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No description available.

In this thesis, we have explicitly calculated all lower order terms for the n-correlation of zeros of certain families of L-functions. These calculations follow from Conrey and Snaith's similar work for the Riemann zeta function. Katz and Sarnak have argued that the zero statistics of families of L-functions have an underlying symmetry relating to certain ensembles of random matrices. With this in mind, we have looked at a family with orthogonal symmetry (even twists of the Hasse-Weil L-function of a given elliptic curve) and a family with symplectic symmetry (Dirichlet Lfunctions) . Assuming the ratios conjectures of Conrey, Farmer, and Zirnbauer, we prove a formula which explicitly gives all of the lower order terms in the n-correlation . For the families relating to elliptic curves, this formula agrees with the known results of Huynh, Keating and Snaith for n = 1 and as the conductor tends to infinity the 2-correlation matches that of eigenangles of random orthogonal matrices under Haar measure. The method used in this thesis works by first calculating n-correlation of eigenangles of SO(2N) and USp(2N) via ratios of characteristic polynomials. In a similar manner to Conrey and Snaith's work on U(N), we can identify which terms remain in the n-correlation of eigenangles of random orthogonal or symplectic matrices when restrictions are placed on the support of the test function. It is hoped that this will allow for an easier way of checking results with L-functions match those predicted by random matrix theory.

It is an important problem in analytic number theory to estimate mean values of the Riemann zeta-function and other L-functions. The study of moments of L-functions has some important applications, such as to give information about the maximal order of the Riemann zeta-function on the critical line, the Lindelof Hypothesis for L-functions and non-vanishing results. Furthermore, according to the Katz-Sarnak philosophy [Katz-Sar99a, Katz-Sar99b] it is believed that the understanding of mean values of different families of L-functions may reveal the symmetry of such families. The analogy between characteristic polynomials of random matrices and L- functions was first studied by Keating and Snaith [Kea-SnaOOa, Kea-SnaOOb]. For example, they were able to conjecture asymptotic formulae for the moments of L- functions in different families. The purpose of this thesis is to study moments of L-functions over function fields, since in this case the L-functions satisfy a Riemann Hypothesis and one may give a spectral interpretation for such L-functions as the characteristic polynomial of a unitary matrix. Thus, we expect that the analogy between characteristic polynomials and L-functions can be further understood in this scenario. In this thesis, we study power moments of a family of L-functions associated with hyperelliptic curves of genus 9 over a fixed finite field lFq in the limit as g-->∞, which is the opposite limit considered by the programme of Katz and Sarnak. Specifically, we compute some average value theorems of L-functions of curves and we extend to the function field setting the heuristic for integral moments and ratios of L-functions previously developed by Conrey et. al [CFKRS05, Conr-Far-Zir] for the number field case.

Let Q be a simply laced Dynkin quiver, Db(Q) the bounded derived category of the path algebra associated to Q and C(Q) the TΣ2-orbit category ofDb( Q), where T is the Auslander- Reiten-translation and Σ is the shift functor. Note that C( Q) is a triangulated category [57]. In this thesis we study two classes of representation-theoretic objects in C ( Q): maximal Hom- free objects, also known as Horn-configurations, and maximal rigid (i.e. Ext-free) objects. Horn-configurations (in the derived category) were used by Riedtrnann [65] in order to classify selfinjective algebras of finite-representation type. Riedtmann proved that these objects are invariant under the autoequivalence TΣ2 of Db(Q). This is the reason why we consider the orbit category C( Q). We establish a bijection between Horn-configurations and noncrossing partitions of the Coxeter group associated to Q which are not contained in any proper standard parabolic subgroup. These noncrossing partitions are said to be positive, because they are proved to be in 1-1 correspondence with the positive clusters in the corresponding cluster algebra (cf. [62]). The bijection between Horn-configurations and positive noncrossing partitions generalizes a result of Riedtmann which states that Horn-configurations in Db (Q), where Q is of type An' are in bijection with classical noncrossing partitions of the set {1, ... , n}. Riedtmann's bijection allows us to construct a geometrical model for C(Q) in type A. Using this geometrical setup, and inspired by the classification of the cluster-tilting objects in the cluster category of type An in terms of triangulations of a polygon, we classify, also in type An, the maximal rigid objects in C(Q) in terms of certain noncrossing bipartite graphs. In addition, we describe the corresponding endomorphism algebras in terms of quivers with relations. We also give a natural notion of mutation of Horn-configurations in C(Q) and we present some partial results and conjectures about the mutation graph and the representation-theoretic description of these mutations

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This PhD thesis consists of five papers dealing with problems in various branches of Diophantine approximation. The results obtained contribute to the theory of twisted, weighted, multiplicative and mixed approximation. In Paper I a twisted analogue of the classical set of badly approximable linear forms is introduced. We prove that its intersection with any suitably regular fractal set is of maximal Hausdorff dimension. The linear form systems investigated are natural extensions of irrational rotations of the circle. Even in the latter one-dimensional case, the results obtained are new. The main result of Paper II concerns a weighted version of the classical set of badly approximable pairs. We establish a new characterization of this set in terms of vectors that are well approximable in the twisted sense. This naturally generalizes a classical result of Kurzweil. In Paper II we also study the metrical theory associated with a weighted variant of the set introduced in Paper I. In particular, we provide a sufficient condition for this variant to have full Hausdorff dimension. This result is extended in Paper III to imply the stronger property of 'winning'. Paper IV addresses various problems associated with the Mixed Littlewood Conjecture. Firstly, we solve a version of the conjecture for the case of one p-adic value and one pseudo-absolute value with bounded ratios. Secondly, we deduce the answer to a related metric question concerning numbers that are well approximable in the mixed multiplicative sense. This provides a mixed analogue to a classical theorem of Gallagher. In Paper V we develop the metric theory associated with the mixed Schmidt Conjecture. In particular, a Khintchine-type criterion for the 'size' of the natural set of mixed well approximable numbers is established. As a consequence we obtain a combined mixed and weighted version of the classical Jarník-Besicovich Theorem.

This thesis investigates the causal theory of properties (CTP). CTP states that properties must be understood via the complicated network of causal relations to which a property can contribute. If an object instantiates the property of being 900C, for instance, it will burn human skin on contact, feel warm to us if near, etc. In order to best understand CTP, I argue that we need to distinguish between properties and particular instances of them. Properties should be analysed via the causal relations their instances stand in, it is this oven’s being 900C which causes my skin to burn, etc. The resulting CTP offers an illuminating analysis of properties. First, it provides a criterion of identity for properties, their identity being analysed via the causal roles property instances realise. It also offers an account of how property instances are sorted into genuine kinds, in cases of determinables and determinates. I show how we can distinguish between genuine and non-genuine similarity via the property instances of objects. The implications of CTP for an analysis of causation are then investigated. I argue that the proposed CTP offers a plausible causal ontology. The fine-grainedness of property instances enables us to capture the subtleties involved in questions concerning what causes what. But, even more importantly, CTP enables us to reconcile two highly attractive theses concerning the causal relation. The first of these is the generalist’s thesis. This states that causal relations are part of more general patterns. The second of these is the singularist’s thesis. This states that the causal connection between two entities, doesn’t depend upon anything extraneous to that relation. I argue that by combining CTP with an ontology of tropes, we can thereby respect what is driving both singularism and generalism.