On the correlation of the Thue–Morse sequence

The Ramanujan Journal - Let $$s_{q}$$ denote the sum of digits function in base q. The aim of this work is to estimate the exponential sums involving the sum of digits of shifted integers, namely...     

New series of acridines and phenanthrolines: synthesis and characterization

New series of acridines and phenanthrolines have been prepared from β-chlorovinyl aldehydes and various aniline derivatives allowing the installation of valuable substituents, such as ketone, nitro or amino groups at the heterocyclic core. X-ray analyses confirmed the structures of acridines and phenanthrolines as well as the presence of partially hydrogenated rings and their crucial impact on the overall shape of the acridine-based architectures. ¹H NMR revealed the helical behaviour of several acridine motives. Comparison of UV data between architectures and influence of number of partially hydrogenated rings is also described.     

Electrochemical Capacitive K+ EMIS Chemical Sensor Based on the Dibromoaza[7]helicene as an Ionophore for Potassium Ions Detection

A K⁺‐sensitive capacitive electrolyte‐membrane‐insulator‐semiconductor (EMIS) based on a novel dibromoaza[7]helicene ionophore has been developed. An ion‐sensitive membrane based on polyvinylchloride (PVC) doped with the ionophore was deposited on the Si₃N₄/SiO₂/Si‐p/Cu‐Al transducer. The properties of the K⁺‐EMIS chemical sensor were investigated by electrochemical impedance spectroscopy (EIS). All the developed devices upon being tested have shown good sensitivity and linearity responses within the range 10^?6 M to 10^?1 M of potassium activity, with good selectivity over a wide variety of other cations (Na⁺, Li⁺, Cu²⁺, Ca²⁺, and Mg²⁺). To our knowledge, this is the first time that a capacitive field‐effect sensor has been fabricated using helicene as a carrier for K⁺‐detection, combined with the structure: Si₃N₄/SiO₂/Si‐p/Cu‐Al as a transducer.     

Enhanced functional properties of chitosan films cross-linked by biosourced dicarboxylic acids

Chitosan-based films were developed using different biosourced dicarboxylic acid solutions (succinic acid, adipic acid, suberic acid and sebacic acid). The effect of incorporating these nontoxic solutions on water vapor barrier, tensile and antimicrobial properties of chitosan films was evaluated. Fourier Transform Infrared (FTIR) and cross polarization magic angle spinning (CP/MAS) nuclear magnetic resonance (NMR) analyses were also performed to investigate functional groups interactions between chitosan and dicarboxylic acids. Acetic acid-chitosan films showed significantly higher water vapor permeability (WVP) and lower tensile strength (TS) and elongation at break (%E) than dicarboxylic acid-prepared films (p < 0.05). Using either adipic acid or suberic acid solutions to fabricate chitosan films reduced WVP by 85% and enhanced TS by 21 to 27% and %E by more than 60% when compared with the acetic acid-prepared films. Chitosan films modified with either adipic acid or suberic acid exhibited antimicrobial activity against all tested microorganisms.     

Chebyshev polynomials of the second kind via raising operator preserving the orthogonality

It is well known that monic orthogonal polynomial sequences \(\{T_n\}_{n\ge 0}\) and \(\{U_n\}_{n\ge 0}\), the Chebyshev polynomials of the first and second kind, satisfy the relation \(DT_{n+1}=(n+1)U_n\) (\(n\ge 0\)). One can also easily check that the following “inverse” of the mentioned formula holds: \({\mathcal {U}}_{-1}(U_n)=(n+1)T_{n+1}\) (\(n\ge 0\)), where \({\mathcal {U}}_\xi =x(xD+{\mathbb {I}})+\xi D\) with \(\xi \) being an arbitrary nonzero parameter and \({\mathbb {I}}\) representing the identity operator. Note that whereas the first expression involves the operator D which lowers the degree by one, the second one involves \({\mathcal {U}}_\xi \) which raises the degree by one (i.e. it is a “raising operator”). In this paper it is shown that the scaled Chebyshev polynomial sequence \(\{a^{-n}U_n(ax)\}_{n\ge 0}\) where \(a^2=-\xi ^{-1}\), is actually the only monic orthogonal polynomial sequence which is \({\mathcal {U}}_\xi \)-classical (i.e. for which the application of the raising operator \({\mathcal {U}}_\xi \) turns the original sequence into another orthogonal one).     

Stereoselective 1,2-cis glycosylation of 2-O-allyl protected thioglycosides

The technique of intramolecular aglycon delivery (IAD), whereby a glycosyl acceptor is temporarily appended to a hydroxyl group of a glycosyl donor is an attractive method that can allow the synthesis of 1,2-cis glycosides in an entirely stereoselective fashion. 2-O-Allyl protected thioglycoside donors are excellent substrates for IAD, and may be glycosylated stereoselectively through a three-step reaction sequence. This sequence consists of quantitative yielding allyl bond isomerisation, to produce vinyl ethers that can then undergo N-iodosuccinimide mediated tethering of the desired glycosyl acceptor, and subsequent intramolecular glycosylation, to yield either alpha-glucosides or beta-mannosides accordingly. Although attempted one-pot tethering and glycosylation is hampered by competitive intermolecular reaction with excess glycosyl acceptor, this problem can be simply overcome by the use of excess glycosyl donor. Allyl mediated IAD is a widely applicable practical alternative to other IAD approaches for the synthesis of beta-mannosides, that is equally applicable for alpha-gluco linkages. It is advantageous in terms of both simplicity of application and yield, and in addition has no requirement for cyclic 4,6-protection of the glycosyl donor.