Study investigates classification with unknown label noise in non-compact feature spaces.
problem Classification in the presence of unknown class-conditional label noise in non-compact feature spaces.
method Determines minimax optimal learning rates and presents an adaptive algorithm for classification.
result Optimal learning rates differ from those without label noise, displaying interesting threshold behavior.
Paper proves geodesic ball maximizes second Robin eigenvalue in non-compact symmetric spaces.
problem Maximizing the second Robin eigenvalue in non-compact rank-1 symmetric spaces.
method Quantitative spectral inequality for the second Robin eigenvalue.
result Geodesic ball maximizes the second Robin eigenvalue among domains of the same volume.
We study integral geometric properties of non-compact harmonic spaces.
The paper studies spaces of non-compact real algebraic curves and their uniformisation.
problem Understanding the spaces of non-compact real algebraic curves and their uniformisation.
method Construction of spaces of non-compact real algebraic curves and description of their connected components using Fuchsian groups.
result Any connected component of the spaces of non-compact real algebraic curves is homeomorphic to a quotient of a finite-dimensional real vector space by a discrete group.
New examples of isoparametric families on non-compact symmetric spaces.
problem Constructing isoparametric families with non-austere focal sets.
method New extension method from Euclidean spaces to symmetric spaces of non-compact type.
result First examples of isoparametric families on non-compact symmetric spaces.
The paper studies HKKN stratifications for non-compact spaces and proves convexity properties.
problem Proving convexity properties of moment maps for non-compact subsets.
method Algebraic and analytical study of HKKN stratifications for a vector space and compact Kähler manifold, then applying to non-compact subsets.
result Convexity properties of moment maps for invariant subsets are proven.
Volume comparison theorem for rank 1 symmetric spaces proved.
problem Volume comparison for symmetric spaces of non-compact type.
method Normalized Ricci--DeTurck flow to analyze volume functional and derive monotonicity properties.
result Volume comparison theorem established for rank 1 symmetric spaces of non-compact type.
Logarithmic Sobolev inequality proven for non-compact self-shrinkers.
problem Establishing a logarithmic Sobolev inequality for non-compact self-shrinkers.
method Using Alexandrov-Bakelman-Pucci (ABP) method to prove the inequality for Euclidean space, then applying this method to non-compact self-shrinkers.
result Optimal logarithmic Sobolev inequality for complete, non-compact, properly embedded self-shrinkers.
The paper studies how certain surfaces evolve in space-time.
problem Preserving the space-like condition of non-compact hypersurfaces.
method Prescribed mean curvature flow in generalized Robertson-Walker spaces.
result The flow preserves space-like condition and exists for infinite time.
Bonahon's method for compactifying Teichmüller space extended to non-compact surfaces.
problem Compactifying Teichmüller space for non-compact finite area surfaces.
method Using geodesic currents, extending Bonahon's construction.
result The method applies to surfaces of finite area.
The paper reformulates regression in infinite dimensions as an inverse problem, showing it's equivalent to compact inverse problems.
problem Learning a linear operator between Hilbert spaces from empirical observations.
method Reformulates regression as an inverse problem, proving equivalence to compact inverse problems under specific conditions.
result The inverse problem is equivalent to compact inverse problems in terms of spectral properties and regularisation theory.
Polynomial growth bounds for eigenfunctions on non-compact spaces.
problem Growth of eigenfunctions on non-compact spaces with Gaussian weight.
method Analyzing non-compact manifolds with Gaussian weight and drift Laplacian.
result Showed same polynomial growth bounds as Euclidean space for general tensors.
The paper shows neural networks can approximate functions over non-compact domains with non-polynomial activation.
problem Approximating functions over non-compact domains using neural networks.
method Using single-hidden-layer feedforward neural networks with non-polynomial activation functions over non-compact subsets of Euclidean spaces.
result Neural networks can approximate functions in weighted Ck-spaces and weighted Sobolev spaces over unbounded domains. In this paper, we prove that full irreducible curvature-adapted isoparametric submanifolds of codimension greater than one in a symmetric space of non-compact type are principal orbits of Hermann actions on the symmetric spaces under certain conditions.
In this work we extend the ODE Maximum principle of Hamilton to non-compact hypersurfaces using the Omari-Yau maximum principle at infinity. As an application of this result, we investigate Inverse Mean Curvature Flow (IMCF) of non-compact hypersurfaces in hyperbolic space. Specifically, we look at bounded graphs over …
We study the Cauchy data spaces of the strongly Callias-type operators using maximal domain on manifolds with non-compact boundary, with the aim of understanding the Atiyah-Patodi-Singer index and elliptic boundary value problems.
