Geodesic currents in strongly hyperbolic spaces are dense.
problem Characterizing geodesic currents with strongly hyperbolic dual pseudometrics.
method Combining finite-cover argument and boundary data characterization.
result Dense subset of geodesic currents with strongly hyperbolic dual pseudometrics.
Let D− and D+ be properly immersed closed locally convex subsets of a Riemannian manifold with pinched negative sectional curvature. Using mixing properties of the geodesic flow, we give an asymptotic formula as t→+∞ for the number of common perpendiculars of length at most t from D− to D+, count…
Enhanced Bishop-Gromov theorem for homogeneous and inhomogeneous spaces.
problem Bounding the volume growth of geodesic balls in spaces.
method Introducing coefficient shuffling and using the Raychaudhuri equation, geodesic flow conservation, and the full spectrum of Ricci curvature.
result Upper bounds on the average rate of growth of geodesics for finite-volume inhomogeneous spaces.
Researchers analyze the structure of token spaces in large language models.
problem Understanding the geometric and topological structure of token spaces in large language models.
method Developed estimators for the dimension and Ricci scalar curvature of the token subspace and applied them to three large language models.
result The token subspace is not a manifold but a stratified manifold with significantly negative Ricci curvature.
New manifolds with negative curvature limit to one with negative curvature.
problem Preserving nonnegative scalar curvature under intrinsic flat convergence.
method Constructing sequences of manifolds with positive scalar curvature.
result Intrinsic flat limit of manifolds with negative scalar curvature.
Construct Kähler metrics with negative holomorphic bisectional curvature on compact relative Kähler fibrations.
problem Constructing Kähler metrics with negative holomorphic bisectional curvature on compact relative Kähler fibrations.
method Constructing Kähler metrics with negative holomorphic bisectional curvature on compact relative Kähler fibrations using specific curvature conditions.
result Explicit construction of Kähler metrics with negative holomorphic bisectional curvature on compact relative Kähler fibrations.
New metrics found without topological restrictions.
problem Finding metrics with constant negative scalar-Weyl curvature.
method Extended Aubin's construction to prove existence.
result Every manifold admits a metric with constant negative scalar-Weyl curvature.
New compact K-E manifolds with negative curvature found.
problem Constructing compact Kähler-Einstein manifolds with negative curvature.
method Created compact Kähler-Einstein manifolds of dimension n with negative sectional curvature.
result Found compact Kähler-Einstein manifolds of negative curvature not covered by the ball.
The paper extends a theorem about Kähler manifolds with quasi-negative curvature to almost quasi-negative curvature.
problem Understanding the ampleness of canonical line bundles for Kähler manifolds with specific curvature properties.
method Introducing a new notion of almost quasi-negative holomorphic sectional curvature and extending the theorem to this setting.
result The theorem is extended to compact Kähler manifolds with almost quasi-negative holomorphic sectional curvature, and a gap-type theorem is derived.
The paper explores the relationship between joint mixability and negative dependence structures.
problem Understanding the connection between joint mixability and various negative dependence concepts.
method Analyzes the properties of joint mixes and their relation to negative dependence structures.
result Derives necessary and sufficient conditions for a joint mix to be negatively dependent.
Constructs two types of Eguchi-Hanson metrics with negative scalar curvature.
problem Creating metrics with negative scalar curvature.
method Constructed two types of Eguchi-Hanson metrics.
result Found metrics with negative scalar curvature.
The study examines symmetries in spaces with positive or non-negative curvature.
problem Understanding symmetries in spaces with curvature constraints.
method Survey of existing results for Riemannian manifolds with specified curvature properties and symmetries.
result Results on symmetries in spaces with curvature bounds.
Non-negative matrix factorization is a basic tool for decomposing data into the feature and weight matrices under non-negativity constraints, and in practice is often solved in the alternating minimization framework. However, it is unclear whether such algorithms can recover the ground-truth feature matrix when the wei…
Solves curvature problems on manifolds with negative curvature.
problem Prescribed curvature problems on closed manifolds with negative curvature.
method Investigates fully nonlinear prescribed curvature problems for modified Schouten tensor on closed Riemannian manifolds with negative curvature.
result Proves solvability of curvature problems under certain conditions.
This study uses local Gaussian correlation to analyze stock return tails, revealing more sensitive network properties.
problem Misleading results from Pearson correlation in financial networks.
method Local Gaussian correlation coefficient for capturing nonlinear dependence and heavy-tailed distributions.
result Local Gaussian correlation network among negative tails is more sensitive to stock market risks.
Sharp inequality in spaces with non-negative Ricci curvature.
problem Proving a sharp isoperimetric inequality in metric measure spaces.
method Using volume entropy in non-compact metric measure spaces with non-negative synthetic Ricci curvature.
result Proved a sharp dimension-free isoperimetric inequality.
