Fundamentals of Riemannian geometry and its evolution : a thesis presented in partial fulfilment of the requirements for the degree of Master of Science in Mathematics at Massey University, Palmerston North, New Zealand

In this thesis we study the theory of Riemannian manifolds: these are smooth manifolds equipped with Riemannian metrics, which allow one to measure geometric quantities such as distances and angles. The main objectives are: (i) to introduce some of the main ideas of Riemannian geometry, the geometry of curved spaces. (ii) to present the basic concepts of Riemannian geometry such as Riemannian connections, geodesics, curvature (which describes the most important geometric features of universes) and Jacobi fields (which provide the relationship between geodesics and curvature). (iii) to show how we can generalize the notion of Gaussian curvature for surfaces to the notion of sectional curvature for Riemannian manifolds using the second fundamental form associated with an isometric immersion. Finally we compute the sectional curvatures of our model Riemannian manifolds - Euclidean spaces, spheres and hyperbolic spaces

Old Wine In A New Bottle? Proposed Sugar Industry Restructuring and Land Conflict in Fiji

The proposed restructure of Fiji's sugar industry has the potential to change fundamentally the incentives facing stakeholders and their interaction. How well the stated objective of 'unity in purpose' is achieved will depend on whether the causes of th

Relativistic x-ray free-electron lasers in the quantum regime

We present a nonlinear theory for relativistic x-ray free-electron lasers in the quantum regime, using a collective Klein-Gordon (KG) equation (for relativistic electrons), which is coupled with the Maxwell-Poisson equations for the electromagnetic and electrostatic fields. In our model, an intense electromagnetic wave is used as a wiggler which interacts with a relativistic electron beam to produce coherent tunable radiation. The KG-Maxwell-Poisson model is used to derive a general nonlinear dispersion relation for parametric instabilities in three space dimensions, including an arbitrarily large amplitude electromagnetic wiggler field. The nonlinear dispersion relation reveals the importance of quantum recoil effects and oblique scattering of the radiation that can be tuned by varying the beam energy

Maternal Morbidity in Rural Andhra Pradesh

Pregnancy constitutes a high risk of morbidity and mortality due to associated physiological stress. Many women do not die of causes related to pregnancy but suffer severe morbidities. In developing countries, pregnancy and childbirth related complications are the leading cause of disability among women aged 15-44 years. The objective of the present paper is to assess the extent of maternal morbidity and assess the factors responsible for it in rural Andhra Pradesh. An attempt is also made to understand the treatment seeking behaviour of the women for the morbidities. The study revealed a high prevalence of maternal morbidity in rural areas of Andhra Pradesh. Especially in the less developed district, nearly 95 percent of the women experienced at least one of the morbidities and in the developed district it is 61 percent. 'Life threatening' and 'Serious' morbidities are experienced by 39 and 54 percent in the less developed; 15 and 46 percent of women in developed district respectively. High incidence of maternal morbidity in rural areas can be attributed to the combination of individual, household, community, medical as well as nutrition variables. Determinants of maternal morbidity at individual level are age, parity, education and work status; household variables are type of family, possession of land, availability of toilet with in the premises of house. In addition poor infrastructure facilities such as access to villages and non-availability of health care facilities further added to the level of maternal morbidity. Government's introduction of RCWHCs is not been effective in bringing down the morbidity levels due to lack of proper staff, particularly lady doctor. In spite of low levels of perception about maternal morbidities, relatively higher utilization of health services during antepartum period should be taken as an advantage for initiating complete antenatal services, i.e. beyond the coverage of women by TT injection and supply of IFA tablets. The governmental initiation of supplying protein-rich food to pregnant and lactating mothers through ICDS programme has significantly lowered the morbidity levels probably by lessening the levels of anaemia.Maternal Morbidity, rural, Andhra Pradesh

The discrete logarithm problem over prime fields: the safe prime case. The Smart attack, non-canonical lifts and logarithmic derivatives

In this brief note we connect the discrete logarithm problem over prime fields in the safe prime case to the logarithmic derivative.Comment: 8 page