New Bianchi-convex sets generalize Ricci flow maximum principle.
problem Generalizing maximum principle for Ricci flow.
method Introducing Bianchi-convex sets.
result Hamilton's maximum principle extended to Bianchi-convex sets.
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48 results for “Tensor Maximum Principle”
Study tensor completion for tensors with sparse CP factors under noisy conditions.
problem Noisy tensor completion for tensors with a sparse CP factor.
method Complexity-regularized maximum likelihood principle and ADMM-type algorithm.
result General theoretical error bounds and validation via experiments.
Study of 3D vacuum static spaces with specific curvature properties.
problem Classifying 3D vacuum static spaces with certain curvature conditions.
method Used generalized maximum principle to classify 3D spaces.
result Gave a complete classification of 3D complete vacuum static spaces.
Study of asymmetric rank-one tensor models with non-Gaussian noise.
problem Analyzing maximum-likelihood estimators for asymmetric rank-one tensor models.
method Spectrally separated branch analysis, resolvent methods, cumulant expansions, Efron-Stein-type variance bounds.
result Asymptotic singular value and mode-wise alignments are robust to non-Gaussian noise.
The paper studies a heat flow for diffeomorphisms on flat surfaces, preserving their structure.
problem Studying a specific heat flow equation on flat surfaces.
method Using a tensor maximum principle and a change of variables, the authors establish bounds and regularity results.
result The heat flow preserves diffeomorphisms, unlike harmonic map heat flow.
Kähler spaces with zero first Chern class: Bochner principle, Albanese map and fundamental groupsmath.AG
Compact Kähler spaces with zero first Chern class have special geometric properties.
problem Characterizing Kähler spaces with zero first Chern class.
method Analyzing holomorphic tensors, decomposing tangent sheaf, and using Bochner principle.
result Spaces with zero first Chern class split off a complex torus and have specific holonomy representations.
We consider the four-dimensional nonholonomic distribution defined by the 4-potential of the electromagnetic field on the manifold. This distribution has a metric tensor with the Lorentzian signature , therefore, the causal structure appears as in the general relativity theory. By means of the Pontryagin's m…
The paper establishes a version of the Hopf boundary point lemma for sections of a vector bundle over a manifold with boundary. This result may be viewed as a counterpart to the tensor maximum principle obtained by R. Hamilton in 1986. Potential applications include the study of various geometric flows and the construc…
We derive some elliptic differential inequalities from the Weitzenböck formulas for the traceless Ricci tensor of a Kähler manifold with constant scalar curvature and the Bochner tensor of a Kähler-Einstein manifold respectively. Using elliptic estimates and maximum principle, some and pinching result…
The paper establishes maximum principles and stochastic completeness for pseudo-Hermitian manifolds.
problem Maximum principles and stochastic completeness for pseudo-Hermitian manifolds.
method Established generalized maximum principles and proved stochastic completeness equivalence.
result Stochastic completeness for the heat semigroup is equivalent to generalized maximum principles.
Rigidity Theorems for Complete Sasakian Manifolds with Constant Pseudo-Hermitian Scalar Curvaturemath.DG
The orthogonal decomposition of the Webster curvature provides us a way to characterize some canonical metrics on a pseudo-Hermitian manifold. We derive some subelliptic differential inequalities from the Weitzenböck formulas for the traceless pseudo-Hermitian Ricci tensor and the Chern-Moser tensor of Sasakian manifol…
Based on a suggestion of Richard Hamilton, we give an alternate proof of his matrix Harnack inequality for solutions of the Ricci flow with positive curvature operator. This Harnack inequality says that a certain endomorphism, consisting of an expression in the curvature and its first two covariant derivatives, of the …
Establishes a boundary maximum principle for varifolds with fixed contact angle.
problem Boundary behavior of varifolds with contact angle constraints.
method Maximum principle for stationary pairs of varifolds with fixed contact angle condition.
result Boundary maximum principle proven for stationary varifolds.
Four-dimensional complete gradient shrinking Ricci solitons with half positive isotropic curvaturemath.DG
Study on 4D Ricci solitons with specific curvature properties.
problem Characterizing 4D complete gradient shrinking Ricci solitons with half positive isotropic curvature.
method Curvature estimates, strong maximum principle, classification arguments.
result New classification results for gradient shrinking Kähler-Ricci solitons and 4D complete gradient shrinking Ricci solitons with half nonnegative isotropic curvature.
