New approach confirms Kruskal-Szekeres extension for Schwarzschild spacetime.
problem Confirming the Kruskal-Szekeres extension for Schwarzschild spacetime.
method Reformulating the problem as an ODE and showing the ODE admits a solution if and only if the horizon is non-degenerate.
result Photon surfaces approaching the Killing horizon must necessarily cross it.
Constructs constant mean curvature foliations in Schwarzschild spacetime.
problem Finding constant mean curvature foliations in Schwarzschild spacetime.
method Constructs T-axisymmetric, spacelike, spherically symmetric, constant mean curvature hypersurfaces. result Mean curvature varies in each slice and ranges from minus infinity to plus infinity.
We construct a Kruskal-Szekeres-type analytic extension of the Emparan-Reall black ring, and investigate its geometry. We prove that the extension is maximal, globally hyperbolic, and unique within a natural class of extensions. The key to those results is the proof that causal geodesics are either complete, or approac…
We solve spacelike spherically symmetric constant mean curvature (SS-CMC) hypersurfaces in Schwarzschild spacetimes and analyze their asymptotic behavior near the coordinate singularity r = 2M. Furthermore, we join SS-CMC hypersurfaces in the Kruskal extension to obtain complete ones and discuss the smooth properties.
Improved algorithm for multidimensional scaling reduces stress.
problem Stress in multidimensional scaling.
method Proposed modifications of the smacof algorithm.
result Convergent majorization algorithm for Kruskal's stress formula two.
Proves special foliation property of SS-CMC hypersurfaces in Schwarzschild spacetime.
problem Characterizing and foliating SS-CMC hypersurfaces in Schwarzschild spacetime.
method Characterization and proof of foliation property using SS-CMC hypersurfaces.
result Verifies part of Malec and Ó Murchadha's conjecture.
Enhances tensor regression for interpretability and performance.
problem Interpreting and modeling multidimensional tensor data with structural heterogeneity.
method Generalized Nonnegative Structured Kruskal Tensor Regression (NS-KTR) with hybrid regularization and nonnegativity constraints.
result NS-KTR outperforms conventional methods in synthetic and real hyperspectral datasets.
Study directed completion of spacetimes, focusing on Schwarzschild spacetime.
problem Characterizing directed completions of spacetimes.
method Directed completion of Lorentzian pre-length spaces, focusing on Schwarzschild spacetime.
result Directed completion of Schwarzschild spacetime coincides with future causal completion.
A new tensor-factor analysis model improves image denoising and classification.
problem Improving image denoising and classification performance.
method Introduces a deep convolutional tensor-factor analysis model for multi-way data.
result Improves PSNR by over 1dB in multi-way denoising and image classification.
Paper identifies latent factors from noisy measurements using tensor decomposition.
problem Identification of latent factors from noisy, correlated measurements.
method Tensor decomposition of third order cross moments, Kruskal theorem, Kotlarski identity, generalized Kruskal rank.
result Full distribution of latent factors and measurement errors identified without injective measurements.
By considering suitable axially symmetric slices on the Kruskal spacetime, we construct counterexamples to a recent version of the Penrose inequality in terms of so-called generalized apparent horizons.
New CSC model extracts EEG signals with low noise sensitivity.
problem Analyzing noisy EEG signals during anesthesia.
method Kruskal CSC model using Kruskal decomposition for low-rank tensor activations.
result TC-FISTA efficiently extracts robust, sparse, and interpretable EEG encodings.
New model encodes multivariate signals more efficiently with sparsity and low-rank constraints.
problem Efficiently encoding multivariate signals with sparsity and low-rank constraints.
method Multivariate convolutional sparse coding with tensor algebra, CP decomposition, and alternating optimization.
result Proves model closely related to Kruskal tensor regression problem with theoretical guarantees.
New tensor decomposition method handles more noise and higher orders.
problem Efficient tensor decomposition for noisy data.
method Two-mode higher-order SVD (HOSVD) with Kruskal's theorem.
result Proves higher noise tolerance and high accuracy.
Study on identifiability of deep polynomial neural networks.
problem Understanding when polynomial neural networks can be uniquely identified.
method Comprehensive analysis including various architectures, using tensor decompositions and Kruskal-type theorems.
result Identifiability conditions for deep PNNs, including layer width and activation degree constraints.
