Faster polynomial multiplication over finite fields

Let p be a prime, and let M_p(n) denote the bit complexity of multiplying two polynomials in F_p[X] of degree less than n. For n large compared to p, we establish the bound M_p(n) = O(n log n 8^(log^* n) log p), where log^* is the iterated logarithm. This is the first known F\"urer-type complexity bound for F_p[X], and improves on the previously best known bound M_p(n) = O(n log n log log n log p)

Les noms de My et Sy (province de Liège, arrondissement de Huy): découverte par l'onomastique d'une structure publique carolingienne dans l'Ourthe moyenne

Peer reviewe

Differential Equations for Algebraic Functions

It is classical that univariate algebraic functions satisfy linear differential equations with polynomial coefficients. Linear recurrences follow for the coefficients of their power series expansions. We show that the linear differential equation of minimal order has coefficients whose degree is cubic in the degree of the function. We also show that there exists a linear differential equation of order linear in the degree whose coefficients are only of quadratic degree. Furthermore, we prove the existence of recurrences of order and degree close to optimal. We study the complexity of computing these differential equations and recurrences. We deduce a fast algorithm for the expansion of algebraic series

Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer

A digital computer is generally believed to be an efficient universal computing device; that is, it is believed able to simulate any physical computing device with an increase in computation time of at most a polynomial factor. This may not be true when quantum mechanics is taken into consideration. This paper considers factoring integers and finding discrete logarithms, two problems which are generally thought to be hard on a classical computer and have been used as the basis of several proposed cryptosystems. Efficient randomized algorithms are given for these two problems on a hypothetical quantum computer. These algorithms take a number of steps polynomial in the input size, e.g., the number of digits of the integer to be factored.Comment: 28 pages, LaTeX. This is an expanded version of a paper that appeared in the Proceedings of the 35th Annual Symposium on Foundations of Computer Science, Santa Fe, NM, Nov. 20--22, 1994. Minor revisions made January, 199

Intraspecific variability in leaf traits strongly affects alder leaf decomposition in a stream

This study assessed the intraspecific variability of senescent leaves of alder (Alnus glutinosa Gaertn.) and the effects of this variability on leaf decomposition in streams. Leaves were collected at five geographically distant locations in Europe. We analyzed 10 batches of leaf samples for seven quantitative leaf traits as well as leaf decomposition rate in coarse and fine mesh bags exposed in a single stream. The geographic origin of leaf samples largely explained the observed variation in litter quality and decomposition rate. Phosphorus (0.034–0.187%) and lignin (3.9–18.7%) concentrations in leaves varied widely. Together, these two traits accurately predicted leaf decomposition rate (r2 = 84.1%). Intraspecific variation in leaf decomposition rate was within a range similar to that reported for interspecific variation among co- occurring riparian plant species in Europe. Our study demonstrates extensive intraspecific variability in leaf traits on a continental scale, which can have enormous effects on major ecosystem processes such as leaf decomposition

A short survey on Kantorovich-like theorems for Newton's method

We survey influential quantitative results on the convergence of the Newton iterator towards simple roots of continuously differentiable maps defined over Banach spaces. We present a general statement of Kantorovich's theorem, with a concise proof from scratch, dedicated to wide audience. From it, we quickly recover known results, and gather historical notes together with pointers to recent articles

Plougonvelin – Les sépultures en coffre du plateau de Bertheaume

Depuis de nombreuses années, la présence de sépultures en coffres est attestée sur le plateau de Bertheaume. Les 14 et 15 avril 1998, des sondages furent réalisés préalablement à des décapages extensifs liés à un projet de lotissement. L’intervention archéologique s’est étendue sur l’ensemble des parcelles concernées par le projet et les sondages ont permis de localiser la concentration de sept sépultures (parcelle 854, section D2). Aménagées à l’aide de plaques de micaschiste local, les tomb..

Leaf diversity influences in-stream litter decomposition through effects on shredders

1. The functioning of many aquatic ecosystems is controlled by surrounding terrestrial ecosystems. In a view of growing interest in linking biodiversity to ecosystem-level processes, we examined whether and how leaf diversity influences litter decomposition and consumers in streams. 2. We tested experimentally the hypothesis that the effects of leaf diversity on decomposition are determined by the responses of leaf consumers to resource–habitat heterogeneity. Leaves from three common riparian trees, beech (Fagus sylvatica), hazel (Corylus avellana) and ash (Fraxinus excelsior), were exposed alone and in all possible mixtures of two and three species in a stream. We analysed individual leaf species for decomposition rate, microbial respiration and mycelial biomass, and we determined the species composition, abundance and biomass of shredders in leaf bags. 3. We found that the decomposition of the fastest decomposing leaves (hazel and ash) was substantially stimulated (up to twofold higher than single species leaf packs) in mixtures containing beech leaves, which are refractory. In contrast, the decomposition of beech leaves was not affected by leaf mixing. Such species-specific behaviour of leaves in species mixtures has been overlooked in previous studies that examined the overall decompo- sition of litter mixtures. 4. The effects of leaf diversity on decomposition varied with the abundance and biomass of shredders but not with microbial parameters. Beech leaves alone were less attractive to shredders than leaf packs made of hazel, ash or any mixture of species. Moreover, the presence of beech leaves in mixtures led to higher shredder abundance and biomass than we had expected from data from single species exposed alone. Lastly, we found that early instars of the caddisfly Potamophylax (the dominant shredder in terms of biomass) almost exclusively used the toughest material (i.e. beech leaves) to construct their cases. 5. Leaf pack heterogeneity may have altered shredder-mediated decomposition. Shredders colonising diverse leaf packs benefited from the stable substratum provided by beech leaves, whereas ash and hazel leaves were primarily used as food. Thus, our findings provide strong evidence for an intimate linkage between the diversity of riparian vegetation and aquatic communities