On 3-Dimensional Lattice Walks Confined to the Positive Octant

Many recent papers deal with the enumeration of 2-dimensional walks with prescribed steps confined to the positive quadrant. The classification is now complete for walks with steps in \({\{0, \pm 1\}^{2}}\): the generating function is D-finite if and only if a certain group associated with the step set is finite. We explore in this paper the analogous problem for 3- dimensional walks confined to the positive octant. The first difficulty is their number: we have to examine no less than 11074225 step sets in \({\{0, \pm 1\}^{3}}\) (instead of 79 in the quadrant case). We focus on the 35548 that have at most six steps. We apply to them a combined approach, first experimental and then rigorous. On the experimental side, we try to guess differential equations. We also try to determine if the associated group is finite. The largest finite groups that we find have order 48 — the larger ones have order at least 200 and we believe them to be infinite. No differential equation has been detected in those cases. On the rigorous side, we apply three main techniques to prove D-finiteness. The algebraic kernel method, applied earlier to quadrant walks, works in many cases. Certain, more challenging, cases turn out to have a special Hadamard structure which allows us to solve them via a reduction to problems of smaller dimension. Finally, for two special cases, we had to resort to computer algebra proofs. We prove with these techniques all the guessed differential equations. This leaves us with exactly 19 very intriguing step sets for which the group is finite, but the nature of the generating function still unclear.     

Two‐step Synthesis of New γ‐Lactones via Cyclization of 7‐Chloro‐2‐(methoxycarbonyl)‐4‐6‐dimethylocta‐(2E,4E,6E)‐trienoic acid

A new rapid synthesis of γ‐lactones, cis fused with a cyclopentenic ring by thermal cyclization of 7‐chloro‐2‐(methoxycarbonyl)‐4‐6‐dimethylocta‐7‐phenyl (or methyl) (2E,4E,6E)‐trienoic acids was reported. The key step implicates an intramolecular cyclization to a cyclopentenyl cation, according to an electrocyclic π_2s + π_2a conrotatory process, published in a recent paper (from the corresponding diacids). We have investigated the thermal behavior of the corresponding half‐esters since; if the cyclization obeys to the proposed mechanism, the diacids, half‐esters must also cyclize in a similar manner. Saponification of these led to γ‐dilactones via intermediary cyclopropanes. Mechanistic pathways were investigated.     

Study of beta-cyclodextrin/fluorinated trimethyl ammonium bromide surfactant inclusion complex by fluorinated surfactant ion selective electrode

The construction and performance of a liquid membrane electrode responsive to N-(1,1,2,2-tetrahydroperfluorooctyl)-N,N,N-trimethylammonium bromide (FTABr) and its use for the study of β-cyclodextrin/fluorinated surfactant inclusion complex is described. The electrode is based on the use of tetrahydroperfluorooctyltrimethylammonium-tetraphenylborate ion pair as electro active material in polyvinyl chloride (PVC) matrix plasticized using 2-Nitrophenyl octyl ether (NPOE). The electrode exhibits a fast, stable, reproducible and “Nernstian” response (59 ± 2 mV) for FTABr over the concentration range of 10⁻⁵ to 2 × 10⁻³ mol L⁻¹ at 298 K. The lowest detection limit is 2 × 10⁻⁶ mol L⁻¹ and the response time is around 20-30 s. The validity of the electrode, for detection of fluorinated surfactant ions and hence to carry out electrochemical measurements to study micellization of fluorinated surfactant, is verified by comparing the critical micelle concentration (cmc) value of FTABr obtained by using the electrode, with that obtained by surface tension measurements. Association constant K for β-cyclodextrin/FTABr complex is evaluated from the potentiometric measurements carried out using this electrode and is observed to be ∼1.26 × 10⁵. The results suggest that β-cyclodextrin forms an equimolar association complex with the FTA⁺ surfactant ion.     

Polyethylene glycol adsorption on silica: from bulk phase behavior to surface phase diagram

A characterization of the bulk-phase diagram from literature data and new NMR and DSC measurements provided us with valuable elements that are helpful for gaining, from aqueous solution, better insight into the surface behavior of polyethylene glycol on Aerosil 200. Adsorption isotherms built further to measurements by a depletion method showed a strong and temperature-dependent variation of the isotherm shape in agreement with the variations of interactions already evidenced in the bulk. In temperature-concentration areas, where water is behaving as a helix-promoting solvent, the finding of positive PEG adsorptions and stairlike isotherms agrees with observations reported in the literature. We identified some of the vertical parts as corresponding to the formation of monolayers of helix-shaped PEG molecules. In poor-solvent zones, adsorptions were null or negative, and the isotherms exhibited oscillations suggesting very different surface behavior. Our data analysis evidenced the presence of a much greater amount of water than in the previous surface states; however, the similar analysis of PEG behavior remains relevant. Indeed, the occurrence of first-order transitions in the surface layer implies some water reorganization, permitting the PEG molecules to move closer to the surface and become helix-shaped to rearrange in a monolayer. The surface phase diagram confirmed this analysis in a very satisfying way.     

