Related papers: Higher order reduction theorems for general linear…
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…
The cut and join operations play important roles in tensor models in general. We introduce a generalization of the cut operation associated with the higher order variations and demonstrate how they generate operators in the Aristotelian…
We discuss the extent to which it is necessary to include higher-derivative operators in the effective field theory of general scalar-tensor theories. We explore the circumstances under which it is correct to restrict to second-order…
We develop a reduction theory for $G$-invariant Lagrangian field theories defined on the higher-order jet bundle of a principal $G$-bundle, thus obtaining the higher-order Euler-Poincar\'e field equations. To that end, we transfer the…
We extend the well-known formula for the Euler class of a real oriented even-dimensional vector bundle in terms of the curvature of a metric connection to the case of a general linear connection provided a metric is present. We rewrite the…
In this paper we prove reducibility of classes of linear first order operators on tori by applying a generalization of Moser's theorem on straightening of vector fields on a torus. We consider vector fields which are a $C^\infty$…
We consider natural differential operations acting on sections of tensor vector bundles. Arrising problems can be reformulated as invariant theoretical problems (the IT-reduction). We give examples of usage of the IT-reduction. In…
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…
For symmetric classical field theories on principal bundles there are two methods of symmetry reduction: covariant and dynamic. Assume that the classical field theory is given by a symmetric covariant Lagrangian density defined on the first…
Tree tensor networks (TTNs) provide a compact and structured representation of high-dimensional data, making them valuable in various areas of computational mathematics and physics. In this paper, we present a rigorous mathematical…
We propose a conceptually economical and computationally tractable completion of the foundations of gauge theory on quantum principal bundles \`{a} la Brzezi\'{n}ski--Majid to the case of general differential calculi and strong bimodule…
We prove that, in the setting of noncommutative differential geometry, a system of higher order connections is equivalent to a suitable generalization of the notion of phase space quantization (in the sense of Moyal star products on the…
We explicitly establish a unitary correspondence between spherical irreducible tensor operators and cartesian tensor operators of any rank. That unitary relation is implemented by means of a basis of integer-spin wave functions that…
The geometry of graded principal bundles is discussed in the framework of graded manifold theory of Kostant-Berezin-Leites. In particular, we prove that a graded principal bundle is globally trivial if and only if it admits a global graded…
Motivated by obtaining a consistent mathematical description for the radiation reaction of point charged particles in linear classical electrodynamics, a theory of generalized higher order tensors and differential forms is introduced. The…
Any set of truth-functional connectives has sequent calculus rules that can be generated systematically from the truth tables of the connectives. Such a sequent calculus gives rise to a multi-conclusion natural deduction system and to a…
We address the recently introduced notions of generalized principal bundle and generalized principal connection by keeping track of global geometric properties through local coordinate transformation laws. This approach leads us to…
Dimensionality reduction for high-order tensors is a challenging problem. In conventional approaches, higher order tensors are `vectorized` via Tucker decomposition to obtain lower order tensors. This will destroy the inherent high-order…
Elimination of the fibre coordinate dependence from the connection form transformation rule for a bundle with a coset manifold standard fibre reduces the structure group. The nonlinear SU(4) action on an $S^7$ bundle is applied to the…
We prove Behrend's conjecture on the rationality of the canonical reduction of principal bundles and reductive group schemes for classical groups and give new bounds for the conjecture for exceptional groups. However we find a…