Computing the Loewner Driving Process of Random Curves in the Half Plane

We simulate several models of random curves in the half plane and numerically compute the stochastic driving processes that produce the curves through the Loewner equation. Our models include models whose scaling limit is the Schramm-Loewner evolution (SLE) and models for which it is not. We study several tests of whether the driving process is Brownian motion, as it is for SLE. We find that testing only the normality of the process at a fixed time is not effective at determining if the random curves are an SLE. Tests that involve the independence of the increments of Brownian motion are much more effective. We also study the zipper algorithm for numerically computing the driving function of a simple curve. We give an implementation of this algorithm which runs in a time O(N ^1.35) rather than the usual O(N ²), where N is the number of points on the curve.     

Experimental analysis of phase conjugation properties of four-wave mixing in an SOA after probe broadening due to fibre dispersion

Experimental results are presented which analyze the phase conjugation properties of four-wave mixing signals generated due to the beating between probe pulses broadened by a length of fibre and narrow pump pulses. This results in four-wave mixing pulses significantly narrower than the injected probe pulses albeit with reduced phase conjugation properties, which are examined. The pulses are completely characterized using the second-harmonic generation frequency resolved optical gating technique. The probe pulse is initially broadened due to propagation through 40 m of dispersion compensating fibre. This causes the probe pulse to be much wider than the injected pump pulse, in contrast to previously reported results. The four-wave mixing signal is therefore both wavelength converted and compressed, due to the gating properties of four-wave mixing, with respect to the initial probe signal. The phase conjugation properties of the converted signal are discussed and this signal is then passed through a second length of dispersion compensating fibre in order to further compress the pulse and to examine in more detail the phase conjugation properties.     

Synthesis,Structure, and DFT Analysis of the THF Solvate of 2-Picolyllithium: A 2-Picolyllithium Solvate with Significant Carbanionic Character

Previous studies of different solvates of 2-methylpyridyllithium (2-picolyllithium) have uncovered electronic structures corresponding to aza-allyl and enamido resonance forms of the metallated pyridine-based compounds. Here, we report the synthesis and characterization of [2-CH₂Li(THF)₂C₅H₄N], a new THF solvate. X-ray crystallographic studies reveal a dimeric arrangement featuring a non-planar eight-membered [NCCLi]₂ ring, in which the primary cation-anion interaction is between the central Li atom and the C atom of the deprotonated methyl group [length, 2.285(2) Å], suggesting a new carbanionic resonance structure for this 2-picolyllithium series. The significant carbanionic character of [2-CH₂Li(THF)₂C₅H₄N] was confirmed by gas-phase DFT calculations [B3LYP/6-311+G(d)] with the calculated electron density interrogated by means of quantum theory of atoms in molecules (QTAIM) and natural bond orbital (NBO) analyses. For comparison these computational analyses were also performed on the literature structures of [2-CH₂Li(2-Picoline)C₅H₄N] and [2-CH₂Li(PMDETA)C₅H₄N]. In a reactivity study, [2-CH₂Li(THF)₂C₅H₄N] was found to undergo nucleophilic addition to pyridine to generate dipyridylmethane in a good yield.     

Boosting Conjugate Addition to Nitroolefins Using Lithium Tetraorganozincates: Synthetic Strategies and Structural Insights

We report the first transition metal catalyst- and ligand-free conjugate addition of lithium tetraorganozincates (R₄ZnLi₂) to nitroolefins. Displaying enhanced nucleophilicity combined with unique chemoselectivity and functional group tolerance, homoleptic aliphatic and aromatic R₄ZnLi₂ provide access to valuable nitroalkanes in up to 98 % yield under mild conditions (0 °C) and short reaction time (30 min). This is particularly remarkable when employing β-nitroacrylates and β-nitroenones, where despite the presence of other electrophilic groups, selective 1,4 addition to the C=C is preferred. Structural and spectroscopic studies confirmed the formation of tetraorganozincate species in solution, the nature of which has been a long debated issue, and allowed to unveil the key role played by donor additives on the aggregation and structure of these reagents. Thus, while chelating N,N,N’,N’-tetramethylethylenediamine (TMEDA) and (R,R)-N,N,N’,N’-tetramethyl-1,2-diaminocyclohexane (TMCDA) favour the formation of contacted-ion pair zincates, macrocyclic Lewis donor 12-crown-4 triggers an immediate disproportionation process of Et₄ZnLi₂ into equimolar amounts of solvent-separated Et₃ZnLi and EtLi.     