Note establishes a local maximum principle for Ricci flow under curvature conditions.
problem Preserving nonnegativity of curvature along Ricci flow with unbounded curvature.
method Combining scaling invariant curvature condition with Dirichlet heat kernel estimates.
result Unified and more direct proof of localized maximum principle.
Study on curvature tensors, discovering new Osserman tensors.
problem Investigate properties of curvature tensors and their relations.
method Introduce quasi-Clifford curvature tensors and analyze their properties.
result Discovered an Osserman curvature tensor not satisfying the duality principle.
We present a new statistical learning paradigm for Boltzmann machines based on a new inference principle we have proposed: the latent maximum entropy principle (LME). LME is different both from Jaynes maximum entropy principle and from standard maximum likelihood estimation.We demonstrate the LME principle BY deriving …
Study proves Maximum Principles for unbounded Riemannian domains.
problem Proving Maximum Principles for unbounded Riemannian domains.
method Examines both ambient manifold and differential operator assumptions.
result Valid Maximum Principles established for unbounded domains.
In this paper we present a proof of a Neumann type maximum principle for the Laplace operator on compact Riemannian manifolds. A key p oint is the simple geometric nature of the constant in the a priori estimate of this maximum principle. In particular, this maximum principle can be applied to manifolds with Ricci curv…
Study extends ODE Maximum Principle to non-compact hypersurfaces in hyperbolic space.
problem Analyzing long-term behavior of IMCF on non-compact hypersurfaces.
method Extends ODE Maximum Principle to non-compact hypersurfaces using Omari-Yau maximum principle at infinity.
result Showed long-time existence and asymptotic convergence of IMCF to horospheres.
A principle for minimal surfaces in any dimension and codimension.
problem Boundary maximum principle for minimal submanifolds.
method Established a boundary maximum principle for minimal submanifolds in arbitrary codimension.
result Generalized boundary maximum principle to any dimension and codimension.
Proves a principle for one-phase Bernoulli problem minimizers.
problem One-phase Bernoulli problem minimizers.
method Strong maximum principle, Alt-Caffarelli functional, Hardt-Simon-type foliation.
result Constructs a foliation for global minimizers.
Derives a parabolic maximum principle for mean curvature flow.
problem Maximum principle for mean curvature flow in curved spaces.
method Derives a parabolic Omori-Yau maximum principle.
result Maximum principle preserves image of Gauss map and self-shrinkers.
In this paper we characterize the degenerate elliptic equations F(D^2u)=0 whose viscosity subsolutions, (F(D^2u) \geq 0), satisfy the strong maximum principle. We introduce an easily computed function f(t) for t > 0, determined by F, and we show that the strong maximum principle holds depending on whether the integral …
Study on maximum principles for nonlinear equations on Riemannian manifolds.
problem Investigating strong maximum principles for fully nonlinear equations on Riemannian manifolds.
method Analyzing scaling conditions and applying to various nonlinear operators.
result Established new strong comparison principles for second order uniformly elliptic problems.
Study strong maximum principles for mean curvature operators on subriemannian manifolds.
problem Investigate strong maximum principles for mean curvature operators on subriemannian manifolds.
method Analyze subriemannian manifolds including Heisenberg groups and cylinders, under Hormander type conditions.
result Show strong maximum principles for horizontal (p-) mean curvature operator and p-(sub)laplacian operator under certain conditions.
The time-dependent maximum principle for systems of parabolic equations subject to an avoidance setmath.DG
In this paper we prove two extensions of Hamilton's maximal principle for systems pf parabolic equations which sould be useful for the study of the Ricci flow and some other geometric evolution equations. One extension is a time-dependent maximum principle and the other is a time-dependent maximum principle subject to …
In this work we consider viscosity solutions to second order partial differential equations on Riemannian manifolds. We prove maximum principles for solutions to Dirichlet problem on a compact Riemannian manifold with boundary. Using a different method, we generalize maximum principles of Omori and Yau to a viscosity v…
This article studies a discrete geometric structure on triangulated manifolds and an associated curvature flow (combinatorial Yamabe flow). The associated evolution of curvature appears to be like a heat equation on graphs, but it can be shown to not satisfy the maximum principle. The notion of a parabolic-like operato…
Extends geometrical description of tensor manifolds in tree-based formats.
problem Geometrical description of tensor manifolds in tree-based formats.
method Provided a new geometrical description of manifolds of tensors in tree-based format.
result Geometrical description compatible with Tucker format.
Extends variational principle to tensor Banach spaces.
problem High-dimensional partial differential equations and minimization problems.
method Describes tensor product as disjoint connected components, each modeled as a Banach manifold.
result Extension of Dirac-Frenkel variational principle to topological tensor spaces.
