The paper explores geometric influences on PDE solutions on manifolds.
problem Qualitative behavior of solutions to quasilinear PDEs on Riemannian manifolds.
method Investigates strong and weak maximum principles, compact support principles, and Liouville theorems.
result Identifies thresholds involving curvatures or volume growth to guarantee properties under Keller-Osserman conditions.
We prove a Paley-Wiener Theorem for a class of symmetric spaces of the compact type, in which all root multiplicities are even. This theorem characterizes functions of small support in terms of holomorphic extendability and exponential type of their (discrete) Fourier transforms. We also provide three independent new p…
Study semiclassical measures on complex hyperbolic quotients, identifying measure supports.
problem Understanding Laplacian eigenfunctions on complex hyperbolic quotients.
method Combining fractal uncertainty principle and Ratner theory to analyze measure supports.
result Semiclassical measures support is either cosphere bundle or a compact submanifold.
Study on hyperbolic manifolds finds measures of Laplace eigenfunctions restricted to cosphere bundles.
problem Analyzing semiclassical measures on hyperbolic manifolds.
method Adapting Dyatlov and Jin's argument to higher dimensions and using Ratner theory.
result Semiclassical measures' support contains the cosphere bundle of a compact totally geodesic submanifold.
The paper proves a 'long neck principle' for Riemannian spin manifolds with positive scalar curvature.
problem Establishing a 'long neck principle' for Riemannian spin manifolds with boundary.
method Developed index theory on compact Riemannian spin manifolds with boundary and applied it to prove the 'long neck principle'.
result The distance between the support of the differential of a strictly area decreasing map and the boundary of a manifold is bounded.
Abstract perspective on quadratic programming for optimal portfolio allocation.
problem Optimal allocation problems in long portfolio theory.
method Using maximum principles and distinguished boundaries in reproducing kernel Hilbert spaces.
result Support of an optimal distribution lies in a variety intersecting a distinguished boundary.
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.
Study inverse mean curvature flow on non-compact hypersurfaces, proving long-term existence and characterizing maximal time.
problem Evolution of non-compact convex hypersurfaces in Rn+1 by inverse mean curvature. method Establish long-term existence via pointwise mean curvature estimate and viscosity solutions for strict convexity.
result Characterization of maximal time of existence in terms of tangent cone at infinity.
Compact support steady flow in 3D Euler equations found.
problem Finding a nontrivial steady Euler flow with compact support.
method Constructing a nontrivial smooth steady incompressible Euler flow in three dimensions with compact support.
result A nontrivial smooth steady incompressible Euler flow in three dimensions with compact support constructed.
Formula derived for FUP exponent in quasi-Fuchsian groups.
problem Quantifying the fractal uncertainty principle in higher dimensions.
method Explicit formula derivation for FUP exponent, dependence on porosity parameter quantified.
result Explicit essential spectral gap for quasi-Fuchsian groups in higher dimensions.
Generative model uses random weighted support points for interpretable data sampling.
problem Creating diverse and interpretable sample sets from large datasets efficiently.
method Random weighted support points from Dirichlet process and Bayesian bootstrap.
result High-quality and diverse outputs at lower computational cost.
Develops a method to deform metrics on manifolds with non-compact boundaries.
problem Creating metrics with positive scalar curvature on manifolds with boundary.
method General deformation principle for Riemannian metrics on manifolds with non-compact boundaries.
result Non-existence of metrics with positive scalar curvature and mean convex boundary.
A new method for support vector regression using a data-driven insensitive parameter.
problem Determining an optimal insensitive parameter in support vector regression.
method A data-driven approach to approximate the insensitive parameter by minimizing a generalized loss function based on the likelihood principle.
result The proposed method outperforms traditional support vector regression methods and has lower computational costs.
Study proves rigidity for solitons with uncertainty principle.
problem Rigidity of shrinking Ricci solitons with uncertainty principle.
method Proves rigidity theorems for shrinking gradient Ricci solitons.
result Proves rigidity theorems with sharp constant in R^n.
A support theorem for the horocycle Radon transform f \to \hat{f} is a property of the form \hat{f} of compact support \rightarrow f of compact support. Here we prove a variation of this result where support (\hat{f}) is outside a fixed horocycle in hyperbolic space.
