On the Maximum Arc Length of Monotonic Functions

International audienceWe revisit a problem solved in 1963 by Zaanen & Luxemburg in this monthly: what is the largest possible length of the graph of a monotonic function on an interval? And is there such a function that attains this length? This is an interesting and intriguing problem with a somewhat surprising answer, that should be of interest to a broad spectrum of mathematicians starting with upper level undergraduates. The proof given by Zaanen & Luxemburg is very short and elegant but not accessible to an undergraduate. We give here a longer, but elementary, proof

Maximal product of primes whose sum is bounded

41 pagesInternational audienceIf n is a positive integer, let h(n) denote the maximal value of the product of distinct primes whose sum does not exceed n. We give some properties of this function h and describe an algorithm able to compute h(n) for large values of n

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

Une propriété arithmétique équivalente à l'hypothèse de Riemann

International audienceLet h(n) denote the largest product of distinct primes whose sum is n. The main result of this article is that the property " for all n > 0, we have log h(n) 0 log h(n) < li^-1(n) " où li^-1 est la fonction réciproque du logarithme intégral est équivalent à l' l' Hypothèse de Riemann

First tests of a 800 kJ HTS SMES

SMES using high critical temperature superconductors are interesting for high power pulsed sources. Operation at temperatures above 20 K makes cryogenics easier, enhances stability and improves operation as pulsed power source. In the context of a DGA (Delegation Generate pour l'Armement) project, we have designed and constructed a 800 kJ SMES. The coil is wound with Nexans conductors made of Bi-2212 PIT tapes soldered in parallel. The coil consists in 26 superposed simple pancakes wound and bonded on sliced copper plates coated with epoxy. The rated current is 315 A for an energy of 814 kJ. The external diameter of the coil is 814 mm and its height 222 mm. The cooling at 20 K is only performed by conduction from cryocoolers to make cryogenics very friendly and invisible for the SMES users. The cooling down has been successfully carried out and the thermal system works as designed. After a brief description of the SMES design and construction, some tests will be presented. From a current of 244 A, the SMES delivered 425 kJ to a resistance with a maximum power of 175 kW.Comment: 5 page

Short Polynomial Representations for Square Roots Modulo p

Abstract. Let p be an odd prime number and a a square modulo p. It is well known that the simple formula a p+1 4 mod p gives a square root of a when p ≡ 3 mod 4. Let us write p − 1 = 2 n s with s odd. A fast algorithm due to Shanks, with n steps, allows us to compute a square root of a modulo p. It will be shown that there exists a polynomial of at most 2 n−1 terms giving a square root of a. Moreover, if there exists a polynomial in a representing a square root of a modulo p, it will be proved that this polynomial would have at least 2 n−1 terms, except for a finite set P n of primes p depending on n. Résumé. Soit p un nombre premier impair et a un carré modulo p. La formule très simple a p+1 4 mod p fournit une valeur de la racine carrée de a lorsque p ≡ 3 mod 4. Plus généralement, si l&apos;onécrit p − 1 = 2 n s avec s impair, un algorithme dûà Shanks, comprenant nétapes, permet de calculer la racine carrée de a modulo p. Nous montrerons qu&apos;il existe un polynôme d&apos;au plus 2 n−1 termes et dont la valeur est une racine carrée de a pour tout carré a. De plus, pour n fixé, nous démontrons que tout polynôme en a représentant la racine carrée de a modulo p a au moins 2 n−1 termes, excepté pour un ensemble fini P n de nombres premiers p ≡ 1 (mod 2 n )

FAME : A new beamline for X-ray absorption investigations of very-diluted systems of environmental, material and biological interests

International audienceFAME is the French Absorption spectroscopy beamline in Material and Environmental sciences at the ESRF (France), operational since September 2002. Technically speaking, the source is a 0.85 T bending magnet and the main optical element is a two-crystals monochromator using either Si(111) or Si(220) monocrystals so that the available energy ranges from 4 to 40 keV. The first crystal is liquid nitrogen cooled in order to avoid a thermal bump and thus preserve the energy resolution. The second crystal is dynamically bent during the energy scan in order to focus the beam in the horizontal plane. Two bendable mirrors are located before and after the monochromator, for beam-collimation (to optimize the energy resolution) and vertical focalization, respectively. During scans, the beam on the sample is kept constant in position and size (around 150 × 200 μm2, V × H). The high flux on the sample combined with the sensitivity of our 30-elements fluorescence detector allow to decrease the detection limit down to 10 ppm or around less than a monolayer. Moreover, quick-EXAFS acquisition is operational: the acquisition time may be reduced down to 30s