Study shows nonextendibility of warped spacelike singularities in specific spacetimes.
problem Nonextendibility of warped spacelike singularities in specific spacetimes.
method Establishes a local obstruction through integrability conditions and radial compression.
result Imply C0-inextendibility for the one-horizon Birmingham-Kottler family. Study examines C0-inextendibility of spacetimes, focusing on FLRW models.
problem Examining the extendibility of spacetimes, particularly focusing on FLRW models.
method Reviewing Sbierski's methodology and applying similar techniques to new observations.
result Certain open FLRW spacetimes admit C0 extensions through the big bang. Machine learning models predict crash rates on narrow lanes.
problem Impact of narrow lanes on arterial road vehicle crashes.
method Applied random forest and least squares boosting machine learning algorithms to crash data.
result Random forest model identified as best for studying narrow lanes' safety impact.
The geometry of five-dimensional Kerr black holes is discussed based on geodesics and Weyl curvatures. Kerr-Star space, Star-Kerr space and Kruskal space are naturally introduced by using special null geodesics. We show that the geodesics of AdS Kerr black hole are integrable, which generalizes the result of Frolov and…
Study measures uncertainty in MST identification across different correlation networks.
problem Uncertainty in MST identification across various correlation-based market networks.
method Developed a framework using random variable networks (RVN) to measure uncertainty of MST identification.
result FDR is the most appropriate measure for MST identification reliability.
When response variables are nominal and populations are cross-classified with respect to multiple polytomies, questions often arise about the degree of association of the responses with explanatory variables. When populations are known, we introduce a nominal association vector and matrix to evaluate the dependence of …
The paper characterizes chordal graphs via edge deletions and finds a local minimum spanning tree algorithm.
problem Characterizing chordal graphs and finding efficient minimum spanning trees.
method Focus on exposed edges, characterize chordal graphs via deletions, and use local properties to modify Kruskal's algorithm.
result A modified Kruskal's algorithm for weighted chordal graphs is local and efficient.
This paper is an expository account of the development of soliton mathematics, from its inception in famous numerical experiments of Fermi-Pasta-Ulam and Zabusky-Kruskal to the recent synthesis of Terng-Uhlenbeck (dg-ga/9707004) that explains hidden symmetries of soliton equations in terms of loop-groups acting by dres…
We study graph estimation and density estimation in high dimensions, using a family of density estimators based on forest structured undirected graphical models. For density estimation, we do not assume the true distribution corresponds to a forest; rather, we form kernel density estimates of the bivariate and univaria…
Framework for nonparametric graph estimation with prior info.
problem Estimating graphical models without distributional assumptions.
method Bayesian approach to forest density estimation, incorporating prior distributions.
result Proposed methods outperform parametric methods and improve predictive power.
This paper identifies and estimates the label noise transition matrix without ground truth labels.
problem Learning with noisy labels and identifying the noise transition matrix.
method Building on Kruskal's identifiability results, the paper characterizes the identifiability of the label noise transition matrix for the generic case at the instance level.
result The necessity of multiple noisy labels in identifying the noise transition matrix for the generic case at the instance level.
New tensor-based method for estimating stock correlation matrices.
problem Choosing a proper sample period for estimating correlation matrices.
method Slice-Diagonal Tensor (SDT) factorization technique.
result The new method produces a stable correlation matrix unaffected by the sample period.
This paper corrects ReLU attribution and compares activation functions in deep learning.
problem Historical misattribution and performance comparison of activation functions.
method Historical tracing and empirical comparison of ReLU, Tanh, and Sigmoid across tasks.
result ReLU outperforms Sigmoid and Tanh in deep learning tasks.
Continuous MDS embeds sequences of dissimilarities in Euclidean space.
problem Embedding sequences of dissimilarities as n increases. method Continuous MDS reformulates MDS for sequences of dissimilarity matrices.
result Uniform convergence of interpolated embeddings.
Study on Ricci flow on modified Riemann extensions, finding conditions for their preservation.
problem Properties of modified Riemann extensions under Ricci flow.
method Analysis of necessary and sufficient conditions for modified Riemann extensions to remain as such under Ricci flow.
result Obtained conditions for modified Riemann extensions to stay as modified Riemann extensions under Ricci flow.
Classifies central extensions of Hamiltonian vector fields Lie algebra.
problem Classifying central extensions of Lie algebras.
method Determined universal central extension of Hamiltonian vector fields Lie algebra.
result Classified central extensions of various Lie algebras.
