Study Dirac operators on incomplete cusp edge spaces, proving self-adjointness and Fredholm properties.
problem Analyzing Dirac operators on complex geometric spaces.
method Construct heat kernel, prove self-adjointness and Fredholm properties, establish index formula.
result Proved Dirac operators are essentially self-adjoint and Fredholm.
Study on eta and rho invariants on incomplete edge spaces.
problem Existence and properties of eta and rho invariants on incomplete edge spaces.
method Microlocal analysis of heat kernel asymptotics and Atiyah-Patodi-Singer arguments.
result Derivation of an Atiyah-Patodi-Singer index theorem for incomplete edge spaces.
The paper proves local and long-term existence of Ricci de Turck flow on incomplete edge manifolds.
problem Proving existence of Ricci de Turck flow on incomplete edge manifolds.
method Careful analysis of the Lichnerowicz Laplacian and the Ricci de Turck flow equation.
result Local and long-term existence of Ricci de Turck flow on incomplete edge manifolds.
We derive a formula for the index of a Dirac operator on a compact, even-dimensional incomplete edge space satisfying a "geometric Witt condition". We accomplish this by cutting off to a smooth manifold with boundary, applying the Atiyah-Patodi-Singer index theorem, and taking a limit. We deduce corollaries related to …
The article proves long-time existence and convergence of the edge Yamabe flow.
problem Analyzing the normalized Yamabe flow on incomplete edge singularities.
method Novel maximum principle results and uniform bounds established without barrier functions or Krylov-Safonov estimates.
result Long-time existence and convergence of the edge Yamabe flow.
Incomplete cusp edges model the behavior of the Weil-Petersson metric on the compactified Riemann moduli space near the interior of a divisor. Assuming such a space is Witt, we construct a fundamental solution to the heat equation, and using a precise description of its asymptotic behavior at the singular set, we prove…
We consider the heat operator acting on differential forms on spaces with complete and incomplete edge metrics. In the latter case we study the heat operator of the Hodge Laplacian with algebraic boundary conditions at the edge singularity. We establish the mapping properties of the heat operator, recovering and extend…
Let (M,g) be a compact oriented Riemannian manifold with an incomplete edge singularity. This article shows that it is possible to evolve g by the Yamabe flow within a class of singular edge metrics. As the main analytic step we establish parabolic Schauder-type estimates for the heat operator on certain Hölder spaces …
Let (X,g) be a compact Riemannian stratified space with simple edge singularity. Thus a neighbourhood of the singular stratum is a bundle of truncated cones over a lower dimensional compact smooth manifold. We calculate the various polynomially weighted de Rham cohomology spaces of X, as well as the associated spac…
We construct the biharmonic heat kernel for a suitable self-adjoint extension of the bi-Laplacian on a manifold with incomplete edge singularities. We employ a microlocal description of the biharmonic heat kernel to establish mapping properties of the corresponding biharmonic heat operator on certain Banach spaces. Thi…
Framework predicts logical queries on incomplete knowledge graphs.
problem Complex logical queries involving multiple unobserved edges and entities.
method Low-dimensional embeddings and learned geometric operations.
result Efficient predictions with linear time complexity in query variables.
We consider the Hodge Laplacian on manifolds with incomplete edge singularities, with infinite dimensional von Neumann spaces and intricate elliptic boundary value theory. We single out a class of its algebraic self-adjoint extensions. Our microlocal heat kernel construction for algebraic boundary conditions is guided …
GRCN improves GCNs by predicting missing edges and revising weights.
problem Sub-optimal solutions due to incomplete and noisy real-world graphs.
method Introduces a GCN-based graph revision module for predicting missing edges and revising weights.
result GRCN consistently outperforms strong baseline methods, especially on incomplete graphs.
Defines odd Pfaffian form for odd-dimensional manifolds, proving Chern-Gauss-Bonnet formula.
problem Proving Chern-Gauss-Bonnet formula for incomplete edge singularities.
method Defining odd Pfaffian form through curvature tensor, proving formula for various metrics.
result Proves intrinsic Chern-Gauss-Bonnet formula for edge singularities and complete manifolds.
Study shows long-term flow on special manifolds with positive Yamabe constant.
problem Analyzing long-time behavior of Yamabe flow on singular spaces.
method Formulated axioms for long-time existence, used parabolic Moser iteration for bounds.
result Established long-time existence of normalized Yamabe flow on specified manifolds.
Defines products for fibered corners manifolds, generalizing resolutions.
problem Resolving fibered corners manifolds.
method Introduces a category of fibered corners manifolds with products and transverse fiber products, defining the 'ordered product' for wedge metrics.
result The 'ordered product' is a natural product for wedge metrics.
