Lipschitz and path isometric embeddings of metric spaces

We prove that each sub-Riemannian manifold can be embedded in some Euclidean space preserving the length of all the curves in the manifold. The result is an extension of Nash C ¹ Embedding Theorem. For more general metric spaces the same result is false, e.g., for Finsler non-Riemannian manifolds. However, we also show that any metric space of finite Hausdorff dimension can be embedded in some Euclidean space via a Lipschitz map.     

Hausdorff measures of subgroups of \mathbb{R }/\mathbb{Z } and \mathbb{R }

For a sequence $\underline{u}=(u_n)_{n\in \mathbb{N }}$ of integers, let $t_{\underline{u}}(\mathbb{T })$ be the group of all topologically $\underline{u}$ -torsion elements of the circle group $\mathbb{T }:=\mathbb{R }/\mathbb{Z }$ . We show that for any $s\in ]0,1[$ and $m\in \{0,+\infty \}$ there exists $\underline{u}$ such that $t_{\underline{u}}(\mathbb{T })$ has Hausdorff dimension $s$ and $s$ -dimensional Hausdorff measure equal to $m$ (no other values for $m$ are possible). More generally, for dimension functions $f,g$ with $f(t)\prec g(t), f(t)\prec \!\!\!\prec t$ and $g(t)\prec \!\!\!\prec t$ we find $\underline{u}$ such that $t_{\underline{u}}(\mathbb{T })$ has at the same time infinite $f$ -measure and null $g$ -measure.     

Time-division multiplexing of the orbital angular momentum of light

We present an optical setup for generating a sequence of light pulses in which the orbital angular momentum (OAM) degree of freedom is correlated with the temporal one. The setup is based on a single q plate within a ring optical resonator. By this approach, we demonstrate the generation of a train of pulses carrying increasing values of OAM, or, alternatively, of a controlled temporal sequence of pulses having prescribed OAM superposition states. Finally, we exhibit an "OAM-to-time conversion" apparatus that divides different input OAM states into different time bins. The latter application provides a simple approach to digital spiral spectroscopy of pulsed light.     

Chemical and Mechanical Characterization of Licorice Root and Palm Leaf Waste Incorporated into Poly(urethane-acrylate) (PUA)

A poly(urethane-acrylate) polymer (PUA) was synthesized, and a sufficiently high molecular weight starting from urethane-acrylate oligomer (UAO) was obtained. PUA was then loaded with two types of powdered ligno-cellulosic waste, namely from licorice root and palm leaf, in amounts of 1, 5 and 10%, and the obtained composites were chemically and mechanically characterized. FTIR analysis of final PUA synthesized used for the composite production confirmed the new bonds formed during the polymerization process. The degradation temperatures of the two types of waste used were in line with what observed in most common natural fibers with an onset at 270 °C for licorice waste, and at 290 °C for palm leaf one. The former was more abundant in cellulose (44% vs. 12% lignin), whilst the latter was richer in lignin (30% vs. 26% cellulose). In the composites, only a limited reduction of degradation temperature was observed for palm leaf waste addition and some dispersion issues are observed for licorice root, leading to fluctuating results. Tensile performance of the composites indicates some reduction with respect to the pure polymer in terms of tensile strength, though stabilizing between data with 5 and 10% filler. In contrast, Shore A hardness of both composites slightly increases with higher filler content, while in stiffness-driven applications licorice-based composites showed potential due to an increase up to 50% compared to neat PUA. In general terms, the fracture surfaces tend to become rougher with filler introduction, which indicates the need for optimizing interfacial adhesion.     

Electrochemically controlled formation/dissociation of phosphonate-cavitand/methylpyridinium complexes

The phosphorus-bridged cavitand 1 self-assembles very efficiently in CH2Cl2 with either the monopyridinium guest 2+ or the bispyridinium guest 3(2+). In the first case a 1:1 complex is obtained, whereas in the second case both 1:1 and 2:1 host-guest complexes are observed. The association between 1 and either one of the guests causes the quenching of the cavitand fluorescence; in the case of the adduct between 1 and 3(2+), the fluorescence of the latter is also quenched. Cavitand complexation is found to affect the reduction potential values of the electroactive guests. Voltammetric and spectroelectrochemical measurements show that upon one-electron reduction both guests are released from the cavity of 1. Owing to the chemical reversibility of such redox processes, the supramolecular complexes can be re-assembled upon removal of the extra electron from the guest. Systems of this kind are promising for the construction of switchable nanoscale devices and self-assembling supramolecular materials, the structure and properties of which can be reversibly controlled by electrochemical stimuli.     

FTIR and WAXS analysis of periodate oxycellulose: Evidence for a cluster mechanism of oxidation

Cellulose paper (Whatman no.1, chromatographic grade) was oxidised with 0.1 M sodium metaperiodate at different oxidation levels (0–11 days at room temperature), and analysed with FTIR before and after chemical treatments (reduction, further oxidation, alkaline hydrolysis). The deconvolution of infrared data allowed us to verify that periodate oxidises cellulose in isolated domains, leading to the decrease of crystallinity, in agreement with the results of wide angle X-ray scattering (WAXS). In particular, the alkaline hydrolysis (β-alkoxy fragmentation) followed by mild acid treatment removed the oxidised groups and recovered most of the crystallinity of cellulose, as determined by the FTIR crystallinity index.     

