On Orlicz-Sobolev capacities

This thesis consists of three articles on Orlicz-Sobolev capacities. Capacity is a set function which gives information of the size of sets. Capacity is useful concept in the study of partial differential equations, and generalizations of exponential-type inequalities and Lebesgue point theory, and other topics related to weakly differentiable functions such as functions belonging to some Sobolev space or Orlicz-Sobolev space. In this thesis it is assumed that the defining function of the Orlicz-Sobolev space, the Young function, satisfies certain growth conditions. In the first article, the null sets of two different versions of Orlicz-Sobolev capacity are studied. Sufficient conditions are given so that these two versions of capacity have the same null sets. The importance of having information about null sets lies in the fact that the sets of capacity zero play similar role in the Orlicz-Sobolev space setting as the sets of measure zero do in the Lebesgue space and Orlicz space setting. The second article continues the work of the first article. In this article, it is shown that if a Young function satisfies certain conditions, then two versions of Orlicz-Sobolev capacity have the same null sets for its complementary Young function. In the third article the metric properties of Orlicz-Sobolev capacities are studied. It is usually difficult or impossible to calculate a capacity of a set. In applications it is often useful to have estimates for the Orlicz-Sobolev capacities of balls. Such estimates are obtained in this paper, when the Young function satisfies some growth conditions.

Coherent quantum Boltzmann equations from cQPA

We reformulate and extend our recently introduced quantum kinetic theory for interacting fermion and scalar fields. Our formalism is based on the coherent quasiparticle approximation (cQPA) where nonlocal coherence information is encoded in new spectral solutions at off-shell momenta. We derive explicit forms for the cQPA propagators in the homogeneous background and show that the collision integrals involving the new coherence propagators need to be resummed to all orders in gradient expansion. We perform this resummation and derive generalized momentum space Feynman rules including coherent propagators and modified vertex rules for a Yukawa interaction. As a result we are able to set up self-consistent quantum Boltzmann equations for both fermion and scalar fields. We present several examples of diagrammatic calculations and numerical applications including a simple toy model for coherent baryogenesis.

Modelling initial survival and growth of radiata pine in New Zealand.

A sensitive framework has been developed for modelling young radiata pine survival, its growth and its size class distribution, from time of planting to age 5 or 6 years. The data and analysis refer to the Central North Island region of New Zealand. The survival function is derived from a Weibull probability density function, to reflect diminishing mortality with the passage of time in young stands. An anamorphic family of trends was used, as very little between-tree competition can be expected in young stands. An exponential height function was found to fit best the lower portion of its sigmoid form. The most appropriate basal area/ha exponential function included an allometric adjustment which resulted in compatible mean height and basal area/ha models. Each of these equations successfully represented the effects of several establishment practices by making coefficients linear functions of site factors, management activities and their interactions. Height and diameter distribution modelling techniques that ensured compatibility with stand values were employed to represent the effects of management practices on crop variation. Model parameters for this research were estimated using data from site preparation experiments in the region and were tested with some independent data sets.