On the Conjecture of Lehmer, limit Mahler measure of trinomials and asymptotic expansions

International audienceLet $n ≥ 2$ be an integer and denote by $\theta_n$ the real root in $(0, 1)$ of the trinomial$G_{n}(X) = −1 + X + X^n$ . The sequence of Perron numbers $(\theta_{n}^{−1} )_{n≥2}$ tends to 1. We prove thatthe Conjecture of Lehmer is true for $\{\theta_{n}^{−1} | n ≥ 2\}$ by the direct method of Poincar\'e asymptoticexpansions (divergent formal series of functions) of the roots $\theta_n , z_{j,n}$, of $G_{n}(X)$ lying in $|z| <1$, as a function of $n, j$ only. This method, not yet applied to Lehmer’s problem up to theknowledge of the author, is successfully introduced here. It first gives the asymptotic expansionof the Mahler measures ${\rm M}(G_n) = {\rm M}(\theta_{n}) = {\rm M}(\theta_{n}^{-1})$ of the trinomials $G_n$ as a function of $n$only, without invoking Smyth’s Theorem, and their unique limit point above the smallest Pisotnumber. Comparison is made with Smyth’s, Boyd’s and Flammang’s previous results. Bythis method we obtain a direct proof that the conjecture of Schinzel-Zassenhaus is true for$\{\theta_{n}^{−1} | n ≥ 2\}$, with a minoration of the house \house\{\theta_{n}^{−1}\}= \theta_{n}^{−1} , and a minoration of the Mahler measure${\rm M}(G_n)$ better than Dobrowolski’s one for $\{\theta_{n}^{−1} | n ≥ 2\}$ . The angular regularity of the roots of $G_n$ , near the unitcircle, and limit equidistribution of the conjugates, for n tending to infinity (in the sense of Bilu,Petsche, Pritsker), towards the Haar measure on the unit circle, are described in the context ofthe Erd\H{o}s-Tur\'an-Amoroso-Mignotte theory, with uniformly bounded discrepancy functions

Pattern Recognition

Pattern recognition is a very wide research field. It involves factors as diverse as sensors, feature extraction, pattern classification, decision fusion, applications and others. The signals processed are commonly one, two or three dimensional, the processing is done in real- time or takes hours and days, some systems look for one narrow object class, others search huge databases for entries with at least a small amount of similarity. No single person can claim expertise across the whole field, which develops rapidly, updates its paradigms and comprehends several philosophical approaches. This book reflects this diversity by presenting a selection of recent developments within the area of pattern recognition and related fields. It covers theoretical advances in classification and feature extraction as well as application-oriented works. Authors of these 25 works present and advocate recent achievements of their research related to the field of pattern recognition

Annual report 2015

Accessions considered in the study. Overview of the material considered in this study. For all materials, the GenBank identifier, the accession and species name as used in this study (Species) as well as their species synonyms used in the donor seed banks or in the NCBI GenBank (Material source/Reference) are provided. The genome symbol, and the country of origin, where the material was originally collected are given. The ploidy level measured in the scope of this study and the information if a herbarium voucher could be deposited in the herbarium of IPK Gatersleben (GAT) is given. Genomic formulas of tetraploids and hexploids are given as “female x male parent”. The genomes of Aegilops taxa follow Kilian et al. [74] and Li et al. [84]. Genome denominations for Hordeum follow Blattner [107] and Bernhardt [12] for the remaining taxa. (XLS 84 kb

Reports of accomplishments of planetology programs, 1975 - 1976

Abstracts of reports which summarize work conducted by Planetology Program Principal Investigators are presented. Full reports of selected abstracts were presented to the annual meeting of Planetology Program Principal Investigators at the Center for Astrogeology, U.S. Geological Survey, Flagstaff, Arizona, March 8, 9, 19, 1976