The imaginary part of the retarded self energy is of particular interest as it contains a lot of physical information about particle interactions. In higher order loop diagrams the calculation become extremely tedious and if we have to do the same at finite temperature, it includes an extra dimension to the difficulty. In such a condition we require to switch between bases and select the best basis for a particular diagram. We have shown in our calculation that in higher order loop diagrams, at #12;finite temperature, the R/A basis is most convenient on summing over the internal vertices and very efficient on calculating some particular diagrams while the result is most easily interpretable in the Keldysh basis for most other complex diagrams.

Tropical geometry is an area of mathematics that interfaces algebraic geometry and combinatorics. The main object of study in tropical geometry is the tropical variety, which is the combinatorial counterpart of a classical variety. A classical variety is converted into a tropical variety by a process called tropicalization, thus reducing the problems of algebraic geometry to problems of combinatorics. This new tropical variety encodes several useful information about the original variety, for example an algebraic variety and its tropical counterpart have the same dimension. In this thesis, we look at the some of the computational aspects of tropical algebraic geometry. We study a generalization of Grobner basis theory of modules which unlike the standard Grobner basis also takes the valuation of coefficients into account. This was rst introduced in (Maclagan & Sturmfels, 2009) in the settings of polynomial rings and its computational aspects were first studied in (Chan & Maclagan, 2013) for the polynomial ring case. The motivation for this comes from tropical geometry as it can be used to compute tropicalization of varieties. We further generalize this to the case of modules. But apart from that it has many other computational advantages. For example, in the standard case the size of the initial submodule generally grows with the increase in degree of the generators. But in this case, we give an example of a family of submodules where the size of the initial submodule remains constant. We also develop an algorithm for computation of Grobner basis of submodules of modules over Z=p`Z[x1; : : : ; xn] that works for any weight vector. We also look at some of the important applications of this new theory. We show how this can be useful in efficiently solving the submodule membership problem. We also study the computation of Hilbert polynomials, syzygies and free resolutions.

The theoretical part is focused on the definition of the basic concepts of the assets, cash and accrual bases. In addition to profit, its components and its differences from the tax base. Much of the work deals with different concepts of costs and revenues in accordance with accounting regulations and tax law. The work deals with the various costs and revenues in terms of tax efficiency and impact on adjustments to the tax base.The practical part is devoted to empirical research, the proportion of tax on profit for the sampled companies.

Since the competence provided in the CFSP and the AFSJ areas in certain situation can overlap, issues have arisen in the recent case law and literature concerning the choice of legal basis. The provisions of the two policy areas concern important international areas and the institutional balance, which is based on what legal basis is chosen, leads to a number of institutional consequences such as division of power between the Member states and the Union’s institutions. The idea behind this paper is to display an ample and just picture of a complicated situation in order to highlight the issues at hand that have arisen due to the complexness of the legal framework. If one of the policy areas are chosen as the correct legal basis, different rules in the treaty applies, which leads to different distribution of power since different institutions in EU play different roles depending on legal basis. The purpose of this work is therefore to analyse the external dimension of the AFSJ and the CFSP rules and examine how the correct legal basis can be determined by the legislator at the same time as the Member States and the EU’s ability to fulfil their objectives and goals is not undermined.

Let $V$ be a finite-dimensional vector space over a field $\mathbb{K}$, and let \text{End}$(V)$ be the set of all $\mathbb{K}$-linear transformations from $V$ to $V$. A {\em Leonard system} on $V$ is a sequence \[(\A ;\B; \lbrace E_i\rbrace_{i=0}^d; \lbrace E^*_i\rbrace_{i=0}^d),\] where $\A$ and $\B $ are multiplicity-free elements of \text{End}$(V)$; $\lbrace E_i\rbrace_{i=0}^d$ and $\lbrace E^*_i\rbrace_{i=0}^d$ are orderings of the primitive idempotents of $\A $ and $\B$, respectively; and for $0\leq i, j\leq d$, the expressions $E_i\B E_j$ and $E^*_i\A E^*_j$ are zero when $\vert i-j\vert > 1$ and nonzero when $\vert i-j \vert = 1$. % Leonard systems arise in connection with orthogonal polynomials, representations of many nice algebras, and the study of some highly regular combinatorial objects. We shall use the construction of Leonard pairs of classical type from finite-dimensional modules of $\mathit{sl}_2$ and the construction of Leonard pairs of basic type from finite-dimensional modules of $U_q(\mathit{sl}_2)$. Suppose $\Phi:=(\A ;\B; \lbrace E_i\rbrace_{i=0}^d; \lbrace E^*_i\rbrace_{i=0}^d)$ is a Leonard system. For $0 \leq i \leq d$, let \[ U_i = (E^*_0V+E^*_1V+\cdots + E^*_iV)\cap (E_iV+E_{i+1}V+\cdots + E_dV). \] Then $U_0$, $U_1$, \ldots, $U_d$ is the {\em split decomposition of $V$ for $\Phi$}. % The split decomposition of $V$ for $\Phi$ gives rise to canonical matrix representations of $\A$ and $\B$ in terms of useful parameters for the Leonard system. %These canonical matrix representations for $\A$, $\B$ are respectively lower bidiagonal and upper bidiagonal. In this thesis, we consider when certain Leonard systems share a split decomposition. We say that Leonard systems $\Phi:=(\A ;\B; \lbrace E_i\rbrace_{i=0}^d; \lbrace E^*_i\rbrace_{i=0}^d)$ and $\hat{\Phi}:=(\hat{\A} ;\hat{\B}; \lbrace \hat{E}_i\rbrace_{i=0}^d; \lbrace \hat{E^*}_i\rbrace_{i=0}^d)$ are {\em friends} when $\A = \hat{\A}$ and $\Phi$, $\hat{\Phi}$ have the same split decomposition. % We obtain Leonard systems which share a split decomposition by constructing them from closely related module structures for either $\mathit{sl}_2$ or $U_q(\mathit{sl}_2)$ on $V$. We then describe friends by a parametric classification. In this manner we describe all pairs of friends of classical and basic types. In particular, friendship is not entirely a property of isomorphism classes.

