Elliptic theory explains indicial weights for non-linear geometry problems.
problem Understanding indicial weights for elliptic operators on non-compact manifolds.
method Developed an elliptic theory for indicial weights, proving Fredholm conditions.
result An elliptic theory exists even when the weight is indicial.
The study computes indicial roots and metric convergence orders for Ricci-flat conifolds.
problem Understanding the convergence behavior of Ricci-flat conifolds.
method Analyzing the Lichnerowicz Laplacian and tensor fields on cones, computing indicial roots and metric convergence orders.
result Lower bounds for metric convergence orders on Ricci-flat conifolds.
We analyze the indicial roots of the self-dual deformation complex on a cylinder (R×Y3,dt2+gY), where Y3 is a space of constant curvature. An application is the optimal decay rate of solutions on a self-dual manifold with cylindrical ends having cross-section Y3. We also resolve a conjectu…
We study the eleven dimensional supergravity equations which describe a low energy approximation to string theories and are related to M-theory under the AdS/CFT correspondence. These equations take the form of a non-linear differential system, on B7×S4 with the characteristic degeneracy at t…
An analytic index is defined for a family of cusp pseudodifferential operators, Pb, on a fibration with fibres which are compact manifolds with boundaries, provided the family is elliptic and has invertible indicial family at the boundary. In fact there is always a perturbation Qb by a family of cusp operators of…
New boundary conditions for knot polynomials solved via elliptic KW equations.
problem Computing Jones polynomial coefficients from knot solutions.
method Formulated generalized Nahm pole boundary conditions for KW equations, proved ellipticity, analyzed polyhomogeneity.
result KW equations with generalized Nahm pole conditions are elliptic and solutions are polyhomogeneous.
Classifies solutions to critical sixth order equations with a singularity.
problem Classifying entire positive singular solutions to critical sixth order equations.
method Integral sliding methods, qualitative analysis of ODEs, topological two-parameter shooting technique.
result Solutions are given by a singular radial factor times a periodic solution to a sixth order IVP with constant coefficients.
The paper introduces a new family of instanton-invariants for four-manifolds and connects them to Khovanov homology.
problem Understanding the relationship between gauge-theoretic instanton invariants and Khovanov homology.
method Develops a one-parameter family of Haydys-Witten instanton Floer homology groups and investigates dimensional reductions of the Haydys-Witten equations.
result Establishes a connection between the new instanton invariants and Khovanov homology, particularly for geometric blow-ups of three-manifolds.
A new weighted MCC measure improves classifier performance evaluation.
problem Lack of measures sensitive to observation weights in multiclass classification.
method Proposes weighted versions of Pearson-Matthews Correlation Coefficient (MCC) for binary and multiclass classification.
result Weighted MCC values are higher for classifiers that perform better on highly weighted observations.
Develops theory of weightings for Lie groupoids and algebroids.
problem Understanding differential geometry of weightings for Lie groupoids and algebroids.
method Extending work on weighted manifolds, defining weighted submanifolds, and developing theories of linear weightings and multiplicative weightings.
result Characterizes infinitesimally multiplicative weightings for Lie algebroids and classifies multiplicative weightings of Lie groupoids.
Stability of weighted extremal manifolds proven through blowups.
problem Stability of weighted extremal manifolds.
method Blowup technique to analyze weighted extremal Kähler manifolds.
result Proves weighted extremal manifolds are relatively weighted K-polystable.
The paper extends spin geometry to weighted manifolds and defines a new mass for Ricci flow.
problem Generalizing spin geometry to weighted manifolds and defining a new mass.
method Investigates spectral properties of the weighted Dirac operator and defines a new mass.
result Defines a new mass for weighted asymptotically Euclidean manifolds and shows its monotonicity under Ricci flow.
Paper generalizes CR Obata theorem to weighted Sasakian manifolds.
problem Deriving eigenvalue estimates for weighted Kohn Laplacian.
method Derived weighted CR Reilly's formula and applied to Sasakian manifolds.
result CR Obata theorem proven for weighted Sasakian manifolds.
New mass and staticity concepts derived from weighted curvature maps.
problem Deriving mass and staticity concepts for weighted manifolds.
method Developed a weighted curvature map and its adjoint, leading to weighted mass and static metrics.
result Equivalence and uniqueness theorems for weighted static manifolds and Penrose inequality.
The study explores weightings on submanifolds and their geometric properties.
problem Understanding weightings on submanifolds and their geometric implications.
method Detailed exploration of weighted normal bundles, weighted deformation spaces, and weighted blow-ups.
result A description of weightings in terms of subbundles of higher tangent bundles, leading to new concepts for Lie algebroids and groupoids.
