Small quotients in Euclidean algorithms

Numbers whose continued fraction expansion contains only small digits have been extensively studied. In the real case, the Hausdorff dimension ?? _M of the reals with digits in their continued fraction expansion bounded by M was considered, and estimates of ?? _M for M???? were provided by Hensley (J. Number Theory 40:336?C358, 1992). In the rational case, first studies by Cusick (Mathematika 24:166?C172, 1997), Hensley (In: Proc. Int. Conference on Number Theory, Quebec, pp. 371?C385, 1987) and Vallée (J. Number Theory 72:183?C235, 1998) considered the case of a fixed bound M when the denominator N tends to ??. Later, Hensley (Pac. J. Math. 151(2):237?C255, 1991) dealt with the case of a bound M which may depend on the denominator N, and obtained a precise estimate on the cardinality of rational numbers of denominator less than N whose digits (in the continued fraction expansion) are less than M(N), provided the bound M(N) is large enough with respect to N. This paper improves this last result of Hensley towards four directions. First, it considers various continued fraction expansions; second, it deals with various probability settings (and not only the uniform probability); third, it studies the case of all possible sequences M(N), with the only restriction that M(N) is at least equal to a given constant M ₀; fourth, it refines the estimates due to Hensley, in the cases that are studied by Hensley. This paper also generalises previous estimates due to Hensley (J. Number Theory 40:336?C358, 1992) about the Hausdorff dimension ?? _M to the case of other continued fraction expansions. The method used in the paper combines techniques from analytic combinatorics and dynamical systems and it is an instance of the Dynamical Analysis paradigm introduced by Vallée (J. Théor. Nr. Bordx. 12:531?C570, 2000), and refined by Baladi and Vallée (J. Number Theory 110:331?C386, 2005).     

Gradient elution method in centrifugal partition chromatography for the separation of a complex sophorolipid mixture obtained from Candida bombicola yeasts

Sophorolipids represent an important class of natural surfactants with a variety of environmental, cosmetic, and pharmaceutical applications. Despite their promising physicochemical and biological properties, the use of sophorolipids is hampered by the lack of information regarding their individual structure‐activity relationships. The major difficulty in isolating pure sophorolipids arises from the high complexity of crude fermentation media composition and from their strong structural similarities. In this work, a centrifugal partition chromatography method was developed in an original gradient elution mode for the separation of sophorolipids produced by the yeast Candida bombicola. Experiments were realized by using three sets of solvent systems composed of n‐heptane, ethyl acetate, n‐butanol, methanol, and water in different proportions. The separation was performed at 5 mL/min in the ascending mode by increasing progressively the polarity of the organic mobile phase. In these conditions, more than 80% of the sophorolipids present in the initial crude fermentation extract were eluted successively from the most hydrophobic lactone forms to the most hydrophilic acid forms. The structures of the isolated sophorolipids were further elucidated by HPLC and NMR analyses.     

Using the Population‐Shift Mechanism to Rationally Introduce “Hill‐type” Cooperativity into a Normally Non‐Cooperative Receptor

Allosteric cooperativity, which nature uses to improve the sensitivity with which biomolecular receptors respond to small changes in ligand concentration, could likewise be of use in improving the responsiveness of artificial biosystems. Thus motivated, we demonstrate here the rational design of cooperative molecular beacons, a widely employed DNA sensor, using a generalizable population‐shift approach in which we engineer receptors that equilibrate between a low‐affinity state and a high‐affinity state exposing two binding sites. Doing so we achieve cooperativity within error of ideal behavior, greatly steepening the beacon’s binding curve relative to that of the parent receptor. The ability to rationally engineer cooperativity should prove useful in applications such as biosensors, synthetic biology and “smart” biomaterials, in which improved responsiveness is of value.     

Self-adaptable catalysts: substrate-dependent ligand configuration

Pd(II) allyl and Pd(0) olefin complexes containing the configurationally labile ligand 1,2-bis-[4,5-dihydro-3H-dibenzo[c-e]azepino]ethane were studied as models for intermediates in Pd-catalyzed allylic alkylations. According to NMR and DFT studies, the ligand prefers C(s) conformation in both eta3-1,3-diphenylpropenyl and eta3-cyclohexenyl Pd(II) complexes, whereas in Pd(0) olefin complexes it adopts different conformations in complexes derived from the two types of allyl systems in both solution and, as verified by X-ray crystallography, in the solid state. These results demonstrate that the Pd complex is capable of adapting its structure to the reacting substrate. The different structural preferences also provide an explanation for the behavior of 1,3-diphenyl-2-propenyl acetate and 2-cyclohexenyl acetate in Pd-catalyzed allylic alkylations using pseudo-C2 and pseudo-C(s) symmetric ligands.     

