Fourier analysis of the approximation power of principal shift-invariant spaces

The approximation order provided by a directed set {S _h } _h>0 of spaces, each spanned by thehZ ^d -translates of one function, is analyzed. The nearoptimal approximants of [R2] from eachs _h to the exponential functions are used to establish upper bounds on the approximation order. These approximants are also used on the Fourier transform domain to yield approximations for other smooth functions, and thereby provide lower bounds on the approximation order. As a special case, the classical Strang-Fix conditions are extended to bounded summable generating functions.The second part of the paper consists of a detailed account of various applications of these general results to spline and radial function theory. Emphasis is given to the case when the scale {s _h } is obtained froms ₁ by means other than dilation. This includes the derivation of spectral approximation orders associated with smooth positive definite generating functions.     

Long wave theory of absorption and reflection in molecular crystals

The theory of long waves originally developed by Born and Huang for ionic crystals and then adapted for molecular crystals with one molecule per unit cell by Hall is extended to include molecular crystals with two molecules per unit cell. A study is made of the non-analytic behaviour of the energy bands as the wave vector tends to zero and it is found that the energies of the optically excited levels will be different from the values usually assumed. The dispersion relation is obtained and used to discuss metallic reflectivity. The finite lifetime of the excited states can also be included in the theory and this has the effect of changing the position of maximum absorption as well as reducing the reflectivity. A comparison is made between the long wave theory, the simple form of Davydov theory and experiment.     

Type II Anion Relay Chemistry: Exploiting Bifunctional Weinreb Amide Linchpins for the One‐Pot Synthesis of Differentiated 1,3‐Diketones,Pyrans, and Spiroketals

The design, synthesis, and validation of new highly effective bifunctional linchpins for type II anion relay chemistry (ARC) has been achieved. The mechanistically novel negative‐charge migration that comprises the Brook rearrangement is now initiated by a stabilized tetrahedral intermediate, which is generated by nucleophilic addition to a Weinreb amide, rather than by a simple oxyanion that is generated from an epoxide. As a result, the linchpin preserves the carbonyl functionality in the ARC adducts, thus permitting access to functionally complex systems in a single flask without the need for further chemical manipulations. This tactic was validated with the one‐pot preparation of monoprotected 1,3‐diketones as well as pyran and spiroketal scaffolds, depending on the choice of nucleophile, electrophile, and work‐up conditions.     

Developing Eco-Friendly and Cost-Effective Porous Adsorbent for Carbon Dioxide Capture

To address the issue of global warming and climate change issues, recent research efforts have highlighted opportunities for capturing and electrochemically converting carbon dioxide (CO₂). Despite metal doped polymers receiving widespread attention in this respect, the structures hitherto reported lack in ease of synthesis with scale up feasibility. In this study, a series of mesoporous metal-doped polymers (MRFs) with tunable metal functionality and hierarchical porosity were successfully synthesized using a one-step copolymerization of resorcinol and formaldehyde with Polyethyleneimine (PEI) under solvothermal conditions. The effect of PEI and metal doping concentrations were observed on physical properties and adsorption results. The results confirmed the role of PEI on the mesoporosity of the polymer networks and high surface area in addition to enhanced CO₂ capture capacity. The resulting Cobalt doped material shows excellent thermal stability and promising CO₂ capture performance, with equilibrium adsorption of 2.3 mmol CO₂/g at 0 °C and 1 bar for at a surface area 675.62 m²/g. This mesoporous polymer, with its ease of synthesis is a promising candidate for promising for CO₂ capture and possible subsequent electrochemical conversion.     

Facile Synthesis of Phosphaamidines and Phosphaamidinates using Nitrilium Ions as an Imine Synthon

Readily accessible nitrilium triflates are convenient imine building blocks for the expedient synthesis of a novel class of 1,3‐P,N ligands as demonstrated for the reaction with primary phosphanes. This procedure allows variation of all substituents. X‐ray crystal structures are reported for nitrilium ions, phosphaamidines, and three phosphaamidinate complexes. The lithium phosphaamidinate is N coordinated and its reaction with [AuCl(tht)] (tht=tetrahydrothiophene) gives a unique P‐bridged gold trimer, while a P,N‐bidentate complex results from [{RhCl(cod)}₂]. The nitrilium ion methodology allows extension of the 1,3‐P,N motive to bis(imino)phosphanes, which are the neutral phosphorus analogues of the valuable β‐diketiminate ligand.     

Density functionals with asymptotic-potential corrections are required for the simulation of spectroscopic properties of materials

Five effects of correction of the asymptotic potential error in density functionals are identified that significantly improve calculated properties of molecular excited states involving charge-transfer character. Newly developed materials-science computational methods are used to demonstrate how these effects manifest in materials spectroscopy. Connection is made considering chlorophyll-a as a paradigm for molecular spectroscopy, 22 iconic materials as paradigms for 3D materials spectroscopy, and the V_N⁻ defect in hexagonal boron nitride as an example of the spectroscopy of defects in 2D materials pertaining to nanophotonics. Defects can equally be thought of as being “molecular” and “materials” in nature and hence bridge the relms of molecular and materials spectroscopies. It is concluded that the density functional HSE06, currently considered as the standard for accurate calculations of materials spectroscopy, should be replaced, in most instances, by the computationally similar but asymptotically corrected CAM-B3LYP functional, with some specific functionals for materials-use only providing further improvements.

Spectroscopic transitions in materials that involve charge transfer require asymptotically corrected density functionals. As most transitions do have some charge transfer character, use of such methods are generally warranted.     

Syntheses of (-)-oleocanthal, a natural NSAID found in extra virgin olive oil, the (-)-deacetoxy-oleuropein aglycone, and related analogues

Phenolic compounds extracted from extra virgin olive oil have attracted considerable recent attention. One of the components, (-)-oleocanthal (1), an inhibitor of the COX-1 and COX-2 enzymes, possesses similar potency as the NSAID ibuprofen. In this, a full account, we disclose the first- and now second-generation syntheses of both enantiomers of the oleocanthals, as well as the first synthesis of the closely related (-)-deacetoxy-oleuropein aglycone and a series of related analogues for structure activity studies. To demonstrate the utility of the second-generation synthesis, multigram quantities of (-)-oleocanthal were prepared in 10 steps (14% overall yield) from commercially available D-lyxose.