arXiv research

A new tensorial metric describes geometry in 4D space.

Using the tractor calculus to study smooth metric measure spaces, we adapt results of Gover and Nurowski to give sharp metric obstructions to the existence of quasi-Einstein metrics on suitably generic manifolds. We do this by introducing an analogue of the Weyl tractor $W$ to the setting of smooth metric measure space…

Smooth deformations of a Minkowski type metric in a four-dimensional space-time manifold are considered. Deformations of the basic spin-tensorial fields associated with this metric are calculated and their application to calculating the energy-momentum tensor of a massive spin 1/2 particle is shown.

Traditional Recurrent Neural Networks assume vectorized data as inputs. However many data from modern science and technology come in certain structures such as tensorial time series data. To apply the recurrent neural networks for this type of data, a vectorisation process is necessary, while such a vectorisation leads…

Proves energy estimates for tensorial wave equations, decoupling components for stability proof.

Paper analyzes online tensorial ICA convergence with stochastic approximation.

We give a tensorial description of the Turaev cobracket on any genus 0 compact surface through the standard group-like expansion, where the Bernoulli numbers appear.

The paper defines constraints for commuting endomorphisms in generalized tangent bundles.

Study peels tensor equations on Schwarzschild spacetime.

In a preceding work it is determined when a centrally symmetric convex body in $\mathbb{R}^d,$ $d=d_1\cdots d_l,$ is the closed unit ball of a reasonable crossnorm on $\mathbb{R}^{d_1}\otimes\cdots\otimes\mathbb{R}^{d_l}.$ Consequently, the class of tensorial bodies is introduced, an associated tensorial Banach-Mazur d…

The problem of characterizing conformally Einstein manifolds by tensorial conditions has been tackled recently in papers by M. Listing, and in work by A. R. Gover and P. Nurowski. Their results apply to metrics satisfying a "non-degeneracy" condition on the Weyl tensor \W. We investigate the geometry of the foliations …

Developed a new symmetric hyperbolic formulation for Einstein-Yang-Mills system.

Paper shows invertibility of tensor X-ray transform on certain manifolds.

Let $(\tilde Q,g)$ be a para-quaternionic Hermitian structure on the real vector space $V$. By referring to the tensorial presentation $(V, \tilde{Q},g) \simeq (H^2 \otimes E^{2n}, \mathfrak{sl}(H),ω^H \otimes ω^E)$, we give an explicit description, from an affine and metric point of view, of main classes of subspaces…

This paper consists of two parts. In the first part we show that in odd dimension, as well as in even dimension below the critical weight (i.e. half the dimension), the logarithmic singularities of Schwartz kernels and Green kernels of conformal invariant pseudodifferential operators are linear combinations of Weyl con…

A new geometric method approximates slow invariant manifolds without explicit time-scale separation.

Riemannian Manifolds may be $C^{1,1}$ and the geometry of these manifolds is investigated in \cite{Groah1}. Here, a similar analysis is given for pseudohermitian, torsion-free manifolds whereby, instead of assuming that the metric is parallel, it is assumed that the metric is pseudohermitian, a condition adopted by Ein…

Study finds conditions for operator fields to be in strictly upper triangular form in small dimensions.

This research examines anisotropic conformal transformations of pseudo-Finsler surfaces.

Topology of isometric classes and flows of geometric structures

Extended spinor connections associated with composite spin-tensorial bundles are considered. Commutation relationships for covariant and multivariate differentiations and corresponding curvature spin-tensors are derived.

Graph classification is a significant problem in many scientific domains. It addresses tasks such as the classification of proteins and chemical compounds into categories according to their functions, or chemical and structural properties. In a supervised setting, this problem can be framed as learning the structure, f…

Kosmann-Lie derivatives in the bundle of Weyl spinors are considered. It is shown that the basic spin-tensorial fields of this bundle are constants with respect to these derivatives.

Proves stability of Minkowski space-time for Einstein-Yang-Mills equations.

TACE unifies scalar and tensorial modeling in Cartesian space for accurate, stable, and efficient atomistic predictions.

The paper solves linearized Ricci curvature equations on compact manifolds.

