The intersection of free submonoids of a free monoid is free

Several necessary and sufficient conditions exist for a submonoid of a free monoid to be free. For a few, see [1] through [6]. This short note uses one of these conditions, due to Schützenberger (see [2], p. 119, or [4], Theorem 1.4) to establish that the intersection of free submonoids of a free monoid M is again free. Schützenberger's condition is also proven in this note.     

Decomposition and complexity of finite semigroups

The Prime Decomposition Theorem for finite transformation semigroups of Krohn and Rhodes and the resultant theory of complexity of finite semigroups are the subjects of this survey. All concepts used are defined in the text, and proofs are given if they are accessible without a large amount of preparation. This article is an attempt to lead the reader to the essential ideas of the theory in the shortest possible time and effort.     

Ab initio pseudopotential study of vacancies and self-interstitials in hcp titanium

By means of an ab initio plane-wave pseudopotential method, monovacancy, divacancy and self-interstitials in hcp titanium are investigated. The calculated monovacancy formation energy is 1.97 eV, which is in excellent agreement with other theoretical calculations, and agrees qualitatively with published experimental results. The relaxation of the atoms around a single vacancy is observed to be small. Two divacancy configurations, the in-plane and the off-plane, have also been shown to be equally stable. With regards to the interstitials, of the eight configurations studied, two (octahedral and basal octahedral) have relatively lower formation energies and are, thus, the most likely stable configurations. We find small energy differences between them, suggesting their possible co-existence. It is also observed that the tetrahedral configuration decays to a split dumbbell configuration, whereas both the basal tetrahedral and the basal pseudocrowdion interstitials decay to the basal octahedral configuration. Using the nudged elastic band method (NEB), we determine a possible minimum energy path (MEP) for the diffusion of self-interstitial titanium atoms from an octahedral site to the nearest octahedral site. The energy barrier for this migration mechanism is shown to be about 0.20 eV.     

Exact relativistic kinetic theory of an electron-beam-plasma system: hierarchy of the competing modes in the system-parameter space

The stability analysis of an electron-beam-plasma system is of critical relevance in many areas of physics. Surprisingly, decades of extensive investigation have not yet resulted in a realistic unified picture of the multidimensional unstable spectrum within a fully relativistic and kinetic framework. All attempts made so far in this direction were indeed restricted to simplistic distribution functions and/or did not aim at a complete mapping of the beam-plasma parameter space. The present Letter comprehensively tackles this problem by implementing an exact linear model. Three kinds of modes compete in the linear phase, which can be classified according to the direction of their wave number with respect to the beam. We determine their respective domain of preponderance in a three-dimensional parameter space and support our results with multidimensional particle-in-cell simulations.     

A rapid dissolution procedure to aid initial nuclear forensics investigations of chemically refractory compounds and particles prior to gamma spectrometry

A rapid and effective preparative procedure has been evaluated for the accurate determination of low-energy (40–200 keV) gamma-emitting radionuclides (²¹⁰Pb, ²³⁴Th, ²²⁶Ra, ²³⁵U) in uranium ores and uranium ore concentrates (UOCs) using high-resolution gamma ray spectrometry. The measurement of low-energy gamma photons is complicated in heterogeneous samples containing high-density mineral phases and in such situations activity concentrations will be underestimated. This is because attenuation corrections, calculated based on sample mean density, do not properly correct where dense grains are dispersed within a less dense matrix (analogous to a nugget effect). The current method overcomes these problems using a lithium tetraborate fusion that readily dissolves all components including high-density, self-attenuating minerals/compounds. This is the ideal method for dissolving complex, non-volatile components in soils, rocks, mineral concentrates, and other materials where density reduction is required. Lithium borate fusion avoids the need for theoretical efficiency corrections or measurement of matrix matched calibration standards. The resulting homogeneous quenched glass produced can be quickly dissolved in nitric acid producing low-density solutions that can be counted by gamma spectrometry. The effectiveness of the technique is demonstrated using uranium-bearing Certified Reference Materials and provides accurate activity concentration determinations compared to the underestimated activity concentrations derived from direct measurements of a bulk sample. The procedure offers an effective solution for initial nuclear forensic studies where complex refractory minerals or matrices exist. It is also significantly faster, safer and simpler than alternative approaches.     

Total Synthesis,Stereochemical Assignment,and Field‐Testing of the Sex Pheromone of the Strepsipteran Xenos peckii

The sex pheromone of the endoparasitoid insect Xenos peckii (Strepsiptera: Xenidae) was recently identified as (7E,11E)‐3,5,9,11‐tetramethyl‐7,11‐tridecadienal. Herein we report the asymmetric synthesis of three candidate stereostructures for this pheromone using a synthetic strategy that relies on an sp³–sp² Suzuki–Miyaura coupling to construct the correctly configured C7‐alkene function. Comparison of ¹H NMR spectra derived from the candidate stereostructures to that of the natural sex pheromone indicated a relative configuration of (3R*,5S*,9R*). Chiral gas chromatographic (GC) analyses of these compounds supported an assignment of (3R,5S,9R) for the natural product. Furthermore, in a 16‐replicate field experiment, traps baited with the synthetic (3R,5S,9R)‐enantiomer alone or in combination with the (3S,5R,9S)‐enantiomer captured 23 and 18 X. peckii males, respectively (mean±SE: 1.4±0.33 and 1.1±0.39), whereas traps baited with the synthetic (3S,5R,9S)‐enantiomer or a solvent control yielded no captures of males. These strong field trapping data, in combination with spectroscopic and chiral GC data, unambiguously demonstrate that (3R,5S,9R,7E,11E)‐3,5,9,11‐tetramethyl‐7,11‐tridecadienal is the X. peckii sex pheromone.