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David Harvey

Publications and source records attributed to David Harvey.

85 records · Page 5Linked to original sources
arXiv2011

Fast computation of Bernoulli, Tangent and Secant numbers

We consider the computation of Bernoulli, Tangent (zag), and Secant (zig or Euler) numbers. In particular, we give asymptotically fast algorithms for computing the first n such numbers in O(n^2.(log n)^(2+o(1))) bit-operations. We also give very short in-place algorithms for computing the first n Tangent or Secant numbers in O(n^2) integer operations. These algorithms are extremely simple, and fast for moderate values of n. They are faster and use less space than the algorithms of Atkinson (for Tangent and Secant numbers) and Akiyama and Tanigawa (for Bernoulli numbers).

math.CO↗
arXiv2010

Characterizing projective spaces on deformations of Hilbert schemes of K3 surfaces

We seek to characterize homology classes of Lagrangian projective spaces embedded in irreducible holomorphic-symplectic manifolds, up to the action of the monodromy group. This paper addresses the case of manifolds deformation-equivalent to the Hilbert scheme of length-three subschemes of a K3 surface. The class of the projective space in the cohomology ring has prescribed intersection properties, which translate into Diophantine equations. Possible homology classes correspond to integral points on an explicit elliptic curve; our proof entails showing that the only such point is two-torsion.

math.AG↗
arXiv2010

An in-place truncated Fourier transform and applications to polynomial multiplication

The truncated Fourier transform (TFT) was introduced by van der Hoeven in 2004 as a means of smoothing the "jumps" in running time of the ordinary FFT algorithm that occur at power-of-two input sizes. However, the TFT still introduces these jumps in memory usage. We describe in-place variants of the forward and inverse TFT algorithms, achieving time complexity O(n log n) with only O(1) auxiliary space. As an application, we extend the second author's results on space-restricted FFT-based polynomial multiplication to polynomials of arbitrary degree.

cs.DS↗
arXiv2009

Irregular primes to 163 million

We compute all irregular primes less than 163,577,356. For all of these primes we verify that the Kummer-Vandiver conjecture holds and that the lambda-invariant is equal to the index of irregularity.

math.NT↗
arXiv2009

Faster exponentials of power series

We describe a new algorithm for computing exp(f) where f is a power series in C[[x]]. If M(n) denotes the cost of multiplying polynomials of degree n, the new algorithm costs (2.1666... + o(1)) M(n) to compute exp(f) to order n. This improves on the previous best result, namely (2.333... + o(1)) M(n).

cs.SC↗
arXiv2009

Faster algorithms for the square root and reciprocal of power series

We give new algorithms for the computation of square roots and reciprocals of power series in C[[x]]. If M(n) denotes the cost of multiplying polynomials of degree n, the square root to order n costs (1.333... + o(1)) M(n) and the reciprocal costs (1.444... + o(1)) M(n). These improve on the previous best results, respectively (1.8333... + o(1)) M(n) and (1.5 + o(1)) M(n).

cs.SC↗
arXiv2008

A cache-friendly truncated FFT

We describe a cache-friendly version of van der Hoeven's truncated FFT and inverse truncated FFT, focusing on the case of `large' coefficients, such as those arising in the Schonhage--Strassen algorithm for multiplication in Z[x]. We describe two implementations and examine their performance.

cs.SC↗
arXiv2008

A multimodular algorithm for computing Bernoulli numbers

We describe an algorithm for computing Bernoulli numbers. Using a parallel implementation, we have computed B(k) for k = 10^8, a new record. Our method is to compute B(k) modulo p for many small primes p, and then reconstruct B(k) via the Chinese Remainder Theorem. The asymptotic time complexity is O(k^2 log(k)^(2+epsilon)), matching that of existing algorithms that exploit the relationship between B(k) and the Riemann zeta function. Our implementation is significantly faster than several existing implementations of the zeta-function method.

math.NT↗
arXiv2007

Faster polynomial multiplication via multipoint Kronecker substitution

We give several new algorithms for dense polynomial multiplication based on the Kronecker substitution method. For moderately sized input polynomials, the new algorithms improve on the performance of the standard Kronecker substitution by a sizeable constant, both in theory and in empirical tests.

cs.SC↗
arXiv2007

Efficient computation of p-adic heights

We analyse and drastically improve the running time of the algorithm of Mazur, Stein and Tate for computing the canonical cyclotomic p-adic height of a point on an elliptic curve E/Q, where E has good ordinary reduction at p >= 5.

math.NT↗
arXiv2005

Superhumps in Cataclysmic Binaries. XXV. q_crit, epsilon(q), and Mass-Radius

We report on successes and failures in searching for positive superhumps in cataclysmic variables, and show the superhumping fraction as a function of orbital period. Basically, all short-period systems do, all long-period systems don't, and a 50% success rate is found at P_orb=3.1+-0.2 hr. We can use this to measure the critical mass ratio for the creation of superhumps. With a mass-radius relation appropriate for cataclysmic variables, and an assumed mean white-dwarf mass of 0.75 M_sol, we find a mass ratio q_crit=0.35+-0.02. We also report superhump studies of several stars of independently known mass ratio: OU Virginis, XZ Eridani, UU Aquarii, and KV UMa (= XTE J1118+480). The latter two are of special interest, because they represent the most extreme mass ratios for which accurate superhump measurements have been made. We use these to improve the epsilon(q) calibration, by which we can infer the elusive q from the easy-to-measure epsilon (the fractional period excess of P_superhump over P_orb). This relation allows mass and radius estimates for the secondary star in any CV showing superhumps. The consequent mass-radius law shows an apparent discontinuity in radius near 0.2 M_sol, as predicted by the disrupted magnetic braking model for the 2.1-2.7 hour period gap. This is effectively the "empirical main sequence" for CV secondaries.

astro-ph↗
arXiv2002

Superhumps in Cataclysmic Binaries. XXIII. V442 Ophiuchi and RX J1643.7+3402

We report the results of long observing campaigns on two novalike variables: V442 Ophiuchi and RX J1643.7+3402. These stars have high-excitation spectra, complex line profiles signifying mass loss at particular orbital phases, and similar orbital periods (respectively 0.12433 and 0.12056 d). They are well-credentialed members of the SW Sex class of cataclysmic variables. Their light curves are also quite complex. V442 Oph shows periodic signals with periods of 0.12090(8) and 4.37(15) days, and RX J1643.7+3402 shows similar signals at 0.11696(8) d and 4.05(12) d. We interpret these short and long periods respectively as a "negative superhump" and the wobble period of the accretion disk. The superhump could then possibly arise from the heating of the secondary (and structures fixed in the orbital frame) by inner-disk radiation, which reaches the secondary relatively unimpeded since the disk is not coplanar. At higher frequencies, both stars show another type of variability: quasi-periodic oscillations (QPOs) with a period near 1000 seconds. Underlying these strong signals of low stability may be weak signals of higher stability. Similar QPOs, and negative superhumps, are quite common features in SW Sex stars. Both can in principle be explained by ascribing strong magnetism to the white dwarf member of the binary; and we suggest that SW Sex stars are borderline AM Herculis binaries, usually drowned by a high accretion rate. This would provide an ancestor channel for AM Hers, whose origin is still mysterious.

astro-ph↗