The paper classifies solutions to semilinear equations on curved spaces.
problem Classifying solutions to semilinear equations on manifolds with nonnegative Ricci curvature.
method Proving classification results for subcritical and critical semilinear elliptic equations.
result Strong rigidity results for nontrivial solutions in the critical case.
The paper studies CR Yamabe solutions on Sasakian manifolds with nonnegative curvature.
problem Characterizing CR Yamabe solutions on Sasakian manifolds with nonnegative curvature.
method Analyzes solutions to the CR Yamabe equation in noncompact (2n+1)-dimensional Sasakian manifolds with nonnegative curvature. result The Heisenberg group H1 is the only (complete) Sasakian space with nonnegative Tanaka-Webster scalar curvature admitting a (nontrivial) positive solution. Study finite curvature solutions on surfaces with nonnegative Gauss curvature.
problem Finite total curvature solutions of Liouville equation on surfaces with nonnegative Gauss curvature.
method Analyzes asymptotic behavior of solutions on complete surfaces.
result Two extremal cases identified: Euclidean plane or flat cylinder, with specific decay conditions.
Ancient solutions to Kähler Ricci flow classified completely.
problem Ancient solutions to Kähler Ricci flow with nonnegative bisectional curvature.
method Complete classification of κ-noncollapsed, complete ancient solutions.
result Classification of all ancient solutions to Kähler Ricci flow with nonnegative bisectional curvature.
The paper splits manifolds with nonnegative Ricci curvature using inequalities from a Poisson equation.
problem Splitting complete Riemannian manifolds with nonnegative Ricci curvature.
method Analysis of pointwise inequalities of Modica type for bounded solutions to a semilinear Poisson equation.
result The existence of a nonconstant bounded solution leading to splitting results and solution classification.
We introduce a new entropy functional for nonnegative solutions of the heat equation on a manifold with time-dependent Riemannian metric. Under certain integral assumptions, we show that this entropy is non-decreasing, and moreover convex if the metric evolves under super Ricci flow (which includes Ricci flow and fixed…
Ancient solutions of Ricci flow with Type I growth are classified.
problem Understanding ancient solutions of Ricci flow with specific curvature growth.
method Analyzing ancient solutions with Type I curvature growth in arbitrary dimensions.
result Ancient solutions with Type I growth are classified into specific types.
Gradient bounds and Liouville theorems for quasi-linear equations on manifolds with nonnegative Ricci curvature.
problem Establishing bounds and theorems for solutions to quasi-linear elliptic equations on compact manifolds with nonnegative Ricci curvature.
method Gradient bounds, Liouville-type theorems, local splitting theorem, Harnack-type inequality, ABP estimate.
result Gradient bounds and Liouville-type theorems for solutions to quasi-linear equations on compact manifolds with nonnegative Ricci curvature.
In this paper we prove classification results for gradient shrinking Ricci solitons under two invariant conditions, namely nonnegative orthogonal bisectional curvature and weakly PIC1, without any curvature bound. New results on ancient solutions for the Ricci and Kähler-Ricci flow are also obtained. The main new featu…
MaxVol NMF maximizes the volume of H in NMF for better sparse and interpretable solutions.
problem Finding interpretable and unique NMF solutions.
method Dual approach to MinVol NMF, maximizing the volume of H. result MaxVol NMF solutions correspond to clustering columns in disjoint clusters.
In this paper we discuss the asymptotic entropy for ancient solutions to the Ricci flow. We prove a gap theorem for ancient solutions, which could be regarded as an entropy counterpart of Yokota's work. In addition, we prove that under some assumptions on one time slice of a complete ancient solution with nonnegative c…
Prove rigidity and classification results for quasilinear Liouville equation on manifolds with nonnegative Ricci curvature.
problem Quasilinear Liouville equation on manifolds with nonnegative Ricci curvature.
method Prove rigidity and classification results for the quasilinear Liouville equation associated with the n-Laplacian on complete noncompact Riemannian manifolds with nonnegative Ricci curvature. result Under a sharp logarithmic lower bound, the ambient manifold must be isometric to the Euclidean space and the solution must be one of the standard bubbles.
Paper proves translating solutions for a specific flow in a product manifold.
problem Existence of translating solutions for nonparametric mean curvature flow with Neumann boundary data.
method Proves existence using product manifold MnimesR with specific conditions. result Existence of translating solutions for the flow in the product manifold.
We prove short time existence for the Ricci flow on open manifolds of nonnegative complex sectional curvature. We do not require upper curvature bounds. By considering the doubling of convex sets contained in a Cheeger-Gromoll convex exhaustion and solving the singular initial value problem for the Ricci flow on these …
Study classifies solutions to specific equations on half-space and ball.
problem Classifying nonnegative solutions to Q-flat and constant T-curvature equations. method Introduced a biharmonic Poisson kernel and derived its explicit representation formula.
result Established classification theorems for solutions on R+n+1 and Bn+1. The paper proves estimates for a specific flow on compact manifolds.
problem Proving estimates for the Ricci-Bourguignon flow.
method Hamilton-Ivey estimates for the Ricci-Bourguignon flow on compact manifolds with n=3 and ρ<0. result Compact ancient solutions have nonnegative sectional curvature for all negative ρ. Established concavity principle for curved spaces.
problem Solving equations on curved spaces with nonnegative curvature.
method Applied concavity principle to elliptic and parabolic equations on locally symmetric spaces with nonnegative curvature.
result First general concavity principle on spaces with non-constant sectional curvature.