Metal-mediated transformation of a triazinephenanthridinium ligand leading to a {Pd5} coordination complex observed crystallographically and by cryospray mass spectrometry

The formation of a pentanuclear palladium(II) complex with a phenanthridinonetriazine-based ligand system, which itself is formed by a metal-mediated rearrangement of a triazinephenanthridinium proligand, is described.     

Pyrazolo[1,5-a]pyrimidine-Dioxaborinine Hybrid Dyes: Synthesis and Substituent Effect in the Photophysical Properties

A novel family of pyrazolo[1,5-a]pyrimidine-dioxaborinine (PP-DB) hybrid dyes was synthesized by the direct construction of the dioxaborinine (DB) fragment on the pyrazolo[1,5-a]pyrimidine (PP) ring, which implies the formation of four new bonds in a one-pot manner. The dyes’ optical properties were investigated and compared with the starting pyrazolo[1,5-a]pyrimidines; a study evidencing large fluorescence quantum yields in products (φ_f up to 69 %) due to an intramolecular charge transfer (ICT) process from the PP core to a ring of DB (PP→DB) that is absent in precursors (φ_f=0.03–0.30). Time-dependent density functional theory (TD-DFT) calculations confirmed the fluorescence process involved in the novel dyes, where their ICT limits the non-radiative process due to the restricted rotation in the D−A system. The present work provides insight into how phenyl and DB ring incorporation impact the optical properties of this new group of hybrids dyes based on PP-DB.     

Three Complementary One-Pot Four-Component Reaction Sequences for Rapid,General and Direct Spiropyran Synthesis

Three one-pot three-step four-component reaction sequences for spiropyran synthesis have been developed, based around three different methods for in situ generation of 1,2,3-trisubstituted indoles (indole N-alkylation, Fischer indolisation and indole C-alkylation). The reaction sequences give access to a broad swathe of spiropyran structures. Each sequence is operationally straightforward, rapid and high yielding. We have synthesized 58 structurally diverse spiropyrans bearing a wide range of useful functional groups, and their utility were demonstrated in the targeted synthesis of potential ratiometric fluorescence probes for metal ions and spiropyran-cholesterol conjugates. Finally, we showed that our one-pot N-alkylation-C-alkylation-condensation sequence can underpin combinatorial spiropyran synthesis through generation of a 16-member spiropyran library.     

Reaction Dynamics of Weakly-Bound Few-Body Nuclei at Energies Around the Coulomb Barrier

We present a quantum reaction approach that unambiguously quantifies the complete and incomplete fusion of weakly-bound few-body nuclei. Calculations carried out within a simple model for \({\text{}^{6}}\)Li + \({\text{}^{209}}\)Bi at energies near the Coulomb barrier show that converged probabilities for the total, complete and incomplete fusion as well as for the scattering process can be obtained with the time-dependent wave-packet dynamics.     

Letter: characterisation and identification of spermine and spermidine derivatives in Microdesmis keayana and Microdesmis puberula roots by electrospray ionisation tandem mass spectrometry and high-performance liquid chromatography/electrospray ionisation tandem mass spectrometry

Three new N(1),N(5),N(14)-tris(4- hydroxycinnamoyl)spermines were identified in hydromethanolic root extracts of Microdesmis keayana J. Léonard and Microdesmis puberula Hook f. The electrospray ionisation tandem mass spectrometry (ESI-MS/MS) technique with specific nuclear magnetic resonance analysis of hydrolysed products made it possible to identify N(1),N(5),N(14)-tris(p-coumaroyl)spermine, N(1)-feruloyl,N(5),N(14)-di(p-coumaroyl)spermine and N(1),N(5),N(14)-tris(feruloyl)spermine, named keayanines B, C and D, respectively. ESI-MS/MS analysis most effectively provided structural data although high-performance liquid chromatography/electrospray ionisation tandem mass spectrometry was also used to characterise four other compounds from Microdesmis puberula-keayanidines A, B, C and keayanine A-which had already been identified in M. keayana. This chemical data is the first to be published for M. puberula which is a commonly used plant in Central African traditional medicine.