Iterated compositions of linear operations on sets of positive upper density

Starting from a result of Stewart, Tijdeman and Ruzsa on iterated difference sequences, we introduce the notion of iterated compositions of linear operations. We prove a general result on the stability of such compositions (with bounded coefficients) on sets of integers having a positive upper density

Orbital measures on SU(2)/SO(2)

We let U=SU(2) and K=SO(2) and denote N_{U}(K) the normalizer of K in U. For a an element of U\ N_{U} (K), we let \mu_{a} be the normalized singular measure supported in KaK. For p a positive integer, it was proved that \mu_{a}^{( p)}, the convolution of p copies of \mu_{a}, is absolutely continuous with respect to the Haar measure of the group U as soon as p>=2. The aim of this paper is to go a step further by proving the following two results : (i) for every a in U\ N_{U} (K) and every integer p >=3, the Radon-Nikodym derivative of \mu_{a}^{(p)} with respect to the Haar measure m_{U} on U, namely d\mu_{a}^{(p)}/d m_{U}, is in L^{2}(U), and (ii) there exist a in U\ N_{U} (K) for which d\mu_{a}^{(2)}/ dm_{U} is not in L^{2}(U), hence a counter example to the dichotomy conjecture. Since L^{2} (G) \subseteq L^{1} (G), our result gives in particular a new proof of the result when p>2

K5(7,3)⩽100

AbstractOne of the main aims in the theory of covering codes is to obtain good estimates on Kq(n,R), the minimal cardinality of an R-covering code over the nth power of an alphabet with q elements. This paper reports on the new bound K5(7,3)⩽100, obtained by an improved computer search based on Östergård and Weakley's method. In particular, the code leading to this bound has a group of automorphisms quite different from the one Östergård and Weakley used. This new upper bound significantly improves the former record (which was 125)

Multi-Target Vectorization with MTPS C++ Generic Library

International audienceThis article introduces a C++ template library dedicated at vectorizing algorithms for different target architectures: Multi-Target Parallel Skeleton (MTPS). Skeletons describing the data structures and algorithms are provided and allow MTPS to generate a code with optimized memory access patterns for the choosen architecture. MTPS currently supports x86-64 multicore CPUs and CUDA enabled GPUs. On these architectures, performances close to hardware limits are observed

Optimally small sumsets in finite abelian groups

AbstractLet G be a finite abelian group of order g. We determine, for all 1⩽r,s⩽g, the minimal size μG(r,s)=min|A+B| of sumsets A+B, where A and B range over all subsets of G of cardinality r and s, respectively. We do so by explicit construction. Our formula for μG(r,s) shows that this function only depends on the cardinality of G, not on its specific group structure. Earlier results on μG are recalled in the Introduction

Marek E. Jasinski, Oleg V. Ovsjannikov, Vzgljad na Evropejskuju Arktiku

L’Institut d’histoire de la civilisation matérielle de l’Académie des sciences de Russie (Saint-Pétersbourg) et l’Institut d’archéologie de la faculté de sciences naturelles et de technologie de l’université de Norvège publient, avec le soutien du Conseil norvégien de la recherche, deux gros volumes de Marek Jasinski et Oleg Ovsjannikov sur « l’Europe arctique », c’est-à-dire le « nord arkhangelskien ». La zone étudiée va d’est en ouest, du golfe de la Petchora et de l’Oural à la mer Blanche ..