Classifies singular foliations of a specific type and studies their extensions.
problem Classifying singular foliations of a particular type and understanding their extensions.
method Introduces and classifies singular foliations of bk+1-type, proving they are encoded by k-th order foliations. result Singular foliations of bk+1-type are encoded by k-th order foliations, and these groupoids fiber over certain character stacks. Derives a formula for the k-th covariant derivative of tensor fields.
problem Finding a formula for the k-th covariant derivative of tensor fields.
method Introducing symbols P and Q depending on Christoffel symbols, deriving a formula (3.1).
result Derives a formula for the k-th covariant derivative of tensor fields.
This article presents the further steps of the previously done studies taking into consideration the k-th order extensions of a complex manifold. In the previous studies higher order vertical and complete lifts of structures on the complex manifold were introduced. Presently, k-th extended spaces of a product manifold …
This paper develops a new method for constructing splines on Lie groups using Poisson equation solutions.
problem Existing methods for constructing splines on Lie groups have limitations and assumptions that may not reflect actual curves.
method The paper introduces a new approach using solutions of the Poisson equation on Lie groups to construct splines.
result The new method allows for global splines with arbitrary initial conditions, improving curve reconstruction.
New set class preserves Fourier series terms for planar ovals, leading to isoperimetric inequalities.
problem Investigate geometric properties of kth Order Preserving Sets and ovals. method Introduce and analyze kth Order Preserving Sets and Midpoint Sets; study geometric properties and isoperimetric inequalities. result Established an isoperimetric-type inequality relating perimeter and area of ovals and their associated sets.
Proposes a new method for estimating non-pathwise differentiable functional parameters.
problem Estimating dose-response curves for continuous exposure.
method Targeted Highly Adaptive Lasso (HAL) for non-pathwise differentiable functional parameters.
result The Targeted HAL-MLE achieves dimension-free rates up to log(n) factors and outperforms other methods in simulations.
The paper improves bounds on how many squares can fit in a rectangle and still have stable homology.
problem Homological stability in the space direction of square configurations.
method Analyzing the ordered configuration space of squares in a rectangle.
result Most rectangles can be almost entirely filled with squares and still have stable homology.
New GAN loss functions improve image generation quality and stability.
problem Improving the performance of GANs in generating high-quality images.
method Introducing least kth-order GAN (LkGAN) and Rényi-centric GAN loss functions. result The proposed loss functions lead to better image quality and stability.
Motivated by the theory of isoparametric hypersurfaces, we study submanifolds whose tubular hypersurfaces have some constant "higher order mean curvatures". Here a k-th order mean curvature Qk (k≥1) of a hypersurface Mn is defined as the k-th power sum of the principal curvatures, or equivalently, of the…
The study of higher-order homology embeddings for manifold topology.
problem Understanding the structure of higher-order homology embeddings to disclose geometric or topological information.
method Analysis of the null space of the k-th order Laplacian and proposing an algorithm to factorize the homology embedding. result The proposed spectral loop detection algorithm is more efficient and effective on various data types.
New equations for pseudo-spherical surfaces found, with unique isometric immersions.
problem Finding isometric immersions for pseudo-spherical surfaces described by k-th order evolution equations.
method Investigating the relationship between pseudo-spherical surfaces and k-th order evolution equations, proving the existence of unique isometric immersions.
result There is only one type of k-th order evolution equations that admit local isometric immersions, with universal coefficients of the second fundamental form.
Optimal control trajectories have limited irregularities.
problem Regularity of time-optimal control trajectories in control-affine systems.
method Generic conditions on drift and controlled vector field are used to prove smoothness out of a countable set of times, up to K-th order iterated singularities.
result Control trajectories are smooth out of a countable set of times, with singularities limited to K-th order iterated.
We investigate the eigenvalues of the buckling problem of arbitrary order on compact domains in Euclidean spaces and spheres. We obtain universal bounds for the kth eigenvalue in terms of the lower eigenvalues independently of the particular geometry of the domain.
We generalize reduction theorems for classical connections to operators with values in k-th order natural bundles. Using the first reduction theorem in order two we classify all (0,2)-tensor fields on the cotangent bundle of a manifold with a linear (non-symmetric) connection.
This paper studies eigenvalues of the buckling problem of arbitrary order on bounded domains in Euclidean spaces and spheres. We prove universal bounds for the k-th eigenvalue in terms of the lower ones independent of the domains. Our results strengthen the recent work in [28] and generalize Cheng-Yang's recent estimat…
In this paper, we computed the first three coefficients of the asymptotic expansion of Zelditch. We also proved that in general, the k-th coefficient is a polynomial of the curvature and its derivative of weight k.
Taylorized training improves neural network training at finite width.
problem Understanding and improving neural network training at finite width.
method Training the k-th order Taylor expansion of the neural network at initialization.
result Taylorized training agrees with full neural network training better as k increases and can significantly close the performance gap.
