This thesis centers around a generalization of the classical discrete Fourier transform. We first present a general diagrammatic approach to the construction of efficient algorithms for computing the Fourier transform of a function on a finite...
'Orthogonal modular forms are algebraic modular forms arising from lattices in quadratic spaces. In this thesis, we define orthogonal modular forms, establish their basic properties, and then apply them to a case of ternary quadratic spaces to...
For a local field K, we study the affine buildings Ξ n and Δ n naturally associated to SL n ( K ) and Sp n ( K ), respectively. Since Sp n ( K ) is a subgroup of SL 2 n ( K ), we investigate properties of a natural embedding of Δ n in Ξ 2 n ....
In this thesis, we consider several problems relating to cyclic subgroups of the group [mathematical equation]. Each element of [mathematical equation] has a unique representative in one of the two intervals [mathematical equation] and...
In this thesis, we characterize and enumerate the permutations which are realized by the orbits of signed shifts, a family of discrete dynamical systems on words. The permutations, which are called patterns of the signed shifts, are given by the...
Visual searches for a conjunction of features (e.g., a particular combination of color and shape) are ordinarily slow and difficult. However, search efficiency for a particular conjunction can improve dramatically within a few hundred practice...
Siegel domains. Modular groups. Hecke algebras. Forms
In the 1960s Satake proved the existence of an isomorphism between the local Hecke algebra and the ring of polynomials invariant under the action of the signed permutation group W n (the Weyl group associated to Sp n over a local field)....
Let K be the function field over a finite field of odd order, and let H be a definite quaternion algebra over K. If Α is an order of level M in H , we define theta series for each ideal I of Α using the reduced norm on H. Using harmonic analysis...
In the late 1960s, Ihara began work that led to the Ihara zeta function, a zeta function which is defined on a finite graph. This function is an interesting graph invariant which gives information on expansion properties of the graph. It also...
This thesis investigates the embedding theory of orders in central simple algebras, placing a particular emphasis on the role that the phenomenon known as selectivity plays in the theory. Although the notion of selectivity is completely algebraic,...