Fundamental solutions to Pell equation with prescribed size

We prove that the number of parameters D up to a fixed x ≥ 2 such that the fundamental solution ? _D to the Pell equation T ² ? DU ² = 1 lies between \(D^{\tfrac{1}{2} + \alpha _1 }\) and \(D^{\tfrac{1}{2} + \alpha _2 }\) is greater than \(\sqrt x \log ^2 x\) up to a constant as long as α ₁ < α ₂ and α ₁ < 3/2. The starting point of the proof is a reduction step already used by the authors in earlier works. This approach is amenable to analytic methods. Along the same lines, and inspired by the work of Dirichlet, we show that the set of parameters D ≤ x for which log ? _D is larger than D ^¼ has a cardinality essentially larger than x ^¼ log² x.     

Photo‐Cross‐Linking Probes for Trapping G‐Quadruplex DNA

We have developed a straightforward synthetic pathway to a set of six photoactivatable G‐quadruplex ligands with a validated G4‐binding motif (the bisquinolinium pyridodicarboxamide PDC‐360A) tethered through various spacers to two different photo‐cross‐linking groups: benzophenone and an aryl azide. The high quadruplex‐versus‐duplex selectivity of the PDC core was retained in the new derivatives and resulted in selective alkylation of two well‐known G‐quadruplexes (human telomeric G4 and oncogene promoter c‐myc G4) under conditions of harsh competition. The presence of two structurally different photoactivatable functions allowed the selective alkylation of G‐quadruplex structures at specific nucleobases and irreversible G4 binding. The topology and sequence of the quadruplex matrix appear to influence strongly the alkylation profile, which differs for the telomeric and c‐myc quadruplexes. The new compounds are photoactive in cells and thus provide new tools for studying G4 biology.     

Conservation of ultrafast photoprotective mechanisms with increasing molecular complexity in sinapoyl malate derivatives

Sinapoyl malate is a natural plant sunscreen molecule which protects leaves from harmful ultraviolet radiation. Here, the ultrafast dynamics of three sinapoyl malate derivatives, sinapoyl L-dimethyl malate, sinapoyl L-diethyl malate and sinapoyl L-di-t-butyl malate, have been studied using transient electronic absorption spectroscopy, in a dioxane and methanol solvent environment to investigate how well preserved these dynamics remain with increasing molecular complexity. In all cases it was found that, upon photoexcitation, deactivation occurs via a trans-cis isomerisation pathway within ∼20–30 ps. This cis-photoproduct, formed during photodeactivation, is stable and longed-lived for all molecules in both solvents. The incredible levels of conservation of the isomerisation pathway with increased molecular complexity demonstrate the efficacy of these molecules as ultraviolet photoprotectors, even in strongly perturbing solvents. As such, we suggest these molecules might be well-suited for augmentations to further improve their photoprotective efficacy or chemical compatibility with other components of sunscreen mixtures, whilst conserving their underlying photodynamic properties.     

On the use of Kronecker's algorithm in the generalized rational interpolation problem

Kronecker's algorithm can be used to solve the generalized rational interpolation problem. In order to present the algorithm, rational forms are used here instead of too restrictive rational fractions. The proposed algorithm is reliable as soon as the functionals that characterize the problem satisfy two precise conditions. These conditions are fulfilled in the modified Hermite rational interpolation problem and, as a consequence, in the special case of the Cauchy problem and of the Padé approximation problem. This reliability covers two properties: on one hand, every rational form resulting from the algorithm is a solution of the problem whereas, on the other hand, every solution of the problem is found by the algorithm (with the exception of a possible reduction of the rational form). However, if the algorithm yields a non-reduced rational form, then the corresponding rational fraction is not a solution of the problem.     

Volume entropy,weighted girths and stable balls on graphs

We prove new isoperimetric inequalities on graphs involving quantities linked with concepts from differential geometry. First, we bound from above the product of the volume entropy (defined as the log of the exponential growth rate of the universal cover) and the girth of weighted graphs in terms of their cyclomatic number. In a second part, we study a natural polyhedron associated to a weighted graph: the stable ball. In particular, we relate the volume of this polyhedron, the weight of the graph and its cyclomatic number. © 2007 Wiley Periodicals, Inc. J Graph Theory 55: 291–305, 2007     

Cleavable Binary Dyads: Simplifying Data Extraction and Increasing Storage Density in Digital Polymers

Digital polymers are uniform macromolecules that store monomer‐based binary sequences. Molecularly stored information is usually extracted from the polymer by a tandem mass spectrometry (MS/MS) measurement, in which the coded chains are fragmented to reveal each bit (i.e. basic coded monomer unit) of the sequence. Here, we show that data‐extraction can be greatly simplified by favoring the formation of MS/MS fragments containing two bits instead of one. In order to do so, digital poly(alkoxyamine phosphodiester)s, containing binary dyads in each repeat unit, were prepared by an orthogonal solid‐phase approach involving successive phosphoramidite and radical‐radical coupling steps. Three different sets of monomers were considered to build these polymers. In all cases, four coded building blocks—two hydroxy‐nitroxides and two phosphoramidite monomers—were required to build the dyads. Among the three studied monomer sets, one combination allowed synthesis of uniform sequence‐coded polymers. The resulting polymers led to clear dyad‐containing fragments in MS/MS and could therefore be efficiently decoded. Additionally, an algorithm was created to detect specific dyad fragments, thus enabling automated sequencing.     

Accurate Ab Initio Calculations for LiH and its Ions, LiH+ and LiH−

Ab initio calculations were performed for LiH using a pseudopotential approach with CPP corrections and huge basis sets on both atoms. A wide range of ^1,3Σ⁺ electronic adiabatic states have been investigated, from the ground state up to those dissociating into Li(5p)+H. Permanent and transition electric dipole moments are also considered for the first few excited states. Comparison with experiments and recent all-electron calculations, reveals an excellent global accuracy, only the bottom of the ground state being better described by all-electron approaches. Using almost identical basis sets, coupled cluster all-electron calculations are performed for the ground states of LiH⁺, LiH⁻ and LiH. High care has been given to the correct relative position of the asymptotes, allowing for this rather complete set of accurate ab initio data to be useful for further molecular physics studies.     

Structure of Diacetato bis (2,4,6 Trimethyl Pyridine) Copper(II)

(2.4.6 trimethyl pyridine)₂ Cu(CH₃COO)₂ has been obtained as violet needles by slow evaporation. The crystals belong to the monoclinic system, space group P2_1/C with two molecules in a unit cell of dimension: a = 7.842 (1), b = 16.384 (2), c = 8.320 (2) Å, β = 101.91 (1)°. The structure was solved by heavy atom method and refined by a least squares method (R = 0.029). The coordination of copper is 4 + 2. The atom arrangements of this structure is very near to the copper acetate one solvated by one water molecule and two 3.4 dimethyl pyridine molecules.