New principle for supersymmetric localization on Lie groups.
problem Computing supertrace of non-supersymmetric observables.
method Invariant supersymmetric deformations and fermionic zero modes.
result Path integral localizes to periodic orbits.
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…
Geometrically calculates multiplicities of K-types in tempered representations.
problem Calculating multiplicities of K-types in tempered representations of Lie groups.
method Geometric formula based on Kirillov's orbit method and quantisation commutes with reduction.
result Geometric expression for multiplicities of K-types in tempered representations.
New Einstein manifolds split into symmetric and compact parts.
problem Understanding Einstein manifolds with unimodular isometry groups.
method Theory of polar actions, Lie-theoretic arguments, and maximum principles.
result Negative Einstein manifolds split into symmetric and compact parts.
Constructs Poisson structures with compact support on manifolds.
problem Creating Poisson structures with compact support on manifolds.
method Explicit construction of Poisson structures with polynomial coefficients and modification outside open balls.
result Even-dimensional manifolds can be equipped with Poisson structures that vanish to infinite order at codimension one subsets.
Extends strong comparison principle for p-harmonic functions in Carnot-Caratheodory spaces.
problem Proving strong comparison principle for p-harmonic functions in specific geometric settings.
method Extends Bony's propagation of support argument to C^1 solutions of sub-elliptic p-Laplacian.
result Proves strong maximum and comparison principles for p-harmonic functions.
Localized Multidirectional Correction improves non-refusal target-response behavior in foundation models.
problem Controlled post-training refusal suppression in routed MoE and hybrid-MoE foundation models.
method Introduce Localized Multidirectional Correction (LoMC), a support-gated intervention framework.
result Substantially improves non-refusal target-response behavior while maintaining general capability under a compact intervention footprint.
Support theorem proved for X-ray transform on certain non-compact manifolds.
problem Support theorem for X-ray transform on non-compact manifolds with conjugate points.
method Use of plane covers and support theorem for simple manifolds by Krishnan.
result Support theorem proved for simply connected 2-step nilpotent Lie groups and some non-homogeneous 3D manifolds.
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.
By adapting the Cheeger-Simons approach to differential cohomology, we establish a notion of differential cohomology with compact support. We show that it is functorial with respect to open embeddings and that it fits into a natural diagram of exact sequences which compare it to compactly supported singular cohomology …
Injectivity and support theorem for X-ray transform on specific Lie groups.
problem Injectivity and support theorem for X-ray transform on 2-step nilpotent Lie groups.
method General reduction principle for manifolds with uniformly escaping geodesics.
result Injectivity and support theorem for X-ray transform on 2-step nilpotent Lie groups.
Solves Monge-Ampère equations on compact Hessian manifolds using the Perron method.
problem Solving Monge-Ampère equations on compact Hessian manifolds.
method Perron method, compactness properties of normalized quasi-convex functions, local and global comparison principles for twisted Monge-Ampère operators.
result Solves Monge-Ampère equations involving arbitrary probability measures.
Study shows Sobolev functions on non-compact manifolds can't be approximated by smooth compactly supported functions.
problem Sobolev functions on non-compact manifolds cannot be approximated by smooth compactly supported functions.
method Analysis of Sobolev spaces on non-compact manifolds.
result Proves the failure of the density of smooth compactly supported functions in Sobolev spaces on non-compact manifolds.
FedSLIM optimizes compact pattern models across distributed databases without sharing raw data.
problem Privacy-preserving federated descriptive analytics for data silos.
method Federated MDL-based framework using SLIM principle.
result FedSLIM variants preserve high-quality compression structure and recover globally informative patterns.
The paper proves the regularity and compactness of stable CMC integral varifolds in codimension 1.
problem Stable CMC integral varifolds of codimension 1.
method Structural conditions and variational hypotheses.
result The support of the varifold is an immersed constant-mean-curvature (CMC) hypersurface under given conditions.
The paper proves a large deviation principle for Gibbs measures on Polish spaces and applies it to specific cases.
problem Large deviation principles for Gibbs measures on Polish spaces.
method General Laplace principle for non-normalized Gibbs measures, applied to conditional Gibbs measures, Coulomb gases, and Fekete points.
result The Laplace principle is proven and applied to specific cases, providing a deterministic version of Γ-convergence. MPWTSVM improves multi-view learning by reducing redundancy and enhancing accuracy.
problem Improving multi-view learning models for better accuracy and efficiency.
method Proposes MPWTSVM, which combines WLTSVM's strengths with multi-view learning principles.
result Demonstrates better accuracy and efficiency compared to existing multi-view classification models.