Researchers derived heat kernel expansions for non-compact spaces using Witten deformation.
problem Heat kernel expansions on non-compact spaces, especially for Witten Laplacians.
method Introduced parabolic distance and used it to derive asymptotic expansions.
result Derived an asymptotic expansion of trace of heat kernel for small-time t. The paper develops new methods to study sharp isoperimetric properties on complex spaces.
problem Sharp isoperimetric comparison on non-collapsed spaces with lower Ricci bounds.
method Original argument to estimate first and second variation of the area for isoperimetric sets, avoiding regularity theory.
result Generalizes results for smooth and non-compact manifolds, Alexandrov spaces, and convex bodies.
We investigate Inverse Mean Curvature Flow (IMCF) of non-compact hypersurfaces in hyperbolic space. Specifically, we look at bounded graphs over horospheres in Hn+1 and show long time existence of the flow. Along the way many important local estimates as well as global estimates are obtained. In addition,…
New spectral analysis on non-compact spaces.
problem Analyzing pseudo-Riemannian locally symmetric spaces.
method Initiating spectral analysis beyond classical settings.
result Recent results in non-compact spaces.
We construct new explicit toric scalar-flat K{ä}hler ALE metrics on weighted projective spaces of non-compact type, which we use to obtain smooth extremal K{ä}hler metrics on appropriate resolutions of orbifolds. In particular, we obtain new extremal metrics certain resolutions of weighted projective spaces of compact …
Random feature models approximate functions in Banach spaces efficiently.
problem Approximating functions in Banach spaces efficiently.
method Randomly initialized feature maps and linear readout training.
result Universal approximation in Bochner spaces for Banach space-valued models.
In this paper, we assume that all isoparametric submanifolds have flat section. The main purpose of this paper is to prove that, if a full irreducible complete isoparametric submanifold of codimension greater than one in a symmetric space of non-compact type admits a reflective focal submanifold and if it of real analy…
Constructs real algebraic functions with both compact and non-compact preimages.
problem Finding real algebraic functions with specific preimage properties.
method Explicit construction of real algebraic functions.
result Demonstrates real algebraic functions on non-compact manifolds with non-compact preimages.
We show that compact, locally symmetric spaces of non-compact type have positive simplicial volume. This gives a positive answer to a question that was first raised by Gromov in 1982. We provide a summary of results that are known to follow from positivity of the simplicial volume.
We study solutions for the Hodge laplace equation Δu=ω on p forms with Lr estimates for r>1. Our main hypothesis is that Δ has a spectral gap in L2. We use this to get non classical Lr Hodge decomposition theorems. An interesting feature is …
The purpose of this article is to prove existence of mass minimizing integral currents with prescribed possibly non-compact boundary in all dual Banach spaces and furthermore in certain spaces without linear structure, such as injective metric spaces and Hadamard spaces. We furthermore prove a weak∗-compactness theo…
No spin structures found in a hyperbolic 4D space.
problem Finding hyperbolic 4-manifolds without spin structures.
method Constructed a non-compact, orientable, hyperbolic 4-manifold.
result Demonstrated existence of a hyperbolic 4D space without spin structures.
Analytic torsion defined for non-compact Lie groups and discrete subgroups.
problem Defining and calculating analytic torsion for non-compact Lie groups and their discrete subgroups.
method Localised analytic torsion and relative analytic torsion defined for Lie groups of type I, using representations and discrete subgroups.
result Relative analytic torsion of (G,Γ) coincides with Lott L2 analytic torsion of a covering space. Develops sublinear Morse theory in symmetric spaces.
problem Understanding sublinear Morse properties in symmetric spaces.
method Theory of sublinearly Morse boundary and lemma in higher rank symmetric spaces.
result Proves sublinear Morse lemma in higher rank symmetric spaces.
Study eta invariant on non-compact manifolds with positive scalar curvature.
problem Proving geometric formulas and index theorems for uniformly positive scalar curvature metrics.
method Using Dirac-Schrödinger operators and relative eta invariant.
result New geometric formula for spectral flow and index formula for uniformly positive scalar curvature metrics.
Study differential operators on non-compact harmonic manifolds, finding conditions for radial fundamental solutions and dense heat-semigroups.
problem Conditions for differential operators on non-compact harmonic manifolds to have specific properties.
method Analyzing the algebra of differential operators, their commutation properties, and using geometric averages.
result Algebra of differential operators on non-compact harmonic manifolds has specific properties related to radial fundamental solutions and dense heat-semigroups.
The paper generalizes spectral section concepts to non-compact spaces.
problem Generalizing spectral sections to non-compact base spaces.
method Generalization to arbitrary base spaces, applications to cobordism theorems, investigation of Riesz continuity.
result If a family of operators has a spectral section, it is Riesz continuous.
We formulate a conjecture that arithmetic locally symmetric manifolds have simple homotopy type, and prove it for the non-compact case. More precisely, we show that, for any symmetric space S of non-compact type without Euclidean de Rham factors, there are constants a=a(S) and d=d(S) such that any non-compact arithmeti…
The Hodge decomposition is well-known for compact manifolds. The result has been extended by Kodaira to include non-compact manifolds and L2 forms. We further extend the Hodge decomposition to the Sobolev space H1 for general k-forms on non-compact manifolds of nonpositive constant sectional curvature. As a res…
Constructs explicit p-harmonic functions on specific Lie groups.
problem Finding explicit p-harmonic functions on a specific class of Lie groups.
method Constructs explicit p-harmonic functions on rank-one Lie groups of Iwasawa type.
result Proves existence of proper p-harmonic functions on these groups.
In this paper, we obtain a Cartan type identity for curvature-adapted isoparametric hypersurfaces in symmetric spaces of compact type or non-compact type. This identity is a generalization of Cartan-D'Atri's identity for curvature-adapted(=amenable) isoparametric hypersurfaces in rank one symmetric spaces. Furthermore,…
In this paper, we develop a loop group description of harmonic maps F:M→G/K ``of finite uniton type", from a Riemann surface M into inner symmetric spaces of compact or non-compact type. This develops work of Uhlenbeck, Segal, and Burstall-Guest to non-compact inner symmetric spaces. To be mo…
The paper extends rigidity results to non-compact domains and infinite energy maps.
problem Rigidity of maps in non-compact domains with specific conditions.
method Analyzing homomorphisms and pluriharmonic maps between different geometric spaces.
result Existence of ρ-equivariant pluriharmonic maps of infinite energy under certain conditions. In this paper, we introduce a deformation analysis of index theory over non compact manifolds, by use of new functional spaces which are the reduced version of Sobolev spaces. It allows to construct Fredholm theory for elliptic differential operators over non compact spaces which possibly do not have closed range with …
New coordinates show Toda flow is Morse-Smale.
problem Toda flow on full flags of Lie algebras.
method Introduced atlas and local coordinates.
result Toda flow on full flags is Morse-Smale.
For 1≤q≤n−2, we provide explicit examples to demonstrate non-compactness of the Neumann operator for the Kohn Laplacian acting on L2 (0,q)-forms on the unit ball in (2n+1)-dimensional Heisenberg space.
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.
Assigns compact set distance-like functions to non-compact geodesic spaces.
problem Assigning distance-like functions to compact sets in non-compact geodesic spaces.
method Assigns each compact set a distance-like function and studies the pseudo-metric on the space of compact subsets.
result Obtains a pseudo-metric on the space of compact subsets that is less than the Hausdorff distance.
In the first part of the paper we investigate some geometric features of Moser-Trudinger inequalities on complete non-compact Riemannian manifolds. By exploring rearrangement arguments, isoperimetric estimates, and gluing local uniform estimates via Gromov's covering lemma, we provide a Coulhon, Saloff-Coste and Varopo…
New example shows non-compact manifolds can lack Lp-Calderón-Zygmund inequalities.
problem Exploring Lp-Calderón-Zygmund inequalities on non-compact manifolds. method Developed a concrete example using local deformations of metrics.
result Found a non-compact manifold without Lp-Calderón-Zygmund inequalities. Develops Gaussian processes on non-compact Lie groups.
problem Invariance to symmetries in non-Euclidean spaces.
method Constructive techniques for stationary Gaussian processes.
result Makes non-Euclidean Gaussian processes compatible with standard software.
Paper develops a new method for harmonic maps into symmetric spaces.
problem Tackles harmonic maps of finite uniton number into symmetric spaces.
method Uses loop group description and DPW theory to transform results.
result Establishes a 1-1 relation between harmonic maps and normalized potentials.