Polystyryl-supported TBD as an efficient and reusable catalyst under solvent-free conditions

Polystyryl supported-TBD (PSTBD) is an efficient and reusable heterogeneous basic catalyst under solvent-free conditions for a variety of organic transformations such as 1,2-epoxide ring-opening, aldol-type condensation and Michael addition.     

Unexpected reactivity between aromatic nitro compounds and PCl3/AlCl3. A new one-pot synthesis of phenazines

The reaction between 4-nitroalkoxybenzenes 7 and PCl₃/AlCl₃, when carried out in appropriate molar ratio, gives a prevalent formation of diazenes 8 and 2,7-dialkoxyphenazines 9 with their new chlorinated derivatives 10-13. These compounds are obtained, in satisfactory yield, in a one-pot procedure, in mild conditions, from commercially available and safe starting materials. In this reaction both the reagents PCl₃ and AlCl₃ play a fundamental role in obtaining the products, and this method might be applicable to other 4-alkoxynitrobenzenes.     

Evaluation of teicoplanin chiral stationary phases of 3.5 and 5 microm inside diameter silica microparticles by polar-organic mode capillary electrochromatography

Different types of fused-silica capillaries of 75 microm inside diameter (ID) were packed, namely type A and B, and evaluated for the direct resolution of racemates of several basic compounds by enantioselective capillary electrochromatography (e-CEC). Type A was packed with a chiral stationary phase (CSP) containing teicoplanin (TE) mixed with silica microparticles (3:1 w/w) while type B contained only the TE-CSP. In both cases, particles of different sizes (3.5 and 5 microm ID) were employed. A polar-organic mobile phase containing methanol-acetonitrile (60-40% v/v and 0.05% w/v ammonium acetate was used. Several beta-blockers (alprenolol, oxprenolol, metoprolol, pindolol, salbutamol, propranolol, atenolol, acebutolol) were baseline-enantioresolved with both capillary types, in very short times.     

An orbifold relative index theorem

In this paper we prove a relative index theorem for pairs of generalized Dirac operators on orbifolds which are the same at infinity. This generalizes to orbifolds a celebrated theorem of Gromov and Lawson.     

Crystal structures of the novel hydrated borates Ba2B5O9(OH), Sr2B5O9(OH) and Li2Sr8B22O41(OH)2

Three novel hydrated borates Ba₂B₅O₉(OH) (1), Sr₂B₅O₉(OH) (2) and Li₂Sr₈B₂₂O₄₁(OH)₂ (3) have been synthesized hydrothermally and their structures determined. Compounds (1) and (2) are isostructural, crystallizing in space group P2₁/c and having lattice parameters of a=6.6330(13) Å, b=8.6250(17) Å, c=14.680(3) Å, β=93.46(3)° and a=6.4970(13) Å, b=8.4180(17) Å, c=14.177(3) Å, β=94.35(3)°, respectively. Compound (3) crystallizes in P-1 with lattice parameters of a=6.4684(13) Å, b=8.4513(17) Å, c=14.881(3) Å, α=101.21(3)°, β=93.96(3)°, γ=90.67(3)°. While similar in their axis lengths, (3) differs greatly in structure and formulation from (1) and (2). The structure of (1) and (2) is contrasted to that of the well-known mineral hilgardite (Ca₂B₅O₉Cl·H₂O).     

A tension approach to controlling the shape of cubic spline surfaces on FVS triangulations

We propose a parametric tensioned version of the FVS macro-element to control the shape of the composite surface and remove artificial oscillations, bumps and other undesired behaviour. In particular, this approach is applied to C¹ cubic spline surfaces over a four-directional mesh produced by two-stage scattered data fitting methods.     

Euler's partition theorem and the combinatorics of ℓ-sequences

Euler's partition theorem states that the number of partitions of an integer N into odd parts is equal to the number of partitions of N in which the ratio of successive parts is greater than 1. It was shown by Bousquet-Mélou and Eriksson in [M. Bousquet-Mélou, K. Eriksson, Lecture hall partitions II, Ramanujan J. 1 (2) (1997) 165–185] that a similar result holds when “odd parts” is replaced by “parts that are sums of successive terms of an ℓ-sequence” and the ratio “1” is replaced by a root of the characteristic polynomial of the ℓ-sequence. This generalization of Euler's theorem is intrinsically different from the many others that have appeared, as it involves a family of partitions constrained by the ratio of successive parts.In this paper, we provide a surprisingly simple bijection for this result, a question suggested by Richard Stanley. In fact, we give a parametrized family of bijections, that include, as special cases, Sylvester's bijection and a bijection for the lecture hall theorem. We introduce Sylvester diagrams as a way to visualize these bijections and deduce their properties.In proving the bijections, we uncover the intrinsic role played by the combinatorics of ℓ-sequences and use this structure to give a combinatorial characterization of the partitions defined by the ratio constraint. Several open questions suggested by this work are described.     

Strong convergence rates for the estimation of a covariance operator for associated samples

Let X _n , n ≥ 1, be a strictly stationary associated sequence of random variables, with common continuous distribution function F. Using histogram type estimators we consider the estimation of the two-dimensional distribution function of (X ₁,X _k+1) as well as the estimation of the covariance function of the limit empirical process induced by the sequence X _n , n ≥ 1. Assuming a convenient decrease rate of the covariances Cov(X ₁,X _n+1), n ≥ 1, we derive uniform strong convergence rates for these estimators. The condition on the covariance structure of the variables is satisfied either if Cov(X ₁,X _n+1) decreases polynomially or if it decreases geometrically, but as we could expect, under the latter condition we are able to establish faster convergence rates. For the two-dimensional distribution function the rate of convergence derived under a geometrical decrease of the covariances is close to the optimal rate for independent samples.      

Novel Bisphosphonates Derived from 1H‐Indazole, 1H‐Pyrazolo[3,4‐b]Pyridine,and 1H‐Pyrazolo[3,4‐b]Quinoline

Novel tetraethyl ethylene‐1,1‐bisphosphonate esters derived from 1H‐indazole, 1H‐pyrazolo[3,4‐b]pyridine, and 1H‐pyrazolo[3,4‐b]quinoline were synthesized by a Michael addition reaction of tetraethyl ethylidene‐1,1‐bisphosphonate with the corresponding heterocycle, using conventional heating and microwave‐assisted methods. The microwave‐assisted method provides shorter reaction times and better yields. The hydrolysis of bisphosphonates afforded the corresponding bisphosphonic acids or salt, using concentrated hydrochloric acid or TMSBr/collidine, respectively. All new compounds were fully characterized, and their structures were assigned using ¹H, ³¹P, and ¹³C NMR and IR spectroscopies and mass spectrometry. The molecular structure of compound 6 was confirmed by X‐ray diffraction studies.