Every monotone graph property has a sharp threshold

In their seminal work which initiated random graph theory Erdös and Rényi discovered that many graph properties have sharp thresholds as the number of vertices tends to infinity. We prove a conjecture of Linial that every monotone graph property has a sharp threshold. This follows from the following theorem. Let denote the Hamming space endowed with the probability measure defined by , where . Let be a monotone subset of . We say that is symmetric if there is a transitive permutation group on such that is invariant under . Theorem. For every symmetric monotone , if then for . ( is an absolute constant.)

     

Coupled-column liquid chromatography method with photochemically induced derivatization for the direct determination of benzoylureas in vegetables

The coupled-column (LC-LC) system, consisting of a first column packed with internal surface reversed phase (ISRP) (50 x 4.6mm ID) and a Chrompack C18 (100 x 4.6 mm ID) as second column, allowed the simultaneous determination of five benzoylurea insecticides in dichloromethane (CH2Cl2) extracts of vegetable samples without any clean-up step. This system was combined with a photochemically induced fluorescence (PIF) post-column derivatization in order to provide strongly fluorescent photoproducts from the non-fluorescent benzoylureas. Limits of detection ranged from 0.21 to 0.98 microg L(-1) of pesticide (equivalent to 0.14-0.65 microg kg(-1) in vegetable samples) and limits of determination ranged from 4.0 to 10.0 microg L(-1) (equivalent to 2.7-6.7 microg kg(-1)). Linearity of the method was established between 2 and 1800 microg L(-1), depending upon the compound. Validation of the total method was performed by randomly analyzing recoveries of four vegetable samples (aubergine, cucumber, green bean, and tomato) spiked at two levels of concentration (10.0 and 33.3 microg kg(-1)). The combination of the LC-LC system with PIF detection provides a sensitive, selective, and rapid method for the determination of pesticides in vegetable samples at levels lower than the maximum residue levels (MRLs) established for these compounds by Spanish legislation.