Jean-Marc Deshouillers

On Bounds for the Concentration Function II

AbstractWe derive an upper bound for the concentration of the sum of i.i.d. random variables with values in

New Experimental Results Concerning the Goldbach Conjecture

\mathbb{Z}^d

Sums of the digits in bases 2 and 3

by appealing to functions of positive type and the structure theory of set addition.

Integral Points on a Very Flat Convex Curve

Mitochondrial pathologies are a heterogeneous group of metabolic disorders that are frequently characterized by anomalies of oxidative phosphorylation, especially in the respiratory chain. The identification of these anomalies may involve many investigations, and biochemistry is a main tool. However, considering the whole set of biochemical data, the interpretation of the results by the traditionally used statistical methods remains complex and does not always lead to an unequivocal conclusion about the presence or absence of a respiratory chain defect. This arises from three main problems: (a) the absence of an a priori-defined control population, because the determination of the control values are derived from the whole set of investigated patients, (b) the small size of the population studied, (c) the large number of variables collected, each of which creates a wide variability. To cope with these problems, the principal component analysis (PCA) has been applied to the biochemical data obtained from 35 muscle biopsies of children suspected of having a mitochondrial disease. This analysis makes it possible for each respiratory chain complex to distinguish between different subsets within the whole population (normal, deficient, and, in between, borderline subgroups of patients) and to detect the most discriminating variables. PCA of the data of all complexes together showed that mitochondrial diseases in this population were mainly caused by multiple deficits in respiratory chain complexes. This analysis allows the definition of a new subgroup of newborns, which have high respiratory chain complex activity values. Our results show that the PCA method, which simultaneously takes into account all of the concerned variables, allows the separation of patients into subgroups, which may help clinicians make their diagnoses.

Density Modulo 1 of a Sequence Associated with a Multiplicative Function Evaluated at Polynomial Arguments

The Goldbach conjecture states that every even integer ≥ 4 can be written as a sum of two prime numbers. It is known to be true up to 4 × 1011. In this paper, new experiments on a Cray C916 supercomputer and on an SGI compute server with 18 R10000 CPUs are described, which extend this bound to 1014. Two consequences are that (1) under the assumption of the Generalized Riemann hypothesis, every odd number ≥7 can be written as a sum of three prime numbers, and (2) under the assumption of the Riemann hypothesis, every even positive integer can be written as a sum of at most four prime numbers. In addition, we have verified the Goldbach conjecture for all the even numbers in the intervals [105i , 105i +108], for i=3, 4,..., 20 and [1010i , 1010i + 109], for i=20,21,..., 30.

How Often is n ! a Sum of Three Squares?

The mitochondrial pathologies are a heterogeneous group of metabolic disorders that are characterized by anomalies of oxidative phosphorylation, especially in the respiratory chain. The diagnosis of these pathologies involves many investigations among which biochemical study is at present the main tool. However, the analysis of the results obtained during such study remains complex and often does not make it possible to conclude clearly if a patient is affected or not by a biochemical and/or bioenergetic deficiency. This arises from two main problems: 1) The determination of control values from the whole set of variable values (affected and unaffected people). 2) The small size of the population studied and the large number of variables collected which present a rather large variability. To cope with these problems, the principal component analysis method is applied to the results obtained during our biochemical studies. This analysis makes it possible for each respiratory chain complex, to distinguish clearly two subsets of the whole population (affected and unaffected people) as well as to detect the variables which are the most discriminative.

On the density of sums of three cubes

Let b ≥ 2 be an integer and let s b (n) denote the sum of the digits of the representation of an integer n in base b. For sufficiently large N , one has Card{n ≤ N : |s 3 (n) − s 2 (n)| ≤ 0.1457205 log n} > N 0.970359. The proof only uses the separate (or marginal) distributions of the values of s 2 (n) and s 3 (n).

Revisiting the pseudorandom number generator ran 1 from the NUMERICAL RECIPES

The second named author studied in 1988 the possible relations between the length \(\ell \), the minimal radius of curvature r and the number of integral points N of a strictly convex flat curve in \(\mathbb {R}^2\), stating that \(N = O(\ell /r^{1/3})\) (*), a best possible bound even when imposing the tangent at one extremity of the curve; here flat means that one has \(\ell = r^{\alpha } \) for some \(\alpha \in [2/3, 1)\). He also proved that when \(\alpha \le 1/3\), the quantity N is bounded. In this paper, the authors prove that in general the bound (*) cannot be improved for very flat curves, i.e. those for which \(\alpha \in (1/3, 2/3)\); however, if one imposes a 0 tangent at one extremity of the curve, then (*) is replaced by the sharper inequality \(N \le \ell ^2/r +1\).