Essential self-adjointness proven for powers of first-order differential operators on non-compact manifolds with low-regularity metrics.
problem Essential self-adjointness of powers of first-order differential operators on non-compact manifolds with low-regularity metrics.
method Demonstrates the equivalence between essential self-adjointness and a negligible boundary property for first-order differential operators with locally bounded measurable coefficients. For higher regularity coefficients, essential self-adjointness of higher powers is shown under the same condition.
result Essential self-adjointness of first-order differential operators and their higher powers proven under specific conditions on non-compact manifolds with low-regularity metrics.
We extend the Besicovitch-Federer projection theorem to transversal families of mappings. As an application we show that on a certain class of Riemann surfaces with constant negative curvature and with boundary, there exist natural 2-dimensional measures invariant under the geodesic flow having 2-dimensional supports s…
We study the Gaffney Laplacian on a vector bundle equipped with a compatible metric and connection over a Riemannian manifold that is possibly geodesically incomplete. Under the hypothesis that the Cauchy boundary is polar, we demonstrate the self-adjointness of this Laplacian. Furthermore, we show that negligible boun…
Schwarzschild 3-manifold stability proven for 3D Penrose inequality.
problem Stability of the Schwarzschild 3-manifold in the context of the 3D Riemannian Penrose inequality.
method Pointed measured Gromov-Hausdorff topology, negligible domains and boundary area perturbations.
result Schwarzschild 3-manifold stability proven for 3D Penrose inequality.
New method shows Hessian estimator from random samples converges to true Hessian on complex manifolds.
problem Uncertainty in Hessian estimator accuracy on complex manifolds with boundaries and nonuniform sampling.
method Locally fitting quadratic polynomials, rigorous theoretical analysis under mild conditions.
result The Hessian estimator asymptotically converges to the true Hessian, even near boundaries.
Binary linear classifiers are the most explainable up to negligible sets.
problem Measuring the explainability of machine learning classifiers.
method Introducing pointwise coverage to measure explainability and proving the binary linear classifier is the most explainable up to negligible sets.
result The binary linear classifier is uniquely the most explainable classifier up to negligible sets.
The Zumbach effect is significant under rough Heston but negligible in classical Heston.
problem Identifying the Zumbach effect in stochastic volatility models.
method Explicit computations of the Zumbach effect under rough Heston model.
result The Zumbach effect is negligible in the classical Heston model but significant under rough Heston.
We describe typical degenerations of quadratic differentials thus describing ``generic cusps'' of the moduli space of meromorphic quadratic differentials with at most simple poles. The part of the boundary of the moduli space which does not arise from ``generic'' degenerations is often negligible in problems involving …
Arithmetic spaces' thin parts are negligible, impacting Betti numbers.
problem Understanding the structure of arithmetic locally symmetric spaces.
method Analyzing thin parts and deducing asymptotic results on Betti numbers.
result Arithmetic spaces' thin parts are negligible, impacting Betti numbers.
Paper proposes a visual tool for analyzing financial markets.
problem Rational negligence in financial markets during the 2008 crisis.
method Visual declarative language based on port-graph rewriting.
result Visual tool for analyzing asset-backed securities.
Simple technique turns any adversarial attack into a universal one using few test examples.
problem Creating universal adversarial attacks with minimal data.
method Universalization technique using few adversarial test examples and spectral properties.
result Simple universalization technique achieves comparable fooling rates to state-of-the-art methods.
New property ensures non-looseness of ribbon boundaries.
problem Non-looseness of ribbon boundaries for Legendrian graphs.
method Define and prove the Tight Reattachment Property.
result Ribbon boundaries of Legendrian graphs with the Tight Reattachment Property are non-loose.
Boundary properties of hyperbolic groups are invariant under a maximization procedure.
problem Proving boundary properties of hierarchically hyperbolic groups are invariant.
method Proving boundary invariance under a maximization procedure.
result Boundary properties of hierarchically hyperbolic groups are invariant under maximization.
Characterizes groups with specific boundary properties.
problem Groups with Schottky set boundaries.
method Study relatively hyperbolic group pairs with Schottky boundaries.
result Groups with boundaries where Schottky sets have 1 or 2 component incidence graphs.
Theory developed for complex Hessian measures on Hermitian manifolds.
problem Defining and analyzing complex Hessian measures on Hermitian manifolds.
method Potential theory for m-subharmonic functions with respect to a Hermitian metric.
result Equivalence between polar sets and negligible sets for m-subharmonic functions.
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.
