International Journal of Geometric Methods in Modern Physics

In this paper, with the study of combinatorial Dyson–Schwinger equations at the level of the universal Hopf algebra of renormalization and with the extension of the universality of this specific Hopf algebra and also category of flat equisingular vector bundles to the level of these equations, we are going to consider the notion of a geometric description from nonperturbative theory.

We extend the Falceto–Zambon version of Marsden–Ratiu Poisson reduction to Poisson quasi-Nijenhuis structures with background on manifolds. We define gauge transformations of Poisson quasi-Nijenhuis structures with background, study some of their properties and show that they are compatible with reduction procedure. We use gauge transformations to construct Poisson quasi-Nijenhuis structures with background.

In this paper, we introduce and investigate a general transformation or change of Finsler metrics, which is referred to as a generalized β-conformal change:

In this paper, we study the operator forms of the classical Yang–Baxter equation (CYBE) in Lie superalgebras. We introduce the notion of super -operators and give their relationship with the (standard) tensor form. There are close relationships between the CYBE in Lie superalgebras and left-symmetric superalgebras which can be interpreted through the super -operators. In particular, there are natural and explicit solutions of the CYBE in Lie superalgebras constructed from left-symmetric superalgebras.

In the context of Covariant Quantum Mechanics for a spin particle, we classify the "quantum vector fields", i.e. the projectable Hermitian vector fields of a complex bundle of complex dimension 2 over spacetime. Indeed, we prove that the Lie algebra of quantum vector fields is naturally isomorphic to a certain Lie algebra of functions of the classical phase space, called "special phase functions". This result provides a covariant procedure to achieve the quantum operators generated by the quantum vector fields and the corresponding observables described by the special phase functions.

The age-independent deterministic model for the transmission of a constant rumor, recently proposed by Kawachi, is shown to admit a bi-Hamiltonian structure, structure that is leading to its complete integrability.

In this paper, we discuss possible extensions of the concept of the Cartan form of classical mechanics to higher-order mechanics on manifolds, higher-order field theory on jet bundles and to parametric variational problems on slit tangent bundles and on bundles of nondegenerate velocities. We present a generalization of the Cartan form, known as a Lepage form, and basic properties of the Lepage forms. Both earlier and recent examples of differential forms generalizing the Cartan form are reviewed.

The main purpose of this article is a proposal of non(anti)commutative super twistor space by making the odd coordinates θ not anticommuting, but satisfying Clifford algebra relations. Despite the deformation, we can introduce a deformed associative product which is globally defined on P^3|N.

In this paper, we present a geometric version of several examples of partial differential equations which appear in field theories. These examples are described using the k-symplectic and k-cosymplectic Lagrangian formalism.

Kerr–Schild metrics have been introduced as a linear superposition of the flat spacetime metric and a squared null-vector field, say k, multiplied by some scalar function, say H. The basic assumption which led to Kerr solution was that k be both geodesic and shearfree. This condition is relaxed here and Kerr–Schild Ansatz is revised by treating Kerr–Schild metrics as exact linear perturbations of Minkowski spacetime. The scalar function H is taken as the perturbing function, so that Einstein's field equations are solved order-by-order in powers of H. It turns out that the congruence must be geodesic and shearfree as a consequence of third- and second-order equations, leading to an alternative derivation of Kerr solution.

We present a construction transforming a general connection Γ on a fibered manifold Y → M and a classical connection Λ on its base M into a classical connection on the total space Y by means of a vertical parallelism Φ and an auxiliary linear connection Δ. The relations to the theory of gauge-natural operators are discussed.