Paper proves new theorems about curvature in weighted manifolds.
problem Understanding curvature in weighted manifolds.
method Proved spectral comparison and splitting theorems for infinity-Bakry-Emery Ricci curvature.
result Results extend existing theorems and provide new supplements.
Proves existence of Poisson equation on certain non-compact manifolds.
problem Existence of solutions to Poisson equation on non-compact Riemannian manifolds.
method Proves existence result using weighted Poincaré inequalities outside compact sets.
result Proves existence for a broad class of manifolds including non-parabolic ones.
We study stability properties of f-minimal hypersurfaces isometrically immersed in weighted manifolds with non-negative Bakry-Emery Ricci curvature under volume growth conditions. Moreover, exploiting a weighted version of a finiteness result and the adaptation to this setting of Li-Tam theory, we investigate the top…
Study complete manifolds with weighted Poincaré inequality and Ricci curvature bounds.
problem Understanding the structure of complete manifolds with specific curvature and inequality conditions.
method Analyzing manifolds with weighted Poincaré inequality and Ricci curvature bounds.
result Obtained splitting results for manifolds with non-zero weight function limit at infinity.
Paper proves Koszul duality for weighted A-infinity algebras.
problem Koszul duality for weighted A-infinity algebras.
method Constructs new box tensor product for weighted A-infinity bimodules and verifies correspondence between maps and bimodules.
result Proves Koszul duality result between weighted A-infinity algebras.
The paper models and deforms A-infinity structures for bordered knot algebras.
problem Understanding A-infinity structures for bordered knot algebras.
method Combinatorial model and weighted deformation of A-infinity structures.
result Explicit combinatorial model for bordered knot algebras' A-infinity structure.
New Einstein metrics created by modifying hyperbolic infinity.
problem Creating new Einstein metrics.
method Perturbing conformal infinity of geometrically finite hyperbolic metrics and applying inverse function theorem.
result Construct new examples of Einstein metrics.
We establish Carleman inequalities for the weighted laplacian associated to an expanding gradient Ricci soliton. As a consequence, a unique continuation at infinity is proved for asymptotically Ricci flat Ricci expanders. The obstruction at infinity is a symmetric 2-tensor defined on the link of the corresponding asymp…
New Calabi-Yau metrics constructed with detailed geometry at infinity.
problem Constructing complete Calabi-Yau metrics with specific properties.
method Weighted blow-up and Hölder spaces for Laplacian analysis.
result Examples of Calabi-Yau metrics with conical singularities and non-uniqueness of tangent cones.
We prove structure theorems for complete manifolds satisfying both the Ricci curvature lower bound and the weighted Poincaré inequality. In the process, a sharp decay estimate for the minimal positive Green's function is obtained. This estimate only depends on the weight function of the Poincaré inequality, and yields …
Invariant for 3-manifolds with torus boundary defined.
problem Defining invariants for 3-manifolds with specific boundaries.
method Module over a weighted A-infinity algebra associated to a torus.
result Invariant constructed for bordered 3-manifolds with torus boundary.
Proves global existence for quasilinear wave equations with weak null condition.
problem Global existence for quasilinear wave equations with weak null condition.
method p-weighted energy method, hierarchical structure in semilinear terms, robust methods.
result Proves global existence for a larger class of quasilinear wave equations.
Classifies ancient and expanding Ricci flows with specific groups.
problem Classifying ancient and expanding Ricci flows with certain groups.
method Uses a renormalized λALE-functional to control the large-scale behavior of Perelman's μ-functional.
result Identifies hyperkähler ALE metrics as the only spin ancient Ricci flows with specific groups.
This paper compares gradient estimators in importance-weighted VI and justifies the superiority of DREP over REP.
problem Understanding the impact of gradient estimators on importance-weighted VI algorithms.
method Unified theoretical comparison of reparameterized and doubly-reparameterized gradient estimators tied to IWAE, VR, and VR-IWAE bounds.
result Formally justifies the superiority of doubly-reparameterized gradient estimators over reparameterized ones in importance-weighted VI.
Estimates for Poisson equation on manifolds with weighted Poincare inequality.
problem Existence and estimates of Poisson equation solutions on manifolds.
method Develops Green's function estimate using weighted Poincare inequality and Ricci curvature.
result Proves Liouville property for finite energy holomorphic functions on Kähler manifolds.
