arXiv research

Abstract reviews recent Lagrangian analysis on immersions into higher dimensions.

For a smooth manifold $M$, it was shown in \cite{BPH} that every affine connection on the tangent bundle $TM$ naturally gives rise to covariant differentiation of multivector fields (MVFs) and differential forms along MVFs. In this paper, we generalize the covariant derivative of \cite{BPH} and construct covariant deri…

Defines a bundle map for currents on manifolds using higher covariant derivatives.

We show that solutions to certain higher-order intrinsic geometric flows on a compact manifold, including some flows generated by the ambient obstruction tensor, are unique. With the goal of providing a complete self-contained proof, details surrounding map covariant derivatives and a careful application of the DeTurck…

We obtain a complete time expansion of the pull-back operator generated by a real analytic flow of real analytic automorphisms acting on analytic tensor sections of a manifold. Our expansion is given in terms of multiple Lie derivatives. Motivated by this expansion, we provide a rather simple and explicit estimate for …

WeSpeR speeds up non-linear shrinkage for high-dimensional weighted covariance.

A regular normal parabolic geometry of type $G/P$ on a manifold $M$ gives rise to sequences $D_i$ of invariant differential operators, known as the curved version of the BGG resolution. These sequences are constructed from the normal covariant derivative $\na^\om$ on the corresponding tractor bundle $V,$ where $\om$ is…

The paper generalizes Cartan Geometry using Polacek and Siegel's approach.

We consider the gradient flow of hypersurfaces immersed in the Euclidean space associated to geometric energy functionals. We show that for particular functionals depending by higher covariant derivatives of the curvature, singularities in finite time cannot occur during the evolution. Such geometric functionals are re…

The very definition of an Einstein metric implies that all its geometry is encoded in the Weyl tensor. With this in mind, in this paper we derive higher-order Bochner type formulas for the Weyl tensor on a four dimensional Einstein manifold. In particular, we prove a second Bochner type formula which, formally, extends…

The aim of this work is to develop a systematic manner to close overdetermined systems arising from conformal Killing tensors (CKT). The research performs this action for 1-tensor and 2-tensors. This research makes it possible to develop a new general method for any rank of CKT. This method can also be applied to other…

We introduce a Hilbert $A$-module structure on the higher oscillatory module, where $A$ denotes the $C^*$-algebra of bounded endomorphisms of the basic oscillatory module. We also define the notion of an exterior covariant derivative in an $A$-Hilbert bundle and use it for a construction of an $A$-elliptic complex of d…

Proposes a method to represent high-dimensional covariates for causal inference.

The paper studies Stein-Weiss operators on symmetric tensors, extending previous work.

Deep learning extracts terrain texture covariates for geostatistical modeling.

We consider the variational complex on infinite jet space and the complex of variational derivatives for Lagrangians of multidimensional paths and study relations between them. The discussion of the variational (bi)complex is set up in terms of a flat connection in the jet bundle. We extend it to supercase using a part…

Improved LDA using a nonlinear covariance estimator for better performance.

Invariant covariant derivatives on homogeneous spaces are characterized.

The Killing tensor equation is a first order differential equation on symmetric covariant tensors that generalises to higher rank the usual Killing vector equation on Riemannian manifolds. We view this more generally as an equation on any manifold equipped with an affine connection, and in this setting derive its prolo…

A simple theory of the covariant derivatives, deformed derivatives and relative covariant derivatives of multivector and multiform fields is presented using algebraic and analytical tools developed in previous papers.

The paper introduces new boundary operators and proves higher order CR Sobolev trace inequalities for Siegel domain and complex ball.

The paper shows objective derivatives are covariant derivatives on Riemannian metrics.

Constructs covariant derivatives for Ehresmann connections.

Extends magnetic flow theory results to higher dimensions.

Unified normative modeling for neuroimaging phenotypes using denoising diffusion models.

The paper refines classical covariance asymptotics using geometric information geometry.

Homotopy equivalence between formalities with different covariant derivatives.

Scattering representations simplify SBI for images without extra compression.

In this study, we tested the interaction effect of multimodal datasets using a novel method called the kernel method for detecting higher order interactions among biologically relevant mulit-view data. Using a semiparametric method on a reproducing kernel Hilbert space (RKHS), we used a standard mixed-effects linear mo…

The paper introduces boundary operators for Poincaré-Einstein manifolds and proves higher order trace inequalities.

The paper analyzes multivariate payments in multi-state life insurance using Markovian state processes.

Derives a formula for the k-th covariant derivative of tensor fields.

We present a unified derivation of covariant time derivatives, which transform as tensors under a time-dependent coordinate change. Such derivatives are essential for formulating physical laws in a frame-independent manner. Three specific derivatives are described: convective, corotational, and directional. The covaria…

Formulae for non-symmetric connections derived from covariant derivatives.

This paper derives an explicit formula for Branson's Q-curvature in even-dimensional conformal geometry. The ingredients in the formula come from the Poincare metric in one higher dimension; hence the formula is called holographic. When specialized to the conformally flat case, the holographic formula expresses Q-curva…

The paper connects higher-dimensional mechanics to Lie n-algebroids.

Sharp inequalities for radial functions on hyperbolic spaces without boundary conditions.

Study covariant derivatives of eigenfunctions on curved spaces, proving they are scalar multiples of the functions.

The algebra of differential geometry operations on symmetric tensors over constant curvature manifolds forms a novel deformation of the sl(2,R) [semidirect product] R^2 Lie algebra. We present a simple calculus for calculations in its universal enveloping algebra. As an application, we derive generating functions for t…

Paper analyzes holdout cross-validation for large non-Gaussian covariance estimation.

We present a derivation and theoretical investigation of the Adams-Bashforth and Adams-Moulton family of linear multistep methods for solving ordinary differential equations, starting from a Gaussian process (GP) framework. In the limit, this formulation coincides with the classical deterministic methods, which have be…

The Gauss-Bonnet curvature of order $2k$ is a generalization to higher dimensions of the Gauss-Bonnet integrand in dimension $2k$, as the usual scalar curvature generalizes the two dimensional Gauss-Bonnet integrand. In this paper, we evaluate the first variation of the integrals of these curvatures seen as functionals…

This is the first of a series of two technical papers devoted to the analysis of holonomy invariants in strict higher gauge theory with end applications in higher Chern--Simons theory. For a flat 2--connection, we define the 2-holonomy of surface knots of arbitrary genus and determine its covariance properties under 1-…

We consider complete Riemannian manifolds with a controlled growth of the covariant derivatives of Ricci curvatures up to order $k-2$ and a controlled decay of the injectivity radii. On such manifolds we construct distance-like functions with a control on covariant derivatives up to order $k$. Alternatively, the assump…

A new algorithm COVA-FC improves subgroup-fair clustering efficiency.

The geometrical structure known as Tulczyjew triple has been used with success in analytical mechanics and first order field theory to describe a wide range of physical systems including Lagrangian/Hamiltonian systems with constraints and/or sources, or with singular Lagrangian. Starting from the first principles of th…

Study on stochastic covariant derivatives in curved space-time.

Estimates for covariant derivatives and Riesz transforms on differential forms.