n this thesis the analytic properties of power series over a class of non-Archimedean field extensions of the real numbers, a representative of which will be denoted by F, are investigated. In Chapter 1 we motivate the interest in said fields by recalling work done by K. Shamseddine and M. Berz . We first review some properties of well-ordered subsets of the rational numbers which are used in the construction of such a field F. Then, we define operations + and * which make F a field. Then we define an order under which F is non-Archimedean with infinitely small and infinitely large elements. We embed the real numbers as a subfield; and the embedding is compatible with the order. Then, in Chapter 2, we define an ultrametric on F which induces the same topology as the order on the field. This topology will allow us to define continuity and differentiability of functions on F which we shall show are insufficient conditions to ensure intermediate values, extreme values, et cetera. We shall study convergence of sequences and series and then study the analytical properties of power series, showing they have the same smoothness properties as real power series; in particular they satisfy the intermediate value theorem, the extreme value theorem and the mean value theorem on any closed interval within their domain of convergence. / October 2016

M.Comm. / This study analyzes the effect of the subprime crisis on portfolio allocation from the perspective of dependence structure. Empirical evidence has proved that the multivariate normal distribution is inadequate to model portfolio asset return distribution - firstly because the empirical marginal distributions of asset returns are skewed and fat tailed; and secondly because it does not consider the possibility of extreme joint co-movement of asset returns (Fama and French, 1993; Richardson and Smith, 1993; Géczy, 1998; Longin and Solnik, 2001; Mashal and Zeevi, 2002). This study employs Archimedean copulas to capture both the dependence structure and the asymmetry of asset returns in the tails of the empirical distributions.

The hypertoric variety M_A defined by an arrangement A of affine hyperplanes admits a natural tropicalization, induced by its embedding in a Lawrence toric variety. In this thesis, we explicitly describe the polyhedral structure of this tropicalization and calculate the fibers of the tropicalization map. Using a recent result of Gubler, Rabinoff, and Werner, we prove that there is a continuous section of the tropicalization map.

In this thesis we are interested in the calculation of economic capital for the to- tal loss which is the sum of partial dependent losses, whose dependence structure is described by Archimedean and hierarchical Archimedean copulas. Firstly, the concept of economic capital and the ways of its aggregation are introduced. Then the basic definitions and properties of copulas are listed, as well as the depen- dence measures. After that we work with definition and properties of Archimedean copulas and their simulation. We also mention the most popular families of Ar- chimedes copulas. Next, hierarchical Archimedean copulas are defined, as well as the algorithm for their sampling. Finally, we present methods for estimating the parameters of copulas and the recursive algorithm for estimating the hierarchical Archimedean copula structure. In the last chapter we perform simulation studies of selected models using hierarchical Archimedes copulas. 1

This thesis investigates some properties of valuations on fields. Basic definitions and theorems assumed are stated in Capter I. Chapter II introduces the concept of a valuation on a field. Real valuations and non-Archimedean valuations are presented. Chapter III generalizes non-Archimedean valuations. Examples are described in Chapters I and II. A result is the theorem stating that a real valuation of a field K is non-Archimedean if and only if $(a+b) < max4# (a), (b) for all a and b in K. Chapter III generally defines a non-Archimedean valuation as an ordered abelian group. Real non-Archimedean valuations are either discrete or nondiscrete. Chapter III shows that every valuation ring identifies a non-Archimedean valuation and every non-Archimedean valuation identifies a valuation ring.

The purpose of this thesis is to study the concept of completeness in an ordered field. Several conditions which are necessary and sufficient for completeness in an ordered field are examined. In Chapter I the definitions of a field and an ordered field are presented and several properties of fields and ordered fields are noted. Chapter II defines an Archimedean field and presents several conditions equivalent to the Archimedean property. Definitions of a complete ordered field (in terms of a least upper bound) and the set of real numbers are also stated. Chapter III presents eight conditions which are equivalent to completeness in an ordered field. These conditions include the concepts of nested intervals, Dedekind cuts, bounded monotonic sequences, convergent subsequences, open coverings, cluster points, Cauchy sequences, and continuous functions.

The field of real numbers does not permit a direct representation of the (improper) delta functions used for the description of impulsive (instantaneous) or concentrated (localized) sources. Of course, within the framework of distributions, these concepts can be accounted for in a rigorous fashion, but at the expense of the intuitive interpretation. The existence of infinitely small numbers and infinitely large numbers in the Levi-Civita field allows us to have well-behaved delta functions which, when restricted to the real numbers, reduce to the Dirac delta function. Here we develop the foundations for a mathematically rigorous theory of localized and instantaneous sources that has a clear and unambiguous way of specifying a mathematically concentrated source. We use the already existent one variable measure and integration theory on Levi-Civita field to construct the foundations of a measure and integration theory in two and three dimensions. First we construct measurable sets using sets with boundaries that can be expressed as analytic functions and we show the the resulting measure is Lebesgue-like. In particular we prove the measurability of countable sets, the countable union of measurable sets, and the finite intersection of measurable sets. Following that we use analytic functions to construct a larger class of measurable functions, we then define the integral of a measurable function over a measurable set. We prove several propositions regarding measurable functions and the associated integration theory including that the set of measurable functions is closed under multiplication and addition, and that integration is linear. This allows for a wide range of applications for the delta function in one, two, and three dimensions and sets the course for a more extensive study of this topic in the future.

Šio darbo tema yra jungčių (angl. copulas) panaudojimas ryšiams tarp daugiamačių atsitiktinių dydžių modeliuoti. Jungtis yra funkcija, kuri sujungia kelių atsitiktinių dydžių marginalinius skirstinius į bendrą daugiamatę funkciją. Jungties sąvoka pirmą kartą statistikoje įvesta 1959 m. Šiame darbe aprašomos pagrindinės jungčių savybės, keletas jungčių šeimų, išskiriant atskirą šeimą - Archimedo jungtis, taip pat priklausomumo matai tarp atsitiktinių dydžių. Vėliau tinkamos jungties pritaikymo turimam duomenų rinkiniui procedūra iliustruojama nagrinėjant transporto priemonių valdytojų civilinės atsakomybės privalomojo draudimo žalų ir išlaidų žaloms administruoti duomenis. / In this Master work the concept of copulas as a tool for modeling relationships among multivariate outcomes is introduced. A copula is a function that links univariate margins to their multivariate distribution. Copulas were introduced in 1959. The literature on the statistical properties and application of copulas has been developing rapidly in recent years. In this Master work basic properties of copulas are described, then several families of copulas and relationships to measures of dependences. Later procedure for selecting the parametric family of Archimedean copulas is illustrated by using Lithuanian Motor Third Party Liability insurance data losses and expenses. For these data it is shown how to fit copulas according to nonparametric procedure which was proposed by Genest and Rivest.

In this thesis, we classify up to conjugacy the maximal elliptic toral subgroups of all special orthogonal groups SO(V), where (q,V) is a 4-dimensional quadratic space over a non-archimedean local field of odd residual characteristic. Our parameterization blends the abstract theory of Morris with a generalization of the practical work performed by Kim and Yu for Sp(4). Moreover, we compute an explicit Witt basis for each such torus, thereby enabling its concrete realization as a set of matrices embedded into the group. This work can be used explicitly to construct supercuspidal representations of SO(V).

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