Free Boolean algebras over unions of two well orderings

Given a partially ordered set P there exists the most general Boolean algebra which contains P as a generating set, called the free Boolean algebra over P. We study free Boolean algebras over posets of the form P=P₀∪P₁, where P₀, P₁ are well orderings. We call them nearly ordinal algebras.Answering a question of Maurice Pouzet, we show that for every uncountable cardinal κ there are κ² pairwise non-isomorphic nearly ordinal algebras of cardinality κ.Topologically, free Boolean algebras over posets correspond to compact 0-dimensional distributive lattices. In this context, we classify all closed sublattices of the product (ω₁+1)×(ω₁+1), showing that there are only ℵ₁ many types. In contrast with the last result, we show that there are ℵ₁² topological types of closed subsets of the Tikhonov plank (ω₁+1)×(ω+1).     

Ultrasonic elastography using sector scan imaging and a radial compression

Elastography is an imaging technique based on strain estimation in soft tissues under quasi-static compression. The stress is usually created by a compression plate, and the target is imaged by an ultrasonic linear array. This configuration is used for breast elastography, and has been investigated both theoretically and experimentally. Phenomena such as strain decay with tissue depth and strain concentrations have been reported. However in some in vivo situations, like prostate or blood vessels imaging, this set-up cannot be used. We propose a device to acquire in vivo elastograms of the prostate. The compression is applied by inflating a balloon that covers a transrectal sector probe. The 1D algorithm used to calculate the radial strain fails if the center of the imaging probe does not correspond to the center of the compressor. Therefore, experimental elastograms are calculated with a 2D algorithm that accounts for tangential displacements of the tissue. In this article, in order to gain a better understanding of the image formation process, the use of ultrasonic sector scans to image the radial compression of a target is investigated. Elastograms of homogeneous phantoms are presented, and compared with simulated images. Both show a strain decay with tissue depth. Then experimental and simulated elastograms of a phantom that contains a hard inclusion are presented, showing that strain concentrations occur as well. A method to compensate for strain decay and therefore to increase the contrast of the strain elastograms is proposed. It is expected that such information will help to interpret and possibly improve the elastograms obtained via radial compression.     

Darstellung und eigenschaften von und reaktionen mit metallhaltigen heterocyclen XXII. Synthese, eigenschaften und kristallstruktur des sechsgliedrigen heterocyclus [(h-C5H5)NiSP(CH3)2]2 und dessen verhalten gegenüber alkinen

The heterocycle [(h⁵-C₅H₅)NiSP(CH₃)₂]₂ is obtained by treatment of (h⁵-C₅H₅)₂Ni with (CH₃)₂HPS in toluene and crystallizes monoclinic in the space group P2₁/c with Z = 2. The highly reactive three-membered ring (h⁵-C₅H₅)$\text{NiSP}$(CH₃)₂ which is a dissociation product of [(h⁵-C₅H₅)NiSP(CH₃)₂]₂, can be trapped with bis(methoxycarbonyl)acetylene to give the PS containing nickelacyclopentadiene (h⁵-C₅H₅)$\text{NiSP(CH}{}_3\text{)}{}_2\text{CRC}$R (R  CO₂CH₃).     

A convenient reagent for generation and stabilization of cationic methallylpalladium complexes containing nitrogen ligands

A convenient synthesis of the easily handled, air stable methallyloxyphosphonium salt [CH₂C(Me)CH₂⁻O-P(NMe₂)₃]⁺[(3,5-(CF₃)₂-C₆H₃)₄B]⁻5 is described. The utility of this reagent in the generation and stabilization of cationic η³-methallyl palladium complexes through oxidative addition reactions is illustrated by the preparation of the stable salts of [(η³-C₄H₇)Pd(NN)]⁺[(3,5-(CF₃)₂-C₆H₃)₄B]⁻ from Pd₂(dba)₃. The molecular structure of one of them has been determined by a single-crystal X-ray diffraction.     

Synthesis and resolution of a new helically chiral azahelicene

Helically chiral azahexahelicene 3 was prepared in four steps using the Mizoroki-Heck coupling followed by classical oxidative photodehydrocyclisation. Resolution of this new chiral system was achieved through separation by HPLC providing (−)- and (+)-3 in high optical purity. The absolute configurations of (−)- and (+)-3 were assigned as M and P, respectively, by means of circular dichroism. Each of the hexacyclic systems (M)-(−)- and (P)-(+)-3 was reacted with boron tribromide to provide the corresponding helical pyridophenols in good yields.     

