On the integrated density of states for crystals with randomly distributed impurities

In the present paper, we discuss spectral properties of a periodic Schrödinger operator which is perturbed by randomly distributed impurities; such operators occur as simple models for crystals (or semi-conductors) with impurities. While the spectrum itself is independent of the concentrationp of impurities, for 0<p<1, we focus our attention on the limiting behavior of the integrated density of states _p of the random Schrödinger operator, inside a spectral gap of the periodic operator, asp0. Denoting byU ₀ the set of eigenvalues (in the gap) of the reference problem having precisely one impurity (located at the origin, say), we show that the integrated density of states concentrates around the points ofU ₀, in the sense that _p (U ) is of orderp, for any fixed -neighborhoodU ofU ₀, while _p (K)C·p ², for any compact subsetK of the gap which does not intersectU .Research partially supported by Deutsche Forschungsgemeinschaft     

Statistical evaluation of the influence of several sample pretreatment methods on the mercury content detectable by chemical analysis of contaminated soil samples under practical conditions

The estimation of the environmental risk of contaminated sites caused by hazardous components may be obtained, for instance, by means of a soil survey. There unavoidable errors by sampling, sample preparation and chemical analysis occur. Furthermore, in case of mercury contaminations, the mercury content detectable by chemical analysis can be falsified, if between sampling, on the one hand, and sample preparation and sample decomposition for chemical analysis, on the other hand, volatile components or elementary metallic mercury escape from the sample. Thus, in these cases, handling of samples such as air drying, storing in plastic bags or thermal evaporation, generally termed sample pretreatment, is a further source of error in evaluating a material. However, the measuring results are influenced not only by sampling, sample pretreatment, sample preparation by homogenization and splitting, and chemical analysis; they must also reflect the intrinsic properties of the soil sample subject to both global fluctuations and local heterogeneities. The present work shows by example of a non-uniformly contaminated site to what extent the analytically detectable mercury content is changed by the method of handling of soil samples in the period between sampling and chemical analysis. A hierarchical experimental design was realized in order to separately quantify the different sources of variation of the measured mercury contents, which are caused by global variations, local heterogeneities, sample preparation, sample pretreatment as well as chemical analysis. As turned out by variance analysis, the variance portion contributed to the total variance by sample pretreatment is highly significant and lies in the same order of magnitude as the variance caused by local heterogeneities of the soil. That means that the type of sample pretreatment influences the analytical results essentially. In order to quantify the effect of a definite pretreatment method in comparison with the mercury content of the unchanged original soil sample, the probable systematic error of a method was introduced. Investigations were only carried out at two sampling locations of the contaminated site because of the relatively high labour; the mean values and variances obtained cannot be immediately transferred to other sites. However, the general knowledge can be used as methodical basis for further investigations. Particularly, the consequence arises that the regulations existing for the treatment of mercury-contaminated samples between sampling and chemical analysis must be revised to obtain comparable criteria of evaluation.Symbols used level of error in statistical tests, - _i random effect of the i-th sampling location with respect to the mercury content, effect of global fluctuations - _ij random effect of the j-th primary sample (composite sample) at the i-th sampling location, effect of local heterogeneity, sampling error - _ijk random effect of the k-th subsample within the respective j-th primary sample (composite sample) at the i-th sampling location, sample preparation error - _ijkl random effect of the l-th pretreated sample belonging to the respective k-th subsample of the j-th primary or composite sample at the i-th sampling location, sample pretreatment effect - _1– probable relative systematic error of a pretreatment methods as compared with the original material at the coverage probability 1– - overall expected mercury content of the sampling results - _i expected mercury content of the i-th pretreated composite sample at a fixed location - û_{1.1 –} lower limit of a confidence interval for the unknown expected mercury content of unchanged original material at the confidence probability 1– - _i intraclass correlation coefficient - ² , ² , ² , ² , _Z ² variance components caused by global variability, local heterogeneity, sample preparation, sample pretreatment and analytical error, respectively - _total ² total variance of mercury content - D² operator of the variance of a random variable - E operator of the mathematical expectation of a random variable - F_i mean squared sum quotient of the Fisher's F-distribution - F_1–/2(fg_i, fg_j) critical value of the F-distribution at fg_i and fg_j degrees of freedom, respectively - fg_i degree of freedom of a (mean) sum of squared deviations - H_i hypothesis regarding a statistical law - I_1– confidence interval for the unknown expected value difference of two methods compared at the probability 1–, probable systematic error of a pretreatment method as compared with the unchanged original material at the coverage probability 1– - MQ_i, mq_i mean squared deviations as explained in the context - n_i number of sampling locations, primary samples per location, subsamples per primary sample, pretreatment methods and measuring values per pretreated sample, respectively - S _i ² , s _i ² variance estimation obtained by chemical analysis - SQ_i, sq_i sum of squared deviations - S(|Y₁–Y_i|) mean error of the absolute mean value difference |Y₁–Y_i|) - t_1–/2;m critical value of the Student's t-distribution at m degrees of freedom and the probability 1–/2 - Y_ijklm mercury content obtained on the m-th final sample of the l-th pretreated sample belonging to the k-th subsample of the j-th primary sample at the i-th sampling location - Y_i, y_i,... mean values obtained by sampling and chemical analysis - Z_ijklm random error of the ijklm-th measuring value (final sample value) Y_ijklm.     

On-line determination of mercury in river water at the German monitoring station Schnackenburg/Elbe

A monitor is described which provides the on-line determination of mercury in river water at concentrations from 20 to 1000 ng/L. The measurement includes an on-line digestion with Br-/BrO3- and UV-radiation. Each determination is controlled by an on-line addition of 50 and 100 ng/L mercury carried out by pre-dilution of a 500 and 1000 ng/L stock solution using sequential injection analysis (SIA). One cycle of analysis takes 20 min and results in nine signals. A five days stand-alone operation has been performed successfully. Details are also published at web page: "http/www.rzbd.fh-hamburg.de/-prmercol".     

Strategies to improve the sensitivity in capillary electrophoresis for the analysis of drugs in biological fluids

Capillary electrophoresis (CE) is a useful method to quantify drugs in biological fluids. However, especially for blood or plasma samples, the sensitivity is not sufficient to quantify drugs and their metabolites as they often need to be quantified in the lower microg/L range. To overcome this limitation and to increase the sensitivity, two strategies are applied: first, to increase the amount of analyte added to the capillary and, second, to increase the sensitivity on the detector site. To improve the sensitivity on the detector site, alternative detection techniques to UV detection, e.g., laser-induced fluorescence detection (LIF) or mass spectroscopy (MS), can be applied. However, LIF detection can only be used for fluorescent analytes and the current equipment for CE-MS coupling provides only small improvements in sensitivity compared to UV detection. The detection window for UV detection can be enhanced using capillaries with an extended light path (bubble cell) or Z-shaped capillaries. Sensitivity improvements up to a factor of 10 have been reported. Increasing the amount of analyte in the capillary can be done either by chromatographic or by electrokinetic methods. Chromatographic methods such as on-capillary membrane preconcentration have been used for several analytes. However, no validated application has been reported to date. In contrast, several validated examples can be found in which electrokinetic techniques like sample stacking have been applied to achieve limits of quantification in the lower microg/L range. In conclusion, to date, electrokinetic techniques such as field-amplified sample injection offer the most promising results in achieving a sufficient sensitivity to quantify drugs in biological fluids.     

Determination of purines including 2,8-dihydroxyadenine in urine using capillary electrophoresis

For the purpose of rapid drug monitoring, methods have been developed for the determination of 2,8-dihydroxyadenine, allopurinol, oxypurinol, adenine, hypoxanthine, hippuric acid and xanthine in urine with and without sodium dodecyl sulfate as additive in sodium tetraborate running buffer. No sample preparation is necessary. 6-methylmercaptopurine and etofylline have been used as the internal standards. The limit of detection is 5 microM and the range of quantification stretches from 20 to 2000 microM. The capillary electrophoresis methods are simple, fast and robust.     

Orientation reversing involutions and the first betti number for finite coverings of 3-manifolds

LetMS ³,P ³ be a closed, orientable irreducible 3-manifold which admits an orientation reversing involution :MM. If dim(Fix )=0, suppose ₁ (M) has a subgroup of even index. We show thatM has finite coverMMM} with ₁(M<0). As an application we show that the hyperbolic dodecahedral space has a finite cover with positive 1st betti number.