The Le numbers of the square of a function and their applications

Lê numbers were introduced by Massey with the purposeof numerically controlling the topological properties of familiesof non-isolated hypersurface singularities and describing thetopology associated with a function with non-isolated singularities.They are a generalization of the Milnor number for isolatedhypersurface singularities. In this note the authors investigatethe composite of an arbitrary square-free f and z². They geta formula for the Lê numbers of the composite, and considertwo applications of these numbers. The first application isconcerned with the extent to which the Lê numbers areinvariant in a family of functions which satisfy some equisingularitycondition, the second is a quick proof of a new formula forthe Euler obstruction of a hypersurface singularity. Severalexamples are computed using this formula including any X definedby a function which only has transverse D(q, p) singularitiesoff the origin.     

High-frequency nanostructured magnetic materials for integrated inductors

Electroplating is compatible with the deposition of relatively thick layers i.e. several μm to 10s- of μm. However eddy current losses mean that thick layers will have an inferior frequency response. We have used a low content phosphorous bath together with pulse reverse plating to generate Co-rich and Co-deficient multi-nanolayers of CoP having improved saturation magnetisation and a better frequency response. The plating parameters have been optimised in order to produce a material with low loss, and a high permeability of around 700 retained up to 103 MHz for a sample with a thickness of 1.7 μm and a resistivity of 136 μΩ cm.     

The primes contain arbitrarily long polynomial progressions

We establish the existence of infinitely many polynomial progressions in the primes; more precisely, given any integer-valued polynomials P ₁, …, P _k  ∈ Z[m] in one unknown m with P ₁(0) = … = P _k (0) = 0, and given any ε > 0, we show that there are infinitely many integers x and m, with 1 \leqslant m \leqslant x^e1 \leqslant m \leqslant x^\varepsilon, such that x + P ₁(m), …, x + P _k (m) are simultaneously prime. The arguments are based on those in [18], which treated the linear case P _j  = (j − 1)m and ε = 1; the main new features are a localization of the shift parameters (and the attendant Gowers norm objects) to both coarse and fine scales, the use of PET induction to linearize the polynomial averaging, and some elementary estimates for the number of points over finite fields in certain algebraic varieties.     

Die Schmelz- und Sättigungskurven der binären Systeme: Substituierte Benzoesäuren und Wasser

Ohne Zusammenfassung     

Things That are Right with the Traditional Square of Opposition

The truth conditions that Aristotle attributes to the propositions making up the traditional square of opposition have as a consequence that a particular affirmative proposition such as ‘Some A is not B’ is true if there are no Bs. Although a different convention than the modern one, this assumption remained part of centuries of work in logic that was coherent and logically fruitful.      

An unexpected ring contraction of two nitroaryl pro-drugs: conversion of N-(nitroaryl)-3-chloropiperidine derivatives into N-(nitroaryl)-2-chloromethylpyrrolidines

Treatment of the N-nitroaryl-3-hydroxypiperidine derivatives 12 and 13 with thionyl chloride afforded the corresponding N-aryl-2-chloromethylpyrrolidines 5 and 15 via a ring-contraction process involving an intermediate aziridinium ion.     

Calculation and Measurement of Terahertz Active Normal Modes in Crystalline PETN

The terahertz frequency spectrum of pentaerythritol tetranitrate (PETN) is calculated using Discover 1 with the COMPASS 2 force field, CASTEP^[3] and PWscf. 4 The calculations are compared to each other and to terahertz spectra (0.3–3 THz) of crystalline PETN recorded at 4 K. A number of analysis methods are used to characterise the calculated normal modes.     

Terahertz imaging through self-mixing in a quantum cascade laser

We demonstrate terahertz (THz) frequency imaging using a single quantum cascade laser (QCL) device for both generation and sensing of THz radiation. Detection is achieved by utilizing the effect of self-mixing in the THz QCL, and, specifically, by monitoring perturbations to the voltage across the QCL, induced by light reflected from an external object back into the laser cavity. Self-mixing imaging offers high sensitivity, a potentially fast response, and a simple, compact optical design, and we show that it can be used to obtain high-resolution reflection images of exemplar structures.     

In vitro and in vivo studies of androst‐4‐ene‐3,6,17‐trione in horses by gas chromatography–mass spectrometry

This paper describes the application of gas chromatography–mass spectrometry (GC‐MS) for in vitro and in vivo studies of 6‐OXO in horses, with a special aim to identify the most appropriate target metabolite to be monitored for controlling the administration of 6‐OXO in racehorses. In vitro studies of 6‐OXO were performed using horse liver microsomes. The major biotransformation observed was reduction of one keto group at the C3 or C6 positions. Three in vitro metabolites, namely 6α‐hydroxyandrost‐4‐ene‐3,17‐dione (M1), 3α‐hydroxyandrost‐4‐ene‐6,17‐dione (M2a) and 3β‐hydroxyandrost‐4‐ene‐6,17‐dione (M2b) were identified. For the in vivo studies, two thoroughbred geldings were each administered orally with 500 mg of androst‐4‐ene‐3,6,17‐trione (5 capsules of 6‐OXO_®) by stomach tubing. The results revealed that 6‐OXO was extensively metabolized. The three in vitro metabolites (M1, M2a and M2b) identified earlier were all detected in post‐administration urine samples. In addition, seven other urinary metabolites, derived from a further reduction of either one of the remaining keto groups or one of the remaining keto groups and the olefin group, were identified. These metabolites included 6α,17β‐dihydroxyandrost‐4‐en‐3‐one (M3a), 6,17‐dihydroxyandrost‐4‐en‐3‐one (M3b and M3c), 3β,6β‐dihydroxyandrost‐4‐en‐17‐one (M4a), 3,6‐dihydroxyandrost‐4‐en‐17‐one (M4b), 3,6‐dihydroxyandrostan‐17‐one (M5) and 3,17‐dihydroxyandrostan‐6‐one (M6). The longest detection time observed in urine was up to 46 h for the M6 metabolite. For blood samples, the peak 6‐OXO plasma concentration was observed 1 h post administration. Plasma 6‐OXO decreased rapidly and was not detectable 12 h post administration. Copyright © 2009 John Wiley & Sons, Ltd.