Morse theory for some lower-C 2 functions in finite dimension

Using inf-regularization methods, we prove that Morse inequalities hold for some lower-C ² functions. For this purpose, we first recall some properties of the class of lower-C ² functions and of their Moreau-Yosida approximations. Then, we establish, under some qualification conditions on the critical points, that it is possible to define a Morse index for a lower-C ² functionf. This index is preserved by the Moreau-Yosida approximation process. We prove in particular that the Moreau-Yosida approximations are twice continuolusly differentiable around such a critical point which is shown to be a strict local minimum of the restriction off and of its approximations to some affine space. In a last step, Morse inequalities are written for Moreau-Yosida approximations and with the aid of deformation retractions we prove that these inequalities also hold for some lower-C ² functions.     

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‐Photon Polarity Probes Built from Octupolar Fluorophores: Synthesis,Structure–Properties Relationships,and Use in Cellular Imaging

A series of octupolar fluorophores built from a triphenylamine (TPA) core connected to electron‐withdrawing (EW) peripheral groups through conjugated spacers has been synthesized. Their photoluminescence, solvatochromism, and two‐photon absorption (2PA) properties were systematically investigated to derive structure–property relationships. All derivatives exhibit two 2PA bands in the 700–1000 nm region: a first band at low energy correlated with a core‐to‐periphery intramolecular charge transfer that leads to an intense 1PA in the blue‐visible range, and a second more intense band at higher energy due to an efficient coupling of the branches through the TPA core. Increasing the strength of the EW end groups or the length of the conjugated spacers and replacing triple‐bond linkers with double bonds induces both enhancement and broadening of the 2PA responses, thereby leading to cross‐sections up to 2100 GM at peak and higher than 1000 GM over the whole 700–900 nm range. All derivatives exhibit intense photoluminescence (PL) in low‐ to medium‐polarity environments (with quantum yields in the 0.5–0.9 range) and display a strong positive solvatochromic behavior (with Lippert–Mataga specific shifts ranging from 15 000 to 27 500 cm^?1), triple bonds, and phenyl moieties in the conjugated spacers, thereby leading to larger sensitivities than those of double bonds and thienyl moieties. More hydrophilic derivatives were also shown to be biocompatible, to retain their 2PA and PL properties in biological conditions, and finally to be suitable as polarity sensors for multiphoton cell imaging.     

Octupolar merocyanine dyes: a new class of nonlinear optical chromophores

A set of new octupolar merocyanine chromophores was designed and synthesized. These compounds were prepared from the reaction of 1,3,5-triformyl-2,4,6-trihydroxybenzene with heterocyclic nucleophiles. Octupolar dyes were formed exclusively in their open-dye form. The one- and two-photon-absorption spectra of the dyes consist of two bands: The long-wavelength band in the two-photon absorption spectrum (a few hundreds GM above 1000?nm) matches well with the intense, long-wavelength-absorption band that is located in the visible region in the linear spectrum. Interestingly, an additional, much-more-intense TPA band in the NIR region is observed at higher energy, which corresponds to a weakly allowed one-photon electronic transition. Changing the peripheral heterocyclic moieties allows tuning of the optical properties to approach the cyanine limit (i.e., polymethine state), thus resulting in a red-shift of the low-energy one-photon-absorption band as well as to the rise of an intense two-photon-absorption band in the NIR region. To the best of our knowledge, this is the first synthesis and TPA characterization of octupolar merocyanine chromophores with typical low-bond-length alternation.     

Triaryl-1,3,5-triazinane-2,4,6-triones (Isocyanurates) Peripherally Functionalized by Donor Groups: Synthesis and Study of Their Linear and Nonlinear Optical Properties

The linear optical (LO) and nonlinear optical (NLO) properties of a series of isocyanurates functionalized by donor arms at the periphery are reported herein. These octupolar derivatives were obtained in a straightforward way from commercial isocyanate derivatives and were fully characterized. Although several of these compounds are known, those that exhibited the largest NLO activities are all new compounds. In terms of second-order activity, several of these derivatives exhibit remarkable activity/transparency tradeoffs. In terms of third-order activity, the longer derivatives with the stronger donor groups (X=NH(2) , NMe(2) , or NPh(2) ) were shown to possess significant two-photon absorption cross sections. These strongly luminescent derivatives exhibit two-photon absorption cross sections up to 410?GM. DFT computations were also conducted to unravel their electronic structures and to rationalize their NLO properties. To our knowledge, the present study is the first concerned with the nonlinear optical properties of these original cyclotrimers.     

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.     

Étude de réactions à l'état solide entre l'arbre de Saturne et des sels contenant le groupement XO3

Résumé Le plomb utilisé sous forme d'arbre de Saturne possède une grande réactivité et permet de réduire les nitrates en nitrites. Il s'agit d'une réaction à l'état solide, débutant dès la température ordinaire. On a pu caractériser ainsi la présence de nitrites en mélangeant le plomb avec les nitrates de lithium, sodium, potassium, césium, cuivre, argent, magnésium, calcium, strontium, baryum, zinc, thallium, plomb, cobalt et nickel.

---

Summary Lead used in the form of the lead tree has great reactivity and is able to reduce nitrates to nitrites. This involves a solid state reaction and starts at room temperature. In this way it is possible to detect the presence of nitrites after mixing the lead with the following nitrates — Li, Na, K, Cs, Cu, Ag, Mg, Ca, Sr, Ba, Zn, Tl, Pb, Co, Ni.

---

Zusammenfassung In der Form des Bleibaumes ist Pb sehr reaktionsfähig und reduziert Nitrat zu Nitrit. Dabei handelt es sich um eine Festkörperreaktion, die bei gewöhnlicher Temperatur einsetzt. Nach Vermengen von Blei mit den Nitraten von Li, Na, K, Cs, Cu, Ag, Mg, Ca, Sr, Ba, Zn, Th, Pb, Co und Ni konnte die Anwesenheit von Nitrit festgestellt werden.

---

---

En l'honneur du ProfesseurA. A. Benedetti-Pichler, pour son 70^e anniversaire.     

Enantioselective aptameric molecular recognition material: Design of a novel chiral stationary phase for enantioseparation of a series of chiral herbicides by capillary electrochromatography

A chiral stationary phase derived from an L-RNA aptamer is evaluated for the enantiomer separation of a series of herbicide molecules (aryloxypropionic, aryloxyphenoxypropionic, and aminopropionic acid) by CEC after binding to biotin and grafting upon streptavidin-modified porous glass beads. We demonstrated that the aptamer capillary was stable in term of efficiency and retention during a long period. The influences of the mobile phase constitution and its flow-velocity on the enantioseparation were also investigated. The results suggest that the interactions of the enantiomer during CEC are solely based on chromatographic mechanisms and that the electrophoresis plays only a minor role. The separation efficiency and peak shape could be improved by Mg2+ divalent cation that stabilized the aptamer secondary structure and thus enhanced the mass transfer kinetics during the ligand-aptamer binding process. In addition, it was demonstrated that the determination of the enantiomerization barrier of flamprop was possible using this chiral stationary phase.