<p style='text-indent:20px;'>In this paper, dedicated to the memory of Kirill Mackenzie, I relate origins and early development theory graded Lie brackets, first in publications on differential geometry Schouten, Nijenhuis, Frölicher–Nijenhuis, then work Gerstenhaber Nijenhuis–Richardson cohomology theory.</p>

Let (M, I) be an almost complex 6-manifold. The obstruction to integrability of almost complex structure (socalled Nijenhuis tensor) N : Λ0,1(M)−→ Λ(M) maps a 3-dimensional bundle to a 3-dimensional one. We say that Nijenhuis tensor is non-degenerate if it is an isomorphism. An almost complex manifold (M, I) is called nearly Kähler if it admits a Hermitian form ω such that ∇(ω) is totally antis...

In this paper, we first study infinitesimal deformations of a Lie conformal algebra and with module (called an [Formula: see text] pair), which lead to the notions Nijenhuis operator on structure pair, respectively. Then, by adding compatibility conditions between structures text]-operators, introduce notion text]-structure pair show that gives rise hierarchy pairwise compatible text]-operators...

We introduce Lie-Nijenhuis bialgebroids as Lie endowed with an additional derivation-like object. They give a complete infinitesimal description of Poisson-Nijenhuis groupoids, and key examples include manifolds, holomorphic flat bialgebra bundles. To achieve our goal we develop theory "generalized derivations" their duality, extending the well-established derivations on vector

In this paper we discuss the structure of recursion operators. We show that recursion operators of evolution equations have a nonlocal part that is determined by symmetries and cosymmetries. This enables us to compute recursion operators more systematically. Under certain conditions (which hold for all examples known to us) Nijenhuis operators are well defined, i.e., they give rise to hierarchi...

background and aims: to obtain maximum advantages from orthodontic fixed appliances and to achieve the desired final occlusion, correct bracket positioning is highly necessary. in order to measure and determine bracket’s height, diverse gauges have been used. the objective of the present study is to determine and compare bracket positioning accuracy by using height bracket positioning gauge (hb...

We prove that for compatible weakly nonlocal Hamiltonian and symplectic operators, hierarchies of infinitely many commuting local symmetries and conservation laws can be generated under some easily verified conditions no matter whether the generating Nijenhuis operators are weakly nonlocal or not. We construct a recursion operator of the two dimensional periodic Volterra chain from its Lax repr...