We prove a Poincaré lemma for a set of r smooth functions on a 2n-dimensional smooth manifold satisfying a commutation relation determined by r singular vector fields associated to a Cartan subalgebra of sp(2r, R). This result has a natural interpretation in terms of the cohomology associated to the infinitesimal deformation of a completely integrable system.

We introduce the notion of bipolar fuzzy soft Lie subalgebras and investigate some of their properties. We also introduce the concept of an (∈,∈ ∨q)-bipolar fuzzy (soft) Lie subalgebra and present some of its properties.

A Cartan subalgebra of a finite-dimensional Lie algebra L is a nilpotent subalgebra H of L that coincides with its normalizer NL H . Such subalgebras occupy an important place in the structure theory of finite-dimensional Lie algebras and their properties have been explored in many articles (see, e.g., Barnes, 1967; Benkart, 1986; Wilson, 1977; Winter, 1969). In general (more precisely, when th...

The space of linear differential operators on a smooth manifold M has a natural one-parameter family of Diff(M) (and Vect(M))-module structures, defined by their action on the space of tensor densities. It is shown that, in the case of second order differential operators, the Vect(M)-module structures are equivalent for any degree of tensordensities except for three critical values: {0, 1 2 , 1...

The notion of multilinear quantum Lie operation appears naturally in connection with a different attempts to generalize the Lie algebras. There are a number of reasons why the generalizations are necessary. First of all this is the demands for a ”quantum algebra” which was formed in the papers by Ju.I. Manin, V.G. Drinfeld, S.L. Woronowicz, G. Lusztig, L.D. Faddeev, and many others. These deman...

Transcriptional activation and repression of the vascular smooth muscle (VSM) alpha-actin gene in myoblasts and fibroblasts is mediated, in part, by positive and negative elements contained within an approximately 30-bp polypurine-polypyrimidine tract. This region contains binding sites for an essential transcription-activating protein, identified as transcriptional enhancer factor I (TEF-1), a...

Abstract Dedicated to the memory of Jan Saxl Let $G$ be a finite primitive permutation group on set $\Omega $ with nontrivial point stabilizer $G_{\alpha }$. We say that is extremely if }$ acts primitively each its orbits in \setminus \{\alpha \}$. These groups arise naturally several different contexts, and their study can traced back work Manning 1920s. In this paper, we determine almost simp...

This article concerned on the study of signature submanifolds for curves under Lie group actions SE(2), SA(2) and for surfaces under SE(3). Signature submanifold is a regular submanifold which its coordinate components are differential invariants of an associated manifold under Lie group action, and therefore signature submanifold is a key for solving equivalence problems.

W. Plesken found a simple but interesting construction of a Lie algebra from a finite group. Cohen and Taylor posed themselves the question of what the Plesken Lie algebra, which is the Lie subalgebra of the group algebra k[G] generated by the elements g − g−1, could be. The result is very fascinating: It turns out that the Lie algebra decomposition of the Plesken Lie algebra into simple Lie al...

the attempt to find out if someone is telling a lie has been of human's interest. one of the techniques which is believed to be efficient in future is fmri (functional magnetic resonance imaging), using which the changes of regional blood flow during the designed paradigm can be recorded. this method has been considered as a better technique for studying lie and deception. for investigatin...