Equivalence of laws and null controllability for SPDEs driven by a fractional Brownian motion

We obtain necessary and sufficient conditions for equivalence of law for linear stochastic evolution equations driven by a general Gaussian noise by identifying the suitable space of controls for the corresponding deterministic control problem. This result is applied to semilinear (reaction-diffusion) equations driven by a fractional Brownian motion. We establish the equivalence of continuous dependence of laws of solutions to semilinear equations on the initial datum in the topology of pointwise convergence of measures and null controllability for the corresponding deterministic control problem.     

Impact of metal cation complexation and protonation on tautomeric and resonance forms of the oxaalkyl Schiff bases derived from 5-substituted salicylaldehyde and 2-hydroxy-1-naphthlaldehyde

Structures of the main resonance and tautomeric forms of three Schiff bases of 2-hydroxy-1-naphthaldehyde (1OXA), 5-bromosalicylaldehyde (2OXA) and 5-nitrosalicylaldehyde (3OXA) with 1,8-diamine-3,6-dioxaoctane, before and after protonation and complexation of monovalent metal cations, have been studied by FTIR, ¹H, ¹³C and ¹⁵N NMR methods. The spectroscopic investigations provided clear evidence that the Schiff bases exist in acetonitrile solution as three different tautomers: 1OXA in enamine-oxo, 2OXA in imine-hydroxy and 3OXA in a low-barrier O···H···N imine-oxo forms. It was found on the basis of multinuclear NMR studies, that in solid state, the enamine-oxo and imine-hydroxy tautomers are formed exclusively, but not the untypical imine-oxo tautomer, which requires strong stabilisation by solvent molecules in solution. MOG-PM6 calculations of the different tautomers allowed visualisation of their energetically the most favourable structures. Protonation of 1OXA–3OXA Schiff bases leads to formation of common forms, i.e. protonated imine-hydroxy structure, irrespectively of the structure of tautomer before protonation. In turn, coordination of monovalent metal cations implies common formation of zwitterionic forms within all studied ligands in solution. Application of FT-IR and NMR titrations in combination with ESI MS spectrometry revealed the nature and the structure of OXA complexes, whose formation is impeded by the intra- and intermolecular interactions. MOG-PM6 calculations allowed visualisation of Li⁺ and Na⁺ metal coordination sphere geometries within structure of all investigated complexes, stabilised by intermolecular interactions with solvent molecules.     

N-Alkylhydrazones of aliphatic ketones in the synthesis of 1,3,4-trisubstituted non-symmetric pyrazoles

Reactions of N-alkylhydrazones of aliphatic ketones with the Vilsmeier–Haack reagent (DMF–POCl₃) were evaluated as a promising approach toward the synthesis of trisubstituted non-symmetric pyrazoles. It was found that either 1,3,4-trialkylpyrazoles or 1,3-dialkylpyrazole-4-carbaldehydes could be obtained in these transformations in high yields (72–83%), in a regioselective manner.     

Simulated pressure response of crystalline indole

The isostatic pressure response of crystalline indole up to 25 GPa was investigated through static geometry optimization using Tkatchenko-Scheffler dispersion-corrected density functional theory method. Different symmetries were identified in the structural evolution with increased pressure, but no motif transition was observed, owing to the stability of the herringbone (HB) motif for small polycyclic aromatic hydrocarbons. Hirshfeld surface analysis determined that there was an increase in the fraction of H···π and π···π contacts within the high pressure structures, while the fraction of H···H contacts was lowered via geometric rearrangements. It was found that isostatic pressure alone, up to 25 GPa, was not sufficient to induce a chemical reaction due to the poor π-orbital overlap existing within the HB motif. However, the applied pressure sets the stage for an activated chemical reaction when the molecules approach each other along the long molecular axis, with a reaction energy and reaction barrier of 1.05 eV and 1.80 eV per molecular unit, respectively.     

Using Student Reasoning to Inform the Development of Conceptual Learning Goals: The Case of Quadratic Functions

Despite the proliferation of mathematics standards internationally and despite general agreement on the importance of teaching for conceptual understanding, conceptual learning goals for many K-12 mathematics topics have not been well-articulated. This article presents a coherent set of five conceptual learning goals for a complex mathematical domain, generated via a method of systematic empirical analysis of students' reasoning. Specifically, we compared the reasoning of pairs of students who performed differentially on the same task and inferred the pivotal intermediate conceptions that afforded one student deeper engagement with the task than another student. In turn, each pivotal intermediate conception framed an associated conceptual learning goal. While the empirical analysis of student reasoning is typically used to understand how students learn, we argue that such analysis should also play an important role in determining what concepts students should learn.