In [10], R. Hamilton established a differential Harnack inequality for solutions to the Ricci flow with nonnegative curvature operator. We show that this inequality holds under the weaker condition that M x R^2 has nonnegative isotropic curvature.
The paper proves that under certain conditions, solutions to a specific differential inequality are nonnegative.
problem Preserving positivity of solutions to a differential inequality on Riemannian manifolds.
method Analytic approach using Llocp norms and growth conditions over geodesic balls. result Nonnegative solutions to the inequality −Δu+λu≥0 are preserved under suitable growth conditions. In this paper, we prove that any non-flat ancient solution to Kähler-Ricci flow with bounded nonnegative bisectional curvature has asymptotic volume ratio zero. We also prove that any gradient shrinking solitons with positive bisectional curvature must be compact. Both results generalize the corresponding earlier resul…
Paper introduces a novel matrix-wise sparse MNNLS formulation and algorithm.
problem Sparse nonnegative least squares with multiple right-hand sides.
method Matrix-wise sparsity constraint, two-step algorithm.
result More accurate results compared to state-of-the-art methods.
Sharp Liouville theorem for minimal graphs on manifolds with nonnegative Ricci curvature.
problem Characterizing smooth solutions to minimal hypersurface equations on manifolds with nonnegative Ricci curvature.
method Gradient estimate for minimal graphs over Σ with small linear growth of the negative parts of graphic functions via iteration. result Every smooth solution u to minimal hypersurface equation on Σ is a constant provided u has sublinear growth for its negative part. We consider dimension reduction for solutions of the Kähler-Ricci flow with nonegative bisectional curvature. When the complex dimension n=2, we prove an optimal dimension reduction theorem for complete translating Kähler-Ricci solitons with nonnegative bisectional curvature. We also prove a general dimension reducti…
Researchers solve porous medium equation on noncompact manifolds with Ricci curvature.
problem Solving porous medium equation on noncompact manifolds with nonnegative Ricci curvature.
method Constructing a space X of functions larger than L1, in which the Green function on M appears as a weight, to solve the PME.
result The porous medium equation admits a solution in the weak dual sense for certain initial data.
The paper explores partial identifiability in nonnegative matrix factorization under specific conditions.
problem Identifying specific columns of the matrices in nonnegative matrix factorization.
method Mathematical rigor and geometric interpretation to analyze partial identifiability of columns in nonnegative matrix factorization.
result The partial uniqueness of a single column of C or S can be guaranteed under certain sparsity and algebraic conditions. This paper classifies complete self-shrinkers in R^(n+1) with nonnegative constant scalar curvature.
problem Classifying self-shrinkers with specific curvature conditions.
method Analyzing the mean curvature flow and using geometric properties.
result Complete classifications of n-dimensional self-shrinkers in R^(n+1) with nonnegative constant scalar curvature.
Study ancient Ricci flows with nonnegative Ricci curvature and their asymptotic geometry.
problem Understanding the asymptotic geometry of ancient Ricci flows with nonnegative Ricci curvature.
method Analyze tangent flows at infinity and use estimates for noncollapsed F-limit metric solitons.
result Two dichotomy theorems for ancient Ricci flows: either the asymptotic volume ratio is zero or every tangent flow is a Ricci flat cone.
Study 4D solitons with specific curvature properties, proving curvature bounds and classifying solutions.
problem Investigate 4D gradient solitons with specific curvature properties.
method Analyze 4D gradient steady and shrinking solitons with nonnegative or half nonnegative isotropic curvature.
result Prove 2-nonnegativity of Ricci curvature and bound the curvature tensor for ancient solutions.
New method reduces computational cost for nonnegative low rank matrix approximation.
problem Efficiently compute nonnegative low rank matrix approximation for nonnegative matrices.
method Alternating projections onto tangent spaces of fixed rank matrices manifold and nonnegative matrix manifold.
result Sequence converges linearly to optimal solutions, showing better performance in terms of computational time and accuracy.
