Calibration of piezoelectric positioning actuators using a reference voltage-to-displacement transducer based on quartz tuning forks

We use a piezoelectric quartz tuning fork to calibrate the displacement of ceramic piezoelectric scanners that are widely employed in scanning probe microscopy. We measure the static piezoelectric response of a quartz tuning fork and find it to be highly linear, nonhysteretic and with negligible creep. These performance characteristics, close to those of an ideal transducer, make quartz transducers superior to ceramic piezoelectric actuators. Furthermore, quartz actuators in the form of a tuning fork have the advantage of yielding static displacements comparable to those of local probe microscope scanners. We use the static displacement of a quartz tuning fork as a reference to calibrate the three axis displacement of a ceramic piezoelectric scanner. Although this calibration technique is a nontraceable method, it can be more versatile than using calibration grids because it enables characterization of the linear and nonlinear response of a piezoelectric scanner in a broad range of displacements, spanning from a fraction of a nanometer to hundreds of nanometers. In addition, the creep and the speed dependent piezoelectric response of ceramic scanners can be studied in detail.     

Analyzing the electric response of molecular conductors using “electron deformation” orbitals and occupied‐virtual electron transfer

The concept of “electron deformation orbitals” (EDOs) is used to investigate the electric response of conducting metals and oligophenyl chains. These orbitals and their eigenvalues are obtained by diagonalization of the deformation density matrix (difference between the density matrices of the perturbed and unperturbed systems) and can be constructed as linear combinations of the unperturbed molecular orbitals within “frozen geometry” conditions. This form of the EDOs allows calculating the part of the electron deformation density associated to an effective electron transfer from occupied to virtual orbitals (valence to conduction band electron transfer in the band model of conductivity). It is found that the “electron deformation” orbitals pair off, displaying the same eigenvalue but opposite sign. Each pair represents an amount of accumulation/depletion of electron charge at different molecular regions. In the oligophenyl systems investigated only one pair contributes effectively to the charge flow between molecular ends, resulting from the promotion of electrons from occupied orbitals to close in energy virtual orbitals of appropriate symmetry and overlapping. Analysis of this pair along explains the differences in conductance of olygophenyl chains based on phenyl units. © 2014 Wiley Periodicals, Inc.     

The Reaction of the Bis-spirocyclic Phosphazene [N 3 P 3 (O 2 C 12 H 8 ) 2 Cl 2 ] (O 2 C 12 H 8 = 2,2′-Dioxybiphenyl) with Thiophenols,in the Presence of Alkali Carbonates

The reactions of the spirocyclic phosphazene [N 3 P 3 (O 2 C 12 H 8 ) 2 Cl 2 ] (O 2 C 12 H 8 = 2,2'-dioxybiphenyl) with the thiophenols HS--C 6 H 4 --R and M 2 CO 3 (M = K or Cs) in refluxing acetone gave respectively the spirocyclic substituted derivatives [N 3 P 3 (O 2 C 12 H 8 ) 2 (SC 6 H 4 --R) 2 ] R = H ( 2a ), Br ( 2b ), OMe ( 2c ), NO 2 ( 2d ). The reaction is a two-step process the second of which is much faster than the first and the monosubstituted intermediate [N 3 P 3 (O 2 C 12 H 8 ) 2 (SC 6 H 4 --R)Cl] cannot be detected. By contrast, in the analogous reactions with the phenols HO--C 6 H 4 --R and M 2 CO 3 (M = K or Cs) in acetone or THF, to give the known derivatives [N 3 P 3 (O 2 C 12 H 8 ) 2 (OC 6 H 4 --R) 2 ], the first step is faster although both are very dependent on R, M and the solvent. Thus, in the case of the phenol HO--C 6 H 4 --OMe the reaction conditions could be adjusted to give the useful synthetic intermediate monosubstituted derivative [N 3 P 3 (O 2 C 12 H 8 ) 2 (OC 6 H 4 --OMe)Cl] ( 3 ). The reaction of [N 3 P 3 (O 2 C 12 H 8 ) 2 Cl 2 ] with the bifunctional reagent mercaptophenol HS--C 6 H 4 --OH was not specific and led to mixtures of cyclic and oligomeric products.     

On the massive sine-Gordon equation in the first few regions of collapse

We study the ultraviolet stability problem for the two-dimensional Yukawa interaction :cos:d in the region , where . The results have a natural Coulomb gas interpretation, because the counterterms do not depend on the field.Partially supported by C.N.R.S. trough the University of Paris VI     

Exploring Mathematics in Imaginative Places: Rethinking What Counts as Meaningful Contexts for Learning Mathematics

This paper explores what happens when students engage with mathematical tasks that make no attempt to be connected with students' everyday life experiences. The investigation draws on the work of educators who call for a broader view of what might count as real and relevant contexts for studying mathematics. It investigates students' experiences with two imaginative tasks and reports on the students' intellectual and emotional engagement. This engagement is examined and described in terms of the character and quality of the class and group discussions generated. Findings suggest that students can indeed engage productively with mathematics when it is explored in imaginative settings and that such contexts can help students support and sustain their engagement with the mathematics in the task.     

