Minimal Matrix Representations for Six-Dimensional Nilpotent Lie Algebras

This paper is concerned with finding minimal dimension linear representations for six-dimensional real, indecomposable nilpotent Lie algebras. It is known that all such Lie algebras can be represented in gl(6, R). After discussing the classification of the 24 such Lie algebras, it is shown that only one algebra can be represented in gl(4, R). A Theorem is then presented that shows that 13 of the algebras can be represented in gl(5, R). The special case of filiform Lie algebras is considered, of which there are five, and it is shown that each of them can be represented in gl(6, R) and not gl(5, R). Of the remaining five algebras, four of them can be represented minimally in gl(5, R). That leaves one difficult case that is treated in detail in an Appendix

Lie Symmetries of the Canonical Geodesic Equations for Four-Dimensional Lie Groups

For each of the four-dimensional indecomposable Lie algebras the geodesic equations of the associated canonical Lie group connection are given. In each case a basis for the associated Lie algebra of symmetries is constructed and the corresponding Lie brackets are written down

Symmetry Algebras of the Canonical Lie Group Geodesic Equations in Dimension Three

For each of the two and three-dimensional indecomposable Lie algebras the geodesic equations of the associated canonical Lie group connection are given. In each case a basis for the associated Lie algebra of symmetries is constructed and the corresponding Lie brackets are written down

Classification of Symmetry Lie Algebras of the Canonical Geodesic Equations of Five-Dimensional Solvable Lie Algebras

In this investigation, we present symmetry algebras of the canonical geodesic equations of the indecomposable solvable Lie groups of dimension five, confined to algebras A_{5,7}^{abc} to A_{18}^a. For each algebra, the related system of geodesics is provided. Moreover, a basis for the associated Lie algebra of the symmetry vector fields, as well as the corresponding nonzero brackets, are constructed and categorized

Bundle theory for symplectic and contact geometry with applications to Lagrangian and Hamiltonian mechanics

This work is primarily concerned with various aspects of Lagrangian and Hamiltonian mechanics. These different aspects are related in a somewhat complicated way, which is clarified by the introduction of the unifying concept of a jet bundle. Here, bundles are first considered in some generality and then jet bundles are introduced and their bundle structure investigated especially with a view to formulating Lagrangian and Hamiltonian mechanics invariantly. The second chapter is devoted to a discussion of the two Schouten brackets and it is explained why each is of importance in mechanics. The symmetric Schouten bracket enables the notion of a killing tensor to be defined on a Riemannian or pseudo-Riemannian manifold. A theorem is proved which gives an upper bound on the dimension of the space of killing tensors of a fixed rank. In chapter 3 Hamilton-Jacobi theory and a version of Noether's theorem are presented from a m o d e m viewpoint. Then conditions are obtained which entail the existence of a constant of motion which is polynomial in momenta. These conditions are used to construct several classical Hamiltonians with "hidden" symmetries - the usage of the latter term is briefly justified. Most importantly, all systems with two degrees of freedom which have a quadratic integral independent of the Hamiltonian are characterized. In chapter 4 the geometrical properties of TM and the closely related space J1 (IR,M) are investigated. It is shown that J1(IR,M) has an intrinsically defined 1-1 tensor in analogy to TM. The behaviour of these tensors under diffeomorphisms is investigated. The chapter concludes with a discussion of the various notions of symmetry in Lagrangian theory. Chapter 5 begins with a review of symplectic geometry and continues with the definition and examination of contact manifolds in the same spirit as symplectic geometry. In particular, contact diffeomorphisms are described on the space J1(IR,M). Finally, two theorems are given which endeavour to explain the general principle enabling the Lie algebra of a subalgebra of vector fields to be transferred to a collection of forms

Symmetries of the Canonical Geodesic Equations of Five-Dimensional Nilpotent Lie Algebras

In this paper, symmetries of the canonical geodesic equations of indecomposable nilpotent Lie groups of dimension five are constructed. For each case, the associated system of geodesics is provided. In addition, a basis for the associated Lie algebra of symmetries as well as the corresponding non-zero Lie brackets are listed and classified. This is a joint work with Ryad Ghanam and Gerard Thompson

Development and validation of a firm-level vertical and horizontal internationalization metric

The lack of valid and reliable measures of firm-level vertical and horizontal internationalization is impeding the development and testing of hypothesized relationships between these respective dimensions of internationalization and a range of important MNE characteristics, actions, and effects. Through a series of qualitative and quantitative studies using data collected from senior MNE executives (total N=3,146), we develop and validate a scale to measure both vertical and horizontal firm-level internationalization. Subscales for each type of internationalization prove to be unidimensional, reliable, temporally stable, and to have predictive, cross-cultural, cross-sectoral, and discriminant validity