Study geometric bounds on generalized Ricci flow.
problem No specific problem stated; focuses on bounds.
method Analogous geometric quantities and bounds proven.
result Geometric and analytic bounds established.
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48 results for “geometric bounds”
New geometric SDEs and discretizations on Riemannian manifolds with error bounds.
problem Modeling diffusion processes on Riemannian manifolds with geometric SDEs.
method Introduced a new construction of geometric SDEs and provided non-asymptotic error bounds.
result First non-asymptotic error bound for geometric Euler-Murayama discretization.
We show that some hyperbolic 3-manifolds which are tessellated by copies of the regular ideal hyperbolic tetrahedron embed geodesically in a complete, finite volume, hyperbolic 4-manifold. This allows us to prove that the complement of the figure-eight knot geometrically bounds a complete, finite volume hyperbolic 4-ma…
Paper finds new 3D shapes that can be inside a 4D space.
problem Finding new 3D shapes with specific properties.
method Examined arithmetic hyperbolic 3-manifolds and their homology.
result Discovered infinitely many 3D shapes that are rational homology spheres and can bound geometrically.
Lower bounds for Hodge-Laplacian spectrum on orbifolds.
problem Finding bounds for the essential spectrum of Hodge-Laplacian.
method Deriving lower bounds for the essential spectrum of the Hodge-Laplacian on geometrically finite orbifolds and their suborbifolds.
result Lower bounds for the essential spectrum of the Hodge-Laplacian.
A finite-volume hyperbolic 3-manifold geometrically bounds if it is the geodesic boundary of a finite-volume hyperbolic 4-manifold. We construct here an example of non-compact, finite-volume hyperbolic 3-manifold that geometrically bounds. The 3-manifold is the complement of a link with eight components, and its volume…
New bound on neural network generalization error using geometric complexity.
problem Understanding the generalization capabilities of deep neural networks.
method Derive a new upper bound on generalization error using margin-normalized geometric complexity.
result Empirical validation of the bound for ResNet-18 on CIFAR-10 and CIFAR-100 datasets.
Abstract: Geometrically reformulates estimation theory for finite-dimensional C*-algebras.
problem Estimation theory for finite-dimensional C*-algebras.
method Geometrical formulation of estimation theory.
result Derivation of Cramer-Rao and Helstrom bounds.
Study wall singularities in spaces with upper curvature bounds.
problem Understanding singularities in spaces with curvature constraints.
method Geometric structure theorem and geometric characterization for codimension one and two.
result Necessary and sufficient conditions for singular sets to be of codimension at least two.
The paper proves geometric bordisms for specific hyperbolic surfaces.
problem Proving geometric bordisms for Accola-Maclachlan, Kulkarni, and Wiman surfaces.
method Explicit geodesic embeddings and geometric proofs for specific surfaces.
result The surfaces bound geometrically compact hyperbolic 3-manifolds.
We establish a lower bound on the complexity orientable locally orientable geometric 3-orbifolds in terms of Delzant's T-invariants of their orbifold-fundamental groups, generalizing previously known bounds for complexity of 3-manifolds.
The paper classifies group-actions on surfaces of small genus, focusing on bounding and geometrically bounding cases.
problem Classifying group-actions on surfaces of small genus, particularly focusing on bounding and geometrically bounding cases.
method Analyzing large group-actions on surfaces of genus 3, distinguishing between bounding and geometrically bounding cases.
result Identifies which large group-actions on surfaces of genus 3 are bounding or geometrically bounding.
We show that any two geometric triangulations of a closed hyperbolic, spherical or Euclidean manifold are related by a sequence of Pachner moves and barycentric subdivisions of bounded length. This bound is in terms of the dimension of the manifold, the number of top dimensional simplexes and bound on the lengths of ed…
The paper explores symplectic foliations and their leaves on manifolds.
problem Which manifolds can be realized as leaves of codimension-1 symplectic foliations?
method Observations and deformations of symplectic structures; examples of manifolds.
result Examples of manifolds that can be realized as leaves but not as symplectic leaves.
In this note, we show that there exist cusped hyperbolic -manifolds that embed geodesically, but cannot bound geometrically. Thus, being a geometric boundary is a non-trivial property for such manifolds. Our result complements the work by Long and Reid on geometric boundaries of compact hyperbolic -manifolds, and…
Study eigenvalues on quaternion-Kähler manifolds with geometric bounds.
problem Estimating eigenvalues on quaternion-Kähler manifolds.
method Lower bounds derived from modulus of continuity estimates for heat equation solutions and Laplace comparison theorem.
result Established bounds for first nonzero eigenvalues in terms of dimension, diameter, and scalar curvature.
This work addresses the classic machine learning problem of online prediction with expert advice. A new potential-based framework for the fixed horizon version of this problem has been recently developed using verification arguments from optimal control theory. This paper extends this framework to the random (geometric…
New invariant for hyperbolic surfaces, geometric criterion for domains.
problem Geometric criterion for bounded domains in complex plane.
method Renormalized volume type invariant on hyperbolic surfaces.
result New geometric criterion for bounded domains in complex plane.
Optimal geometric estimates for Kähler manifolds with bounded Nash entropy
problem Optimal geometric estimates for compact Kähler manifolds
method Proving Sobolev-type inequality and local volume noncollapsing with optimal exponents
result Uniformly bounded -Nash entropy
We explain some interesting relations in the degree three bounded cohomology of surface groups. Specifically, we show that if two faithful Kleinian surface group representations are quasi-isometric, then their bounded fundamental classes are the same in bounded cohomology. This is novel in the setting that one end is d…
We show that the number of isometry classes of cusped hyperbolic -manifolds that bound geometrically grows at least super-exponentially with their volume, both in the arithmetic and non-arithmetic settings.
New bounds for geometric flows of Hermitian metrics established.
problem Regularity of geometric flows of Hermitian metrics.
method Establishing a a priori bound for smooth curves of Hermitian metrics.
result New regularity result for Hermitian curvature flows, including the second Chern-Ricci flow.
Proves upper bounds for heat kernels evolving on manifolds.
problem Bounding heat kernels on evolving manifolds.
method Logarithmic Sobolev inequalities and ultracontractivity estimates.
result Gaussian upper bounds for heat kernels are derived.
The paper bounds Pachner moves and systoles in hyperbolic 3-manifolds.
problem Bounding Pachner moves and systoles in cusped hyperbolic 3-manifolds.
method Using geometric ideal triangulations and dihedral angles, the paper gives bounds on Pachner moves and systoles.
result Lower bounds on systole length and Pachner move sequence length.
Geometric tempering fails for Langevin dynamics, proving convergence limits.
problem Proving convergence and limitations of geometric tempering for Langevin dynamics.
method Theoretical investigation of geometric tempering using Langevin dynamics.
result Geometric tempering can lead to exponential time convergence and poor functional inequalities.
Bootstrap bounds on Einstein manifolds using semidefinite programming.
problem Bounding geometric data of closed Einstein manifolds.
method Semidefinite programming applied to consistency conditions of geometric data.
result Bootstrap bounds translate to constraints on Kaluza-Klein modes.
Study geometric structures in transfer learning to avoid negative transfer.
problem Understanding information-theoretic limits of transfer learning without exploiting domain geometry.
method Integrates geometric structure into linear regression models, using Gram matrices of source and target domains.
result Proposes an interpolation estimator that matches minimax lower bound and outperforms existing methods.
The paper shows bounds for a geometric flow related to Type IIB string theory.
problem Establishing derivative bounds for a geometric flow in non-Kähler geometry.
method Unified formulation of the flow with Ricci flow, proving bounds from metric and torsion 1-form uniform bounds.
result Derivative bounds follow from uniform metric and torsion 1-form bounds.
The paper examines torsional rigidity bounds under geometric flows.
problem Torsional rigidity behavior under geometric flows.
method Bounds on torsional rigidity derived under Ricci Flow and Inverse Mean Curvature Flow.
result Inequalities of comparison with the flat disk for torsional rigidity.
Estimates mean curvature flow with geometric bounds.
problem Controlling mean curvature flow dynamics.
method Pointwise estimate using initial geometry and jHAj bound.
result Extension theorem and blowup rate estimate of HA.
Uniform Sobolev, interpolation and geometric Calderón-Zygmund inequalities for graph hypersurfacesmath.DG
The study shows that close hypersurfaces have uniformly bounded inequalities.
problem Bounding inequalities for close hypersurfaces.
method Analyzing families of smooth hypersurfaces close to a fixed one.
result Uniformly bounded constants in Sobolev, Gagliardo-Nirenberg, and geometric Calderón-Zygmund inequalities.
Study sets lower bounds for Kähler manifolds' Laplacian eigenvalues.
problem Finding bounds for eigenvalues on Kähler manifolds.
method Establishes lower bounds using geometric data like dimension, diameter, and curvature.
result Proves bounds for Laplacian eigenvalues on Kähler manifolds.