Fluorinated heterocyclic compounds. An expedient route to 5-perfluoroalkyl-1,2,4-triazoles via an unusual hydrazinolysis of 5-perfluoroalkyl-1,2,4-oxadiazoles: first examples of an ANRORC-like reaction in 1,2,4-oxadiazole derivatives

The hydrazinolysis reaction of 5-perfluoroalkyl-1,2,4-oxadiazoles has been investigated. Nucleophilic addition of the reagent to the C(5)-N(4) double bond of the oxadiazole ring, followed by ring-opening and then ring-closure involving the beta-nitrogen atom of the hydrazino moiety and the C(3) of the oxadiazole ring, explains the formation of 5-perfluoroalkyl-1,2,4-triazoles as final products. Useful applications in synthesis of this uncommon hydrazinolysis can be claimed.     

[Gd(AAZTA)]− Derivatives with n-Alkyl Acid Side Chains Show Improved Properties for Their Application as MRI Contrast Agents**

Herein, the synthesis and an extensive characterization of two novel Gd(AAZTA) (AAZTA=6-amino-6-methylperhydro-1,4-diazepine tetra acetic acid) derivatives functionalized with short (C₂ and C₄) n-alkyl acid functions are reported. The carboxylate functionality is the site for further conjugations for the design of more specific contrast agents (CAs). Interestingly, it has been found that the synthesized complexes display enhanced properties for use as MRI contrast agents on their own. The stability constants determined by using potentiometric titration and UV/Vis spectrophotometry were slightly higher than the one reported for the parent Gd(AAZTA) complex. This observation might be accounted for by the larger sigma-electron donation of the acyl substituents with respect to the one provided by the methyl group in the parent complex. As far as concerns the kinetic stability, transmetallation experiments with endogenous ions (e.g. Cu²⁺) implied that the Gd³⁺ ions present in these Gd(AAZTA) derivatives show somewhat smaller susceptibility to chemical exchange towards these ions at 25 °C, close to the physiological condition. The ¹H NMR spectra of the complexes with Eu^III and Yb^III displayed a set of signals consistent with half the number of methylene protons present on each ligand. The number of resonances was invariant over a large range of temperatures, suggesting the occurrence of a fast interconversion between structural isomers. The relaxivity values (298 K, 20 MHz) were consistent with q=2 being equal to 8.8 mm ⁻¹ s⁻¹ for the C₂ derivative and 9.4 mm ⁻¹ s⁻¹ for the C₄ one, that is, sensibly larger than the one reported for Gd(AAZTA) (7.1 mm ⁻¹ s⁻¹). Variable-temperature (VT)-T₂ ¹⁷O NMR measurements showed, for both complexes, the presence of two populations of coordinated water molecules, one in fast and one in slow exchange with the bulk water. As the high-resolution ¹H NMR spectra of the analogs with Eu^III and Yb^III did not show the occurrence of distinct isomers (as frequently observed in other macrocyclic lanthanide(III)-containing complexes), we surmised the presence of two fast-interconverting isomers in solution. The analysis of the ¹⁷O NMR VT-T₂ profiles versus temperature allowed their relative molar fraction to be established as 35 % for the isomer with the fast exchanging water and 65 % for the isomer with the water molecules in slower exchange. Finally, ¹H NMRD profiles over an extended range of applied magnetic field strengths have been satisfactory fitted on the basis of the occurrence of the two interconverting species.     

Finite Dimensional Hopf Algebras Over the Dual Group Algebra of the Symmetric Group in Three Letters

We classify finite-dimensional Hopf algebras whose coradical is isomorphic to the algebra of functions on 𝕊₃. We describe a new infinite family of Hopf algebras of dimension 72.     

Planar Point Sets With Large Minimum Convex Decompositions

We show the existence of sets with $n$ points ( $n\ge 4$ ) for which every convex decomposition contains more than $\frac{35}{32}n-\frac{3}{2}$ polygons, which refutes the conjecture that for every set of $n$ points there is a convex decomposition with at most $n+C$ polygons. For sets having exactly three extreme points we show that more than $n+\sqrt{2(n-3)}-4$ polygons may be necessary to form a convex decomposition.     

Challenging Preservice Teachers' Mathematical Understanding: The Case of Division by Zero

Preservice elementary school teachers' fragmented understanding of mathematics is widely documented in the research literature. Their understanding of division by 0 is no exception. This article reports on two teacher education tasks and experiences designed to challenge and extend preservice teachers' understanding of division by 0. These tasks asked preservice teachers to investigate division by 0 in the context of responding to students' erroneous mathematical ideas and were respectively structured so that the question was investigated through discussion with peers and through independent investigation. Results revealed that preservice teachers gained new mathematical (what the answer is and why it is so) and pedagogical (how they might explain it to students) insights through both experiences. However, the quality of these insights were related to the participants' disposition to justify their thinking and (or) to investigate mathematics they did not understand. The study's results highlight the value of using teacher learning tasks that situate mathematical inquiry in teaching practice but also highlight the challenge for teacher educators to design experiences that help preservice teachers see the importance of, and develop the tools and inclination for, mathematical inquiry that is needed for teaching mathematics with understanding.