PEG400 novel phase description in water

The behavior of hydroxyl-terminated PEG400 in water was investigated by surface tension measurements and 13C NMR as a function of concentration and temperature. PEG400 exhibited a critical aggregative concentration (cac) that evidenced both its amphiphilic character and its aggregation capacity. Moreover, the chemical shifts of the different carbons of the PEG were followed by NMR versus concentration at various temperatures. We observed a plateau between 20 and 35 degrees C at concentrations above 0.2 mol L(-1) and ascribed it to the aggregation process. A good correlation was found between the NMR spectra in the region of aggregation and the cac region in the phase diagram. Our investigations were also focused on the solid-liquid region of the phase diagram at lower temperatures. These experimental data, together with conclusions available in the literature, led us to propose explanations for the conformation/hydration/aggregation in the PEG400-water solutions phenomena.     

Effects of nanorod structure and conformation of fatty acid self-assembled layers on superhydrophobicity of zinc oxide surface

Superhydrophobic surfaces have been prepared from nanostructured zinc oxide layers by a treatment with fatty acid molecules. The layers are electrochemically deposited from an oxygenated aqueous zinc chloride solution. The effects of the layer's structure, from a dense film to that of a nanorod array, as well as that of the properties of the fatty acid molecules based on C18 chains are described. A contact angle (CA) as high as 167 degrees is obtained with the nanorod structure and the linear saturated molecule (stearic acid). Lower values are found with molecules having an unsaturated bond on C9, in particular with a cis conformation (140 degrees ). These results, supplemented by infrared spectroscopy, indicate an enhancement of the sensitivity to the properties of the fatty acid molecules (conformation, flexibility, saturated or not) when moving from the flat surface to the nanostructured surface. This is attributed to a specific influence of the structure of the tops of the rods and lateral wall properties on the adsorption and organization of the molecules. CA measurements show a very good stability of the surface in time if stored in an environment protected from UV radiations.     

Combination of Cyanine Behaviour and Giant Hyperpolarisability in Novel Merocyanine Dyes: Beyond the Bond Length Alternation (BLA) Paradigm

Merocyanine dyes that exhibit antithetic cyanine‐like behaviour and giant first‐order hyperpolarisability (β) values have been designed. These cyanine‐type dyes open up an intriguing route towards molecular‐based electro‐optic materials as well as new second‐harmonic generation dyes for imaging.     

A sensitivity analysis to assess the completion time deviation for multi-purpose machines facing demand uncertainty

This paper addresses multi-purpose machine configuration in an uncertain context through sensitivity analysis. The so-called configuration is the machine??s ability to process products, and the uncertain context is modelled as a demand variation affecting the forecast demand. Given a configuration, this work aims at assessing the completion time deviation when the workshop demand is subject to perturbation. Such quantitative information can be used in a robustness approach for selecting the most appropriate configuration. To do so, the configuration impact on the completion time value that can be reached by solving the attached scheduling problem is first investigated. Then, the completion time deviation is written as a piecewise linear function of the magnitude of demand variation. The proposed approach, which is based on the solution of a set of linear programs, is illustrated through a detailed example. It is shown to be polynomial, and fast enough for addressing real-world instances. Finally, how to compare two configurations on the basis of completion time deviation in an uncertain context is demonstrated.     

Optical electron transfer through 2,7-diethynylfluorene spacers in mixed-valent complexes containing electron-rich "(η2-dppe)(η5-C5Me5)Fe" endgroups

We report in this communication the study of the intramolecular electron transfer through a 2,7-diethynylfluorenyl spacer in the Fe(II)/Fe(III) mixed-valent (MV) complex [(η(2)-dppe)(η(5)-C(5)Me(5))FeC≡C(2,7-C(21)H(24))C≡CFe(η(5)-C(5)Me(5))(η(2)-dppe)][PF(6)] (1[PF(6)]). The complex is generated in situ by comproportionation from its homovalent dinuclear Fe(II) and Fe(III) parents (1 and 1[PF(6)](2)). It is shown that electronic delocalization is much more effective through a 2,7-fluorenyl than through a 4,4'-biphenyl bridging unit.     

Colour coding the co-conformations of a [2]rotaxane flip-switch

Fine-tuning the charge transfer chromophores in a series of [2]rotaxane flip-switches yields a unique optical signal (purple colour) for one of the interactions allowing for facile determination of the position of the flip-switch equilibrium.