Structure Results for Multiple Tilings in 3D

We study multiple tilings of 3-dimensional Euclidean space by a convex body. In a multiple tiling, a convex body $P$ is translated with a discrete multiset $\Lambda $ in such a way that each point of ${\mathbb {R}}^d$ gets covered exactly $k$ times, except perhaps the translated copies of the boundary of $P$ . It is known that all possible multiple tilers in ${\mathbb {R}}^3$ are zonotopes. In ${\mathbb {R}}^2$ it was known by the work of Kolountzakis (Discrete Comput Geom 23(4):537–553, 2000) that, unless $P$ is a parallelogram, the multiset of translation vectors $\Lambda $ must be a finite union of translated lattices (also known as quasi periodic sets). In that work (Kolountzakis, Discrete Comput Geom 23(4):537–553, 2000) the author asked whether the same quasi-periodic structure on the translation vectors would be true in ${\mathbb {R}}^3$ . Here we prove that this conclusion is indeed true for ${\mathbb {R}}^3$ . Namely, we show that if $P$ is a convex multiple tiler in ${\mathbb {R}}^3$ , with a discrete multiset $\Lambda $ of translation vectors, then $\Lambda $ has to be a finite union of translated lattices, unless $P$ belongs to a special class of zonotopes. This exceptional class consists of two-flat zonotopes $P$ , defined by the Minkowski sum of two 2-dimensional symmetric polygons in ${\mathbb {R}}^3$ , one of which may degenerate into a single line segment. It turns out that rational two-flat zonotopes admit a multiple tiling with an aperiodic (nonquasi-periodic) set of translation vectors $\Lambda $ . We note that it may be quite difficult to offer a visualization of these 3-dimensional non-quasi-periodic tilings, and that we discovered them by using Fourier methods.     

Frobenius Problem and the Covering Radius of a Lattice

Let and let be relatively prime integers. The Frobenius number of this N-tuple is defined to be the largest positive integer that cannot be expressed as are non-negative integers. The condition that implies that such a number exists. The general problem of determining the Frobenius number given N and is NP-hard, but there have been a number of different bounds on the Frobenius number produced by various authors. We use techniques from the geometry of numbers to produce a new bound, relating the Frobenius number to the covering radius of the null-lattice of this N-tuple. Our bound is particularly interesting in the case when this lattice has equal successive minima, which, as we prove, happens infinitely often.     

Trifluoromethylation of alkenyl bromides and iodides (including 5-iodouracils) with (CF3)2Hg and Cu ("trifluoromethylcopper")

Bromo- and iodoalkenes undergo trifluoromethylation efficiently in DMA with "CF3Cu" generated from (CF3)2Hg and Cu. Variable stereochemical inversion is observed with substrates having a gem-carbonyl group. Substrates having gem-hydrogen, -alkyl, or -alkenyl groups give products with stereochemical retention.     

Copper and iron interactions with angiotensin-converting enzyme inhibitors. A study by fast-atom bombardment tandem mass spectrometry

Fast atom bombardment, combined with high-energy collision-induced tandem mass spectrometry, has been used to investigate gas-phase metal-ion interactions with captopril, enalaprilat and lisinopril, all angiotensin-converting enzyme inhibitors.Suggestions for the location of metal-binding sites are presented. For captopril, metal binding occurs most likely at both the sulphur and the nitrogen atom. For enalaprilat and lisinopril, binding preferably occurs at the amine nitrogen. Copyright 1999 John Wiley & Sons, Ltd.     

Substituent Effect on the Electronic Properties and Morphologies of Self‐Assembling Bisphenazine Derivatives

Tuning electronic properties and morphologies : We report a unique design platform of n‐type organic semiconductors based on asymmetrically substituted bisphenazines that enable tuning of both electronic properties and morphologies of 1D nanostructures (see figure) by using small substituents with various sizes and electronic demands.

     

Diazocinones: synthesis and conformational analysis

1,2,4,5-Tetrazines (prepared from aryl nitriles) condense with isoxazolylcyclobutanones (prepared from 3-benzenesulfonyl-3-vinylcyclobutanol) in methanolic KOH to give conformationally restricted 6-isoxazol-5-yl-6,7-dihydro-5H-[1,2]diazocin-4-ones. The solution 1H NMR spectra of dihydrodiazocinone 1a with phenyl moieties at C3 and C8 reveal two conformations of the eight-membered heterocycle that are non-interconverting on the NMR time scale at ambient temperature. The kinetics of the conversion process, followed by 1H NMR between 21 and 70 degrees C in DMSO solution, yield an activation energy of approximately 21 kcal/mol relative to the kinetic conformer and show an equilibrated ratio of approximately 5:1 of the thermodynamic to the kinetic conformers. The electronic structure calculations on a model dihydrodiazocinone predict geometries for the two conformations. One of these geometries agrees with the X-ray crystallographic analysis of the thermodynamic conformation of 1a.     

Regiospecific N9 alkylation of 6-(heteroaryl)purines: shielding of N7 by a proximal heteroaryl C-H1

Purine alkylations have been plagued with formation of mixtures of N9 (usually desired), N7, and other regioisomers. We have developed methods for synthesis of 6-(azolyl)purine derivatives whose X-ray crystal structures show essentially coplanar conformations of the linked azole-purine rings. Such ring orientations position the C-H of the azole above N7 of the purine, which results in protection of N7 from alkylating agents. Treatment of 6-(2-butylimidazol-1-yl)-2-chloropurine (9) with sodium hydride in DMF followed by addition of ethyl iodide resulted in exclusive formation of 6-(2-butylimidazol-1-yl)-2-chloro-9-ethylpurine (10), whereas identical treatment of 2-chloro-6-(4,5-diphenylimidazol-1-yl)purine (11) produced a regioisomeric mixture 12/13 (N9/N7, approximately 5:1). The linked imidazole and purine rings are coplanar in 9 (the butyl side chain is extended away from the purine ring and C-H is over N7) but are rotated approximately 57 degrees in 11, and the more bulky azole substituent in 11 did not prevent formation of the minor N7 regioisomer 13. Access to various regioisomerically pure 9-alkylpurines is now readily available.     

Regioisomers in Vorbrüggen''s guanine nucleoside synthesis; N9 selectivity with a glucosamine derivative and 2-N-acetyl-6-O-diphenylcarbamoylguanine

Vorbrüggen coupling of trimethylsilylated 2-N-acetylguanine with pentofuranose derivatives gives N7/N9 glycosyl mixtures. The N9 isomers can be obtained selectively with 2-N-acetyl-6-O-diphenylcarbamoylguanine, and even with a less reactive glucosamine derivative.     

Synthesis of 5-(β-D-ribofuranosyl)-1,2,4-oxadiazole-3-carboxamide

The title compound was synthesized in three steps from ethoxycarbonylformamide oxime (5) and 3,4,6-tri-O-acetyl-2,5-anhydroallonyl chloride (4b) in 62% overall yield. An acid catalyzed de-esterification was required to prevent a facile base catalyzed elimination reaction.