Pseudo-trees and Boolean algebras

We consider Boolean algebras constructed from pseudo-trees in various ways and make comments about related classes of Boolean algebras.     

The scattering of electromagnetic waves by a perfectly conducting infinite cylinder

We consider the scattering of a plane time-harmonic electromagnetic wave by a perfectly conducting infinite cylinder with axis in the direction k , where k is the unit vector along the z axis. Suppose the incident wave propagates in a direction perpendicular to the cylinder. For a given observation angle θ, let F_D(θ, α) k be the far-field pattern of the electric field corresponding to an incident wave with direction angle α polarized perpendicular to the axis and let F_N(θ; α) k be the far-field pattern of the magnetic field corresponding to an incident wave with direction angle α polarized parallel to the z axis. Let {α_n}_n=1^∞ be a distinct set of angles in [ ? π, π] and μ a complex number. Then, necessary and sufficient conditions are given for the set {(1 ? μ)F_D(θ;α_n) + μF_N(θ;α_n)}_{n = 1}^∞ to be complete in L²[ ? π, π]. Applications, together with numerical examples, are given to the inverse scattering problem of determining the shape of the cylinder from a knowledge of the far-field data.     

A finite element method for approximating the time-harmonic Maxwell equations

Summary In this paper we study the use of Nédélec's curl conforming finite elements to approximate the time-harmonic Maxwell equations on a bounded domain. The analysis is complicated by the fact that the bilinear form is not coercive, and the principle part has a very large null-space. This difficulty is circumvented by using a discrete Helmholtz decomposition of the error vector. Numerical results are presented that compare two different linear elements.Research supported in part by grants from AFOSR and NSF     

γ-Lactam analogues of penicillanic and carbapenicillanic acids

Synthesis and biological activity of γ-lactam analogues of penicillanic and carboapenicillanic acids, and the sodium periodate mediated rearrangement of pyrrolidine-2,3-diones are described. 1,3- Dipolar addition of cyclic nitrone (6) and methyl acrylate afforded the bicyclic adducts (7a) and (7b). Reductive cleavage of the N-O bond and subsequent cyclisation of a regioisomer (11a) gave the γ-lactams (12a) and (12b) in a ratio of 85 : 15. They are transformed to the carbapenam analogues (1)-(4). Their stereochemistry was assigned according to the X-ray structure of the γ-lactam (12b). Benzyl 6-oxopenicillanate (20) was directly transformed to the γ-lactam analogue (5) $\text{via}$ a novel ring expansion. These synthetic analogues did not show antibiotic activity or β-lactamase inhibition. Treatment of pyrro1idine-2,3-diones (25a) and (25b) with sodium periodate gave ring contracted β-lactams (26a) and (26b) respectively. Similar treatment of (27) followed by diazomethane afforded an unexpected spiro epoxide.     

A very rigid Boolean algebra

A Boolean algebra is constructed having only those endomorphisms corresponding to prime ideals, which are present in any BA. The BA constructed is of powerc, has 2 ^c endomorphisms, and is not rigid in Bonnet’s sense.     

A combined LC-MS/MS and LC-MS3 multi-method for the quantification of iodothyronines in human blood serum

We report here a novel approach for the extraction and analysis of thyroid hormones (TH) and their metabolites (THM) from human serum samples. Our method features a compact, 96-well micro-titre plate-based pre-analytic extraction/clean-up workflow combined with an isotope dilution LC-MS/MS-MS³ analytical method. In particular, these features make possible the detection of iodothyronines at their endogenous concentrations in serum differing by a factor of ca. 10⁴, with potential to semi-automate the pre-analytics. The method was validated by the assessment of linearity, lower limits of quantification and detection (LLOQ and LLOD respectively), intra- and inter-day accuracy, precision, process efficiency (PE), matrix effect (ME) and relative recovery (RE). Calibration curves were linear in the concentration range in sample matrix from 0.1–250 nM for T₃, rT₃, T₄ and 3-T₁AM and from 0.005–1 nM for 3,5-T₂ and 3,3′-T₂. Using a 200-μL sample volume, the analyte dependant LLOQ were in the range 0.005 (3,5-T₂) to 0.25 (T₄) nM and LLOD were between 0.002 (3,5-T₂) and 0.052 nM (T₄). We applied the LC-MS/MS-MS³ method to the analysis of a cross section of patients with disorders of the thyroid hormone axis. T₄, T₃ and rT₃ concentrations (± standard deviation) were 120 ± 18, 1.9 ± 0.4 and 0.45 ± 0.09 nM respectively. 3,3′-T₂ concentrations (± standard deviation) were 0.079 ± 0.022 nM; 3,5-T₂ concentrations were below the LLOQ and/or LLOD in all but a single sample (0.013 nM). This method expands the analytical spectrum to endogenous thyroid hormone metabolites such as 3,5-T₂ which exert biological actions and rT₃ which may act as surrogate markers for disturbed thyroid hormone metabolism.

Graphical abstract