Moving frames of various kinds are used to derive bi-Hamiltonian operators and associated hierarchies of multi-component soliton equations from groupinvariant flows of non-stretching curves in constant curvature manifolds and Lie group manifolds. The hierarchy in the constant-curvature case consists of a vector mKdV equation coming from a parallel frame, a vector potential mKdV equation coming ...

Abstract We study four-dimensional second Chern-Einstein almost-Hermitian manifolds. In the compact case, we observe that under a certain hypothesis, Riemannian dual of Lee form is Killing vector field. use observation to describe locally conformally symplectic manifolds, and give some examples such Finally, problem on unimodular almost-abelian Lie algebras, classifying those admit left-invaria...

The Economic Load Dispatch (ELD) problems in power generation systems are to reduce the fuel cost by reducing the total cost for the generation of electric power. This paper presents an efficient Modified Firefly Algorithm (MFA), for solving ELD Problem. The main objective of the problems is to minimize the total fuel cost of the generating units having quadratic cost functions subjected to lim...

In the setting of step two Carnot groups we give an explicit representation of Taylor polynomial in terms of a suitable basis of the real vector space of left invariant differential operators, acting pointwisely on monomials like the ordinary Euclidean iterated derivations.

The Castelnuovo-Mumford regularity r of a variety V ⊆ Pn C is an upper bound for the degrees of the hypersurfaces necessary to cut out V . In this note we give a bound for r when V is left invariant by a vector field on

We consider the pair of degenerate compatible antibrackets satisfying a generalization of the axioms imposed in the triplectic quantization of gauge theories. We show that this actually encodes a Lie group structure, with the antibrackets being related to the left-and right-invariant vector fields on the group. The standard triplectic quantization axioms then correspond to Abelian Lie groups.

For any nite volume hyperbolic 3-manifold M we use ideal tri-angulation to deene an invariant (M) in the Bloch group B(C). It actually lies in the subgroup of B(C) determined by the invariant trace eld of M. The Chern-Simons invariant of M is determined modulo rationals by (M). This implies rationality and | assuming the Ramakrishnan conjecture | ir-rationality results for Chern Simons invariants.

We provide the set of left-invariant minimal unit vector fields on the semi-direct product Rn oP R, where P is a nonsingular diagonal matrix and on the 7 classes of 4-dimensional solvable Lie groups of the form R3 oP R which are unimodular and of type (R).

In this paper we discuss function spaces on a general noncompact Lie group, namely the scales of Triebel–Lizorkin and Besov spaces, defined in terms sub-Laplacian with drift. The is written as (negative) sum squares collection left-invariant vector fields satisfying Hormander’s condition. These were recently introduced by authors. prove norm characterization finite differences, density test fun...

Abstract. Let G be a step-two nilpotent group of H-type with Lie algebra G = V ⊕ t. We define a class of vector fields X = {Xj} on G depending on a real parameter k ≥ 1, and we consider the corresponding p-Laplacian operator Lp,ku = divX(|∇Xu| ∇Xu). For k = 1 the vector fields X = {Xj} are the left invariant vector fields corresponding to an orthonormal basis of V , for k = 2 and G being the He...