Conformational behavior of pyrazine-bridged and mixed-bridged cavitands: a general model for solvent effects on thermal "vase-kite" switching

The controllable switching of suitably bridged resorcin[4]arene cavitands between a "vase" conformation, with a cavity capable of guest inclusion, and a "kite" conformation, featuring an extended flattened surface, provides the basis for ongoing developments of dynamic molecular receptors, sensors, and molecular machines. This paper describes the synthesis, X-ray crystallographic characterization, and NMR analysis of the "vase-kite" switching behavior of a fully pyrazine-bridged cavitand and five other mixed-bridged quinoxaline-bridged cavitands with one methylene, phosphonate, or phosphate bridge. The pyrazine-bridged resorcin[4]arene cavitand displayed an unexpectedly high preference for the kite conformation in nonpolar solvents, relative to the quinoxaline-bridged analogue. This observation led to extensive solvent-dependent switching studies that provide a detailed picture of how solvent affects the thermal vase-kite equilibration. As for any thermodynamic process in the liquid phase, the conformational equilibrium is affected by how the solvent stabilizes the two individual states. Suitably sized solvents (benzene and derivatives) solvate the cavity of the vase form and reduce the propensity for the vase-to-kite transition. Correspondingly, the kite geometry becomes preferred in bulky solvents such as mesitylene, incapable of penetrating the vase cavity. As proposed earlier by Cram, the kite form is preferred at low temperatures due to the more favorable enthalpy of solvation of the enlarged surface. Furthermore, the kite conformation is more preferred in solvents with substantial hydrogen-bonding acidity: weak hydrogen-bonding interactions between the mildly basic quinoxaline and pyrazine nitrogen atoms and solvent molecules are more efficient in the open kite than in the closed vase form. Vase-to-kite conversion is entirely absent in dipolar aprotic solvents lacking any H-bonding acidity. Thermal vase-kite switching requires fully quinoxaline- or pyrazine-bridged cavitands, whereas pH-controlled switching is also applicable to systems incorporating only two or three such bridges.     

Aqueous cholesteric liquid crystals using uncharged rodlike polypeptides

The aqueous, lyotropic liquid-crystalline phase behavior of the alpha-helical polypeptide, poly(N(epsilon)-2-[2-(2-methoxyethoxy)ethoxy]acetyl-lysine) (1), has been studied using optical microscopy and X-ray scattering. Solutions of optically pure 1 were found to form cholesteric liquid crystals at volume fractions that decreased with increasing average chain length. At very high volume fractions, the formation of a hexagonal mesophase was observed. The pitch of the cholesteric phase could be varied by a mixture of enantiomeric samples L-1 and D-1, where the pitch increased as the mixture approached equimolar. The cholesteric phases could be untwisted, using either magnetic field or shear flow, into nematic phases, which relaxed into cholesterics upon removal of field or shear. We have found that the phase diagram of 1 in aqueous solution parallels that of poly(gamma-benzyl glutamate) in organic solvents, thus providing a useful system for liquid-crystal applications requiring water as solvent.     

Competitive Gold‐Promoted Meyer–Schuster and oxy‐Cope Rearrangements of 3‐Acyloxy‐1,5‐enynes: Selective Catalysis for the Synthesis of (+)‐(S)‐γ‐Ionone and (−)‐(2S,6 R)‐cis‐γ‐Irone

We report a simple, highly stereoselective synthesis of (+)‐(S)‐γ‐ionone and (‐)‐(2S,6R)‐cis‐γ‐irone, two characteristic and precious odorants; the latter compound is a constituent of the essential oil obtained from iris rhizomes. Of general interest in this approach are the photoisomerization of an endo trisubstituted cyclohexene double bond to an exo vinyl group and the installation of the enone side chain through a [(NHC)Au^I]‐catalyzed Meyer–Schuster‐like rearrangement. This required a careful investigation of the mechanism of the gold‐catalyzed reaction and a judicious selection of reaction conditions. In fact, it was found that the Meyer–Schuster reaction may compete with the oxy‐Cope rearrangement. Gold‐based catalytic systems can promote either reaction selectively. In the present system, the mononuclear gold complex [Au(IPr)Cl], in combination with the silver salt AgSbF₆ in 100:1 butan‐2‐one/H₂O, proved to efficiently promote the Meyer–Schuster rearrangement of propargylic benzoates, whereas the digold catalyst [{Au(IPr)}₂(μ‐OH)][BF₄] in anhydrous dichloromethane selectively promoted the oxy‐Cope rearrangement of propargylic alcohols.     

Effect of the π Bridge and Acceptor on Intramolecular Charge Transfer in Push–Pull Cationic Chromophores: An Ultrafast Spectroscopic and TD‐DFT Computational Study

Three (donor–π–acceptor)⁺ systems with a methyl pyridinium or quinolinium as the electron‐deficient group, a dimethyl amino as the electron‐donor group, and an ethylene or butadiene group as the spacer have been investigated in a joint spectroscopic and TD‐DFT computational study. A negative solvatochromism has been revealed in the absorption spectra, which implies a solution color change, and interpreted by considering the variation in the permanent dipole moment modulus and orientation upon photoexcitation. The fluorescence efficiency decreases upon increasing solvent polarity, in agreement with the excited‐state optimized geometries (planar in low‐polarity media and twisted in high‐polarity media). Femtosecond transient absorption has revealed the occurrence of a fast photoinduced intramolecular charge transfer (ICT) and the molecular factors that determine an efficient ICT. Considering the crucial role of the ICT in tuning the nonlinear optical (NLO) properties, these compounds can be considered promising NLO materials.