Moving toward shared realities through empathy in mathematical modeling: An ecological systems theory approach

The mathematics education community has routinely called for mathematics tasks to be connected to the real world. However, accomplishing this in ways that are relevant to students’ lived experiences can be challenging. Meanwhile, mathematical modeling has gained traction as a way for students to learn mathematics through real-world connections. In an open problem to the mathematics education community, this paper explores connections between the mathematical modeling and the nature of what is considered relevant to students. The role of empathy is discussed as a proposed component for consideration within mathematical modeling so that students can further relate to real-world contexts as examined through the lens of Ecological Systems Theory. This is contextualized through a classroom-tested example entitled “Tiny Homes as a Solution to Homelessness” followed by implications and conclusions as they relate to mathematics education.     

Vibrational state-dependent predissociation dynamics of ClO (A (2)Pi(3/2)): Insight from correlated fine structure branching ratios

We have studied the v'-dependent predissociation dynamics of the ClO A (2)Pi(3/2) state using velocity-map ion-imaging. Experimental final correlated state branching ratios, i.e. Cl((2)P(J=3/2,1/2)) + O((3)P(J=2,1,0)) channels, have been measured for v' = 6-11. We find that the branching ratios are highly variable and depend strongly on v', providing a window into the v'-dependent predissociation mechanism. A comparison of the experimental results with the recent model of Lane et al. (I. C. Lane, W. H. Howie and A. J. Orr-Ewing, Phys. Chem. Chem. Phys., 1999, 1, 3087) in both the diabatic and adiabatic limits suggests that the dynamics are closer to the diabatic limit. The overall Cl((2)P(J)) branching ratios are in good agreement with the diabatic model results. There are significant differences, however, between theory and experiment at the correlated state level, demonstrating the sensitivity of correlated measurements to the role of the exit channel coupling in the predissociation dynamics. The results highlight the need for more sophisticated quantum dynamical calculations to describe the correlated fine structure branching ratios in this system.     

Dimeric phenalenyl-based neutral radical molecular conductors.

We report the preparation, crystallization, and solid-state characterization of ethyl (3)- and butyl (4)-substituted spiro-biphenalenyl radicals. Both of these compounds are found to be conducting face-to-face pi-dimers in the solid state but with different room-temperature magnetic ground states. At room temperature, 4 exists as a diamagnetic pi-dimer (interplanar separation of approximately 3.1 A), whereas 3 is a paramagnetic pi-dimer (interplanar separation of approximately 3.3 A), and both compounds show phase transitions between the paramagnetic and diamagnetic forms. Electrical resistivity measurements of single crystals of 3 and 4 show that the transition from the high-temperature paramagnetic pi-dimer form to the low-temperature diamagnetic pi-dimer structure is accompanied by an increase in conductivity by about 2 orders of magnitude. This behavior is unprecedented and is very difficult to reconcile with the usual understanding of a Peierls dimerization, which inevitably leads to an insulating ground state. We tentatively assign the enhancement in the conductivity to a decrease in the on-site Coulombic correlation energy (U), as the dimers form a super-molecule with twice the amount of conjugation.     

Anion receptors: a new class of amide/quaternized amine macrocycles and the chelate effect

A new class of tetraamide macrocyclic receptors for anions with two quaternized amine functionalities exhibited higher affinities for anions compared with the corresponding neutral amides. In two crystal structures of halide complexes of the prototypes with phenyl and pyridine spacers, the anions are held by hydrogen bonding with the amide hydrogens. The pyridine analogues display higher affinities in general than the phenyl systems, a phenomenon which is attributed to the anion version of the chelate effect.     

Novel iminium ion equivalents prepared through C-H oxidation for the stereocontrolled synthesis of functionalized propargylic amine derivatives

Access to stereochemically complex, polyfunctionalized amine derivatives is made possible using novel oxathiazinane N,O-acetal starting materials. These heterocycles are prepared through intramolecular sulfamate ester C-H insertion with a Rh(2+)-carboxylate catalyst and PhI(OAc)(2) as the terminal oxidant. Such compounds function as unique iminium ion equivalents to which nucleophilic alkynylzinc reagents add smoothly in the presence of BF(3).OEt(2). The coupled products are isolated in high yield (63-92%) and with good levels of diastereoinduction (6 --> 20:1). The alkyne-substituted oxathiazinanes serve as versatile building blocks and may be further manipulated through nucleophilic ring-opening reactions of the sulfamate core. The efficient construction of 1,7,8-trihydroxyindolizidine in six steps and in 34% overall yield highlights the power of these combined methods for synthesis.     