The paper discusses thresholds and bounds for accuracy in binary classification systems.
problem The accuracy of binary classification systems and its dependence on prevalence.
method Analyzing the precision-prevalence curve and negative predictive value-prevalence curve to find thresholds and bounds.
result Thresholds (φe and φn) bound various accuracy metrics (Fβ, F1, FM, MCC) and the ratio of maximum accuracy to prevalence. Study on quaternionic bisectional curvature for quaternion-Kähler manifolds.
problem Characterize quaternionic bisectional curvature on quaternion-Kähler manifolds.
method Analyzing properties of quaternionic bisectional curvature on specific manifolds.
result Non-negative quaternionic bisectional curvature is only on quaternionic projective space.
Constructs metrics with negative constant scalar curvature.
problem Negative constant scalar curvature metrics.
method One-parameter family of complete metrics.
result Verifies positive energy conjecture for these metrics.
The Wu-Yau theorem is proven for Sasakian manifolds with specific curvature conditions.
problem Proving properties of Sasakian manifolds with negative transverse holomorphic sectional curvature.
method Analyzing the curvature properties and applying the Wu-Yau theorem.
result Compact Sasakian manifolds with negative transverse holomorphic sectional curvature have negative transverse Ricci curvature.
A new method helps escape saddle points in non-convex optimization.
problem Escaping saddle points in non-convex optimization problems.
method CNC-SCSG method using a separate SGD step to help escape from strict saddle points.
result The method converges to a second-order stationary point with a rate of O(ε−2log(1/ε)). Study the structure of Kähler foliations with negative Ricci curvature.
problem Characterize the structure of Kähler foliations with negative Ricci curvature.
method Prove a de Rham type theorem decomposition on the leaf space.
result Characterize each factor in the decomposition of the leaf space.
Study on rigidity with non-negative intermediate curvature on low-dimensional manifolds.
problem Extending non-existence theorem of positive scalar curvature to product manifolds.
method Introduced intermediate curvature and studied rigidity conditions.
result Rigidity when intermediate curvature is non-negative in low dimensions.
We show that gamma distributions provide models for departures from randomness since every neighbourhood of an exponential distribution contains a neighbourhood of gamma distributions, using an information theoretic metric topology. We derive also the information geometry of the 3-manifold of McKay bivariate gamma dist…
Proves CLT for Brownian paths on pinched negative curvature manifolds.
problem Distribution of Brownian paths on pinched negative curvature manifolds.
method Proof of central limit theorem for distances and Green functions.
result Central limit theorem holds for Brownian paths in pinched negative curvature.
Study Kähler metrics on complex tori with almost non-negative scalar curvature.
problem Stability of Kähler metrics on complex tori.
method Proved convergence of non-collapsing subsequence of Kähler metrics to flat torus.
result Kähler metrics with almost non-negative scalar curvature on complex tori converge to flat torus.
Two remarks on curvature properties of Kähler manifolds.
problem Curvature properties of Kähler manifolds.
method Analyzing semi-positive holomorphic sectional curvature and quasi-negative k-Ricci curvature. result For semi-positive holomorphic sectional curvature, the rational dimension of the MRC fibration equals the number of non-truly-flat directions. For quasi-negative k-Ricci curvature, the canonical bundle is ample. Study analyzes factors affecting capital adequacy in Bangladesh's banks.
problem Factors influencing capital adequacy in commercial banks in Bangladesh.
method Fixed Effect, Random Effect, and Pooled Ordinary Least Square (POLS) methods.
result Several independent variables significantly affect capital adequacy, with specific relationships between leverage, liquidity risk, and other factors.
Optimal diameter estimates for 3D spaces with non-negative Ricci curvature.
problem Estimating the diameter of 3D spaces with non-negative Ricci curvature.
method Proving positive scalar curvature passes to Ricci limit spaces of non-negative curvature.
result Optimal Bonnet-Myers upper bound for 3D spaces.
Study finds solutions to flows by negative curvature powers.
problem Curvature flows with negative powers.
method Closed self-similar solutions in warped product manifolds, proving non-strict convexity.
result Proves self-similar solutions are slices of warped product manifolds.
The study proves the non-existence of certain Kähler metrics with specific curvature properties.
problem Non-existence of complete Kähler metrics with negatively pinched holomorphic sectional curvature.
method Construction of a Kähler metric with negatively pinched holomorphic sectional curvature and application of equivalence of invariant metrics.
result The dichotomy of completeness and non-existence of Kähler metrics with negatively pinched holomorphic sectional curvature.
Proves existence and uniqueness of metrics with negative curvature and singularities on compact surfaces.
problem Existence and uniqueness of conformal metrics with negative curvature and singularities.
method Proves existence and uniqueness of conformal metrics with negative curvature and singularities on compact surfaces.
result Existence and uniqueness of conformal metrics with negative curvature and singularities on compact surfaces.
The study proves the existence of complete Kähler metrics with negative holomorphic bisectional curvature in specific domains.
problem Proving the existence of complete Kähler metrics with negative holomorphic bisectional curvature in certain domains.
method Analyzing bounded domains in Cn with specific curvature properties. result Strictly pseudoconvex bounded domains and domains with squeezing function tending to 1 at boundary points admit complete Kähler metrics with negative holomorphic bisectional curvature everywhere.
