The monotonic property of the generalized root mean square

The link between a differential equation and its initial conditions is not quite as simple as usually imagined. Several examples illustrate the complications that can arise.     

Triply Existentially Complete Triangle‐Free Graphs

A triangle‐free graph G is called k‐existentially complete if for every induced k‐vertex subgraph H of G, every extension of H to a ‐vertex triangle‐free graph can be realized by adding another vertex of G to H. Cherlin  11 , 12 asked whether k‐existentially complete triangle‐free graphs exist for every k. Here, we present known and new constructions of 3‐existentially complete triangle‐free graphs.     

Chlorine Dioxide as an Electron-Transfer Oxidant of Olefins. Preliminary Communication

The kinetics and product studies of oxidation of eight olefins 1 - 8 by ClO₂ in H₂O in the pH range 3-7 are described. The reaction is faster as the pH decreases. At pH < 4, ClO₂ reacts equimolarly with olefins to yield isomeric mixtures of chlorohydrines and 1,2-dioxygenated products, following the equation: The order of reactivity is: (E)-stilbene > indene > β-methylstyrene > acenaphthylene > α-methylstyrene > styrene > cyclohexene > allylbenzene. A multi-stage radical-cation mechanism is proposed, in which an initial reversible protonation: is followed by an electron-transfer stage (rate-determining): The cation-radical thus produced, adds rapidly an additional ClO₂ to form dioxygenated products. The chlorohydrines most likely arise from HClO additions to the olefinic double bonds, which, in turn, generate from dismutation of 2 HClO₂ into HClO + H⁺ + ClO.     

Development of ion-sensitive field-effect transistor-based sensors for benzylphosphonic acids and thiophenols using molecularly imprinted TiO2 films

Molecularly imprinted polymeric membranes, containing artificial recognition sites for a number of benzylphosphonic acid derivatives, were prepared by the polymerization of titanium(IV) butoxide in the presence of a titanium(IV) phosphonate complex. Reference polymers were prepared in the same manner but in the absence of the phosphonate template. FTIR spectroscopy was used to follow the formation of a benzylphosphonic acid-Titanium(IV) oxide complex during the imprinting process, and upon the association of the substrate in the imprinted TiO₂ thin film. The imprinted polymers were examined as sensing membranes in an ion-sensitive field-effect transistors (ISFETs). The sensors reveal selectivity towards analyzing the imprinted substrates, yet the recognition ability of the sensors strongly depends on the substituents associated with the phosphonic acid structures. The response time of the sensors is ca. 45 s, and the sensors reveal unaffected stability for at least 2 weeks. Also, imprinted TiO₂ films for thiophenol, and para-nitrothiophenol were generated on ISFET devices, and the respective substrates were selectively sensed by the functional devices.     

Coherent backscattering of singular beams

The phenomenon of coherent backscattering depends on both the statistical characteristics of a random scattering medium and the correlation features of the incident field. Imposing a wavefront singularity on the incident field offers a unique and very attractive way to modify the field correlations in a deterministic manner. The field correlations are found to act as a path-length filter which modifies the distribution of different contributions to the enhancement cone. This effect is thoroughly discussed and demonstrated experimentally for the case of single scale scattering systems.     

A novel method for the facile synthesis of depsipeptides

A one pot, high yield synthesis of depsipeptides is described involving room temperature condensation of hindered substrates (N-Boc amino acids and benzyl α-hydroxy esters) in the presence of DCC and a dialkylaminopyridine. In this manner previously inaccessible depsipeptides were prepared.     

The writhe of permutations and random framed knots

We introduce and study the writhe of a permutation, a circular variant of the well‐known inversion number. This simple permutation statistics has several interpretations, which lead to some interesting properties. For a permutation sampled uniformly at random, we study the asymptotics of the writhe, and obtain a non‐Gaussian limit distribution. This work is motivated by the study of random knots. A model for random framed knots is described, which refines the Petaluma model, studied with Hass, Linial, and Nowik (Discrete Comput Geom, 2016). The distribution of the framing in this model is equivalent to the writhe of random permutations. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 51, 121–142, 2017     

On the abundance of chiral crystals

A common perception of many chemists is that non-biological chiral crystals comprise a small fraction of all crystals, as is the case of chiral non-biological molecules (~10%). We show that the proportion of non-biological chiral crystals is as high as 23%; and only ~6% of these are labelled as chiral.