We show how the theory of multiplier ideals can be developed and discuss several applications of this theory. In the second section the same theory in the analytic setting is developed and several...Show moreWe show how the theory of multiplier ideals can be developed and discuss several applications of this theory. In the second section the same theory in the analytic setting is developed and several applications are given. Let X be a smooth algebraic variety and D an effective Q-divisor. We associate to D (or to the pair (X, D)) an ideal sheaf I(D) which controls the behavior of the fractional part of D and determines how close it is to have an simple normal crossing support. Other applications can be treated such as singularities of projective hypersurfaces and characterization of divisors. In the former case a result of Esnault-Viehweg concerning the least degree of hypersurfaces with multiplicity greater than or equal to a given positive integer at each point of a finite set is explained and proved in two different ways. A slight generalization is also given. Several vanishing and non-vanishing results including a global generation theorem are treated which will be used to prove the results about singularities. In the second section the analytic analogues of the materials in section one are given and the characterization of analytic nef and good divisors are explained.Show less
Economical data collected by Statistics Netherlands usually contains missing items. Various imputation methods are available to fill in these gaps, so that completed datasets can be analyzed using...Show moreEconomical data collected by Statistics Netherlands usually contains missing items. Various imputation methods are available to fill in these gaps, so that completed datasets can be analyzed using standard statistical tools. One of the methods often used, the ratio imputation method, appears not to perform very well if we want the completed data to satisfy certain restrictions. This is our motivation to investigate other imputation methods. We look at several methods that we subdivide over two groups. The first group consists of methods based on models that assume a joint distribution for all variables for an individual, and that these variables are all independent. Here we will discuss methods that assumes the data are truncated normally distributed, or exponentially distributed. We propose the proportional variance method, and investigate various possible underlying models. The second group is made up of methods that only specify certain conditional distributions. Here we will investigate the commonly used ratio imputation method and both the classical and the Bayesian variants of sequential regression imputation methods. After we have discussed these methods, we repeatedly apply them to a dataset provided by Statistics Netherlands in which we make a missing pattern ourselves. We use the results of these simulations to assess the performance of the methods on several criteria.Show less
Suppose A is an order of some number field K. In this thesis, we will present some results related to the Galois group and the discriminant under some special condition on A. We apply this to some...Show moreSuppose A is an order of some number field K. In this thesis, we will present some results related to the Galois group and the discriminant under some special condition on A. We apply this to some f ∈ Z[x] with Z[x]/(f, f0 ) cyclic. By studying the trinomial f = x n + axl + b, we solve some exponential Diophantine equations. At last, Selmer’s trinomial is used to illustrate our main theorem.Show less
