The Littlewood-Gowers problem

The paper has two main parts. To begin with, suppose that G is a compact abelian group. Chang’s Theorem can be viewed as a structural refinement of Bessel’s inequality for functions ƒ ∈ L ²(G). We prove an analogous result for functions ƒ ∈ A(G), where A(G) is the space endowed with the norm , and generalize this to the approximate Fourier transform on Bohr sets. As an application of the first part of the paper, we improve a recent result of Green and Konyagin. Suppose that p is a prime number and A ⊂ ℤ/pℤ has density bounded away from 0 and 1 by an absolute constant. Green and Konyagin have shown that ‖χ _A ‖ _A(ℤ/pℤ) ≫ ɛ (log p)^1/3−ɛ; we improve this to ‖χ _A ‖ _A(ℤ/pℤ) ≫ ɛ (log p)^1/2−ɛ. To put this in context, it is easy to see that if A is an arithmetic progression, then ‖χ _A ‖ _A(ℤ/pℤ) ≪ log p.     

On spanning trees and walks of low maximum degree

This article uses the discharging method to obtain the best possible results that a 3‐connected graph embeddable on a surface of Euler characteristic χ ≤ −46 has a spanning tree of maximum degree at most and a closed, spanning walk meetting each vertex at most times. Each of these results is shown to be best possible. © 2001 John Wiley & Sons, Inc. J Graph Theory 36: 67–74, 2001     

Metalloporphyrin Dendrimers with Folding Arms

Rigid and flexible linkers are combined in the new dendrimer shown schematically in the picture, which contains nine metalloporphyrin units (Porph). The construction is such that the four “arms” of the dendrimer can fold in a cooperative and predetermined manner in response to added 1,4-diazabicyclo[2.2.2]octane (DABCO).     

Solid-state NMR, crystallographic, and computational investigation of bisphosphonates and farnesyl diphosphate synthase-bisphosphonate complexes

Bisphosphonates are a class of molecules in widespread use in treating bone resorption diseases and are also of interest as immunomodulators and anti-infectives. They function by inhibiting the enzyme farnesyl diphosphate synthase (FPPS), but the details of how these molecules bind are not fully understood. Here, we report the results of a solid-state (13)C, (15)N, and (31)P magic-angle sample spinning (MAS) NMR and quantum chemical investigation of several bisphosphonates, both as pure compounds and when bound to FPPS, to provide information about side-chain and phosphonate backbone protonation states when bound to the enzyme. We then used computational docking methods (with the charges assigned by NMR) to predict how several bisphosphonates bind to FPPS. Finally, we used X-ray crystallography to determine the structures of two potent bisphosphonate inhibitors, finding good agreement with the computational results, opening up the possibility of using the combination of NMR, quantum chemistry and molecular docking to facilitate the design of other, novel prenytransferase inhibitors.     

Efficient and mild microwave-assisted stepwise functionalization of naphthalenediimide with alpha-amino acids

Microwave dielectric heating proved to be an efficient method for the one-pot and stepwise syntheses of symmetrical and unsymmetrical naphthalenediimide derivatives of alpha-amino acids. Acid-labile side chain protecting groups are stable under the reaction conditions; protection of the alpha-carboxylic group is not required. The stepwise condensation of different amino acids resulted in high yields of unsymmetrical naphthalenediimides. The reaction proceeds without racemization and is essentially quantitative.