Influence of initial order on the microscopic mechanism of electric field induced alignment of block copolymer microdomains

We investigate the mechanism of microdomain orientation in concentrated block copolymer solutions exposed to a dc electric field by in situ synchrotron small-angle X-ray scattering (SAXS). As a model system, we use concentrated solutions of a lamellar polystyrene-b-polyisoprene block copolymer in toluene. We find that both the microscopic mechanism of reorientation and the kinetics of the process strongly depend on the initial degree of order in the system. In a highly ordered lamellar system with the lamellae being aligned perpendicular to the electric field vector, only nucleation and growth of domains is possible as a pathway to reorientation and the process proceeds rather slowly. In less ordered samples, grain rotation becomes possible as an alternative pathway, and the process proceeds considerably faster. The interpretation of our finding is strongly corroborated by dynamic self-consistent field simulations.     

Determination of biogenic amines in food by automated pre-column derivatization with 2-naphthyloxycarbonyl chloride (NOC-CI)

Summary The fluorogenic reagent 2-naphthyloxycarbonyl chloride (NOC-Cl) has been used for the automated precolumn derivatization of biogenic amines (BAs) at ambient followed by liquid-chromatographic separation of the derivatives formed. For optimized derivatization samples in 0.5 M borate buffer (pH 9.0) were derivatized with 5 mM NOC-Cl in acetonitrile (MeCN) for 3 minutes. Excess of reagent was scavenged by addition of 20 mM glycine in water. For HPLC a Superspher^? RP-18e column and gradient elution using 0.1 M sodium acetate buffer (pH 4.4) and MeCN were used. The NOC-derivatives were detected by fluorescence at an emission wavelength of 335 nm at an excitation wavelength of 274 nm. This method allows the detection of BAs (2-phenylethylamine, putrescine, histamine, cadaverine, tyramine, spermidine, spermine) found in food and beverages (fruit juices, wines, various vinegars, fermented cabbage juice, and salmon). Detection limit of BAs are approximately 49–113 μg kg⁻¹ with the exception of histamine (747 μg kg⁻¹) (injected amounts: 18–41 pg histamine 267 pg), at a signalto-noise ratio of 3:1. The limits of determination are approximately 82–189 μg kg⁻¹ (histamine 1245 μg kg⁻¹) at a signal-to-noise ratio of 5:1. The correlation coefficients of linearity are 0.9910–0.9976. Recoveries from different matrices range from 65 to 109%, depending on the sample investigated. Presented at the 21^st ISC held in Stuttgart, Germany, 15^th–20^th September, 1996.     

Enantioselective Michael addition to vinyl phosphonates via hydrogen bond-enhanced halogen bond catalysis

An asymmetric Michael addition of malononitrile to vinyl phosphonates was accomplished by hydrogen bond-enhanced bifunctional halogen bond (XB) catalysis. NMR titration experiments were used to demonstrate that halogen bonding, with the support of hydrogen-bonding, played a key role in the activation of the Michael acceptors through the phosphonate group. This is the first example of the use of XBs for the activation of organophosphorus compounds in synthesis. In addition, the iodo-perfluorophenyl group proved to be a better directing unit than different iodo- and nitro-substituted phenyl groups. The developed approach afforded products with up to excellent yields and diastereoselectivities and up to good enantioselectivities.

An asymmetric Michael addition of malononitrile to vinyl phosphonates was accomplished by hydrogen bond-enhanced bifunctional halogen bond (XB) catalysis.     

Time-integrated monitoring of polycyclic aromatic hydrocarbons (Pahs) in groundwater using the ceramic dosimeter passive sampling device

Passive sampling relies on the uptake of contaminants into appropriate sampling devices along a diffusion gradient without using pumps or bailers. Thus, for example, in groundwater sampling, changes to flow due to pumping can be avoided. If the diffusion gradient can be maintained for extended periods, contaminants can be sampled continuously over time without any action, allowing to determine time-weighted average contaminant concentrations. We here show that the Ceramic Dosimeter, a solid receiving phase passive sampler using a ceramic membrane as sorbent container and diffusion barrier, can be used without calibration for the long-term monitoring of polycyclic aromatic hydrocarbons (PAHs) in groundwater.     

Synthesis of the anti and syn isomers of thieno[f,f ']bis[1]benzothiophene. Comparison of the optical and electrochemical properties of the anti and syn isomers

We report isomer-pure synthesis of thieno[2,3-f:5,4-f ']bis[1]benzothiophene and thieno[3,2-f:4,5-f ']bis[1]benzothiophene, the anti and syn isomers of a pentacyclic compound consisting of alternating thiophene and benzene rings. The optical and electrochemical properties of both are reported. In the anti isomer, the ribbonlike embedding of three thiophene units leads to a near-planar molecule with favorable pi-pi stacking behavior in the solid state as shown by X-ray crystal structure analysis.