published_or_final_version / Business / Doctoral / Doctor of Philosophy

縮基法(RBM) 是對參數化的曲線求逼近解的一個方法，基本上乃使用投影法將解曲線投射到解空間的一子空間中，如此一來，可將原問題轉換成一較小的系統，並經由數值計算出小系統的解，來求得大系統的一逼近解。在本篇論文中主要的乃探討RBM在常微分方程組初始值問題上的應用，並發展一套含有誤差控制的演算法。 本篇論文中所採用的ODE Solver 乃由Gordon 和Shampine 基於Adams PECE方法所發展的。在求解的過程中，對於計算解誤差的控制我們除了利用ODE Solver 的誤差估計，另外我們又發展對縮基解(reduced basis solution) 的後(Aposteriori) 誤差估計，以確保數值計算解的準確性。我們所考慮使用的子空間有三種Taylor, Lagrange , Hermite 。同時為了要增加數值的特定性及簡化小系統的求解工作，我們先行將子空間的基底直交化。因此，除了誤差的控制外，我們也討論了roundoff error 對向量直交化及形成小系統時所造成的影響，並設立誤差標準以判別何時誤差過大到嚴重影響縮基解的準確度。 本篇論文的目的是希望利用RBM發展出一套解常微分方程組初始值問題的求算法，以期計算解能在較短的時間內準確的被計算出來。 / The reduced basis method(RBM) is a scheme for approximating parametric solution curves. The basic technique of RBM is projection. By applying the method, we can find an approximate solution of the original system which satisfies a system of smaller size. In this paper, we mainly concern the applications of RBM for ODE initial value problems and develop an algorithm which contains a set of error controls. The ODE solver used in this paper is developed by Gordon and Shampine based on Adams PECE formulas. To assure the accuracy of the reduced basis approximation, we set up an appropriate automatic error control in calling GS solver and develop an a posteriori error estimate to keep the reduction error under control. The subspaces considered are Taylor, Lagrange and Hermite subspaces.In the meantime, in order to improve the numerical stability and simplify the computation of the reduced basis solution, we orthogonalize the generators of reduced subspaces. We also discuss the roundoff errors in the orthogonalization process and build up a criterion for identifying the case the accuracy of the reduced basis solution up a criterion for identifying the case the accuracy of the reduced basis solution is destroyed by the errors. The aim of this paper is to develop an algorithm to solve the ODE initial value problems efficiently.

Weather index insurance has become a popular subject in agricultural risk management. Under these policies farmers receive payments if they experience adverse weather for their crops. Spatial basis risk is the risk that weather observed at stations does not correspond to the weather experienced by the farmer. The objective of this research is to determine to what extent spatial basis risk can be impacted by the interpolation technique used to estimate weather conditions. Using forage crops from Ontario, Canada, as an example, a temperature based insurance index is developed. Seven different interpolation methods are used to estimate indemnities for forage producers. Results show that the number of weather stations in the interpolation area has a larger impact on spatial basis risk than the choice of interpolation technique. For insurers wishing to implement this type of insurance, more focus should be placed on increasing the number of available weather stations. / October 2016

Integer Programming problems are difficult to solve. The goal is to find an optimal solution that minimizes cost. With the help of Groebner based algorithms the optimal solution can be found if it exists. The application of the Groebner based algorithm and how it works is the topic of research. The Algorithms are The Conti-Traverso Algorithm and the Original Conti-Traverso Algorithm. Examples are given as well as proofs that correspond to the algorithms. The latter algorithm is more efficient as well as user friendly. The algorithms are not necessarily the best way to solve and integer programming problem, but they do find the optimal solution if it exists.

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