arXiv research

For a system of second order differential equations we determine a nonlinear connection that is compatible with a given generalized Lagrange metric. Using this nonlinear connection, we can find the whole family of metric nonlinear connections that can be associated with a system of SODE and a generalized Lagrange struc…

Explains linearizing a nonlinear connection on a pullback bundle.

The main purposes of this article are to extend our previous results on homogeneous sprays to arbitrary (generalized) sprays, to show that locally diffeomorphic exponential maps can be defined for any (generalized) spray, and to give a (possibly nonlinear) covariant derivative for any (possibly nonlinear) connection. I…

Higher order anisotropic superspaces are constructed as generalized vector superbundles provided with compatible nonlinear connection, distinguished connection and metric structures.

We show that, for mechanical system with external forces, the equations of deviations of solution curves of the corresponding Lagrange equations,determine a nonlinear connection on the second order osculator (second order tangent) bundle. In particular, Jacobi equations in Finsler and Riemann spaces determine such a no…

Introduces nonlinear splittings on fibre bundles for generalizing connections.

In this paper we study some geometrical objects (d-tensors, multi-time semisprays of polymomenta and nonlinear connections) on the dual 1-jet vector bundle $J^{1*}(\cal{T}, M)\to \cal{T}\times M$. Some geometrical formulas, which connect the last two geometrical objects, are also derived. Finally, a canonical nonlinear…

A linear connection is associated to a nonlinear connection on a vector bundle by a linearization procedure. Our definition is intrinsic in terms of vector fields on the bundle. For a connection on an affine bundle our procedure can be applied after homogenization and restriction. Several applications in Classical Mech…

The aim of this paper is to generalize the theory of nonlinear connections of Grifone ([3] and [4]). We adopt the point of view of Anona [1] and continue developing the approach established by the first author in [10]. The first part of the work is devoted to the problem of associating to each $L$-regular linear connec…

Study adapts $β$-TCVAE for fMRI to recover nonlinear brain components.

Optimistic estimate predicts best fitting performance of nonlinear models.

In this paper we study the geometrical structures on the cotangent bundle using the notions of adapted tangent structure and regular vector fields. We prove that the dynamical covariant derivative on $T^{*}M$ fix a nonlinear connection for a given $\mathcal{J}$-regular vector field. Using the Legendre transformation in…

To a system of second order ordinary differential equations (SODE) one can assign a canonical nonlinear connection that describes the geometry of the system. In this work we develop a geometric setting that allows us to assign a canonical nonlinear connection also to a system of higher order ordinary differential equat…

Study compares bundles with connections to prehomogeneous geometries.

Adopting the pullback approach to global Finsler geometry, the aim of the present paper is to provide new intrinsic (coordinate-free) proofs of intrinsic versions of the existence and uniqueness theorems for the Cartan and Berwald connections on a Finsler manifold. To accomplish this, the notions of semispray and nonli…

We outline a new geometric method of constructing exact solutions of gravitational field equations parametrized by generic off-diagonal metrics, anholonomic frames and possessing, in general, nontrivial torsion and nonmetricity. The formalism of nonlinear connections is elaborated for (pseudo) Riemannian and Einstein-C…

In our previous work, we have defined a nonlinear connection of Finsler manifold which preserves the Finsler metric $L=L(x,dx)$. To make the method easier and more useful in applications, moving frame (vielbein) $θ^a={e^a}_μdx^μ$ formalism for the nonlinear connection is newly considered. We derive formulae to calculat…

In this paper we present a method by which is obtained a sequence of $k$-semisprays and two sequences of nonlinear connections on the $k$-tangent bundle $T^kM$, starting from a given one. Interesting particular cases appear for Lagrange and Finsler spaces of order $k$.

The main purpose of this article is to introduce a comprehensive, unified theory of the geometry of all connections. We show that one can study a connection via a certain, closely associated second-order differential equation. One of the most important results is our extended Ambrose-Palais-Singer correspondence. We ex…

New neural architectures with multivariate nonlinearities are optimal in function space.

The theory of locally anisotropic superspaces (supersymmetric generalizations of various types of Kaluza--Klein, Lagrange and Finsler spaces) is laid down. In this framework we perform the analysis of construction of the supervector bundles provided with nonlinear and distinguished connections and metric structures. Tw…

Extends Higgs fields theory to complex fiber bundles.

We review the geometric setting of the field theory with locally anisotropic interactions. The concept of locally anisotropic space is introduced as a general one for various type of extensions of Lagrange and Finsler geometry and higher dimension (Kaluza--Klein type) spaces. The problem of definition of spinors on gen…

Newton's method tackles nonlinear mappings into vector bundles with connections and retractions.

New methods for constructing submersions between different types of spaces.