We consider the higher order buckling eigenvalues of the following Dirichlet poly-Laplacian in the unit sphere (−Δ)pu=Λ(−Δ)u with order p(≥2). We obtain universal bounds on the (k+1)th eigenvalue in terms of the first kth eigenvalues independent of the domains. In particular, for p=2, our result is shar…
In this paper, we obtain the sharp k-th order Sobolev inequalities in the hyperbolic space ${\H}^n$ for all k=1,2,3,⋯. This gives an answer to an open question raised by Aubin in [5, p.176-177] for $W^{k,2}({\H}^n)$ with k>1. In addition, we prove that the associated Sobolev constants are optimal.
The paper studies constant kth-mixed curvature on Hermitian manifolds and finds self-duality and Kähler conditions.
problem Investigating constant kth-mixed curvature on Hermitian manifolds. method Analyzing Hermitian manifolds with convex combinations of Chern Ricci curvature and holomorphic sectional curvature.
result Compact Hermitian surfaces with constant kth-mixed curvature are self-dual, and if k=2, the metric is Kähler. Quantitative estimates for Q-curvature near minimizing metrics on Riemannian manifolds.
problem Estimating the Q-curvature near minimizing metrics on Riemannian manifolds. method Proving quantitative estimates for the total k-th order Q-curvature functional near minimizing metrics. result Existence of quantitative estimates for the Q-curvature deficit controlling higher powers of the distance to the minimizing set. In this paper, we establish some sharp inequalities between the volume and the integral of the k-th mean curvature for k+1-convex domains in the Euclidean space. The results generalize the classical Alexandrov-Fenchel inequalities for convex domains. Our proof utilizes the method of optimal transportation.
The article derives some novel independence measures and contrast functions for Blind Source Separation (BSS) application. For the kth order differentiable multivariate functions with equal hyper-volumes (region bounded by hyper-surfaces) and with a constraint of bounded support for k>1, it proves that equality …
Given a set S of n points in general position, we consider all k-th order Voronoi diagrams on S, for k=1,...,n, simultaneously. We deduce symmetry relations for the number of faces, number of vertices and number of circles of certain orders. These symmetry relations are independent of the position of the sites in S. As…
The paper explores Lagrangians with simplified Euler-Lagrange equations.
problem Variational problems with Euler-Lagrange equations of reduced order.
method Geometrical construction to derive a family of Lagrangians.
result Lagrangians with reduced-order Euler-Lagrange equations are polynomials in the highest-order derivatives.
Lower bounds for Steklov eigenvalues derived from Markov operators and mass concentration.
problem Finding bounds for Steklov eigenvalues in various geometric settings.
method Developed techniques involving accelerated Markov operators and mass concentration deformations.
result Proved higher order Cheeger type inequalities for Steklov eigenvalues.
Let S be a compact connected oriented surface with one boundary component, and let P be the fundamental group of S. The Johnson filtration is a decreasing sequence of subgroups of the Torelli group of S, whose k-th term consists of the self-homeomorphisms of S that act trivially at the level of the k-th nilpotent quoti…
We study trend filtering, a recently proposed tool of Kim et al. [SIAM Rev. 51 (2009) 339-360] for nonparametric regression. The trend filtering estimate is defined as the minimizer of a penalized least squares criterion, in which the penalty term sums the absolute kth order discrete derivatives over the input points…
In this paper, we obtain a sharp upper bound for the sum of the first k-th eigenvalues for this Dirichlet problem of poly-Laplacian with any order, which is viewed as an extension of the result due to Cheng and Wei (Journal of Differential Equations, 255 (2013), 220-233). In particular, if l=2 and k is large enou…
Proves conditions for Fourier transforms in rank 1 symmetric spaces.
problem Understanding Fourier transform bounds in symmetric spaces.
method Proves sufficient and necessary conditions using Lipschitz and Fourier type integral conditions.
result Establishes bounds for Fourier transforms in rank 1 symmetric spaces with specific moduli of continuity.
The paper confirms a conjecture about Hardy-Sobolev-Maz'ya inequalities and Green's functions on hyperbolic spaces.
problem Sharp constant in Hardy-Sobolev-Maz'ya inequalities and Green's functions on hyperbolic spaces.
method Fourier analysis techniques on hyperbolic spaces and Green's function estimates.
result The sharp constant in the 2n−1-th order Hardy-Sobolev-Maz'ya inequality in the upper half space of dimension n coincides with the best 2n−1-th order Sobolev constant when n is odd and n≥9. Constructs graded jet bundles for Z-graded manifolds and vector bundles.
problem Generalizing jet manifolds to Z-graded structures for differential equations.
method Directly constructs the sheaf of sections of the k-th order jet bundle of a Z-graded vector bundle.
result Establishes a graded version of Atiyah Lie algebroid.
