Robustness of Tube Formula Based Confidence Bands

Abstract

Simultaneous confidence bands of a regression curve may be used to quantify the uncertainty of an estimate of the curve. The tube formula for volumes of tubular neighborhoods of a manifold provides a very powerful method for obtaining such bands at a prescribed level, when errors are Gaussian. This article studies robustness of the tube formula for non-Gaussian errors. The formula holds without modification for an error vector with a spherically symmetric distribution. Simulations are used for a variety of independent non-Gaussian error distributions. The results are acceptable for contaminated and heavy tailed error distributions. The formula can break down in some extreme cases for discrete and highly skewed errors. Computational issues involved in applying the tube formula are also discussed.     

Some novel heterocyclic systems derived from 7-amino-2,3-dihydro-8-nitro-1H-pyrrolo[1,2-α]benzimidazole and the corresponding diamine

The preparation of 7-amino-2,3-dihydro-8-nitro-1H-pyrrolo[1,2-a]benzimidazole from 1,4-diacetamido-2,3-dinitrobenzene is described. Reaction of this compound with 2,5-dimethoxytetrahydrofuran produces 2,3-dihydro-8-nitro-7-N-pyrrolo-1H-pyrrolo[1,2-a]benzimidazole, which can be cyclised to produce two new heterocyclic ring systems, 9,10-dihydro-8H-pyrrolo[1,2-a]pyrrolo(1′,2′:1,2]imidazo[5,4-f]quinoxaline and 9,10-dihydro-8H-pyrrolo[2,1-c]pyrrolo[1′,2′:1,2]imidazo[4,5-h][1,2,4]benzotriazine. The corresponding diamine, 7,8-diamino-2,3-dihydro-1H-pyrrolo[1,2-a]benzimidazole undergoes a variety of condensation reactions to produce several new heterocyclic systems, for example, with formic acid, 1,7,8,9-tetrahydroimidazo-[4,5-e]pyrrolo[2,1-6]benzimidazole is formed and with diacetyl, 9,10-dihydro-2,3-dimethyl-8H-pyrrolo-[1′,2′:1,2]imidazo[5,4-y]quinoxaline is obtained.     

A New Approach for the Synthesis of Highly Substituted Aromatic Rings: The Alkyne‐Mediated Approach

Pentasubstituted aromatic rings serve as templates for drug design and can be conveniently prepared by the thermolysis of suitably substituted alkynes under microwave conditions.     

Insights into the Crystallisation Process from Anhydrous,Hydrated and Solvated Crystal Forms of Diatrizoic Acid

Diatrizoic acid (DTA), a clinically used X‐ray contrast agent, crystallises in two hydrated, three anhydrous and nine solvated solid forms, all of which have been characterised by X‐ray crystallography. Single‐crystal neutron structures of DTA dihydrate and monosodium DTA tetrahydrate have been determined. All of the solid‐state structures have been analysed using partial atomic charges and hardness algorithm (PACHA) calculations. Even though in general all DTA crystal forms reveal similar intermolecular interactions, the overall crystal packing differs considerably from form to form. The water of the dihydrate is encapsulated between a pair of host molecules, which calculations reveal to be an extraordinarily stable motif. DTA presents functionalities that enable hydrogen and halogen bonding, and whilst an extended hydrogen‐bonding network is realised in all crystal forms, halogen bonding is not present in the hydrated crystal forms. This is due to the formation of a hydrogen‐bonding network based on individual enclosed water squares, which is not amenable to the concomitant formation of halogen bonds. The main interaction in the solvates involves the carboxylic acid, which corroborates the hypothesis that this strong interaction is the last one to be broken during the crystal desolvation and nucleation process.     

Synthesis of 6-Hydroxy-2-methyl-3-thioxo-2H-1,2,4-triazin-5-one

The title compound has been prepared in three steps from 2-methylthiosemicarbazide following acylation, cyclisation and cation exchange chromatography. It was fully characterised by the usual spectroscopic means as well as by ¹⁵N nmr spectroscopy and X-ray crystallographic analysis.