Nucléosides de synthése XXII: Obtention,comme synthons oligoxylonucléotidiques,de d-xylofurannosyl adenines spécifiquement substitués

This work describes the synthesis of certain nucleoside synthons which are necessary in order to obtain oligoxylonucleotides. The utilization of a non-symmetrically substituted xyloside substituted at the 2-position permits one to obtain, in satisfactory yields, the described nucleoside which is conveniently protected. All of the compounds described have been fully characterized using the usual analytical methods.     

On SA, CA, and GA numbers

Gronwall’s function G is defined for n>1 by $G(n)=\frac{\sigma(n)}{n \log\log n}$ where σ(n) is the sum of the divisors of n. We call an integer N>1 a GA1 number if N is composite and G(N)≥G(N/p) for all prime factors p of N. We say that N is a GA2 number if G(N)≥G(aN) for all multiples aN of N. In (Caveney et al. Integers 11:A33, 2011), we used Robin’s and Gronwall’s theorems on G to prove that the Riemann Hypothesis (RH) is true if and only if 4 is the only number that is both GA1 and GA2. In the present paper, we study GA1 numbers and GA2 numbers separately. We compare them with superabundant (SA) and colossally abundant (CA) numbers (first studied by Ramanujan). We give algorithms for computing GA1 numbers; the smallest one with more than two prime factors is 183783600, while the smallest odd one is 1058462574572984015114271643676625. We find nineteen GA2 numbers ≤5040, and prove that a GA2 number N>5040 exists if and only if RH is false, in which case N is even and >10⁸⁵⁷⁶.     

The sum of divisors function and the Riemann hypothesis

The Ramanujan Journal - Let $$\sigma (n)=\sum _{d\mid n}d$$ be the sum of divisors function and $$\gamma =0.577\ldots $$ the Euler constant. In 1984, Robin proved that, under the Riemann...     

Hermite Subdivision Schemes and Taylor Polynomials

We propose a general study of the convergence of a Hermite subdivision scheme ℋ of degree d>0 in dimension 1. This is done by linking Hermite subdivision schemes and Taylor polynomials and by associating a so-called Taylor subdivision (vector) scheme . The main point of investigation is a spectral condition. If the subdivision scheme of the finite differences of is contractive, then is C ⁰ and ℋ is C ^d . We apply this result to two families of Hermite subdivision schemes. The first one is interpolatory; the second one is a kind of corner cutting. Both of them use the Tchakalov-Obreshkov interpolation polynomial.      

Optimal Multisections in Interval Branch-and-Bound Methods of Global Optimization

In this paper we define multisections of intervals that yield sharp lower bounds in branch-and-bound type methods for interval global optimization. A so called 'generalized kite', defined for differentiable univariate functions, is built simultaneously with linear boundary forms and suitably chosen centered forms. Proofs for existence and uniqueness of optimal cuts are given. The method described may be used either as an accelerating device or in a global optimization algorithm with an efficient pruning effect. A more general principle for decomposition of boxes is suggested.     

Subpicosecond pulse generation from a 1.56 μm mode-locked VECSEL

Near-transform-limited subpicosecond pulses at 1.56 μm were generated from an optically pumped InP-based vertical-external-cavity surface-emitting laser (VECSEL) passively mode-locked at 2 GHz repetition rate with a fast InGaAsNSb/GaAs semiconductor saturable absorber mirror (SESAM). The SESAM microcavity resonance was adjusted via a selective etching of phase layers specifically designed to control the magnitude of both the modulation depth and the intracavity group delay dispersion of the SESAM. Using the same VECSEL chip, we observed that the mode-locked pulse duration could be reduced from several picoseconds to less than 1 ps with a detuned resonant SESAM.