Short Character Sums and Their Applications

In analytic number theory, the most natural generalisations of the famous Riemann zeta function are the Dirichlet L-functions. Each Dirichlet L-function is attached to a q-periodic arithmetic function for some natural number q, known as a Dirichlet character modulo q. Dirichlet characters and L-functions encode with them information about the primes, especially in reference to their remainders modulo q. For this reason, number theorists are often interested in bounds on short sums of a Dirichlet character over the integers. For one, such sums appear as an intermediate step in partial summation bounds for Dirichlet L-functions. However, short character sum estimates may also be used directly to tackle other number theoretic problems. In this thesis, we wish to examine the application of character sum estimates in some specific settings. There are three main estimates that we are interested in: the trivial bound, Burgess’ bound, and the Pólya–Vinogradov inequality. Of these, we will focus primarily on Burgess’ bound; the main result of this thesis will be the computation of explicit constants versions of Burgess’ bound for a variety of parameters, improving upon the work of Treviño [54]. The interplay between Burgess’ bound and the Pólya–Vinogradov inequality is vital in this explicit setting, and we will dedicate a portion of this thesis to investigating these interactions. This makes precise the work of Fromm and Goldmakher [21], who demonstrated a counterintuitive influence the Pólya–Vinogradov inequality has on Burgess’ bound. Once we have established improvements to Burgess’ bound in the explicit setting, we will “test” these improvements by tackling several applications where Burgess’ bound has been used previously. Primary among these is an explicit bound for L-functions across the critical strip. We also include applications to norm-Euclidean cyclic fields and least kth power non-residues

Ahlfors circle maps and total reality: from Riemann to Rohlin

This is a prejudiced survey on the Ahlfors (extremal) function and the weaker {\it circle maps} (Garabedian-Schiffer's translation of "Kreisabbildung"), i.e. those (branched) maps effecting the conformal representation upon the disc of a {\it compact bordered Riemann surface}. The theory in question has some well-known intersection with real algebraic geometry, especially Klein's ortho-symmetric curves via the paradigm of {\it total reality}. This leads to a gallery of pictures quite pleasant to visit of which we have attempted to trace the simplest representatives. This drifted us toward some electrodynamic motions along real circuits of dividing curves perhaps reminiscent of Kepler's planetary motions along ellipses. The ultimate origin of circle maps is of course to be traced back to Riemann's Thesis 1851 as well as his 1857 Nachlass. Apart from an abrupt claim by Teichm\"uller 1941 that everything is to be found in Klein (what we failed to assess on printed evidence), the pivotal contribution belongs to Ahlfors 1950 supplying an existence-proof of circle maps, as well as an analysis of an allied function-theoretic extremal problem. Works by Yamada 1978--2001, Gouma 1998 and Coppens 2011 suggest sharper degree controls than available in Ahlfors' era. Accordingly, our partisan belief is that much remains to be clarified regarding the foundation and optimal control of Ahlfors circle maps. The game of sharp estimation may look narrow-minded "Absch\"atzungsmathematik" alike, yet the philosophical outcome is as usual to contemplate how conformal and algebraic geometry are fighting together for the soul of Riemann surfaces. A second part explores the connection with Hilbert's 16th as envisioned by Rohlin 1978.Comment: 675 pages, 199 figures; extended version of the former text (v.1) by including now Rohlin's theory (v.2

Regular Hierarchical Surface Models: A conceptual model of scale variation in a GIS and its application to hydrological geomorphometry

Environmental and geographical process models inevitably involve parameters that vary spatially. One example is hydrological modelling, where parameters derived from the shape of the ground such as flow direction and flow accumulation are used to describe the spatial complexity of drainage networks. One way of handling such parameters is by using a Digital Elevation Model (DEM), such modelling is the basis of the science of geomorphometry. A frequently ignored but inescapable challenge when modellers work with DEMs is the effect of scale and geometry on the model outputs. Many parameters vary with scale as much as they vary with position. Modelling variability with scale is necessary to simplify and generalise surfaces, and desirable to accurately reconcile model components that are measured at different scales. This thesis develops a surface model that is optimised to represent scale in environmental models. A Regular Hierarchical Surface Model (RHSM) is developed that employs a regular tessellation of space and scale that forms a self-similar regular hierarchy, and incorporates Level Of Detail (LOD) ideas from computer graphics. Following convention from systems science, the proposed model is described in its conceptual, mathematical, and computational forms. The RHSM development was informed by a categorisation of Geographical Information Science (GISc) surfaces within a cohesive framework of geometry, structure, interpolation, and data model. The positioning of the RHSM within this broader framework made it easier to adapt algorithms designed for other surface models to conform to the new model. The RHSM has an implicit data model that utilises a variation of Middleton and Sivaswamy (2001)’s intrinsically hierarchical Hexagonal Image Processing referencing system, which is here generalised for rectangular and triangular geometries. The RHSM provides a simple framework to form a pyramid of coarser values in a process characterised as a scaling function. In addition, variable density realisations of the hierarchical representation can be generated by defining an error value and decision rule to select the coarsest appropriate scale for a given region to satisfy the modeller’s intentions. The RHSM is assessed using adaptions of the geomorphometric algorithms flow direction and flow accumulation. The effects of scale and geometry on the anistropy and accuracy of model results are analysed on dispersive and concentrative cones, and Light Detection And Ranging (LiDAR) derived surfaces of the urban area of Dunedin, New Zealand. The RHSM modelling process revealed aspects of the algorithms not obvious within a single geometry, such as, the influence of node geometry on flow direction results, and a conceptual weakness of flow accumulation algorithms on dispersive surfaces that causes asymmetrical results. In addition, comparison of algorithm behaviour between geometries undermined the hypothesis that variance of cell cross section with direction is important for conversion of cell accumulations to point values. The ability to analyse algorithms for scale and geometry and adapt algorithms within a cohesive conceptual framework offers deeper insight into algorithm behaviour than previously achieved. The deconstruction of algorithms into geometry neutral forms and the application of scaling functions are important contributions to the understanding of spatial parameters within GISc