Automatic lineament analysis techniques for remotely sensed imagery

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Eisenstein series and automorphic representations

We provide an introduction to the theory of Eisenstein series and automorphic forms on real simple Lie groups G, emphasising the role of representation theory. It is useful to take a slightly wider view and define all objects over the (rational) adeles A, thereby also paving the way for connections to number theory, representation theory and the Langlands program. Most of the results we present are already scattered throughout the mathematics literature but our exposition collects them together and is driven by examples. Many interesting aspects of these functions are hidden in their Fourier coefficients with respect to unipotent subgroups and a large part of our focus is to explain and derive general theorems on these Fourier expansions. Specifically, we give complete proofs of Langlands' constant term formula for Eisenstein series on adelic groups G(A) as well as the Casselman--Shalika formula for the p-adic spherical Whittaker vector associated to unramified automorphic representations of G(Q_p). Somewhat surprisingly, all these results have natural interpretations as encoding physical effects in string theory. We therefore introduce also some basic concepts of string theory, aimed toward mathematicians, emphasising the role of automorphic forms. In addition, we explain how the classical theory of Hecke operators fits into the modern theory of automorphic representations of adelic groups, thereby providing a connection with some key elements in the Langlands program, such as the Langlands dual group LG and automorphic L-functions. Our treatise concludes with a detailed list of interesting open questions and pointers to additional topics where automorphic forms occur in string theory

Q(sqrt(-3))-Integral Points on a Mordell Curve

We use an extension of quadratic Chabauty to number fields,recently developed by the author with Balakrishnan, Besser and M ̈uller,combined with a sieving technique, to determine the integral points overQ(√−3) on the Mordell curve y2 = x3 − 4

Tannakian Categories of Perverse Sheaves on Abelian Varieties

We study Tannakian categories attached to perverse sheaves on abelian varieties with respect to the convolution product. The construction of these categories is closely intertwined with a cohomological vanishing theorem which is an analog of Artin's affine vanishing theorem and contains the generic vanishing theorems of Green and Lazarsfeld as a special case. To illustrate the geometric relevance of the developed notions, we determine the Tannaka group of the theta divisor on a general principally polarized complex abelian variety of arbitrary dimension and explain its relationship with the Schottky problem in genus 4. Here the convolution square of the theta divisor describes a family of surfaces of general type, and a detailed study of this family leads to a variation of Hodge structures with monodromy group W(E6) which has a natural interpretation in terms of the Prym map. In the final chapter we take a closer look at convolutions of curves inside Jacobian varieties and provide a recursive formula for the generic rank of Brill-Noether sheaves

Hyfs: design and implementation of a reliable file system

Building reliable data storage systems is crucial to any commercial or scientific applications. Modern storage systems are complicated, and they are comprised of many components, from hardware to software. Problems may occur to any component of storage systems and cause data loss. When this kind of failures happens, storage systems cannot continue their data services, which may result in large revenue loss or even catastrophe to enterprises. Therefore, it is critically important to build reliable storage systems to ensure data reliability. In this dissertation, we propose to employ general erasure codes to build a reliable file system, called HyFS. HyFS is a cluster system, which can aggregate distributed storage servers to provide reliable data service. On client side, HyFS is implemented as a native file system so that applications can transparently run on top of HyFS. On server side, HyFS utilizes multiple distributed storage servers to provide highly reliable data service by employing erasure codes. HyFS is able to offer high throughput for either random or sequential file access, which makes HyFS an attractive choice for primary or backup storage systems. This dissertation studies five relevant topics of HyFS. Firstly, it presents several algorithms that can perform encoding operation efficiently for XOR-based erasure codes. Secondly, it discusses an efficient decoding algorithm for RAID-6 erasure codes. This algorithm can recover various types of disk failures. Thirdly, it describes an efficient algorithm to detect and correct errors for the STAR code, which further improves a storage system\u27s reliability. Fourthly, it describes efficient implementations for the arithmetic operations of large finite fields. This is to improve a storage system\u27s security. Lastly and most importantly, it presents the design and implementation of HyFS and evaluates the performance of HyFS

BASEFIELD TRANSFORMS DERIVED FROM CHARACTER TABLES

We show that it is possible to de ne Hartley-like transforms for (generalized) character tables of nite groups. This large class of transforms include Hartley transforms for discrete Fourier transforms over abelian groups and Hartley-like transforms for the discrete cosine transform of type I

Basefield Transforms Derived From Character Tables

We show that it is possible to define Hartley-like transforms for (generalized) character tables of finite groups. This large class of transforms include Hartley transforms for discrete Fourier transforms over abelian groups and Hartley-like transforms for the discrete cosine transform of type I. INTRODUCTION Calculating the Discrete Fourier Transform (DFT) of a real signal vector usually affords complex arithmetic. One can dispense with the complex arithmetic using the Discrete Hartley Transform (DHT) to compute the DFT. Similar ideas can be applied to DFTs over arbitrary fields. Typically, computing the DFT of signal vectors over a basefield F amounts to a field extension, since in general the basefield F does not contain the required roots of unity. However, it is possible to define Hartley-like transforms (sharing many properties with the associated DFTs) that do not require an extension of the basefield. Such transforms were first introduced in [4] under the name Algebraic Discr..