The Reactivity of 2‐Fluoro‐ and 2‐Chloropyridines toward Sodium Ethoxide: Factors Governing the Rates of Nucleophilic (Het)Aromatic Substitutions

The relative displacement rates of the halide substituent from 2‐fluoro‐ and 2‐chloropyridines by EtONa in EtOH at +25° were assessed by competition kinetics. The 2‐fluoropyridine reacts 320 times faster than the chloro analog. A CF₃ group increases the reactivity more than single halogen atoms do, whatever the element, and the latter are superior to Me₃Si groups. Substituents accommodated at the 4‐position operate through their inductive effect, whereas at the 3‐position, this action may be attenuated by steric hindrance. Almost all 5‐substituents enhance the rate of the nucleophilic substitution occurring at the 2‐position. The sole exception concerns the F‐atom at the 5‐position which retards the reaction, presumably by lone‐pair/lone‐pair repulsion with the negative charge building up at the central C‐atom of the intermediate Meisenheimer complex. The substituent effects are additive. Therefore, by using the increments derived from the present work, the rates of future reactions should be predictable with fair accuracy.     

Classification of edges using compactly supported shearlets

We analyze the detection and classification of singularities of functions $f={\chi }_S$, where $S?{\mathbb{R}}^d$ and $d=2,3$. It will be shown how the set ?S can be extracted by a continuous shearlet transform associated with compactly supported shearlets. Furthermore, if ?S is a $d?1$ dimensional piecewise smooth manifold with $d=2$ or 3, we will classify smooth and non-smooth components of ?S. This improves previous results given for shearlet systems with a certain band-limited generator, since the estimates we derive are uniform. Moreover, we will show that our bounds are optimal. Along the way, we also obtain novel results on the characterization of wavefront sets in 3 dimensions by compactly supported shearlets. Finally, geometric properties of ?S such as curvature are described in terms of the continuous shearlet transform of f.     

Compactly supported shearlets are optimally sparse

Cartoon-like images, i.e., C² functions which are smooth apart from a C² discontinuity curve, have by now become a standard model for measuring sparse (nonlinear) approximation properties of directional representation systems. It was already shown that curvelets, contourlets, as well as shearlets do exhibit sparse approximations within this model, which are optimal up to a log-factor. However, all those results are only applicable to band-limited generators, whereas, in particular, spatially compactly supported generators are of uttermost importance for applications.In this paper, we present the first complete proof of optimally sparse approximations of cartoon-like images by using a particular class of directional representation systems, which indeed consists of compactly supported elements. This class will be chosen as a subset of (non-tight) shearlet frames with shearlet generators having compact support and satisfying some weak directional vanishing moment conditions.     

A new identity for Parseval frames

In this paper we establish a surprising new identity for Parseval frames in a Hilbert space. Several variations of this result are given, including an extension to general frames. Finally, we discuss the derived results.

     

Inversion of the Pieri formula for Macdonald polynomials

We give the explicit analytic development of Macdonald polynomials in terms of “modified complete” and elementary symmetric functions. These expansions are obtained by inverting the Pieri formula. Specialization yields similar developments for monomial, Jack and Hall-Littlewood symmetric functions.     

A nonterminating 8φ7 summation for the root system Cr

Using multiple q-integrals and a determinant evaluation, we establish a nonterminating ₈φ₇ summation for the root system C_r. We also give some important specializations explicitly.     

Hydroxid-bewirkte Halogenwasserstoff-Abspaltungen aus vic-Dihalogeniden unter heterogenen Bedingungen: Reaktionssteuerung dank überraschender Solvens- und Metall-Einflüsse

Hydroxide-Promoted Dehydrohalogenation of vic-Dihalides under Heterogeneous Conditions: Control Over the Reaction Outcome Due to Surprising Solvent and Metal Effects Selective elimination of either one or two molecules of hydrogen halide from vic-dihalides can be conveniently achieved by employing solid sodium or potassium hydroxide in the presence of solvents such as glycol dimethylether or tetraethyleneglycol dimethylether.