Globally Linked Pairs of Vertices in Equivalent Realizations of Graphs

A two-dimensional framework (G,p) is a graph G = (V,E) together with a map p: V → ℝ². We view (G,p) as a straight line realization of G in ℝ². Two realizations of G are equivalent if the corresponding edges in the two frameworks have the same length. A pair of vertices {u,v} is globally linked in G if %and for all equivalent frameworks (G,q), the distance between the points corresponding to u and v is the same in all pairs of equivalent generic realizations of G. The graph G is globally rigid if all of its pairs of vertices are globally linked. We extend the characterization of globally rigid graphs given by the first two authors [13] by characterizing globally linked pairs in M-connected graphs, an important family of rigid graphs. As a byproduct we simplify the proof of a result of Connelly [6] which is a key step in the characterization of globally rigid graphs. We also determine the number of distinct realizations of an M-connected graph, each of which is equivalent to a given generic realization. Bounds on this number for minimally rigid graphs were obtained by Borcea and Streinu in [3].     

Highly edge‐connected detachments of graphs and digraphs

Let G = (V,E) be a graph or digraph and r : V → Z₊. An r‐detachment of G is a graph H obtained by ‘splitting’ each vertex ν ∈ V into r(ν) vertices. The vertices ν₁,…,ν_r(ν) obtained by splitting ν are called the pieces of ν in H. Every edge uν ∈ E corresponds to an edge of H connecting some piece of u to some piece of ν. Crispin Nash‐Williams 9 gave necessary and sufficient conditions for a graph to have a k‐edge‐connected r‐detachment. He also solved the version where the degrees of all the pieces are specified. In this paper, we solve the same problems for directed graphs. We also give a simple and self‐contained new proof for the undirected result. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 67–77, 2003     

On-line on-chip post-column derivatisation reactions for pre-ionisation of analytes and cluster analysis in gradient micro-liquid chromatography/electrospray mass spectrometry

A system is presented that demonstrates the principle of on-line and on-chip post-column derivatisation reactions in micro-high-performance liquid chromatography (micro-HPLC) hyphenated to electrospray time-of-flight mass spectrometry (ESI-TOFMS). In this micro-HPLC-chip-MS set-up, the analytes are separated using gradient micro-HPLC and subsequently derivatised on-chip and detected. One of the major limitations of MS detection is its dependency on the degree of ionisation, which is widely variable and compound-specific. Optimising and controlling the degree of ionisation in a simple manner would allow MS detection to be truly generic. One way of achieving this is by pre-ionisation of analytes using simple derivatisation procedures that are both rapid and quantitative. Performing this in situ on the system described here overcomes issues of sample handling and efficiency losses when time-consuming "bench chemistry" is necessary prior to analysis. The power of the system is demonstrated by the separation of primary and secondary amines, which are subsequently derivatised with a positively charged phosphonium complex and detected in an enhanced manner. Typically, molecular cations (M(+)) are detected showing that the ionisation process is dominated by the phosphonium species, leading to more constant ionisation for a variety of compounds. In addition, stable isotopically labelled ((12)C/(13)C)-phosphonium reagent is used for the reactions, allowing for inherent signal/noise (S/N) improvement and automated data processing using cluster analysis. A similar reaction scheme is used for the derivatisation of ketones and aldehydes, also demonstrating dramatic increases in sensitivity, especially with increasing temperature. Minimal loss in chromatographic fidelity in terms of retention times is observed by the introduction of the micromixer chip into the system. Optimal flow rates in micro-HPLC and ESI-MS are compatible with flow rates for the chip as well as a multitude of in-line optical detectors including UV and fluorescence. In addition, the micromixer chip can be positioned pre-column if preferred. The system is robust, easily fully automated and applicable to a wide variety of reactions. The system has a major advantage in its simple robust connection to the "normal scale" outside world.     

Synthesis and Structure of Mixed Ba12F19ClδBr5–δ Crystals

In the structure Ba₁₂F₁₉Cl₅ [hexagonal space group P6 2m] the two chlorides on the sites Cl(1) and Cl(2) can partially be replaced by bromide ions. Single crystals of the type Ba₁₂F₁₉Cl_δBr_5–δ with a chloride to bromide ratio up to 2 : 3 could be obtained by cooling a flux of 75 mol% BaF₂ and 25 mol% BaX₂ with X = Cl, Br. The crystal quality decreases with increasing bromide concentration. Structural parameters of five selected single crystals with different chloride/bromide ratio were studied by single crystal X-ray diffraction methods. The refined total Cl^?/Br^? population ratio in the crystals is close to the one of the flux. The lattice parameters and interatomic distances change in various ways, when the smaller chloride ion is replaced by the bigger bromide ion. The refinements show a statistical disorder on the halide sites with preferential bromide substitution on site Cl(1).     