Proves existence and uniqueness of weighted metrics for smooth spaces.
problem Existence and uniqueness of weighted metrics for smooth metric measure spaces.
method Proves existence and uniqueness using weighted ambient metrics and Poincaré metrics.
result Existence and uniqueness of weighted metrics for smooth metric measure spaces.
Defines and proves properties of weighted renormalized volume coefficients.
problem None explicitly stated; focuses on mathematical definitions and proofs.
method Defines weighted renormalized volume coefficients and proves their variational nature and polynomial representation.
result Weighted renormalized volume coefficients are variational and can be expressed as polynomials of specific tensors.
A new weighted FDA method improves face recognition accuracy.
problem Equal treatment of all class pairs in FDA leads to suboptimal performance.
method Cosine-weighted and automatically weighted FDA methods are proposed.
result Improved face recognition accuracy through weighted FDA.
New invariants help solve existence of weighted cscK metrics.
problem Existence of weighted cscK metrics in K-stability.
method Introduced weighted analytic delta invariant and beta invariant.
result Sufficient condition for existence of weighted cscK metrics.
Proves positive mass theorem for non-spin weighted manifolds.
problem Proving the positive mass theorem for non-spin weighted manifolds.
method Establishing density theorem and generalizing Geroch conjecture.
result Proves positive weighted mass theorem for non-spin weighted manifolds.
Derives integral formulae on weighted manifolds.
problem No specific problem stated; focuses on mathematical derivations.
method Introduces weighted mean sigma-r curvature and uses weighted Newton transformations.
result Derives integral formulae generalizing previous work.
Method measures weight similarity in neural networks using normalization and statistical inference.
problem Quantifying weight similarity in non-convex neural networks.
method Chain normalization rule and hypothesis-training-testing statistical inference.
result Weights of identical neural networks converge to similar local solutions.
The paper studies weighted Ricci curvatures and characterizes Randers metrics.
problem Characterizing Randers metrics with weighted Ricci curvatures.
method General weighted Ricci curvatures and characterization of Randers metrics.
result Characterization of Randers metrics with almost isotropic weighted Ricci curvatures.
Study on stable minimal hypersurfaces under Ricci curvature constraints.
problem Stability of weighted minimal hypersurfaces under Ricci curvature bounds.
method Derive geometric consequences and prove a Schoen-Yau type criterion.
result Structure theorem for three-dimensional weighted manifolds of non-negative Ricci curvature.
The paper classifies vertices in weighted networks using spectral embedding and edge weight distributions.
problem Classifying vertices in weighted networks where edge weights and adjacencies encode class membership.
method Introduced a edge weight distribution matrix to the K-Block Stochastic Block Model for weighted networks. Developed classification procedures based on spectral embedding of the unweighted adjacency matrix under two assumptions on edge weight distributions.
result Proposed classifiers outperform quadratic discriminant analysis on transformed weighted networks.
Explains weightings along submanifolds, focusing on Lie groupoids.
problem None explicitly stated; focuses on theory review.
method Reviews basic notions and emphasizes multiplicative weightings.
result Provides a comprehensive overview of weightings along submanifolds.
The study establishes comparison theorems for weighted Finsler manifolds and spacetimes.
problem Analyzing weighted Finsler manifolds and spacetimes with curvature conditions.
method Using weight function and ε-range, the Bonnet-Myers theorem, Laplacian comparison theorem, and Bishop-Gromov volume comparison theorem are formulated. result New comparison theorems for weighted Finsler manifolds and spacetimes are derived, including those for weighted Riemannian manifolds.
A new method compresses deep neural networks by predicting and quantizing weights between layers.
problem Resource constraints in deep neural networks.
method Inter-Layer Weight Prediction (ILWP) and quantization based on Smoothly Varying Weight Hypothesis (SVWH).
result The method achieves higher weight compression rates at the same accuracy level.
The paper generalizes K-stability results to singular and weighted settings.
problem Generalizing K-stability to singular and weighted settings.
method Generalization of results in \cite{Li22a} to singular and weighted settings.
result The \(\mathbb{G}\)-uniform weighted K-stability for models implies \(\mathbb{G}\)-coercivity of the weighted Mabuchi functional.
The study analyzes weighted manifolds with curvature bounds, proving eigenvalue estimates and inequalities.
problem Analyzing geometric properties of weighted manifolds under Ricci curvature bounds.
method Develops geometric analysis techniques on weighted Riemannian manifolds with lower 0-weighted Ricci curvature bounds. result Proves eigenvalue estimates for Steklov and ABP inequalities on weighted manifolds.
Adaptive learning of sample weights for better model performance.
problem Overfitting to biased training data with corrupted labels or class imbalance.
method Adaptive learning of an explicit weighting function using a meta-weight-net.
result Improves model accuracy in class imbalance and noisy label cases.