Extension formulae on almost complex manifolds studied with applications.
problem Understanding almost complex manifolds through extension formulae.
method Provided extension formulae and decompositions for almost complex manifolds.
result Studied (n,0)-forms, (n,0)-Dolbeault cohomology group, and (n,q)-forms. The paper proves extension theorems for holomorphic sections from divisors.
problem Extension of holomorphic sections from reduced unions of strata of divisors.
method Proves an Ohsawa--Takegoshi type extension theorem.
result Qualitative results on extension from snc divisors and generic global generation of vector bundles.
Generalizes Nielsen equivalence theorem to hyperbolic group extensions.
problem Tackles Nielsen equivalence in hyperbolic group extensions.
method Generalizes a theorem by Juan Souto to a broader class of hyperbolic extensions.
result Includes all hyperbolic extensions of surfaces groups and free groups by Out$(F_n).
The paper connects group extensions, cochains, and spectral sequences.
problem Understanding the relationship between group extensions and spectral sequences.
method Using connection cochains, the paper derives a formula for the extension class.
result A formula clarifies the relation among connection cochains, extension classes, and the LHS spectral sequence.
Proves HNN extensions of nilpotent groups are left-orderable, constructs non-left-orderable examples.
problem Characterizing left-orderability in HNN extensions of groups.
method Analyzes HNN extensions of torsion-free nilpotent groups and left-orderable groups.
result Constructs examples of non-left-orderable HNN extensions of left-orderable groups.
Review of central extensions for symplectic and divergence-free vector fields.
problem Comparing central extensions of symplectic and divergence-free vector fields.
method Analyzing universal central extensions and integrability.
result Similar results for both types of vector fields.
Paper analyzes mathematical theory behind out-of-sample DR extensions.
problem Developing a solid mathematical foundation for out-of-sample DR extensions.
method Utilizes RKHS theory to treat DR extension as an extension of the identity on RKHS defined on X.
result Shows Nyström-type DR extension as an orthogonal projection and provides conditions for exact DR extension.
Examines differential smoothness in a specific skew PBW extension family.
problem Differential smoothness in skew PBW extensions.
method Investigates a specific family of skew PBW extensions.
result Results on differential smoothness of the family.
New insights into identifying mixtures of product distributions using Hadamard extensions.
problem Identifying mixtures of product distributions on binary variables.
method Analysis of Hadamard extensions of matrix products.
result Conditions for full column rank of Hadamard extensions.
A spacetime can be embedded in an enveloping space with all its extensions.
problem Existence and uniqueness of C0-maximal extensions in globally hyperbolic conformally flat spacetimes.
method Proving conformal embedding into an enveloping space containing all extensions.
result Existence and uniqueness of C0-maximal extensions proven.
We generalize the prequantization central extension of a group of diffeomorphisms preserving a closed 2-form ω(ω-invariant diffeomorphisms) to an abelian extension of a group of diffeomorphisms preserving a closed vector valued 2-form ω, up to a linear isomorphism (ω-equivariant diffeomorphisms). Every abelian extensio…
We give a new variant of L2-extension theorem for the jets of holomorphic sections and discuss the relation between the extension problem of singular Hermitian metrics with semipositive curvature.
Study on flux homomorphism and its extension in symplectic group of a disk.
problem Understanding the flux homomorphism and its extension in symplectic group.
method Defined and analyzed the flux homomorphism and its extension, determined the Euler class, and investigated its relation to group 2-cocycle and Calabi invariant.
result Determined the Euler class of the flux extension and investigated its relation to group 2-cocycle and Calabi invariant.
The purpose of this paper is to show how central extensions of (possibly infinite-dimensional) Lie algebras integrate to central extensions of étale Lie 2-groups. In finite dimensions, central extensions of Lie algebras integrate to central extensions of Lie groups, a fact which is due to the vanishing of π_2 for each …
Analytic linearization and holomorphic extensions for proper groupoids.
problem Analytic linearization and holomorphic extensions of proper groupoids.
method Establish analytic linearization around invariant submanifolds and apply to holomorphic extensions.
result Proper groupoids admit holomorphic extensions.
Geometrically constructs central extensions of Poisson Lie algebra.
problem Integrability of central extensions of Poisson Lie algebra.
method Prequantization and geometric construction of central S^1-extensions.
result Explicit description of integrable cocycles and nontrivial central extensions.
The paper examines differential smoothness in skew PBW extensions over polynomial rings.
problem Differential smoothness in skew PBW extensions over polynomial rings.
method Investigation of skew PBW extensions over commutative polynomial rings.
result Results on differential smoothness for skew PBW extensions over polynomial rings.