‎In this paper‎, ‎we study translation invariant surfaces in the‎ ‎3-dimensional Heisenberg group $rm Nil_3$‎. ‎In particular‎, ‎we‎ ‎completely classify translation invariant surfaces in $rm Nil_3$‎ ‎whose position vector $x$ satisfies the equation $Delta x = Ax$‎, ‎where $Delta$ is the Laplacian operator of the surface and $A$‎ ‎is a $3 times 3$-real matrix‎.

Abstract—In this article, a new method is proposed for the measuring of well-being inequality through a model composed of superimposing satisfaction waves. The displacement of households’ satisfactory state (i.e. satisfaction) is defined in a satisfaction string. The duration of the satisfactory state for a given period is measured in order to determine the relationship between utility and tota...

In this paper, we investigate critical Gagliardo–Nirenberg, Trudinger-type and Brezis–Gallouet–Wainger inequalities associated with the positive Rockland operators on graded Lie groups, which include cases of [Formula: see text], Heisenberg, general stratified groups. As an application, using Gagliardo–Nirenberg inequality, existence least energy solutions nonlinear Schrödinger type equations i...

Based on properties of vector fields, we prove Hardy inequalities with remainder terms in the Heisenberg group and a compact embedding in weighted Sobolev spaces. The best constants in Hardy inequalities are determined. Then we discuss the existence of solutions for the nonlinear eigenvalue problems in the Heisenberg group with weights for the psub-Laplacian. The asymptotic behaviour, simplicit...

The study of partial differential operators constructed from noncommutative vector fields satisfying the Hörmander condition 1 has hadmuch development. We refer to 2, 3 and the references therein for a systematic account of the study. Recently there have been considerable interests in studying the sub-Laplacians as square sums of vector fields that are not invariant or do not satisfy the Hörman...

We study the secondary structure of RNA determined by Watson-Crick pairing without pseudo-knots using Milnor invariants of links. We focus on the first non-trivial invariant, which we call the Heisenberg invariant. The Heisenberg invariant, which is an integer, can be interpreted in terms of the Heisenberg group as well as in terms of lattice paths. We show that the Heisenberg invariant gives a...

Let H be a Heisenberg superalgebra. In this paper, the definition of biderivations and the properties of biderivations on Lie superalgebras are introduced. And some properties of biderivations on Heisenberg superalgebras are introduced by the definition of Heisenberg superalge-bras.

Definition. A Heisenberg group for a symplectic vector space (V, ω) is the Lie group with the underlying manifold V ×R and the multiplication (u, s)(v, t) = (u+ v, s+ t+ ω(u, v)/2) where u, v ∈ V and s, t ∈ R. The map t 7→ (0, t) is a Lie group homomorphism from R to the Heisenberg group. Its image coincides with the center of the Heisenberg group. The dimension of the Heisenberg group equals 2...

Abstract The concentration-compactness principle of Lions type in Euclidean space relies on the Pólya-Szegö inequality, which is not available non-Euclidean settings. first proof setting, such as Heisenberg group, was given by Li et al. using a symmetrization-free nonsmooth truncation argument. In this article, we study second-order Adams’ inequality Lorentz-Sobolev <m:math xmlns:m="http://www....