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The thesis consists of four chapters. First chapter is introductory. In Chapter 2, we recall some basic facts from the singularity theory of algebraic varieties (see Section 2.2) and the theory of minimal models (see Section 2.3), which will be used throughout the rest of the thesis. We also make some conventions on the notions and notation used in the thesis (see Section 2.1). Each Chapter 3 and 4 starts with some preliminary results (see Sections 3.1 and 4.1, respectively). Each Chapter 3 and 4 ends with some corollaries and conclusive remarks (see Sections 3.7 and 4.4, respectively). In Chapter 3, we prove Theorem 1.2.7, providing the complete description of Halphen pencils on a smooth projective quartic threefold X in P4. Let M be such a pencil. Firstly, we show that M ⊂ | − nKX | for some n ∈ N, and the pair (X,1n M) is canonical but not terminal. Further, if the set of not terminal centers CS(X, 1 ) (see Remark 2.2.8) does not contain points, we show that n = 1 (see Section 3.2). Finally, if there is a point P ∈ CS(X, n M), in Section 3.1 we show first that a general M ∈ M has multiplicity 2n at P (cf. Example 1.2.3). After that, analyzing the shape of the Hessian of the equation of X at the point P , we prove that n = 2 and M coincides with the exceptional Halphen pencil from Example 1.2.6 (see Sections 3.3-3.6). In Chapter 4, we prove Theorem 1.2.11, which shows, in particular, that a general smooth K3 surfaces of type R is an anticanonical section of the Fano threefold X with canonical Gorenstein singularities and genus 36. In Section 4.2, we prove that X is unique up to an isomorphism and has a unique singular point, providing the geometric quotient construction of the moduli space F in Section 4.3 (cf. Remark 1.2.12). Finally, in Section 4.3 we prove that the forgetful map F −→ KR is generically surjective.

This thesis consists of two parts for the boundary approximation methods(BAMs). Part I is devoted to the Laplace equations with theory and computation; Part II to the biharmonic equations only with computation.

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<p>This dissertation argues that Emmanuel Levinas' s is first and foremost a philosopher of subjectivity. I argue that the themes of restlessness of singularity and restlessness govern Levinas' s account of subjectivity and that these themes directly inform the account of the relationship with the Other found in his mature works. Chapter I presents Levinas' s early reflections on identity and escape as arising from his encounter with Husserl's and Heidegger's respective philosophies. The first chapter establishes restlessness and singularity as central themes in Levinas's philosophy. The second chapter argues that Levinas' s account of the relationship with existence found in From Existence to Existence further develops these themes by establishing the subject as originating from a pre-cognitive event that Levinas calls "hypostasis." Chapter III turns to Totality and Infinity and argues that the notion of the "anterior posterior" condition, by which Levinas means a "logical" condition that precedes a "chronological one," is conceptually similar to the idea of "hypostasis" found in the earlier works and that it allows him to develop the theme of singularity as an ethical category. Chapter IV focuses solely on the connection between singularity and the Face, and argues that Levinas's notion of the Face follows from his earlier accounts of singularity but that Levinas fails to address the precise relationship between the different forms of meaning that make up interior life and the relationship with transcendence, and that these problems lead to further questions concerning the roles of politics and justice in his later philosophy. The final chapter will show that in his later works Levinas rehabilitates the theme of restlessness, which is absent in Totality and Infinity, and that it allows him to show that the singularity of the self anci the singularity of the Other are both bound in the same moment. Thus, Levinas returns to the theme of restlessness as a way of addressing the problems that I find in Totality and Infinity. These later developments lead to further questions concerning the role of context in Levinas's idea of the "political." The last chapter concludes by arguing that Levinas is unable to address everyday moral decisionmaking because of his account of the ethical as a "meaning without context."</p> / Thesis / Doctor of Philosophy (PhD)

Cracked beam problem is an elliptic boundary value problem with singularity. It is often used as a testing model for numerical methods. We use numerical and symbolic boundary approximation methods and boundary collocation method to compute its extremely high accurate solution with global error $O(10^{-100})$. This solution then can be regarded as the exact solution. On the other hand, we vary the boundary conditions of this problem to obtain several related models. Their numerical solutions are compared to those of cracked beam and Motz problems, the prototypes of singularity problems. From the comparison we can conclude the advantage of each model and decide the best testing model for numerical methods.

The particular solutions of the Laplace equations and their singularities are fundamental to numerical partial di erential equations in both algorithms and error analysis. We first review the explicit solutions of Laplace¡¦s equations on sectors with the Dirichlet and the Neumann boundary conditions. These harmonic functions clearly expose the solution¡¦s regularity/singularity at the vertex. So we can analyze the singularity of the Laplace¡¦s solutions on polygons at di erent domain corners and for various boundary conditions. By using this knowledge we can designed many new testing models with di erent kind of singularities, like discontinuous and mild singularities, beside the popular singularity models, Motz¡¦s and the cracked beam problems, We use the boundary approximation method, i.e. the collocation Tre tz method in engineering literatures, to solve the above testing models of Laplace boundary value problems on polygons. Suppose the uniform particular solutions are chosen in the entire domain. When there is no singularity on all corners, this method has the exponential convergence. However, its rate of convergence will deteriorate to polynomial if there exist some corner singularities. From experimental data, we even have three type of convergence, i.e. exponential, polynomial or their mixed types. We will study these convergent behaviors and their causes. Finally, we will uncover the relation between the order of convergence and the intensity of corner singularities.

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In this work we apply the Adomian decomposition method to an orientation aggregation problem modelling the time distribution of filaments. We find analytical solutions under certain specific criteria and programmatically implement the Adomian method to two variants of the orientation aggregation model. We extend the utility of the Adomian decomposition method beyond its original capability to enable it to converge to more than one solution of a nonlinear problem and further to be used as a corrector in path following bifurcation problems.