This paper addresses the following question of neural network identifiability: Does the input-output map realized by a feed-forward neural network with respect to a given nonlinearity uniquely specify the network architecture, weights, and biases? Existing literature on the subject Sussman 1992, Albertini, Sontag et al…

The aim of this paper is to construct a natural Riemann-Lagrange differential geometry on 1-jet spaces, in the sense of nonlinear connections, generalized Cartan connections, d-torsions, d-curvatures, jet electromagnetic fields and jet electromagnetic Yang-Mills energies, starting from some given nonlinear evolution OD…

The theory of spinors is developed for locally anisotropic (la) spaces, in brief la-spaces, which in general are modeled as vector bundles provided with nonlinear and distinguished connections and metric structures (such la-spaces contain as particular cases the Lagrange, Finsler and, for trivial nonlinear connections,…

We present an introduction to the geometry of higher order vector and co-vector bundles (including higher order generalizations of the Finsler geometry and Kaluza--Klein gravity) and review the basic results on Clifford and spinor structures on spaces with generic local anisotropy modeled by anholonomic frames with ass…

We establish a one-parameter family of Harnack inequalities connecting the constrained trace Li-Yau differential Harnack inequality for a nonlinear parabolic equation to the constrained trace Chow-Hamilton Harnack inequality for this nonlinear equation with respect to evolving metrics related to Ricci flow on a 2-dimen…

We present an introduction to the geometry of higher order vector and co--vector bundles (including higher order generalizations of the Finsler geometry and Kaluza--Klein gravity) and review the basic results on Clifford and spinor structures on spaces with generic local anisotropy modeled by higher order nonlinear con…

We propose a new framework for constructing geometric and physical models on nonholonomic manifolds provided both with Clifford -- Lie algebroid symmetry and nonlinear connection structure. Explicit parametrizations of generic off-diagonal metrics and linear and nonlinear connections define different types of Finsler, …

Dual optimization connects ERM-fDR to normalization function.

The geometry of nonholonomic bundle gerbes, provided with nonlinear connection structure, and nonholonomic gerbe modules is elaborated as the theory of Clifford modules on nonholonomic manifolds which positively fail to be spin. We explore an approach to such nonholonomic Dirac operators and derive the related Atiyah-S…

Adopting the pullback approach to global Finsler geometry, the aim of the present paper is to provide intrinsic (coordinate-free) proofs of the existence and uniqueness theorems for the Chern (Rund) and Hashiguchi connections on a Finsler manifold. To accomplish this, we introduce and investigate the notions of semispr…

The aim of this paper is to open the problem of construction of a nonlinear connection $Γ=(M^{(i)}_{(α)β}, N^{(i)}_{(α)j})$ on the jet bundle of first order $J^1(T,M)$, which to be canonically produced by a Kronecker product vertical metrical d-tensor $G^{(α)(β)}_{(i)(j)}=h^{αβ}g_{ij}$, possibly provided by multi-time …

We survey the geometry of Lagrange and Finsler spaces and discuss the issues related to the definition of curvature of nonholonomic manifolds enabled with nonlinear connection structure. It is proved that any commutative Riemannian geometry (in general, any Riemann--Cartan space) defined by a generic off--diagonal metr…

Paper connects contrastive learning to MI maximization and establishes robust methods for nonlinear ICA and subspace estimation.

We formulate an approach to the geometry of Riemann-Cartan spaces provided with nonholonomic distributions defined by generic off-diagonal and nonsymmetric metrics inducing effective nonlinear and affine connections. Such geometries can be modelled by moving nonholonomic frames on (pseudo) Riemannian manifolds and desc…

Graph theory criterion for Hodge theory to match linearly.

Consider $L$ a regular Lagrangian, $S$ the canonical semispray, and $h$ the horizontal projector of the canonical nonlinear connection. We prove that if the Lagrangian is constant along the integral curves of the Euler-Lagrange equations then it is constant along the horizontal curves of the canonical nonlinear connect…

Alternative proof of flatness for Ricci-pinched 3-manifolds.

Structural equation models (SEMs) and vector autoregressive models (VARMs) are two broad families of approaches that have been shown useful in effective brain connectivity studies. While VARMs postulate that a given region of interest in the brain is directionally connected to another one by virtue of time-lagged influ…

A consistent theory of quantum gravity (QG) at Planck scale almost sure contains manifestations of Lorentz local symmetry violations (LV) which may be detected at observable scales. This can be effectively described and classified by models with nonlinear dispersions and related Finsler metrics and fundamental geometri…

Nonlinear RNNs' memory capacity varies widely, making it impractical.

We construct an algebra of nonlinear generalized tensor fields on manifolds in the sense of J.-F. Colombeau, i.e., containing distributional tensor fields as a linear subspace and smooth tensor fields as a faithful subalgebra. The use of a background connection on the manifold allows for a simplified construction based…

In sub-Riemannian geometry the coefficients of the Jacobi equation define curvature-like invariants. We show that these coefficients can be interpreted as the curvature of a canonical Ehresmann connection associated to the metric, first introduced in [Zelenko-Li]. We show why this connection is naturally nonlinear, and…

Study on a nonlinear elliptic equation on compact Hermitian manifolds.