Engel graph associated with a group

Let G be a non-Engel group and let L(G) be the set of all left Engel elements of G. Associate with G a graph as follows: Take G L(G) as vertices of and join two distinct vertices x and y whenever [x,_ky]≠1 and [y,_kx]≠1 for all positive integers k. We call , the Engel graph of G. In this paper we study the graph theoretical properties of .     

Roots of C*-algebra elements under numerical range condition

Let be a C*-algebra with unit 1. For each a ∈ , the C*-algebra numerical range is defined by V(a) = {φ(a): φ ∈ , φ ≥ 0,φ(1) = 1}. In a 2003 paper Li, Rodman and Spitkovsky have found the ω-th roots of elements in C*-algebra under a numerical range condition, when ω ∈ [1,∞). In this paper, we will give a short proof of the above result in the case of ω is a positive integer number. We also give a simple proof for ω-th root of an element a ∈ , when ω ∈ [1,∞) and V(a)∩ {z ∈ ℂ: z ≤ 0} = . The first author was supported by the Shiraz university Research Council Grant No. 86-GRSC-32.     

Spontaneous emission in the near field of two-dimensional photonic crystals

We show theoretically that photonic crystal membranes cause large variations in the spontaneous emission rate of dipole emitters, not only inside but also in the near field above the membranes. Our three-dimensional finite-difference time-domain calculations reveal an inhibition of more than five times and an enhancement of more than ten times for the spontaneous emission rate of emitters with select dipole orientations and frequencies. Furthermore, we demonstrate theoretically the potential of a nanoscopic emitter attached to the end of a glass fiber tip as a local probe for mapping the large spatial variations of the photonic crystal local radiative density of states. This arrangement is promising for on-command modification of the coupling between an emitter and the photonic crystal in quantum optical experiments.     

High resolution capillary electrophoresis of carbon nanotubes

Purification of single-walled carbon nanotubes by capillary electrophoresis (CE) is demonstrated. Real-time Raman spectroscopy of the separation process and single-wavelength UV/vis detection show the ability of CE to provide high-resolution separations of nanotube fractions with baseline separation. AFM images of collected fractions demonstrate that separations are based on tube length. The separation method is suggested to be based on alignment of the nanotubes along the separation field.     

Design of plasmonic nanoantennae for enhancing spontaneous emission

We apply two- and three-dimensional numerical calculations to study optical nanoantennae made of two coupled gold nanostructures, enclosing a single emitter in their gap. We show that, using structures manufacturable with today's nanotechnology, it is possible to increase the radiative decay rate by three orders of magnitude while keeping a quantum efficiency larger than 80% in the near-infrared regime. We examine the competition between the radiative and nonradiative processes in the presence of the antennae as a function of wavelength and antenna geometry. Our results hold great promise for improving the quantum efficiency of poor emitters such as silicon nanocrystals or carbon nanotubes.     

H-point standard addition method for simultaneous determination of Fe(II), Co(II) and Cu(II) in micellar media with simultaneous addition of three analytes

H-point standard addition method, HPSAM, with simultaneous addition of three analytes is proposed for the resolution of ternary mixtures. It is a modification of the previously described H-point standard addition method that permits the resolution of three species from a unique calibration set by making the simultaneous addition of the three analytes. The method calculates the analyte concentration from spectral data at two wavelengths where the two species selected as interferents present the same absorbance relationship. These wavelength pairs are easily found, and can be selected to give the most precise results. Diethyldithiocarbomate (DDC) in a cationic micellar solution of cetyltrimethylammonium bromide (CTAB) was used for determination of Fe(II), Co(II) and Cu(II) at pH 5.50. The results showed that simultaneous determination of Fe(II), Co(II) and Cu(II) could be preformed in the range of 0.0–6.0, 0.0–8.0 and 0.0–12.0 μg ml⁻¹, respectively. The proposed method was successfully applied to the simultaneous determination of Fe(II), Co(II) and Cu(II) in several synthetic mixtures containing different concentration of Fe(II), Co(II) and Cu(II).     

Simultaneous determination of Fe(II) and Fe(III) by kinetic spectrophotometric H-point standard addition method

The H-point standard addition method (HPSAM) for simultaneous determination of Fe(II) and Fe(III) is described. The method is based on the difference in the rate of complex formation of iron in two different oxidation states with Gallic acid (GA) at pH 5. Fe(II) and Fe(III) can be determined in the range of 0.02–4.50 μg ml⁻¹ and 0.05–5.00 μg ml⁻¹, respectively, with satisfactory accuracy and precision in the presence of other metal ions, which rapidly form complexes with GA under working conditions. The proposed method was successfully applied for simultaneous determination of Fe(II) and Fe(III) in several environmental and synthetic samples with different concentration ratios of Fe(II) and Fe(III).     

Application of the H-point standard addition method to the speciation of Fe(II) and Fe(III) with chromogenic mixed reagents

Simultaneous determination of Fe(II) and Fe(III) or selective determination of each oxidation state of iron in the presence of the other one by H-point standard addition method (HPSAM) is described. Mixed reagents of 1,10-phenanthroline and salicylic acid was used as a selective chromogenic system for speciation of Fe(II) and Fe(III). It was shown that with appropriate selection of wavelength pairs, both Fe(II) and Fe(III) can be considered as analyte. The results showed that Fe(II) and Fe(III) can be determined simultaneously with the concentration ratios of Fe(II) to Fe(III) varying from 10:1 to 1:20 in the mixed sample. The accuracy and precision of the method are all satisfactory.     

Rh(II)-Catalyzed Isomerizations of Cyclopropenes Evidence for Rh(II)-Complexed Vinylcarbene Intermediates

The thermocatalytic rearrangements of cyclopropenes 1-4 have been investigated in the presence of Rh(IT) perfiuorobutyrate. 1, 2, 3-Triphenylcyclopropene ( 1a ) undergoes rearrangement lo diphenylindene 5a or, with alkoxycyclopropene derivatives, to α, β-unsaturated ketone 6. Furan formation occurs with 2, 3-diphenylcyclo-propenecarboxylate 2, but the diethyl counterpart 3 rearranges to dienoate 8. 2-Alkylcyclopropenecarboxylates 4 afforded (E)-methylidenecyclopentane derivatives 9 as the only isolable product in yields of ca. 35 %. A mechanism involving regio- and stereospecific cyclopropene ring opening to a Rh-complexed vinylcarbene and insertion of the latter into the C? H bond to give 9 is proposed. An analogous mechanism should account for the rearrangement products of 1 to 3 .