Deep learning detects cloud changes due to human aerosols.
problem Uncertainty in the effect of anthropogenic aerosols on cloud properties and Earth's energy balance.
method Deep convolutional neural networks to analyze cloud images.
result Identified and characterized specific cloud perturbations due to human aerosols.
New varifolds with capillary boundary properties studied.
problem Understanding varifolds with specific boundary conditions.
method Introducing a Radon measure on a Grassmannian bundle as a capillary boundary.
result Structural properties, monotonicity inequality, and integral compactness proved.
New method learns community properties robustly against adversaries.
problem Learning community properties in noisy networks.
method Nonparametric, unsupervised, scalable graph scan procedure.
result Asymptotically correct estimation of normal user activity.
Study geometric properties of cuspidal edges with boundary.
problem Differential geometric properties of cuspidal edges with boundary.
method Analysis of differential geometric invariants and their relations.
result Relation between boundary behavior and other invariants.
The paper explores the geometry and topology of DNN decision boundaries.
problem Understanding the geometric and topological properties of DNN decision boundaries.
method Differential geometry and the Gauss-Bonnet-Chern theorem.
result Computed the Euler characteristics of compact decision boundaries.
The 2008 financial crisis revealed banking consolidation paradoxically increased systemic fragility and global financial contagion with negligible spatial decay.
problem Fundamental vulnerabilities in interconnected banking systems during the 2008 financial crisis were inadequately addressed by existing frameworks.
method Developed a unified spatial-network framework using spectral analysis of network Laplacian operators combined with spatial difference-in-differences identification.
result Banking consolidation paradoxically increased systemic fragility and global financial contagion with negligible spatial decay.
In this article, we discuss the quasiconformal structure of boundaries of right-angled hyperbolic buildings using combinatorial tools. In particular we exhibit some examples of buildings of dimension 3 and 4 whose boundaries satisfy the combinatorial Loewner property. This property is a weak version of the Loewner prop…
New Dynkin condition for manifolds with boundary yields bi-Lipschitz equivalence and spectral properties.
problem Characterizing smooth Riemannian manifolds with boundary.
method Introducing a Dynkin-type condition and proving its equivalence to a weighted manifold.
result Bi-Lipschitz equivalence and various spectral properties of manifolds with boundary.
Study on neuron dynamics for XOR classification with zero-margin.
problem Understanding neural network training dynamics in zero-margin classification problems.
method Analysis of Gaussian XOR problem, focusing on neuron block dynamics and generalization without margin assumptions.
result Neurons cluster into four directions and block-level signals evolve coherently, essential for reliable prediction in the Gaussian setting.
Study the boundary operator property on simplicial complexes, proving essential properties for Hodge theory.
problem Characterize the boundary operator property ∂∂=0 on simplicial complexes. method Characterization in ℓ2 terms of recurrence of links, defining relative cohomology, and proving harmonic eigenforms. result Essential properties for Hodge theory, including weak decomposition and existence of harmonic eigenforms.
Analyzes constant heat flow properties on Riemannian manifolds.
problem Classify manifolds with constant heat flow property.
method Analyzes overdetermined parabolic problems and uses isoparametric tubes.
result Analytic manifolds with constant flow property are isoparametric tubes.
Optimal benchmark design varies based on costs in financial manipulation.
problem Manipulation of price benchmarks in finance.
method Analyzes empirical pattern and cost structures to determine optimal benchmark design.
result The optimal benchmark depends on the relative sizes of fixed and variable costs.
As demonstrated by Croke and Kleiner, the visual boundary of a CAT(0) group is not well-defined since quasi-isometric CAT(0) spaces can have non-homeomorphic boundaries. We introduce a new type of boundary for a CAT(0) space, called the contracting boundary, made up rays satisfying one of five hyperbolic-like propertie…
This paper formalizes Uniswap v3 using PTA and FST for rigorous analysis.
problem Formal modeling of Uniswap v3's concentrated liquidity for rigorous analysis.
method Formal state machine models using PTA and FST, proving rounding bounds.
result Formal justification of Uniswap v3's ε-slack and rounding safety. The paper proves stability of positive mass theorem for flat 3-manifolds.
problem Stability of positive mass theorem for uniformly asymptotically flat 3-manifolds.
method Analyzing sequences of 3-manifolds with nonnegative scalar curvature and zero ADM mass, subtracting open subsets and using Gromov-Hausdorff convergence.
result Convergence of (Mi∖Zi,gi,pi) to Euclidean space (R3,gE,0) in specific topologies. Generalizes Bestvina's Z-boundaries to coarse Z-boundaries.
