Thesis (M. Ed.)--University of Hong Kong, 1996. / Includes bibliographical references (leaf 93-96).

Thesis (M.Ed.)--University of Hong Kong, 1996. / Includes bibliographical references (leaves 93-96). Also available in print.

No description available.

This paper will investigate the geometry that results when Euclid's fifth postulate is replaced by the Lobachevskian assumption. The investigations will be limited to the plane.

Master of Science / Department of Mathematics / Roman Fedorov / The purpose of this report is to serve as an introduction to the language of sheaves and schemes via algebraic geometry. The main objective is to use examples from algebraic geometry to motivate the utility of the perspective from sheaf and scheme theory. Basic facts and definitions will be provided, and a categorical approach will be frequently incorporated when appropriate.

published_or_final_version / Mathematics / Master / Master of Science

Let M denote a 2n-dimensional globally defined orientable manifold from which is constructed the product space N = M x R. It is assumed that N is endowed with a set of 2n independent smooth 1-forms {π(h),πʰ:h = 1,..,n}. Certain conditions are imposed on {π(h),πʰ} which imply the existence of local coordinates {qʰ,p(h)} on M and a function H(qʰ,p(h),t) on N, where t is the single coordinate on R, such that dπ = π(h) ∧ πʰ, where π has the structure of a Cartan form on N. The assumption that the function h = p(h)∂H/∂p(h)-H is non-zero on a region D ⊂ N, implies that π has maximal class on D. This construction gives rise to a local adapted contact structure on N and a local symplectic structure on M. A vector field X on N is said to be a contact field if there exists a smooth function σ : N → R such that ₤ₓπ = σπ. A vector field Z on N is called a canonical vector field if it admits the representation Z = ∂/∂t + (H, ) where (,) denotes the Poisson bracket on M. For a given function σ, a prescription is given for the construction of the space c(σ)(N) of contact fields in terms of solutions F of the p.d.e. Z<F> = σh. The vector space (UNFORMATTED EQUATION FOLLOWS) c(N) = ∪ (σ∊C)(∞)c(σ)(N) (END UNFORMATTED EQUATION) is shown to have the structure of a Lie sub-algebra of the Lie algebra of vector fields on N. It is shown that the associated subspace A(π) = {X:X˩π = 0} is such that c(σ)(N) ∩ A(π) = {0}. This implies that there is an X in c(σ)(N) such that X˩π ≠ 0. Thus, if the function H that appears in the Cartan form π is such that H = X˩π for any X ∊ c(σ)(N) it is possible to deduce that ∂H/∂t ≠ 0, which shows that such vector fields may be of relevance to the theory of non-conservative systems. A different set of 2n 1-forms {π(h),πʰ} is introduced on N that are subject to analogous conditions which ensure the existence of local coordinates (qʰ,p(h)) on M and a function K(qʰ,p(h),t) that gives rise to a new Cartan form π on N such that dπ= π(h) ∧ πʰ. By definition, the fundamental invariant of a parameter-dependent canonical transformation on N is dπ = dπ. In this new setting a contact field X satisfies the ₤ₓπ = σπ for some function σ: N to R. The relationship between the contact vector fields X and X is investigated in depth.

No description available.

No description available.

The purpose of this paper is to develop a geometry based on fourteen axioms and four undefined terms.