The tautological ring of strata of differentials is characterized based on the presence of poles.
problem Characterizing the tautological ring of strata of differentials.
method Using the κ, ψ, and η classes of moduli spaces of pointed smooth curves.
result The tautological ring is generated by η only if there are no poles of order k, otherwise by the ψ classes corresponding to the poles of order k.
A new derivation is given of Branson's factorization formula for the conformally invariant operator on the sphere whose principal part is the k-th power of the scalar Laplacian. The derivation deduces Branson's formula from knowledge of the corresponding conformally invariant operator on Euclidean space (the k-th power…
Study convex capillary hypersurfaces with prescribed curvature in a spherical cap.
problem Prescribed curvature problem for convex capillary hypersurfaces.
method Reformulated as Hessian quotient equation with Robin boundary condition.
result Existence of strictly convex capillary hypersurface with prescribed curvature.
The paper proves rigidity for complex Kleinian groups.
problem Characterizing hyperconvex subgroups of complex Kleinian groups.
method Analyzing critical exponents and using representations of PSL(2, C).
result Uniform lattices in PSL(2, C) are the only (d-k)-hyperconvex subgroups with a specific critical exponent.
In this paper, we investigate the sample size requirement for a general class of nuclear norm minimization methods for higher order tensor completion. We introduce a class of tensor norms by allowing for different levels of coherence, which allows us to leverage the incoherence of a tensor. In particular, we show that …
Characterizes Euclidean balls with lower bounded k-th mean curvature.
problem Characterizing convex bodies with lower bounded k-th mean curvature.
method New isoperimetric-type inequality and sharp characterizations.
result Euclidean balls uniquely satisfy the condition.
On a closed weighted Riemannian manifold with nonnegative Bakry-Émery Ricci curvature, it is shown that the ratio of the k-th to first eigenvalues of the weighted Laplacian is dominated by 641k2, using an argument via the Cheeger constant. While improving the previous exponential upper bound, the order of k here…
The paper studies curvature flows in hyperbolic space and proves geometric inequalities.
problem Proving geometric inequalities in hyperbolic space using curvature flows.
method Locally constrained curvature flows, h-convexity, and shifted principal curvatures.
result Established new sharp geometric inequalities comparing curvature integrals to quermassintegrals.
A new graph convolution framework for node and graph-centric tasks.
problem Graph modeling challenges, especially for high-order and adaptive structures.
method Introduces k-th order convolution operator and adaptive filtering module for graph neural networks. result HA-GCN outperforms state-of-the-art models in node classification and molecule property prediction.
New findings show margins are not sufficient for explaining gradient boosting performance.
problem The inadequacy of margin explanations in explaining the performance of gradient boosting.
method Demonstrated and proved a stronger margin-based generalization bound for boosted classifiers.
result Proved a stronger margin-based generalization bound that explains the performance of modern gradient boosters.
We study eigenvalues of polyharmonic operators on compact Riemannian manifolds with boundary (possibly empty). In particular, we prove a universal inequality for the eigenvalues of the polyharmonic operators on compact domains in a Euclidean space. This inequality controls the kth eigenvalue by the lower eigenvalues,…
This paper demonstrates the power of the calculus developed in the two previous parts of the series for all real forms of the almost Hermitian symmetric structures on smooth manifolds, including e.g. conformal Riemannian and almost quaternionic geometries. Exploiting some finite dimensional representation theory of sim…
The paper examines affine properties of differential strata on curves.
problem Characterizing affine structures in strata of differentials.
method Analyzing affine varieties and Teichmüller dynamics.
result Affine strata do not contain complete curves with certain pole orders.
Paper proves stability of quermassintegral inequalities using inverse curvature flow.
problem Stability of quermassintegral inequalities for nearly spherical sets.
method Inverse curvature flow with special rescaling to study quermassintegral inequalities.
result Decreasing rate of k-th quermassintegral is faster than Fraenkel asymmetry for nearly spherical sets.
Study soap bubbles with almost constant higher-order mean curvature, proving unique limits under certain conditions.
problem Understanding the asymptotic behavior of soap bubbles with almost constant higher-order mean curvature.
method Analyzing sequences of bounded C2-domains in Rn+1 converging in volume and perimeter, with k-th mean curvature functions converging in L1. result Finite unions of mutually tangent balls are the only possible limits under natural mean convexity and L∞-control on the mean curvature outside a set of vanishing area. On any compact manifold of dimension n≥3 with boundary, we prescibe any finite part of the Steklov spectrum whithin a given conformal class. In particular, we prescribe the multiplicity of the first eigenvalues. On a compact surface with boundary, we show that the multiplicity of the k-th eigenvalue is bounded i…