problem Establishing properties of Z-boundaries for groups. method Introducing a new concept of a 'coarse Z-boundary' and proving theorems about it. result Admitting a coarse Z-boundary is a pure quasi-isometry invariant. New proof shows Mazur-type manifolds with specific boundaries are simple.
problem Characterizing Mazur-type manifolds with L-space boundaries. method Using Heegaard Floer homology and irreducibility properties.
result Mazur-type manifolds with L-space boundaries are the 4-ball and 3-sphere. Let M be a Riemannian manifold and Ω a compact domain of M with smooth boundary. We study the solution of the heat equation on Ω having constant unit initial conditions and Dirichlet boundary conditions. The purpose of this paper is to study the geometry of domains for which, at any fixed value of time, the nor…
Study properties of contact structures on symplectic disk bundles with concave boundaries.
problem Understanding the geometric properties of contact structures on concave boundaries of symplectic disk bundles.
method Use tools from toric geometry and algebraic torsion measurements from embedded contact homology.
result All such contact manifolds have a global contact toric structure, and can be tight or overtwisted.
Study on topological properties and boundaries of RCD(K,N) spaces.
problem Understanding the topology and boundaries of non-collapsed RCD(K,N) spaces.
method Established topological regularity and stability, introduced boundary concept.
result Properties of boundaries and behavior under convergence studied.
Study on curvature measures and boundary properties of convex sets in hyperbolic space.
problem Characterizing the boundary structure of convex sets in hyperbolic space.
method Analysis of curvature measures and normal points.
result Generalized Gauss equation and characterizations of Gaussian curvature for convex surfaces.
Various structural properties are developed for non-orientable surfaces in link spaces. The Möbius band tree is described to represent genus growth of one-sided surfaces in solid tori. The structure of the Tree allows various insights into the change of genus under boundary slope, which are not possible using the exist…
Random surfaces with boundary have predictable properties.
problem Understanding the statistical properties of random surfaces.
method Generating surfaces by gluing polygons and analyzing their genus and boundary components.
result Genus and boundary components of random surfaces follow a bivariate normal distribution.
Investment project break-even point analyzed as discount rate changes.
problem Determining the break-even point for a simple investment project.
method Closed expression derived for break-even point Qf as a function of parameters.
result Qf is strictly increasing and convex in r, with strong influence of p and Cv.
Constructs a support-preserving homotopy for differential forms with boundary decay estimates.
problem Non-uniqueness of chain homotopies in de Rham complexes with boundary decay properties.
method Constructs a specific chain homotopy with desirable support propagation and boundary decay estimates.
result Obtains a support-preserving right inverse of the divergence operator with optimal decay estimates.
Faster neural networks with approximate tensor operations reduce computation and training time.
problem Efficiently training deep neural networks with reduced computational and communication costs.
method Sample-based approximation of tensor operations (matrix multiplications and convolutions) to reduce computation and communication.
result Up to 66% reduction in computations and up to 1.37x faster training time with negligible accuracy loss.
The study explores scalar curvatures on manifolds with boundary properties.
problem Understanding scalar curvatures on manifolds with boundary constraints.
method Presentation of problems and results related to scalar curvatures and mean curvatures of boundaries.
result Exploration of natural and artificial constructions in scalar curvature studies.
Hamiltonian properties of earthquakes on surfaces with boundary lengths are proven.
problem Hamiltonian properties of earthquakes on surfaces with boundary lengths.
method Provided a Hamiltonian function extending the classical length map, proving Hamiltonian sum of infinitesimal earthquakes.
result Any sum of infinitesimal earthquakes on a surface with boundary lengths is Hamiltonian.
The study characterizes hypersurfaces in weighted cylinders and generalizes confinement properties.
problem Characterizing hypersurfaces in weighted Riemannian products.
method Analyzing parabolic hypersurfaces with boundary in weighted cylinders.
result Generalized confinement properties of hypersurfaces in weighted cylinders.
Paper studies minimal surfaces in curved spaces, proving existence and properties.
problem Existence and properties of minimal surfaces with free boundaries.
method Degree theory, Banach manifold construction, Fredholm map analysis.
result Space of partially free boundary minimal half disks is a Banach manifold.
We prove that random groups in the Gromov density model, at any density, satisfy property (FA), i.e. they do not act non-trivially on trees. This implies that their Gromov boundaries, defined at density less than 1/2, are Menger curves.
New hyperbolic 3-pseudomanifolds with unique properties.
problem Understanding cubulable groups in hyperbolic 3-manifolds.
method Constructing compact hyperbolic 3-manifolds with specific boundary conditions.
result Found groups that are word hyperbolic but not cubulable.