Conditions ensure constant curvature in negatively curved manifolds.
problem Ensuring constant curvature in negatively curved manifolds.
method Intrinsic conditions on horospheres' geometry.
result Sectional curvature is constant under given conditions.
Paper proves global convergence of NCELM model.
problem Ensuring global convergence of NCELM model.
method Two-stage process: random base learners and NCL penalty term updates.
result Global convergence of NCELM proved using Banach theorem.
Study on Kähler manifolds with non-negative mixed curvature, proving splitting and structure theorems.
problem Investigating properties of Kähler manifolds with specific curvature conditions.
method Conformal perturbation method.
result Established structure and splitting theorems for Kähler manifolds with non-negative mixed curvature.
The study confirms a conjecture about Kähler manifolds with quasi-negative curvature.
problem Confirming a long-standing conjecture about Kähler manifolds with quasi-negative curvature.
method Introducing (ε,δ)--quasi-negativity and applying gap-type theorems. result Obtained gap-type theorems for ∫Xc1(KX)n>0 in terms of real bisectional curvature and weighted orthogonal Ricci curvature. The paper bends Riemannian manifolds to achieve metrics with negative Ricci curvature.
problem Achieving metrics with negative Ricci curvature on closed Riemannian manifolds.
method Solving a fully nonlinear equation to conformally bend the manifold.
result Metrics of quasi-negative Ricci curvature are conformal to metrics with negative Ricci curvature.
Study on G2-structures with negative Ricci curvature on closed and noncompact manifolds.
problem Existence and properties of closed G2-structures with negative Ricci curvature. method Analyzing existence and nonexistence of closed G2-structures with negative Ricci curvature on closed and noncompact manifolds. result No closed manifold admits a closed G2-structure with negative Ricci curvature. For noncompact manifolds, restrictions on lengths of geodesics are found. Lower bound on minimum vertex degree for non-negative Lin-Lu-Yau curvature on graphs.
problem Determining the minimum vertex degree for non-negative Lin-Lu-Yau curvature.
method Investigation of Ollivier-Ricci curvature and Lin-Lu-Yau modification on locally finite graphs.
result Lower bound on minimum vertex degree ensuring non-negative Lin-Lu-Yau curvature.
On curved spaces, viscous fluids reach equilibrium quickly.
problem Thermalization of viscous fluids on negatively curved manifolds.
method Stochastic Navier-Stokes equations with kinematically selected deformation Laplacian.
result Exponential thermalization rate of $2νλ_\Def$.
Stock correlations is crucial to asset pricing, investor decision-making, and financial risk regulations. However, microscopic explanation based on agent-based modeling is still lacking. We here propose a model derived from minority game for modeling stock correlations, in which an agent's expected return for one stock…
Study on prescribing curvature on specific manifolds with negative Gauduchon degree.
problem Prescribing Chern scalar curvatures on compact Hermitian manifolds with negative Gauduchon degree.
method Analysis of geometric flow convergence to obtain existence results.
result Existence results for curvature functions that are nonzero and nonpositive, and sign-changing cases.
Sharp inequality for submanifolds in manifolds with non-negative Ricci curvature.
problem Establishing a Fenchel-Willmore inequality for submanifolds in manifolds with non-negative Ricci curvature.
method Analyzing submanifolds in manifolds with non-negative intermediate Ricci curvature and Euclidean volume growth.
result Sharp Fenchel-Willmore inequality for submanifolds in manifolds with non-negative intermediate Ricci curvature.
We develop a ``canonical Wick rotation-rescaling theory in 3-dimensional gravity''. This includes: (a) A simultaneous classification that shows how generic maximal globally hyperbolic spacetimes of constant curvature, which admit a complete Cauchy surface (in particular a compact one), as well as complex projective str…
Constructs conformal metrics with negative curvature on manifolds with boundary.
problem Creating conformal metrics with negative curvature on manifolds with boundary.
method Using Morse functions to construct conformal metrics and proving results for compact 3-manifolds with boundary.
result Any Riemannian metric on compact 3-manifolds with boundary is conformal to a compact metric of negative sectional curvature.
The Bergman metric on symmetrized bidisc has negative curvature properties.
problem Holomorphic curvature properties of the Bergman metric on symmetrized bidisc.
method Analysis of holomorphic sectional and bisectional curvatures.
result Negative pinched holomorphic sectional curvature and non-positive holomorphic bisectional curvature.
The paper estimates Betti numbers for graphs with specific curvatures, proving bounds and characterizing rigidity.
problem Estimating Betti numbers for graphs with non-negative curvatures.
method Establishing Betti number estimates for graphs with non-negative Ollivier and Bakry-Émery curvatures.
result Upper bounds on the first Betti number for graphs with non-negative curvatures, with characterizations of rigidity.