This note introduces a duality principle for nonlinear equations.
problem Maximum principles at infinity for nonlinear equations.
method Ahlfors property and Khas'minskii potentials.
result Unified framework for various maximum principles.
We prove that for an algebraic curvature tensor on a pseudo-Euclidean space, the Jordan-Osserman condition implies the Rakić duality principle, and that the Osserman condition and the duality principle are equivalent in the diagonalisable case.
New principle for harmonic maps helps study higher-dimensional submanifolds.
problem Understanding unboundedness of totally geodesic projections in higher codimension.
method Introducing a flexible notion of convexity and applying it to harmonic and conformal maps.
result New maximum principle for harmonic maps applicable to various geometric settings.
Study compares nodal sets of solutions to the Allen-Cahn equation.
problem Comparing nodal sets of solutions to the Allen-Cahn equation with conical asymptotics.
method Maximum principle for linearized operator on unbounded domains.
result Positive phase uniquely determines the solution and enforces global ordering.
Based on works by Hopf, Weinberger, Hamilton and Evans, we state and prove the strong elliptic maximum principle for smooth sections in vector bundles over Riemannian manifolds and give some applications in Differential Geometry. Moreover, we use this maximum principle to obtain various rigidity theorems and Bernstein …
New geometric approach to optimal control theory using Stokes Theorem.
problem Optimal control theory, specifically the Mayer problem.
method Geometric unfolding based on the Stokes Theorem.
result Derives a new necessary and sufficient condition for optimal solutions.
Based on ideas of L. Alías, D. Impera and M. Rigoli developed in "Hypersurfaces of constant higher order mean curvature in warped products", we develope a fairly general weak/Omori-Yau maximum principle for trace operators. We apply this version of maximum principle to generalize several higher order mean curvature est…
New estimator for tensor weights with improved bias.
problem Estimating tensor weights from noisy data.
method Random matrix theory and KKT conditions.
result Asymptotically unbiased estimator for tensor rank.
Ancient Ricci flows with bounded girth found in 3D and higher.
problem Finding ancient Ricci flows with bounded girth in dimensions 3 and higher.
method Invariant conditions on curvature and its derivatives under symmetry, proving Ricci flow invariance.
result Construction of new ancient Ricci flows with positive curvature operator and bounded girth.
Maximum principles for boundary-degenerate second-order linear elliptic differential operatorsmath.AP
We prove weak and strong maximum principles, including a Hopf lemma, for smooth subsolutions to equations defined by linear, second-order, partial differential operators whose principal symbols vanish along a portion of the domain boundary. The boundary regularity property of the smooth subsolutions along this boundary…
According to a classical result of E.~Calabi any hyperbolic affine hypersphere endowed with its natural Hessian metric has a non-positive Ricci tensor. The affine hyperspheres can be described as the level sets of solutions to the "hyperbolic" toric Kähler-Einstein equation on proper convex cones. We…
We consider geometric flow equations for contracting and expanding normal velocities, including powers of the Gauss curvature, of the mean curvature, and of the norm of the second fundamental form, and ask whether - after appropriate rescaling - closed strictly convex surfaces converge to spheres. To prove this, many a…
The well known maximum-entropy principle due to Jaynes, which states that given mean parameters, the maximum entropy distribution matching them is in an exponential family, has been very popular in machine learning due to its "Occam's razor" interpretation. Unfortunately, calculating the potentials in the maximum-entro…
Pontryagin's Maximum Principle is an outstanding result for solving optimal control problems by means of optimizing a specific function on some particular variables, the so called controls. However, this is not always enough for solving all these problems. A high order maximum principle (Krener, 1977) must be used in o…
Proposes a new method for high-dimensional density estimation.
problem Estimating high-dimensional probability density functions efficiently.
method Tensorizing flow method combining tensor-train and flow-based generative modeling.
result Efficiently constructs an approximate density in tensor-train form and trains a flow model to match empirical distribution.
MEP-Net uses MEP to generate solutions from limited data.
problem Generating solutions to scientific problems with incomplete information.
method Combines MEP with neural networks to learn complex distributions from moment constraints.
result Demonstrates MEP-Net's effectiveness in modeling biochemical reaction networks and generating complex distributions.
Two methods improve tensor recovery in Ising models, revealing gene interactions.
problem Improving tensor recovery in Ising models for complex data structures.
method Pseudolikelihood and interaction screening approaches for tensor learning.
result Both methods achieve tensor recovery with sample size logarithmic in nodes, exponential in strength and degree.