Regulators of rank one quadratic twists

We investigate the regulators of elliptic curves with rank 1 in some families of quadratic twists of a fixed elliptic curve. In particular, we formulate some conjectures on the average size of these regulators. We also describe an efficient algorithm to compute explicitly some of the invariants of an odd quadratic twist of an elliptic curve (regulator, order of the Tate-Shafarevich group, etc.) and we discuss the numerical data that we obtain and compare it with our predictions.Comment: 28 pages with 32 figure

Finite-State Complexity and the Size of Transducers

Finite-state complexity is a variant of algorithmic information theory obtained by replacing Turing machines with finite transducers. We consider the state-size of transducers needed for minimal descriptions of arbitrary strings and, as our main result, we show that the state-size hierarchy with respect to a standard encoding is infinite. We consider also hierarchies yielded by more general computable encodings.Comment: In Proceedings DCFS 2010, arXiv:1008.127

On the p-adic Beilinson conjecture for number fields

International audienceWe formulate a conjectural p-adic analogue of Borel's theorem relating regulators for higher K-groups of number fields to special values of the corresponding zeta-functions, using syntomic regulators and p-adic L-functions. We also formulate a corresponding conjecture for Artin motives, and state a conjecture about the precise relation between the p-adic and classical situations. Parts of the conjectures are proved when the number field (or Artin motive) is abelian over the rationals, and all conjectures are verified numerically in some other cases

Index of efficiency for strength-grading machines for solid wood

The settings of strength-grading machine for structural pieces of wood are checked according to the EN 14081 standard. However, different machines have different performances depending on the accuracy of the estimation of the board’s properties, and there is no easy way to compare the efficiency of these machines especially if the machine does not use the same sampling. In this paper, we introduce an index called index of efficiency for grading machines. This parameter is in the range of 0–100% and allows to compare performances of different machines for a given set of grades. The computation of this index is based on the cost matrix method of the EN 14081 and requires to have the size matrix of a setting to be computed

Counting Primes in Residue Classes

International audienceWe explain how the Meissel-Lehmer-Lagarias-Miller-Odlyzko method for computing π(x), the number of primes up to x, can be used for computing efficiently π(x,k,l), the number of primes congruent to l modulo k up to x. As an application, we computed the number of prime numbers of the form 4n±1 less than x for several values of x up to 10^20 and found a new region where π(x,4,3) is less than π(x,4,1) near x=10^18