Extended symmetric union with multiple tangle regions and Alexander polynomial properties.
problem Characterizing knots with multiple tangle regions.
method Generalizing the symmetric union construction to include multiple tangle regions and analyzing the Alexander polynomial.
result The Alexander polynomial of the constructed knot is the product of the Alexander polynomials of the tangles and the square of the partial knot's Alexander polynomial.
Paper offers a framework for estimating symmetric properties efficiently.
problem Estimating symmetric properties of distributions from samples.
method General framework using profile maximum likelihood (PML) distribution.
result Optimal sample complexity for many properties, practical algorithms.
We show the existence of isometric (or Ford) fundamental regions for a large class of subgroups of the isometry group of any rank one Riemannian symmetric space of noncompact type. The proof does not use the classification of symmetric spaces. All hitherto known existence results of isometric fundamental regions and do…
The paper defines axis systems for link projections and characterizes them for twist knots.
problem Defining and characterizing axis systems for link projections.
method Defined axis systems of link projections and characterized them for twist knots.
result Characterized axis systems of standard projections of twist knots.
A new method, Residual-Permuted Sums, improves confidence region construction for linear regression models.
problem Constructing reliable confidence regions for linear regression models with non-symmetric noise.
method Residual-Permuted Sums (RPS) method, which permutes residuals instead of perturbing their signs.
result RPS provides exact finite sample coverage probabilities and is uniformly strongly consistent.
Tanaka shows amphichiral symmetric unions of the unknot are trivial.
problem Understanding amphichiral symmetric unions and their Jones polynomials.
method Analyzing the Jones polynomial of amphichiral symmetric unions of the unknot and generalizing to other knots.
result Amphichiral symmetric unions of any knot with one twist region are trivial.
Paper shows regions close to negatively curved metrics are minimal fillings and rigid.
problem Boundary rigidity and minimality of metrics near negatively curved ones.
method Generalizes previous work on filling volume minimality and boundary rigidity for almost hyperbolic metrics.
result Regions with metrics close to a negatively curved symmetric metric are strict minimal fillings and boundary rigid.
Characterizes the local diffeomorphism structure of the exponential in the set of skew-symmetric matrices.
problem Characterizing the local diffeomorphism structure of the exponential in the set of skew-symmetric matrices.
method Introduce the diffeomorphic logarithm of special orthogonal matrices and an efficient algorithm.
result The region containing the principal logarithm has a special multiplicity structure.
We completely characterize isoperimetric regions in R^n with density e^h, where h is convex, smooth, and radially symmetric. In particular, balls around the origin constitute isoperimetric regions of any given volume, proving the Log-Convex Density Conjecture due to Kenneth Brakke.
In this work we classify the stable regions (second order minima of perimeter under an area constraint) in tori of revolution with piecewise continuous decreasing Gauss curvature from the longest parallel and with a horizontal symmetry. Some applications to isoperimetric problems are also given.
Study open orbits in causal flag manifolds with applications in AQFT.
problem Understanding open orbits in causal flag manifolds for applications in AQFT.
method Analyzing open orbits of symmetric subgroups on causal flag manifolds, focusing on invariant causal structures and modular flows.
result Determine the positivity regions of modular flows and their global hyperbolicity for different types of open orbits.
A short proof for a theorem about composite knots.
problem Proving a theorem about composite knots with symmetric union presentations.
method Presenting a concise proof of Tanaka's theorem.
result Composite knots with symmetric union presentations have non-trivial connected summands.
Study of caustics in Einstein-dust system, showing spacetime singularities and diverging curvature.
problem Understanding caustics and singularities in the Einstein-dust system.
method Established local existence result for spherically symmetric spacetimes containing caustics, constructed from solutions to a PDE problem.
result Obtained spherically symmetric spacetimes with diverging curvature and singular boundary.
New method quantifies uncertainty in kernel models without distributional assumptions.
problem Uncertainty quantification in kernel methods without strong distributional assumptions.
method Gradient perturbation to extract uncertainty information.
result Exact, non-asymptotic confidence regions for kernel models.
Stable capillary surfaces in weighted balls are disks.
problem Finding the shape of isoperimetric regions in weighted balls.
method Stability analysis and Hsiang symmetrization.
result Interior boundaries of isoperimetric regions in weighted balls are disks.
