40 research outputs found
Cocyclic Hadamard Matrices: An Efficient Search Based Algorithm
This dissertation serves as the culmination of three papers. “Counting the decimation classes of binary vectors with relatively prime fixed-density presents the first non-exhaustive decimation class counting algorithm. “A Novel Approach to Relatively Prime Fixed Density Bracelet Generation in Constant Amortized Time presents a novel lexicon for binary vectors based upon the Discrete Fourier Transform, and develops a bracelet generation method based upon the same. “A Novel Legendre Pair Generation Algorithm expands upon the bracelet generation algorithm and includes additional constraints imposed by Legendre Pairs. It further presents an efficient sorting and comparison algorithm based upon symmetric functions, as well as multiple unique Legendre Pairs
Computationally efficient recursions for top-order invariant polynomials with applications
The top-order zonal polynomials Ck(A),and top-order invariant polynomials Ck1,...,kr(A1,...,Ar)in which each of the partitions of ki,i = 1,..., r,has only one part, occur frequently in multivariate distribution theory, and econometrics - see, for example Phillips (1980, 1984, 1985, 1986), Hillier (1985, 2001), Hillier and Satchell (1986), and Smith (1989, 1993). However, even with the recursive algorithms of Ruben (1962) and Chikuse (1987), numerical evaluation of these invariant polynomials is extremely time consuming. As a result, the value of invariant polynomials has been largely confined to analytic work on distribution theory. In this paper we present new, very much more efficient, algorithms for computing both the top-order zonal and invariant polynomials. These results should make the theoretical results involving these functions much more valuable for direct practical study. We demonstrate the value of our results by providing fast and accurate algorithms for computing the moments of a ratio of quadratic forms in normal random variables.
Discrete Sampling and Interpolation: Universal Sampling Sets for Discrete Bandlimited Spaces
We study the problem of interpolating all values of a discrete signal f of
length N when d<N values are known, especially in the case when the Fourier
transform of the signal is zero outside some prescribed index set J; these
comprise the (generalized) bandlimited spaces B^J. The sampling pattern for f
is specified by an index set I, and is said to be a universal sampling set if
samples in the locations I can be used to interpolate signals from B^J for any
J. When N is a prime power we give several characterizations of universal
sampling sets, some structure theorems for such sets, an algorithm for their
construction, and a formula that counts them. There are also natural
applications to additive uncertainty principles.Comment: 24 pages, 5 figures, Accepted for publication in IEEE Transactions on
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