We introduce a natural extension of the concept of gradient Ricci soliton: the Ricci almost soliton. We provide existence and rigidity results, we deduce a-priori curvature estimates and isolation phenomena, and we investigate some topological properties. A number of differential identities involving the relevant geome…
One of the main purposes of this paper is to prove that on a complete Kähler manifold of dimension m, if the holomorphic bisectional curvature is bounded from below by -1 and the minimum spectrum λ1(M)≥m2, then it must either be connected at infinity or diffeomorphic to R×N, where N is a compa…
Sharp heat equation gradient estimates on compact manifolds.
problem Gradient estimates for positive solutions on weighted manifolds.
method Proving sharp gradient estimates for positive solutions to the weighted heat equation.
result Refined gradient estimates and Liouville theorems for ancient solutions.
We prove that the dimension of the space of primitive Vassiliev invariants of degree n grows - as n tends to infinity - faster than Exp(c Sqrt(n)) for any c < Pi Sqrt (2/3). The proof relies on the use of the weight systems coming from the Lie algebra gl(N). In fact, we show that our bound is - up to multiplication wit…
Study Gaussian approximation for deep neural networks with random weights.
problem Understanding the distribution of deep neural networks with random weights.
method Established Gaussian approximation bounds in Wasserstein-1 norm.
result Convergence rates of order n−(1/6)L−1+ε for deep networks with proportional layer widths. In this paper, we classify the class of constant weighted curvature curves in the plane with a log-linear density, or in other words, classify all traveling curved fronts with a constant forcing term in R2. The classification gives some interesting phenomena and consequences including: the family of curves conv…
Consider a weighted or unweighted k-nearest neighbor graph that has been built on n data points drawn randomly according to some density p on R^d. We study the convergence of the shortest path distance in such graphs as the sample size tends to infinity. We prove that for unweighted kNN graphs, this distance converges …
We study geometry of complete Riemannian manifolds endowed with a weighted measure, where the weight function is of quadratic growth. Assuming the associated Bakry-Emery curvature is bounded from below, we derive a new Laplacian comparison theorem and establish various sharp volume upper and lower bounds. We also obtai…
New proof of splitting theorem and finite ends of minimal hypersurfaces in nonnegative curvature manifolds.
problem Proving splitting theorem and finite ends of minimal hypersurfaces in nonnegative curvature manifolds.
method New proof of splitting theorem and construction of weighted minimizing geodesics at infinity.
result Minimal hypersurfaces with finite index in manifolds with nonnegative biRic curvature must have finite ends.
The paper studies gravitational instantons with flat limits and finds elliptic regularity estimates.
problem Analyzing gravitational instantons with flat limits using elliptic analysis.
method Establishing elliptic regularity estimates and showing uniform constants for a family of metrics.
result The Laplacian is Fredholm and an isomorphism between specific weighted spaces.
In this note we identify the leading terms of the (reduced) K-energy map with a universal linear combination of the principal and subdominant coefficients of the weight of the mth Hilbert point. This shows that the weight F1(λ;X) introduced by Donaldson in [SKD02] is just the weight of the CM-polarisation.The eq…
We show that the explicit ALE Ricci-flat Kahler metrics constructed by Eguchi-Hanson, Gibbons-Hawking, Hitchin and Kronheimer, and their free quotients are metrics obtained by Tian-Yau techniques. The proof relies on a construction of good compactifications of Q-Gorenstein deformations of quotient surface singularities…
A novel Bayesian computation method using importance weighting improves numerical stability and performance.
problem Bayesian computation stability and performance issues.
method Nonparametric approach via feature means, importance weighting, and kernel Bayes' rule.
result Importance weighted kernel Bayes' rule yields superior numerical stability and performance.
In this paper we show how techniques coming from stochastic analysis, such as stochastic completeness (in the form of the weak maximum principle at infinity), parabolicity and Lp-Liouville type results for the weighted Laplacian associated to the potential may be used to obtain triviality, rigidity results, and scal…
Study regularity of Schrödinger eigenfunctions with Coulomb-type potentials.
problem Regularity of eigenfunctions for Schrödinger operators with singular potentials.
method Blow-ups of manifolds with corners and Lie manifolds.
result Proves regularity estimates in weighted Sobolev spaces for eigenfunctions.
Develops Hodge theory on ALG∗ manifolds, proving existence and vanishing results.
problem Existence and vanishing of certain cohomology groups on ALG∗ manifolds. method Fredholm Theory for Hodge Laplacian in weighted spaces on ALG∗ manifolds. result Non-existence of ALG∗ manifolds with non-negative Ricci curvature at infinity. Study shows how feature weighting affects neural network regularization.
problem Understanding how feature weighting influences neural network regularization.
method Derived equivalence paths connecting different weighting matrices and ridge regularization levels.
result Ridge estimators trained on weighted features are asymptotically equivalent when evaluated against test vectors.