A symmetric matrix invariant is defined for oriented link diagrams.
problem Defining an invariant for oriented link diagrams.
method Defining a symmetric map τD from regions of an oriented link diagram to Z[x], corrected by the writhe. result The negative signature of τD, corrected by the writhe, conjecturally equals twice the Tristram-Levine signature function. The paper studies billiards in symmetric tables and finds a measure bound for maximizing orbits.
problem Understanding the measure of maximizing orbits in symmetric billiard tables.
method Introduced a closed invariant set of locally maximizing orbits and gave an effective bound on its measure.
result An effective bound on the measure of the invariant set in terms of the isoperimetric defect of the curve.
Study modular geodesics and wedge domains in non-compactly causal symmetric spaces.
problem Understanding the geometric implementation of modular group in symmetric spaces.
method Analyzing the flow generated by Euler elements and their geometric properties.
result The wedge region W is connected and coincides with the observer domain under certain conditions.
We study the isoperimetric problem for Euclidean space endowed with a continuous density. In dimension one, we characterize isoperimetric regions for a unimodal density. In higher dimensions, we prove existence results and we derive stability conditions, which lead to the conjecture that for a radial log-convex density…
Generalizing previous work by two of us, we prove the non-existence of certain stationary configurations in General Relativity having a spatial reflection symmetry across a non-compact surface disjoint from the matter region. Our results cover cases such that of two symmetrically arranged rotating bodies with anti-alig…
Extended symmetric unions extend properties of Alexander polynomials.
problem Properties of Alexander polynomials of symmetric unions.
method Constructing pairs of knots with specific epimorphisms.
result Alexander polynomials of constructed knots exhibit extended properties.
Geodesic spheres in certain symmetric spaces are quantitatively stable under small perturbations.
problem Stability of geodesic spheres in symmetric spaces under perturbations.
method Quantitative stability analysis using spectral gap of the Laplacian on geodesic spheres.
result Geodesic spheres are uniformly stable with respect to small C1-volume preserving perturbations. The SPS method constructs confidence regions for true parameters with optimal sample complexity.
problem Constructing exact, non-asymptotic confidence regions for true system parameters.
method Sign-Perturbed Sums (SPS) method, generalized to various types of problems.
result High probability upper bounds for SPS confidence regions show optimal shrinkage rate.
We study the stability of the Positive Mass Theorem using the Intrinsic Flat Distance. In particular we consider the class of complete asymptotically flat rotationally symmetric Riemannian manifolds with nonnegative scalar curvature and no interior closed minimal surfaces whose boundaries are either outermost minimal h…
AMP algorithm analyzes SCAD nonconvex regularization for sparse regression.
problem Sparse regression with nonconvex SCAD regularization under Gaussian data.
method Approximate message passing (AMP) algorithm for SCAD-AMP, stability and asymptotic analysis.
result SCAD-AMP achieves optimal performance and identifies phase transitions.
Model predicts growth competition on curved surfaces.
problem Growth dynamics of two subsets on Riemannian manifolds.
method Modeling growth rates on spherically symmetric Riemannian manifolds.
result Conditions for bounded or unbounded growth on different manifolds.
The paper improves confidence ellipsoids for ridge regression with PAC bounds.
problem Uncertainty quantification in ridge regression for insufficiently exciting inputs.
method Extension of SPS EOA algorithm to ridge regression with PAC bounds.
result Explicitly shows how regularization parameter affects region sizes and provides tighter bounds.
New model accounts for scale variation and noise in pairwise comparisons.
problem Nonreciprocal pairwise comparisons in decision analysis.
method Additive model with structured matrix and random perturbation.
result Explicit estimators and probability assessments of admissible ranking regions.
We show that a smooth radially symmetric solution u to the graphic Willmore surface equation is either a constant or the defining function of a half sphere in R3. In particular, radially symmetric entire Willmore graphs in R3 must be flat. When u is a smooth radial solution over a puncture…
We consider rotationally symmetric spaces with low regularity, which we regard as integral currents spaces or manifolds with Sobolev regularity and are assumed to have nonnegative scalar curvature. Relying on the flat distance and on Sobolev norms, we establish several nonlinear stability estimates about the ``distance…
Very recently Ben Andrews and Haizhong Li showed that every embedded cmc torus in the three dimensional sphere is axially symmetric. There is a two-parametric family of axially symmetric cmc surfaces; more precisely, for every real number H and every C > 2 (H+\sqrt{1+H^2}) there is an axially symmetry surface Σ_{H,C} w…
The SCMU algorithm computes cone factorizations for symmetric cones, improving upon existing methods.
problem Computing cone factorizations for symmetric cones in optimization.
method Introduces and analyzes the symmetric-cone multiplicative update (SCMU) algorithm.
result The SCMU algorithm non-decreases the squared loss objective.