Algorithm reconstructs spreading model parameters from incomplete data.
problem Reconstructing unknown transmission probabilities from partial observation data.
method Dynamic message-passing algorithm for incomplete spreading data.
result Efficient algorithm reconstructs parameters of spreading models.
Improved prediction accuracy in matrix factorization using graph-based priors.
problem Graph side-information may not align with latent-feature relations in matrix completion.
method Identify and remove 'contested' edges using graphical lasso approximation, maintaining linear scalability.
result Improved prediction accuracy with fewer graph edges, demonstrating the often inaccurate nature of graph side-information.
Bayesian method infers transition matrices from incomplete graph data with topological constraints.
problem Inference of transition matrices from incomplete graph data with topological constraints.
method Bayesian approach using repeated interactions and a topological prior.
result Higher accuracy in inferring transition probabilities, improving downstream tasks.
Proposes learning graph structure and GCN parameters together.
problem Inability of GNNs to use graphs that are incomplete or corrupted.
method Solves a bilevel program to learn a discrete graph structure.
result Outperforms related methods in experiments.
Incomplete pressure metric on bordered surface Teichmüller space proved.
problem Incompleteness of pressure metric on Teichmüller space of bordered surfaces.
method Using moduli space of metric graphs and fat graphs associated to bordered surfaces.
result Pressure metric is not a constant multiple of Weil-Petersson metric.
Proves limit curve theorem for incomplete metric spaces, applies to null distance in Lorentzian manifolds.
problem Control of Lorentzian lengths of limit curves in incomplete metric spaces.
method Proves limit curve theorem for incomplete metric spaces and applies to null distance.
result Strong control on Lorentzian lengths of limit curves in Sormani and Vegas' null distance.
Paper explores exact recovery of communities in weighted graphs using Gaussian and exponential distributions.
problem Exact recovery of communities in weighted graphs with Gaussian and exponential distributions.
method Introduces a new semi-metric to describe conditions for exact recovery and analyzes conditions for both complete and incomplete graphs.
result Necessary and sufficient conditions for exact recovery are asymptotically tight and applicable to both complete and incomplete graphs.
A fast framework for image deconvolution with incomplete observations.
problem Deconvolution with unknown boundaries is slow and artifacts-prone.
method Diagonalization of convolution operators, iterative pixel estimation and deconvolution.
result Framework allows fast deconvolution with unknown boundaries.
Formula derived for G2-manifolds, showing moduli spaces are incomplete.
problem Incompleteness of moduli spaces for G2-manifolds. method Derived a formula for the energy of paths in moduli spaces, provided conditions for finite energy and length.
result Compact G2-manifolds produced by the generalised Kummer construction have incomplete moduli spaces. The paper proves pseudolocality theorems for Ricci flows on incomplete manifolds.
problem Pseudolocality of Ricci flows on incomplete manifolds.
method Proves pseudolocality theorems for Ricci flows under specific curvature and isoperimetric conditions.
result Constructs solutions of Ricci flow in balls with pseudolocality property.
Extends cohomology to incomplete Riemannian manifolds.
problem Cohomology of harmonic forms on incomplete Riemannian manifolds.
method Abstract setting of Hilbert complexes.
result Geometric applications to Thom-Mather spaces.
Tian initiated the study of incomplete Kähler-Einstein metrics on quasi-projective varieties with cone-edge type singularities along a divisor, described by the cone-angle 2π(1−α) for α∈(0,1). In this paper we study how the existence of such Kähler-Einstein metrics depends on α. We show that in the negative s…
HSACC improves multi-view clustering of incomplete data.
problem Challenges in clustering incomplete multi-view data.
method Hierarchical Semantic Alignment and Cooperative Completion framework.
result HSACC outperforms state-of-the-art methods on benchmark datasets.
We generalise work of Young-Eun Choi to the setting of ideal triangulations with vertex links of arbitrary genus, showing that the set of all (possibly incomplete) hyperbolic cone-manifold structures realised by positively oriented hyperbolic ideal tetrahedra on a given topological ideal triangulation and with prescrib…
The paper studies cohomology on incomplete manifolds and stratified spaces.
problem Analyzing cohomology groups on incomplete Riemannian manifolds and stratified spaces.
method Proves injective/surjective maps between Lp and L2 cohomology groups under certain conditions. result Injective/surjective maps between Lp and L2 cohomology groups are established. A new method for state estimation in state-space models using incomplete data.
problem State estimation in nonlinear state-space models with incomplete observations.
method Statistical analysis of incomplete observations, score function, observed information matrices, EM-gradient-particle filtering.
result Maximum likelihood estimation of state-vector with explicit form of observed information matrix.