Combinatorial approach to compute satellite knot invariants using graph theory.
problem Computing knot invariants for satellite knots using bordered Heegaard Floer homology.
method Construct weighted A∞-modules using decorated planar graphs and prove their isomorphism. result Combinatorial proof of A∞ structure relations for the constructed modules. Flow preserves curvature sharpness on weighted graphs.
problem Curvature flow on weighted graphs.
method Adapting Bakry-Émery calculus for Markovian preservation and analyzing limits.
result Flow limits to curvature sharp weighted graphs.
Geometric approach clusters intersecting manifolds with high probability.
problem Clustering intersecting d-dimensional manifolds.
method Compute locality graph on d-simplices using dihedral angles, then compute LAPD to separate manifold components.
result The method separates manifold components with high probability under random sampling.
The paper analyzes SBL pruning criteria under weakened assumptions.
problem Sparse Bayesian learning hyperparameter divergence and pruning.
method Analyzing marginal likelihood function under weakened Gaussian assumptions.
result Conditions for finite vs infinite hyperparameters lead to F-SBL pruning.
A model of random walk on knot diagrams is used to study the Alexander polynomial and the colored Jones polynomial of knots. In this context, the inverse of the Alexander polynomial of a knot plays the role of an Ihara-Selberg zeta function of a directed weighted graph, counting with weights cycles of random walk on a …
Let (Mn,g,e−fdv) be a smooth metric measure space of dimensional n. Suppose that v is a positive weighted p-eigenfunctions associated to the eigenvalues λ1,p on M, namely efdiv(e−f∣∇v∣p−2∇v)=−λ1,pvp−1. in the distribution sense. We first give a local gradient estimat…
Consider a family of portfolio strategies with the aim of achieving the asymptotic growth rate of the best one. The idea behind Cover's universal portfolio is to build a wealth-weighted average which can be viewed as a buy-and-hold portfolio of portfolios. When an optimal portfolio exists, the wealth-weighted average c…
The paper shows how random ReLU networks converge to smooth splines.
problem Understanding the behavior of shallow ReLU neural networks with random weights.
method Mathematical analysis of L2-regularized regression and gradient descent.
result Random ReLU networks converge to smooth splines as the number of hidden nodes increases.
As a model problem for clustering, we consider the densest k-disjoint-clique problem of partitioning a weighted complete graph into k disjoint subgraphs such that the sum of the densities of these subgraphs is maximized. We establish that such subgraphs can be recovered from the solution of a particular semidefinite re…
Study of long-time behavior of solutions on negatively curved manifolds.
problem Long-time behavior of solutions to the Porous Medium Equation on Cartan-Hadamard manifolds with negative curvature.
method Analysis of long-time behavior, proving existence and uniqueness of solutions, using comparison principles.
result Unexpected separate-variable behavior, reminiscent of Dirichlet problems on bounded Euclidean domains.
Wide neural networks become linear, but adding bottlenecks makes them bilinear or multilinear.
problem Understanding the transition of neural networks from linearity to higher-order functions.
method Analyzing the behavior of randomly initialized wide neural networks with and without bottleneck layers.
result Bottleneck layers transform the network's function from linear to bilinear or multilinear.
We develop a solution theory for a generalized electro-magneto static Maxwell system in an exterior domain with anisotropic coefficients converging at infinity with a certain rate towards the identity. Our main goal is to treat right hand side data from some polynomially weighted Sobolev spaces and obtain solutions whi…
The paper proves weighted monotonicity theorems in various spaces and applies them to minimal surfaces.
problem Proving weighted monotonicity theorems in different spaces.
method Proving weighted monotonicity theorems for functions proportional to the metric tensor in Riemannian manifolds.
result Weighted monotonicity theorems in hyperbolic space imply unweighted theorems, leading to bounds on minimal surface areas.
Study connects weighted isoperimetric problems to nonlocal elliptic operator extensions.
problem Sharp inequalities for weighted Poisson integrals and their extremizers.
method Formulates variational problem on conformal metric measure space.
result Sharp inequalities are linked to variational problem on CCE manifolds.
Paper proves Sobolev-Poincaré inequality for CR-manifolds with integrable Q-curvature.
problem Proving Sobolev-Poincaré inequality for CR-manifolds with integrable Q-curvature.
method Analyzes CR-manifolds with contact structure conformal to Heisenberg group, proving volume form is a strong A_infinity weight.
result Proves Sobolev-Poincaré inequality for CR-manifolds with integrable Q-curvature.
Study extends eigenvalue formulas to weighted manifolds and proves global rigidity theorems.
problem Eigenvalue formulas and rigidity theorems for weighted manifolds.
method Extends variational formulae to weighted manifolds, proving global rigidity theorems.
result Global rigidity theorems for critical domains in Gaussian half-space.