Copper(I)-doped Wyoming’s montmorillonite for the synthesis of disubstituted 1,2,3-triazoles

A new catalytic system based on copper(I)-doped Wyoming’s montmorillonite was elaborated. This system was studied and was found to catalyze the formation of 1,4-disubstituted 1,2,3-triazoles when starting from azides and alkynes. It was used for the synthesis of a few triazole derivatives to show its usefulness.     

Numerical analysis of the azimuthal periodicity in the two lowest order modes of microstructured fibres applied to the precise determination of the cutoff wavelength

The azimuthal far-field analysis technique (AFFAT) is one of the only two methods reported to date, able to measure the cutoff wavelength λ_c in microstructured fibres (MOFs). The measurement is operated through a parameter D which is related to the modulation of the power crossing a rotating slit at the output of the tested fibre. In standard fibres, the criterion of the CEI/IEC 793-1-C7A normalised method (i.e., 2.25% of the output power carried by the second mode) is expressed by the unchanged −16.4 dB value of D(λ_c). In this paper, we show that this criterion is expressed in MOFs by a value of D which strongly depends on the geometrical parameters of the fibre and on the launching conditions at the input. However, we demonstrate that, with a slightly relaxed criterion corresponding to 10% of the output power carried by the second mode at λ_c, D remains close to −10 dB whatever the considered MOF. Thus, this constitutes a universal decision criterion allowing to determine λ_c in any classical MOF, by means of the AFFAT, with a very limited uncertainty.     

Nonlinearities effects on optical MIMO systems

Optical Review - Multimode fibers (MMF) are enjoying a renewed attention boosted by recent advances in multimode complex nonlinear optics. Reason behind nonlinearity is occurred either due to...     

SNAr reactions of substituted pyridines with secondary amines in aqueous solution: Kinetic and reactivity indices in density functional theory

A kinetic study is reported for the reactions of 2-methoxy-3-nitropyridine 1a and 2-methoxy-5-nitropyridine 1b with three secondary amines 2a–c (morpholine, piperidine, and pyrrolidine) in aqueous solution at 20°C. The Brønsted-type plots are linear with β_nuc = 0.52 and 0.55 for pyridines 1a and 1b , respectively, indicating that the reaction proceeds through a S_NAr mechanism in which the first step is the rate-determining step. Additional theoretical calculations using the DFT/B3LYP method confirm that the C-2 carbon being the most electrophilic center for the both pyridines 1a and 1b . The second-order rate constants have been used to evaluate the electrophilicity parameters E of 1a and 1b according to the linear free energy relationship log k (20°C) = s_N (N + E). The E parameters thus derived are compared with the electrophilic reactivities of a large variety of anisoles. The validity of these E values has been satisfactorily verified by comparison of calculated and experimental second-order rate constants for the reactions of pyridines 1a and 1b with anion of ethyl benzylacetate.     

Phase transitions in (C7H12N2)2[SnCl6]Cl2·1.5H2O crystal,studied by NMR and infrared spectroscopy

The (C₇H₁₂N₂)₂[SnCl₆]Cl₂·1.5H₂O complex is a new member of the family of hybrid organic–inorganic perovskite compounds. It exhibits two order–disorder phase transitions with changes in the conformation of aromatic cations at the two transition temperatures 360 and 412 K. Differential scanning calorimetry, nuclear magnetic resonance (NMR), and Fourier-transform infrared (FT-IR) spectroscopy were used to investigate these phase transitions. These transition mechanisms were investigated in terms of the spin–lattice relaxation times T1 for ¹H static NMR and the chemical shifts for ¹³C CP–MAS. The temperature dependence of T1(¹H) and ¹³C chemical shifts are changed near T_C1 and T_C2. Furthermore, the splitting for ¹³C NMR signals in Phases (II) and (III) indicated a ferroelastic characteristic of the compound. In addition, FT-IR results indicate that the ordered conformational structure of aromatic cations undergoes a remarkable disorder with increasing temperature. The NMR and FT-IR studies suggest that the phase transition mechanisms are related to the reorientational motion of [C₇H₁₂N₂]²⁺ cations as a whole. Phase transition was examined in light of the interesting optical properties of this material.