Study on deformation of weighted scalar curvature, proving geometric results and stability.
problem Deformation of weighted scalar curvature and related geometric properties.
method Linearization of weighted scalar curvature, studying kernel of formal adjoint.
result Definition and study of weighted vacuum static spaces, stability results on flat spaces.
This paper reviews weighted clustering ensemble methods.
problem Improving clustering results from individual methods.
method Different types of weights and approaches to determining weight values.
result Unified framework for selecting appropriate weighting mechanisms.
New algorithms avoid weight transport, outperforming current deep learning methods.
problem Current deep learning algorithms rely on weight transport, which is biologically implausible.
method Two mechanisms: weight mirror and modified Kolen-Pollack algorithm, using random feedback weights.
result These mechanisms outperform feedback alignment and other methods on visual recognition tasks.
Reverse-weighted portfolios outperform in commodity futures markets.
problem Efficiency of commodity futures markets.
method Permutation-weighted portfolios, rank-based methods.
result Reverse-weighted portfolio outperforms price-weighted portfolio.
The Penrose theorem and Hawking's topology theorem are extended to weighted spacetimes.
problem Extending Penrose's singularity theorem and Hawking's topology theorem to weighted spacetimes.
method Using weighted null energy condition and synthetic dimension to generalize the theorems.
result Generalized versions of the Penrose and Hawking theorems hold under a weighted null energy condition.
A new method to improve deep neural networks using weight rescaling.
problem Overfitting and sensitivity to hyperparameters in weight decay.
method Weight rescaling (WRS) to control weight norm and prevent overfitting.
result WRS outperforms weight decay and other methods in various applications.
Paper extends trigonometric summation formula with weights.
problem Trigonometric summation formula by Grigor'yan, Lin and Yau.
method Weighted trigonometric summation formula derivation.
result Extension of trigonometric summation formula.
A new method trains deep networks by separating weight locations from values.
problem Training deep networks efficiently and effectively.
method Lookahead Permutation (LaPerm) to train DNNs by reconnecting weights.
result LaPerm can train DNNs with random and dense, sparse, or single-valued initial weights.
The paper predicts edge weights in weighted directed networks using metric geometry.
problem Predicting edge weights in weighted directed networks.
method Introducing new types of weighted directed networks (AWDNs), constructing metrics, and proposing modified kNN and SVM methods.
result The proposed methods outperform traditional approaches in predicting edge weights.
The paper compares isoperimetric quotients and capacities in weighted manifolds.
problem Comparing isoperimetric quotients and capacities in weighted manifolds.
method Analysis of weighted Laplacian of the distance function and techniques for non-compact submanifolds.
result Parabolicity and hyperbolicity criteria for weighted manifolds.
Optimal weight windows are found by projecting the origin onto a convex polytope.
problem Finding the best weight windows for a weighted moving average smoother.
method Formulated as a quadratic program and projection onto a convex polytope.
result Optimal weight windows are symmetrical and decrease in weight away from the center.
Optimizes weights for better model performance in shifting data.
problem Improper importance weighting leads to poor model performance in data shifts.
method Interprets weights as a bias-variance trade-off and optimizes them simultaneously with model parameters.
result Optimizing weights significantly improves model generalization performance.
Study of weighted nonlinear flags in symplectic geometry.
problem Understanding the geometry of weighted nonlinear flags.
method Generalizing weighted nonlinear Grassmannians to Frechet manifolds and using them to describe coadjoint orbits.
result Description of coadjoint orbits of Hamiltonian diffeomorphisms using weighted isotropic nonlinear flags.
The CR Obata theorem is extended to weighted Sasakian manifolds.
problem Extending the CR Obata theorem to weighted Sasakian manifolds.
method Deriving the weighted CR Reilly's formula and first eigenvalue estimate for a weighted sub-Laplacian.
result The CR Obata theorem is proven in compact weighted Sasakian manifolds.
Study on convergence rate of weighted Yamabe flow.
problem Weighted Yamabe problem on smooth metric measure spaces.
method Weighted Yamabe flow and its convergence rate analysis.
result Study and analysis of convergence rate of the weighted Yamabe flow.
Boosts neural network performance by improving weight separability.
problem Improving the separability of weight vectors in neural networks.
method Proposes a new evaluation metric and feed-backward reconstruction loss to encourage weight separability.
result Improves visual recognition performance across various tasks.
Estimates essential spectrum of weighted Laplacian on noncompact manifolds.
problem Finding bounds for essential spectrum of weighted Laplacian.
method Volume growth of geodesic balls and spheres, examples of equality.
result Upper estimates for the bottom of essential spectrum.