Corrigendum to “(p,1)-total labelling of graphs” [Discrete Math. 308 (2008) 496–513]

We give a corrected proof of Proposition 51 of the paper cited in the title.     

Partitions in squares

In this paper we study primarily partitions in different squares. A complete characterization of the least number of terms needed in different cases is given. The asymptotic number of partitions in squares and in different squares is deduced by use of numerical results obtained from extensive computer runs. Some other related problems are also discussed.     

Landau-Level Quantization of the Yellow Excitons in Cuprous Oxide

Lately, the yellow series of P-excitons in cuprous oxide could be resolved up to the principal quantum number n = 25. Adding a magnetic field, leads to additional confinement normal to the field. Thereby, the transition associated with the exciton n is transformed into the transition between the electron and hole Landau levels with quantum number n, once the associated magnetic length becomes smaller than the related exciton Bohr radius. The magnetic field of this transition scales roughly as n^–3. As a consequence of the extended exciton series, we are able to observe Landau level transitions with unprecedented high quantum numbers of more than 75.     

Colliding Solitons for the Nonlinear Schrödinger Equation

We study the collision of two fast solitons for the nonlinear Schrödinger equation in the presence of a slowly varying external potential. For a high initial relative speed ||v|| of the solitons, we show that, up to times of order ||v|| after the collision, the solitons preserve their shape (in L ²-norm), and the dynamics of the centers of mass of the solitons is approximately determined by the external potential, plus error terms due to radiation damping and the extended nature of the solitons. We remark on how to obtain longer time scales under stronger assumptions on the initial condition and the external potential.     

Comparison of alignment free string distances for complete genome phylogeny

In this paper, we compare the accuracy of four string distances on complete genomes to reconstruct phylogenies using simulated and real biological data. These distances are based on common words shared by raw genomic sequences and do not require preliminary processing steps such as gene identification or sequence alignment. Moreover, they are computable in linear time. The first distance is based on Maximum Significant Matches (MSM). The second is computed from the frequencies of all the words of length k (KW). The third distance is based on the Average length of maximum Common Substrings at any position (ACS). The last one is based on the Ziv–Lempel compression algorithm (ZL). We describe a simulation process of evolution to generate a set of sequences having evolved according to a random tree topology T. This process allows both base substitution and fragment insertion/deletion, including horizontal transfers. The distances between the generated sequences are computed using the four formulas and the corresponding trees T′ are reconstructed using Neighbor-Joining. T and T′ are compared according to topological criteria. These comparisons show that the MSM distance outperforms the others whatever the parameters used to generate sequences. Finally, we test the MSM and KW distances on real biological data (i.e. prokaryotic complete genomes) and we compare the NJ trees to a Maximum Likelihood 16S + 23S RNA tree. We show that the MSM distance provides accurate results to study intra-phylum relationships, much better than those given by KW.     

Persistence stability for geometric complexes

In this paper we study the properties of the homology of different geometric filtered complexes (such as Vietoris–Rips, ?ech and witness complexes) built on top of totally bounded metric spaces. Using recent developments in the theory of topological persistence, we provide simple and natural proofs of the stability of the persistent homology of such complexes with respect to the Gromov–Hausdorff distance. We also exhibit a few noteworthy properties of the homology of the Rips and ?ech complexes built on top of compact spaces.     

Second mode transition in microstructured optical fibers: determination of the critical geometrical parameter and study of the matrix refractive index and effects of cladding size

We carried out a numerical study of the second mode transition in finite-sized, microstructured optical fibers (MOFs) for several values of the matrix refractive index. We determined a unique critical geometrical parameter for the second mode cutoff that is valid for all the matrix refractive indices studied. Finite size effects and extrapolated results for infinite structures are described. Using scaling laws, we provide a generalized phase diagram for solid-core MOFs that is valid for all refractive indices, including those of the promising chalcogenide MOFs.