In this paper, we consider various tensorial estimates in geometric Besov-type norms on a one-parameter foliation of surfaces with evolving geometries. Moreover, we wish to do this with only very weak control on these geometries. Several of these estimates were established in previous works by S. Klainerman and I. Rodn…

New metrics derived from geodesics simplify semi-Riemannian geometry.

We derive a representation formula for the tensorial wave equation $\Box_\bg φ^I=F^I$ in globally hyperbolic Lorentzian spacetimes $(\M^{2+1}, \bg)$ by giving a geometric formulation of the method of descent which is applicable for any dimension.

According to a classical result of E.~Calabi any hyperbolic affine hypersphere endowed with its natural Hessian metric has a non-positive Ricci tensor. The affine hyperspheres can be described as the level sets of solutions to the "hyperbolic" toric Kähler-Einstein equation $e^Φ = \det D^2 Φ$ on proper convex cones. We…

Study non-homogeneous operators in 1+0 systems, classifying and analyzing their geometric properties.

Defines strongest integrability condition for skew-symmetric endomorphisms.

In general relativity, an IDEAL (Intrinsic, Deductive, Explicit, ALgorithmic) characterization of a reference spacetime metric $g_0$ consists of a set of tensorial equations $T[g]=0$, constructed covariantly out of the metric $g$, its Riemann curvature and their derivatives, that are satisfied if and only if $g$ is loc…

In this article we introduce a framization of the Hecke algebra of type B. For this framization we construct a faithful tensorial representation and two linear bases. We finally construct a Markov trace on these algebras and from this trace we derive isotopy invariants for framed and classical knots and links in the so…

The extended constraint equations arise as a special case of the conformal constraint equations that are satisfied by an initial data hypersurface $Z$ in an asymptotically simple spacetime satisfying the vacuum conformal Einstein equations developed by H. Friedrich. The extended constraint equations consist of a quasi-…

Casting neural networks in generative frameworks is a highly sought-after endeavor these days. Contemporary methods, such as Generative Adversarial Networks, capture some of the generative capabilities, but not all. In particular, they lack the ability of tractable marginalization, and thus are not suitable for many ta…

This work takes place over a conformally flat spin manifold (M,g). We prove existence and uniqueness of the conformally equivariant quantization valued in spinor differential operators, and provide an explicit formula for it when restricted to first order operators. The Poisson algebra of symbols is realized as a space…

Proposes a new kernel technique for tensor data in SVM.

We apply the theory of Weyl structures for parabolic geometries developed by A. Cap and J. Slovak in to compute, for a quaternionic contact (qc) structure, the Weyl connection associated to a choice of scale, i.e. to a choice of Carnot-Carathéodory metric in the conformal class. The result of this computation has appli…

A conformal structure on a manifold $M^n$ induces natural second order conformally invariant operators, called Möbius and Laplace structures, acting on specific weight bundles of $M$, provided that $n\ge 3$. By extending the notions of Möbius and Laplace structures to the case of surfaces and curves, we develop here th…

We propose tensorial neural networks (TNNs), a generalization of existing neural networks by extending tensor operations on low order operands to those on high order ones. The problem of parameter learning is challenging, as it corresponds to hierarchical nonlinear tensor decomposition. We propose to solve the learning…

In a fibre bundle, natural derivatives of a section are defined as tangent vector fields on the image of a section of the fibre bundle. A local extension to vector fields in the tangent bundle leads to a direct proof of the formula expressing the curvature of a connection in terms of covariant derivatives. The result i…

Study fourth-order geometric flow of shape operator for co-dimension one immersions.

Study proves global existence and decay for complex wave equations.

We introduce the concept of a base conformal warped product of two pseudo-Riemannian manifolds. We also define a subclass of this structure called as a special base conformal warped product. After, we explicitly mention many of the relevant fields where metrics of these forms and also considerations about their curvatu…

Tensor measures chirality for curves, even those with rough edges.

In this paper we characterize sprays that are metrizable by Finsler functions of constant flag curvature. By solving a particular case of the Finsler metrizability problem we provide the necessary and sufficient conditions that can be used to decide whether or not a given homogeneous system of second order ordinary dif…

New algebroids allow studying various geometries simultaneously.