Commutative Lie Group Research Articles

Lie group analysis and novel solutions for the generalized variable-coefficients Sawada-Kotera equation

In this paper, the generalized fifth-order variable-coefficients Sawada-Kotera equation arising in coastal seas, fjords, lakes, and the atmospheric boundary layer is studied by using the symmetry method. As a result, four-vector fields are obtained and a commutative Lie group of transformations. Then, by using suitable combinations of the Lie vector fields three distinct similarity reductions in the form of nonlinear ordinary differential equations are yielded. By solving the reduced equations using the known techniques and the Jacobi expansion method many novel periodic and solitary wave solutions are considered. From a physical point of view, the dynamic behavior of two distinct wave structures, periodic and kink soliton, was investigated for different choices of the variable coefficients and it was clear that the wave propagation shape is affected by the change of the variable function.

Natural and conjugate mates of Frenet curves in three-dimensional Lie group

In this study, we introduce the natural mate and conjugate mate of a Frenet curve in a three dimensional Lie group $ \mathbb{G} $ with bi-invariant metric. Also, we give some relationships between a Frenet curve and its natural mate or its conjugate mate in $ \mathbb{G} $. Especially, we obtain some results for the natural mate and the conjugate mate of a Frenet curve in $ \mathbb{G} $ when the Frenet curve is a general helix, a slant helix, a spherical curve, a rectifying curve, a Salkowski (constant curvature and non-constant torsion), anti-Salkowski (non-constant curvature and constant torsion), Bertrand curve. Finally, we give nice graphics with numeric solution in Euclidean 3-space as a commutative Lie group.

Connected components of definable groups, and o-minimality II

In this sequel to [3] we try to give a comprehensive account of the “connected components” G00 and G000 as well as the various quotients G/G00, G/G000, G00/G000, for G a group definable in a (saturated) o-minimal expansion of a real closed field. Key themes are the structure of G00/G000 and the problem of “exactness” of the G↦G00 functor. We prove that the examples produced in [3] are typical, and that for any G, G00/G000 is naturally the quotient of a connected compact commutative Lie group by a dense finitely generated subgroup (where we allow the trivial Lie group).

Holomorphic bundles on diagonal Hopf manifolds

We show that every stable holomorphic bundle on the Hopf manifold with , where is a diagonal linear operator with all eigenvalues satisfying , can be lifted to a -equivariant coherent sheaf on , where is a commutative Lie group acting on and containing . This is used to show that all bundles on are filtrable, that is, admit a filtration by a sequence of coherent sheaves with all subquotients of rank 1.

On Fatou maps into compact complex manifolds

We will deal with two topics about normality of the iterates of holomorphic self-maps in compact varieties. First, for a holomorphic self-map f of an irreducible compact variety Z, we show that if at least one subsequence of itself is a normal family and the full set of the limit maps is finite or has the structure of a compact commutative Lie group. Second, in the case when Z is non-singular, we deal with the dynamics on forward-invariant compact subsets outside the closures of the post-critical sets. We will describe the semi-repelling structure of the dynamics in terms of repelling points and neutral Fatou discs (center manifolds). In particular, in the case of holomorphic self-maps of projective spaces, we will obtain a stronger result.

P-Adic abelian integrals and commutative lie groups

The aim of this paper is to propose an ``elementary approach to Coleman's theory of p-adic abelian integrals. Our main tool is a theory of commutative p-adic Lie groups (the logarithm map); we use neither dagger analysis nor Monsky-Washnitzer cohomology theory. Notice that we also treat the case of bad reduction. A preliminary version of this paper appeared as Expos\'e 9 dans ``Problemes Diophantiens 88-89 (D. Bertrand, M. Waldschmidt), Publ. Math. Univ. Paris VI 90(1990), 15 pp.

Homotopically trivial actions on aspherical spaces and topological rigidity of free actions

Let ρ : G → Homeo (X) be a homotopically trivial action of a compact commutative Lie group on a connected, finitistic, aspherical topological space. We associate with ρ a certain set of homotopical invariants. Using them we introduce the notion of π 1-freeness, π 1-conjugacy and π 1-effectiveness. We check that ρ is free if and only if it is π 1-free. Applying the rigidity theorems of F.T. Farrell and L. Jones we prove that π 1-conjugate, homotopically trivial, smooth, and free actions of G on appropriate aspherical manifolds are topologically conjugate. Using this we show that the number of topological conjugacy classes of free and smooth Z k -actions that are homotopic to a given free Z k -action on a closed infrasolvmanifold M it is not greather than k rank Z(π 1( M)) Z k .

On "Asymptotically Flat" Space-Times with G2-Invariant Cauchy Surfaces

In this paper we study space-times which evolve out of Cauchy data (Σ, 3 g, K) invariant under the action of a two-dimensional commutative Lie group. Moreover, (Σ 3 g, K) are assumed to satisfy certain completeness and asymptotic flatness conditions in spacelike directions. We show that asymptotic flatness and energy conditions exclude all topologies and group actions except for a cylindrically symmetric R 3, or a periodic identification thereof along the z-axis. We prove that asymptotic flatness, energy conditions, and cylindrical symmetry exclude the existence of compact trapped surfaces. Finally, we show that the recent results of Christodoulou and Tahvildar-Zadeh concerning global existence of a class of wave-maps imply that strong cosmic censorship holds in the class of asymptotically flat cylindrically symmetric electro-vacuum space-times.

Variables of the action-angle type on symplectic manifolds stratified by coisotropic tori

A symplectic manifold is considered under the assumption that a smooth symplectic action of a commutative Lie group with compact coisotropic orbits is defined on it. The problem of existence of variables of the action-angle type is investigated with a view to giving a detailed description of flows in Hamiltonian systems with invariant Hamiltonians. We introduce the notion of a nonresonance symplectic structure for which the problem of recognition of resonance and nonresonance tori is solved.