This paper explores and ties together three themes. The first is to establish regularity of a metric tensor, on a manifold with boundary, on which there are given Ricci curvature bounds, on the manifold and its boundary, and a Lipschitz bound on the mean curvature of the boundary. The second is to establish geometric c…
Last SGD iterate bounds for overparameterized linear regression.
problem Analyzing the last iterate risk bounds of SGD with decaying stepsize for overparameterized linear regression.
method Problem-dependent analysis of last iterate risk bounds of SGD with geometrically decaying stepsize.
result Proved nearly matching upper and lower bounds on the excess risk for last iterate SGD with geometrically decaying stepsize.
In this paper we prove two results, one semi-historical and the other new. The semi-historical result, which goes back to Thurston and Riley, is that the geometrization theorem implies that there is an algorithm for the homeomorphism problem for closed, oriented, triangulated 3-manifolds. We give a self-contained proof…
In this paper we provide a Bonnesen-style inequality which gives a lower bound for the isoperimetric deficit corresponding to a closed convex curve in terms of some geometrical invariants of this curve. Moreover we give a geometrical interpretation for the case when equality holds.
We show that a Laplace isospectral family of two dimensional Riemannian orbifolds, sharing a lower bound on sectional curvature, contains orbifolds of only a finite number of orbifold category diffeomorphism types. We also show that orbifolds of only finitely many orbifold diffeomorphism types may arise in any collecti…
A closed connected hyperbolic -manifold bounds geometrically if it is isometric to the geodesic boundary of a compact hyperbolic -manifold. A. Reid and D. Long have shown by arithmetic methods the existence of infinitely many manifolds that bound geometrically in every dimension. We construct here infinitely …
Geometric bounds for low Steklov eigenvalues on hyperbolic surfaces with boundaries.
problem Finding lower bounds for low Steklov eigenvalues of hyperbolic surfaces with geodesic boundaries.
method Analysis of eigenfunction behavior on an adapted thick-thin decomposition for hyperbolic surfaces with geodesic boundaries.
result Sharp geometric lower bounds for low Steklov eigenvalues that depend on the shortest multi-geodesic disconnecting the surfaces.
On geometrically finite hyperbolic manifolds , including those with non-maximal rank cusps, we give upper bounds on the number of resonances of the Laplacian in disks of size as . In particular, if the parabolic subgroups of satisfy a certain Diophantine condition, the bou…
Study introduces indecomposability for varifolds, leading to geometric consequences.
problem Understanding the structure of varifolds and their connectedness properties.
method Introducing indecomposability and related concepts for varifolds.
result Substantial geometric consequences derived from the connectedness properties of varifolds.
In this paper we research the differential geometric and algebro-geometric proper- ties of the noncollasping limit in the conical continuity equation.
It is well known that an arbitrary closed orientable -manifold can be realized as the unique boundary of a compact orientable -manifold, that is, any closed orientable -manifold is cobordant to zero. In this paper, we consider the geometric cobordism problem: a hyperbolic -manifold is geometrically bounding…
How Many Links Fit in a Box?math.GT
New upper bound for non-trivial links in a cube, geometric approach.
problem Upper limit for disjoint non-trivial links in a unit cube.
method Geometric argument, no multi-linear properties used.
result Proves an upper bound for all non-trivial links.
We study Riemannian metrics on compact, torsionless, non-geometric -manifolds, i.e. whose interior does not support any of the eight model geometries. We prove a lower bound "à la Margulis" for the systole and a volume estimate for these manifolds, only in terms of an upper bound of entropy and diameter. We then ded…
Researchers find a way to bound the complexity of certain subgroup geometric invariants.
problem Understanding the geometric invariants of subgroups of direct products of free groups.
method Generalizing techniques for 'pushing fillings' into normal subgroups.
result Finitely presented subgroups of direct products of three free groups and subgroups of finiteness type in a direct product of free groups have Dehn functions bounded above by .
Constructs flows on manifolds with small curvature, proving Euclidean topology.
problem Geometric structure of manifolds with unbounded curvature.
method Distance like functions with integral hessian bound, Ricci flows.
result Manifolds with Ricci lower bound, non-negative scalar curvature, bounded entropy, Ahlfors -regular and small curvature concentration are topologically Euclidean.
We consider the problem of model selection in Gaussian Markov fields in the sample deficient scenario. In many practically important cases, the underlying networks are embedded into Euclidean spaces. Using the natural geometric structure, we introduce the notion of spatially stationary distributions over geometric grap…