bernstein-type theorems.
result Compact hypersurfaces are stable if and only if they are a truncated Wulff shape.
RCLA reduces noise in topological data analysis, preserving essential structure.
problem Noise in large datasets obscures topological features in persistent homology.
method Grid-based RCLA integrates data reduction and denoising with a threshold parameter.
result RCLA provides a theoretical guarantee and automatic parameter selection.
Establishes relationships between prudence and stability properties of risk functionals.
problem Stability properties of risk functionals
method General relationships and preservation of prudence under cash-additive hulls and inf-convolutions
result General methods for constructing prudent risk measures
Some elementary considerations are presented concerning Catenoids and their stability, separable minimal hypersurfaces, minimal surfaces obtainable by rotating shapes, determinantal varieties, minimal tori in S3, the minimality in Rnk of the ordered set of k orthogonal equal-length n-vectors, and U(1)-invariant minimal…
Sharp stability of Alexandrov's theorem for C1 domains in the small-excess regime
problem Stability of Alexandrov's theorem for C1 domains in the small-excess regime method Combines a BV 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
Momentum affects optimization differently at small vs large batch sizes near instability.
problem Understanding how momentum impacts optimization near the edge of stability.
method Demonstrated through batch-size dependent behavior of SGD with momentum.
result Momentum operates in two distinct regimes: amplifying stochastic fluctuations at small batch sizes and stabilizing at large batch sizes.
The paper maps two types of hyperkähler manifolds and identifies their symplectic structures.
problem Mapping and identifying symplectic structures of two types of hyperkähler manifolds.
method Produced a map from star-shaped quiver varieties to Higgs bundle moduli spaces, verified stability, and showed it is a homeomorphism.
result Identified natural holomorphic symplectic structures on the two spaces.
Enhanced 3D shape analysis using information geometry.
problem Challenges in comparing 3D point clouds due to their unstructured nature and complex geometry.
method Information geometric framework for 3D point cloud shape analysis using Gaussian Mixture Models (GMMs) on a statistical manifold. Proposed MSKL divergence with upper and lower bounds.
result MSKL provides stable and monotonically varying values that directly reflect geometric variation, outperforming traditional distances and existing KL approximations.
Survey on stability of Minkowski spacetime in relativity.
problem Nonlinear stability of Minkowski spacetime in general relativity.
method Decay assumptions, geometric foliations, energy identities, and gauge choices.
result Understanding of decay, dispersion, and geometry-analysis interplay.
We present a new algorithm for boosting generalized additive models for location, scale and shape (GAMLSS) that allows to incorporate stability selection, an increasingly popular way to obtain stable sets of covariates while controlling the per-family error rate (PFER). The model is fitted repeatedly to subsampled data…
Unified framework for robust causal directionality in quantum systems under MNAR observation.
problem Determining causal directionality in quantum systems under MNAR observation.
method Integrates CVAE-based latent constraints, MNAR-aware selection models, GEE-stabilized regression, penalized empirical likelihood, and Bayesian optimization.
result Achieves lower bias and variance, near-nominal coverage, and superior quantum-specific diagnostics.
We introduce the notion of multiscale covariance tensor fields (CTF) associated with Euclidean random variables as a gateway to the shape of their distributions. Multiscale CTFs quantify variation of the data about every point in the data landscape at all spatial scales, unlike the usual covariance tensor that only qua…
Study characterizes bladder motion using dynamic MRI and statistical analysis.
problem Limited volume coverage in dynamic MRI sequences hinders 3D shape reconstruction.
method 3D dense velocity measurements, LDDMM framework, statistical characterization, mean curvature changes, surface deformation analysis.
result Stable shape descriptor for characterizing bladder surface dynamics.
Study handles in 3D shapes, applies to material patterns.
problem Understanding 3D shapes through handlebody decompositions.
method Introduced handlebody decompositions, showed stability, applied to materials.
result Stable equivalence of handlebody decompositions in 3-manifolds.
Online algorithms stabilize in feedback loops of performative prediction.
problem Feedback loops in algorithmic predictions influence data distributions.
method Martingale argument and randomization to avoid distributional assumptions.
result No-regret algorithms converge to performatively stable equilibria.
An affine hypersurface is said to admit a pointwise symmetry, if there exists a subgroup of the automorphism group of the tangent space, which preserves (pointwise) the affine metric h, the difference tensor K and the affine shape operator S. In this paper, we deal with positive definite affine hypersurfaces of dimensi…
This paper introduces TDA and TSI for better business analytics.
problem Nonlinear, multi-scale business datasets under-represented by traditional tools.
method Topological Data Analysis (TDA) and Topological Stability Index (TSI).
result TSI reveals structural variability in business data.
Paper stabilizes persistent homology rank functions for statistical inference.
problem Stability issues in persistent homology rank functions.
method Derive stability results for rank functions under FDA metrics.
result Rank functions stabilize, improving statistical inference.
VL finds flatter solutions at edge of stability, matching theory with practice.
problem Understanding implicit regularization in deep learning.
method Edge of Stability framework, controlling variational posterior shape and sample number.
result VL finds even flatter solutions than gradient descent.
See http://www.youtube.com/watch?v=izbGXdjvK_I for a YouTube video showing part of the results in this paper.We will consider surfaces whose mean curvature at a point is a linear function of the square of the distance from that point to the vertical axis. We restrict ourselves here to surfaces which are cylinders over …
New method estimates extreme outcomes in heavy-tailed data, breaking circular dependence.
problem Estimating outcomes for extreme events in heavy-tailed data.
method Proposes an ADRF estimator that includes a structured tail-shape output and a diagnostic to evaluate tail shape.
result Successfully reduces MAE in deep-tail and conditional-shortfall predictions.
In constant curvatures spaces, there are a lot of characterizations of geodesic balls as optimal domain for shape optimization problems. Although it is natural to expect similar characterizations in rank one symmetric spaces, very few is known in this setting. In this paper we prove that, in a non-compact rank one symm…
Echo state networks are powerful recurrent neural networks. However, they are often unstable and shaky, making the process of finding an good ESN for a specific dataset quite hard. Obtaining a superb accuracy by using the Echo State Network is a challenging task. We create, develop and implement a family of predictably…
Unified framework for stability and generalization of Push-Sum in decentralized learning over directed graphs.
problem Understanding stability and generalization of Push-Sum in decentralized learning over directed networks.
method Developed a unified uniform-stability framework for SGP algorithm, incorporating imbalance-aware consistency bounds.
result Established finite-iteration stability and optimization guarantees for convex and non-convex objectives.
We introduce a new fundamental domain for the cusp stabilizer of a Hilbert modular group over a real quadratic field K=Q(sqrt n). This is constructed as the union of Dirichlet domains for the maximal unipotent group, over the leaves in a foliation of the biplane. The region is the Cartesian product of the positive real…
Study how firm liquidation regimes affect shareholder value and stability.
problem Balancing shareholder value and financial stability during firm liquidation.
method Modelled forced liquidation in reduced form, solved singular stochastic control problem.
result Combining distress regions below and above ruin threshold improves both shareholder value and firm survival.
New findings on maximizing noise stability in partitions of Gaussian space.
problem Maximizing noise stability in partitions of Gaussian space.
method Analyzing the correlation between sets and their noise stability, proving conditional conjectures and hardness results.
result Hyperstable partitions maximize noise stability and have specific properties.
Given a positive function F on Sn which satisfies a convexity condition, we define the r-th anisotropic mean curvature function HrF for hypersurfaces in Rn+1 which is a generalization of the usual r-th mean curvature function. Let X:M→Rn+1 be an n-dimensional closed hypersu…