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…
Curves in Carnot groups avoid compact sets, growing at least t1/s.
problem Existence of periodic normal geodesics in subFinsler Carnot groups.
method Analysis of curves satisfying Pontryagin Maximum Principle.
result Normal curves in subFinsler Carnot groups leave every compact set.
Study on compact and finite-type support in mapping class group homology.
problem Understanding non-trivial classes supported on compact or finite-type subsurfaces.
method Use of shiftable subsurfaces and homological stability for finite-type surfaces.
result Almost-complete answer for surfaces with positive genus, partial answer for zero genus.
The strong maximum principle is proved to hold for weak (in the sense of support functions) sub- and super-solutions to a class of quasi-linear elliptic equations that includes the mean curvature equation for C0 spacelike hypersurfaces in a Lorentzian manifold. As one application a Lorentzian warped product splittin…
Study shows most symmetric and 3-symmetric spaces lack solvable Clifford-Klein forms.
problem Existence of compact solvable Clifford-Klein forms in homogeneous spaces.
method Combination of Hirzebruch-Kobayashi-Ono proportionality principle and syndetic hull theory.
result Almost all symmetric and 3-symmetric spaces do not admit solvable compact Clifford-Klein forms.
In this short note we extend Chow and Lu's advanced maximum principles for parabolic systems on closed manifolds to the case of compact manifolds with boundary, which also generalizes a Hopf type theorem of Pulemotov.
When applying the support vector machine (SVM) to high-dimensional classification problems, we often impose a sparse structure in the SVM to eliminate the influences of the irrelevant predictors. The lasso and other variable selection techniques have been successfully used in the SVM to perform automatic variable selec…
The formal principle holds for certain globally generated vector bundles on Fano manifolds and smooth rational curves.
problem Proving the formal principle for globally generated vector bundles on compact complex manifolds.
method Applying Cartan's equivalence method to a differential system on the universal family of the Douady space.
result The formal principle is true for Fano manifolds and smooth rational curves under specific conditions.
A geometric version of the Poincaré Lemma is established for the topological vector space of differential chains. In particular, every differential k-cycle with compact support in a contractible open subset U of a smooth n-manifold M is the boundary of a differential (k+1) -chain with compact support in U. Applications…
In this paper, we study the evolution of L2 one forms under Ricci flow with bounded curvature on a non-compact Rimennian manifold. We show on such a manifold that the L2 norm of a smooth one form with compact support is non-increasing along the Ricci flow with bounded curvature. The L∞ norm is showed to…
Extends weak continuity of Yang-Mills connections to a broader class.
problem Weak compactness of Ω-Yang-Mills connections. method Compensation compactness argument applied to Yang-Mills fields.
result Weak continuity result extended to Ω-Yang-Mills connections. Sparse Gaussian processes with compact kernels for faster inference.
problem Efficient Gaussian process inference with high computational complexity.
method Parametric families of compactly-supported kernels for sparse matrix representations.
result Sub-quadratic inference complexity and improved performance on real-world tasks.
When studying the causal propagation of a field in a globally hyperbolic spacetime M, one often wants to express the physical intuition that it has compact support in spacelike directions, or that its support is a spacelike compact set. We compare a number of logically distinct formulations of this idea, and of the com…
Gradient estimate proved for Donaldson's equation on Kähler manifolds.
problem Proving gradient estimates for Donaldson's equation on compact Kähler manifolds.
method Using uniform upper bounds for trωχφ and Alexandrov-Bakelman-Pucci (ABP) maximum principle. result Gradient estimate for Donaldson's equation derived from uniform bounds.
Derives generalizations of the long neck principle and spectral width inequality.
problem Understanding the spectral width of geodesic collar neighborhoods.
method Spinorial Callias operator approach and relative Gromov-Lawson pair.
result Generalizations of the long neck principle and spectral width inequality.
In this paper we extend to non-compact Riemannian manifolds with boundary the use of two important tools in the geometric analysis of compact spaces, namely, the weak maximum principle for subharmonic functions and the integration by parts. The first one is a new form of the classical Ahlfors maximum principle whereas …