© Hindawi Publishing Corp. PARA-f-LIE GROUPS

Special para-f-structures on Lie groups are studied. It is shown that every para-f-Lie group G is the quotient of the product of an almost product Lie group and a Lie group with trivial para-f-structure by a discrete subgroup.

1. Para-f -structures.The notion of a para-f -structure on a differentiable manifold was introduced and studied in [2].
Let M be an n-dimensional differentiable manifold of class C ∞ .The set of all vector fields on M will be denoted by χ(M) and the tangent space of M at a point m ∈ M by T m M. Definition 1.1.Let M be an n-dimensional differentiable manifold.If ϕ is an endomorphism field of constant rank k on M satisfying then ϕ is called a para-f -structure on M and M is a para-f -manifold.
Definition 1.2.A para-f -structure ϕ on M is integrable if there exists a coordinate system in which ϕ has constant components where I is the unit matrix and p + q = k.
Proposition 1.3.A para-f -structure ϕ on M is integrable if and only if its Nijenhuis tensor field N ϕ vanishes, that is, where X, Y ∈ χ(M).
For a para-f -structure ϕ on M, let be the kernel and image of ϕ, respectively, where are the kernel and image of ϕ at any point m ∈ M, respectively.

Para-f -Lie groups.
In this section, the notion of a para-f -Lie group is introduced.Some properties of its Lie algebra are established.Finally, its special decomposition in terms of an almost product Lie group and a Lie group with trivial para-f -structure is proved.
Let G be a Lie group and g its Lie algebra.As usual, we define (2.1) Definition 2.1.Let G be a Lie group with a para-f -structure ϕ.If both L g and R g are para-f -maps, then ϕ is said to be bi-invariant .
for all X, Y ∈ g.
Hence, by a standard result in Lie groups, or, for any Y ∈ g, Letting t → 0 in (2.5) gives us the desired result.
Proposition 2.4.A bi-invariant para-f -structure ϕ on a Lie group G is integrable.
Proof.From Proposition 2.3, the Nijenhuis tensor of a bi-invariant paraf -structure ϕ must vanish at the unity e of G.

Corollary 2.5. A Lie group G with a bi-invariant para
, where I is the unit matrix.Then ϕ is a bi-invariant para-f -structure on G. Proposition 2.7.Let G be a para-f -Lie group with a para-f -structure ϕ.Then its Lie algebra g is expressed as the direct sum (as a Lie algebra), where V k = (ker ϕ) e and V i = (im ϕ) e are subalgebras of g, and e ∈ G is the unity of G.
Proof.From Proposition 1.6, V k ∩ V i = {0}.Therefore, g is the direct sum (as a vector space) of V k and V i .It is clear, from Proposition 2.3, that both V k and V i are Lie subalgebras of g.Furthermore, if Theorem 2.8.Every para-f -Lie group G is the quotient of the product of an almost product Lie group and a Lie group with trivial para-f -structure by a discrete subgroup.
Proof.Let V k and V i be subalgebras (defined in Proposition 2.7) of the Lie algebra g of a para-f -Lie group G. From Proposition 2.7, g is the Lie algebra direct sum of V k and V i .Using Propositions 1.4 and 1.5, we obtain the theorem from [4].
Remark 2.9.Since a para-f -structure with parallelizable kernel [2] is an almost r -paracontact structure [1], some examples of almost r -paracontact structures are used in [3] to illustrate para-f -Lie groups.

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.
Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable: