Normal generation and \ell ^2-Betti numbers of groups

The normal rank of a group is the minimal number of elements whose normal closure coincides with the group. We study the relation between the normal rank of a group and its first $\ell ^2$ -Betti number and conjecture the inequality $\beta _1^{(2)}(G) \le \mathrm{nrk}(G)-1$ for torsion free groups. The conjecture is proved for limits of left-orderable amenable groups. On the other hand, for every $n\ge 2$ and every $\varepsilon >0$ , we give an example of a simple group $Q$ (with torsion) such that $\beta _1^{(2)}(Q) \ge n-1-\varepsilon $ . These groups also provide examples of simple groups of rank exactly $n$ for every $n\ge 2$ ; existence of such examples for $n> 3$ was unknown until now.     

Thickness-Two Graphs Part Two: More New Nine-Critical Graphs,Independence Ratio,Cloned Planar Graphs,and Singly and Doubly Outerplanar Graphs

There are numerous means for measuring the closeness to planarity of a graph such as crossing number, splitting number, and a variety of thickness parameters. We focus on the classical concept of the thickness of a graph, and we add to earlier work in [4]. In particular, we offer new 9-critical thickness-two graphs on 17, 25, and 33 vertices, all of which provide counterexamples to a conjecture on independence ratio of Albertson; we investigate three classes of graphs, namely singly and doubly outerplanar graphs, and cloned planar graphs. We give a sharp upper bound for the largest chromatic number of the cloned planar graphs, and we give upper and lower bounds for the largest chromatic number of the former two classes.     

On the rate of convergence of some difference schemes for second order elliptic equations

The purpose of this paper is to estimate the rate of convergence for some natural difference analogues of Dirichlet's problem for uniformly elliptic differential equations, $$\begin{gathered} \sum\limits_{j,k = 1}^N {\frac{\partial }{{\partial x_j }}} \left( {a_{jk} \frac{{\partial u}}{{\partial x_k }}} \right) = F in R, \hfill \\ u = f on B, \hfill \\ \end{gathered}$$ in aN-dimensional domainR with boundaryB. These schemes will in general not be of positive type, and the analysis will therefore be carried out in discreteL ²-norms rather than in the maximum norm. Since our approximation of the boundary condition is rather crude, we will only arrive at a rate of convergence of first order for smoothF andf. Special emphasis will be put on appraising the dependence of the rate of convergence on the regularity ofF andf.     

An improved implementation of the McCracken and Yanosik nine point finite difference procedure

The grid orientation phenomenon present in numerical models of fluid flow in a porous media can give rise to unrealistic predictions when simulating adverse mobility displacements. McCracken and Yanosik [11] proposed a nine point finite difference scheme for approximating the solution of the continuity equations that has the potential of eliminating many of the unrealistic predictions that are observed when using five point finite difference operators. Coats and Ramesh [5] have implemented this scheme in a steamflooding model, and have noted that serious grid effects are present in the simulation of an inverted seven spot pattern. Potempa [12,13] has described a procedure which exhibits minimal grid effects for the problem described by Coats and Ramesh [5]. This paper describes modifications to the McCracken and Yanosik procedure which allow for realistic simulations of this inverted seven spot pattern under a steam drive. These modifications are based upon an approximation scheme that has been previously reported [12,13], and affect the incorporation of upstream weighting in a similator.     

Hyphenation of quadrupole MS to GC and comprehensive two-dimensional GC for the analysis of suspected allergens: review and improvement

This survey reviews papers that have been previously published on the quantitative analysis of suspected allergens. The routine gas chromatography (GC)-mass spectrometry (MS) method allows their evaluation in most fragrances, but the application of an automated data treatment sometimes leads to over- or underestimations when target compounds are coeluted or shifted because of the presence of other fragrance ingredients. In such cases, an appropriate retreatment of data generated by the routine analysis is proposed to better estimate these shifted or coeluted peaks. A second and more sophisticated approach, based on comprehensive bidimensional GC hyphenated to quadrupole MS, overcomes coelution problems. However, its use is still time consuming because of the lack of a commercial program. In this work, a software prototype is tested to reprocess the data. It dramatically shortens the data treatment and offers good quantitative results.     

On-line coupling of size exclusion and capillary zone electrophoresis via a reversed-phase C18 trapping column for the analysis of structurally related enkephalins in cerebrospinal fluid

On-line coupled analytical techniques can be advantageous in the assay of smaller peptides in complex biological matrices such as plasma, cerebrospinal fluid (CSF) and tissues. The present study shows the feasibility of a recently developed system, consisting of a size-exclusion chromatographic (SEC) separation followed by a trapping procedure on an RP18 microcolumn with subsequent elution of the trapped fraction and separation by capillary zone electrophoresis (CZE) for the quantification of structural-related peptides in biological matrices, as demonstrated for a number of enkephalins in CSF. After SEC separation of the enkephalins from large proteins present in CSF a heart-cut of 200 nuL, containing the enkephalin peak, is taken, concentrated on the RP18 microcolumn and, after elution of the enkephalins with 80% acetonitrile, a fraction of the eluate is electrokinetically injected into the CZE system, where stacking and separation is achieved. The degradation of the peptides, caused by endogenous peptidases in the matrix, is sufficiently inhibited with imipramine HCl. The assay has a satisfactory linearity and intraday (9.70-16.3%) precision considering the complexity of this multidimensional separation system. The sensitivity of the method, with a concentration limit of quantification of 2.5 nug/mL, is comparable with other CZE assays for peptides and sufficient for the quantification of peptide drugs in biological matrices.     

Computational investigation of irreversible inactivation of the zinc-dependent protease carboxypeptidase A

Zinc proteases are ubiquitous and the zinc ion plays a central function in the catalytic mechanism of these enzymes. A novel class of mechanism-based inhibitors takes advantage of the zinc ion chemistry in carboxypeptidase A (CPA) to promote covalent attachment of an inhibitor to the carboxylate of Glu-270, resulting in irreversible inhibition of the enzyme. The effect of the active site zinc ion on irreversible inactivation of CPA was probed by molecular orbital (MO) calculations on a series of active site models and the Cl(-) + CH(3)Cl S(N)2 reaction fragment. Point charge models representing the active site reproduced energetics from full MO calculations at 12.0 A separation between the zinc and the central carbon of the S(N)2 reaction, but at 5.0 A polarization played an important role in moderating barrier suppression. ONIOM MO/MO calculations that included the residues within 10 A of the active site zinc suggest that about 75% of the barrier suppression arises from the zinc ion and its ligands. A model of the pre-reactive complex of the 2-benzyl-3-iodopropanoate inactivator with CPA was constructed from the X-ray structure of l-phenyl lactate bound in the active site of the enzyme. The model was fully solvated and minimized by using the AMBER force field to generate the starting structure for the ONIOM QM/MM calculations. Optimization of this structure led to the barrierless S(N)2 displacement of the iodide of the inhibitor by Glu-270, assisted by interaction of the zinc ion with the leaving group. The resulting product is in good agreement with the X-ray structure of the covalently modified enzyme obtained by irreversible inhibition of CPA by 2-benzyl-3-iodopropanoate.     

Native defect changes in cds single crystal platelets induced by vacuum heat treatments at temperatures up to 600°C

Physical properties of selected CdS single crystal platelets as-grown and after vacuum heat treatments at temperatures up to 600°C have been studied using u.v. excited edge emission, mass spectrometry, electrical resistivity and electron paramagnetic resonance (EPR). It was found that sulfur leaves the crystal at temperatures as low as 100°C creating a depletion layer. The native defect changes were monitored by edge emission studies at 4.2°K in combination with etch treatments. The defect structure throughout the crystal is not only dependent upon the temperature and atmosphere of the treatments, but is also strongly dependent upon the cooling rate.