Performance of nonrelativistic and quasi-relativistic hybrid DFT for the prediction of electric and magnetic hyperfine parameters in 57Fe Mössbauer spectra

57Fe electric and magnetic hyperfine parameters were calculated for a series of 10 iron model complexes, covering a wide range of oxidation and spin states. Employing the B3LYP hybrid method, results from nonrelativistic density functional theory (DFT) and quasi-relativistic DFT within the zero-order regular approximation (ZORA) were compared. Electron densities at the iron nuclei were calculated and correlated with experimental isomer shifts. It was shown that the fit parameters do not depend on a specific training set of iron complexes and are, therefore, more universal than might be expected. The nonrelativistic and quasi-relativistic electron densities gave fit parameters of similar quality; the ZORA densities are only shifted by a factor of 1.32, upward in the direction of the four-component Dirac-Fock value. From a correlation of calculated electric field gradients and experimental quadrupole splittings, the value of the 57Fe nuclear quadrupole moment was redetermined to a value of 0.16 barn, in good agreement with other studies. The ZORA approach gave no additional improvement of the calculated quadrupole splittings in comparison to the nonrelativistic approach. The comparison of the calculated and measured 57Fe isotropic hyperfine coupling constants (hfcc's) revealed that both the ZORA approach and the inclusion of spin-orbit contributions lead to better agreement between theory and experiment in comparison to the nonrelativistic results. For all iron complexes with small spin-orbit contributions (high-spin ferric and ferryl systems), a distinct underestimation of the isotropic hfcc's was found. Scaling factors of 1.81 (nonrelativistic DFT) and 1.69 (ZORA) are suggested. The calculated 57Fe isotropic hfcc's of the remaining model systems (low-spin ferric and high-spin ferrous systems) contain 10-50% second-order contributions and were found to be in reasonable agreement with the experimental results. This is assumed to be the consequence of error cancellation because g-tensor calculations for these systems are of poor quality with the existing DFT approaches. Excellent agreement between theory and experiment was found for the 57Fe anisotropic hfcc's. Finally, all of the obtained fit parameters were used for an application study of the [Fe(H2O)6]3+ ion. The calculated spectroscopic data are in good agreement with the Mossbauer and electron paramagnetic resonance results discussed in detail in a forthcoming paper.     

Determination of flux from a saddle field fast-atom bombardment gun

The flux or beam density (equivalent current/area) of xenon atoms striking the sample target from a saddle field fast-atom bombardment (FAB) gun has been compared with that from a cesium ion gun mounted on the same instrument. A shielded Faraday cup mounted on the end of a solids probe was used to measure directly the flux of the Cs⁺ beam. Samples of methylene blue in glycerol solution were then exposed to the ion beam at different fluxes and the extents of reduction were measured. The extent of reduction varied linearly with flux up to a value of about 1.16 × 10¹³ particles s^?1 cm^?2 (1.85 μ cm^?2); above this level, the reduction effect appeared to saturate. FAB spectra were obtained from the same dye solution by using varying settings of the FAB gun. By comparing the extents of reduction of the dye from the two guns, the flux from the atom gun could be estimated. Observation of luminescence from a CsI-coated target allowed estimation of the area of the atom beam. The atom beam “equivalent current” could then be calculated by multiplying the flux times the area. It was noted that for given settings, the flux from the atom gun depended on the physical condition of the gun electrodes. With new electrodes, a flux ≥ 1.16 × 10¹³ particles s^?1 cm^?2 was obtained with nominal gun emission currents of 0.60–1.0 mA. Electrodes used extensively, but freshly cleaned, provided a flux of ～ 8 × 10¹² particles s^?1 cm^?2 at nominal emission currents of 0.40–1.0 mA. With dirty electrodes this flux could only be achieved at the highest (1.0 mA) emission current. This decline in performance occurs over a matter of months as a result of contamination and erosion of the electrodes during use. Such behavior can adversely affect spectral reproducibility even when nominal FAB gun voltage and emission current are carefully reproduced.     

Determination of low-level residual ethylene oxide by using solid-phase microextraction and gas chromatography

Current methods of analysis for ethylene oxide (EO) in medical devices include headspace and simulated-use extractions followed by gas chromatography with either a packed or a capillary column. The quantitation limits are about 0.5-1.0 microg/g for a packed column and about 0.1-0.2 microg/g for a capillary column. The current allowable levels of EO on medical devices sterilized with EO gas as outlined in International Organization for Standardization (ISO) 10993-7 may be significantly reduced from current levels by applying the ISO Draft International Standard 10993-17 method for establishing allowable limits. This may require EO test methods with detection and quantitation limits that are much lower than those of the currently available methods. This paper describes a new method that was developed for the determination of low-level EO by solid-phase microextraction using the direct-immersion method. Factors such as temperature and stirring were found to affect absorption efficiency and absorption time. A low extraction temperature (about 6 degrees C) was found to be more efficient than room-temperature extraction. Stirring was found to reduce absorption time by about 50%. Under these conditions, detection and quantitation limits of 0.002 and 0.009 microg/g, respectively, were obtained by using a capillary column. As a result, this method makes compliance with lower EO limits feasible.