Visible-Light Photoswitching by Azobenzazoles

Three visible-light responsive photoswitches are reported, azobis(1-methyl-benzimidazole) ( 1 ), azobis(benzoxazole) ( 2 ) and azobis(benzothiazole) ( 3 ). Photostationary distributions are obtained upon irradiation with visible light comprising approximately 80 % of the thermally unstable isomer, with thermal half-lives up to 8 min and are mostly invariant to solvent. On protonation, compound 1 H⁺ has absorption extending beyond 600 nm, allowing switching with yellow light, and a thermal half-life just under 5 minutes. The two isomers have significantly different pK_a values, offering potential as a pH switch. The absorption spectra of 2 and 3 are insensitive to acid, although changes in the thermal half-life of 3 indicate more basic intermediates that significantly influence the thermal barrier to isomerization. These findings are supported by high-level ab initio calculations, which validate that protonation occurs on the ring nitrogen and that the Z isomer is more basic in all cases.     

The Effects of a Mathematics Infusion Curriculum on Middle School Student Mathematics Achievement

Increasing mathematical competencies of American students has been a focus for educators, researchers, and policy makers alike. One purported approach to increase student learning is through connecting mathematics and science curricula. Yet there is a lack of research examining the impact of making these connections. The Mathematics Infusion into Science Project, funded by the National Science Foundation, developed a middle school mathematics‐infused science curriculum. Twenty teachers utilized this curriculum with over 1,200 students. The current research evaluated the effects of this curriculum on students' mathematics learning and compared effects to students who did not receive the curriculum. Students who were taught the infusion curriculum showed a significant increase in mathematical content scores when compared with the control students.     

Comparison Between the Mean-Variance Optimal and the Mean-Quadratic-Variation Optimal Trading Strategies

ABSTRACT

We compare optimal liquidation policies in continuous time in the presence of trading impact using numerical solutions of Hamilton–Jacobi–Bellman (HJB) partial differential equations (PDEs). In particular, we compare the time-consistent mean-quadratic-variation strategy with the time-inconsistent (pre-commitment) mean-variance strategy. We show that the two different risk measures lead to very different strategies and liquidation profiles. In terms of the optimal trading velocities, the mean-quadratic-variation strategy is much less sensitive to changes in asset price and varies more smoothly. In terms of the liquidation profiles, the mean-variance strategy is much more variable, although the mean liquidation profiles for the two strategies are surprisingly similar. On a numerical note, we show that using an interpolation scheme along a parametric curve in conjunction with the semi-Lagrangian method results in significantly better accuracy than standard axis-aligned linear interpolation. We also demonstrate how a scaled computational grid can improve solution accuracy.     

Heavy Alkali Metal Manganate Complexes: Synthesis,Structures and Solvent-Induced Dissociation Effects

Rare examples of heavier alkali metal manganates [{(AM)Mn(CH₂SiMe₃)(N^‘Ar)₂}_∞] (AM=K, Rb, or Cs) [N^‘Ar=N(SiMe₃)(Dipp), where Dipp=2,6-iPr₂-C₆H₃] have been synthesised with the Rb and Cs examples crystallographically characterised. These heaviest manganates crystallise as polymeric zig-zag chains propagated by AM⋅⋅⋅π-arene interactions. Key to their preparation is to avoid Lewis base donor solvents. In contrast, using multidentate nitrogen donors encourages ligand scrambling leading to redistribution of these bimetallic manganate compounds into their corresponding homometallic species as witnessed for the complete Li - Cs series. Adding to the few known crystallographically characterised unsolvated and solvated rubidium and caesium s-block metal amides, six new derivatives ([{AM(N^‘Ar)}_∞], [{AM(N^‘Ar)⋅TMEDA}_∞], and [{AM(N^‘Ar)⋅PMDETA}_∞] where AM=Rb or Cs) have been structurally authenticated. Utilising monodentate diethyl ether as a donor, it was also possible to isolate and crystallographically characterise sodium manganate [(Et₂O)₂Na(ⁿBu)Mn[(N^‘Ar)₂], a monomeric, dinuclear structure prevented from aggregating by two blocking ether ligands bound to sodium.