Study critical quasilinear equations on Riemannian manifolds with curvature constraints.
problem Investigate critical quasilinear elliptic equations on Riemannian manifolds with nonnegative Ricci curvature.
method Utilize a new nonlinear Kato inequality and Cheng-Yau type gradient estimates for positive solutions.
result Classify positive solutions to the critical p-Laplace equation and show rigidity concerning the ambient manifold. The nonnegativity of the CR Paneitz operator plays a crucial role in three-dimensional CR geometry. In this paper, we prove this nonnegativity for embeddable CR manifolds. This result and previous works give an affirmative solution of the CR Yamabe problem for embeddable CR manifolds. We also show the existence of a co…
Nonnegative matrix factorization (NMF) has become a very popular technique in machine learning because it automatically extracts meaningful features through a sparse and part-based representation. However, NMF has the drawback of being highly ill-posed, that is, there typically exist many different but equivalent facto…
In this paper, we propose a provably correct algorithm for convolutive nonnegative matrix factorization (CNMF) under separability assumptions. CNMF is a convolutive variant of nonnegative matrix factorization (NMF), which functions as an NMF with additional sequential structure. This model is useful in a number of appl…
In this paper we generalize the neck-stability theorem of Kleiner-Lott to a special class of four-dimensional nonnegatively curved Type I κ-solutions, namely, those whose asymptotic shrinkers are the standard cylinder S3×R. We use this stability result to prove a rigidity theorem: if a four-…
In our previous work we showed that for an ancient solution to the Ricci flow with nonnegative curvature operator, assuming bounded geometry on one time slice, bounded entropy implies noncollapsing on all scales. In this paper we prove the implication in the other direction, that for an ancient solution with bounded no…
For any complete noncompact Ka¨hler manifold with nonnegative and bounded holomorphic bisectional curvature,we provide the necessary and sufficient condition for non-ancient solution to the Ricci flow in this paper.
We study sequences of 3-dimensional solutions to the Ricci flow with almost nonnegative sectional curvatures and diameters tending to infinity. Such sequences may arise from the limits of dilations about singularities of Type IIb. In particular, we study the case when the sequence collapses, which may occur when dilati…
In this paper, we introduce and provide a short overview of nonnegative matrix factorization (NMF). Several aspects of NMF are discussed, namely, the application in hyperspectral imaging, geometry and uniqueness of NMF solutions, complexity, algorithms, and its link with extended formulations of polyhedra. In order to …
Sharp inequalities and eigenvalue problems on Finsler manifolds with nonnegative Ricci curvature.
problem Establishing sharp Morrey-Sobolev inequalities and eigenvalue problems on Finsler manifolds.
method Combining sharp isoperimetric inequality and anisotropic symmetrization argument.
result Existence and multiplicity of solutions for eigenvalue problems and elliptic PDEs.
Study nonnegative solutions on Riemannian manifolds using fractional porous medium equation.
problem Analyzing solutions to fractional porous medium equation on noncompact Riemannian manifolds.
method Existence and smoothing estimates for weak solutions in L1 and weighted spaces. result Results hold for Euclidean and hyperbolic spaces, including larger data classes.
Given a matrix M (not necessarily nonnegative) and a factorization rank r, semi-nonnegative matrix factorization (semi-NMF) looks for a matrix U with r columns and a nonnegative matrix V with r rows such that UV is the best possible approximation of M according to some metric. In this paper, we study th…
Study finds solutions to curvature equation with boundary conditions.
problem Finding solutions to curvature equation with boundary conditions.
method Established local C2 estimates and used blow-up analysis. result Existence of conformal metrics with prescribed curvature and boundary conditions.
Proves existence and uniqueness of CMC solutions in product manifolds.
problem Existence and uniqueness of solutions to CMC equation with Neumann boundary data.
method Analyzes product manifold MnimesR with specific curvature conditions. result Proves existence and uniqueness of solutions.
Study surfaces with nonnegative curvature in spectral sense, proving inequalities and bounds.
problem Closed orientable surfaces with nonnegative curvature in spectral sense.
method Spectral condition and associated conformal metrics to prove inequalities and bounds.
result Isoperimetric inequalities, area growth theorems, and diameter bounds for surfaces.
We derive a local Gaussian upper bound for the f-heat kernel on complete smooth metric measure space (M,g,e−fdv) with nonnegative Bakry-Émery Ricci curvature, which generalizes the classic Li-Yau estimate. As applications, we obtain a sharp Lf1-Liouville theorem for f-subharmonic functions and an Lf1-u…
The paper proves the existence of finite-energy solutions to a specific elliptic equation on Riemannian manifolds.
problem Existence of finite-energy solutions to a singular elliptic equation on Riemannian manifolds.
method ε-regularized problems, mountain pass arguments, and limiting procedures.
result Existence of nonnegative finite-energy supersolutions under certain conditions.
Study on solutions to conformally invariant fourth order equations, classifying their properties.
problem Classify qualitative properties of solutions to conformally invariant fourth order equations.
method Analyze two cases: removable and non-removable singularities, using Pohozaev-type invariant.
result Non-existence of semi-singular solutions, classifying them as multiples of the Emden--Fowler solution.
New theorem shows curvature concentration depends linearly on volume ratio.
problem Gap theorem for nonnegative Ricci curvature manifolds with small curvature concentration.
method Exhibited Ricci flow solution with faster than 1/t curvature decay.
result Curvature concentration depends linearly on asymptotic volume ratio.