Latent diffusion improves robustness in missing data imputation.
problem Missing data imputation under MCAR corruption.
method Two-stage framework: VAE for latent feature learning, diffusion model in latent space.
result Latent diffusion maintains high quality and stability up to 50% missingness.
Gradient descent near stability threshold exhibits sharpness oscillations.
problem Understanding sharpness behavior near stability threshold in non-Euclidean norms.
method Interpreted EoS through Directional Smoothness and generalized sharpness under arbitrary norms.
result Non-Euclidean GD with generalized sharpness shows sharpness oscillations near 2/η. New risk measures for incomplete markets without lattice structures.
problem Risk measures on incomplete markets without lattice structures.
method Study of risk measures without lattice structures, focusing on tractable dual representations and solid superspaces.
result Existence of a tractable dual representation equivalent to a Fatou-like property, and extension theorems under certain conditions.
Study dualities of geometric invariants on cuspidal edges in hyperbolic and de Sitter spaces.
problem Computing and understanding dualities of geometric invariants on cuspidal edges.
method Analyzing differential geometric invariants of cuspidal edges in hyperbolic and de Sitter spaces.
result Identified dualities of invariants on cuspidal edges.
Study special Lagrangian submanifolds with edge singularities using elliptic theory.
problem Characterize the moduli space of deformations of special Lagrangian submanifolds with edge singularities.
method Use elliptic theory for edge-degenerate differential operators on singular manifolds.
result Obtain a general theorem describing the local structure of the moduli space.
Along cuspidal edge singularities on a given surface in Euclidean 3-space, which can be parametrized by a regular space curve, a unit normal vector field ν is well-defined as a smooth vector field of the surface. A cuspidal edge singular point is called generic if the osculating plane of the cuspidal edge (as a regul…
In L^3, cuspidal edges can have bounded mean curvature under specific conditions.
problem Understanding cuspidal edges with bounded mean curvature in Lorentz-Minkowski 3-space.
method Investigated cuspidal edges and generalized cuspidal edges, analyzing their singular points and principal curvatures.
result Cuspidal edges with bounded mean curvature in L^3 occur only when the singular set is a light-like curve.
Study on functional inequalities on simple edge spaces.
problem Whether classical functional inequalities hold in simple edge spaces.
method Analyzing Sobolev and Poincaré inequalities, proving optimality of Sobolev constant.
result Optimality result concerning the B-constant of the Sobolev inequality.
Study on convex surfaces in Minkowski space, proving completeness and incompleteness conditions.
problem Characterizing isometric embeddings of hyperbolic plane in Minkowski 3-space.
method Analysis of null support function and curvature conditions.
result Conditions for completeness and incompleteness of convex surfaces.
Kernel clustering algorithm improved for large datasets using incomplete Cholesky factorization.
problem Large memory usage in kernel-based clustering for large-scale datasets.
method Approximate the kernel matrix using incomplete Cholesky factorization and apply linear k-means clustering. result The proposed method achieves similar performance to kernel k-means clustering but handles large-scale datasets efficiently. Researchers develop multi-utility representations for incomplete preferences linked to risk measures.
problem Handling incomplete preferences induced by set-valued risk measures.
method Established dual representations of set-valued risk measures to create parsimonious and well-behaved multi-utility representations.
result Unified dual representations of set-valued risk measures, linking them to scalar risk measures.
Study finds optimal investment strategies in incomplete markets.
problem Optimal investment in non-concave, non-smooth utility functions.
method Dynamic programming and measurable selection arguments.
result Existence of optimal portfolio in genericly incomplete markets.
New method clusters strong and weak views effectively, improving performance by up to 40%.
problem Clustering incomplete multi-view data with unbalanced incompleteness.
method View evolution scheme and weighted multi-view subspace clustering.
result Improves clustering performance by up to 40% on three metrics.
Holomorphic foliations found in ball space with unique properties.
problem Finding holomorphic foliations in the ball space.
method Proving existence of nonsingular holomorphic foliations by closed complex hypersurfaces.
result First example of a holomorphic foliation with complete and incomplete leaves.
Optimal hedging strategy found in markets with incomplete pricing kernels.
problem Finding optimal hedging in markets with incomplete pricing kernels.
method Demonstrated existence of an optimal hedge portfolio using an expected least squared-error criterion.
result Existence of an optimal hedge portfolio in Lévy-Ito markets.
Study on planar graph braid groups' second homology.
problem Characterize the second homology of planar graph braid groups.
method Analyzing configuration spaces of planar graphs under specific operations.
result The second homology is generated by three specific graphs.