Given a collection of N solutions of the (3+1) vacuum Einstein constraint equations which are asymptotically Euclidean, we show how to construct a new solution of the constraints which is itself asymptotically Euclidean, and which contains specified sub-regions of each of the N given solutions. This generalizes earlier…
Sparse symmetric tensor regression reduces brain connectivity complexity.
problem Complex brain connectivity analysis in neuroimaging.
method Sparse symmetric tensor regression model for functional connectivity.
result Superior performance in Alzheimer's disease detection.
A new complexity measure for neural networks improves upon classical methods.
problem Lack of a refined complexity measure for comparing different neural network architectures, especially permutation-invariant ones.
method Introduced an equivalence relation among linear functions and counted them relative to this relation.
result The new complexity measure clearly distinguishes between different models and increases exponentially with depth.
CP4SBI improves the calibration of credible sets in SBI models.
problem Inaccurate credible sets in SBI models lead to underestimation of true parameters.
method Develops a local conformal calibration framework for SBI models.
result Improves the quality of uncertainty quantification for neural posterior estimators.
Study shows apparent horizons emerge from gravitational collapse centers.
problem Understanding apparent horizons in gravitational collapse.
method Solving Einstein vacuum equations, constructing marginally outer trapped surfaces, applying critical trapped surface formation criterion.
result Emergence of apparent horizons from gravitational collapse centers.
We investigate a potential obtained as the convolution of a radially symmetric function and the characteristic function of a body (the closure of a bonded open set) with exterior cones. In order to restrict the location of a maximizer of the potential into a smaller closed region contained in the interior of the body, …
Study of naked singularities in Einstein vacuum equations using new self-similarity.
problem Mathematical study of naked singularities in the Einstein vacuum equations.
method Introduction of new self-similarity and geometric twisting for singularity formation.
result Construction of solutions corresponding to the exterior region of a naked singularity.
Dreaming neural networks learn and consolidate patterns during sleep.
problem Maximizing information storage and critical capacity in neural networks.
method Daily routine of learning during awake state and consolidation during sleep, using Guerra's interpolation techniques.
result The network achieves perfect retrieval regime after sleep, storing the same number of patterns as neurons.
Proves stability of spacetime Penrose inequality for spherical symmetric initial data.
problem Stability of the Penrose inequality for spherical symmetric spacetimes.
method Formulated and proved stability statement using spherical symmetry and asymptotically flat initial data.
result Initial data must arise from an isometric embedding into a static spacetime close to Schwarzschild spacetime.
Theory explains symmetry and saddle points in nonconvex optimization landscapes.
problem Understanding the optimization landscape of nonconvex matrix factorization problems.
method Characterizing stationary points and null spaces via invariant groups.
result Identifies infinitely many nonisolated strict saddle points and global minima.
We study the spaces of polynomials stratified into the sets of polynomial with fixed number of roots inside certain semialgebraic region Ω, on its border, and at the complement to its closure. Presented approach is a generalisation, unification and development of several classical approaches to stability problems in …
Efficient algorithms find solutions in a rare well-connected cluster at low constraint densities.
problem Finding solutions in the symmetric binary perceptron at low density.
method Formal proof of existence of a subdominant connected cluster and application of an efficient multiscale majority algorithm.
result An efficient algorithm can find solutions in a subdominant connected cluster with high probability.
Given a complete non-compact surface embedded in R^3, we consider the Dirichlet Laplacian in a layer of constant width about the surface. Using an intrinsic approach to the layer geometry, we generalise the spectral results of an original paper by Duclos et al. to the situation when the surface does not possess poles. …
Paper constructs non-symmetric collapsing spacetimes without symmetries.
problem Forming non-symmetric collapsing spacetimes in vacuum.
method Modified Christodoulou's a priori estimates and gluing construction.
result Past geodesic completeness and asymptotic Minkowski space.
Study on detecting a single spike in high-dimensional data matrices.
problem Detecting a single unknown spike in high-dimensional rectangular data matrices.
method Analysis of likelihood ratio between spiked and null models, using Gaussian fluctuations and Talagrand's interpretation of cavity method.
result Asymptotic Gaussian fluctuations of the likelihood ratio below the BBP threshold, with open maximal parameter region.
Constructs initial data leading to apparent horizons and tests Penrose Inequality.
problem Testing Penrose Inequality in dynamical spacetimes.
method Scale critical initial data for Einstein vacuum system, constructing Cauchy data.
result Penrose Inequality holds in an open region of the future of initial data.