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Computations of irregular primes and associated cyclotomic invariants were extended to all primes up to 12 million using multisectioning/convolution methods and a novel approach which originated in the study of Stickelberger codes (Shokrollahi, 1996). The latter idea reduces the problem to that of finding zeros of a polynomial over Fp of degree < (p− 1)/2(More)
The Madelung constant-essentially the Coulomb energy density of a crystal-is usually calculated via Ewald error function expansions or, for the simpler cubic structures, by the 'cosech' series of modern vintage. By considering generalised functional equations for multidimensional zeta functions, we provide explicit expansions for the spatial potential and(More)
The calculation of crystal structure from X-ray diffraction data requires that the phases of the “structure factors” (Fourier coefficients) determined by scattering be deduced from the absolute values of those structure factors. Motivated by a question of Herbert Hauptman, we consider the problem of determining phases by direct algebraic means in the case(More)
We consider the problem of orienting the edges of the n-dimensional hypercube so only two different in-degrees a and b occur. We show that this can be done, for two specified in-degrees, if and only if an obvious necessary condition holds. Namely, there exist non-negative integers s and t so that s + t = 2n and as + bt = n2. This is connected to a question(More)
Abstract. Let h(`) denote the class number of the maximal totally real subfield Q(cos(2π/`n)) of the field of `n-th roots of unity. The goal of this paper is to show that (speculative extensions of) the Cohen-Lenstra heuristics on class groups provide support for the following conjecture: for all but finitely many pairs (`, n), where ` is a prime and n is a(More)
Many applications of fast Fourier transforms (FFT’s), such as computer tomography, geophysical signal processing, high-resolution imaging radars, and prediction filters, require high-precision output. An error analysis reveals that the usual method of fixed-point computation of FFT’s of vectors of length 2 leads to an average loss of /2 bits of precision.(More)